Oliver Heaviside (1850-1925) - Physical MathematicianD. A. EDGE. AFIMAThe Mary Erskine School, Edinburgh

4m-; JL

Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latterBold and Broad. To have to stop to formulate rigorous demonstrations would put a stopto most physico-mathematical enquiries. Am I to refuse to eat because I do not fullyunderstand the mechanism of digestion?

Oliver Heaviside

PrefaceToo little attention is paid on educational courses inmathematics and science to the personalities of thegreat innovators. When teaching at Paisley Collegeof Technology I was very impressed with the tape-slide facilities in the library and I thought that herewas an ideal way of presenting biographicalmaterial. Not only would well-prepared material beinstructive but also it would be entertaining.

I was inspired to produce a tape-slide presentationon Oliver Heaviside in particular by an excellentradio talk by Dr. D. M. A. Mercer.1 The essay belowis along the lines of my script and is presented with aview to interesting others in Heaviside and perhapsin the idea mentioned at the outsetSummary

There are many misconceptions about OliverHeaviside. He had a delightful and noble characterthough these features were obscured by his appa-rently hermit-like way of life. He was self-taught andretired from work as a telegraphist in his earlytwenties to devote himself to experimentation andwriting. He made no money from his epoch-makingdiscoveries and lived and died in near-poverty.

He is responsible for Maxwell's Equations as weknow them and he extended the theory of electro-magnetic wave propagation. He established themathematical theory of telegraphy and telephonyand formulated the condition for distortionless trans-mission of speech. He found the mathematics of histime unsatisfactory for solving many importantproblems and consequently invented the operationalcalculus which he used to great effect. He predictedthe possibility of a reflecting layer in the upperatmosphere (the Kennelly-Heaviside layer) and hewas very interested in- terminology and coined anddenned many new words (e.g., inductance).Volume 2 No. 2, 1983

DetailsOliver Heaviside was the youngest of four sons of

Thomas Heaviside, then at 55 King Street, CamdenTown, London. His mother was formerly RachelWest, who had been a governess in the Spottiswoodefamjly and whose sister was the wife of Sir CharlesWheatstone, the telegraphist and prolific inventor.Thomas Heaviside was a skilful wood-engraver whomoved to London from Teesside with his wife andyoung sons in about 1849. Oliver was born on May18th, 1850.

It has been suggested2 that the Heaviside familymoved to London because the development ofphotography was making wood-engravers redun-dant and there were better prospects of employmentin the capital. It seems that the family experiencedhard times during Oliver's youth and for a while hismother ran a school and subsequently let rooms.3

The high quality of Thomas Heaviside's work isevident in an engraving of a drawing by Godwin(reproduced in reference 2). A brother of Thomaswas an artist and engraver still more capable, andfrom two of Oliver's drawings (at age 11)reproduced in reference 2 he was at least a buddingartist.

Oliver Heaviside attended Camden House Schooland did well in his examinations, taking a prize fortop place in natural sciences. However he got a badmark in geometry - probably because he disagreedwith the way it was taught to small boys. He left theschool in 1866 but his uncle, Professor Wheatstone,advised him to continue to study (in particularFrench, German, Danish and Natural Sciences).Also the youthful Heaviside did experimental workon electromagnetism and taught himself MorseCode.

In 1868 he started work with a Danish telegraph55

company at Fredericia in Denmark. In September ofthat year the first Anglo-Danish cable was laid andHeaviside was involved in various tests on the line.He observed a number of anomalous effects whichhe confessed he could not understand at all - butthen neither could anyone else. In 1870 he waspromoted and based at Newcastle-upon-Tyne, theGreat Northern Company having taken over theDanish company. Between 1870 and 1874Heaviside was at Newcastle and he cooperated withhis brother Arthur who was an engineer then withthe Post Office at Newcastle.

