Scottish School Reforms of 1870s.

The report of the Royal Commission on Education in Scotland of 1867 presents a vivid and detailed picture of the educational conditions of the whole country at that time. One can see from this that Scotland had been in possession of a National system of Education for nearly two hundred years. The Education Act of 1872 gave effect to the recommendations of the Commission, and the Act improved primary education. Its object was to provide education for "the whole people of Scotland" and not merely for the labouring classes as was implied in the English measure.

In Scotland there were basically two kinds of schools. The parish schools, which originally were purely elementary, were encouraged to provide at least the elements of secondary education. These schools played this role so well, that the Argyle Commission in its report of 1868 reported that over fifty per cent of the students attending the four Scottish universities came direct from parish schools.

The burgh or grammar schools, which were the true secondary schools, owing to the competition of the parish schools, were compelled to open their doors to primary pupils who were prepared to pay increased fees for the privilege. It is in this way that both types of schools became universal education providers, and gave to Scotland an education system far removed from the highly specialised character of continental schools. The general effect of this policy was to depress secondary education in the higher class reaches, but greatly to raise the level for the whole country. Through it, indeed, Scotland possessed for more than two hundred years the most democratic education system in the world, and to a considerable extent in consequence of this it has enjoyed an influence and importance in the world altogether out of proportion to its size and population.

Since 1872 repeated efforts were made to remedy the more glaring defects in the original Act, some of them successful. As an example, The Education Act of 1878 empowered the Education Department to conduct inspection of all higher class schools, but state inspection was not carried out because of financial difficulties until after the reorganisation of the Scottish Education Department in 1885. The Scottish Education Department was set up in 1839, but it only became effective from 1872 when a separate committee of the privy council was set up to administer the Scottish Education Department.

In 1882 The Educational Endowments (Scotland) Act was passed, under which such inspection was extended to all endowed schools and a commission consisting of seven commissioners was established, with Lord Balfour of Burleigh being chairman. Alexander Gibson, was appointed to be the secretary to the commissioners. The commissioners were provided full powers necessary for their job.

Chrystal was not a commissioner, but was a friend of the secretary of the commission Alexander Gibson. Many questions the commission considered were frequently discussed by Gibson and Chrystal, sometime their friend Professor Robertson Smith, the famous theologist who often in Edinburgh, being part of the discussions. Gibson always found their opinions helpful and worthy of the consideration of his commission. These discussions gave Chrystal a renewed interest in educational reform. In 1877 Chrystal had remarked with concern that secondary education had not kept pace with primary education, but had, on the whole taken a step backwards. He said that secondary schools were dying while even those with money were far from efficient. The universities, Chrystal said, were "wholesomely prosperous", their standard, like that of secondary schools was:-

... below the level of the cultured nations of Europe.
His thoughts soon led him to become more and more an advocate of extending the policy of state aid to secondary schools. With the advent of an independent Education Department, Scottish Education once more resumed its onward course. As a result of this administrative change Chrystal, along with some other Scottish professors, took part in the inspection of secondary schools over a period of several years. The inspectors appointed by the Education Department in 1886 to investigate the conditions in the higher class schools presented a somewhat depressing report: the staff were found to be inadequate and underpaid, the curricula far behind the times, and the methods antiquated and ineffective.

The Royal Society Of Edinburgh and the purchase of 22-24 George Street

Professor George Chrystal's close connection with the Royal Society of Edinburgh (RSE) began with his appointment to the chair of Mathematics at the University of Edinburgh and continued throughout the rest of his life. He was asked to address the Society in 1879, the year of his appointment to Edinburgh, then he was elected a Fellow of the Royal Society of Edinburgh at its meeting on Monday, 2nd February 1880. In November of that year he was elected to the Council of the Society for the first of three terms of office: 1880-3; 1884-7; and 1895-1901. On the death of Professor Tait in 1901, Chrystal succeeded him as General Secretary of the RSE. He showed great loyalty to the Society and most of his original contributions to science appear in its publications.

The most important achievement of Chrystal's time as General Secretary was to acquire for the RSE a permanent site for its offices and library at 22-24 George Street, Edinburgh, accommodation it still occupies today although it has recently acquired the adjacent property in addition. The RSE had occupied for eighty years the building in Princess Street known as the Royal Institution, the west wing of which had been planned for the RSE when the building was erected. The RSE had been one of the bodies to propose this building in 1821 and, since the completion of the building, had shared it with the Society of Antiquaries and the Royal Institution for the Encouragement of the Fine Arts. For those familiar with Edinburgh today and interested in identifying the Royal Institution, we should say that it has been named the Royal Scottish Academy since 1911.

By the beginning of the 20th century, the need of more accommodation for the RSE's unique library required some change in accomodation at the Royal Institution. By this stage the RSE were the sole occupants of the Royal Institution, the Society for the Promotion of Fine Arts having ceased to exist and the Society of Antiquaries having left in 1892. In 1903, through the initiative of Sir John Murray, a scheme was proposed to secure the whole of the Royal Institution building for the RSE. The RSE would, under this scheme, have been able to accommodate its valuable library, and it was also intended that other scientific societies share the accommodation. The proposal was therefore one which would give the whole of the Royal Institution building over to scientific use. A representative committee was formed, and the Secretary of State for Scotland approached with the proposal. Mr Graham Murray, later Lord Dunedin, met the deputation, and spoke very sympathetically in favour of the whole idea. In 1906 a Liberal Government was elected and one of its first acts was to introduce the National Galleries of Scotland Bill. The Bill was proposed by Mr John Sinclair, later Lord Pentland, the Secretary for Scotland, and introduced into the House of Commons. The Bill was intended to promote the Arts in Scotland and set up an art gallery, one of the National Galleries of Scotland, in the Royal Institution building. There was no provision in the National Galleries of Scotland Bill to provide accommodation for the RSE. That legislation would be framed with such an obvious omission was probably no surprise to Chrystal who had already shown his opinion of legislators when he wrote earlier:-

We all have a great respect for the integrity of British legislators, whatever doubts may haunt us occasionally as to their capacity in practical affairs. The ignorance of many of them regarding some of the most elementary facts that bear on everyday life is surprising. Scientifically speaking, uneducated themselves, they seem to think that they will catch the echo of the fact or the solution of an arithmetical problem by putting their ears to the sounding-shell of uneducated public opinion.

The Royal Society of Edinburgh now had to fight for recognition that it required accommodation and Chrystal, as its General Secretary, had to organise the Society's case. The Secretary of State for Scotland received a first deputation, which consisted of Fellows of the Society led by its President Lord Kelvin, in Edinburgh. In his reply, while sympathising with the Society's aims, the Secretary of State expressed the opinion that the Society was not supported by members of parliament or other public bodies and, worst of all as far as a politician was concerned:-

... it was not supported by the body of public opinion.

