LIFE AND SCIENTIFIC WORK OF PETER GUTHRIE TAIT CAMBRIDGE UNIVERSITY PRESS aonDon: FETTER LANE, E.G. C. F. CLAY, MANAGER ffiinburgl) : 100, PRINCES STREET Berlin: A. ASHER AND CO. ItiDMS: F. A. BROCKHAUS gfto Hork : G. P. PUTNAM'S SONS Bomfcan 8n6 Calcutta: MACM1LLAN AND CO., LTD. Ail rifkts reserved LIFE AND SCIENTIFIC WORK OF PETER GUTHRIE TAIT SUPPLEMENTING THE TWO VOLUMES OF SCIENTIFIC PAPERS PUBLISHED IN l8g8 AND by CARGILL GILSTON KNOTT D.Sc. ; one of the Secretaries R.S.E. ; Order of the Rising Sun, Empire of Japan (Class IV) ; Assistant to Professor Tail from 1879 to 1883; Professor of Physics, Imperial University of Japan from 1883 to 1891 ; Lecturer on Applied Mathematics in Edinburgh University from 1892 Cambridge : at the University Press 1911 Cambrttge: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS PREFACE AT the time of his death in 1901 Professor P. G. Tait had just finished editing the Second Volume of his Collected Scientific Papers. The series is now completed by this Memorial Volume whose preparation I undertook at the request of Mrs Tait, who kindly placed a great deal of material at my disposal, and who, together with the other members of the family, has been closely in touch with the work as it proceeded. Professor Crum Brown, the late Professor's brother-in-law and colleague for over 30 years, closely associated himself with the work. His knowledge and judgement were always at my service. Lord Kelvin at the outset afforded me much useful information generally about events of an early date, especially certain facts connected with the preparation of "The Treatise on Natural Philosophy," a work unfortunately never completed. The proofs have been read by Dr A. W. Ward, Master of Peterhouse, and Mr J. D. Hamilton Dickson, Fellow and Tutor of Peterhouse, to both of whom I am deeply indebted for many valuable criticisms and suggestions ; and for similar helpful services my sincere thanks are also due to Professor J. G. MacGregor and Professor W. Peddie. The interest expressed by others among Professor Tail's friends and students has greatly encouraged me in my work. Their reminiscences of the Natural Philosophy Class Room or Laboratory, and their memory of the stimulating character of the teaching, will be found reflected in the pages which follow. In arranging the material I have been influenced largely by one considera- tion the convenience of the reader. The opening chapter, including the description of Professor Tait on holiday in St Andrews, for which I am indebted to Mr J. L. Low, gives simply the main facts of the Life. The various aspects of the Scientific Work are taken up, in more or less detail, in the succeeding chapters. 280250 vi PREFACE The care with which Professor Tait preserved the letters he received from his scientific correspondents has enabled me greatly to enrich the pages of the Memoir by the inclusion of letters from Sir William Rowan Hamilton, Professor Cayley, Lord Kelvin, and Professor Clerk Maxwell. Introduced as far as possible in its immediate setting, the correspondence brings out interesting points of history, and shows how heartily all these great men helped one another in their scientific investigations. It is much to be regretted that Professor Tail's own letters to Clerk Maxwell are not now available. Professor Tail's foreign correspondence was carefully arranged and annotated by Dr J. S. Mackay, to whom I am greally indebted for thus enabling me rapidly to choose what was serviceable for the purposes of the Memoir. Several of the old students having suggested that the controversy between Professor Tait and Mr Herbert Spencer would prove interesling, I have given ihe delails at some length. It seemed advisable to bring the real points at issue clearly before the reader's mind, more especially as Mr Spencer had given his own views at great length in a published pamphlet and in the appendix to subsequent editions of his First Principles. On looking into the matter I found myself forced to begin with what preceded Professor Tail's share in the controversy ; and in ihis conneclion I wish lo lhank Lord Justice Fletcher Moulton for his help in presenling an accurate accounl of ihe stages of a lively debate which had its origin in his review of Mr Spencer's Work. The original photograph of Professor Tait writing a note in his retiring- room is the property of ihe Rev. L. O. Crilchley, M.A., who most willingly granted the inclusion of the portrait in the present volume. To him also special thanks are due. I wish also to record my thanks to ihe Editors of Nature, of the Philosophical Magazine, and of ihe Badminton Magazine, for permission lo reprinl articles contribuled by Professor Tait ; and to ihe Council of ihe Royal Sociely of Edinburgh for certain diagrams and figures which have been reproduced. CARGILL GILSTON KNOTT EDINBURGH UNIVERSITY February 1911 TABLE OF CONTENTS CHAPTER I MEMOIR PETER GUTHRIE TAIT Early life in Edinburgh, 1-7; life in Cambridge, 8-n; life in Belfast, 12-15; l ater ^ e m Edinburgh, 16-52; Tail as lecturer, 17-22; contributions to Chambers 1 Encyclopaedia, 23; sketch of literary work, 23-24; the physics of golf, 25-28; Royal Society of Edinburgh, 28-30; Kelvin's visits to Edinburgh, 31-32 ; favourite authors, 33 ; social meetings, 33-34; GifTord Lectures, 35 ; views on religion and politics, 35-37 ; the South African War, 37 ; retirement and last illness, 3941; obituary notices, 42-46; colleagues in Senatus, 46-47; portraits, 47-50; Tail Prize at Peterhouse, 50; Tail Memorial, 50-51; Sir John Jackson and Sir James Dewar, 51; Tail at St Andrews (contributed by J. L. Low), 52-63; "The Morning Round," 55; phosphorescent golf balls, 57; theory of the golf ball flight, 59-60; "The Bulger," 61-62; Freddie and his Father, 63. pp. 1-63 CHAPTER II EXPERIMENTAL WORK Visit to Edinburgh in 1859, 65; enthusiasm over Thomson's galvanometers and electro- meters, 67-68; Vortex rings, 68-69; Sir David Brewster, 69-70; Physical Laboratory, 70-71; W. Robertson Smith, 71-72; Robert Louis Stevenson, 72-74; James Lindsay, 74; Fox Talbot, 76; Thermoelectricity, 77-80; Crooke's Radiometer, 81-82; the "Challenger" thermo- meters, 82-85 ; expansion of laboratory, 86-87 > hygrometry on Ben Nevis, 87-88 ; impact, 88-90; fog horns, 91; rotatory polariscope, 91-92; diathermancy of water vapour, 92-93; rhyming correspondence with Maxwell, 93-95 ; general estimate, 95-97. pp. 64-97 CHAPTER III MATHEMATICAL WORK Brachistochrones, 99-100; Maxwell writes on spherical harmonics, 100-102; golf-match problem, 102-104; Maxwell writes on vortex rings, 106 ; Knots, 106-109; Mirage, 109; kinetic theory of gases, 109-113; Maxwell writes on viscosity, quaternions, entropy, the Second Law, etc., 114116; golf-ball trajectory, 116-117; Josephus' problem, 118. pp. 98-118 viii CONTENTS CHAPTER IV QUATERNIONS Introduction to Hamilton, 119; correspondence with Hamilton, 119-141; differentials, 120-124; envelopes, 124; wave-surface, 124-126; early studies in quaternions, 126-128; wave-surface, 129-134; linear vector function, 135-138; electrodynamics, 138; misunder- standings, 139-140; analysis of correspondence, 140-141; letter to Herschel, 141-142; correspondence with Maxwell on Nabla, 143-152; Maxwell's report on Tait's quaternion work, 149150; correspondence with Cayley on quaternions and matrices, 152-165; Maxwell's indebtedness to Tait, 166-167; the scientific world's indebtedness to Tait, 167-168; Treatise on Quaternions, 169-170; Maxwell's Tyndallic Ode, 171-173; Maxwell's report of Brit. Assoc. Meeting at Belfast, 174-175. PP- "9-i?5 CHAPTER V THOMSON AND TAIT "T AND T," OR THOMSON AND TAIT'S NATURAL PHILOSOPHY Tait resolves to write Text-book, 177; joined by Thomson, 177; early draft of contents, 178-179; facsimile of Tait's manuscript, 181 ; correspondence with Thomson, 180-184; quaternions excluded, 185; reception of book, 186187; description of scope and contents of "T and T'," 187194; Maxwell's criticisms, 195; German translation, 195-197; early pamphlets, 197200; "Little T and T'," 201-202; Second Edition of Treatise, 202-203; Maxwell's review, 203-204. pp. 176-204 CHAPTER VI OTHER BOOKS "Tait and Steele," 205-207; Thermodynamics, 208-228; controversy with Tyndall, 209; articles in North British Review, 209-210; estimate of Mayer's work, 211; translates Mohr's paper, 212; Maxwell's account of his "demons," 213-215; letter from Maxwell, 215-216; correspondence with Helmholtz, 216-217; correspondence with Thomson, 218-220; Maxwell reviews Thermodynamics, 220-221; Thomson writes on the Entropy integral, 223-225; charge of Chauvinism, 225-226; Recent Advances, 227-228; Heat, 228; Light, 229; Properties of Matter, 229-230; reviews by Rayleigh and Stewart, 229-230; Dynamics, 230-232; Newton's Laws of Motion, 233-234; correspondence with Cayley on Laws of Motion, 234-236; The Unseen Universe, 236-240; review by Clifford, 240; Paradoxical Philosophy, 241-242; review by Maxwell, 241-242 ; Maxwell's Paradoxical Ode, 242-244 ; Headstone's soliloquy, by Maxwell, 244-245. pp. 205-245 CONTENTS ix CHAPTER VII ADDRESSES, REVIEWS, AND CORRESPONDENCE Nebulae and comets, 246-247; addresses to graduates, 247-251; an ideal university, 248; evils of Cram, 249-251; the Rede Lecture, 251-252; lectures to Industrial Classes, 252; lecture on Force, 252-253; Maxwell's metrical version, 253-255; lecture on Thunder- storms, 255; Sensation and Science, 256; de Morgan's Budget of Paradoxes, 257-258; Maxwell's Electricity and Magnetism, 258-260; Maxwell's Matter and Motion, 260-261; life and work of Maxwell, 261-264; scientific work of Stokes, 264-265; Stokes' Mathematical and Physical Papers, 265-266; Stokes' Burnett Lectures, 267-269; Clifford's Dynamic, 270-272; Clifford's Common Sense of the Exact Sciences, 272-273; Poincare > 's 2 ' hermodynamique, 273-276; McAulay's Utility of Quaternions, 276-278; Tay Bridge disaster, 278; controversy with Herbert Spencer, 279-288; Balfour Stewart, 289-291; Robertson Smith, 291-292; Religion and Science, 293-295. pp. 246-295 CHAPTER VIII POPULAR SCIENTIFIC ARTICLES Thunderstorms, 296-320 ; state of the atmosphere which produces the forms of mirage observed by Vince and by Scoresby, 321-328; Long Driving, 329-344; J. J. Thomson's illustration of golf-ball trajectories by streams of electrified particles, 344-345. ADDENDA : J. D. Hamilton Dickson's extension of Tail's thermoelectric theory and diagram, 345-346 ; The Evening Club, 347-349 ; Recent Advances in Physical Science, 349-350. pp. 296-350 BIBLIOGRAPHY pp. 351-365 INDEX pp. 367-379 PORTRAITS Reproduced from a photograph taken by the late Dr Adamson of St Andrews. Date, 1870. Frontispiece Peterhouse Group of Graduates; from a faded photograph. P. G. Tail and W. J. Steele are the first and third, reckoning from the left. Date, 1852 . . . To face page II Reproduced from photograph of Professor Tail and Mr Thomas Lindsay, taken in the University Retiring Room by Rev. L. O. Critchley, M.A., then a laboratory student. Date, 1895, To face page 89 Reproduced from photograph by Mr Marshall Wane, Edinburgh. The photograph was taken for the Album of portraits which the Contributors to the Challenger Expedition Reports presented to Sir John Murray in 1895 To fate page 181 Reproduced from the Replica Portrait painted by Sir George Reid and placed in the Hall of Peterhouse, Cambridge. The original Portrait in the Council Room of the Royal Society of Edinburgh was painted in 1891. The mathematical expressions on the board in the back-ground exist only in the Peterhouse Replica, and are copied from the early portrait of 1882. They refer to the Mirage problem ..... To face page 325 ADDENDA AND CORRIGENDA p. 22, footnote. In connection with early laboratories, those begun by Professor Clifton in Oxford, and by Professor Grylls Adams in King's College, London, should also have been mentioned. p. 48, 1. 2 from fool for "rauschend" read "rauchend." p. 96, 1. 6 from top for "Reader in" read "Professor of." CHAPTER I MEMOIR PETER GUTHRIE TAIT OF all human activities and developments none are more characteristic of the Victorian Era than those clustering round the word Science. Scientific theory and its application to the growing needs of mankind advance hand in hand. On the one side are the developments of steam power, and the practical creations of Electric Telegraphy, Telephony and Dynamo-electric machinery ; on the other the framing of new theories of Heat and Electricity. Practical engineers and scientific men of all types and degrees of ability and talent have had their share in this great development, which within two generations has transformed the whole aspect of human life. i> But of far greater import to the philosophical student than the dove- tailed features of this development is the apprehension of the broad principle of Energy which has unified the various branches of science. The biography of any of the outstanding natural philosophers of the latter half of the Nineteenth Century must, indeed, be to a large extent a history of Energetics, to use Rankine's convenient nomenclature. These minds, trained under masters of an older school who knew of no such guiding principle, grew with the scientific environment which they were themselves creating. It is not easy for us, who are the heirs of the rich legacy of thought which our immediate predecessors bequeathed to us, fully to realise the greatness of the transformation which they effected. We may be able to note here and there the subtle manner in which, not always consciously to themselves, they acted and reacted one upon the other ; but we are perhaps too near the age of transition to see clearly the interplay of all that made for progress. Each of us has had his own peculiar training, his own personal contact with the mighty ones of the immediate past ; and this forms as it were a telescopic tube determining limits to our field of vision. No doubt we may range the whole horizon ; but after all we look from our own point of vantage. What may appear x. i 2 PETER GUTHRIE TAIT as a towering peak to one may seem but an ordinary eminence to another. Nevertheless, incomplete and historically partial though it must be, a sketch of the career of a leader of scientific thought who lived his strenuous mental life through this formative time cannot be without its value as a contribution to the history of the growth of ideas. Such a one, pre-eminently, was Professor Tail of Edinburgh University. He was the personal friend of Hamilton, Andrews, Stokes, Joule, Kelvin, Maxwell, Stewart, Helmholtz, Cayley, Sylvester to name a few of the more outstanding of those who have passed away. These contemporaries were to him personalities and not mere writers of papers or of books. He got much from them and he gave much to them. As a historian of contemporary developments he takes high rank ; and to him we owe in a manner which can only now be clearly recognised the very existence of Thomson and Tail's Natural Philosophy and of Hamilton's Elements of Quaternions, In tracing his career I have received every help possible from Mrs Tait and the other members of the family. My own recollections of his tales of earlier days have been corroborated and supplemented by evidence from letters written contemporaneously with the events they describe. His Scrap Book, a fascinating collection of all kinds of letters and cuttings bearing upon his own work and the work of others that touched him closely, has been of unique value. I feel it a great honour to have had confided to me the privilege of preparing this memorial volume. My sole endeavour has been to give a faithful picture of Professor Tait as teacher, investigator, author, and friend. To this end I have reproduced a few of his more popular scientific articles as well as numerous quotations from letters, addresses, and reviews. The picturesque account of the St Andrews holiday life of Professor Tait is from the pen of Mr John L. Low, the author of F. G. Tait, a Record, being the biography of Professor Tail's soldier son, Lieutenant in the Black Watch, who lost his life in the South African War. EDINBURGH. 1837-48 Peter Guthrie Tait was born at Dalkeith on 28 April, 1831. He was educated in his very early years at the Dalkeith Grammar School. On his father's death his mother came to Edinburgh with her young family of two girls and one boy; and after a year or two at Circus Place School, Tait entered the Academy at the age of ten. He and his sisters finally lived with their uncle, John Ronaldson, in an old-fashioned roomy house called Somerset Cottage, which is still occupied by the Misses Tait. Mr Ronaldson was a banker by profession, but was keenly interested in many scientific pursuits. He would take his nephew geological rambles in the long summer days, and study the planets and stars through his telescopes during the dark winter nights ; or he would dabble in the mysteries of photography which had just been invented by Daguerre and Talbot. There is little doubt that the receptive mind of the young lad must have been greatly influenced by his uncle's predilection for scientific study. A small room on the left of the hall as one enters Somerset Cottage contains to this day the stand and tube of a Newtonian reflector, and a good serviceable refractor of two-inch aperture. The room has been long used by Miss Tait for storing her canvasses and artistic materials ; but the scientific contents of the apartment have never been disturbed since 1854, when P. G. Tait definitely made his home in Belfast. On his return to Edinburgh in 1860 his interests were in other directions than observational astronomy, and the old telescopes and theodolite were left in undisturbed possession. Nevertheless, his early appreciation of astronomical instruments declared itself from time to time when he purchased a beautiful speculum or a complete reflector for the Natural Philosophy Museum. In his Scrap Book Tait preserved a neatly constructed chart of date 1844, showing graphically the positions of Jupiter's satellites on successive nights from Sept. 18 to Sept. 31. These "Observa- tions on Jupiter" were made by himself when he was a little over thirteen years of age. Probably they were interrupted by bad weather. The environment amid which Tait spent his schooldays is well described in the Chronicles of the Gumming Club, a remarkable book printed for private circulation in 1887. Written by the late Lt.-Col. Alexander I 2 4 PETER GUTHRIE TAIT Fergusson, it places on record the life history of a class of boys which began its corporate existence in the winter of 1841. Peter Guthrie Tait was one of this class, which at the start numbered some sixty lads all about ten years of age. The reason for this great gathering of the first year or " Geits 1 " class was the popularity of the master, James Gumming. According to the custom then holding in Edinburgh Academy, each master began in rotation with the first year's scholars and carried them on for four years under his exclusive instruction in classical studies. For the remaining three years of the regular curriculum the boys, although coming directly under the care of the Rector, still continued to spend some hours of tuition with the master who had trained them from the first. When in accordance with the routine of the school the time came for Mr Gumming to start the new first year, his fame as a teacher drew an unusually large number of boys. Of the members of this particular Gumming class as many as twenty- seven entered "the Services at an important juncture in the history of our country," and won thirty-nine military honours including six British and Foreign Knightly Orders. This was the class in which Tait was through- out his schooldays the "permanent dux." In 1850 the surviving members of the class formed themselves into a club called the Gumming Club, which met for good fellowship year by year. In Colonel Fergusson's brightly written chronicle we find a perfect picture of the school life in Edinburgh during the early part of last century. Especially are we introduced to the masters who helped to mould the mind of P. G. Tait. Tait himself had many reminiscences of his schoolmasters; and for James Gumming, the classical master, and James Gloag, who gave him his first acquaintance with mathematics, he retained always the greatest admiration and respect. So thoroughly was Tait taught the classics that (as he once told me) he never required to turn up a Greek Lexicon all the time he was at school. This no doubt was largely due to the pupil's own extraordinary verbal memory ; but the master who could teach with such results must have been to the manner born. Gloag was a teacher of strenuous character and quaint originality a type familiar enough in Scotland before School Boards and Leaving Certificates cooperated to mould teachers after the same type. With him mathematics 1 In Jamieson's Dictionary of the Scottish Language, geit, gett, gyte, variously spelt, is denned as "a contemptuous name for a child." Compare modern "kid." EDINBURGH ACADEMY 5 was a mental and moral discipline. How keenly Gloag enjoyed exposing the superficial knowledge of a boy who thought he knew ! A very characteristic story is told in the Chronicles of the way in which, in the presence of the Rector, Gloag demanded a proof from one of the Rector's classical pets. After the Rector in a foolish assumption of knowledge had for some time encouraged the boy with such remarks as "Why, my boy, don't you see it? Think a moment! It's quite easy, don't you know perfectly simple ! " Gloag in a moment of supreme triumph exclaimed " Naw, Mr Ractor, Sir, it's nott easy the thing's imp5ssible, Sir it's gross non- sense, Sir!" Such was the teacher who first led Tait's mind in the paths in which ere long he was to gain the highest distinction. Lewis Campbell and James Clerk Maxwell were also Edinburgh Academy boys ; and in Campbell's Life of Maxwell an interesting account is given of the school. They were a year ahead of Tait and were not therefore members of the Gumming Club. Fleeming Jenkin, the first Professor of Engineering in Edinburgh University, was a classmate of Tait, as were also Sir Patrick Heron Watson the eminent surgeon, Sir Edward Harland of Harland and Wolff, Belfast, A. D. Stewart, C.E., who selected he plans for the Forth Bridge, Andrew Wilson, traveller and author of The Abode of Snow, General Cockburn, General Sherriff, Frederick Pitman, W.S., one of the early Secretaries of the Gumming Club, Dr Thomas Wright Hall, a well-known physician for many years resident in Brazil, and many others whose careers are sketched in the Roll Call of the Chronicles of the Gumming Club. Tait himself preserved in printed form the result of the examination held in 1846 to determine the winner of the Edinburgh Academical Club Prize. The competition was open to all the Rector's classes, namely, the Fifth to the Seventh. Lewis Campbell came out first over all and gained the prize. Tait was third, being the only Fifth Class boy who was named in the list, and Maxwell was sixth. In the department of mathematics, however, the order of merit was Tait, Campbell, Maxwell, the others named being far behind. On the classical and linguistic side Tait naturally fell behind the more widely read scholars of the higher classes. In the competition for the Academical Club Prize in 1847, Tait was again third, but Maxwell, now in the Seventh Class, was second on the whole. In mathematics, Maxwell was first and Tait was second. 6 PETER GUTHRIE TAIT Tail's skill in Latin verses is specially recorded in the School Reports, and a good specimen of his efforts in versification will be found by the curious in the Edinburgh Academy Report for 1845. To the end of his life he remembered hundreds of lines of Greek and Latin poetry. His children remember how he used to declaim Odes of Horace and long passages of Homer when the fancy struck him. German ballads also were among his stock in trade for apt quotation. A favourite time for such outpourings was on St Andrews Links before breakfast, when he was still young enough to cover the ground without trouble at a good five miles an hour. It may be doubted if anyone whose classical studies ended when he was little more than fifteen years old ever carried away such a store of poetry, or found in it such a never-failing source of pleasure. He frequently spoke of Archdeacon Williams, the Rector of the Academy with whom he read Homer, as a born teacher. "A gentleman, every inch of him," was his emphatic verdict a few weeks before his death. In the Rector's report for the year 1851-2, when Tait's position as Senior Wrangler added glory to his old school, it is stated that Tait gained eight medals, six as dux of his class for the successive years 1841-47, and two for mathematical excellence in the Fifth and Sixth classes. Tait left the Academy in 1847, and then spent a session at Edinburgh University under the tutelage of Kelland and Forbes. He enrolled himself in the two highest of Kelland's three mathematical classes and attended all the examinations. He secured high positions in both, but was distanced in the competition by several of his fellow students. In the highest class he was third in the honours list. There was only one class in Natural Philosophy ; but this was divided by Forbes into three divisions. All members of the class attended the same lectures, on the subject matter of which they were periodically examined. The home reading, on which there were special examinations, varied with the division. A student usually entered the third or lowest division, passing into the higher divisions if he enrolled himself in the class more than once. Tait boldly entered himself for the first division. There is a tradition that Forbes in his most dignified manner tried to induce Tait to be content with the second division. This was the course Clerk Maxwell took, in spite of the fact that he was certainly as advanced in his mathematical studies as Tait, and had moreover already published a mathematical paper of distinct originality. Neither Maxwell nor Tait STUDENT AT EDINBURGH UNIVERSITY 7 markedly excelled in comparison with the best of their fellow students. Tait was third in the honours list of the five men who formed the first division. The Gold Medal, which was awarded to the student who made most marks in the special examinations in the highest division, was gained by James Sime, one of the most brilliant students of his day, and well known in Edinburgh educational circles throughout a long and active life. In the examinations on Newton's Principia (first three sections) and Airy's Tracts (probably that on the undulatory theory of light), Sime gained twice as many marks as Tait. In the ordinary examinations on the Class Lectures Tait had a slight advantage, although a wrong addition in the class book makes him a mark or two behind Sime. The prize was, however, gained by Maxwell. It is not a little curious that the Gold Medal was not won by Balfour Stewart in 1846, nor by Tait in 1848, nor by Maxwell in 1849; and yet Edinburgh University can claim no greater names in physical science than these three. An interesting fact which I learned from Tait himself is worth recording. On one occasion when, in preparation for a lecture on statics, I was arranging and admiring the models of catenaries of various forms which belong to the Natural Philosophy Museum of Edinburgh University, Tait remarked, " I helped Forbes to make these when I was a young student here." The models were constructed of beautifully turned disks of wood of suitable form, suitably strung together, and represented the common catenary, the circular arc catenary and the catenaries of parabolic form and of uniform strength. I pointed to the last word " strength " which was misspelled, the penultimate letter being dropped probably from want of room, and said in joke, "Is this an example of your accuracy?" "Ah," he rejoined, " I was responsible only for the calculations of the sizes of the disks, not for anything else." Clerk Maxwell spent three sessions in Edinburgh University before he decided to go to Cambridge ; but Tait was content with one session, and began his mathematical training in Cambridge before he was eighteen. CAMBRIDGE. 