**Betty Ponting**published a two-part article on the history of Mathematics at Aberdeen in

*The Aberdeen University Review.*Both parts appeared in volume

**XLVIII**(1979-80), the first part on pages 26-35 and the concluding part on pages 162-176. We give a version of the article below. We have divided the article into four web pages. On this page we give the second part of the second of the two articles. Here are links to the other parts:

Mathematics at Aberdeen, Part 1

Mathematics at Aberdeen, Part 2

Mathematics at Aberdeen, Part 3

**Developments, Characters and Events, 1717-1860 - Continued**

**Robert Hamilton**

Hamilton, born in 1743, the eighth son of an Edinburgh bookseller, came to Aberdeen after experience in banking, the family paper mill and ten years as rector of Perth Academy, at a time when the need for mathematics was being increasingly recognized. Conditions for the award of a degree, laid down in 1781, required regular attendance at the four standard classes and the first two mathematical classes. In 1784 mathematics joined the other subjects for public examination in February. It was common for students, not otherwise members of the University, to attend the mathematics and natural philosophy classes. The Governors of Robert Gordon's College were granted the right, in 1781, to send four boys each year to join one or both of these classes, in recognition of a gift of fifty guineas for astronomical instruments. A letter of 1787 recorded an instance of *'a common carpenter at Aberdeen'* who applied to attend the lowest mathematical class. On examination *'they found he had taught himself all they could teach him, and instead of receiving him as a student, they gave him a degree'.*

A former rector, John Gray, had left money in 1770 for a bursary, to be awarded annually by examination, to a regular student who had already taken the first two mathematical classes. For the next two years the holder was to study *'all such parts as form the compleat mathematician'.* The legacy also provided for a gold medal, if a bursar should produce outstanding original work. When it was first awarded in 1795, for solutions of questions in geometry, the dies were made larger than intended, but, according to William Knight (a later Professor of Natural Philosophy), this was *'better, as there is less temptation in future time to give away a large than a small medal'.* It was not awarded again for nearly thirty years. As the Gray fund, managed by Copland and Hamilton from 1780, prospered, the value of the bursary was increased. The costs of apparatus for Mathematics and Natural Philosophy, together with other less justifiable expenses, were also charged to the fund; for a period of eighteen years from 1801 it paid the annual tavern bill for both King's and Marischal.

The syllabus which Hamilton taught was little changed from that of 1753. Throughout he was concerned to emphasize the applications of mathematics to such practical matters as mensuration, surveying, navigation, astronomy and dialling. To avoid the time wasting and inaccuracies of note taking he published a detailed summary of this part of his course. It is perhaps surprising to find in the section on Astronomy *'There is much reason from analogy to believe that the Planets are inhabited'.* In his classes the professor, a scholarly and unassuming eccentric, faced constant disorder despite disciplinary backing from the Senatus and a system of fines ranging from two pence for writing at the wrong time to half a crown for disturbance, doubled *'if anything be thrown'.* Donald Sage, who attended the classes, later published a vivid and sympathetic description of Hamilton's mannerisms which *'all taken together really held out a premium to every student, from the lightest to the gravest, to look on and laugh'.*

Outside the classroom Hamilton, a man of wide interests, took his full part in college, church and local affairs. He was prominent in the attempt of 1786 to unite the two Universities. In 1792 he and Copland were made free burgesses of guild for work on Aberdeen's water Supply. His publications included works on Merchandise, Peace and War, The Management of the Poor and The National Debt. This last, in which he exposed fallacies in Pitt's sinking fund, brought him lasting fame as a political economist. At seventy he obtained Senatus approval to employ an assistant but at least two people refused his offer before John Cruickshank agreed to take over the first and second classes. When, three years later in 1817, Hamilton and Copland finally exchanged chairs Cruickshank was officially appointed Hamilton's assistant and successor. Hamilton continued to teach the higher classes until 1824 and remained active in the Senatus until a few weeks before his death in 1829. His name is commemorated today in Aberdeen in Hamilton Place.