This clearly required Chrystal to set to work to have such support on the side of the RSE before a second deputation was to be received in London on Thursday, 22 November 1906. Chrystal approached Scottish members of parliament from all political parties. He looked for support from the Royal Society of London and wrote to Sir Joseph Larmor on 13 November 1906 [Royal Society of London Library (MSS. Lm263).',3)">3]:-

I hope you will by your presence support a deputation next week to the Secretary for Scotland to make a last appeal for justice to the Royal Society of Edinburgh in the matter of its accommodation. We are to be expelled without any previous consultation from the rooms that were built for us and which we have occupied for eighty years; and now government proposes to put us into a miserably inconvenient house in a bad situation and to give for our installation a sum which at the highest computation is less than half of what is necessary. The only compensation being a shadowy promise to remedy another grievance of thirty years standing by giving us a publication grant of pounds300. Which amounts to enlarging the blanket by cutting off the top and sewing it to the tail. The result of course would be the financial ruin of the Society. Our annual deficit is already about pounds300 and we are paying for our publications partly out of capital.

The whole thing is the result of an intrigue by some friends of the R. S. A. You will have a similar trick played on you in London some day.

Two days latter in another letter to Sir Joseph Larmor he wrote [Royal Society of London Library (MSS. Lm264).',4)">4]:-

The deputation is fixed for Thursday, 22nd at 12.30 in Dover House. I can not see that you have no locus standi in a matter affecting one of the oldest scientific societies in the kingdom. If I heard of a proposal to evict the Royal Society of London to make way for the R. S. A. and to transplant the former to inferior rooms, say, in Bloomsbury or London Tower, I should certainly come up to London and join a deputation to Government to protest against such an enormity, although I am not a member of the R. S. L. and not an Englishman.

We are going to the Secretary who is not likely to know you by right. All we want is the outward expression of your sympathy; and that would be of great value to us just now. I hope you will think better of it. I shall be in London from Tuesday morning early till Thursday night, and my address will be my son's rooms 78 St George's Square.

The next day Chrystal again wrote to Sir Joseph Larmor, replying to Larmor's letter of 15 November 1906 [Royal Society of London Library (MSS. Lm265).',5)">5]:-

Many thanks for your kind letter of 15th and promise to countenance our demonstration. Thursday 22nd at 12.30 in the Scottish Office Whitehall is the hour and place. It is kind of the Royal Society of London to take up our cause. It is taking its proper place in so doing for it is the mother of scientific societies and therefore leader of them all.

The second deputation, again headed by Lord Kelvin, consisted of Fellows of the Royal Society of Edinburgh, Scottish members of parliament, Fellows of the Royal Society of London and other eminent scientists. The deputation achieved all that had been hoped for and the National Galleries Bill of Scotland had a clause added to it allocating sums of money for the purchase and equipment of buildings for the RSE. The Bill, with this amendment, was passed and became law on 21 December 1906.

Chrystal discovered that 22-24 George Street could be purchased by the Society. Sir William Turner persuaded [The Royal Society of Edinburgh 1783-1983 (Edinburgh, 1983).',1)">1]:-

... the Secretary for Scotland, Lord Pentland, that the Treasury granted the necessary pounds25 000 for purchase of 22-24 George Street and pounds3 000 to cover the cost of removal and equipment.

John Horne in The Student, 11 July 1916, stated:-

The present rooms may not inaptly be regarded as a monument to two distinguished men, Chrystal and Turner.

Lord Kelvin, the President of the RSE, died in 1907 and Sir William Turner was elected President. In his Presidential Address on 8 November 1907, Turner said:-

It is due to Lord Pentland that we should record our sense of his courtesy at our interviews with him, as well as our hearty thanks for the effective advocacy of our claim to obtain the requisite funds from the Treasury, both for the purchase and equipment of our habitation and for an annual grant of pounds600 to assist in the discharge of our scientific work. We are now, therefore, no longer tenants-at-will of apartments, to be dispossessed on short notice; we sit rent free in a handsome and commodious building, and with our occupancy ensured by a parliamentary title. We have a lecture theatre equipped with modern appliances for the illustration of the subjects from time to time discussed in our meetings; we are provided with ample library accommodation and with storage and safety for our publications and manuscripts of value. We have reading and other rooms for the use of the Fellows and the officials, and a house of residence for the caretaker. It is sometimes said that history repeats itself, a saying which in one particular applies to that of our Society. In 1810 the Society purchased No.40 George Street, in which house it was accommodated until 1826, when it removed to the Royal Institution Buildings. George Street again provides us with a home, larger, more dignified, and more fully adapted to our present needs than the house purchased by the Society a hundred years ago, and with much more accommodation than was at our disposal in the rooms in the Royal Institution which we have just vacated...

Other MacTutor references

History of the Royal Society of Edinburgh

References (5 books/articles)

Royal Society of Edinburgh

The Royal Society of Edinburgh was founded in 1783. It has played an important role in the scientific and literary life of Scotland in the years since its foundation and, with the recent re-establishment of a Scottish Parliament, it is now becoming even more important as an advisory body. In this article we discuss the way in which the Society came into being and, in particular, we examine the way in which some of the mathematicians whose biographies are in this archive were involved with the Society.

Although Maclaurin died nearly forty years before the founding of the Royal Society of Edinburgh he played a very important part. The Society for the Improvement of Medical Knowledge was founded in Edinburgh in 1731 and Maclaurin became one of its members. Not happy with such restricted topics, he worked to expand the Medical Society of Edinburgh into a wider society to include other branches of learning. Perhaps he already had in mind a society broadly similar to the Royal Society of London of which he had been a Fellow since 1719.

In 1737 the broader Society was formed with the full title "Edinburgh Society for Improving Arts and Sciences and particularly Natural Knowledge". As one might expect this was far too long a title for people to use, and the Society was known as the Philosophical Society of Edinburgh. Maclaurin himself acted as one of the two secretaries of this expanded Society and at the monthly meetings he often read a paper of his own or a letter from a foreign scientist on the latest developments in some topic of current interest.

The Philosophical Society of Edinburgh was not the only Edinburgh Society of which Maclaurin was a member. He also belonged to the Rankenian Club which met in Ranken's Inn in Edinburgh. This Club was founded in 1716 nearly ten years before Maclaurin was appointed to the University of Edinburgh, and it was a Club which suited Maclaurin with its mixture of congenial fellowship and the aim of its members in pursuing knowledge. It is generally accepted that the Philosophical Society of Edinburgh was the major player in the move towards the establishment of the Royal Society of Edinburgh, but undoubtedly the Rankenian Club played its part.

Shapin describes the events which led to the founding of the Royal Society of Edinburgh in [2], see also his paper [4]. The year 1782 was crucial in its foundation. That was in the year in which the Society of Antiquaries of Scotland, at this time quite a new Society, decided to seek a Royal Charter. The Society had fairly broad aims which it stated as:-

... the cultivation of both antiquities and many aspects of general knowledge.

Both the University of Edinburgh and the Faculty of Advocates worried about this proposal by the Society of Antiquaries for they saw an expanded Society with a Royal Charter as a competitor to them in certain of its functions. In a move to counter this proposal the Professor of History at the University of Edinburgh, John Walker, proposed that the University of Edinburgh, the Faculty of Advocates, the Philosophical Society of Edinburgh, and the Society of Antiquaries of Scotland should seek a Royal Charter to establish the Royal Society of Edinburgh. The proposal drawn up by Walker was entitled:-

Proposal for establishing at Edinburgh, a Society for the advancement of Learning and Useful Knowledge.

