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A Brief History of Algebra and Computing:
An Eclectic Oxonian View
Jonathan P. Bowen
Oxford University Computing Lab oratory Wolfson Building, Parks Road, Oxford OX1 3QD
Dedicated to Prof. C.A.R. Hoare, FRS, James Martin Professor of Computing at the Oxford University Computing Lab oratory. Completed on his 60th birthday, 11th January 1994.
That excel len t woman knew no more about Homer than she did about Alge- bra, but she was quite contented with Pen's arrangements... and felt perfectly con dent that her dear boy would get the place which he merited.
Pendennis (1848{50), by Wil liam Makepeace Thackeray (1811{1863) (The story of the progress of an Oxford student.)
Oh heck|another hour of algebra!
Opposite side Hypoteneuse
= sine;
Adjacent side Hypoteneuse
= cosine;
Opposite side Adjacent side
= tangent
1 The Origins of Algebra
The following problem on the Rhind Papyrus in the British Museum, London, was written in ab out 1650 BC:
Divide 100 loaves among 10 men including a b oatman, a foreman and a do or- keep er, who receive double p ortions. What is the share of each?
This can of course b e solved using algebra. The rst treatise on algebra was written by Diophantus of Alexandria in the 3rd century AD. The term derives from the Arabic al-jabr or literally \the reunion of broken parts." As well as its mathematical meaning, the word also means the surgical treatment of fractures. It gained widespread use through the title of a b o ok ilm al-jabr wa'l-mukabala|the science of restoring what is missing and equating like with like|written by the mathematician Abu Ja'far Muhammad (active c800{847), who subsequently has b ecome know as al-Kharazmi,
the man of Kwarazm (now Khiva in Uzb ekistan). He intro duced the writing down of calculations in place of using an abacus. Algorism (the Arabic or decimal system of writing numb ers) and algorithm b oth derive from his name. Algebra was brought from ancient Babylon, Egypt and India to Europ e via Italy by the Arabs.
2 Early English Algebra
In the rst half of the 16th century, Cuthb ert Tonstall (1474{1559) and Rob ert Recorde (1510?{1558 ) were two of the foremost English mathematicians [2]. They were the rst mathematicians at the University of Cambridge whose lives have b een recorded in any de- tail and as such may b e considered founders of one of the most imp ortant centres of math- ematics in the world. Both migrated to Oxford University during their careers. Rob ert Recorde, p erhaps the more imp ortant of the two, b ecame a Fellow of All Souls College at Oxford in 1531. The earliest use of the word algebra may b e found in Recorde's Pathway of Know ledge (1551) in which he wrote:
Also the rule of false p osition, with dyvers examples not onely vulgar, but some app ertayning to the rule of Algebra.
In 1557 he intro duced the equality sign =' in his Whetstone of Witte, chosen \bicause no e 2 thynges can b e moare equalle" (than two parallel lines of the same length). The sym- b ols+' and ` ' were intro duced for the rst time in print in John Widmann's Arithmetic (Leipzig, 1489), but only came into general use in England after Recorde's Whetstone:
There b e other 2 signes in often use of which the rst is made thus -+- and b etokeneth more: the other is thus made | and b etokeneth lesse.
These symb ols may have originated from marks chalked on chests of merchandise in Ger- man warehouses, indicating the variation from some standard weight. They app eared in a work by Stifel in 1544. Thomas Hariot (1560{162 1), a native of Oxford, at St. Mary Hall (subsequently incor- p orated with Oriel College), invented the signs of inequality <' and>' [3]. He continued the work of Recorde, helping to give algebra its mo dern form. His work on algebra was pub- lished ten years after his death, although it was probably written around 1610, including Hariot's law of Signs concerning ro ots. William Oughtred (1574{1660), a private tutor to Oxford students, worked on math- ematics at a country vicarage and extended the use of the multiplication sign ' in his Clavis Mathematica (1631), previously used in a rather di erent manner by Recorde in his Grounde of Artes (1542). He also intro duced the trigonometrical terms sine, cosine and tangent. The division sign' was used by J. H. Rahn in 1659 and was intro duced into England by John Pell in 1668.