**John Cruickshank**

Born in the parish of Rothiemay in 1787, Cruickshank was a Marischal graduate and Gray bursar. In 1814 he was a part-time divinity student, supporting himself as a private tutor. After some initial difficulties with the insubordinate mathematics classes, he soon established a reputation for good discipline and excellent teaching, keeping his classes alert with a system of daily oral tests. A student rhyme of 1815 summed up his attitude:

The instruction in elementary arithmetic needed by his first class was a symptom of the generally low standards of the early nineteenth century. From about 1765 the publicly defended thesis required for a degree had been replaced by the repetition of previously dictated answers to a few questions on Logic and Rhetoric. Class examinations were a mere formality. Cruickshank became a leader of reform, an advocate of concentrated work and strict examinations. It was his motion, passed in 1823, which reduced the students' Christmas vacation from three days to one, a reduction he considered necessary since the holidayTo fining this class I'm very unwilling,

But give me an answer or down with a shilling.

*'tends to withdraw their attention from their studies and to lead them into amusements which are not always relinquished'.*Degree examinations on all subjects of the Arts course, extending over six days, were introduced in 1825. From 1826 the bursars had to pass entrance examinations each year, including one in arithmetic for the first mathematical class, and these were extended a year later to all except private students. Degree results listed

*'particularly distinguished'*candidates from 1828.

Details of university life at the time are given in the report of the Royal Commission of 1826-30. A student spent up to three hours a day on Civil and Natural History and an hour each on Classics and Mathematics in his semi year, three hours on Natural Philosophy and an hour in the second mathematical class in his tertian year. Cruickshank, like his predecessors, stressed the applications of mathematics, but had made dialling an optional extra, due to lack of time. He did not start fluxions until the small third class whose students were given many astronomical and nautical problems, subjects continued by the Gray bursar in his last year. The content of the Natural Philosophy course was restricted by lack of mathematics, the professor needing to anticipate results in conics and to avoid fluxions. Although desirable, higher entrance requirements were beyond many of the parochial schools. Cruickshank favoured extending the session to six or seven months, but definitely no more holidays, students only went to the country and caught colds. He supported the union of the Universities while wanting to retain separate Arts classes, maintaining that amalgamation would produce classes too large for efficient teaching.

While the new examinations were generally resulting in higher standards, the test for the Gray Bursary was becoming excessively lengthy, eventually extending throughout the night. After an appeal to the Senatus in 1835 it was spread over two days, restricted to 10 a.m. to 12 P.M. on the first day, 10 a.m. to 10 p.m. on the second . A one year Boxill bursary, founded in 1847 for a student in his magistrand year, gave further stimulus to advanced mathematics. From 1848 the first three books of Euclid and elementary algebra became a prerequisite for the first mathematical class, enabling Cruickshank to introduce Leibniz' differential and integral calculus, which had replaced fluxions, in his second class. Arithmetic was included in the entrance bursary competition from 1851.

The University owed much to Cruickshank as an administrator. He acted as secretary and procurator of public funds from 1821, later organized building work and was librarian from 1844 to 1860. Outside, he took an active interest in banking and insurance, being an early advocate of the decimalization of money. He gave service as a school inspector and to many charitable organizations, work which he continued after his retirement in 1860 until his death in 1875.

His daughter, Anne, ensured the perpetuation of the family name in the University by setting up a trust fund in 1898 to purchase and support the Botanical Gardens in memory of her brother Alexander and in 1906 erected a stained glass window in memory of her father in the library of Marischal College. This window, which was over the entrance gateway was dismantled about 1970 and is now in store. The trust was further augmented on Anne Cruickshank's death to support several other University institutions including a lectureship in Astronomy, the Science Library and a prize in the Faculty of Law.

**KING'S COLLEGE**

**1717-1800**

At King's, where lack of money and dilapidated buildings were the predominating problems, no attempt was made to replace Bower. In 1719, the New Aberdeen council discharged the Old Aberdeen magistrates from their contract to collect the tax on liquor which had contributed to the professor's salary and three years later took legal advice *'how the Town shall be secured against any professor of Mathematicks that shall be hereafter presented'.* The teaching of mathematics remained the responsibility of the class regents. It was not until October 1732 that the Senatus *'taking unto their serious consideration that this Society hath been at a great loss for the want of a publick Professor of Mathematicks'* nominated Alexander Rait. Two months earlier, Rait, a graduate of King's who had *'taught Mathematicks privatly in this college with great approbation',* had been appointed to assist the ageing sub principal, Alexander Fraser, whose class was about to start its semi year. Rait held both offices until he became a regent two years later. His successor as Fraser's assistant, Thomas Gordon (who was to hold various college posts for nearly sixty-five years), recorded that Rait had been given his professorial rank *'in order to gain him more respect from the students, ... but without any salarie; nor did he teach any but Mr Fraser's class'.* The title was not used again until 1800.