1848-54 It was a curious fate which brought to Peterhouse in 1848 the two young mathematicians, P. G. Tait and W. J. Steele, the one from Scotland, the other from Ireland by way of Glasgow 1 . They "coached" with the famous private mathematical tutor of those days, also a Peterhouse man, William Hopkins, another of whose pupils a few years earlier was William Thomson, afterwards Tail's lifelong friend. Tait and Steele at once became marked out as future high wranglers ; but one would hardly have dared to prophesy that they would come out respectively first and second in the Tripos. Tail's method of preparing for the great contest is preserved in his own hand-writing on three quarto sheets afterwards pasted into the Scrap Book. From Dec. 16, 1851, to Jan. 5, 1852, each day (Sundays excepted) is marked off for revision of definite subjects of study, morning and evening. When the work is accomplished, the subject is scored out and the time taken marked in the margin. Four hours are the most he gives at one sitting, and on no day does his time of study exceed 6\ hours, usually much less. Opposite Jan. 6, Tuesday, is printed by hand the words " Senate House." Then comes an irrelevant note of a lunar eclipse which occurred on Jan. 7, and below this appears in large letters right across the sheet the word " Porgatorio." The three days of Purgatory past, the time schedule begins again on Jan. 8 (evening) with "Brief Respite from Torment"; and during the succeeding eight working days the morning and evening tasks are again portioned out. But the work is more serious now. Tait never gives less than 5^ hours a day, and on one occasion reaches 7^ hours. Beneath the last date "January 19, Monday and subsequent" he prints across the page in huge capital letters "L'ENFER!" The guiding principle seems to have been not greatly to exceed in sustained work during any one day the time allotted for the examination. 1 In Kelvin's early paper on the Absolute Thermometric Scale (Cambridge Phil. Trans., June 1848, Phil. Mag., Oct. 1848) William Steele is mentioned as having assisted in comparing the proposed scale with that of the air thermometer (see Math, and Phys. Papers, Vol. I, p. 105). SENIOR WRANGLER 9 Steele seems to have been generally ahead of Tait in the College examina- tions, so that Tail's winning of the Senior Wranglership came somewhat as a surprise to those who deemed they knew. The story of this day, famous in the annals of Peterhouse, is well told by J. D. Hamilton Dickson in the Magazine of the Peterhouse Sexcentenary Club for the Michaelmas Term, 1902. " How the old gyp's face used to light up as he told the story of that January morning when the Tripos list was read. One gyp was in the Senate House to hear the list, and as soon as Steele's name came out as Senior Wrangler he was to rush out and make a signal by stretching out his arms like a big T; another gyp near the ' Bull ' was to repeat the signal ; and a third at the College gate was to rush in with the news. When that list was read and Tait's name came first the gyp nearly collapsed, but hearing Steele's name next he recovered, and noting only that Peterhouse was first, rushed out, made the signal, and fled with all speed to College to correct the pardonable error he had telegraphed." Tait telegraphed home " Tait Senior, Steele second, tell Gloag." How Gloag received the news is told in a footnote in the Chronicles of the Gumming Club. "When intelligence reached the Academy of the great event, Gloag was 'raised' and out of himself with excitement. ' Have ye hard the news aboot Tait ? ' he asked of everybody he met, M among others. ' No,' answered M , ' he's got a Bishopric, I suppose, or something of that sort.' ' No, Sir, it's not Archibald Cam'ell Tait it's Peter Guthrie Tait, a vara different parson 1 Senior Wrangler, Sir,' and off he went to spread the news." Through the kindness of Sir Doyle Money Shaw, at that time president of the Gumming Club, Mr Beatson Bell, for many years Secretary of the Club, was able to show me the brief note in which Tait told of his success. COLL : Div : PET : CANT. Jany. y.st 1852. My dear Doyle, I'm all in a flutter I scarcely can utter, &c., as the song has it: I AM SENIOR WRANGLER! Tell it to the Gumming Club &c. &c. and believe me yours very sincerely PETER GUTHRIE TAIT, B.A. 1 So Gloag pronounced "person." T. 2 io PETER GUTHRIE TAIT Tail's achievement was made the occasion of a special meeting of the Gumming Club. It was (to quote from the Chronicles) " felt to be an honour conferred on the Academy, the Masters Gloag in particular the Class, and the Club. Consequently they could do no less than offer to their old friend and Dux a banquet specially designed to do him worship. And right well they did it "For once the exclusive rule of the Club was broken through, and invitations scattered with a lavish hand amongst those and they were many who beyond the limits of the Class, held kindly memories of Tait and of the Academy " It was a high occasion for them all. Gloag could hardly divest himself of the idea that he was the hero of the occasion, such credit did he take to himself. " Festive conversation was at fullest swing that is to say, many talkers, few listeners when suddenly the scene of revelry was broken in upon by an ominous 'boom.' Tongues were still for a moment, but only for a moment. " Then once again, clearer, deadlier than before, the ' boom ' is heard above the clatter of tongues. " In a moment the mystery is solved. The President, Doyle Shaw, ever active for good, or evil, from his end of the table as it approached the gallery, had observed peeping over the edge of this gallery, at an inviting angle, the rim of a big drum. Straightway the idea arose that by well directed vertical fire this tempting object might be reached. The first orange discharged hit the mark unobserved by the company, but the second ' boom ' discovered all. "The idea was hailed as a brilliant one that only needed development. The entire dessert, oranges and apples, was soon expended. Then the thought occurred to Doyle Money Shaw to improve on his original idea. While the practice was still going on he managed cleverly to ' swarm ' up one of the pillars with the intention of capturing the big drum. But on arriving at the spot and with a shout of ecstasy he announced to those below that the entire band instruments were there. Without a moment's loss of time these were handed down, and from hand to hand; and nothing would serve these festive spirits but the ' Conquering Hero ' in Tait's honour." Steele was evidently a man after Tait's own heart. They were close friends throughout their College life, and when Fellows of the same college they collaborated in the production of a treatise on the Dynamics of a Particle. The book was planned and to some extent written during a holiday they spent together after they took their degree. Unfortunately Steele's health gave way, and his early death left his portion of the work unfinished. With the true chivalry of his nature Tait issued the book in 1856 under the joint names of Tait and Steele; and "Tait and Steele" is still its familiar title. The character of the book will be discussed later. The MS was presented to Peterhouse by Mrs Tait, and is now preserved in the College FELLOW OF PETERHOUSE n Library. The accompanying picture of the group containing Tait and Steele, who are respectively first and third reckoning from the left, has been reproduced from a somewhat faded photograph. Its probable date is 1852. Having taken his degree as Senior Wrangler and First Smith's Prizeman, Tait was elected a Fellow of his College and began to establish himself as a "coach." To quote from an address he gave to the Edinburgh Graduates fourteen years later, he became one of those who, " eagerly scanning examination papers of former years, and mysteriously finding out the peculiarities of the Moderators and Examiners under whose hands their pupils are doomed to pass, spend their lives in discovering which pages of a text-book a man ought to read and which will not be likely to ' pay.' The value of any portion as an intellectual exercise is never thought of; the all-important question is Is it likely to be set? I speak with no horror of or aversion to such men ; I was one of them myself, and thought it perfectly natural, as they all do. But I hope that such a system may never be introduced here." His hopes, it is to be feared, are being only partially realised. Tait's experience as a coach was fortunately very limited. During the two and a half years he continued to reside at Peterhouse he had hardly time to establish a reputation. There is indeed a story 1 of "Tait's one Pupil," who had begun to read with Hopkins. So unsatisfactory was his progress that Hopkins advised him to seek another tutor. Naturally the pupil protested and said he would do his utmost not to keep the others back. But Hopkins was obdurate. Accordingly the aspirant to Wrangler honours became Tait's one pupil, and was taught to such good purpose that when the Tripos list came out he was one place above Hopkins' best man. When congratulated upon the success of his pupil Tait is said to have remarked, "Oh, that's nothing I could coach a coal scuttle to be Senior Wrangler." Tait, however, was not a man to let time hang on his hands. He read widely and thoroughly in all branches of mathematical physics. During these years also he learned to read Italian with ease and made himself master of the French and German languages. 1 The story is given with full details in a letter from W. A. Porter, whose authority was C. B. Clarke, 3rd Wrangler in 1856, and Mathematical Lecturer in Queens', 1857-65. 2 2 BELFAST. 1854-60 On September 14, 1854, P. G. Tail was appointed Professor of Mathe- matics in Queen's College, Belfast. Among his colleagues were Thomas Andrews, the famous experimenter on the liquefaction of gases, Wyville Thomson, afterwards of Edinburgh and the scientific leader of the Challenger Expedition, James Thomson (Lord Kelvin's brother), subsequently professor of Engineering in Glasgow and the discoverer of the lowering of the melting point of ice by pressure, and James MCosh, afterwards President of Princeton. The Right Hon. Thomas Sinclair, of Belfast, who as senior scholar in mathematics in 1857 assisted Tail in tutoring the junior men, mentions that in addition to conducting his official classes in mathematics Tait supplemented Professor Stevelly's lectures in Natural Philosophy by starting a voluntary class for Honours men in the more advanced treatment of dynamics. This was a great boon to those studying for honours. The voluntary class is mentioned in a footnote in the Calendar, but there is no indication that the class was carried on by the professor of mathematics. We can well imagine the delight with which Tait would escape from the comparative dreariness of Pure mathematics into the satisfying realities of Applied. Tait proved an admirable teacher, clear and systematic in his treatment of the various branches taught. In addition to the regular lectures, he gave tutorial instruction to his pupils, setting them exercises and problems and helping each individually in turn. In these years he continued to practise on the flute on which he was a skilled performer. In Cambridge he had been a member of the amateur orchestra, and we hear of him appearing at a concert in Belfast to play a flute obligato to a distinguished local soprano singer. The two great scientific facts of his life in Belfast were his association with Dr Andrews in experimental work and his study of Hamilton's calculus of Quaternions. Often in conversation Tait expressed his indebtedness to Andrews for initiating him into certain lines of experimentation. Their joint papers on Ozone are published in Andrews' memorial volume. The original conception of the investigation was due to the older man who had ASSOCIATION WITH PROFESSOR ANDREWS 13 already published important work on the same subject. Tait gave efficient aid, more particularly in the calculations involved, and in the construction of much of the apparatus used. He proved such an apt pupil in the art of glass blowing that ere long Andrews gave that part of the manipulation over to his eager and energetic companion. Tait used to speak with intense admiration of the extreme care and patience with which Andrews carried out all his researches. Each difficulty or discrepancy as it arose had to be disposed of before progress could be reported and the investigation advanced a stage. At times indeed the patient care of the skilled experimenter must have chafed somewhat the brilliant young mathematician ever eager to get to the heart of things ; but no amount of argument or theorising on Tait's part could move the master from the steady tenor of his way. Years after when Andrews in his failing health visited Edinburgh Physical Laboratory to inspect a set of his own apparatus for the liquefaction of gases it was at once a privilege and an inspiration to witness the deep affection and admiration with which Tait regarded his whilom colleague. In his letter to Mrs Andrews immediately after the death of her husband, Tait expresses his feelings and regard in these words : "It does not become me to speak of the irreparable loss which you and your family have suffered. But it may bring some consolation to you to be assured that there are many, in many lands, whose sympathies are sincerely with you ; and who lament, with you, the loss of a great man and a good man. " For my own part, I feel that I cannot adequately express my obligation to him whether as instructor or example. I have always regarded it as one of the most important determining factors in my own life (private as well as scientific) and one for which I cannot be sufficiently thankful, that my appointment to the Queen's College at the age of 23 brought me for six years into almost daily association with such a friend." Hamilton's first book, Lectures on Quaternions, was published in 1853. We learn from the inscription on the title page of Tait's copy that he bought it the same year while still a resident at Peterhouse. As he explained in the preface to his own Treatise (ist edition, 1867) Tait was attracted to the study of quaternions by the promise of usefulness in physical applications. Yet in Hamilton's Lectures very few pages indeed touch upon dynamical problems. Tait used to tell how his faith in the new calculus was put to a severe test as he read through these remarkable so-called lectures of Hamilton. Lecture after Lecture he carefully perused, wearied though he was with Hamilton's extraordinary prolixity in laying strong and deep the foundations H PETER GUTHRIE TAIT of his calculus. He seemed to be making no progress. Did the fault lie with the author, or with Tail's own inability to understand the system ? Such were his feelings through the first six " Lectures." But perseverance had its reward when he came to Lecture VII. Here, after a few sections of recapitulation, Hamilton revels in the wealth of geometrical applications fitted to display the power of the calculus. This so-called Seventh Lecture occupies 356 pages in a book of which the other six Lectures occupy 380! Tait was one of very few who really appreciated the immense value of Hamilton's work. Many who with gay confidence began to read the Lectures lost heart and fell back from Quaternion heights into Cartesian valleys, where the paths seemed easier in their artificial symmetry. Now, however, the early hopes of Hamilton and Tait are being realised in the growing use of vector methods and symbolism, especially in their physical applications. Hamilton's and Tail's theorems have been rediscovered by later workers, some of whom, under the domination of new notations for the quantities and functions which Hamilton made familiar, think the novelty extends to the functions and quantities themselves ! During his undergraduate days Tait made the acquaintance of William Archer Porter and James Porter, brothers from Belfast. William, Third Wrangler in 1849, was for a time Tutor of Peterhouse, and after being called to the English bar became Principal of Combaconum College in India, and subsequently Tutor and Secretary to the Maharajah of Mysore. James Porter was Seventh Wrangler in 1851 and was elected a Fellow immediately after graduating. He was for some years mathematical professor of the Collegiate Institute in Liverpool, but returned ere long to Peterhouse, first as Tutor then as Master (1876-1901). He was endowed with a great activity both mental and physical, which found expression on the one hand in a keen participation in athletic sports, and on the other in whole-hearted efforts to promote the highest interests of the University. In Dr T. A. Walker's History of Peterhouse (1906) the Rev. James Porter is described as a " man of notable business qualifications and of a rare generosity of spirit." When Tait went to Belfast he became closely intimate with the Porter family, and on October 13, 1857, he married one of the sisters of his Peterhouse friends. As Kelvin expressed it : " During these bright years in Belfast he found his wife and laid the foundation of a happiness which lasted as long as his life." The youngest brother, John Sinclair Porter, was one of Tail's most PROJECTED "KNOCKLAYD" EXPERIMENT 15 distinguished students at Queen's College. He entered the Indian Civil Service in 1861 and retired in 1889. There is a good story told of how Tait saved valuable personal property of his colleague Wyville Thomson from the process of arrestment executed upon the landlord's house and goods. When the bailiffs took possession Tait came on the scene and after some conversation got permission for Wyville Thomson and his wife, who were simply lodgers, to fill two boxes with their purely personal goods. The men of law retired to the kitchen to be refreshed for their labours. They looked out occasionally and always saw the two boxes in the hall being filled. But they did not realise that as soon as one box was filled another took its place, a process of substitution which continued for some little time. Meanwhile the landlord's family thought they might be doing similar deeds of saving, and began to pitch things out of the window. A feather bed happened to fall on an onlooker. The consequent excitement roused the bailiffs from their ease, but not until all the valuables of the Thomsons had been removed. Although Tait was professor of pure mathematics in Queen's College, his real interest lay towards the physical side. Writing to his uncle, John Ronaldson, in 1858 he says: " I have got the contoured map of Knocklayd from the Ordnance Office and have done a rough calculation which shows io"'28 as the effect on the plumb line, a very hopeful indication. If Thomson reports as well of the geology we shall commence in earnest next summer." Knocklayd is a conspicuous hill of conical form in County Antrim, and evidently Tait contemplated using it after the manner of the Schiehallion Experiment to measure the mass of the earth. In one of his quarto note books there are tabulations of stars convenient for zenith observations which he purposed making with suitable instruments both at Belfast and at Knocklayd. Beyond these preparations, nothing more definite seems to have been done. Other problems had to be dealt with and the proposed book on Quaternions pushed on ; and before two more summers had passed Tait had bidden farewell to Ireland and had begun his great career in Edinburgh. EDINBURGH. 1860-1901 In 1860 the Chair of Natural Philosophy in Edinburgh University became vacant owing to the retirement of James David Forbes, and Tail offered himself as a candidate. The other candidates were Professor Fuller, King's College, Aberdeen ; the Rev. Cosmo Reid Gordon, Manchester ; Pro- fessor Clerk Maxwell, Marischal College, Aberdeen ; E. J. Routh, Peterhouse, Cambridge ; Edward Sang, Edinburgh ; and Professor Swan, St Andrews. There is no difficulty now about placing these men in their appropriate niches in the Temple of Fame; but in 1860, when the best work of most of them was still to do, it could not have been an easy matter to discriminate among them. In the Edinburgh Courant of the day we find a remarkably sane and prescient discussion of the choice which the Curators had made. Some of the sentences are well worth quoting as showing that even in these days the characteristics of some of the men had been clearly diagnosed. After noting the distinction already gained by Fuller and Routh as eminently successful teachers, the writer disposes of their claims in comparison with those of Maxwell and Tail by the remark that neither " had as yet acquired a reputa- tion for powers of original scientific investigation." With regard to Maxwell and Tait the writer continues "it will be no disrespect to the warmest friends of the successful candidate, and we do not mean to dispute the decision of the curators, by saying, that in Professor Maxwell the curators would have had the opportunity of associating with the University one who is already acknowledged to be one of the remarkable men known to the scientific world. His original investigations on the nature of colours, on the mechanical condition of stability of Saturn's Rings, and many similar subjects, have well established his name among scientific men ; while the almost intuitive accuracy of his ideas would give his connection with a chair of natural philosophy one advantage, namely, that of a sure and valuable guide to those who came with partial knowledge requiring direction and precision. But there is another power which is desirable in a professor of a University with a system like ours, and that is, the power of oral exposition proceeding upon the supposition of a previous imperfect knowledge, or even total ignorance, of the study on the part of pupils. We little doubt that it was the deficiency of this power in Professor Maxwell principally that made the curators prefer Mr Tait With a clear understanding, and talents only second in order to genius, cultivated by persevering industry, he LECTURER AND TEACHER 17 has attained to great and solid scientific acquirements, and to very much of that habitual accuracy which his rival, Mr Maxwell, possesses by a sort of intuition. We have never heard Mr Tait lecture, but we should augur from all we can learn that he will have great powers of impressing and instructing an audience such as his class will consist of, combined with that conscientious industry which is so necessary in a successful professor." Whoever wrote these words or supplied the underlying thoughts had formed a just estimate of the respective strengths of the candidates. Fuller was certainly one of the greatest mathematical teachers any Scottish University ever possessed ; Routh was unsurpassed in Cambridge as a trainer of Senior Wranglers and has, moreover, left his mark on dynamical science ; Maxwell towers as one of the creative geniuses of all time, curiously lacking though he was in the power of oral exposition ; Tait, who possessed, also by intuition, the clearest physical conceptions, has left behind him a great record of research both in mathematics and physics, while, as a teacher and clear exponent of physical laws and principles, he took a foremost place among his contemporaries. He had all the gifts of a born lecturer. His tall form and magnificent head at once impressed the student audiences which gathered year after year on the opening day of the session. The impression was deepened as with easy utterance, clear enunciation, and incisive phrase, he proceeded to indicate the nature of the subject of study. J. M. Barrie in An Edinburgh Eleven gives a graphic picture of Tait lecturing : " Never, I think, can there have been a more superb demonstrator. I have his burly figure before me. The small twinkling eyes had a fascinating gleam in them ; he could