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[PDF page 183] CH. VI] KNIGHT’S PATH ON A CHESS-BOARD. 167 in the figure given above on page 162. Another way of dividing the board into closed circuits which can be connected was given in1843 by Moon*. He divided the board into a a b c d a b c d c d a b c d a b b a A B C D d c d c C D A B b a a b B A D C c d c d D C B A a b b a d c b a d c d c b a d c b a Moon’s Solution. 63 22 15 40 1 42 59 18 14 39 64 21 60 17 2 43 37 62 23 16 41 4 19 58 24 13 38 61 20 57 44 3 11 36 25 52 29 46 5 56 26 51 12 33 8 55 30 45 35 10 49 28 53 32 47 6 50 27 34 9 48 7 54 31 Jaenisch’s Solution. central square containing42 cells and a surrounding annulus (see figure on this page). The annulus may be divided into four closed circuits, each containing12 cells: these are marked respectively with the letters a, b, c, d. The central square may be divided similarly into four closed circuits, each containing 4 cells, denoted by the letters A, B, C, D. We can connect these routes as follows. If we are moving outwards from the central square to the annulus we can go from a cellA either to b or to c or to d (but not to a) and similarly for the other letters. If we are moving inwards from the annulus to the central square we must go froma to D, or d to A, or b to C, or c to B, as the case may be. Thus if the initial cell is ona, we might take either of the cycles a D b C d A c B , or a D c B d A b C . By following these rules we always can connect the routes into one path, but in general it will not be re-entrant. It is convenient to take the cells in each circuit in one and the same direction, but a circuit in the outer annulus must not end in a corner cell, and to avoid this we may have to alter the direction in which a circuit is taken. Moon’s rule can be modified to cover the case of any doubly even square board, and the path can be made to begin and end on any two given squares, but I do not propose to go further into details. * Cambridge Mathematical Journal, 1843, vol.iii, pp. 233–236.

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[PDF page 184] 168 UNICURSAL PROBLEMS. [CH. VI The method which Jaenisch gives as the most fundamental is not very different from that of Roget. It leads to eight forms, similar to that in the diagram printed on the previous page, in which the sum of the numbers in every column and every row is260; but although symmetri- cal it is not in my opinion so easy to reproduce as that given by Roget. It is as yet impossible to say how many solutions of the problem exist. Legendre* mentioned the question, but Minding†was the earliest writer to attempt to answer it. More recent investigations have shown that on the one hand the number of possible routes is less‡ than the number of combinations of168 things taken 63 at a time, and on the other hand is greater than31, 054144—since this latter number is the number of re-entrant paths of a particular type§. There are many similar problems in which it is required to deter- mine routes by which a piece moving according to certain laws (ex. gr.a chess-piece such as a king, knight, &c.) can travel from a given cell over a board so as to occupy successively all the cells, or certain specified cells, once and only once, and terminate its route in a given cell. Euler’s method can be applied to find routes of this kind: for in- stance, he applied it to find a re-entrant route by which a piece that moved two cells forward like a castle and then one cell like a bishop would occupy in succession all the black cells on the board. As another instance, a castle, placed on a chess-board ofn2 cells, can always be moved in such a manner that it shall move successively on to every cell once and only once; moreover, starting on any cell, its path can be made to terminate, ifn be even, on any other cell of a different colour, and, if n be odd, on any other cell of the same colour∥. But it will suffice to have discussed the classical problem of the determination of a knight’s path on an ordinary chess-board, and I need not enter on the discussion of other similar problems. * Théorie des Nombres, Paris, 2nd edition, 1830, vol.ii, p. 165. † Cambridge and Dublin Mathematical Journal, 1852, vol. vii, pp. 147–156; and Crelle’s Journal, 1853, vol.xliv, pp. 73–82. ‡ Jaenisch, vol. ii, p. 268. § Bulletin de la Société Mathématique de France, 1881, vol.ix, pp. 1–17. ∥ L’Intermédiaire des mathématiciens, Paris, July, 1901, p. 153.