The year 1753 was marked by a reorganization of the curriculum similar to that at Marischal. The semi year was *'to be employed (besides reading some Greek as usual) in a Course of Mathematicks both Speculative & Practicall & in ane Introduction to all the Branches of Natural History'.* The tertian class was to have more mathematics along with a course of Natural and Experimental Philosophy. King's Senatus, however, decided to continue the regent system, largely due to the influence of Thomas Reid, a regent who had succeeded Rait in 1751 Reid (one of the famous Gregory family), firmly believed in the advantages to the student of being supervised by a single regent and that a regent should have no difficulty in teaching all subjects. Although later renowned as a philosopher at Glasgow, Reid, who was a friend of Professor John Stewart of Marischal College where they had both graduated in 1726, was himself a capable mathematician with an interest in geometry, which led him to foresee the possibility of non-euclidean systems.

The mathematics syllabus taught by the regents was very similar to that at Marischal. Notes taken from Thomas Gordon's lectures in the seventies show that, starting from the simplest addition tables, he needed to drill his students in arithmetic and elementary algebra, setting them numerous examples to be done during the summer vacation at the end of their semi year. Whereas at Marischal the First Mathematical Class consisted of a single hour a day in addition to the Civil and Natural History which formed the main part of the semi year course, at King's, Mathematics was the principal subject of that year. From the available lecture notes, Natural History seems to have been of dwindling importance, forming only a small part of the course by the end of the century. Since he taught the same students for three years the regent could and sometimes did allow some mathematics from the tertian course to overflow into the magistrand year. With more time available the student in the Old Town progressed considerably further in the subject than his New Town counterpart. He studied conic sections in much greater detail and made considerable progress in fluxions (a topic not touched until the optional third class at Marischal), during his tertian year. In 1800 a regent, Robert Eden Scott, Gordon's grandson, prepared extensive notes for his students on Dialling, Conic Sections and Fluxions *'for want of a suitable elementary treatise on these subjects'.*

In 1798 it was decided as an experiment to fix regents to particular classes. The sub principal Roderick Macleod took the semis and Scott the tertians *'in which department shall be included the higher branches of Mathematics and the whole of natural and experimental Philosophy except Astronomy'.* Astronomy was taught in the magistrand year with the abstract sciences. Two years later, when Macleod became principal, the arrangement was confirmed, the regents of the semi and tertian years taking the titles of Professors of Mathematics and Natural Philosophy respectively. It was however laid down that when a vacancy occurred the regents should have the option of changing classes.

**1800-1860**

The first Professor of Mathematics under the new system was Dr William Jack, the Vice Principal who had graduated from King's in 1785, become a divinity student, then changed to medicine and practised in his native Shetland, before accepting a regent's post at King's in 1794. He taught the Mathematics and Natural History course until 1811, when, exercising his right to change classes, he became Professor of Moral Philosophy. In 1815 he was appointed Principal, a post which he held until his death in 1854. In 1800 the responsibility for advanced mathematics and the rest of the tertian course was allocated to the new regent. Professor Patrick Copland of Marischal was offered the chair, by the casting vote of the Principal, over another nominee Dr Andrew Mackay. After a few weeks consideration Copland declined and William Duncan *'who has for many years taught Mathematics in Aberdeen with the greatest success',* was appointed. Mackay claimed the right to the post and, in spite of the Chancellor's decision in favour of Duncan, took his case, unsuccessfully, to the Court of Session. For the next three sessions the tertian class was taught by substitutes, Duncan being unwilling to give up his school post until the case was settled. For the first two of these sessions the substitute was the minister of Maryculter, William Paul, who was to return again in January 1811 when Duncan was ill. Later that month Duncan became Professor of Mathematics and Paul succeeded to the Natural Philosophy chair. In the following summer Duncan, whose health was failing, proffered his resignation, but the Senatus retained the benefits of his advice and experience for the remaining four years of his life, by appointing him conjunct professor with John Tulloch, a King's graduate *'for several years a respectable teacher in the Academy of Inverness'.*