In 1873 Heaviside obtained a first edition ofMaxwell's famous treatise "Electricity and Mag-netism." He was profoundly impressed and subse-quently became the principal exponent of Maxwell'sideas. Heaviside cleared away the debris of thebattles fought by Maxwell in establishing these ideasagainst the older theories, and he reduced Maxwell'smaze of symbols practically to just two: viz., electricand magnetic force. Thus Heaviside establishedsymmetry throughout the whole of electromagnetismand the present form of Maxwell's Equations is dueto Heaviside. Fitzgerald in reviewing Heaviside'scollected "Electrical Papers" wrote: "Since OliverHeaviside has written the whole subject ofelectromagnetism has been remodelled by his work.No future introduction to the subject will be at allfinal that does not attack the problem from at least asomewhat similar standpoint to the one that he putsforward." On the other hand H. J. Josephs haspointed out4 that although he is of the opinion thatthe so called Maxwell's Equations should be calledHeaviside's equations this is "not merely because theformulation is Heaviside's but also because they areless general than the proper Maxwell's Equationswhich are best expressed in quaternionic notation."

In 1874 Heaviside was elected an AssociateMember of the Society of Telegraph Engineers.(Later, this Society became the Institution ofElectrical Engineers.) He left the Great NorthernCompany and returned to London to live with hisparents at 117 Camden Street, St. Pancras, and itwas here that he did most of his original work.

He liked to study mathematics in the quietest partof the day (from 10.00 pm into the small hours) andduring the day proper he conducted experiments. Hehad begun to publish in 1872, and from 1874onwards he published numerous articles (though notwithout encountering difficulties with editors orreferees).

In 1876 he was fairly well known (though not wellunderstood) and regarded as a prominent scientist.He was elected to the Council of the Society of

Telegraph Engineers (though he did not attend asingle meeting and was not re-elected the followingyear). At this time William Preece was also amember of the Council; Heaviside and Preece wereto cross swords at a later date. In 1881 Heavisidewas unable to pay the subscription of the Society ofTelegraph Engineers and he was struck off theregister. Despite this apparent shortage of money heturned down a well-paid appointment made avail-able by Preece with the Western Union Company.

Early in his studies at Newcastle, Heaviside haddiscovered the method of analysing alternatingcurrent using the rules for direct current circuits andimpedances jasL, 1/jaC, etc. Steinmetz andKennelly made these complex quantities familiar butHeaviside deserves as much credit as these two men.Subsequently he established the method of analysingalternating currents which is in use today. Duringthe early course of his studies he developed theoperational calculus, using the multiplier l/p torepresent integration and the multiplier p differen-tiation. In the course of subsequent analysis he had

to find interpretation for such as 1 / \/p. It seems

that Heaviside was unaware that "fractional diffe-rentiation" was an old subject which had beenintroduced by Leibniz in 1695 and developed bysuch great mathematicians as Euler, Liouville,Gregory and Kelland. Heaviside, it should beremembered, was self-taught and had very limitedaccess to the work of earlier mathematicians.Despite this he developed the subject of fractionaldifferentiation much further in some directions thanany of his illustrious predecessors.4 AlthoughHeaviside was successful the pure mathematiciansof the day did not like his methods.

He had enormous physical insight which he usedfreely: the mathematicians were averse to this, theywanted rigorous analysis. Sir Edmund Whittaker3

quoting a contemporary pure mathematician writes:"there was a sort of tradition that a Fellow of theRoyal Society could print almost anything he likedin the Proceedings without being troubled byreferees: but when Heaviside had published twopapers on his symbolic methods, we felt that the linehad to be drawn somewhere, so we put a stop to i t "Heaviside was deeply offended. He spoke in privateof the w o o d e n - h e a d e d R o y a l S o c i e t ymathematicians and wrote scathingly that goodmathematicians, when they die, go to Cambridge. In1904, after Heaviside's methods had borne suchobviously good fruit, the Council of the RoyalSociety decided to offer him the Hughes Medal. Hedeclined it (privately).

56 Teaching Mathematics and its Applications

By 1887 Heaviside had done his main originalwork and in 1889 he moved with his parents toPaignton, Devon, to live in a house taken by hisbrother Charles who had become quite successful inhelping to run a music business in Torquay. Aboutthis time Heaviside entered the last phase of hisscientific life. This consisted mainly of publishing hiscollected papers in two volumes,5 and his three-volume "Electromagnetic Theory."6

In 1894 his mother died, to be followed in 1896by his father, and the following year he rented ahouse at Newton Abbott (near to both Paignton andTorquay).