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[PDF page 185] PART II. Miscellaneous Essays and Problems. “No man of science should think it a waste of time to learn something of the history of his own subject; nor is the investigation of laborious meth- ods now fallen into disuse, or of errors once com- monly accepted the least valuable of mental disci- plines.” “The most worthless book of a bygone day is a record worthy of preservation. Like a telescopic star, its obscurity may render it unavailable for most purposes; but it serves, in hands which know how to use it, to determine the places of more im- portant bodies.” (De Morgan.) 169

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[PDF page 186] CHAPTER VII. THE MATHEMATICAL TRIPOS. The Mathematical Tripos has played so prominent a part in the history of education at Cambridge and of mathematics in England, that a sketch of its development†may be interesting to general readers. So far as mathematics is concerned the history of the University before Newton may be summed up very briefly. The University was founded towards the end of the twelfth century. Throughout the mid- dle ages the studies were organised on lines similar to those at Paris and Oxford. To qualify for a degree it was necessary to perform var- ious exercises, and especially to keep a number ofacts or to oppose acts kept by other students. An act consisted in effect of a debate in Latin, thrown, at any rate in later times, into syllogistic form. It was commenced by one student, therespondent, stating some proposition, often propounded in the form of a thesis, which was attacked by one or moreopponents, the discussion being controlled by a graduate. The teaching was largely in the hands of young graduates—every master of arts being compelled to reside and teach for at least one year—though no doubt Colleges and private hostels supplemented this instruction in the case of their own students. † The following pages are mostly summarised from my History of the Study of Mathematics at Cambridge, Cambridge 1889. The subject is also treated in Whewell’sLiberal Education, Cambridge, three parts, 1845, 1850, 1853; Wordsworth’sScholae Academicae, Cambridge, 1877; my ownOrigin and His- tory of the Mathematical Tripos, Cambridge, 1880; and Dr Glaisher’s Presiden- tial Address to the London Mathematical Society,Transactions, volxviii, 1886, pp. 4–38. 170

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[PDF page 187] CH. VII] THE MATHEMATICAL TRIPOS. 171 The Reformation in England was mainly the work of Cambridge divines, and in the University the Renaissance was warmly welcomed. In spite of the disorder and confusion of the Tudor period, new studies and a system of professional instruction were introduced. Probably the science (as distinct from the art) of mathematics, save so far as involved in the quadrivium, was still an exotic study, but it was not wholly ne- glected. Tonstall, subsequently the most eminent English arithmetician of his time, migrated, perhaps about 1495, from Balliol College, Ox- ford, to King’s Hall, Cambridge, and in 1530 the University appointed a mathematical lecturer in the person of Paynell of Pembroke Hall. Most of the subsequent English mathematicians of the Tudor period seem to have been educated at Cambridge; of these I may mention Record, who migrated, probably about 1535, from Oxford, Dee, Digges, Blundeville, Buckley, Billingsley, Hill, Bedwell, Hood, Richard and John Harvey, Edward Wright, Briggs, and Oughtred. The Elizabethan statutes re- stricted liberty of thought and action in many ways, but, in spite of the civil and religious disturbances of the early half of the 17th century the mathematical school continued to grow. Horrox, Seth Ward, Foster, Rooke, Gilbert Clerke, Pell, Wallis, Barrow, Dacres, and Morland may be cited as prominent Cambridge mathematicians of the time. Newton’s mathematical career dates from 1665; his reputation, abilities, and influence attracted general attention to the subject. He created a school of mathematics and mathematical physics, among the earliest members of which I note the names of Laughton, Samuel Clarke, Craig, Flamsteed, Whiston, Saunderson, Jurin, Taylor, Cotes, and Robert Smith. Since then Cambridge has been regarded, as in a special sense, the home of English mathematicians, and from 1706 onwards we have fairly complete accounts of the course of reading and work of mathematical students there. Until less than a century ago the form of the method of qualifying for a degree remained substantially unaltered, but the subject-matter of the discussions varied from time to time with the prevalent studies of the place. After the Renaissance some of the statutable exercises were “hud- dled,” that is, were reduced to a mere form. To huddle an act, the proc- tor generally asked some question such asQuid est nomento which the answer usually expected wasNescio. In these exercises considerable license was allowed, particularly if there were any play on the words