The Society of Antiquaries of Scotland of course realised that this was an attempt to head off their proposal and they strongly objected to Walker's document. There were meetings which were designed to smooth the way forward and overcome the objections which were now put up by the Society of Antiquaries. Unable to agree, a meeting between the Rev Robertson, Principal of the University of Edinburgh, and Buchan, the leader of the Society of Antiquaries, was suggested. This meeting between the two men took place but broke up in total disarray with considerable anger on behalf of Buchan.

This left two separate parties who now both aimed to set up a Royal Society in Edinburgh. The Philosophical Society of Edinburgh proposed to the University of Edinburgh that they go it alone in setting up the Society. The Faculty of Advocates and a Member of Parliament for Edinburgh made preliminary enquiries in London and advised the University of Edinburgh to proceed. The authors of [1] write:-

The Senatus [of the University of Edinburgh] met on 30 November 1782, a petition was submitted to the King, and on 29 March 1783 the King's signature was obtained. On 6 May 1783, the Royal Charter and that of the Society of Antiquaries were extended under the Great Seal in Edinburgh.

The first meeting of the new Royal Society of Edinburgh took place in the Old Library of Edinburgh University on Monday 23 June 1783. It was decided that all members of the Philosophical Society of Edinburgh should automatically become Fellows of the Royal Society of Edinburgh. All accepted except, as one might have anticipated, Buchan. Others including professors from the other Scottish universities, were invited to the Fellowship. From its beginnings the Society set itself up as a Scottish wide Society, based in Edinburgh.

The beginning of the Royal Society of Edinburgh was described in the first volume of the Transactions of the Royal Society of Edinburgh published in 1788. It is factually correct but omits any of the drama involved in the setting up of the Society which we have described above:-

About the end of the year 1782 in a meeting of the Professors of the University of Edinburgh, many of whom were likewise members of the Philosophical Society and warmly attached to its interests, a scheme was proposed by the Rev. Dr Robertson, Principal of the University, for the establishment of a Society on a more extended plan, and after the model of some foreign academies, which have for their object the cultivation of every branch of science, erudition, and taste.

It is worth noting that Alexander Wilson was one of the founding members of the Society and John Playfair was a member who:-

... for the first two decades of the Society was the life and soul of the institution.

The Royal Society met in the library of Edinburgh University in its early years. In 1810 the Society purchased 42 George Street, and it occupied this building until 1826 when the Royal Institution Building on Princes Street was completed. For about 80 years it occupied part of the Royal Institution Building. The story of how the Society was ejected from the Royal Institution and their subsequent fight for a new home is recounted in our article The%20Royal%20Society%20Of%20Edinburgh%20and%20the%20purchase%20of%2022-24%20George%20Street.From%20its%20beginnings%20the%20Royal%20Society%20of%20Edinburgh%20was%20not%20exclusively%20a%20scientific%20society.%20It%20was%20originally%20set%20up%20with%20a%20Literary%20Class%20and%20a%20Physical%20Class%20and%20fellows%20were%20elected%20to%20one%20of%20these%20two%20classes.%20Until%201828%20there%20were%20two%20Presidents%20and%20two%20Secretaries,%20one%20for%20each%20Class,%20but%20from%20this%20time%20on%20only%20one%20President%20and%20one%20Secretary%20were%20appointed.%20Then%20four%20years%20later%20the%20Classes%20themselves%20were%20joined.%20By%20this%20time%20the%20Society%20was%20almost%20exclusively%20a%20scientific%20Society%20and%20it%20remained%20so%20until%20more%20modern%20times%20when%20it%20again%20recovered%20the%20balance%20between%20scientific%20and%20literary%20Fellows.We%20note%20that%20some%20scientists%20whose%20biographies%20are%20given%20in%20this%20archive%20held%20high%20office%20in%20the%20Society.%20Thomson was President of the Royal Society of London, then, this time as Lord Kelvin, he was President of the Royal Society of Edinburgh for a third time from 1895 until his death in 1907. Two others from our archive held the office of President: D'Arcy Thompson from 1934 to 1939, followed by Edmund Whittaker from 1939 to 1945.

Three mathematicians from our archive served the Royal Society of Edinburgh as General Secretaries: Playfair from 1798 to 1819, Tait from 1879 to 1901, and Chrystal from 1901 to 1912. It was during Chrystal 's time as General Secretary that the Society was forced out of its rooms in the Royal Institution and the story of this is told in the separate article The%20Royal%20Society%20Of%20Edinburgh%20and%20the%20purchase%20of%2022-24%20George%20Street.Other%20points%20worth%20noting%20which%20relate%20to%20mathematicians%20in%20this%20archive%20are%20that%20Joseph Wedderburn was elected a Fellow in 1903 when he was 21 years of age making him one of the youngest Fellows ever elected. Some years later Herbert Turnbull won both the Keith Medal and Gunning Victoria Jubilee Prize.

There is a link below to some of the mathematicians in this archive who have been elected Fellows of the Royal Society of Edinburgh:

References


  1. N Campbell, R Martin and S Smellie, The Royal Society of Edinburgh 1783-1983 (Edinburgh, 1983).
  2. S A Shapin, The Royal Society of Edinburgh : A study of the social context of Hanoverian science (Doctoral Thesis, University of Pennsylvania, 1971).
  3. M Yousuf, George Chrystal (Ph.D. Thesis, University of St Andrews, 1990).
  4. S A Shapin, The Royal Society of Edinburgh, British J. Hist. Sci. 7 (1974), 1-.

MacTutor links:

Fellows of the Royal Society Of Edinburgh (Alphabetical list)
Fellows of the Royal Society Of Edinburgh (Chronological list)

The Royal Society Of Edinburgh and the purchase of 22-24 George Street

Other Web site:

The RSE home page

A visit to James Clerk Maxwell's house

On a grey November day we [JOC and EFR] made the 50 mile train journey from St Andrews to Edinburgh to visit James Clerk Maxwell's house.

The house where James Clerk Maxwell was born is at 14 India Street, Edinburgh about a fifteen minute walk from the railway station which is in the centre of Edinburgh. The house is now owned by the James Clerk Maxwell Foundation who have restored it almost to its original state by removing partitions which had been erected by previous owners. The International Centre for Mathematical Sciences shares the building and organises mathematical meetings there. The Director of Development of the James Clerk Maxwell Foundation, Professor David S Ritchie, showed us the house and the historical documents and other items owned by the Foundation.

You can see a map of the area at the time Maxwell was there and a picture of the house as it is now and the sign outside marking it as Maxwell's birthplace.

The hall of the house is impressive with two marble pillars giving an instant impression of grandeur. The room, entered from a door on the right, is the former dining room. On the wall facing you as you enter are two large portraits, the rightmost one of James Clerk Maxwell, the left most one being a portrait of his school friend P G Tait.