Heaviside, then, was at Paignton from 1889 to1897. Towards the end of 1957 a collection of hispapers was found under the floorboards of his roomin the house where he lived. The many documentsfilled three sacks and were scrutinised by H. J.Josephs.4 Mr. Josephs found that most of the collec-tion was associated with the publication of Volumes1 and 2 of "Electromagnetic Theory" but there weremany other issues dealt with. For example, harkingback to the dispute about operational calculus andfractional differentiation it seems that Heavisidefurther analysed his own approach and achieved adifferent interpretation of his parameter p whichwould have allowed him to construct a rigorousmathematical basis for his work. He did not bother.

For the first 2 years at Newton Abbott (1897-1899) he retained the elderly housekeeper who hadbeen with his parents at Paignton, but then she had astroke and had to leave him. From 1899 he workedand lived in solitude and suffered from ill-health.Perhaps because of these new circumstances it is tooeasy to get the impression from some sources thatOliver Heaviside was a bitter recluse with a twistedpersonality. This is quite untrue. There are, in theHeaviside collection at the Institution of ElectricalEngineers, Savoy Place, London, the negatives ofsome photographs of Heaviside taken about 1890,shortly after the move to Paignton. They are printedin reference 2 and one of the photographs was usedfor a posthumous portrait by Francis Hodge. Thisportrait hangs in the Heaviside room at Savoy Placeand is reproduced in reference 7, although not incolour. The photographs were taken by Heaviside'snephew Charles, son of his brother of the samename. They show Heaviside to be a strikinglyhandsome man, which according to various reportshe remained to the end of his life. It was about thetime the photographs were taken that Dr. G. F. C.Searle became friendly with Heaviside. Searle, sub-sequently with his wife, visited him on many occa-sions up to the end of his life and it is chiefly to

Searle that we are indebted for an account of the realHeaviside (reference 7, pp. 8-9 and 93-96).

Heaviside and Searle (who was 14 years hisjunior) used to go for cycle rides in the nearby coun-tryside and it seems that Heaviside occasionally toreahead by putting his feet on the handlebars, foldinghis arms and letting the bike race down steep slopes.(In those days braking facilities on bicycles wereprimitive and many lanes would be merely carttracks.)

There is another photograph printed in reference2 which is of interest. It is probably contemporarywith the other photographs and shows a groupwhich is mostly the Heaviside family. (It includesmother, father and brothers, Arthur and Charles.)Also in the picture is Charles' sister-in-law, MissWay, of whom more in a moment. At the back ofthe group, peeping out from a central position, isOliver - bright eyed, casually confident and smokinga pipe. The effect is surreal. Apparently he detestedalcohol in all its forms but liked a pipe of thestrongest tobacco. He liked walking and cycling andwas a good gymnast but when working he was thereverse of a fresh-air fiend. He would close doorsand windows, light a gas fire and an oil stove, andpuff away at his pipe. Visitors found it ratheruncomfortable but he loved it.

In 1908 Heaviside was offered a home by MissWay. This was not a romantic move but a practicalone. Heaviside had been living alone for about 8years, he was an incompetent cook and had sufferedgenerally with his health, having had a bad attack ofjaundice at Christmas 1907.

Miss Way's house was Homefield, LowerWarberry Road, Torquay, and she gave up to himthe upper part of it. He was a paying guest at therate of £100 p.a. At the time he was 58 years oldand she about 66 years of age. He lived at Homefielduntil 5 weeks before he died, at a nearby NursingHome. From 1916 to 1925 he was alone.

He was without doubt an oddity, a strong-mindedindividualist, but never remotely insane nordominated by a sense of bitterness. He had a greatsense of humour and Searle tells of many occasionswhen he demonstrated this.7

He was not quite the recluse that one mightexpect; although he would never leave his home tovisit others (family excluded), he was always deligh-ted to receive visitors himself and would scan thevisitors' lists in the newspapers for names of holiday-makers in the local area. His visitors included manyprominent scientists, by whom he was highlyregarded.

One visitor was J. S. Highfield who was President

Volume 2 No. 2, 1983 57

of the IEE in 1921. This was the 50th year of theInstitution's foundation and one of the events incelebration was the creation of the award of theFaraday Medal. It was decided that Heavisideshould be the first recipient and the President himselfjourneyed to Torquay to make the formal presenta-tion in September 1922. Heaviside was then 72years old.