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[PDF page 188] 172 THE MATHEMATICAL TRIPOS. [CH. VII involved. For example, T. Brasse, of Trinity, was accosted with the question, Quid est aes? to which he answered,Nescio nisi finis ex- aminationis. It should be added that retorts such as these were only allowed in the pretence exercises, and a candidate who in the actual examination was asked to give a definition of happiness and replied an exemption from Payne—that being the name of the moderator then presiding—was plucked for want of discrimination in time and place. In earlier years even the farce of huddling seems to have been unnecessary, for it was said in 1675 that it was not uncommon for the proctors to take “cautions for the performance of the statutable exercises, and ac- cept the forfeit of the money so deposited in lieu of their performance.” In medieval times acts had been usually kept on some scholastic question or on a proposition taken from theSentences. About the end of the fifteenth century religious questions, such as the interpretation of Biblical texts, began to be introduced, some fifty or sixty years later the favourite subjects were drawn either from dogmatic theology or from philosophy. In the seventeenth century the questions were usually philosophical, but in the eighteenth century, under the influence of the Newtonian school, a large proportion of them were mathematical. Further details about these exercises and specimens of acts kept in the 18th century are given in myHistory of Mathematics at Cambridge. Here I will only say that they provided an admirable training in the art of presenting an argument, and in dialectical skill in attack and defence. The mental strain in a contested act was severe. De Morgan, describing his act kept in 1826, wrote*, I was badgered for two hours with arguments given and answered in Latin,—or what we call Latin—against Newton’s first section, Lagrange’s derived functions, and Locke on innate principles. And though I took off everything, and was pronounced by the moderator to have disputed magno honore, I never had such a strain of thought in my life. For the inferior opponents were made as sharp as their betters by their tutors, who kept lists of queer objections drawn from all quarters. HadthelanguageofthediscussionsbeenchangedtoEnglish, aswas repeatedly urged from 1774 onwards, these exercises might have been retained with advantage, but the barbarous Latin and the syllogistic form in which they were carried on prejudiced their retention. About 1830 a custom grew up for the respondent and opponents to meet previously and arrange their arguments together. The discussions * Budget of Paradoxes, by A. De Morgan, London, 1872, p. 305.

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[PDF page 189] CH. VII] THE MATHEMATICAL TRIPOS. 173 then became an elaborate farce, and were a mere public performance of what had been already rehearsed. Accordingly the moderators of 1839 took the responsibility of abandoning them. This action was singularly high-handed, since a report of May 30, 1838, had recom- mended that they should be continued, and there was no reason why they should not have been reformed and retained as a useful feature in the scheme of study. On the result of the acts a list of those qualified to receive degrees was drawn up. This list was not arranged strictly in order of merit, because the proctors could insert names anywhere in it, but by the beginning of the 18th century this power had become restricted to the right reserved to the vice-chancellor, the senior regent, and each proctor to place in the list one candidate anywhere he liked—a right which con- tinued to exist till 1828, though it was not exercised after 1797. Subject to the granting of these honorary degrees, this final list was arranged in order of merit into wranglers and senior optimes, junior optimes, and poll-men. The bachelors on receiving their degrees took seniority according to their order on this list. The titlewrangler is derived from these contentious discussions; the titleoptime from the customary com- pliment given by the moderator to a successful disputant,Domine..., optime disputasti, or even optime quidem disputasti, and the title of poll-man from the description of this class asοÉ πολλοÐ. The final exercises for the B.A. degree were never huddled, and until 1839 were carried out strictly. University officials were responsi- ble for approving the subject-matter of these acts. Stupid men offered some irrefutable truism, but the ambitious student courted reputation by affirming some paradox. Probably all honour men kept acts, but poll-men were deemed to comply with the regulations by keeping op- ponencies. The proctors were responsible for presiding at these acts, or seeing that competent graduates did so. In and after 1649 two examin- ers were specially appointed for this purpose. In 1680* these examiners were appointed by the Senate with the title of moderator, and with the joint stipend of four shillings for everyone graduating as B.A. during their year of office. In 1688 the joint stipend of the moderators was fixed at £40 a year. The moderators, like the proctors, were nominated by the Colleges in rotation. From the earliest times the proctors had the power of questioning * See Grace of October 25, 1680.