You can see a picture of the hall, a picture of the dining room and the portrait of Maxwell and the portrait of Tait.

Between the portraits is a notice giving a brief history:-

James Clerk Maxwell was born on 13th June 1831 in Edinburgh at 14 India Street, a house built for his father in that part of Edinburgh's elegant Georgian New Town which was built after the Napoleonic Wars. Although the family moved to their estate at Glenlair, near Dumfries, shortly afterwards, James returned to Edinburgh to attend school at The Edinburgh Academy. He continued his education at the Universities of Edinburgh and Cambridge.

The room is surrounded by portraits of James Clerk Maxwell's family and a Display Cabinet near the windows contains a fine collection of items associated with Maxwell. In this article we describe Maxwell's early life in Edinburgh and Glenlair and illustrate it with references to items in the house.

James Clerk Maxwell's father, John Clerk Maxwell, had one brother and one sister.

You can see the family tree of James Clerk Maxwell.

His brother Sir George Clerk inherited one part of the families property at Penicuik, south of Edinburgh, while John Clerk Maxwell inherited the Maxwell estate at Middlebie near Dumfries. There were conditions laid down which prevented the holder of the Middlebie estate also holding the property at Penicuik and the brothers had agreed this split of the property while still schoolboys. John Clerk Maxwell's sister Isabella married James Wedderburn and they were living at 31 Heriot Row in Edinburgh. You can see a picture of this house as it is now. When George Clerk moved to the Penicuik estate, John Clerk Maxwell was left at home with his mother in Edinburgh. He arranged to have a house built at 14 India Street so that they could be nearer to Isabella. The substantial terrace house was built in 1820 and documents relating to the purchase of the house are in the Display Cabinet.

After John Clerk Maxwell's mother died, he married Frances Cay. He had met Frances through a friendship with her brother John Cay with whom he shared an interest in designing machinery and attending meetings of the Royal Society of Edinburgh. John Clerk Maxwell and Frances now chose to move to their estate at Middlebie and they had a house built for them at Glenlair on the estate. Their son James Clerk Maxwell was born in the house at 14 India Street and he would eventually inherit the house on the death of his father, retaining the house throughout his life.

You can see a picture of Glenlair as it was when Maxwell finally left it in 1884.

James Clerk Maxwell's mother, Frances, had a sister Jane Cay who lived at 6 Great Stuart Street, Edinburgh. You can see a picture of this house as it is now. A portrait of Jane and Frances as young girls, painted by their mother Elizabeth Cay, now hangs in 14 India Street. Both sides of the family had remarkable artistic talent. Frances's father was Robert Hodsham Cay LLD (1758-1810), a judge to the High Court of Admiralty of Scotland. His portrait hangs on the wall of the former dining room opposite the window.

John Clerk Maxwell's sister Isabella and her husband James Wedderburn had a daughter Jemima Wedderburn who was an outstanding artist. She was eight years older than James Clerk Maxwell and she painted pictures of the family almost every day, some of which are now displayed in 14 India Street. This pictorial diary records many of the events in James Clerk Maxwell's childhood and some of the pictures and events will be described later.

James was born in the first floor room overlooking the stables. Here is the view from the window of this room taken on the day of our visit. By a remarkable coincidence, the family now living in the house above the former stables is descended from Maxwell's friend P G Tait. Shortly after James's birth, the family moved to their house at Glenlair. The earliest sketch of James is today on view in the Display Cabinet. A letter describing the Boy, as he was called by his family, was written to Jane Cay at 6 Great Stuart Street on 25 April 1834 containing this description:-

He is a very happy man, and has improved much since the weather got moderate; he has great work with doors, locks, keys etc., and 'Show me how it doos' is never out of his mouth. He also investigates the hidden course of streams and bell-wires, the way the water gets from the pond through the wall and a pend or small bridge and down a drain ...

Here is a picture of James aged six, clearly more interested in the violin bow than in the dance. Another picture shows him sitting on the table making a basket. At Glenlair James played with Toby the dog, who he also called by a variety of variants of the name Toby such as Tobin, Tobs or Tobit. This picture of James with Toby is when James was slightly older.

As a child James had toys which would prove important as some would later play a role in deep scientific studies he would make. One of these toys was a stroboscope which rotated in front of a mirror and simulated moving pictures from figures drawn in it which he had from about age eight. He also had a stick, his leaping pole, which he took everywhere with him using it to cross streams, jump over rocks and he quickly learnt to use it with amazing dexterity.

There is a watercolour by his cousin Jemima of him at about this age showing him with his father riding a horse.

James Clerk Maxwell's mother died when he was 9 years old and a 16 year old boy was employed as a tutor to James. Here is a picture which shows Jemima holding James's leaping pole, James is sailing in a tub on the duck pond trying to avoid the tutor who has a garden rake and is trying to stop James. Toby looks on.

The arrangement with the tutor did not work well, probably the tutor was unsuitable for such a talented child as James, filled with curiosity at the world around him. It was decided that James should attend the Edinburgh Academy and in November 1841 the family travelled from Glenlair, stopping a few days at his uncles house in Penicuik, and at other relations at Newton, before reaching Isabella Wedderburn's house at 31 Heriot Row on 18 November. James was mostly to live here, but sometimes at Aunt Jane's in Great Stuart Street, while he attended the Edinburgh Academy.
You can see a picture at Glenlair, the stop at Newton and the arrival in Edinburgh all taken from watercolours by Jemima Wedderburn.
You can see an engraving of Edinburgh Academy as it was when James went there.

Tait [Proc. Royal Soc. Edinburgh 10 (1880), 331-339.',5)">5] relates James Clerk Maxwell's early days at the Edinburgh Academy:-

At school he was at first regarded as shy and rather dull. he made no friendships and spent his occasional holidays in reading old ballads, drawing curious diagrams and making rude mechanical models. This absorption in such pursuits, totally unintelligible to his school fellows, who were then totally innocent of mathematics, of course procured him a not very complimentary nickname...

This nickname was "Dafty" and it must have been given to him largely on account of his appearance. At 14 India Street, in the Display Cabinet, we [JOC and EFR] studied newspaper cuttings with the headline "They called him Dafty". James Clerk Maxwell's father had prepared his son well for education in many ways but, to send him to the Academy dressed in the country clothes he would have worn at Glenlair, shows a lack of understanding of how James's fellow pupils would react. The way they reacted on his first day at school was clear from the state in which he arrived back at 31 Heriot Row [The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).',1)">1]:-

... with his tunic in rags ... his neat frill rumpled and torn; himself excessively amused by his experiences, and showing not the smallest sign of irritation.

Progress at school was reasonable in these first years but nothing spectacular. He spent his summers back at Glenlair and, by Jemima's pictures drawn in the summer of 1843, he seems to have slotted back into his old pastimes.

You can see several pictures from this period.

About this time he was given a "devil - on - two - sticks", a toy which he always had with him from that time on when he was on holiday at Glenlair, on holiday in Glasgow, and he even took it with him when he went to study at Cambridge University. James quickly became an expert at making the devil do wonderful tricks, which almost seemed to defy the laws of physics. This toy, and others he was given as a child, must have had a profound effect on Maxwell. Later he was to apply his mathematical skills to a study of the dynamical top, doubtless still trying to understand the behaviour that had filled him with so much wonder as a child.