When he was elected FRS in 1891, Heavisidewould not travel up to London for the installation.Despite this apparent reticence he had a wide scien-tific correspondence with the foremost men of theday. Hertz, for example, was his friend althoughthey never met. A special friend was G. F.Fitzgerald of University College, Dublin - the twomen met only twice, for short periods. Fitzgerald,like Maxwell, died before he reached 50 andHeaviside dedicated Volume III of "ElectromagneticTheory," "In memory of Geroge Francis Fitzgerald,FRS - 'we needs must love the highest when we'know him."

The three volumes of "Electromagnetic Theory"were published in 1893, 1899 and 1912, respec-tively. In the Heaviside Collection at Savoy Placethere is the following poem, in Heaviside's own hand-writing, in pencil on the flyleaf of Volume I.

DEDICATION

To My Dear ChildrenI did not send you any cards,For I had none to send,So now I send you this here book,Whereby to make amend.

The first chap, is for Freddie,And may he always be,A credit to his parents,And an ornament to Torquay.

The next chap, is for Ethel,And may she read it well,And study it, and find it good,Nor think the book a sell.

The third chap, is for Charlie,And may he never be,A terror to his parents,And a torment to Torquay.The fourth chap, is for Rachael,Because it is the best,And may she never never try,To turn it into jestThe preface is for Beatrice,Because it is so short,And may she never never think,It all amounts to nought.

58

The contents are for Pa and Ma,And may they never know,The pangs of tortured conscience,Or the awful depths of woe!

This volume must have been given by Heavisideto his brother Charles' family.

Heaviside kept notebooks concerning papers thathe wrote and these are now preserved at SavoyPlace. The handwriting is neat and the books aretidy; an editor once wrote to him: "No other con-tributor can approach the admirable clearness ofyour copy and the cleanness of your proofs." In thenotebooks are not only precis of the papers but alsocomments about the reception they met with. Forexample, concerning his first paper in thePhilosophical Magazine of February 1873, "On theBest Arrangement of Wheatstone's Bridge"* heobserves that Sir William Thomson (Lord Kelvin)mentioned it when they met at Newcastle a veryshort time after it appeared and so Heaviside senthim a copy. He also sent Maxwell a copy andobserves in the notebook that Maxwell noted it inthe second edition of "Electricity and Magnetism."Heaviside comments that for his first paper in thePhilosophical Magazine it was a good beginning.

In the latter half of the nineteenth century therewas a good deal of interest in terminology in generaland in units in particular. As an innovator Heavisideoften had to think carefully about the use of wordsto describe his ideas. In his writing he used a styleuniquely his own: always precise and oftenrefreshingly humorous.

One of Heaviside's major achievements was toestablish the importance of inductance in telegraphyand telephony, but he had a big struggle convincingother workers that he was right. Inductance is animportant factor in getting the correct balance suchthat the attenuation of all frequencies is the sameand signal distortion avoided. This was the mainbone of contention between himself and WilliamPreece. Unfortunately Preece walked the corridorsof power and Heaviside did not, so it was the latterwho got the worse of the struggle at the time — hispunishment being that he was denied access to thescientific press for a little while.

In June 1887, Heaviside published one of hisarticles in his series "Electromagnetic Induction and

• Heaviside was anything but a sycophantic name-dropper andcertainly would not have chosen his title or his topic because ofits connection with his uncle, Sir Charles Wheatstone. As amatter of fact, prolific inventor though Wheatstone was, thebridge was not invented by him (but by S. H. Christie, in 1833,as Wheatstone more than once pointed out). Wheatstone'sname became associated with the bridge because of hisfrequent use of it.

Teaching Mathematics and its Applications

its Propagation" in The Electrician. He commentedthat the results in this paper were originallyscheduled to appear in a paper by his brother Arthurbut his contribution was blocked "by the eminentscienticulist*, W. H. Preece, FRS, in the Spring of'87" the grounds being "irrelevancy and want ofnovelty." The paper concerns attenuation anddistortion in submarine cables which were by nomeans irrelevant topics and were certainly novel asHeaviside sought to correct Sir William Thomson'stheory, and incidentally some inaccurate estimatesby Preece. (It should be added that Sir WilliamThomson was aware of the limitations of his theory,in the way that all great pioneers are.)