concentrate them until they held the object looked at ; when they flashed round the room he seemed to have drawn a rapier. I have seen a man fall back in alarm under Tail's eyes, though there were a dozen benches between them. These eyes could be merry as a boy's, though, as when he turned a tube of water on students who would insist on crowding too near an experiment " This is good ; but in some other respects Barrie's pen portrait is unsatisfactory if not misleading. For example in the succeeding paragraph he states that " Tail's science weighed him to the earth " a remark almost too grotesque to need refutation. With regard to the real character of the man whose eyes could flash rapier-like glances or scintillate with heartiest merriment Barrie had, indeed, little chance of intimate knowledge. Tait used to speak of T. 3 i8 PETER GUTHRIE TAIT himself as a "lecturing machine" appointed by the University to instruct the youth of our country in the "common sense view of the universe we live in." Students were invited to send in their difficulties in writing before the lecture ; but conditions were not favourable for personal inter- course between teacher and pupil. Tait let nothing interfere with his official duties towards his class, declining on principle to make mention of anything but what had a direct connection with University regulations or College work. Once an enthusiastic secretary approached him with the request that he would announce a meet- ing of the highly important society represented by the petitioner. Tait opened his lecture with the remark that in this class room they met to discuss Natural Philosophy and that he made it a rule to speak only of what concerned the work of the class. A few mornings later there appeared in the public prints the announcement of the birth of his youngest son. As Tait appeared on the platform behind the lecture table he was greeted with a burst of applause, which lasted several minutes. In grim silence he waited till the noise subsided ; then, with a quizzical glance round the full benches, he remarked " Gentlemen, I said the other day that I make it a rule to take notice here only of what affects directly the work of the class." This pertinent sally was received with laughter and a ringing cheer, and then the students settled down to listen attentively to the lecture of the day. To the student who passed through the general class of Natural Philosophy on the way to the ordinary degree Tait was the superb lecturer and nothing more. Those who entered the optional laboratory course or who took the Advanced Class with a view to honours were better able to appreciate his varied gifts ; but a full revelation of the great personality came only to the privileged few who acted as his assistants, or who worked with him or for him in the laboratory. The sterling honesty of the man shone through all he did. As Sir Patrick Heron Watson once said, the charm of Tait was his naturalness and he had known Tait from their boyhood's days. Sincerity was to him the touchstone of a man's character. Strong in his likes he was also strong in his dislikes. With true chivalry he fought for the claims of his friends if these were challenged by others. It was this indeed which led him into controversy. Thus arose the controversies with Tyndall concerning the history of the modern theory of heat and Forbes' glacier work, and the discussion with Clausius in reference ORDINARY CLASS LECTURES 19 to the thermo-dynamic discoveries of Kelvin. His passage at arms with Herbert Spencer Tail himself never regarded as anything else than a big joke. As a lecturer Tail was probably unsurpassed by any of his con- temporaries. His lecture notes were merely jottings of headings with the experiments indicated and important numerical values interspersed. In the original note book, which was in use till 1881, these headings were entered with intervening spaces so as to allow for additions as time went on. In 1 88 1 he rewrote the greater part of the notes in a smaller octavo book, and this he continued to use to the end. These lecture notes had to do with the properties of matter, which largely occupied the attention of the class for the first half of the winter session. Tait regarded this part of the course as a general introduction to the study of Natural Philosophy. He devoted the first few days to a discussion of the nature of the subject and of the means by which we gain knowledge of the physical universe. His treatment of the subjective and objective from the point of view of the natural philosopher was always clear and reasonable. I remember going back with a former classmate to hear Tail's opening lecture. Since we had first sat together in the benches of the Natural Philosophy Class room my friend had pondered deeply on meta- physical themes ; and, as we listened again to Tail's exposition of objective and subjective, he whispered lo me " Beauliful, Berkeley couldn't have done it better. " The conceplions of lime and space, and the realities known as matter and energy, were introduced and placed in their righl selling from ihe physical slandpoint. These preliminaries disposed of, Tait began his syste- malic leclures on the properties of malter. His aim was to build a truly philosophical body of connected irulhs upon the familiar experiences of the race. In ordered sequence ihe various obvious properties of matter were considered, first, in themselves, then in their theorelical setting and their practical applications. Thus, lo take but one example, the discussion of the divisibility of matter led to the consideration of mechanical sub-division and of the elementary principles of the diffraction and inlerference of light, illustrated by colours of soap films, halos and supernumerary rainbows. The fuller explanation of these was, however, reserved for a later date when the laws of physical optics were taken up in more detail. In this way the intelligent studenl was able during the firsl iwo monihs lo gain a general outlook upon 3* so PETER GUTHRIE TAIT physical science. The nature of the course may be inferred from the contents of his book The Properties of Matter ; but no written page could teach like the living voice of the master. After the first few weeks the systematic lectures on the properties of matter were given during not more than three hours each week, Tuesdays and Thursdays being devoted to elementary dynamics. These were supple- mented by some tutorial lectures by the assistant. The properties of matter having been disposed of, the subjects of heat, sound, light and electricity were taken up in turn, the amount of time given to each varying with different years. With the exception of heat Tail's lecture notes on these branches were not prepared with the same affectionate care as had been bestowed upon those dealing with the properties of matter. He had a few systematic notes on geometrical optics but none on physical optics or electricity. Indeed, as time went on, the properties of matter, like the Arab's Camel, encroached more and more on the limited time of the session. This was inevitable. Tait was always adding to his notes either new facts or new illustrations, and he never dropped any part out. His experiments hardly ever failed. They were chosen because they were instructive and elucidated the physical principle under discussion not merely because they were beautiful or sensationally striking. To the intelligent student who had worked through the earlier part of the course namely, dynamics and properties of matter the comparatively meagre treatment of physical optics and electricity was not perhaps of great con- sequence. He had been guided along a highway from which all parts of the great domain could be sighted and some information gained of each secluded region. He had been taught how to look and how to appreciate the view. He had been warned that the senses alone were untrustworthy guides ; that he must illuminate the dark places with the light of reason, with the search light of a scientific imagination. To those of us who came with some knowledge of physical science, Tail's whole method was a reve- lation. But the great majority of those students who knew nothing of natural philosophy till they came under the fascination of his lectures were hardly in a position to appreciate the majestic beauty of the whole presentation. In addition to the task of digesting the lectures the students were expected to do some extra reading on which they were specially examined. The junior division, that is, nearly the whole class, read Herschel's Astronomy; and the senior division, consisting of a few enthusiasts who were strong enough ADVANCED CLASS LECTURES 21 in mathematics, studied the first three sections of Newton's Principia. This home work was however purely voluntary even when, under the later regulations, the attendance of students at the examinations on the Class Lectures became compulsory. To the advanced student able to follow him Tait was not merely a superb lecturer but was also a great natural philosopher and mathematician. The more abstruse the subject the more clearly did Tait seem to expound it. The listener felt that here was a master who could open the secrets of the universe to him. Unfortunately, when deprived of the aid of Tail's lucid exposition, in the easiest of English speech, of the knottiest mathematical or physical problems, the student, now left to himself, felt that his original ignorance was doubled. In the Advanced Class Tait treated dynamical science in the manner of " Thomson and Tait." He does not seem to have kept notes of his course, but simply to have prepared his ideas the night before the lecture. In the earlier days down to about 1876 he used as a guide the elementary treatise known as " Little T and TV Following the sequence of ideas there set down he developed the subject by use of the calculus. After 1876 he used for lecture notes a set neatly written out by his assistant, now Professor Scott Lang of St Andrews ; but later he found his Britannica article on Mechanics with interleaved blank sheets more suitable for his purpose. In the end he lectured along the lines of his own book on Dynamics, which was largely a reprint of the Mechanics article with important additions on Elasticity and Hydrodynamics. One outstanding feature of Tail's style of lecturing was its calm, steady, emphatic strength. He never seemed to hurry; and yet the ground covered was enormous. Was he for example establishing the general equations of hydrodynamics ? Bit by bit the expressions were formed, each added item being introduced and fitted on with the clearest of explanations, until by a process almost crystalline in its beauty the whole formula stood displayed. All was accomplished with the minimum of chalk, but with sufficient slowness to allow of the student adding the running commentary to his copy of the formulae. The equations only and their necessary transformations were put on the black board, the student being credited with sufficient alertness of mind and agility of hand to supply enough of the explanation to make his notes remain intelligible to himself. Though broadly the same, his advanced course varied in detail from year 22 PETER GUTHRIE TAIT to year. For certain parts he had a particular affection, such as, applications of Fourier analysis, Green's theorem, and especially the theory of strains. The last named was, indeed, a subject peculiarly his own, and many of his demonstrations, although given in ordinary Cartesian coordinates, were suggested by the quaternion mode of attack. An important feature of the Natural Philosophy Department since 1868 was the Physical Laboratory, for which Tait had secured a money grant as early as April 1867 but was unable at the time to find accommodation 1 . Lying quite outside any recognised course of study this purely voluntary course of practical physics offered no inducement to the ordinary student intent on getting his Degree. Tail's idea was to attract men who wished to familiarise themselves with methods of research. This he did by giving every encouragement to the man who had thought of some physical question worthy of investigation, or (as was more frequent) by suggesting some line of research to the eager student. Whoever showed real aptitude had all the resources of the Department placed at his disposal ; and beyond the initial fee of two guineas for the first winter session no other charge was made, no matter how long the student continued to work in the laboratory. Those students whose interest in the subject brought them back after the first session of their enrolment, were nicknamed "veterans"; and on their enthusiastic help Tait largely depended for the successful carrying out of his many ideas. This will be brought out in Chapter II on Tail's Experimental Work. Having given a broad outline of Tail's method of instruction I propose now to sketch briefly the main scientific events of his life, the more important of which will, however, be discussed in detail in later chapters. On taking up the duties of the Edinburgh Chair Tait gave his first care to the preparation of his class lectures ; and we get glimpses of the early development of his ideas from his letters to Andrews, for access to which I am indebted to the kindness of the Misses Andrews. The following is his own description of his first lecture given on November 5, 1860. "The Lecture (that is, the formal inaugural lecture) has not yet appeared in public. I began to-day, but, fancying that a dry technical lecture to commence with might perhaps keep off rather than attract amateur students, I gave a set of ' Thomson's laboratory in Glasgow began about 1850; and Carey Foster's in University College, London, was established in 1866. SCIENTIFIC AND LITERARY ACTIVITY 23 experiments the most striking I could muster professedly without any explanation in fact gave them as examples of the objects of Nat. Phil I gave a 20 m. lecture on the nature of the study, and the arrangements for the present session, and then plunged into the paradoxes. I reserved as the last the beautiful one of balls and egg shells suspended on a vertical jet of water, as they cannot be shown without some risk of a wetting to the performer and the nearest of the audience. To-morrow I bring into play the large American induction coil, and show the rotation of a stream of violet light in vacuo round a straight electromagnet. I shall also show an inch spark in air and the discharge by it about 10 times per second of a jar with about 3 square feet of tin foil. There is no self acting break for safety the interruption is made by a toothed wheel worked by hand which for short experiments is much preferable. I shall also show the huge Coin magnet (made under Pliicker's direction) which took six of us to heave it up a gently inclined plane into the class room this afternoon 1 " Outside his official University work his tireless energies were finding other fields for exercise. He wrote most of the longer and more important physical articles as well as the article Quaternions for the first edition of Chambers Encyclopaedia (1859-68) edited by Dr Findlater. His friendship with Findlater had important consequences ; for it was he who first took Tait out to learn the game of golf on the Bruntsfield Links, where they played frequently together. In 1 86 1 he began the writing of Thomson and Tail's Natural Philosophy, while at the same time he was busy strengthening himself in the use of quaternions and preparing his book on the subject. Together with Kelvin he communicated to Good Words in 1862 an article on Energy, which was intended as a corrective to Tyndall's state- ments regarding the historical development of the modern theory of heat. This led to two important articles in the North British Review which finally took shape as his admirable Sketch of Thermodynamics (1868). Some curious speculations by Balfour Stewart as to the thermal equi- librium within an enclosure of a number of radiating bodies moving with different velocities led Balfour Stewart and Tait to plan a series of experi- ments on the heating of a disk by rapid rotation in vacuo. The results were communicated to the Royal Society of London ; but no definite conclusion 1 In these days a roomy platform a few steps above the floor both of the class room and retiring room lay behind the long curving table on which the experiments were arranged. About 1880 the rapidly increasing number of students compelled the addition of two new benches, and this addition was managed by removing the platform, lowering the table and setting it back nearer the wall. The old Natural Philosophy lecture room is now used by the Logic and Psychology departments. 24 PETER GUTHRIE TAIT could be drawn from them. The outstanding difficulty was the uncertainty that all possible sources of heating had been taken account of. After some years of laborious experimenting the research was finally abandoned. When Nature was started in 1869 the Editor, (Sir) Norman Lockyer, secured the services of Tait not only as a reviewer of books but also as a contributor of articles ; and, especially during the seventies, Tait supplied many valuable and at times very racy discussions of scientific developments. In 1871 as President of Section A at the Edinburgh meeting of the British Association, Tait gave a characteristic address (Scientific Papers, Vol. i, p. 164), in which Hamilton's Quaternions and Kelvin's Dissipation of Energy are held up to admiration. The publication in 1873 of Tyndall's Forms of Water, in which the work done by J. D. Forbes in the elucidation of Glacier motion was some- what belittled, roused Tail's indignation and led to a controversy of some bitterness (see Nature, Vol. vm, pp. 381, 399, 431). Tyndall defended himself in the Contemporary Review ; and Tail's final reply, in which Tyndall's quotalions from ihe writings of Forbes are shown to be so in- complete as to lead the reader lo a false conclusion, appeared in the English iranslalion of Rendu's Glaciers of Savoy ediled by Professor George Forbes (Macmillan and Co., 1874). To Good Words of 1874 Tait contributed a series of most readable articles on Cosmical Astronomy, which embodied his lectures delivered to the Industrial Classes in the Museum of Science and Art, now known as the Royal Scottish Museum. At the same time Balfour Stewart and he launched their Unseen Universe upon an astonished world. Before 1880 the Editors of the new edition of the Encyclopaedia Britannica secured Tait as the contributor of the articles Hamilton, Light, Mechanics, Quaternions, and Radiation. The two longest articles namely Light and Mechanics were afterwards published, with additions, as separate books ; while the article on Radiation is practically embodied in his book on Heat. While all this literary work was going on, he was studying the errors of the Challenger Thermometers, writing an elegant paper on Mirage, investigating the intricacies of knots, pushing on his quaternion investigations when leisure permitted and putting together his Properties of Matter (1885). Throughout these years he also took a very practical interest in actuarial mathematics. A great believer in the benefits of Life Assurance he was SCIENTIFIC ACTIVITY 25 for a lengthened period a Director of the Scottish Provident Institution. The Directors of this Company were divided into two standing Committees of Agency and of Investment. Tait naturally served on the former; but he was never happier than when engaged with James Meikle, the well-known actuary, in solving actuarial problems. The two men had, each of them, the greatest confidence in the other's capacity. Very often after Board meetings, Meikle would way-lay the Professor and draw him into his sanctum to discuss some knotty question. The last heavy piece of mathematical investigation which fascinated Tait was the Kinetic Theory of Gases. Prompted by Kelvin, he wrote four important memoirs which by simplifying the mathematical treatment have greatly helped to clear up the difficulties inherent to the theory 1 . Before this work was well off his hands he was mastering the intricacies of the flight of the golf ball and planning experiments in impact and ballistics to elucidate some of the problems requiring solution. Not only did Tait in the end solve the main problem but it was he who first discovered that there was a problem to be solved. For hundreds of years Scotsmen had driven their balls over the historic links of St Andrews, Musselburgh, and Prestwick ; but no one had ever put the question to himself, why does a well driven ball "carry" so far and remain so long in the air? The adept knew by experience that it was not a question of mere muscle, but largely of knack. It was reserved for Tait, however, to find in it a dynamical problem capable of exact statement and approximate solution. From his earliest initiation into Scotland's Royal Game, he began to form theories and make experiments with different forms of club and various kinds of ball ; but not until late in the eighties did he begin to get at the heart of the mystery. Golf had now become a popular British sport, played wherever the English speech was prevalent ; and Tail's second youngest son, Freddie, was rapidly coming to the front as one of the most brilliant of amateur golfers. While the son was surprising and delighting the world by his strong straight driving, his remarkable recoveries from almost unplayable "lies," and his brilliant all-round play with every kind of club, the father was applying his mathematical and physical knowledge to explain the prolonged flight of the golf ball. The practical golfer at first 1 It is interesting to note that the first and second memoirs were translated into Russian by Captain J. Gerebiateffe and published with annotations expanding Tail's mathe- matical processes in the Russian Review of Artillery (1894). T. 4 26 PETER GUTHRIE TAIT smiled in a superior way at this new science of the game ; and Tait was scoffed at when he enunciated the truth that underspin was the great secret in long driving. It is interesting to see how step by step he advanced to the final elucidation of the whole problem or rather set of problems. Not until he had made definite calculations, did Tait or anyone else for a moment imagine that the flight of a golf ball could not be explained in terms of initial speed of projection, initial elevation or direction of projection, and the known resistance of the air. By means of ingenious experiments on the firing of guns, Bash- forth had completely worked out the law of resistance of the air to the passage of projectiles through it. When however Tait tried to make use of the data supplied by Bashforth's tables he found that it was impossible to reach even an approximate agreement between his theoretically calculated path and the path as observed. Two facts were known with fair accuracy the distance travelled by a well-driven ball, and the time it remained in the air ; and a third fact was also with some measure of certainty known, namely, the angle of projection or the elevation. But no reasonable combination of elevation, speed of projection, and resistance of air could give anything like the combined time and " carry " as observed daily on the links. Tait also showed that, on this obvious theory of projection and resistance, very little extra "carry" could be secured by extra effort on the part of the player in giving a stronger stroke with a correspondingly higher speed of projection. The resistance of the air rapidly cut down the initial high velocities. When therefore Freddie Tait on January u, 1893, exceeded far all his previous efforts by a glorious drive of 250 yards' "carry" on a calm day, he deemed that his father's dynamical theory was at fault. How often has the tale been told on Golf links and in the Club-house that Freddie Tait disproved his father's supposed dictum, by driving a ball many yards farther than the maximum distance which mathematical calculation had proved to be possible ! It is no doubt a good story, but very far indeed from the mark, as a glance at Tail's writings on the subject will at once prove. On August 31, 1887, Tait communicated to the Scotsman newspaper an article called " The unwritten Chapter on Golf," reproduced a few weeks later in Nature (Vol. xxxvi, p. 502). In that article he shows clearly that the evils of "slicing," "pulling" and "topping" were all due to the same dynamical cause, namely rotation of the travelling golf ball about a particular axis and in a particular way. The explanation was based on the fact, established THE PHYSICS OF GOLF 27 experimentally by Magnus in 1852 but already made clear by Newton in 1666, that, when a spherical ball is rotating and at the same time advancing in still air, it will deviate from a straight path in the same direction as that in which the front side is being carried by the rotation. Thus (to quote Tait) " in topping, the upper part of the ball is made to move forward faster than does the centre, consequently the front of the ball descends in virtue of the rotation, and the ball itself skews in that direction. When a ball is undercut it gets the opposite spin to the last, and, in consequence, it tends to deviate upwards instead of downwards. The upward tendency often makes the path of a ball (for a part of its course) concave upwards in spite of the effects of gravity " This last sentence contains the germ of the whole explanation ; but it was not developed by Tait till four or five years later. Neither here nor in any of his writings on the subject is any rash statement made as to the greatest possible distance attainable by a well-driven golf ball. In his first article " On the Physics of Golf" (Nature, Vol. XLII, August 28, 1890) Tait calculates by an approximate formula the range of flight of a golf ball for a particular elevation and various speeds of projection, the ball being assumed to have no rotation. In this way by comparison with known lengths of " carry " he finds a probable value for the initial