You can see a photograph of his "diabolo" and of some of his other toys.

Back in Edinburgh James was taken by his father to a meeting of the Royal Society of Edinburgh on 18 December 1843. In October 1844 both James's father and Mrs Wedderburn were ill and James lived at Aunt Jane's house. Here is a picture of James doing his homework at 6 Great Stuart Street and a picture of his Aunt Jane seated in an adjacent room. The chair in which James sat to study while at 31 Heriot Row is now in the former dining room at 14 India Street, recovered with a material with a pattern depicting the digital nature of light waves, to honour one of Maxwell's great pieces of work.

At the Edinburgh Academy things were about to change [The clacken and the slate (London, 1974).',4)">4]:-

The highest position he achieved in class was fourteenth - until he reached Mr Gloag. Then it became obvious that this boy had a brain for mathematics, and his self-confidence grew so much that he also began to do well in Latin and Greek.

James Gloag was the Master of the Arithmetical and Geometrical School. He had a reputation for discipline and Tait describes how Mr Gloag put his mathematical knowledge to practical use:-

To use a well-known cricketing phrase, Gloag could get 'more work' on the tawse than any of the other masters. His secret was in great part a dynamical one.

Magnusson [The clacken and the slate (London, 1974).',4)">4] speaks highly of Mr Gloag:-

Mr Gloag took the most intense pride and delight in his former pupil's successes; but unlike many masters who concentrated almost exclusively on the cleverest boys and let the other languish, Mr Gloag's sense of duty impelled him to try and make mathematicians out of even the most backward. In a way his tyranny was a sign that he cared about them all - the constant vigilance that ensured that he never missed a turn of the head and could switch his attention in a twinkling from a problem in Higher Mathematics at the top of a class to a boy struggling with vulgar fractions on the lowest bench.

Maxwell began to comment on mathematical topics in his letters. He wrote on 19 June 1844:-

I have made a tetrahedron, a dodecahedron and two more hedrons that I don't know the right names for.

By July 1845 Maxwell had won the Mathematics Medal. Tait writes [Proc. Royal Soc. Edinburgh 10 (1880), 331-339.',5)">5]:-

About the middle of his school career however he surprised his companions by suddenly becoming one of the most brilliant among them, gaining prizes and sometimes the highest prizes for scholarship, mathematics, and English verse composition. From this time forward I became very intimate with him, and we discussed together, with schoolboy enthusiasm, numerous schoolboy problems, among which I remember particularly the various plane sections of a ring or tore, and the form of a cylindrical mirror which should show one his own image unperverted.

If Maxwell's progress in mathematics had been outstanding, better was to come. By early 1846 he was working on his own researches on ovals. John Clerk Maxwell writes in his diary for Thursday 26 February 1846:-

Call on Prof. Forbes at the College and see about James's ovals and 3-foci figures and plurality of foci. New to Prof. Forbes, and settle to give him the theory in writing to consider.

It appears that two accounts of this work on ovals by James Clerk Maxwell were written. One by his father, as he states in his diary entry and this is now in the possession of the Royal Society of Edinburgh, the other by James himself and this copy is on view in the Display Cabinet at 14 India Street. The paper was accepted by the Royal Society of Edinburgh and read to the Society on 6 April 1846. The paper was printed in the Proceedings of the Royal Society of Edinburgh and here is Plate XI as it appears in the paper

Tait writes [Proc. Royal Soc. Edinburgh 10 (1880), 331-339.',5)">5]:-

I still possess some of the manuscripts we exchanged in 1846 and early 1847. Those by Maxwell are on 'The Conical Pendulum', Descartes' Ovals', 'Meloid and Apioid', and 'Trifocal curves'. All are drawn up in strict geometrical form, and divided into consecutive propositions. The three latter are connected with his first published paper, communicated by Forbes to this Society and printed in our Proceedings, vol. ii, under the title 'On the description of Oval Curves, and those having a plurality of Foci' (1846). At the time when these papers were written he had received no instruction in mathematics beyond a few books of Euclid and the merest elements of algebra.

On our [JOC and EFR] visit to 14 India Street, we were fascinated when we were given the chance to examine the notebook containing the manuscripts to which Tait referred. The notebook contains work which was written with differing degrees of care.

Here is a page from it.

Much of the manuscripts are written in beautiful calligraphy. Great care has been taken with these parts and pencil lines have been drawn in and then removed to ensure that all lines are straight and are perfectly left justified. Most of the manuscripts are by Tait and signed 'fecit P G Tait' with a date. Some, in particular the ones that Tait refers to above, are by Maxwell and signed and dated by him. For example the manuscript on the Conical Pendulum is signed by Maxwell and dated 25 May 47. There are some parts of the notebook written in ordinary handwriting rather than the calligraphy of much of the notebook.

Other than the topics of Maxwell's described above by Tait, there are also manuscripts by Tait on Vanishing Fractions which is l'Hôpital's rule, a manuscript on Maclaurin's Theorem and On the imaginary roots of negative quantities by the Rt Rev Terrot. This last manuscript is signed and dated by Tait - 27 May 47, this being two days after Maxwell's Conical Pendulum manuscript. There is a manuscript on Fagnano's property of the Ellipse and Finding the mass of Saturn's rings. This last manuscript is particularly interesting given that Maxwell would produce his prize winning Adams prize essay on Saturn's rings ten years later. Other surprises which we noticed looking through the notebook were that e is given to 7 decimal places and π is given to 36 decimal places. With π it looks as if Tait has only stopped writing down the decimal places when he has reached the edge of the page.

You can see a model of Saturn's rings that Maxwell made.
The Display Cabinet in the former dining room contain many other interesting items. There is a fine collection of biographies of Maxwell. Some books are open at a page which shows a portrait of him. Here are a number of further portraits.

Portrait of Maxwell as a child
Portrait as a young man
Portrait
Watching dancing

There is also a copy of one of Maxwell's most famous papers with the remarkable words highlighted:-

This velocity is so nearly that of light, that it seems we have strong reasons to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

Maxwell had his lighter side too. He was fond of writing poetry. For example [The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).',1)">1] he wrote The Song of the Edinburgh Academy in 1848:-

If ony here has got an ear,
He'd better tak' a hand o' me
Or I'll begin, wi' roarin' din,
To cheer our old Academy.

Dear old Academy,
Queer old Academy,
A merry lot we were, I wot,
When at the old Academy.

There's some may think me crouse wi' drink,
And some may think it mad o' me,
But ither some will gladly come
And cheer our old Academy.

Some set their hopes on Kings and Popes,
But, o' the sons of Adam, he
Was first, without the smallest doubt,
That built the first Academy.

Let Pedants seek for scraps of Greek,
Their lingo to Macadamize;
Gie me the sense, without pretence,
That comes o' Scots Academies.

Let scholars all, both grit and small,
Of Learning mourn and sad demise;
That's as they think, but we will drink
Good luck to Scots Academies.