... this day I had a general but only indistinct conception of the p ossibility of making an engine work out algebraic developments... I mean without any ref- erence to the value of the letters... My notion is that as the cards (Jacquards) of the calc. engine direct a series of op erations and then recommence with the rst so it might p erhaps b e p ossible to cause some of the cards to punch others equivalent to any given numb er of rep etitions. But these hole[s] might p erhaps b e small pieces of formula previously made by the rst cards and p ossibly some mo de might b e found of arranging such detached parts.
Next year, on 13 Decemb er 1837, Babbage noted the following in one of his sketchb o oks:
On machinery for the Algebraic Development of Functions Ab out this date the idea of making a development engine arose with consider- able distinctness. It is obvious that if the Calculating Engine could print the substitutions which it makes in an Algebraic form we should arrive at the alge- braic development `it can print all those substitutions which are noted in the composition in some of the notations'|it will however b e b etter to construct a new engine for such purp oses.
I.e., he was considering the construction of an engine for general algebraic op erations and printing formulae resulting from calculations on such a machine. These ideas may b e regarded as an extension of the Analytical Engines, but were not develop ed further by Babbage. Towards the end of his life Babbage asked Harry Wilmot Buxton to write his biography. An early draft, together with other pap ers Babbage lent him whilst undertaking this task, are now held as part of the Buxton collection at the Museum of the History of Science in Oxford.
4 Bo olean Algebra
Leibniz initiated the search for a system of symb ols with rules of their combination in his De Arte Combinatoria of 1666, as well as developing the binary notation. In 1854, George Bo ole (1815{1864), Professor of Mathematics at Cork from 1849 despite having no rst degree, formalised a set of such rules in the seminal work entitled, p erhaps optimistically, An Investigation of the Laws of Thought. Bo ole's aim was to identify the rules of reasoning in a rigorous framework and revolutionised formal logic after thousands of years of little progress. They transformed logic from a philosophical into a mathematical discipline. These rules have subsequently b ecome known as Bo olean algebra and the design of all mo dern binary digital computers has dep ended on the results of this work. These logical op erations, normally implemented as electronic gates, are all that are required to p erform more complicated op erations such as arithmetic. Charles Lutwidge Do dgson (1832{1898) [1], a Mathematics Lecturer at Christ Church, Oxford from 1855 to 1881, was in uenced by the work of Bo ole. He had a general interest in algebra and also teaching. In May 1855 he noted in his diary:
I b egan arranging a scheme for teaching systematically the rst part of Alge- braic Geometry: a thing which no one hitherto seems to have attempted|I nd it exceedingly dicult to do it in anything like a satisfactory way.
He subsequently pro duced works on The Fifth Book of Euclid proved Algebraical ly and A Syl labus of Plan Algebraical Geometry as well as collections of algebraic and arithmetic formulae to aid examination candidates. Much later, in his diaries of 1884 Do dgson noted: \In these last few days I have b een working on a Logical Algebra and seem to b e getting to a simpler notation than Bo ole's." In 1885 he notes: \I have o ccupied myself at Guildford in teaching my new `Logical Algebra' to Loiusa, Margaret and the two b oys." In March 1885 he mentions \A Symb olic Logic, treated by my algebraic metho d." He published a numb er of works on logic, including Symbolic Logic, Part I: Elementary in 1896 under his more famous alias of Lewis Carroll. Unfortunately Part I I never app eared.
5 Algebra and Computing
From the end of the 2nd World War in 1945 the world was set for an exp onential growth in the use of computers. At this time, Leslie Fox (1918{1992 ) [9] moved from the Admiralty Computing Service to the National Physical Lab oratory (NPL) joining a section including Alan Turing [7] and led by E. T. Go o dwin. He was interested in numerical linear algebra and whilst at the NPL he started a line of investigation into using Gaussian elimination to estimate the accuracy with solving linear equations. Prof. Fox went on to b ecome the rst director of the Oxford University Computing Lab oratory in 1957, b ecoming Professor of Numerical Analysis in 1963. He stayed there until his retirement in 1983. J. R. Womersley was the Sup erintendent of the Mathematics Division at the NPL. In 1946 he noted in a rep ort on the prop osed ACE computer that:
... this device is not a calculating machine in the ordinary sense of the word. One do es not need to limit its functions to arithmetic. It is just as much at home in algebra...