speed of projection. He also points out that, to double the " carry," the ball because of atmospheric resistance must set out with nearly quadruple energy. About a year later (Sept. 24, 1891, Nature, Vol. XLIV, p. 497), he treats more particularly of the time of flight. He finds that, although we may approximate to the observed value of the range of a well- driven ball by proper assumptions as to speed and elevation, it is impossible, along those lines, to arrive at anything like the time of flight. The non-rotating golf ball will according to calculation remain in the air a little more than half the time the ball is known from experience to do. "The only way of reconciling the results of calculation with the observed data is to assume that for some reason the effects of gravity are at least partially counteracted. This, in still air, can only be a rotation due to undercutting." Thus he comes back to the rotation of the ball as the feature which not only explains the faults of slicing, pulling and topping, but is the great secret of long driving. When the rotation is properly applied as an underspin about a truly horizontal axis, the ball goes unswervingly towards its goal ; but, when owing to faulty striking the axis of rotation is tilted from the horizontal one way or the other, there is a component spin about a vertical axis and the ball 42 28 PETER GUTHRIE TAIT swerves to right or left according as the axis of rotation tilts down to the right or to the left. The clue was found, and the rest of the investigation was merely a question of overcoming the mathematical difficulties of the calculation. Thus undoubtedly before his son's brilliant drive of 250 yards' "carry," Tait knew well the influence of the underspin in prolonging both the range and time of flight; and before the summer of 1893 he had calculated the effect of the underspin sufficiently to establish the truth of his theory as a complete explanation of the flight of a golf ball. The results are given in the third article "On the Physics of Golf" (Nature, Vol. XLVIII, June 29, 1893), which is an abridgment of his first paper "On the Path of a Rotating Spherical Projectile" (Trans. R. S. E. Vol. xxxvu, Sci. Pap. Vol. u, p. 356). The theory is stated in popular language in an article on " Long Driving " communicated to the Badminton Magazine (March 1896) and reprinted below with slight additions and alterations made by Tait himself. Following up the indications of his theory Tait attempted to improve the driving power of a " cleek " or " iron " by furrowing its face with a number of fine parallel grooves, which by affording a better grip on the ball might be expected to produce a greater amount of underspin. He got several clubs constructed on this principle ; and four form part of the Tait collection of apparatus in Edinburgh University, having been presented to the Natural Philosophy Department by Mrs Tait. One of these is a "universal iron," in which the iron head in addition to being grooved is adjustable to all possible inclinations. The idea was to supply the golfer with one club having a degree of " loft " which could be varied at will. Tait himself found the weapon serviceable enough ; but Freddie would have none of it. The elucidation of the golf ball problem led Tait to another line of research, namely, the investigation of the laws of impact. These experiments and their bearing on the manner of projection of the ball are discussed in a later chapter, and in the article on " Long Driving " already referred to. Outside his University duties Tail's energies were devoted mainly to the interests of the Royal Society of Edinburgh. Elected a Fellow in 1860 he became one of the Secretaries to the ordinary meetings in 1864, and in 1879 succeeded Professor J. H. Balfour as General Secretary. This important post he continued to hold till his last illness. With the exception of his early mathematical papers, his conjoint papers with Andrews and with Balfour Stewart, and a few mathematical notes communicated in later years to the THE ROYAL SOCIETY OF EDINBURGH 29 Mathematical Society of Edinburgh, all Tail's original contributions to Science are to be found either in his own books or in the publications of the Royal Society of Edinburgh. For many a year hardly a month passed without some communication from him bearing on a physical or mathematical problem. But whether he himself had a communication to make or not, he was always in his place to the right of the Chairman guiding the business of the Society and frequently taking part in the discussions. The Royal Society of Edinburgh is no longer tenant under Government of the building in Princes Street known as the Royal Institution, the west wing of which had been planned for the Society when the building was erected. The need of more accommodation for the Society's unique library and for the National Art Galleries of Scotland demanded some change ; and finally, in 1907, by Act of Parliament the Royal Institution was wholly given up to Art and the Royal Society was assigned a more commodious home in George Street. A description of the old Meeting Room, of which now only the outer wall remains, is not inappropriate in the memoir of one who was for fully thirty years the most conspicuous of the Society's permanent officials, and the most active contributor to its literature. The arrangement of the room in which the meetings of the Society were held was certainly not convenient for modern requirements, such as experimental demonstrations or lantern exhibitions ; but there was a peculiar dignity and old-world flavour about it which will long linger in the memory. It is easily pictured an oblong room with doors at the ends flanked by crowded book-shelves. Along the east wall were two low book-cases, separated by fire-place and blackboards, and surmounted by portraits of illustrious Fellows such as Sir Walter Scott, Principal Forbes, Sir Robert Christison, and Professor Tait himself; and along the west wall were five windows looking towards the Castle. The President's Chair stood on a slightly raised platform in the very centre of the west wall before the central curtained window, and in front, running fully half across the width of the room towards the reader's desk, was a large oblong table, round which the members of Council were expected to sit. On this table the reader of the paper of the evening would place his microscopes or specimens or objects of interest. With the exception of the President and the leading officials, the Fellows occupied cushioned benches looking towards the large central table. The three secretaries sat invariably on the right of the Chairman, with their eyes towards the north door through which the members entered the room. 30 PETER GUTHRIE TAIT Occasionally, when the meetings were very full, part of the audience had to cross between the reader's desk and the Council Table and take their seats behind the secretaries' chairs. For modern lecture purposes a worse arrangement could hardly have been devised ; and yet it was quite in keeping with the fundamental idea of a Society whose Fellows met to communicate and discuss subjects of literary and scientific interest. At any rate, the reader or lecturer from his position in front of the blackboard looked across his small table towards the President at the far end of the long table, and addressed the Chair in reality, not contenting himself with the formal phrase which has largely lost its significance. It seems but yesterday when Piazzi Smythe with the peculiar hesitation in his speech uttered his tloge of Leverrier in the quaintly wrought involved sentences of a bygone century. Or it was Kelvin moving eagerly on the soft carpet and putting his gyrostats through their dynamical drill ; or Fleeming Jenkin amusing and instructing the audience with the sounds of the first phonograph which was used scientifically to analyse human speech ; or Lister quaffing a glass of milk which had lain for weeks simply covered up by a lid under which no air germs could creep ; or Turner demonstrating the characteristics of whales or of human skulls ; or Tait himself talking in easy English about strains and mirage, golf ball underspin or kinetic theory of gases. With the exception of the last two years of his life Tait hardly ever failed throughout his long tenure of the Secretaryship to be at the meetings of the Royal Society. There he sat listening courteously it might be to the most wearisome of readers who knew not how to give the broad lines without the details or on the alert for the next bit of inimitable humour with which Lord Neaves when presiding used to delight the Society. No one could enjoy a joke better than Tait ; and who could resist being infected with his whole-hearted laugh or the merry twinkle in the eye which some humorous situation called forth ? To many of the frequenters of the meetings in the seventies and eighties, Tait was in fact the Royal Society ; and there is no doubt that he guided its affairs with consummate skill. At the Council meetings which occurred regularly twice a month during the working session all matters of business were carefully presented by him in due order. It was his duty to conduct the correspondence of the Society, which during his Secretaryship grew steadily with the progress of the years. Lord Kelvin, especially during his various terms of office as President, attended the meetings of the Royal Society of Edinburgh with fair SIR WILLIAM THOMSON 31 regularity ; and on the morning following the Monday evening meetings paid a visit to Tail's Laboratory immediately after the conclusion of Tail's lecture. It was then that we laboratory "veterans" had an opportunity of coming into closer touch with the great Natural Philosopher, who would occasionally pass round the laboratory and inspect the experiments which were in progress. Most instructive discussions would at times arise, Kelvin's mind branching off into some line of thought suggested by, but not really intimately connected with, the experiment. At other times the conversation between Kelvin and Tait turned on the papers which had been communicated the evening before. I remember a lively discussion arising on the statistical effect of light impressions on the eye. The argument was reminiscent of the old tale of the two knights and the shield; for while Tait was laying stress on the time average, Thomson was looking at it from the point of view of the space average. For many years Tail's successive assistants reported the Meetings of the Royal Society to Nature ; and this duty fell to me during the years 1879-83. Al one of these meetings Sir William, as he then was, had in his well-known discursive but infinitely suggestive manner so talked round the subject of the communicalion lhat I had some difficulty in quite under- standing its real essence. Next morning I tried to get enlightenment from Tait. He laughed and said " I had rather not risk it ; but the great man is coming at twelve better tackle him himself." When in due time Sir William was "tackled," he fixed his gaze at infinity for a few moments and then, a happy thought striking him, he said, with a quick gesture betokening release from burden, " Oh, I'll tell you what you should do. Just wait till the Nature Report is published that fellow always reports me well." Tail's merriment was immense as he unfolded the situation, and he chaffed Thomson as to his obvious inability to explain his own meaning. Not a few of both Kelvin's and Tail's communications to the Royal Society of Edinburgh were never writlen oul by ihem ; ihey appear as reporls only in the columns of Nature. Anolher scene, in which Thomson and Tail were the main agenls, rises in the memory. Once on a Saturday morning in summer when two of us were working with electromeler and galvanomeler in ihe Class room Tait arrived in some excitement and said " Thomson will be here in half an hour on his way to London. He wishes to try some experiments with our Gramme machine and will need your cooperation wilh electrometer 32 PETER GUTHRIE TAIT and galvanometer." Sir William soon appeared, and we were immediately commandeered into his service. And then followed the wildest piece of experimenting I ever had the delight of witnessing. The Gramme machine was run at various rates with various resistances introduced, and simultaneous readings of the quadrant electrometer and a shunted mirror galvanometer were taken. The electrometer light-spot danced all over the scale, and I had to bring it to reason by frequent changes in its sensitiveness demanding a continual retesting with a standard cell so as to be able to reduce to the same scale. Full of impatience and excitement Thomson kept moving to and fro between the slabs on which the instruments stood, suggesting new combinations and jotting down in chalk on the blackboard the readings we declared. Tait stood by, assisting and at the same time criticising some of the methods. At length Sir William went to the further side of the lecture table and copied into his note book the columns of figures on the blackboard. After a few hasty calculations he said : " That will do, it is just what I expected." Then off he hurried for a hasty lunch at Tail's before the start for London where during the next week he was to give expert evidence in a law case. As they withdrew Tait looked back at us with a laugh and said " There's experimenting for you ! " Early on Monday morning we were startled by a message from Tait who had just received a telegram asking for the numbers on the blackboard. Thomson had mislaid his note book ! Also the original record had been obliterated ! Fortunately for a man of Thomson's profound physical intuitions the loss would not be irreparable. He had in fact tested his theory as the experiments were in progress. Tait's official position combined with his high reputation as mathe- matician and physicist brought him into touch with many of the great scientific men of the day. More especially was his verdict on questions of scientific history regarded with interest and respect, in spite of the fact that in several instances his views and those of his correspondents diverged considerably. I have quoted in a later chapter from both Helmholtz and Verdet in illustration of this point. Many instances are to be found in his correspondence expressive of the esteem in which he was held by his contemporaries on the Continent of Europe. The letters display a friendliness of tone and a frankness of utterance which show that the writers, one and all, recognised his unfailing honesty of purpose and looked upon him as one whose opinion was worth IN HIS STUDY 33 the asking. The subjects discussed were chiefly scientific, but occasionally matters of purely personal interest were touched upon. Up to the last year of his busy life, Tait's mind was for ever thinking out some new line of attack on the elusive laws of nature or on the properties of quaternion functions, while with ready utterance and facile pen he was teaching hundreds and thousands the grand principles as well as some of the mysteries of his science. As the years increased, he mingled less and less with general society. In his own home he was the most hospitable of hosts, full of story and jest, and alive to all the passing humours of the moment. Possessed of a verbal memory of unusual accuracy he could often suit the occasion with a quotation from one of his favourite authors, Horace, Cervantes, Shakespeare, Scott, Byron, Dumas, Thackeray, Dickens, etc. It mattered not on what he was engaged, he had a ready welcome for his friends in the small study which looked out south across the Meadows. His shaded gas-lamp which stood on the table cast a shadow round the walls, somewhat further dimmed by the wreaths of tobacco smoke which stole slowly from his pipe for though a steady he was not a rapid smoker. There he would sit when alone and work the long night through, rising occasionally to fill his pipe as he once remarked "it is when you are filling your pipe that you think your brilliant thoughts." But let a visitor enter, then, unless there was a batch of examination papers to finish off before a certain early date, he would lay his work aside and clear decks for a social or scientific chat as the case might be. In that den walled with book shelves and furnished with a few chairs, the table littered with journals, with proofsheets and manuscript, with books waiting to be reviewed, or with the most recent gifts of original papers from scientific men in every centre of life and civilisation in that den Tail had entertained the greatest mathematicians and physicists of the age ; Kelvin, Maxwell, Stokes, Helmholtz, Newcomb, Cayley, Sylvester, Clifford, Bierens de Haan, Cremona, Hermite, to name only some of those who are no more with us. There only was it possible to find him at leisure to discuss a scientific question. At college matters were different, the lecture was just about to begin, or it had just ended and some other University work called for attention. There were three outstanding occasions on which Professor Tait and Mrs Tait made their home a lively centre of science and fun, namely, the British Association Meetings of 1871 and 1892 and the University Tercentenary Celebration of 1884. At the later meeting of the British T. 5 34 PETER GUTHRIE TAIT Association, the Natural Philosophy class room was the haunt of Section A ; and Stokes, Helmholtz, Kelvin, and Tait sat side by side on the platform through most of the forenoon sederunts. The afternoons, however, were frequently given up to less formal gatherings. On one such occasion Mrs Tait's drawing room was converted into a lecture hall with lantern and screen ; and C. V. Boys gave a seance of his flash photographs of the aerial disturbance produced by a bullet shot from a pistol. At another gathering which was purely social Mrs Tait, to make sure of the tea being absolutely perfect, had a kettle " singing " merrily on the open fire. Stokes and Kelvin were seated on a couch conversing diligently with a lady whose knowledge of Japan and Japanese students was interesting them when suddenly a sharp hissing sound was heard above the talk and laughter which filled the room. "See" said the calm contemplative Stokes pointing with his finger, " the kettle is boiling over " ; but Kelvin, who was furthest from the fire, leaped forward in his alert eager way, drawing out his handkerchief as he went, and lifted the kettle off just as Mrs Tait herself reached the hearth rug. On another occasion, when the meeting of Section A was in full swing, Tait, wishing to show Helmholtz and Kelvin some of the experimental work which was in progress at the time, led them out quietly through the door into the apparatus room behind the platform and then down to the basement. Here in the large cellar containing the Admiralty Hydraulic Press he had some compression experiments going on ; and in an adjoining cellar I was experimenting on magnetic strains. While Helmholtz and Kelvin were inspecting the arrangements and asking questions about the results a message came from the Secretaries of the Section demanding the presence of the three truants and especially of Lord Kelvin. Kelvin, however, was too eager over the problems of magnetic strains to pay immediate heed to the summons. Meanwhile Section A sat in silence like a Quaker's meeting. After a few minutes, a second and urgent message was sent to the effect that an important discussion in Section A could not be begun until Kelvin re-appeared on the platform. Reluctantly he tore himself away from the fascination of the research room, mounted the long stair, took his seat along with Helmholtz beside the President, and began almost immediately to occupy himself with a model on which he was to discourse an hour later. In 1890 Tait tried his utmost to prevail upon Helmholtz to give the Gifford Lectures on Natural Theology in Edinburgh University. His letter of entreaty was as follows : GIFFORD LECTURES 35 38 GEORGE SQUARE, EDINBURGH, 22/2/90. My dear v. Helmholtz I write to beg that you will give careful consideration to a formal document which will reach you in a day or two. It is to request that you will accept the post of Giffbrd Lecturer in the University of Edinburgh for the next two years. The duties are not onerous, as they consist in giving 10 lectures in each year; and the remuneration is very handsome indeed. You would not require to spend more than a month, each year, in Scotland ; and Glasgow is within such easy reach that you might spend part of the time there. The terms of Lord Gifford's Will are such that the post may be held by any one; and we are particularly anxious that you should accept it, as a representative of so wide a range of thought. You have the inestimable advantage, over such men as Stokes and Thomson, of profound knowledge of Physiology. Besides, it is only a few years since Stokes occupied a somewhat similar (but more restricted) post in Aberdeen : and we are of opinion that, at first at least, we should not appoint to the Gifford Lectureship a Professor (such as Thomson) in a Scottish University. I can assure you of a most hearty welcome here ; and we are sure to profit largely by your unfettered utterances. Helmholtz, however, did not accept the offer ; and Tait, who was anxious to have as Gifford Lecturer a man of recognised scientific reputation instead of the usual philosopher or theologian, prevailed upon Sir George Stokes to take up the burden. During the delivery of one of the second series in 1892 an amusing episode happened. It was a warm close afternoon, and Kelvin had come through from Glasgow to attend an evening meeting of the Royal Society. Wishing to honour his friend he accompanied Stokes to the platform along with Tait, Crum Brown, and other members of the Edinburgh University Senatus. Sir George had occasion to refer in his lecture to some of the views of Kelvin. When he came to the name he looked up with his beautiful smile and said " I little dreamed when I wrote those words some months ago that Lord Kelvin would be listening to me as I read them." The audience applauded heartily ; and Kelvin who had been half dozing roused himself and joined in the applause ! Tait was invited by the Glasgow University Senatus to give the Gifford Lectures in that University ; but he declined on the ground that so long as he had his Class Lectures to deliver he could not think of undertaking extra lecturing duties. When his last grave illness compelled him to resign he was no longer able for the task of preparing twenty lectures on natural theology. His own religious beliefs may easily be 52 36 PETER GUTHRIE TAIT inferred from the attitude of mind exhibited in the Unseen Universe or Physical Speculations on a Future State which Balfour Stewart and he wrote together. A more distinct utterance however is to be found in an article published in the International Review of November, 1878, and named " Does Humanity demand a New Revelation ? " This article was largely polemical, being avowedly a reply to Froude who had communicated to the same Review some articles on " Science and Theology Ancient and Modern." Towards the close of Tail's article these sentences occur : "It would therefore appear, from the most absolutely common-sense view independent of all philosophy and speculation it would appear that the only religion which can have a rational claim on our belief must be one suited equally to the admitted necessities of the peasant and of the philosopher. And this is one specially distinguishing feature of Christianity. While almost all other religious creeds involve an outer sense for the uneducated masses and an inner sense for the more learned and therefore dominant priesthood, the system of Christianity appeals alike to the belief of all ; requiring of all that, in presence of their common Father, they should sink their fancied superiority one over another, and frankly confessing the absolute unworthiness which they can not but feel, approach their Redeemer with the simplicity and confidence of little children. ********* All who approach the subject without bias can see from the New Testament records how some of the most essential features of Christianity were long in impressing themselves on the minds even of the Founder's immediate followers. And we could not reasonably have expected it to be otherwise. The revelation of Himself which the Creator has made by His works we are only, as it were, beginning to comprehend. Are we to wonder that Christianity, that second and complementary revelation, is also, as it were, only beginning to be understood ; or that, in the struggle for light, much that is wholly monstrous has been gratuitously introduced, and requires a Reformation for its removal ? What more likely than that, in the endeavour to frame a document for the stamping out of a particular heresy, over- zealous clergy should carry the process a little too far, and so introduce a new and opposite heresy? But this is no argument against Christianity; rather the reverse. It might in fact be asserted, with very great reason, that a religion which, like any one of the dogmatic systems of particular Christian sects, should be stated to men in a form as precise and definite as was the mere ceremonial law, would be altogether an anomaly inconsistent in character with all the other dealings of God with man and altogether incompatible with that Free Will which every sane man feels and knows himself to possess." Tait was indeed a close student of the sacred records. The Revised Version of the New Testament always lay conveniently to hand on his RELIGION AND POLITICS 37 study table ; and frequently alongside of it lay the Rev. Edward White's book on Conditional Immortality. I am not aware that he distinctly avowed himself a believer in this doctrine, as Stokes did, but he often expressed the high opinion he held of Edward White and his writings. His reverence for the undoubted essentials of the Christian Faith was deep and unmovable ; and nothing pained him so