Maxwell also wrote mathematical poetry. While in his final year of study for the Mathematical Tripos at Cambridge he wrote a poem A Problem in Dynamics [The life of James Clerk Maxwell with selections from his correspondence and occasional writings (London, 1884).',1)">1] which begins:-


An inextensible heavy chain
Lies on a smooth horizontal plane,
An impulsive force is applied at A,
Required the initial motion of K.

Let ds be the infinitesimal link,
Of which for the present we've only to think;
Let T be the tension, and T + dT
The same for the end that is nearest to B.

Let a be put, by common convention,
For the angle at M 'twixt OX and the tension;
.......

In the Display Cabinet one book of poems is open at the following poem which has been written in a style similar to many poems written by Maxwell himself:-

Energies through the ether flow,
Waves travel to and fro,
And with a ratio
Their speed you measure.
Colours yield their secret hue,
And Saturn's rings subdued by you
Suggest that gases
Might be measured too.

Science you freed
From cramping mechanistic creed,
And by your theory brought
The elastic solid ether to naught,
And changed the axiomatic basis
Of scientific thought.

Oh Maxwell! How can I declaim
On such a genius, such a fame,
And speak of one so very wise
Who saw the world through splendid eyes,
And though of such a subtle mind
Was yet so humorous and kind?
Yours was a mind unique and rare
That, nurtured in a northern air,
Struck out new paths in many ways
Through all too short, yet fruitful days.
How can one capture in a line
Something so great, so pure, so fine?

Give thanks,
That such a man drew breath,
And lament with all the world
His early death.

One final comment must be made about our visit to 14 India Street. There is on display the Stairway Gallery of framed engraved portraits of famous mathematicians and physicists. These begin at the bottom of the stairs and are arranged in chronological order as one ascends the two flights. You can see the view looking down the staircase in this picture. Many of these engravings are from the personal collection of Sir John Herschel which was sold at a Sotheby auction on 3-4 March 1958.

We end this article with some quotes concerning the importance of James Clerk Maxwell's work. The first, by Richard P Feynman, is displayed between the portraits of Tait and Maxwell in the former dining room of Maxwell's house:-

From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics.

Also displayed at the same location is the quote by Albert Einstein:-

One scientific epoch ended and another began with James Clerk Maxwell.

Another quote, this time by Sir J J Thomson, concerns one of Maxwell's discoveries [James Clerk Maxwell : A Commemorative Volume 1831-1931 (Cambridge, 1931), 1-44.',6)">6]:-

The discovery of electrical waves has not merely scientific interest though that alone inspired it. ... it has had a profound influence on civilization; it has been instrumental in providing the methods which may bring all inhabitants of the world within hearing distance of each other and has potentialities social, educational and political which we are only beginning to realize.

Actually this quote by Sir J J Thomson, written as long ago as 1931, is remarkable in almost predicting the Internet.

Sir James Jeans wrote [James Clerk Maxwell : A Commemorative Volume 1831-1931 (Cambridge, 1931), 91-108.',3)">3], also in 1931 on the centenary of Maxwell's birth:-

... many think that Maxwell's study of the particles of Saturn's rings led him directly and inevitably into the realm of the kinetic theory of gases, in which so much of his life was spent. However this may be, when he crossed the bridge from Astronomy to Physics he left behind him for ever the prospect of becoming a great astronomer - but only to become the greatest mathematical physicist the world has seen since Newton.



References used in this article

Further references on Maxwell and his work.

You can see a full biography of James Clerk Maxwell and a biography of P G Tait.

References (6 books/articles)

Mathematics in St Andrews to 1700

The University of St Andrews was founded by Bishop Henry Wardlaw. His charter of incorporation is dated 28 February 1412 (1411 according to the Scottish calender which had a year start of 25 March until 1600) and he set up the University partly for prestige but mainly so that students could be educated for the Church. Prior to this bishops in St Andrews had provided funds to send their students to the universities of Bologna, Paris and Oxford but the political situation at the time made it increasingly difficult to continue this practice. A Papal Bull of Foundation was issued on 28 August 1413 by Pope Benedict XIII who wrote:-

... considering also the peace and quietness which flourish in the said city of St Andrews and its neighbourhood, its abundant supply of victuals, the number of its hospices and other conveniences for students, which it is known to possess, we are led to hope that this city, which the devine bounty has enriched with so many gifts, may become the fountain of science...

The first Rector, who was also Dean of the Faculty of Arts, was Laurence of Lindores. He was a scholar of some standing since his commentary on Aristotle's Physics was used in many centres of learning in Europe. Laurence of Lindores was a Nominalist. This means that he believed that the existence of a word describing a property of a set of objects did not mean that there were any objects satisfying this property. The opposing camp consisted of the Realists whose views were based on those of Plato and Aristotle. Albertus had written a Realist commentary on Aristotle which was widely used but the Faculty of Arts in St Andrews banned Albertus's book in 1418.

Little of originality came out of the first 200 years of the University, in fact 250 years was to pass before mathematical research was to flourish. The 4 year degree in Arts in the 15th Century took in pupils of 13 years of age and their main study was Aristotle. His works on Logic, Physics and Natural Philosophy and Metaphysics were studied in the second, third and fourth years respectively.

John Maior, renowned for his work in philosophy, logic and in particular on infinity, lectured in theology at St Andrews from 1531 until 1534 when he became the Provost of St Salvator's, a post he held until his death at the age of 80 in 1550.

Napier entered the University of St Andrews in 1563 at the age of 13. However although St Andrews gave him a versatile and international outlook he seems, like most students, to have learnt more about religion than mathematics at St Andrews. Reforms in 1574 meant that specific duties were allocated to professors in each college and a master in St Mary's College was to act as Professor of Mathematics. In 1579 further reforms, led by Andrew Melville a Protestant extremist, were brought in mainly with the motive of making the University a Presbyterian showpiece. For the first time professors with specialist subjects were appointed, and one of these was in mathematics, now attached to St Salvator's College and not to St Mary's College.

Mathematical research at the University of St Andrews really began in 1668. Sir Robert Moray, a graduate of St Andrews and a founder member of the Royal Society, and the Archbishop of St Andrews James Sharp, influenced the king, Charles II, to found a Regius Chair of Mathematics at St Andrews. James Gregory, who had been elected a Fellow of the Royal Society on 11 June 1668 and was a friend of Moray, was appointed the first holder of the chair. Gregory was not attached to a College, as were the other professors, but given the Upper Hall of the university library as his place of work. The building of the library had begun in 1612 but the work was only completed in 1643. It was, at that time and for the next 200 years, the only university building which was not part of a college so in a sense the only possible place for an unattached professor.

Gregory found that St Andrews was of classical outlook where the mathematical work of Newton and his continental colleagues was unknown. Gregory gave two public lecture each week which were not well received:-

...I am often troubled with great impertinences: all persons here being ignorant of these things to admiration.