In September 1887 Heaviside submitted an article"Mr. W. H. Preece on the self-induction of Wires."It was not published then but had to wait inclusionin his "Electrical Papers" (reference 5, Volume II,pp. 160-165) of 1892. Understandably it was notpublished, for Heaviside did not restrain himself:

" . . . It will be remembered that Mr. Preece, inspite of the well-known influence of resistance inlowering the speed of signalling, was formerly anadvocate of thin wires of high resistance for tele-phony; but that, perhaps taught by costly failuresin his own departmentf, and by the experience ofmore advanced Americans and Continentals whohad signally succeeded with wires of lowresistance, he recently signified his conversion.Along with this however, it will be rememberedthat, although it had been previously shown howvery different the theory of the rapid undulatorycurrents of telephony is from the electrostatictheory of the submarine cable, he adopted ratherpronouncedly what should, it appears, be under-stood to be the electrostatic theory, with fullapplication to telephony. It is not to be presumedthat Mr. Preece meant to deny the existence ofmagnetic induction, but that he meant to assertthat it was of so little moment as to be negligible.It will also be remembered that his views wererather severely criticised by Prof. S. P. Thomp-son, and that Prof. Ayrton and others pointed outthat he had not treated the telephonic problem atalL More recently still, it may be remembered bythe readers of this journal that it has beenendeavoured to explain how and why the

* Scienticulist »was Heaviside's coined word used to referdisparagingly to those who affected to be scientists but werenot truly so.

fW. H. Preece worked for the Post Office becoming Engineer-in-Chief and Electrician 1892-99. He was created KCB in1899.

electrostatic theory has so limited an applicationto telephony (E.M.I, and its P., section XI etseq.) [reference 5, Vol. II, pp. 119-155]. Nothingdaunted, however, Mr. Preece now, although tosome extent modifying his views as regards ironwires, maintains that self-induction is negligible incopper-wire circuits; and in fact, on the basis ofhis latest researches, asks us to believe that theinductance of a copper circuit is several hundredtimes smaller than what it is maintained to be byexperimental theorists, and is really quite neglig-ible in consequence..."Support for Heaviside's views on the importance

of inductance in distortionless transmission camefrom the United States in the person of ProfessorMichael Pupin. In 1900 Pupin published the resultsof some brilliant experiments verifying (13 yearsafter they were made) Heaviside's conjectures. In hispaper* Pupin points out the analogy between theeffect of inductance coils in telephony (when trans-mitting electromagnetic waves) and the effect ofweights on a string (when transmitting displacementwaves), the inductance in the former case corre-sponds to momentum in the latter. Pupin had tocalculate the best positions at which to insert thediscrete inductance coils; this was a mathematicalproblem which hitherto had not been solved. AfterPupin had gone on to demonstrate physically thatHeaviside was correct it became standard practiceto add inductance to (i.e., "to load") the many landlines and submarine cables that were laid at thebeginning of this century. Initially discrete loadingwas the practice but with improvements in cablemanufacturing techniques this was succeeded bycontinuous loading. Modern telecommunicationssystems do not use loading because a loaded linebehaves like a low-pass filter and nowadays high fre-quency transmission systems are in use to cope withthe great volume of traffic.

Heaviside also concerned himself with the theoryof electromagnetic waves in free space and as aresult of his calculations, and in the light ofMarconi's radio transmission across the Atlantic in1901, he suggested the possibility of a reflecting layerin the upper atmosphere. A. E. Kennelly made asimilar suggestion in that same year, 1902, and theirprediction was verified directly some 20 years laterby Edward Appleton.

Although Heaviside concerned himself largelywith practical problems of fairly immediate impor-tance he did at times take interest in what he musthave regarded as purely theoretical problems, e.g.,the rigorous mathematical basis of the operationalcalculus. A slightly different type of theoretical

Volume 2 No. 2, 1983 59

problem that he investigated, which is now a verypractical problem, was the transmission of signalsthrough waveguides and coaxial cables. He made awide range of contributions to the field of electro-magnetism and in addition gave attention, in adisconnected fashion, to a very large number ofancillary topics - so many, in fact, that the merelisting of them would take up a considerable space.Some of the less weighty and less esoteric topicswere: the age of the earth (a subject dear to LordKelvin); what is now known as a low-pass filter; andthe teaching of mathematics. In this last connectionone topic that concerned him deeply was that ofvectors (see Preface to reference 5, pp.xi-xii): andthese especially in preference to quaternions sincethey had no physical interpretation whereas vectorsquite clearly had.