Tulloch and Paul were responsible for the teaching of Mathematics for many years. From 1817, the Humanist taught Natural History and Chemistry for one hour daily, leaving Tulloch three hours a day for elementary mathematics, while, as a legacy of the old regent system, the Professor of Natural Philosophy gave the advanced mathematics course of spherics, conics and fluxions to the tertians. This, which occupied the first period each day (before breakfast!) was reduced to less than an hour by prayers and Paul found it insufficient to cover the requirements of his Natural Philosophy class. In his evidence to the Royal Commission of 1826-30 he complained bitterly of his excessive duties. As at Marischal, the award of the degree of AM had become little more than a formality. When Paul himself graduated in 1788 the regent of the tertian class merely asked each magistrand a few easy oral questions on Mathematics; Paul extended the examination to include Natural Philosophy but failures were unknovm. During the eighteen-twenties there was some tightening of the rules on class examinations and attendance, bursars were required to pass entrance examinations to each year from 1825 and in 1828, on the initiative of the students, facilities were improved by the establishment of a class library. It was not until 1834 that new degree regulations were introduced. These required the candidate to answer a third of the questions set in the examinations at the end of each session, introducing higher or highest distinction for greater overall attainments. In Mathematics the minimum required was on the first six books of Euclid, plane trigonometry and algebra as far as simple and quadratic equations. Paul died in 1834 and in the following year Tulloch started a second mathematical class, which became compulsory for the degree from 1842. A prize of £60, founded in 1841 under the will of John Simpson, awarded by special examination in the magistrand year, encouraged study of Mathematics to a higher level. (The college servants were allocated twenty shillings to recompense them for the additional trouble involved.) By the time Tulloch died in 1851 he had seen the second mathematics class firmly established and a steady increase in the amount of mathematics taught at all stages of the University courses.

One of two nominees for the vacant chair was Frederick Fuller, fourth wrangler at Cambridge in 1842, fellow and tutor of St Peter's College. His appointment depended on the casting vote of Principal Jack. Fuller's opponents tried to delay the appointment by leaving the room, taking advantage of Jack's blindness, but were prevented by the Professor of Hebrew who *'set his back to the door of the Senatus room and drove back the reactionaries to their seats till Fuller was elected'.* The young and energetic new professor was an enthusiast for his subject. In lectures he covered blackboards with great rapidity but was known for his clearness and precision. With David Thomson, the then Professor of Natural Philosophy, he formed a powerful team, rapidly raising standards in Aberdeen and sending a succession of students on to become wranglers at Cambridge. In 1856 entrance examinations for the mathematics classes, hitherto restricted to bursars, were extended to all regular students, that for the first class being on Arithmetic and Euclid Book I, for the second on Plane Trigonometry and Euclid Book VI. The syllabus for this senior class included Spherical Trigonometry, Conic Sections, Analytical Geometry and Differential and Integral Calculus. Another regulation of 1856 allowed a student to obtain classified honours in a selected group of subjects, one being the Mathematical and Physical Sciences, by taking additional examinations *'of a higher nature'* at the beginning and end of the last session. Part of the credit for the increasing success of these subjects must go to David Rennet. For upwards of forty years from 1856 this individualistic private tutor was in great demand for his highly effective coaching in Mathematics and Natural Philosophy at all levels.

**TWO UNIVERSITIES INTO ONE**

By 1860 the Universities of both Old and New Aberdeen were much changed from the easy going institutions of the early years of the century. The average ages of entry at King's and Marischal in 1827 were fourteen and twelve respectively. By 1857 these had increased to seventeen years nine months at King's and sixteen years eight months at Marischal. The evolving examination system now required much higher academic attainments from the students. On 15 September 1860 the fusion of the two Universities into the University of Aberdeen was finally achieved and, in spite of opposition, the Arts classes were united. Professor Cruickshank, then aged 73, was compulsorily if regretfully retired and the responsibility for mathematics, to remain for many years a compulsory subject for an Arts degree, was left in the capable hands of Professor Fuller with one assistant. Today we still have reminders of the development of mathematics in the old institutions. The Gray fund, conjoined with those of Fullerton and Moir, supports post graduate scholarships and in the Mathematics Department the revenues of the Simpson and Boxill foundations provide the first and second prizes awarded to Honours degree candidates.