However Gregory was to carry out much important mathematical and astronomical work during his six years in the Regius chair. He kept in touch with current research by corresponding with John Collins. Gregory preserved all Collins' letters, writing notes of his own on the backs of Collins' letters. These are still preserved in the St Andrews University library and provide a vivid record of how one of the foremost mathematicians of his day made his discoveries.

Collins sent Gregory Barrow's book and, within a month of receiving Barrow's book, Gregory was sending Collins results which would give him an excellent claim to be a co-inventor of the calculus. In February 1671 he discovered Taylor's theorem (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671. The notes Gregory made in discovering this result still exist written on the back of a letter sent to Gregory on 30 January 1671 by an Edinburgh bookseller. Collins wrote back to say that Newton had found a similar result and Gregory decided to wait until Newton had published before he went into print. Had he not been so modest Gregory would have attained even more fame!

The feather of a sea bird was to allow Gregory to make another fundamentally important scientific discovery while he worked in St Andrews. The feather became the first diffraction grating but again Gregory's respect for Newton could have prevented him going further with this work. He wrote:-

Let in the sun's rays by a small hole to a darkened house, and at the hole place a feather (the more delicate and white the better for this purpose), and it shall direct to a white wall or paper opposite to it a number of small circles and ovals (if I mistake them not) whereof one is somewhat white (to wit, the middle which is opposite the sun) and all the rest severally coloured. I would gladly hear Mr Newton's thoughts of it.

The Upper Room of the library had an unbroken view to the south and was an excellent site for Gregory to set up his telescope. A bracket on the wall beside a window, still with its adjustment screw, remains of where Gregory's telescope was set up. He drew a north/south line across the floor (the line was transferred to the present floor). He lined up his telescope on a wooden pillar he built on the ridge of a hill just over a mile to the south. Gregory hung his pendulum clock on the wall beside the same window. The clock, made by Joseph Knibb of London, was purchased in 1673 and originally hung on the library wall. A few years later it was made into a long-cased clock, in which form it remains today. It is important to realise what an early clock this is. Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock.

Here is the upper room of the library.

Gregory's clock and its clockface.

In 1674 Gregory cooperated with colleagues in Paris to make simultaneous observations of an eclipse of the moon and he was able to work out the longitude for the first time. However he had already begun work on an observatory. In 1673 the university allowed Gregory to purchase instruments for the observatory, but (things have not changed much!) told him he would have to make applications and organise collections for funds to build the observatory. In fact Gregory went home to Aberdeen and took a collection outside the church doors for money to build his observatory. On 19 July 1673 Gregory wrote to Flamsteed, the Astronomer Royal, asking for advice and contemplated using St Rule's tower for some of his instruments. St Rule's tower still stands among the ruins of the Cathedral in St Andrews and must look exactly as it did in Gregory's day.

St Rules tower in about 1680 and St Rule's tower today.

A new observatory was built in St Andrews several hundred metres south of the university library, probably to Gregory's specification, but almost certainly Gregory had left St Andrews before it was completed. In fact Gregory left St Andrews for Edinburgh in 1674. His reasons for leaving again paint a sorry picture of prejudice against the brilliant mathematician. Writing after taking up his Edinburgh chair Gregory said:-

I was ashamed to answer, the affairs of the Observatory of St Andrews were in such a bad condition, the reason of which was, a prejudice the masters of the University did take at the mathematics, because some of their scholars, finding their courses and dictats opposed by what they had studied in the mathematics, did mock at their masters, and deride some of them publicly. After this, the servants of the colleges got orders not to wait on me at my observations: my salary was also kept back from me, and scholars of most eminent rank were violently kept from me, contrary to their own and their parents wills, the masters persuading them that their brains were not able to endure it.

William Sanders was appointed to the Chair of Philosophy at St Andrews in 1672. He actively supported Gregory in his plans for the new Observatory and, when Gregory left for Edinburgh, Sanders was appointed in 1674 to the Regius Chair of Mathematics. Sanders, although in no way of Gregory's stature, was at least also a believer in the new scientific ideas as he demonstrated in a book published in in 1674. During his period as Regius Professor he also published The Elements of Geometry in 1686.

Sanders resigned the Regius Chair in 1688 to become a schoolmaster in Dundee. It is unclear why he should have made such a move and it is equally unclear why James Fenton should have been appointed to the Chair in 1689. He is thought to have been a graduate of St Andrews but failed to bring distinction since he appears to have been sacked in 1690 and expelled from the University one year after being appointed. Whether there were then legal problems in filling the Regius Chair is unknown but it remained unfilled until 1707 after Fenton's death.

In 1707 Charles Gregory, the son of James Gregory's brother David, was appointed to the Regius Chair. Charles Gregory, only 26 years old when appointed, held the Chair for 32 years. A graduate of Oxford where he had an excellent reputation, he followed Newton's ideas and was a good teacher, but was far from his uncle in research ability or inclination. In 1714 the University Senate introduced a new course 'Experimental Philosophy' which was to teach the

great improvements made in late years in Natural Philosophy and Mathematics by means of experiments.

New apparatus was specially bought for the course which was taught by Charles Gregory. The Senate, looking to bring in funds to help with the purchase of the apparatus, allowed subscribers to the apparatus fund to attend the experiments free of charge!

The observatory set up by James Gregory was dismantled in 1736 and does not seem to have been used by Charles. Charles Gregory held the Regius Chair until his son was old enough to succeed him, which he did in 1739.
Reference (One book/article)

The History of the American Mathematical Society

It appears that the idea for a mathematical society in the United States came from a visit T S Fiske made to England where he visited Cambridge. He arrived with letters of introduction to Cayley, Glaisher, Forsyth and Darwin. Fiske writes:-

Scientifically I benefited most from my contacts with Forsyth and from my reading with Dr H W Richmond, who consented to give me private lessons. However, from Dr Glaisher, who made me an intimate friend, who spent many an evening with me in heart to heard talks, who took me to meetings of the London Mathematical Society and the Royal Astronomical Society, and entertained me with gossip about scores of contemporary and earlier mathematicians, I gained more in a general way than from anyone else.

Back in New York Fiske organised a meeting on 24 November 1888 to discuss creating the New York Mathematical Society. He invited to the meeting his fellow students Jacoby and Stabler, his professor from Columbia University J H Van Amringe, Professor Rees and a graduate student Maclay.

The Society was set up and had only 11 members in its first year with J H Van Amringe as President. J E McClintock joined the Society in December 1889 and was to become its second President. Only these two were Presidents of the New York Mathematical Society for, in 1894, the Society decided to become a national organisation and change its name accordingly to the American Mathematical Society.

The Bulletin was modelled on other journals. Fiske write:-

The external appearance of the Bulletin, the size of its page, and the color of its cover were copied from Glaisher's The Messenger of Mathematics ... The Bulletin's character, however, was influenced chiefly by Darboux's Bulletin des Sciences Mathématique and the Zeitschrift für Mathematik ...

The decision to publish the Transactions of the American Mathematical Society was taken by Fiske, Eliakim Moore, McClintock, Bôcher, Osgood and Pierpont (the first Colloquium lecturer). The name was suggested by Bôcher.