His approach to mathematics was pragmatic. Heregarded it as an experimental science - in fact hewrote an article with just such a title (reference 6;Volume II, pp. 1-12): first get on in any way possibleand let the logic be left for later work - a view subse-quently expressed by Einstein and G. H. Hardy.Despite the rough handling he received from theCambridge mathematicans of about 1890 his articleon mathematics as an experimental science (firstpublished in 1894) was generous to what one mightcall "the true spirit" of Cambridge. He commentsthat "even men who are not Cambridge mathe-maticians deserve justice, which I fear very muchthey do not always get"

Some writers suggest that Heaviside retired at anearly age because of deafness. This seems not to beentirely credible. His deafness appears to have beenintermittent and certainly did not stop him indulginghis love of music. He owned an "aeolian" whichappears to have been either a pianola or a player-piano made by the Aeolian Co. of America.Heaviside evidently could play the piano and oncetaught himself Beethoven's Opus 90. He wasespecially fond of the "divine Schubert" andprobably possessed rolls of Schubert's music forplaying the "aeolian" mechanically. This is not asgruesome as it sounds: apparently the nearest'modern equivalent is a set of hi-fi equipment, as theAeolian Company took pride in the fact that itsplayer-pianos were excellent instruments in them-selves.

The quality of Heaviside's life as he neared hisdeath left a great deal to be desired. For more than ayear (August 1921-October 1922) he had no gas forheat or light He was very poor and not the besthousekeeper. He owed much to the voluntary effortsof the local "bobby" as he called him, P. C. Brock,

and to Brock's small daughter. But despite his owntroubles he could still take an interest in theproblems of the world at large and his scientificcorrespondence was enlivened by references topassing events with frequent quotations orimpromptu verses. The United States' claim for therepayment of the English debt in 1923 led. to somescathing remarks on American greed and greed ingeneral ending with:

"Eat slowly. Only men in rags or gluttons deep insin mistake themselves for carpet bags and shovethe vittles in."3

Heaviside did meet wkh much opposition to hiswork and ideas, but in later life, starting in 1891 withhis election to the Royal Society, he received fairlywidespread recognition. However it should beremembered that he was never in paid employmentafter the age of 24 and this resulted in some financialproblems.

Many friends were aware of this and tried to help.The first official attempt to give him some financialsupport came from the Royal Society via somefriends (G. F. Fitzgerald, Sir Oliver Lodge and JohnPerry). But he refused the offer of an honorarium.This was early in 1894. In 1896 he was granted aCivil List pension of £120 p.a. by the Government(This was apparently on account of the efforts of thesame three men together with the help of LordRayleigh and Lord Kelvin. Sir Oliver Lodge was apersonal friend of A. J. Balfour, then Leader of theHouse of Commons and a man with scientificinterests. Balfour's uncle, Lord Salisbury, was PrimeMinister and Lord Rayleigh was Balfour's brother-in-law.) This pension was increased by a further£100 p.a. in 1914. Heaviside also had a smallincome from his writings and occasionally he wouldreceive cash from well-wishers. However, he wasextremely touchy about charity and would not seehis friend Searle for a period of many months'towards the end of his life because of his anger abouta financial matter.

In a short presentation it̂ is possible to give onlythe briefest outline of the life and work of OliverHeaviside. It is hoped that a balanced picture hasbeen presented whilst at the same time showingHeaviside to be the truly interesting man that hewas. Undoubtedly a genius, he was also a philan-thropist in his own, unique way. He once wrote an(unpublished) article on the immortality of the soul:

" . . . There are large souls and small souls. Theimmortal soul of the scienticulist is a small affair,scarcely visible. Indeed its existence has beendoubted. That of a Shakespear or a Newton isstupendously big. Such men live the best parts of

60 Teaching Mathematics and its Applications

their lives after they are dead. Maxwell is one ofthese men. His soul will live and grow for long tocome, and hundreds of years hence will shine asone of the brightest stars of the past, whose lighttakes ages to reach us."