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[PDF page 190] 174 THE MATHEMATICAL TRIPOS. [CH. VII a candidate at the end of a disputation, and probably all candidates for a degree attended the public schools on certain days to give an op- portunity to the proctors, or any master that liked, to examine them*, though the opportunity was not always used. Different candidates at- tended on different days. Probably such examinations were conducted in Latin. But soon after 1710† the moderators or proctors began the custom of summoning on one day in January all candidates whom they proposed to question. The examination was held in public, and from it the Senate-House Examination arose. The examination at this time did not last more than one day, and was, there can be no doubt, partly on philosophy and partly on mathematics. It is believed that it was always conducted in English, and it is likely that its rapid development was largely due to this. This introduction of a regular oral examination seems to have been largely due to the fact that when, in 1710, George I gave the Ely library to the University, it was decided to assign for its reception the old Senate-House—now the Catalogue Room in the Library—and to build a new room for the meetings of the Senate. Pending the building of the new Senate-House the books were stored in the Schools. As the Schools were thus rendered unavailable for keeping acts, considerable difficulty was found in arranging for all the candidates to keep the full number of statutable exercises, and thus obtaining opportunities to compare them one with another: hence the introduction of a supplementary oral examination. The advantages of this examination as providing a ready means of testing the knowledge and abilities of the candidates were so patent that it was retained when the necessity for some system of the kind had passed away, and finally it became systematized into an organized test to which all questionists were subjected. In 1731 the University raised the joint stipend of the moderators to £60 “in consideration of their additional trouble in the Lent Term.” This would seem to indicate that the Senate-House Examination had then taken formal shape, and perhaps that a definite scheme for its conduct had become customary. * Ex. gr. see De la Pryme’s account of his graduation in 1694,Surtees Society, vol. liv, 1870, p. 32. † W. Reneu, in his letters of 1708–1710 describing the course for the B.A. degree, makesnomentionoftheSenate-Houseexamination, andIthinkitisareasonable inference that it had not then been established.

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[PDF page 191] CH. VII] THE MATHEMATICAL TRIPOS. 175 As long as the order of the list of those approved for degrees was settled on the result of impressions derived from acts kept by the dif- ferent candidates at different times and on different subjects, it was impossible to arrange the men in strict order of merit, nor was much importance attached to the order. But, with the introduction of an examination of all the candidates on one day, much closer attention was paid to securing a strict order of merit, and more confidence was felt in the published order. It seems to have been consequent on this that in and after 1747 the final lists were freely circulated, and it was further arranged that the names of the honorary optimes should be indicated. In the lists given in the Calendars issued subsequent to 1799 these names are struck out. It is only in exceptional cases that we are acquainted with the true order for the earlier tripos lists, but in a few cases the relative positions of the candidates are known; for example, in 1680 Bentley came out as third though he was put down as sixth in the list of wranglers. Of the detailed history of the examination until the middle of the eighteenth century we know nothing. From 1750 onwards, however, we have more definite accounts of it. At this time, it would seem that all the men from each College were taken together as a class, and questions passed down by the proctors or moderators till they were answered: but the examination remained entirely oral, and technically was regarded as subsidiary to the discussions which had been previously held in the schools. As each class contained men of very different abilities a custom grew up by which every candidate was liable to be taken aside to be questioned by any M.A. who wished to do so, and this was regarded as an important part of the examination. The subjects were mathematics and philosophy. The examination now continued for two days and a half. At the conclusion of the second day the moderators received the reports of those masters of arts who had voluntarily taken part in the examination, and provisionally settled the final list; while the last half- day was used in revising and re-arranging the order of merit. Richard Cumberland has left an account of the tests to which he was subjected when he took his B.A. degree in 1751. Clearly the dis- putations still played an important part, and it is difficult to say what weight was attached to the subsequent Senate-House examination; his reference to it is only of a general character. After saying that he kept