AMS Prizes:

AMS/SIAM Birkhoff Prize
AMS Bôcher Prize
AMS Cole Prize in Algebra
AMS Cole Prize in Number Theory
AMS Conant Prize
AMS Satter Prize
AMS Steele Prize
AMS Veblen Prize
AMS Wiener Prize

Other Web site:

AMS Web site

Mathematics of the Incas

It is often thought that mathematics can only develop after a civilisation has developed some form of writing. Although not easy for us to understand today, many civilisations reached highly advanced states without ever developing written records. Now of course it is difficult for us to know much about such civilisations since there is no written record to be studied today. This article looks at the mathematical achievements of one such civilisation.

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, before the Spanish conquest, was vast. It spread over an area which stretched from what is now the northern border of Ecuador to Mendoza in west-central Argentina and to the Maule River in central Chile. The Inca people numbered around 12 million but they were from many different ethnic groups and spoke about 20 different languages. The civilisation had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design, and administration. Of course even if writing is not required to achieve this level, counting and recording of numerical information is necessary. The Incas had developed a method of recording numerical information which did not require writing. It involved knots in strings called quipu.

The quipu was not a calculator, rather it was a storage device. Remember that the Incas had no written records and so the quipu played a major role in the administration of the Inca empire since it allowed numerical information to be kept. Let us first describe the basic quipu, with its positional number system, and then look at the ways that it was used in Inca society.

The quipu consists of strings which were knotted to represent numbers. A number was represented by knots in the string, using a positional base 10 representation. If the number 586 was to be recorded on the string then six touching knots were placed near the free end of the string, a space was left, then eight touching knots for the 10s, another space, and finally 5 touching knots for the 100s.



586 on a quipu.


For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.

Now it is not quite true that the same knots were used irrespective of the position as would be the case in a true positional system. There seems only one exception, namely the unit position, where different styles of knots were used from those in the other positions. In fact two different styles were used in the units position, one style if the unit were a 1 and a second style if the unit were greater than one. Both these styles differed from the standard knot used for all other positions. The system had a zero position, for this would be represented as no knots in that position. This meant that the spacing had to be highly regular so that zero positions would be clear.

There are many drawings and descriptions of quipus made by the Spanish invaders. Garcilaso de la Vega, whose mother was an Inca and whose father was Spanish, wrote (see for example [The crest of the peacock (London, 1991).',5)">5]):-

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they are all well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger.

Now of course recording a number on a string would, in itself, not be that useful. A quipu had many strings and there had to be some way that the string carrying the record of a particular number could be identified. The primary way this was done was by the use of colour. Numbers were recorded on strings of a particular colour to identify what that number was recording. For example numbers of cattle might be recorded on green strings while numbers of sheep might be recorded on white strings. The colours each had several meanings, some of which were abstract ideas, some concrete as in the cattle and sheep example. White strings had the abstract meaning of "peace" while red strings had the abstract meaning of "war".

As well as the colour coding, another way of distinguishing the strings was to make some strings subsidiary ones, tied to the middle of a main string rather than being tied to the main horizontal cord.



Quipu with subsidiary cords.


We quote Garcilaso de la Vega again [The crest of the peacock (London, 1991).',5)">5]:-

The ordinary judges gave a monthly account of the sentences they imposed to their superiors, and they in turn reported to their immediate superiors, and so on finally to the Inca or those of his Supreme Council. The method of making these reports was by means of knots, made of various colours, where knots of such and such colours denote that such and such crimes had been punished. Smaller threads attached to thicker cords were of different colours to signify the precise nature of the punishment that had been inflicted.

It was not only judges who sent quipus to be kept in a central record. The Inca king appointed quipucamayocs, or keepers of the knots, to each town. Larger towns might have had up to thirty quipucamayocs who were essentially government statisticians, keeping official census records of the population, records of the produce of the town, its animals and weapons. This and other information was sent annually to the capital Cuzco. There was even an official delivery service to take to quipus to Cuzco which consisted of relay runners who passed the quipus on to the next runner at specially constructed staging posts. The terrain was extremely difficult yet the Incas had constructed roads to make the passing of information by quipus surprisingly rapid.

Much information on the quipus comes from a letter of the Peruvian Felipe Guaman Poma de Ayala to the King of Spain, written about eighty years after the Spanish conquest of the Incas. This remarkable letter contains 1179 pages and there are several drawings which show quipus. A fascinating aspect of one of these drawing is a picture of a counting board in the bottom left hand corner of one of them. This is called the yupana and is presumed to be the counting board of the Incas.



This is what the yupana looked like.


Interpretations of how this counting board, or Peruvian abacus, might have been used have been given by several authors, see for example [Bol. Lima No. 11 (1981), 1-15.',9)">9] and [Rev. Integr. Temas Mat. 4 (1) (1986), 37-56. ',11)">11]. However some historians are less certain that this really is a Peruvian abacus. For example [Code of the quipu : A study in media, mathematics, and culture (Ann Arbor, Mich., 1981).',2)">2] in which the authors write:-

It is unclear from Poma's commentary whether it is his version of a device associated with Spanish activities analogous to those of the person depicted or whether he is implying its association with the Incas. In either case, his commentary makes interpretation of the configuration and the meaning of the unfilled and filled holes highly speculative.

It is a difficult task to gain further insights into the mathematical understanding of the Incas. The book [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)">6] by Urton is interesting for it examines the concept of number as understood by the Inca people. As one might expect, their concept of number was a very concrete one, unlike our concept of number which is a highly abstract one (although this is not really understood by many people). The concrete way of conceiving numbers is illustrated by different words used when describing properties of numbers. One example given in [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)">6] is that of even and odd numbers. Now the ideas of an even number, say, relies on having an abstract concept of number which is independent of the objects being counted. However, the Peruvian languages had different words which applied to different types of objects. For example separate words occur for the idea of [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)">6]:-

... the two together that make a pair ...

... the one together with its mate ...

... two - in reference to one thing that is divided into two parts ...

... a pair of two separate things bound intimately together, such as two bulls yoked together for ploughing ...

etc.

This is a fascinating topic and one which deserves much further research. One wonders whether the Incas applied their number system to solve mathematical problems. Was it merely for recording? If the yupana really was an abacus then it must have been used to solve problems and this prompts the intriguing question of what these problems were. A tantalising glimpse may be contained in the writings of the Spanish priest José de Acosta who lived among the Incas from 1571 to 1586. He writes in his book Historia Natural Moral de las Indias which was published in Madrid in 1596:-

To see them use another kind of calculator, with maize kernels, is a perfect joy. In order to carry out a very difficult computation for which an able computer would require pen and paper, these Indians make use of their kernels. They place one here, three somewhere else and eight, I know not where. They move one kernel here and there and the fact is that they are able to complete their computation without making the smallest mistake. as a matter of fact, they are better at practical arithmetic than we are with pen and ink. Whether this is not ingenious and whether these people are wild animals let those judge who will! What I consider as certain is that in what they undertake to do they are superior to us.

What a pity that de Acosta did not have the mathematical skills to give a precise description which would have allowed us to understand this method of calculation by the Incas.

References (12 books/articles)

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