References1. Mercer, D. M. A., "Rewards that Came too Late," BBC

Broadcast on January 22nd and March 2nd, 1976..2. Appleyard, R-, "Pioneers of Electrical Communication,"

Macmillan & Co., 1930: Reprinted 1968, Books for LibrariesPress, N.Y.

3. Whittaker, E. T., Bulletin Calcutta Mathematical Society,1928-9, 20, 199-220: Reprinted as Foreword in Chelseareprint of reference 6. ^

4. Josephs, H. J., "The Heaviside Papers found at Paignton in1957," Proc. IEE, Part C, March 1959, 70-76: Reprinted asAppendix to VoL 3 in Chelsea reprint of reference 6.

5. Heaviside, O., "Electrical Papers," (2 vols.), Macmillan,1892.

6. Heaviside, O., "Electromagnetic Theory," (3 vols.), TheElectrician Printing and Publishing Co., 1893, 1899 and1912. Reprinted by photolitho, E. & F. N. Spon Ltd.,London, 1951 (with a critical and historical introduction byE. Weber). Reprinted, Chelsea Publishing Co., New York,1971 (with a Foreword by Sir Edmund Whittaker - reprint ofreference 3; also Appendix to Vol. 1 "The Work of OliveTHeaviside," B. A. Behrend - reprinted from The ElectricalJournal, January and February 1928, Pittsburgh; andAppendices to Vol. 3 "Some Unpublished Notes of OliverHeaviside," H. J. Josephs, - reprinted from reference 7, and"The Heaviside Papers found at Paignton in 1957," H. J.Josephs - reprint of reference 4).

7. "The Heaviside Centenary Volume," IEE, 1950. (Containsaddresses made at the Heaviside Centenary Meeting, the1078th ordinary meeting of the IEE, 18th May, 1950,together with a number of papers dedicated to the work andmemory of Oliver Heaviside. Approximately 100 pages - awealth of information and some pictures.)

8. Pupin, M. I., "Wave Transmission over Non-Uniform Cablesand Long-Distance Air-Lines," Trans. AIEE, 1900, XVH,445-512.

BibliographyBerg, E. J., "Heaviside's Operational Calculus," McGraw HOI,

2nd edition, 1936. (Includes brief personal sketch, pp.xiii-xv.)

Josephs, H. J., "Heaviside's Electric Circuit Theory," Methuen,1946.

The Electrical World, January-June 1925, 85. (Includes avariety of articles: p. 337; p. 364;pp. 403-405, by Sir Oliver Lodge;pp. 405-407, by B. A. Behrend;p. 475, by Prof. E. J. Berg;p. 513, a letter from J. R. Carson.)

Piaggio, H. T. H., "The Operational Calculus," Nature, 1943,152,93.

Appleyard, R., "Dictionary of National Biography."Fitzgerald, G. F., "Heaviside's Electrical Papers," in "The Scien-

tific Writings of the Late Gco. Francis Fitzgerald,"Longmans Green, 1902, pp. 292-300.

The author obtained his BSc in TheoreticalMechanics from the University of Nottingham in1968. He obtained his PGCE from Keswick HallTeachers' Training College, Norfolk, and spent 1year (on VSO) teaching mathematics in Kenya. Hesubsequently obtained his MPhil (in ElectricalEngineering) from the University of Nottingham.He has held a variety of teaching appointments atplaces including Gresham's School, Paisley Collegeof Technology and Norwich School. Since August1981 he has been Head of the Mathematics Depart-ment at the Mary Erskine School in Edinburgh.

Solutions to Calculator Challenges

rhe Challenges were published in Volume 1,Number 3, p. 100.

11 .3 was displayed as 7.The actual calculations were

5632 + 7343 = 12975

239 x 367 =87713.

12. Tsu Ch'ung-Chih:

1̂113

Volume 2 No. 2,1983

=3.1415929

13. An example of ill-conditioned equations. Did you

think your calculator was on the blink?

x, = 1776 _K, = - 4 1 8 6

x2 = - 7 7 0 y 2 = 1816.

Thought of as the equations of straight lines it willbe seen that their gradients are very nearly equal:

-2.3585 and -2.3581 to 4 d.p.

Thus the slightest change of the gradient of oneline, and the point of intersection of the two lineschanges radically.

61