In 1951, Christopher Strachey, then a teacher at Harrow Scho ol, made contact with Mike Wo o dger at the NPL via a mutual friend. He started to write a draughts program for the Pilot ACE, and so on progressed to the machine b eing develop ed at Manchester University. He obtained a copy of the Programmer's Handbook by Alan Turing and wrote a long letter to Turing on his plans:
... It would b e a great convenience to say the least if the notation chosen were intelligible as mathematics when printed by the output... once the suitable notation is decided, all that would b e necessary would b e to typ e more or less ordinary mathematics and a sp ecial routine called, say, `Programme' would convert this into the necessary instructions to make the machine carry out the op erations indicated. This may sound rather Utopian, but I think it, or something like it, should b e p ossible...
mundane asp ects including asso ciative and commutative (\AC") matching. Mo delling is an imp ortant asp ect of computer science. However many algebraic prop- erties may b e derived from mo dels. To quote Hoare [5]:
A mo del of a computational paradigm starts with choice of a carrier set of p otential direct or indirect observations that can b e made of a computational pro cess. A particular pro cess is mo delled as the subset of observations to which it can give rise. Pro cess comp osition is mo delled by relating observations of a comp osite pro cess to those of its comp onents. Indirect observations play an essential role is such comp ositions. Algebraic prop erties of the comp osition op erators are derived with the aid of the simple theory of sets and relations.
The viability of such approaches has yet to b e seen in widespread industrial practice. As Hoare says [5]:
The construction of a single mathematical mo del ob eying an elegant set of algebraic laws is a signi cant intellectual achievement; so is the formulation of a set of algebraic laws characterising an interesting and useful set of mo dels. But neither of these achievements is enough. We need to build up a large collec- tion of mo dels and algebras, covering a wide range of computational paradigms, appropriate for implementation either in hardware or in software, either of the present day or of some p ossible future.
However the prosp ects for the use of algebraic techniques in the design and veri cation of computer based systems are promising if the techniques can b ecome as familiar to engineers as scho ol algebra is to many to day.
References
[1] J. Gattegno. Lewis Carrol l: Fragments of a Looking Glass. Thomas Y. Crowell Company, New York, 1976. Translated by R. Sheed.
[2] R.T. Gunther. Chemistry, Mathematics, Physics and Surveying, volume I of Early Science in Oxford. Oxford, 1923. Printed for the Oxford Historical So ciety at the Clarendon Press.
[3] R.T. Gunther. Oxford Col leges and their Men of Science, volume XI of Early Science in Oxford. Oxford, 1937. Printed for the author.
[4] C.A.R. Hoare. Re nement algebra proves correctness of compiling sp eci cations. In C.C. Morgan and J.C.P. Wo o dco ck, editors, 3rd Re nement Workshop, Workshops in Computing, pages 33{48. Springer-Verlag, 1991.
[5] C.A.R. Hoare. Algebra and mo dels. ACM Software Engineering Notes, 18(5):1{8, Decemb er 1993.
[6] C.A.R. Hoare et al. Laws of programming. Communications of the ACM, 30(8):672{ 687, August 1987.
[7] A. Ho dges. Alan Turing: The Enigma. Simon & Schuster, New York, 1983.
[8] R.A. Hyman. Charles Babbage: Pioneer of the Computer. Oxford University Press,
[9] N.K. Nichols. Obituary: Professor Leslie Fox, C. Math., FIMA. IMA Bul letin, 29:175{ 176, Novemb er/Decemb er 1993.
[10] A.W. Rosco e and C.A.R. Hoare. Laws of Occam programming. Theoretical Computer Science, 60:177{229, 1988.