much as a flippant use of a quotation from the Gospel writings. I have heard him reduce to astonished silence one guilty of this lack of good taste with the remark, "Come now, that won't do; that kind of thing is 'taboo'." Tail's general outlook upon human affairs was fundamentally conservative. He had a deep distrust of Mr Gladstone as statesman and legislator. His strong political views did not however in any way interfere with his private friendships ; and he refrained on principle from taking any public part in political discussions. He never failed to give his vote at an election ; and was a consistent supporter throughout of the Conservative and latterly the Unionist Governments. When the South African War broke out he rejoiced to be able to send his son as a Lieutenant of the Black Watch to fight for his country and his Queen. But swiftly came the stroke of sorrow as it came to many a family in the dark days of the South African War. Lieutenant F. G. Tait left this land with his regiment on October 24, 1 899 ; on December 1 1 he was wounded at Magersfontein, where the Highland Brigade suffered so terribly ; and after a few weeks in hospital he returned to the front only to meet his death on February 7, 1900, at Koodoosberg. The rumour of the tragic event came first through non-official channels and the uncertainty which hung over it for some days was harder to bear than if the worst had been immediately reported through the War Office. But there was no doubt of it ; and all Scotland mourned the loss of her brilliant soldier golfer as she mourned few others of her warrior sons whose lives were cut short on the African veldt. Tail's scientific work practically ended with his son's death. In December 1899 he communicated to the Royal Society a criticism on the "Claim recently made for Gauss to the Invention (not the Discovery) of Quaternions." It is a fitting finish to the publications of one whose con- troversies were always on behalf of others. Meanwhile he was editing the second volume of his Scientific Papers, published by the Pitt Press, Cambridge. The Preface to the second volume 38 PETER GUTHRIE TAIT is dated January 15, 1900. There is only one later printed statement by him the preface to the seventh edition of Tait and Steele's Dynamics of a Particle. The great physical and mental powers of the man were gradually beginning to fail. The vigour of his long stride was not what it had been. Yet in the keenness of ear and eye there was no abatement. Far beyond the years at which the great majority of normal-sighted men are forced to use spectacles or glasses, Tait was able to read his newspaper without artificial aid. Latterly, in reading an unfamiliar hand-writing he was occasionally compelled to hold it at the extreme stretch of his long arms ; still he could read it a very rare feat for a man of seventy. During the spring and summer of 1900 he carried on his University work and his Secretarial duties at the Royal Society. He never failed to be present at the Council Meetings ; but the general meetings of the Society saw less and less of him. He and his family took their usual summer holiday at St Andrews, whose links in every hole and "hazard" were full of the memories of his son Freddie. But alas, the shadow of death had chilled these golden memories ; and it was no surprise to his friends to learn that Tait returned to Edinburgh in the autumn none the better of his summer rest. As he drew on his gown on the opening day of the session he confessed that for the first time in his experience he felt no desire to meet his new class. He was resolved in his own mind to complete the century at least in harness ; but the task was too great for his waning strength. For nearly two months he carried on his lectures, to the great anxiety of all who knew and loved him best. On December n, 1900, the anniversary of the Magersfontein disaster, he left the University, never again to pass within its portals. He was indeed very ill : yet he himself never desponded, but spoke cheerily of looking in at College some day before the Christmas holidays, just to be able to say that he had completed the century. He was still able for mental work, and occupied himself forecasting his third volume of Scientific Papers and even criticising some of his own later papers published in the second volume. Once or twice in these days, when he was wholly confined to bed, he spoke to me of the linear vector function as something which still awaited development there was a truth in it which had not yet been divined by the mind of man. RETIREMENT FROM CHAIR 39 Tait formally retired from the duties of his chair on March 30, 1901. The Senatus expressed their appreciation of his long services in the following minute : " In taking regretful leave of their eminent and highly valued colleague, Professor P. G. Tait, the Senatus desire to place on record their warm appreciation of the ability and success with which, for the long period of forty-one years, he has discharged the duties and upheld the splendid traditions of the Chair of Natural Philosophy. They recognise with pride that his world-wide reputation as an original thinker and investigator in the domain of Mathematical and Physical Science has added lustre to this ancient university. A master in research, he is not less distinguished as an exponent of the Science with which his name will ever be associated. The zeal which inspired his Professorial work is well known to his colleagues, and has been keenly appreciated by successive generations of pupils, many of whom now risen to distinction have gratefully acknowledged their indebtedness to their teacher. In parting from their colleague, the Senatus would express the hope that he may speedily regain his wonted health and strength and be long spared to enjoy his well-earned leisure. He may be assured that he carries with him into his retirement their brotherly sympathy and affectionate regard." On June 28, 1901, the Senatus resolved that the Honorary LL.D. degree be conferred on Emeritus Professor Tait. The formal intimation of this resolution was never seen by him. Immediately after Tait's retirement a number of his former pupils resident in Edinburgh resolved to prepare an illuminated address, which would be signed by all former students who had made a specialty in laboratory work under his supervision. The address was illuminated by Mrs Traquair, who introduced round the margin illustrations of the various forms of apparatus which Tait had devised or used in carrying out his most important investigations. A portrait of Newton was placed at the top, and was flanked by scrolls, on which were inscribed certain Quaternion formulae and a few of the more characteristic lines of the Thermoelectric Diagram. Interwoven links and knots formed the foundation of the decorative design, and here and there appeared the names of Steele, Andrews, Thomson, Balfour Stewart, and Dewar, with whom he had collaborated in experimental and literary work. Immediately beneath the printed address was a group of curves taken from his papers : and then followed the sixty-three signatures in facsimile of the former students referred to above. Of these nearly thirty fill or have filled professorial appointments in universities and colleges both at home and abroad, while 40 PETER GUTHRIE TAIT among the others we find eminent engineers and scientists, distinguished educationists, successful physicians, and vigorous self-denying clergymen. Tait's constant companion through the weary months of illness was J. L. Low's Record of the life and golfing triumphs of Frederick Guthrie Tait. This finely written memoir gives a perfect picture of the generous hearted athletic Freddie, and traces with a genial literary touch his rise into the front ranks of golfers, among whom to this day his prowess is of undying interest. As Tait read and re-read the story of Freddie's peaceful victories he would live over again the happy rejoicings as medal was added to medal, or a new " record " was established, or another championship won. As the summer of 1901 wore on there was no evidence of returning strength. In the hope that the change might be beneficial Sir John Murray offered his old Friend the use of his house and garden near Granton. Tait was greatly touched not only by the thoughtful care which prompted the act of kindness, but also by the loving solicitude with which Sir John gave all directions for his comfort and welfare. There in the secluded quiet of the garden of Challenger Lodge, carefully shielded from aught that might distract or weary, he passed through the last days of his pilgrimage. At first everything promised well. On July 2 Tait felt able to return to his quaternion studies and covered a sheet of foolscap with brief notes of investigations in the theory of the linear vector function. This he handed to his eldest son, with the request to keep it carefully 1 . But it was the last effort of the keen vigorous mind. Two days later on Thursday, July 4, 1901, the once strong life passed peacefully away. There was cause for lamentation. Edinburgh had lost a son who had early brought fame to one of her oldest schools, and who had for forty years added to the renown of her University. Always strenuous, always devoted, always striving to extend our knowledge of the mysterious universe in which we live, full of interest in all that was best in humanity, and with a true reverence for the highest ideals of the Christ-like life, Peter Guthrie Tait had finished his appointed task. On July 6 a large and representative company of Edinburgh citizens and University graduates assembled for the last sacred rites in St John's Episcopal Church, the Rev. Canon Cowley Brown and the Rev. H. S. Reid, 1 The notes were afterwards published in facsimile by the Royal Society of Edinburgh, with a commentary in which I indicated their relation to his other papers on the same subject. LETTERS OF SYMPATHY 41 Professor Tail's son-in-law, officialing al ihe funeral service. The body was inlerred in the Church Yard immediately to the easl of the church. The pall bearers were Professor Tail's ihree surviving sons, his Iwo brolhers-in-law (Professor Crum Brown and Mr J. S. Porler), Lord Kelvin, Sir Thomas R. Eraser, and Sir John Murray. Among the many letlers of sympathy which Mrs Tait and her family received during ihe sad days which followed Professor Tail's dealh, one may be given in full. 1 1 was from Sir George Slokes, lo whom all ihrough his life Tail looked as lo a masler, and from whom he had frequently laken advice and suggeslions in his scientific work. LENSFIELD, CAMBRIDGE. 9 J u fy> 1901. Dear Mrs Tait, Now that the earth has closed over the remains of one most dear to you, permit me as a very old friend of your husband, and as one who not very long ago sustained a bereavement similar to that which you have just passed through, to express to you a feeling of sincere sympathy. When the last rites are over, and all is quiet again, the feeling of loneliness comes on all the more strongly. But we " sorrow not even as others which have no hope." Your husband was distinguished in the world of science. But it is more consolatory to you now to think of him who, with all that, looked " at the things which are not seen." We can think of him as one of those who in the beautiful language of the first reformed prayer book "are departed hence from us, with the sign of faith, and now do rest in the sleep of peace." Pray do not trouble yourself to make any reply to this letter. Yours very sincerely G. G. STOKES. The following extracls from lellers wrillen by former colleagues in Edinburgh University describe in appropriate language the real character of ihe man : "To me he was a dear friend as well as a colleague, and in his loveable simplicity and warmth of heart one sometimes forgot his great gifts of intellect." And again: " No one could know him without being drawn to him by the warmest ties. My early recollections of him go back far into the past century. He was always so hearty and kindly, so ready to help and so pleased to have his friends around T. 6 42 PETER GUTHRIE TAIT him. We all reverenced his gigantic intellectual power and were proud of all that he did for the advancement of science, but the charm of his buoyant and unselfish nature won our hearts from the very first." Sympathetic letters were received not only from friends but from associations and corporations such as the Master and Fellows of Peter- house, Cambridge, and the Students' Representative Council of Edinburgh University. Full and appreciative notices of Tail's career and scientific work appeared in the leading newspapers, for the most part accurate, although here and there disfigured by some wild imaginings on the part of the writer. The able article in the Glasgow Herald is specially worthy of note. My own contribution to the Scotsman of July 5 was put together at a few hours' notice and was not of course seen by me in proof. I am not aware of anything inaccurate or misleading in the notice, although there were many points necessarily not touched upon. Professor Chrystal's article in Nature (July 25, 1901) gives an admirable sketch of his colleague's life and labours, with a sympathetic reference to the sincerity and honesty of purpose which were so characteristic of the man. Dr G. A. Gibson, who along with Sir Thomas R. Fraser attended him in the last illness, wrote a graceful biographical notice in the Edinburgh Medical Journal (1901). Dr Alexander Macfarlane contributed to the pages of the Physical Review a sympathetic sketch of his old master ; and Dr J. S. Mackay (mathematical master in the Edinburgh Academy) supplied a short biographical note to t Enseignement mathe'matique (January 1905). J. D. Hamilton Dickson's sketch in the Magazine of the Peterhouse Sexcentenary Club for the Michaelmas Term, 1902, gives, in addition to other matter, some interesting Peterhouse details as to Tail's under- graduate days. Appropriate references were minuted by all the important organisations with which he was associated the University, the Royal Society of Edinburgh, the Scottish Meteorological Society, the Gumming Club, the Scottish Provident Institution, etc. After recording the main facts in connection with Professor Tail's labours as an official of the Royal Society of Edinburgh, the Council placed on record the following appreciation : "This is not the occasion for an analysis of Professor Tail's work and influence. That will, no doubt, be given in due time by those specially qualified. What the OBITUARY NOTICES 43 Council now feel is that a great man has been removed, a man great in intellect and in the power of using it, in clearness of vision and purity of purpose, and therefore great in his influence, always for good, on his fellowmen ; they feel that they and many in the Society and beyond it have lost a strong and true friend." The obituary notice in the Proceedings of the Royal Society (Vol. xxni, p. 498) was prepared by Lord Kelvin. It contains, in addition to the customary biographical details, some interesting reminiscences of the days they worked together. Kelvin tells how they became acquainted in 1860 when Tait came to Edinburgh, and how they quickly resolved to join in writing a book on Natural Philosophy. He then continues : " I found him full of reverence for Andrews and Hamilton, and enthusiasm for science. Nothing else worth living for, he said ; with heart-felt sincerity I believe, though his life belied the saying, as no one ever was more thorough in public duty or more devoted to family and friends. His two years as ' don ' of Peterhouse and six of professorial gravity in Belfast had not polished down the rough gaiety nor dulled in the slightest degree the cheerful humour of his student days ; and this was a large factor in the success of our alliance for heavy work, in which we persevered for eighteen years. ' A merry heart goes all the day, Your sad, tires in a mile-a.' The making of the first part of ' T and T' ' was treated as a perpetual joke, in respect to the irksome details of interchange of 'copy,' amendments in type, and final corrections of proofs. It was lightened by interchange of visits between Green- hill Gardens, or Drummond Place, or George Square, and Largs or Arran, or the old or new College of Glasgow ; but of necessity it was largely carried on by post. Even the postman laughed when he delivered one of our missives, about the size of a postage stamp, out of a pocket handkerchief in which he had tied it, to make sure of not dropping it on the way. One of Tail's humours was writing in charcoal on the bare plaster wall of his study in Greenhill Gardens a great table .of living scientific worthies in order of merit. Hamilton, Faraday, Andrews, Stokes, and Joule headed the column, if I remember right. Clerk Maxwell, then a rising star of the first magnitude in our eyes, was too young to appear on the list... After enjoying eighteen years' joint work with Tait on our book, twenty-three years without this tie have given me undiminished pleasure in all my intercourse with him. I cannot say that our meetings were never unruffled. We had keen differences (much more frequent agreements) on every conceivable subject, quaternions, energy, the daily news, politics, quicquid agunt homines, etc., etc. We never agreed to differ, always fought it out. But it was almost as great a pleasure to fight with Tait as to agree with him. His death is a loss to me which cannot, as long as I live, be replaced. The cheerful brightness which I found on our first acquaintance forty-one years ago remained fresh during all these years, till first clouded when news came of the death in battle of his son Freddie in South Africa, on the day of his return to duty 62 44 PETER GUTHRIE TAIT after recovery from wounds received at Magersfontein. The cheerfulness never quite returned." On opening his Divinity class the succeeding session Professor Flint uttered a beautiful tribute to the memory of his friend. This was published shortly afterwards in the Student, the Edinburgh University Magazine, and is now reproduced in full. THE LATE PROFESSOR TAIT AN APPRECIATION BY PROFESSOR FLINT Since we last met here the University has lost through death the teacher who had been longest in her service, who was probably the most widely renowned member of her professorial staff. He was known to almost all of you not only by report but by personal contact and acquaintance, for almost all of you have come directly from his class room to the class rooms in the Divinity Hall. Undoubtedly it was a great advantage for our students here that they should have entered the Hall through that portal, and received the instruction and come under the influence of one universally recognised to have had not only a genius of the first order for research, but rare gifts as a teacher. He was not one whom his students were likely ever to forget, while many of them must have felt that they owed to him far more than they could estimate or express. If you have not learned to be interested in the truths of Natural Philosophy, the fault cannot have been your teacher's, and unless altogether incapable of learning anything, you at least cannot have failed to learn the very important lesson that such a man's mind was immeasurably larger than your own. Our deceased friend was a man of strong, self-consistent individuality. He was "himself like to himself alone." And he had about him the charm inseparable from such a character. He never lost the freshness of spirit which so soon disappears in the majority of men that it is apt to be deemed distinctive of youth. There was to the last a delightful boyishness of heart in him such as is assuredly a precious thing to possess. I am quite aware that great as he was, he had his own limitations, and sometimes looked at things and persons from one-sided and exaggerated points of view, but the consequent aberrations of judgment were of a kind which did no one much harm and only made himself the more interesting. His strong likes and dislikes, although generally in essentials just, were apt to be too strong. Although, like all great physicists, he was not really uninterested in metaphysics, yet he felt and professed the most supreme contempt for all that he called metaphysics. In connection with that I may mention an incident which once afforded much amusement to academic men in St Andrews, but is probably now forgotten even there. Shortly after Tait had delivered the remarkable lectures to which we owe the work entitled Recent Advances in Physical Science, he dined one evening at the house of the Professor of Mathematics in St Andrews, and among other guests PROFESSOR FLINT'S APPRECIATION 45 present was a Glasgow Professor of Theology who had even less esteem for physical science than our dear departed friend had for metaphysics. Tait was very naturally drawn out to talk about the subjects which he had been lecturing on, and he did so largely and to the delight and edification of every one except the worthy and venerable Glasgow Professor, who, when he could stand it no longer, gravely put the question "But, Mr Tait, do you really mean to say that there is much value in such inquiries as you have been speaking about?" After that the subject was changed, and during the rest of the evening the great physicist and great metaphysicist did little else than, as Tulloch expressed it, "glour at each other." Tait was a genius, but a genius whose life was ruled by a sense of duty, and which was shown to be so by the vast amount of work he accomplished, and which is acknowledged by those who are ablest to judge of its worth, to be of the highest value. He was a genius with an immense capability of doing most difficult work, and he faithfully did it. His life was one of almost continuous labour. He faithfully obeyed the injunction, " Work while it is called to-day." And the work which he chose to do was always hard work, work which few could do, work which demands no scattering of one's energies, but the utmost concentration of them. He wasted no portion of his time in trying to keep himself en evidence before the world. He willingly left to others whatever he thought others could do as well or better than himself. But whatever he thought it his duty to undertake he did thoroughly. Thus for the last twenty years at least he was the leading spirit in an institution more closely connected, perhaps, than any other with the University of Edinburgh. I mean the Royal Society of Edinburgh. It is natural for those of us who painfully feel that we shall not see his like again, natural for those who are most deeply deploring his loss, to wish that a longer life had been granted to him. Yet they may well doubt if he himself would have desired a mere prolongation of life. I cannot but think that he would not have cared for a life in which he could not labour. While his friends must sorrow for his loss, they are bound also to acknowledge that God had been very good and gracious to him. He was favoured with many years of health and strength in which to work. His abilities were so conspicuous even in youth that they could not be hid. He could hardly have been earlier placed than he was in the very positions most favourable to the exercise of the gifts which had been bestowed on him. He was a Professor for forty-seven years, a Professor in Edinburgh for forty-one years. He was beloved by his students. His colleagues were proud of him. His country knew his worth. His many contributions are to be published in a suitable form at the cost of his English Alma Mater. He is among the rare few in a generation of whom the memories live through the centuries. Add thereto that his own worth and the value of his work were by none more fully appreciated than by those who were nearest and dearest to him, and that all distracting cares were spared him, and he was wisely left to follow the bent of his own genius. He had, so far as I know, only few great afflictions. The greatest which fell alike on him and his family was the loss of the generous, gallant, brilliant youth, who met a soldier's death near the Modder River, and in that loss a nation sympathised with him. 46 PETER GUTHRIE TAIT Our departed friend had no sympathy with theological dogmatism, and as little with anti-religious scepticism, and consequently held in contempt discussions on the so-called incompatibility of religion and science. At the same time he had a steady yet thoughtful faith in God, and in that universe which no mere eye of sense, aided by any material instrument, can see. That faith must have made his life richer, stronger, and happier than it would otherwise have been. And it must be a comfort to those who have the same faith, and to those who most deeply mourn his loss, to believe that he has entered into that universe which is so much vaster, and which may well have far greater possibilities of progress in truth and goodness in it than there are in the "seen" universe of us the passing creatures of a day. The things that are seen are temporal. The things that are unseen are eternal. For none of his colleagues on the Senatus had Tait a greater esteem and affection than for Professor Flint. Sir Alexander Grant, who was Principal from 1868 to 1885, was regarded by Tait as the ideal tactful President, able to restrain the contending idiosyncrasies of the members of the Senatus and to guide their deliberations with unfailing courtesy. Professor Blackie, who ostentatiously scoffed at all things mathematical, used to ask Tait occasionally to give him some elementary instruction in analytical geometry. Tait drew the x and y axes and expounded their use with his accustomed clearness, and all went well until the teacher pointed out the need of the use of the negative sign, when the irrepressible Grecian broke away with the remark " Humbug, how can a quantity be less than nothing?" On one occasion in the Senate Hall shortly after Blackie had been uttering some strong patriotic sentiments Tait posed him with the conundrum, "What is the difference between an Englishman and a Scot?" The answer was, " Because the one is John Bull and the other is John (Kn)ox." Blackie replied to this chaff by throwing an ink bottle past the head of his tormentor. In 1860, the Senatus numbered thirty, and in 1901 thirty-nine. During the forty years' tenure of his Chair, Tait had met in council with one hundred and seven colleagues, most of whom have left their mark in the history of theology, science, literature, or medicine. Of those who have passed away the following Principals and Professors may be mentioned, the latter in the order of their chairs as officially arranged in the University Calendar : Sir David Brewster, Sir Alexander Grant, Sir William Muir, Kelland, Pillans, Sellar, Goodhart, Blackie, Aytoun, Masson, Simon Laurie, Piazzi Smyth, Copeland, Fleeming Jenkin, Rev. Dr James Robertson, Sir Douglas Maclagan, Fraser Tytler, J. H. Balfour, A. Dickson, Hughes Bennett, HONOURS AND AWARDS 47 W. Rutherford, Laycock, Sir T. Grainger Stewart, Sir John Goodsir, Sir Lyon (Lord) Playfair, Sir J. Y. Simpson, Sir Robert Christison, Allman, Sir Wyville Thomson, Spence, Syme, Annandale, and Sanders. To them we may add the Chancellors, Lord Brougham and Lord Inglis. Tait was awarded the Keith Prize twice (1867-9 and 1871-3) by the Royal Society of Edinburgh, and was the second holder (1887-90) of the Gunning Victoria Jubilee Prize. The Royal Society of London awarded him a Royal Medal in 1886 for his various mathematical and physical researches. The following are the principal recognitions by Societies and Universities: Honorary Member of the Literary and Philosophical Society of Manchester, 1868; Honorary Doctor of Science of the University of Ireland, 1875; Honorary Fellow, Societas Regia Hauniensis (Copenhagen), 1876 ; Honorary Fellow, Soci^te" Batave de Philosophic Experimental (Rotterdam), 1890; Honorary Fellow, Societas Regia Scientiarum (Upsala), 1894; Honorary Fellow of the Royal Irish Academy, 1900; Honorary Doctor of Laws, Glasgow University, 1901. In 1882 some of Tail's many friends in Edinburgh commissioned George Reid (now Sir George) to paint a portrait of the Professor of Natural Philosophy in the act of lecturing. On the blackboard behind is the Curve of Vertices by means of which he elucidated the phenomena of mirage, with the Hamiltonian equation alongside. The general effect of the portrait is well described in the following contemporary criticism of the Exhibition of the Royal Scottish Academy. "In portraits, George Reid's most characteristic effort is a portrait of Professor Tait. The grand domed cranium of the Professor of Physics, and his sagacious, solemnly comical face, seem to surmount a figure more likely to be met with in a Skye crofter's potato plot as a scarecrow, than among the amenities of a Scottish University. But the next look reassures you. The coat, as well as the noble head, is Professor Tait's veriest own the coat, in fact, 'with which he divineth.' Even if the blackboard, and the high mathematical hieroglyphic thereon emblazoned, were silent, the Professor's ' office coat ' is so redolent of chalk and experimental physics, that to old habitues of his class room, it would recount the tale of a hundred fights between the cutting mental gymnastic of the Professor and the mystic powers of mathematical abstraction. Altogether, this is a masterly portrait of a master, who knows no living rival in the sphere which he has made his own. As I stand and look on the characteristic picture I almost fancy that I can catch, on the solemn face of the grim mathematician demonstrator, some faint suspicion of a good-natured smile at the grotesqueness of the toggery in which he has chosen to be handed down to posterity." 