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[PDF page 192] 176 THE MATHEMATICAL TRIPOS. [CH. VII two acts and two opponencies he continues*: The last time I was called upon to keep an act in the schools I sent in three questions to the Moderator, which he withstood as being all mathematical, and required me to conform to the usage of proposing one metaphysical question in the place of that, which I should think fit to withdraw. This was ground I never liked to take, and I appealed against his requisition: the act was accordingly put by till the matter of right should be ascertained by the statutes of the university, and in the result of that enquiry it was given for me, and my question stood.... I yielded now to advice, and paid attention to my health, till we were cited to the senate house to be examined for our Bachelor’s degree. It was hardly ever my lot during that examination to enjoy any respite. I seemed an object singled out as every man’s mark, and was kept perpetually at the table under the process of question and answer. It was found possible by means of the new examination to differen- tiate the better men more accurately than before; and accordingly, in 1753, the first class was subdivided into two, called respectively wran- glers and senior optimes, a division which is still maintained. The semi-official examination by M.A.s was regarded as the more important part of the test, and the most eminent residents in the Uni- versity took part in it. Thus John Fenn, of Caius, 5th wrangler in 1761, writes†: On the following Monday, Tuesday, and Wednesday, we sat in the Senate-house for public examination; during this time I was officially ex- amined by the Proctors and Moderators, and had the honor of being taken out for examination by Mr Abbot, the celebrated mathematical tutor of St John’s College, by the eminent professor of mathematics Mr Waring, of Magdalene, and by Mr Jebb of Peterhouse, a man thoroughly versed in the academical studies. This irregular examination by any master who chose to take part in it constantly gave rise to accusations of partiality. In 1763 the traditional rules for the conduct of the examination took more definite shape. Henceforth the examiners used the disputa- tions only as a means of classifying the men roughly. On the result of their “acts,” and probably partly also of their general reputation, the candidates were divided into eight classes, each arranged in alphabet- ical order. The subsequent position of the men in the class was deter- mined solely by the Senate-House examination. The first two classes * Memoirs of Richard Cumberland, London, 1806, pp. 78, 79. † Quoted by C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 30–31.

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[PDF page 193] CH. VII] THE MATHEMATICAL TRIPOS. 177 comprised all who were expected to be wranglers, the next four classes included the other candidates for honours, and the last two classes con- sisted of poll-men only. Practically anyone placed in either of the first two classes was allowed, if he wished, to take an aegrotat senior optime, and thus escape all further examination: this was called gulphing it. All the men from one College were no longer taken together, but each class was examined separately andvivâ voce; and hence, since all the students comprised in each class were of about equal attainments, it was possible to make the examination more effective. Richard Watson, of Trinity, claimed that this change was made by him when acting as moderator in 1763. He says *: There was more room for partiality...then [i.e. in 1759] than there is now; and I attribute the change, in a great degree, to an alteration which I introduced the first year I was moderator [i.e. in 1763], and which has been persevered in ever since. At the time of taking their Bachelor of Arts’ degree, the young men are examined in classes, and the classes are now formed according to the abilities shown by individuals in the schools. By this arrangement, persons of nearly equal merits are examined in the presence of each other, and flagrant acts of partiality cannot take place. Before I made this alteration, they were examined in classes, but the classes consisted of members of the same College, and the best and worst were often examined together. It is probable that before the examination in the Senate-House began a candidate, if manifestly placed in too low a class, was allowed the privilege of challenging the class to which he was assigned. Perhaps this began as a matter of favour, and was only granted in exceptional cases, but a few years later it became a right which every candidate could exercise; and I think that it is partly to its development that the ultimate predominance of the tripos over the other exercises for the degree is due. In the same year, 1763, it was decided that the relative position of the senior and second wranglers, namely, Paley, of Christ’s, and Frere, of Caius, was to be decided by the Senate-House examination and not by the disputations. Henceforward distinction in the Senate- House examination was regarded as the most important honour open to undergraduates. In 1768 Dr Smith, of Trinity College, founded prizes for mathe- matics and natural philosophy open to two commencing bachelors. The * Anecdotes of the Life of Richard Watson by Himself, London, 1817, pp. 18, 19.