48 PETER GUTHRIE TAIT This portrait was presented by the subscribers to Mrs Tait, who has now gifted it to the Natural Philosophy Department of Edinburgh University. It hangs in the library where the students gather to read the books of reference and study their notes. Nearly ten years later Sir George Reid undertook a second portrait, which was subscribed to by Fellows of the Royal Society of Edinburgh. This portrait is the property of the Royal Society ; but two replicas of it were made by Sir George Reid. One of these is hung in the National Portrait Gallery of Scotland, Queen Street, Edinburgh, and the other in the hall of Peterhouse, Cambridge. It is a three-quarter length portrait, and gives a faithful representation of Tait standing in a thoughtful attitude just in the act of elucidating some difficult point in mathematics or physics. The Peterhouse portrait was unveiled on October 29, 1902, by Lord Kelvin, who gave some interesting reminiscences of how he and Tait worked together. The following report is from the Cambridge Chronicle : " Lord Kelvin said he valued most highly the privilege of being allowed to ask the Master and Fellows of Peterhouse to accept for their College a portrait of Professor Tait. He felt especially grateful for this privilege as a forty-years' comrade, friend, and working ally of Tait. Their friendship began about 1860, when Tait came to Scotland to succeed Forbes as Professor of Natural Philosophy at Edinburgh. He remembered Tait once remarking that nothing but science was worth living for. It was sincerely said then, but Tait himself proved it to be not true later. Tait was a great reader. He would get Shakespeare, Dickens, and Thackeray off by heart. His memory was wonderful. What he once read sympathetically he ever after remembered. Thus he was always ready with delightful quotations, and these brightened their hours of work. For they did heavy mathematical work, stone breaking was not in it. A propos, perhaps, of the agonies (he did not mean pains, he meant struggles) of the mathematical problems which they had always with them, Tait once astonished him with Goethe's noble lines, showing sorrow as raising those who knew it to a higher level of spiritual life and more splendid views all round than it was fashionable to suppose fell to the lot of those who live a humdrum life of happiness. He did not know them, having never read ' Sorrows of WertherV ' Who never ate bread in tears, Who never through long nights of sorrow Sat weeping on his bed, He knows you not, ye heavenly powers." But Tait gave it him in the original German, with just one word changed. ' Wer nie sein Brod mit Thranen ass, Wer nie die kummervolle Nachte An seinem Bette rauschend sass, Der kennt euch nicht, ihr himmlischen Machte.' 1 The passage is from the Lehrjahre, Book II, Chapter xin THE TAIT PRIZE AT PETERHOUSE 49 "Tait hated emotionalism almost as much as he hated evil, and he did hate evil with a deadly hatred. His devotion, not only to his comrades and fellow-workers, but also to older men such as Andrews and Hamilton was a remarkable feature of his life. Tait was a most attractive personality, and its attractiveness would be readily understood when he unveiled the portrait. It gave one the idea of a grand man, a man whom it was a privilege to know. His only fault was that he would not come out of his shell for the last twenty years, and that he never became a Fellow of the Royal Society of London." Tait used to say that when he was young and would have liked to become a Fellow of the Royal Society, he could not afford it, and that later, when he could afford it, he had ceased to care about the distinction. It should be stated that from 1875 onwards Tait was never out of Scotland. His last visits to Cambridge were in 1874 and 1875, when he was Rede Lecturer and Additional Examiner in the Mathematical Tripos. Having, as it were, taken root in Edinburgh, he could have no very keen desire to become a Fellow of a Society whose meetings he would never have attended. For the last twenty-five years of his life he never left Edinburgh, except for a holiday at St Andrews of ten days in the spring and six weeks in the autumn 1 . Hence it came that he was personally unacquainted with most of the younger generation of scientific workers, and in this sense it is true that he did not come " out of his shell." But it must be repeated that the men of science who sought him out in his chosen haunts found the warmest of welcomes ; and Mr Low's sketch will show how far Tait was from the crabbed recluse that the phrase suggests. About 1880 the President of the Royal Society suggested privately to Tait that he should allow his name to be submitted to the Council. Tait, who knew that the name of a valued friend whom he regarded as a genuine man of science had been recently rejected by the Council, replied that he had no pretensions to belong to a Society which was too good for his friend. This humorous excuse not only served its immediate purpose, but also, to Tail's delight, helped to procure for his friend soon afterwards the distinction he sought. In "Quasi Cursores," the gallery of portraits of the Principal and Professors of Edinburgh at the time of the Tercentenary in 1884, the artist, William Hole, R.S.A., although very happy in most of his de- lineations, has not caught Tait quite satisfactorily. The attitude and figure generally are admirable, as are also the accessories of the Holtz machine, 1 Tail's family can only recall one slight exception to this. In January, 1880, he delivered a popular lecture on Thunderstorms in Glasgow. T. 7 50 PETER GUTHRIE TAIT Leyden jars, and blackboard ; but the expression of the face is not altogether suggestive to those who knew him well. Of the likenesses reproduced in this volume one of the most striking is that from the photograph taken by the Rev. L. O. Critchley, when he was a student in the laboratory. He had been assisting Tait in some work requiring the camera ; and, without the knowledge of the Professor he set the camera so as to photograph him in the act of writing a note. Tom Lindsay, the mechanical assistant, who is standing at the side ready to receive the note when finished, was in the secret. The portrait is admirable, giving not only a fine picture of the massive head, but also showing the usual condition of the writing table and general environment of what served as Tail's retiring room. The establishment in 1903 of the "Tait Prize for Physics" at Peter- house, Cambridge, was associated in an interesting manner with the execution of the portrait already referred to. Following up the proposal of Lord Kelvin and Sir James Dewar, the Master and Fellows of Peterhouse commissioned Sir George Reid to paint a replica of the portrait in the possession of the Royal Society of Edinburgh. The portrait was, however, more than a mere replica, for the painter worked into it reminiscences of his own long and intimate friendship with Professor Tait. Through the generosity of Sir George Reid a large portion of the funds contributed was left in the hands of the Treasurer, Mr J. D. Hamilton Dickson, who suggested that an effort should be made, by an appeal to a few other friends, to increase the fund until it should suffice for the establishment of a prize associated with Tail's name, to be given periodically for the best essay on a subject in Mathemalical or Experimental Physics. In this way the fund for the foundation of the Prize was soon raised to two hundred pounds. The idea of establishing a Tait Memorial in connection with the Natural Philosophy Department of the Edinburgh University occurred to many of Tail's pupils and friends. Considerations of general University policy prevented an authorilalive appeal being made at ihe lime. Nevertheless, quile unsolicited, a Tait Memorial Laboratory Fund took shape and began to grow. It has now reached the sum of nearly two thousand pounds. On June 10, 1907, Sir John Jackson founded a Tait Memorial Fund, with ihe object of encouraging physical research in the University of Edinburgh on the lines of the work of the late Professor Tait. It is unnecessary here to give the whole Declaration of Trust, which may be SIR JAMES DEWAR 51 found in the Edinburgh University Calendar ; but as indicative of Sir John Jackson's personal feelings towards Tail the following quotation is of interest : " LASTLY. I desire to place upon record that I have been induced to act in the premises as hereinbefore appearing from a deep sense of the advantages I as a student in the said University have derived from having been a pupil of the late Professor Tait and from a desire to assist instruction on similar lines to those followed by him for the benefit of future students in the said University." In this closing sentence Sir John Jackson expresses the feelings of all who were serious students of Physical Science under Tait's guidance. One of the earliest of these was (Sir) James Dewar. There was then no Physical Laboratory, and I have heard Tait lament that he was unable to make use of Dewar's ability in those very early days. He was also in the habit of saying that one of the greatest services he did to experimental science was recommending him to Lyon Playfair as demonstrator and assistant. While still Professor of Chemistry in the Veterinary College in Edinburgh, Dewar frequently came to discuss physical problems with Tait at the laboratory ; and in later years he never failed when he passed through Edinburgh to call on his old master and renew their fruitful intercourse. Shortly after Tait's death, Sir James was awarded the Gunning Victoria Jubilee Prize by the Council of the Royal Society of Edinburgh, and the sum received by him on this account he at once passed on to the Tait Memorial Fund as an expression of his regard for one to whom he owed so much. Another frequent visitor at the Physical Laboratory was Dr Alexander Buchan, the well-known meteorologist, who could never rest satisfied with his own conclusions until he had sounded Tait on the physics of the problem. I have often heard Buchan express his great indebtedness to Tait for his valuable hints and criticisms. But this feeling of indebtedness was not confined to those only who walked the pleasant paths of science. Many of his old pupils, who are now clergymen, physicians, teachers, lawyers, engineers, merchants, etc., retain not only a lively memory of the clear lecturer but a great deal of the principles of Natural Philosophy which he taught them. Some have even found his experimental illustrations useful in driving home spiritual and religious truths. Others, from their experience, have declared that what Tait taught them of the physical basis of things has been of more 72 52 PETER GUTHRIE TAIT service in their pastoral work than most of the theology and church history they learned in their divinity course ; and one maintains that a science degree in Mathematics and Natural Philosophy is probably more useful to a clergyman than a B.D. degree. This man, however, passed through Tail's laboratory, and was not an average specimen of the divinity student. Before 1892 every Arts student was compelled to take Natural Philosophy as one of the seven sacred subjects ; and even after 1892, although a certain amount of option was allowed to students, the majority who entered for the ordinary degree still passed under the spell of our great interpreter of Natural Law. Nevertheless, partly owing to the severity of the newly established preliminary examinations, partly to this introduction of option, the numbers of those attending the Natural Philosophy Class immediately fell off. From the outset Tait had little sympathy with the details of the New Regula- tions. In the diminished class which he met during the last eight years of his professoriate he saw one bad result of the University Commissioners' handling of the situation, and he never ceased to deplore that many students would hereafter pass out into the world with the degree of Master of Arts who had had no opportunity of learning the grand principles of Natural Philosophy. A great deal might be said in favour of this view of University study, more even now than formerly, when scientific developments bulk so largely in our modern civilisation. The difficulty mainly lies in the multiplicity of subjects now taught, all of them alike valuable as means of culture. When Tait resigned his Chair in 1901 he was teaching the sons of men whom he had taught in the sixties and seventies ; and it was with feelings of laudable satisfaction that he realised how he had served his University for two generations, and had impressed on the minds of fully nine thousand intelligent youths the great truths associated with the names of Archimedes, Newton, Carnot, Faraday, and Joule. TAIT AT ST ANDREWS BY J. L. Low It is the morning of a St Andrews' day in September; the early "haar" which had covered the Links like smoke has given place to sunshine and warmth, and the golfers are glad as they march in well matched parties, each player hopeful that he will make some notable per- THE CLUB-HOUSE, ST ANDREWS 53 formance. The last of the matches has left the first teeing ground, for it is nearing noon ; the golfers are already in grips, and for every idle evening boast they are giving, as best they can, some sort of account. We enter the club-house and at once glance round the great smoking-room. It is deserted, save for the waiters who are gathering up the morning papers, which have had but a short perusal, and are placing them in order on the reading table. The scene is familiar to every golfer who remembers his September mornings, and there comes back quickly with this remembrance a figure which will not easily be severed from Golf and from its Fifeshire home. By the south fireplace on its right-hand side sits in the big arm-chair a venerable gentleman who was the oldest boy and the youngest old man we ever knew. The head is bent as the reader's eye glances quickly over the pages of the Saturday or the Nineteenth, his pipe is in his mouth, and by his side on a small table stands a tankard of small ale which he has ordered to make him not altogether forgetful of his Cambridge days. Here, alone in this big room, we would seem to have come across some recluse who would most strenuously oppose our interruption, and by his silence demand his peace. But in a moment the whole man changes ; in a second he re- bounds from sixty to sixteen, and by the mere raising of the head throws off the garment of his years. The head is the head of the scientist, and the brow, not without its furrows, tells of problems solved and yet to be solved. But the eye, though small, twinkles with an unquenchable boyish- ness which will not grow old, and the fullness which lies beneath it proclaims that sense, whether of measure, of words, or of music, which always accom- panies this peculiarity of feature. Before we can speak he is laughing ; he greets us heartily, and demands, in order that we may laugh with him, that we read some passage he has just been enjoying. It is a dull passage on some subject we do not understand ; but his eye twinkles when he marks that we detect in the writing some absurd incongruity of expression. " What do you think of that, my boy, from a professor of Philosophy?" he exclaims, and then, as if to be quit of the thing, he rises, shakes himself, knocks the tobacco ash off his waistcoat, and adds : " Well, let's go out and meet Freddie, he will be past the turn by now." We, who were but golfers and fellow-sojourners in a city full of golf and professors, called this boy- man " The Professor," and we loved him. J have been asked to add to the content of this biography as it were the 54 PETER GUTHRIE TAIT side glance of the golfer to the all important view of Professor Tait as the great scientist. It was not far from the fitness of things that the Professor who was so full of Scottish character and was so well equipped as a mathematician and a philosopher should have found in the national Scottish game a field agreeable alike to his physical and mental recreation. Some recollection of him from the golfers' standpoint is therefore suggested, and is indeed the object of these reminiscences. The Tait connection with golf is dual ; for the Professor is known to many by the title he was wont, with his keen sense of humour, most to delight in, "The father of Freddie Tait." The Professor was a well- known figure at St Andrews from 1868 onwards to the end, and golf was his favourite recreation long before the prowess of his sons connected the name of Tait so closely with the national game ; but it was not until his sons were beginning to show signs of great aptness for the sport that the father began those experiments which have not only been of importance to the student of natural philosophy but have intimated to the golfer the fact that he was playing a game which was a science as well as an art. It is reported of Freddie that, in reply to a question addressed to him by the Czar of Russia, he stated that he " took seriously to golf when he was eight years old" ; of the Professor it may be said that he never took to the game seriously ; by this I mean that his interest in the game was athletic and philosophical rather than competitive. About 1860 the Professor made his beginning as a golfer in early morning rounds on Bruntsfield Links ; golf is still played on the historic ground but the fair way is intersected by paths, and play is now allowed only at holes of a mean length. However, in these days of the early sixties the course was of sufficient importance to warrant its being the scene of an open tournament, to which came such heroes as the late Hugh and Pat Alexander. For many years after this the Professor was in the habit of taking his pleasure at Musselburgh, and was a member of the Honourable Company of Edinburgh Golfers until that society removed to Muirfield. At Mussel- burgh he was in the habit of playing with Lord Inglis, with Mr A. D. Stewart, and with others who had long been accustomed to fight furiously with feather-stuffed balls in the depths of " Pandy." In 1868 the Professor began those visits to St Andrews which were continued without intermission until the year of his death. Being a regular glutton for play his daily rounds often amounted to five ; and though his strength was equal to the "THE MORNING ROUND" 55 task, he required, needless to say, several caddies to help him during his uncommon performance ; these were at first chosen for him by the late Dr Blackwell, father of illustrious golfing sons. Of these five rounds the one he loved the best had its start at 6.30 a.m. and was not equally popular with the other members of his family. It is indicative of his boyish nature that not only did he play at an hour when birds alone should be playing but even sang about it in an ode which he appropriately signed " The Glutton." THE MORNING ROUND (68 A.M.). AIR " BEAUTIFUL STAR." 1. Beautiful Round ! Superbly played Round where never mistake is made ; Who with enchantment would not bound For the round of the morning, Beautiful Round? 2. Never a duffer is out of bed ; None but the choicest bricks instead On the Links at six can ever be found ; Round of the morning, Beautiful Round. 3. There they lie in a hideous doze Different quite from a golfer's repose That from which he starts with a bound For the round of the morning, Beautiful Round. 4. Agile and light, each tendon strung, With healthy play of each active lung He strides along o'er the dewy ground In the round of the morning, Beautiful Round. 5. Beautiful Round ! most cleverly won Under the gaze of the rising sun, And hailed with a pleasant chuckling sound Round of the morning, Beautiful Round. 6. Beautiful Round ! vain duffers try Thy manifold virtues to deny : They ! ! ! mere specimens of a hound : Round of the morning, Beautiful Round. 7. Beautiful Round in thee is health, The choicest gem of earthly wealth : Hands and face most thoroughly browned; Round of the morning, Beautiful Round. 56 PETER GUTHRIE TAIT 8. Beautiful Round ! to thee is due All the work I am fit to do: Therefore in fancy stand thou crowned Queen of the morning, Beautiful Round. 9. Beautiful Round ! I think of thee Through months of labour and misery : Round thee the strings of my heart are wound, Round of the morning, Beautiful Round. Among those who were his companions on his rounds early or late were Mr Tom Hodge and the Bethunes, Lord Borthwick, Lord Rutherford Clark, and Mr James Balfour; Professor John Chiene and Lord Kingsburgh were also at times his opponents ; and as partners in foursome play he had Old Tom and Young Tom, and the Straths. He always stoutly maintained that given similar conditions Young Tom's play was equal to that of the best of the modern professionals. From the beginning of his golf the Professor used very upright clubs and played with the largest ball he could get " a thirty " as compared with the more common " twenty sevens " and " twenty eights." In 1871 the meeting of the British Association was held in Edinburgh, the Professor being President of Section A. After the proceedings were finished some of the most distinguished members of this assembly accompanied him to St Andrews. Among these were Huxley, Helmholtz, Andrews, and Sylvester. Helmholtz took no interest in golf and "could see no fun in the leetle hole " ; but Huxley played a round every afternoon during his stay of two months. He lived in the house known as Castle- mount, hard by the Castle gate : the house is now occupied by Dr Hunter Paton, whose family was at that time intimate with the Huxleys. In the afternoon round the Professor's eldest son Jack was in the habit of partnering Robertson Smith against Huxley and various people. Jack was only a small boy and no doubt too young to appreciate the excellence of the company in which he found himself; and indeed seems to have taken rather a high-handed position as regards these matters. At St Andrews there used to be an idea that the weaker player should make the easy drive at the first hole and Jack on one occasion was asked to perform this trivial task, but refused, declaring that he was " not the biggest duffer of the party " ; this greatly amused Huxley, who willingly accepted the chastisement and topped his ball gently towards the road. FUN AND JESTING 57 The St Andrews of those days was a city quite other than the fashionable watering place of to-day. The society, though small, was intellectual, and though intellectual yet devoted to the jests which are dictated by humour : the merry parties of the small colony were more than willing to enjoy at the seaside that freedom which is curtailed in the larger cities. The Professor was, from the nature of the man, the leader in everything which tended to humour and gaiety. It is difficult to imagine any man of years who day by day seemed so devoted to what, for lack of a more dignified term, must be called " fun " ; one felt sure that he found jokes in his algebraical symbols, and jests even in his quaternions. It is the dinner hour and the Professor proposes to the company that a round may be played with phosphorescent balls. When proper arrangements have been made the party assemble at the first teeing ground. To this match come the Professor and his lady, Huxley, keen on the humour of the thing, Professor Crum Brown and another friend. The idea is a success ; the balls glisten in the grass and advertise their situation ; the players make strokes which surprise their opponents and apprise themselves of hitherto unknown powers. All goes well till the burn is passed, and Professor Crum Brown's hand is found to be aflame ; with difficulty his burning glove is unbuttoned and the saddened group return to the Professor's rooms, where Huxley dresses the wounds. The pains of the phosphorescent hand having been mitigated by the tender care of the great scientist, it is not difficult to picture the fun which our Professor would derive from the night's adventure. In a nature so strong we cannot but expect to meet an accidental note which gives the theme originality. The Professor was a man of very strong, and as it seemed to some of us, almost unreasonable antipathies ; endowed as he was with a humour which, had he given it vent, could have been magnificently satirical, he dealt by argument with those he did not favour, allowing the joy and humour of his nature to play only on his friends, and more particularly on his own family and his more intimate circle. Of a morning his opening words had relation to a small incident of home life ; he would tell of something that had given him a chance of chaffing Freddie or Alec, or playing a practical joke on some member of the family : one such story must suffice to exemplify. The Taits had a house in Gibson Place overlooking the Links on one side and the old station road on the other. The front door was generally open and an umbrella stand which stood by it seemed to the Professor T. 8 58 PETER GUTHRIE TAIT to offer a too easy prey to the light-handed. Mrs Tait took another view and said that " no one stole in St Andrews " ; but when the Fair day arrived, and she went for her parasol, she found that the stand had been pillaged. She immediately informed the police and went to the railway station to see if the thief was escaping by train. Returning she found the Professor and General Welsh finishing their round, and at once said, "Guthrie, you were quite right, the umbrellas are all gone." The Professor's eye sparkled as he asked, " What steps have you taken ? " On being told he resumed his game ; but when lunch was finished he pulled back the curtains and disclosed the umbrellas. Mrs Tait found herself in a position of some embarrassment as she had to tell the policeman that the affair had been a hoax ; and this worthy, who afterwards became the well-known and respected Inspector, did not in any way relieve the situation by saying that he had suspected the truth from the first. For many years after the incident, Mrs Tait was in the habit of crossing the road rather than meet her late colleague in the cause of justice. With the advancing years the exuberance of the Professor's golf decreased ; the two round limit was never exceeded. In the later eighties he played but little, and after 1892 never a full round; but only the nine outward holes followed by a rapid walk home by way of the new course. This athletic decline on the part of the Professor synchronises with Freddie's advance as a golfer ; it also marks the beginning of the transference of the former's interest to the philosophical side of the game. The Professor's famous experiments were begun in 1887 and reported in Nature, August, 1890, September, 1891 ; and his full theory was complete in 1893. He also wrote articles on "The Pace of a Golf Ball," Golf, Dec. 1890; " Hammering and Driving," Golf, Feb. 19, 1892; "Carry," Golf, August 25, 1893; "Carry and Run," Golf, Sept. 1893; "The Initial Pace of a Golf Ball," Golf, July 17, 1894; and he contributed an important summary of his work in a paper to the Badminton Magazine, March, 1896 (reprinted below). One of the Professor's most interesting pieces of mathematical work deals with the subject of Rotating Spheres and Projectiles ; but as this has been adequately discussed in another part of this biography, it will be sufficient if we glance at the general results as they appeared to the golfer. Prior to the Professor's investigations we imagined that speed of projection, elevation, and the resistance of the air, were the three things which determined the flight of a golf ball. The Professor indeed seems himself to have begun THE VIRTUE OF UNDERSPIN 59 from this standpoint ; and his first discovery was that we were all wrong. He told us that we imagined we knew all the laws under which a golf ball flew, but that these laws were in themselves insufficient to explain the duration of the flight, and that he proposed to find out what was lacking in the sum of our knowledge ; he discovered in fact that there was a problem to solve. Mr H. B. Farnie, and afterwards Sir Walter Simpson had told us of the Art of Golf; the Professor detected that there was a Science of Golf, and afterwards worked out and communicated the problems which he had discovered and solved. There is a story that Freddie demolished his father's arguments by driving a ball further than the limit that had been set by the Professor. Freddie perhaps half believed that he had created this joke against "the Governor," for he never studied his father's articles very closely, as we can judge from the fact that it is not till the end of 1898 that we find him writing to Jack to announce that "the Governor's theory is underspin." The grain of truth that was in the story was made into a good jest by the facile pen of Mr Andrew Lang. The Professor indeed said in Golf, Dec. 1890, that from the theoretical data it appears that to gain ten per cent, of additional carry a long driver must apply nearly fifty per cent, more energy. But this statement must be read with his explanatory remark in Nature, " I shall consider the flight of a golf ball in a dead calm only, and when it has been driven fair and true without any spin." The essence of the Professor's discovery was that without spin a ball could not combat gravity greatly, but that with spin it could travel remarkable distances. In the first place the Professor found that a golf ball combated the attraction of gravity for a period nearly twice as long as he had expected. By floating marked golf balls in strong brine or mercury he found that they did not float truly, but wobbled, and that the marked spots ultimately came to certain fixed positions ; from this he gathered that the centre of gravity of a ball seldom, if ever, coincided with its centre of figure. This fact, taken in conjunction with an assumed rotation, at once explained the violent wobbling in the air occasionally observed. Slicing and pulling proved the existence of spin about an axis not truly horizontal ; and mathematical calculation showed that underspin, by introducing a lifting force, would increase the flight of the ball. The sufficiency of the omitted factor was made clear. This discovery has been of the utmost importance to the golfer, and is in fact the groundwork on which the modern school of scientific play has been built. 82 6o PETER GUTHRIE TAIT The Law which was known to Newton, and investigated by Magnus, viz. "That a sphere rotating and advancing in still air deviates from its straight path in the same direction as that in which the front side is being carried by the rotation," is the law which governs all slicing and pulling, topping and skying. We say that we slice if we stand in some particular position ; but we may stand as we like and slice, if only we make the front side of the ball rotate from left to right during its progress. This knowledge of the power of spin having been placed in the hands of the golfer it became necessary for him to find out how he could make strokes which would cause the ball to turn from the right or left, or to rise in its flight and to stop without running, or to make but a short upward journey and then reach the ground with great power of run. Of the strokes indicated the last named is by those ignorant of the finer points in the game called the " common top " ; but it is very far from this, and is a shot which was brought to great perfection by Freddie and by Mr J. E. Laidlay, and when well played from a suitable situation is a fine thing to see done. When the ball is topped it is struck above its centre and rolls in an irresponsible manner along the ground. In the proper stroke the ball is struck with a lofted club well below the belt, and is thus assured of a definite carry ; but just as the head of the club reaches the ball an upward movement is given which imparts overspin and causes the ball to run after it touches the ground. This is the true overspin stroke, known to experts as the "rising club shot." Another stroke which has been understood through the Professor's discovery is the " long carry " over a hazard. The Professor showed that it was not necessary, or indeed advisable, to start the ball with a high trajectory, and that the low stroke which goes, because of underspin and in spite of gravity, concavely upwards produces the best result. These examples may be sufficient to show how deeply golfers were indebted both practically and intellectually to the increased interest he bequeathed to the game. The Professor's experiments were of course conducted with the gutta ball and some of his conclusions have therefore been modified by the introduction of the more resilient rubber core ball. Speaking very roughly, he arrived at the conclusion that in the case of a full drive at the moment of impact the clubhead was travelling at the rate of 200 feet per second, and the initial velocity of the ball's projection was 300 feet ; with the newer balls the initial velocity will no doubt be greater; and it is also possible that their greater carry may be influenced by their greater willingness to THE "BULGER" 61 receive underspin, and as a consequence to allow of their being struck with a very low trajectory. The Professor, perhaps, laboured his theory of underspin too far, and his sons used to regard rather with amusement his famous underspin iron. This weapon was a very light upright cleek with ridges on the face running parallel to the base of the head. I remember the Professor asking me to have a shot with it and telling me that if I hit the ball fast enough I would drive from the "Sandy Road" over the burn. What he wished to impress upon us was that the speed at which the clubhead was travelling and the proper amount of underspin are the two chief factors in long driving ; but he never looked to see us drive a great distance with this club, for he knew as well as we did that the head was too light to bring out the resilience of the ball, a most important practical factor. The introduction of the " Bulger " of course interested him ; but he was not in favour of the weapon, for it did not assist him in his theory of underspin, since it was intended to obviate the evils of rotation about a vertical not a horizontal axis. The true Bulger, he said, should have its vertical section convex. Over the initials G. H. there appeared in the Scots Observer some verses which the Professor afterwards acknowledged, describing them as "expressive at least, if not wholly elegant," which we reproduce as they have a ring of the author's humorous philosophy. The initials G. H., I believe, represented the name Guthrie Headstone, the play on the words Head and Tait and Peter and Stone being obvious. THE BULGER. 1. From him that heeleth from the Heel, Or toeth from the Toe, The Bulger doth his vice conceal ; His drive straight on doth go. 2. To him who from the Toe doth heel, Or from the Heel doth toe, The Bulger doth his faults reveal, And bringeth grief and woe. 3. And the poor slicer's awful fate, Who doth a-bulging go, Is sad indeed to contemplate ; The Bulger is his foe. 62 PETER GUTHRIE TAIT 4. But whoso plays the proper game, His ball who striketh true, He findeth all clubs much the same ; A goodly thing to do. MORAL. Bulgers, and Mashies, Presidents, Are for weak players made ; As spectacles and crutches be For eyes and limbs decayed. G. H. Returning with the Professor to the club-house, we notice that the golfers freely greet him, as he quietly retires to his accustomed seat, or finds a companion for an afternoon game at billiards. In this community, full of cosmopolitan elements, the great man walked humbly and was accessible to everyone. On a doubtful morning no one started for a round without asking him if an umbrella should form part of the caddie's burden ; and his opinion was always backed against the barometer. The Professor seldom addressed anyone, but of all the notables he was the most easy of approach. No topic of conversation was foreign to his interest ; and the more remote the subject from the beat of his scientific enquiries the more were we astonished by the intimate manner in which he threw himself into the discussion. On politics he held tremendous views ; and his eye glistened as he read a slasher in the Saturday Review. In his Edinburgh home he was not a club man, and I believe he refused to join in any way in club life ; but in his holiday time he loved to mingle with the golfers, and enjoyed greatly his billiards. Although not a great player, his intimate knowledge of angles gave him a fine field for amusement and experiment as he tried almost impossible cannons. To an opponent who had indulged in a very forceful game, I remember him remarking that the play had seemed to be a combination of bagatelle and racquets. But these hours in the billiard room were for him, especially in later days, sources of splendid recreation. Many great men have been drawn to St Andrews, and have gone in and out of the Royal and Ancient Club ; but probably no man so great has ever come so closely in touch with its members. We knew that he knew the mysteries which our minds could not grasp ; but the man as he walked among us put himself, almost with diffidence, on our level and invited our opinion. We, who had not been his pupils, were thus able to THE LAST YEAR 63 guess the cause of that power and fascination which he had exercised over generations of Edinburgh students. The Professor never seemed to be far from any one of us ; he disguised the fact that he was in touch with the immortals. Mixing with all, and always friendly with all, his heart was nevertheless fixed within the circle of his own home ; and, as we write of him, more particularly on Freddie and his doings on the links. What Freddie had done, what match he was playing, what chance he had at the next Championship, or medal, these were the thoughts always near to him. Freddie was his companion in his experiments, making herculean drives against the apparatus prepared by the Professor. Freddie chaffing " the Governor," is still the better loved Freddie. Freddie fighting in South Africa, wounded, but making a good recovery, remains the father's idol. It was little wonder then that in that dark February of 1900, when the bad news came, the Professor, the man of rock, was rent. A few months later, when on my way from St Andrews to Sandwich for the Championship meeting, I dined with the Taits in Edinburgh before starting on the night train. Through dinner the Professor seemed very depressed as though afraid to enter into any conversation which might become reminiscent of the golf which had Freddie for its central figure. I tried to draw him on to subjects which involved no risk ; but a most unnatural heaviness seemed to hang over him. After dinner, in his study at the back of the house, he showed some return of his old boyish nature, and made some pithy remarks about the players who were likely to be at Sandwich. I was looking at some shelves full of old text books while he was attending to some small note he had to answer ; suddenly he turned round and called out, "We have new editions of all these." This pregnant remark was followed by his old laugh ; and until I left his conversation was as bright as in former days. Yet I do not think that he ever got back into his true gait after Freddie's death ; the light seemed to have left the eyes which in repose often wore an expression of weariness. The passings of Father and Son were in striking contrast; Freddie died before his life was fulfilled : the Professor died after he had searched the philosophies and completed his investigations. The Professor's favourite theme was the Law of Continuity. It has been well said that every ultimate fact is but the first in a new series ; the Professor was still a boy when he left us. CHAPTER II EXPERIMENTAL WORK CLEAR indications have already been given that from his early student days Tail's main interest was in physical rather than in pure mathematical science. His first experimental work was done in Belfast under the guidance of Andrews, whom he assisted in the preparation of three papers on Ozone. These appeared in the Proceedings of the Royal Society of London between 1856 and 1857. Already in 1855 he had visited the Paris Exposition, one of his chief objects being the study of scientific apparatus. This we learn from the following letter written from Cambridge : ST PETER'S COLLEGE, CAMBRIDGE, Sept. 21/55. My dear Dr Andrews, I have just received your note. I am sorry it will be impossible for me to revisit Paris this vacation. Everything has been going on so wretchedly here during my absence, so far as regards printing 1 , that even with a month's hard work from this date, I fear not more than of the work will be ready.... I have made attempts to see Ruhmkorff, Soleil, and Tyndall. The former was out of the way, Soleil was in Glasgow, and I believe so was Tyndall. I extracted from the woman in Soleil's shop all the information they could give about the Saccharimeter. I saw the instrument, pr. 260 fr., and bought a description of it and its use by Moigno. I found and examined all the electromagnetic apparatus in the Exposition, and it was my decided opinion that an instrument in Ruhmkorff s stall called " Appareil de Faraday" was the very thing for us.... I hope you agree with me in the matter of the apparatus for Faraday's experiments. The only objection that I could see to it is that possibly it might not be powerful enough ; but of that you will be a much better judge. Not far from Ruhmkorff's there is a collection of clockwork, and along with it a small machine for exhibiting the permanence of the plane of rotation. I have not seen the gyroscope itself this machine seemed to me not only comparatively useless, but even dangerous. 1 The printing of Tail and Steele's Dynamics oj a Particle. ON HOLIDAY IN EDINBURGH 65 SOMERSET COTTAGE, COMELY BANK, EDINBURGH, 21/7/59. My dear Dr Andrews, I was very glad to find from your letter that you had been successful in procuring apparatus in London.... I did not expect more from Faraday than you seem to have obtained, for I thought it scarcely possible that he could suggest at an hour's notice anything that we might have missed for three years. My paper on the Wave-surface has reached me in separate form and I have been asked by several men of note, to whom I have sent copies, to publish an elementary work on Quaternions. Todhunter of Cambridge, about the best authority on matters of that sort, is one of them and I have written to Macmillan (the publisher) to enquire about terms etc.... Sir W. Hamilton has expressed his satisfaction with the project and has only asked me to refrain from laying, or trying to lay, new metaphysical or other foundations for the Theory, wishing to reserve such for himself; and I am quite sure that I shall not feel this in any way a restraint.... I have ordered the addition to the small electrical machine.... There is only one novelty here, so far as I can see, and as it is extremely interesting, I have given an order for one. Its object is the compounding of colours by rapid rotation, and so far it is simple but when used in combination with a looking glass (like the Thaumatrope) it gives some most startling but easily explained and instructive effects.... SOMERSET COTTAGE, COMELY BANK, EDINBURGH, 18/6/60. My dear Andrews, I shall probably leave this for Cambridge on Monday next, and it will not be possible for me to be in Oxford as Hopkins and I are to be engaged in getting up our Ex n Papers just at the time of the Ass n Meeting.... Dr Bennett showed me on Saturday the whole series of frog experiments with a splendid galvanometer from Berlin and German Frogs which he had imported ! But what interested me most was the perfect success of the experiment showing the muscular current in the operator himself, that you remember which we could not repeat and had begun to doubt. Mr Pettigrew, his assistant, produced by contracting his right arm a deflection of 15 E., then by contracting his left arm, one of 35 W. 50 in all. Neither Dr B. nor a Russian who was present could produce more than very uncertain results. I no longer entertain any doubt as to the reality of the phenomenon. The explanation, however, does not seem quite satisfactory. Dr B. told me that Humboldt had skinned his forefinger by raising blisters in order to get rid of the great resistance of the skin, and that then he produced extraordinarily great deflections.... T. 9 66 PETER GUTHRIE TAIT Towards the close of Tail's sojourn in Belfast, Andrews was preparing to attack the problem of the compressibility of gases. In this research Tait was to join him; but his election to the Chair of Natural Philosophy in Edinburgh altered all these plans. The duties of his new Chair compelled him to give still more attention to the experimental than to the mathematical side of Natural Philosophy. In the early years he devoted much time to the preparation of his lectures and lecture experiments. In arranging the experimental illustrations he had the able help of James Lindsay who had served both Sir John Leslie and Professor Forbes as mechanical assistant. His scientific activities are clearly displayed in his letters to Andrews ; and from these a few quotations will show how this kind of work grew upon his hands. A long extract referring to his first lecture has already been given (page 22). On December i, 1860, Tait wrote : My dear Andrews, I am very much obliged to you for your note to Faraday. I enclosed it in a letter to him, telling him that I wished to ask his opinion on a point in the optical effects of magnetism ; and as I sent him a copy of my lecture 1 I ventured to ask him to inform me at his leisure whether I had in it fairly stated the case at issue between him and the pure mathematicians about conservation of force. I got a very kind answer yesterday. He requests me to postpone my question (if a difficult one, and it is so) till after Christmas but about the other matter he says " I thank you for the way in which you have put the Gravitation case. It is just what I mean." He says he has been working at it all summer, but still with negative results and that he had drawn up a new paper for the Royal Society, but that Stokes had advised him not to present it... COLLEGE, EDINBURGH, Jan. 29, 1 86 1. My dear Andrews, I would have written to you sooner, had not my hands been full of the January Examinations, and some experiments which Principal Forbes asked me to make In a paper which is I believe to appear in the Phil. Mag. for February, and which was read some weeks ago at the R. S. E., he states that few people living have ever seen Ampere's experiments for the repulsion of a current on itself and that he had never succeeded in getting it. At his request I tried it, and succeeded with a single cell of Grove's battery. With twelve cells the floating wire almost jumped out of the trough ! As there is some slight objection to this form of the experiment on account of the thermoelectric effects which occur at every change of metal in the circuit, 1 This refers to Tail's inaugural lecture, in which he discussed Faraday's attempts to demonstrate the Conservation of Force in the sense of attraction. THOMSON'S ELECTROMETER AND GALVANOMETERS 67 I devised a floating conductor of glass tube full of mercury to replace the copper wire. The mercury is so much worse a conductor than copper, that it required four cells to give a good effect. 6 GREENHILL GARDENS, EDINBURGH, 18/12/61. My dear Andrews, I find that I cannot manage to visit Belfast at present my simple reason is that I am to bring home from Glasgow (where I am going to stay a day with Thomson) two galvanometers and an electrometer on Saturday next and I must have one galvanometer and electrometer fitted up during the holidays, as I shall just have reached the critical point of Radiant heat when we stop. The new galvanometer works by reflexion, and can therefore be easily shown to a large class, which was impossible with the needle ones besides it is delicate enough to show an effect even by frog-currents. The electrometer also works by reflexion, and gives a deflection of some inches on a scale for ^th of the electromotive of one cell (Daniell). Of course the gold- leaf electroscope must now remain unused on the shelf, or at most be brought out to show what we used to be content with.... This prophecy of Tait's was not fulfilled even by himself during the suc- ceeding forty years of lecturing. There is a simplicity about the gold-leaf electroscope which will ever keep it a prime favourite for purposes of demon- stration, especially now when it is so easy to project the moving and divergent leaves magnified upon a distant screen. GREENHILL GARDENS, EDINBURGH, Jan. 15, 1862. My dear Andrews, Three reasons especially urge me to write to you to-night the first and most pressing I shall detail at once. I wish to know (by return of post if possible) what is the nature of the new ammonia process for procuring cold, and from whom, and at what price, it can be procured. This urgent business having been got over, I can be more easy in my future remarks. You should at once get William Thomson's galvanometers acting by reflexion. I have been lecturing on heat for some 4 weeks back ; and I have shown, to my whole class, not only Melloni's experiments about diathermancy &c., but on a large scale the polarization of dark and bright heat... Next I wish to know where your (and others') results as to Heat of Combination are to be found. As to myself I may say that I have done nothing experimentally for a long time except with a view to familiarising myself with new apparatus.... The beauty of the new galvanometers is such that today I arranged to show in a future lecture the Inductive Effects of the Earth's magnetism on a coil of wire about 30 feet long, coiled 92 68 PETER GUTHRIE TAIT in a circle of about eight inches diameter. Turning that through 90 from a position perpendicular to the dipping needle, I got sufficient deflections of the galvanometer to throw the light off the scale. My own peculiar experiments on light, which you assisted at two years ago, I have arranged to try the very first fine day, and now with some hope of success, although Thomson is not at all sanguine about the idea. I intend to repeat (if true) Tyndall's observation on the Adiathermancy of Ozone with an instrument far superior to his. Perhaps something may come of it. The invention of the Divided Ring Electrometer indeed opened up many new lines of research ; and in 1862 Tait and Wanklyn 1 published a joint paper on the electricity developed during evaporation and during effervescence from chemical action (Proc. R. S. .), in which attention was called to the large charges produced by the evaporation of a drop of bromine and especially a drop of aqueous solution of sulphate of copper, from a hot platinum dish. On January 23, 1862, in a letter mainly taken up with the projected treatise on Natural Philosophy, Tait again got into ecstasy over Thomson's galvanometers and electrometer. " They are splendid instruments. If you are in no hurry I will be over in Belfast in April or May and will set them up for you. It requires some practice, but the gain in visibility to the class is ENORMOUS. I showed by his electrometer today to my whole class (150) in lecture the tension of a cell without condenser or anything of the sort." On July 7 of the same year Tait mentioned the visit of Stas of Brussels to Edinburgh and referred to experiments which he was doing along with Wanklyn. With the preparation of the great treatise on hand, and the consideration of the experiments on the rotation of a disk in vacua which Balfour Stewart and he had begun upon, there was not much time for under- taking any other experimental work on his own account. Tait was moreover at this time working hard at quaternions. One very fruitful piece of experi- mental illustration we owe, however, to this period. As will be more clearly brought out in the chapter on quaternions, Tait was greatly impressed with Helmholtz's famous paper on vortex motion, so much so that for his own private use he took the trouble of making a good English translation of it. Early in 1867 he devised a simple but effective method of producing vortex smoke rings ; and it was when viewing the behaviour of these in Tail's Class Room that Thomson was led to the conception of the vortex atom. In his first paper on vortex atoms presented 1 Dr J. A. Wanklyn was assistant to Lyon Playfair the Professor of Chemistry. He was a well-trained chemist, ingenious and resourceful. SIR DAVID BREWSTER 69 to the Royal Society of Edinburgh on February 18, 1867, Sir William Thomson refers as follows to the genesis of the conception : "A magnificent display of smoke-rings, which he recently had the pleasure of witnessing in Professor Tail's lecture-room, diminished by one the number of assump- tions required to explain the properties of matter, on the hypothesis that all bodies are composed of vortex atoms in a perfect homogeneous liquid. Two smoke-rings were frequently seen to- bound obliquely from one another, shaking violently from the effects of the shock.... The elasticity of each smoke-ring seemed no further from perfection than might be expected in a solid india-rubber ring of the same shape.... " Professor Tait's plan of exhibiting smoke-rings is as follows : A large rectangular box open at one side, has a circular hole of six or eight inches diameter cut in the opposite side.... The open side of the box is closed by a stout towel or piece of cloth, or by a sheet of India-rubber stretched across it. A blow on this flexible side causes a circular vortex to shoot out from the hole on the other side. The vortex rings thus generated are visible if the box is filled with smoke." Then follows a description of one way of producing a cloud of sal- ammoniac, not the way however as generally practised by Tait ; and the paper ends with a description of the effects of collision between vortex rings produced from two boxes. This seems to be the earliest printed account of Tait's experiments on vortex rings which gave the start to Thomson's famous theory of vortex atoms. From 1859 till his death in 1868 Sir David Brewster was Principal of Edinburgh University. In spite of his eighty winters the famous experi- menter still continued his researches, and Tom Lindsay, then a youth training as mechanical assistant under his father, James Lindsay, tells how Brewster made considerable use of the optical facilities of the Natural Philosophy Class Room, and discussed many optical phenomena with the young Professor. Sir David had made his residence at Allerly near Melrose and travelled to and from Edinburgh by train whenever his University or Royal Society duties demanded his presence. Had he lived in Edinburgh, he would no doubt have spent a large part of his time in the Natural Philosophy Department; for Tait, then as ever, cordially welcomed any one who had a physical problem to investigate. Among the subjects which specially occupied Brewster's attention during the later years of his life were the colours of soap films and the pheno- menon which he had discovered in 1814 and had described under the name of the Radiant Spectrum. When a bright small image of the sun, such as may be obtained by reflexion from a convex mirror, is viewed through a prism, there appears in addition to the usual spectrum a bright radiant spot beyond the 70 PETER GUTHRIE TAIT violet. Brewster described his latest experiments in a short communication to the Royal Society of Edinburgh on April 15, 1867, but gave no explanation. At the next meeting, on April 29, when Sir David, as President, was again in the chair, Tait read a very brief communication on the same subject, tracing the phenomenon to the peculiar texture of the membrane covering the cornea and to the effect of parallax. There can be no doubt that the experiments on which Tait based his conclusions were made in conjunction with Brewster, who probably agreed with the explanation brought forward by his colleague. It was just at this time (April, 1867) that Tail's efforts to establish a physical laboratory, in which doubtless he was strongly backed by Sir David Brewster, received formal recognition by a grant of money from the Senatus. The minutes simply record the fact, but give no indication of how long a time was required by Tait to educate his colleagues up to the point of admitting that such a new departure was desirable. But to vote the money was one thing, to find accommodation even for a small laboratory was another. Six months seem to have elapsed before the next step was taken ; and then in a letter of date December 20, 1867, Tait wrote to Andrews : " I am about to get a Laboratory for practical students. The money has been voted. Henderson 1 has been induced to give up his class room (which is situated just over my apparatus room), and during the holidays it will be put in order for work.... I want to ask if you can give me hints as to good subjects of experimental work for practical physical students, not subjects that require a Faraday, still less such as require a Regnault." In his opening lecture of the session 1868-9 Tait was able to make a definite announcement regarding the Physical Laboratory. The following report of part of the lecture is taken from the Scotsman of November 3, 1868. " In several respects the present session may be expected to differ for the better, as regards the class of Natural Philosophy, from at least the last eight during which I have been connected with this University.... From the miserable resources of the University enough has been granted me to make at least a beginning of what will I hope, at no very distant time, form one of the most important features in our physical education. A room has been fitted up as a practical laboratory, where a student may not only repeat and examine from any point of view the ordinary lecture experiments, thereby acquiring for himself an amount of practical information which no mere lecturer can pretend to teach him ; but where he may also attempt original work, and possibly even in his student days make some real addition to scientific knowledge. That this is no delusive expectation is proved by the fact that in Glasgow, 1 The Professor of Pathology at the time, the predecessor of the well-known Professor Sanders. W. ROBERTSON SMITH 71 under circumstances as to accommodation and convenience far more unfavourable than I can now offer, Sir W. Thomson's students have for years been doing excellent work, and have furnished their distinguished teacher with the experimental bases of more than one very remarkable investigation. What has been done under great difficulties in the dingy old buildings in Glasgow, ought to be possible in so much more suitable a place as this." The most complete account given by Tait himself of his method of running a physical laboratory is to be found in his evidence before the University Commission of 1872, which consisted of Professor William Sharpey, Professor G. G. Stokes, and Professor H. J. S. Smith. The following suc- cessive answers to questions form a concise statement of Tail's views. " I have made the laboratory open to all comers, limited of course by the number of students which my assistant and I can look after, and which my space can accom- modate.... They (the students) are free to spend their whole time in the laboratory when it is open each day, and thoroughly to devote themselves to their work.... "There is a small fee of two guineas for each student, but... that does not pay for the mere chemicals and other materials used by each student... With the help of my assistant I put each student as he enters the laboratory through an elementary course of the application of the various physical instruments, the primary ones. For instance, I begin by practising them in measuring time, estimating small intervals of time, then measuring very carefully length, angle, temperature, electric current, electric potential, and so on.... " When I find that they have sufficiently mastered those elementary parts of the subject I allow them to choose the particular branch of natural philosophy to which they wish to devote themselves, and when they have told me that, it is not by any means difficult to assign to them, if they carry it out properly, what may be excessively useful and valuable work." The assistant under whose care the Laboratory first took shape was William Robertson Smith 1 , M.A., afterwards well known as a theologian and Semitic scholar, the final editor of the ninth edition of the Encyclopaedia Britannica, and Librarian of the University of Cambridge. Smith was an Aberdeen graduate who shortly before had gained the Ferguson Scholarship in Mathematics open to the four Scottish Universities. Tait was examiner that year; and, impressed with the brilliant though untrained, indeed "almost uncouth," powers of the young student, he invited him to become his assistant. When Robertson Smith saw that he could combine the duties of the post with his theological studies at the Free Church College, he accepted Tail's offer; and after training himself in physical manipulation 1 A biographical note communicated by Tait to Nature is reprinted below. 72 PETER GUTHRIE TAIT during the summer months of 1868 undertook, the next winter session, the systematic teaching of students in practical physics. In this small upper room stripped of its benches, but with the terraced floor left intact, the men were put through a short course of physical measurements, such as specific gravities, specific heats, electrical resistance, and the like. Any who showed talent were soon utilised by Tait in carrying out original research ; and, to facilitate this kind of work, every possible corner of the old suite of rooms of the Natural Philosophy Department was adapted by means of slate slabs built into the thick steady walls for the installation of galvanometers and electrometers. The small room which Professor Forbes had used as his sanctum became the centre of experimental work. In this room Forbes had made his classical researches in polarisation of heat ; and here also Tait, with the help of successive sets of students, made his novel discoveries in thermoelectricity. The large class room was also used as a research room, especially during the summer session when (at least until well on in the seventies) no class met. Two slate slabs were built into the wall, one on each side of the blackboard ; and on these were placed the mirror galvanometers and electrometers necessary for delicate electrical investigations. Robertson Smith remained with Tait till 1870, and found time to carry through an interesting piece of experimental work on the flow of electricity in conducting sheets. In the paper giving an account of these experiments he considerably simplified the mathematical treatment, which had already engaged the attention of Maxwell and Kirchhoff. Among the students who passed through the Laboratory during the first and second years of its existence were Sir John Murray, Sir John Jackson, and Robert Louis Stevenson. Stevenson was paired off to work with D. H. Marshall, who succeeded Smith as assistant in 1870 and is now Emeritus Professor of Physics of Queen's University, Kingston, Ontario. Marshall of course was keen in all things physical, while Stevenson's preference was for a lively interchange of thought on every thing of human interest except science. When, as frequently happened, Stevenson got weary of reading thermometers or watching the galvanometer light-spot, he easily found some excuse to bring Robertson Smith within hearing and set him and John Murray arguing on the age of the earth and the foundations of Christianity. In some idle moments these lively students broke Tait's walking-stick. In haste and trepidation they commissioned two of their number to buy another as like the shattered one as ROBERT LOUIS STEVENSON 73 possible. Tait who had been attending some Committee meeting returned ere long, and went to the usual corner to take possession of the stick. He paused doubtfully for a moment, then advanced, took the stick in his hand, and felt its weight and surface with considerable uncertainty. He looked at it again, glanced round the room, and then walked off towards the door. Back he came again almost immediately, glanced more carefully into various corners, swung the unfamiliar weapon to and fro, and at length, deciding that it was not what it seemed to be, put it back in the corner, and walked briskly home. Nothing was possible now save a full confession ; and Tait accepted the gift in token of forgiveness. Stevenson's father was Thomas Stevenson, the well-known lighthouse engineer. He hoped that his son would carry on the family traditions, and expressly desired Tait to let him work with optical apparatus. But the future essayist and writer of romances had not the smallest elementary knowledge of the laws of reflexion and refraction. The immediate purposes of the Physical Laboratory were lost on him ; although no doubt what little training he allowed himself to undergo bore some fruit when a few years later he read a paper before the Royal Society of Edinburgh comparing rainfall and temperatures of the air within and without a wood. It was published in the Proceedings : literary critics have, however, left it severely alone. Nevertheless, Stevenson's familiarity with the Physical Department led in after years to the writing of a charming picture of James Lindsay, the mechanical assistant already referred to. In 1886 when the University students held their great Union Bazaar, Stevenson contributed "Some College Memories" to the New Amphion, a beautiful volume (321110.) printed in exquisite old-fashioned style by T. and A. Constable after designs and plans by W. B. Blaikie of that firm. After giving a quaint picture of himself in the third person, Stevenson continues, " But while he is (in more senses than one) the first person, he is by no means the only one I regret, or whom the students of to-day, if they knew what they had lost, would regret also. They have still Tait to be sure long may they have him ! and they have Tait's class-room, cupola and all ; but think of what a different place it was when this youth of mine (at least on roll days) would be present on the benches, and at the near end of the platform, Lindsay senior was airing his robust old age. It is possible my successors may have never even heard of Old Lindsay ; but when he went, a link snapped with the last century. He had something of a rustic air, sturdy and fresh and plain ; he spoke with a ripe east-country accent, which I used to admire ; his reminiscences were all of journeys on foot or highways busy with post-chaises a Scotland before steam ; he had seen the coal fire on the Isle of T. 10 74 PETER GUTHRIE TAIT May, and he regaled me with tales of my own grandfather. Thus he was for me a mirror of things perished ; it was only in his memory that I could see the huge shock of flames of the May beacon stream to leeward, and the watchers, as they fed the fire, lay hold unscorched of the windward bars of the furnace ; it was only thus that I could see my grandfather driving swiftly in a gig along the seaboard road from Pittenweem to Crail, and for all his business hurry drawing up to speak good- humouredly with those he met. And now, in his turn, Lindsay is gone also ; inhabits only the memory of other men, till these shall follow him ; and figures in my reminiscences as my grandfather did in his." James Lindsay retired from his College duties in 1872, after having acted as mechanical assistant since 1819 when Sir John Leslie became Professor of Natural Philosophy. He had for the five previous years acted as Leslie's door- keeper at the mathematical class room. He had thus been connected officially with the University for fifty-seven years ; and his memory went back to the days when Carlyle was still a student. He was a native of Anstruther ; and to quote from an obituary notice which Tait himself supplied to the Scotsman of January 5, 1877 "during the summer months, for at least the half of his life, he pursued the arduous occupation of a fisherman, in order to eke out his scanty income ; and even in later years, when unable to go to sea, the position he had deservedly acquired among the fishing population of the district, led to his being employed during the herring season as an agent in the interests of some of the great fish curers. In this position his punctuality and rectitude were as much displayed at the pier head as in the Natural Philosophy class room." Under Leslie he became wonderfully dexterous in many difficult experimental processes, especially excelling in glass-blowing ; and he rendered most efficient and indeed valuable aid both to Leslie and to Forbes in their experi- mental investigations. For twelve years he continued to assist Tait in the lecture experiments ; and after he had trained his son Thomas to all the duties of the post, he retired to spend his last days in his native village. After his retirement he used occasionally to pay a visit to the scenes of his scientific labours, and I remember him on one such visit expressing great indignation at the careless way in which a box-full of small differential thermometers had been allowed to gather dust in a dark corner. These he had made with his own hand ; and he had not realised that the thermopile and galvanometer had completely displaced the differential thermometer as a delicate instrument of research. The following letter to Thomson touches on several pieces of experimental work which were engaging Tail's mind in the early years of the Laboratory. EXPERIMENTAL ACTIVITY 75 17 D. P. E. 5/7/69. Dear T., I have just heard from T" [i.e. Tyndall] that you are in Largs. I feared you would be in a state of suspense and uselessness at Brest. Do you mean by multiple-arc coils the set which has a separate frame for plugs one in fact into which plugs are to be put, not out of which they must be taken, in order to work them ? If so I shall send them off at once on hearing from you, for I have not even attempted to work with that set. The other set works capitally and I have almost finished my copper wire determinations by its help besides having carefully got the values of the coils of my own set; the unit in which is curiously (purposely?) r$ B.A. units very nearly. You did not answer my query about the equation for heat in a bar. Do so now. d f, df for two similar bars which when heated and left to cool work exactly together Is not kap^ can be worked only at Caius ? See Murphy, Green, O'Brien, Pratt. When I examined here the only men who could do figure of the earth were mild Caius men. All the rest were Prattists if anything. I think a very little mortar would make a desirable edifice out of your article. In selecting the absolute value of the constant coefficient of a harmonic we may go on one of several principles. There then followed a comparison of his own expressions with the cor- responding expressions used by T and T' and by Tait. He continued : The great thing is to avoid confusion. I rather think your value is the best to impress on the mind. It lies between it and ^ (8) which has a certain claim. The diggings in 2 dt ' 1 Nevertheless Tait says in his paper that he was led to the method while engaged in some quaternionic researches. 2 j-=JCM, (Maxwell's initials), one expression for the Second Law of Thermodynamics, as used by Thomson in his early papers, and by Tait in his Historic Sketch, J being Joule's equivalent, C Carnot's function, and M the rate at which heat must be supplied per unit increase of volume, the temperature being constant. 102 PETER GUTHRIE TAIT In a post card to Thomson of date Nov. 5, 1871, Maxwell, after referring to some proof sheets of his book which he had sent to Thomson to revise, remarked : " Laplace has a clear view of the Biaxal harmonic. T' has an excellent discussion of Qi and ^ w and their relations deduced from their definitions and not from their expansions as Murphy does. Murphy is very clever, but not easily appreciated by the beginner." This post card found its way finally to Tait and was duly filed along with the other correspondence. The whole correspondence shows the free inter- change of thought which went on between Maxwell and Tait and the subtle manner in which each helped the other. We can in many cases infer the nature of Tail's letters which Maxwell was obviously replying to ; but the characteristic language in which these must have been expressed is unfortunately irrecoverable. For anything of Hamilton's Tait had a profound respect ; and in the " beautiful invention of the Hodograph " he found on more than one occasion a source of inspiration. His hodograph note communicated to the Royal Society of Edinburgh in 1867 contains an elegant geometrical construction in which the equiangular spiral is used with effect to represent motion in a resisting medium. Maxwell practically introduces the whole investigation into the second volume of his Electricity and Magnetism, when he is dis- cussing the theory of damped vibrations of a swinging magnetic needle. The powerful quaternion papers on the rotation of a rigid body and on Green's theorem were communicated to the Royal Society in 1868 and 1870 respectively. They will be most suitably discussed in the following chapter on quaternions. To this period also belongs a quaternion investigation into the motion of a pendulum when the rotation of the earth is taken into account. This is reproduced in the second edition of his Quaternions. The paper is called an " Abstract " in the Proceedings ; and the closing sentences epitomising other developments imply that Tait had every intention of publishing a complete and elaborate discussion as a Transactions paper. For this however he never found leisure. This habit of printing an abstract, indicating the lines of development in a projected large memoir which never saw the light, was one which grew with the progress of the years. During the early seventies, when the experiments in thermo-electricity were in full swing, nothing very serious was taken up on the mathematical side ; but the game of golf suggested this curious and by no means easy TAIT'S GOLF MATCH PROBLEM 103 problem ; "When a golf-player is x holes 'up' and y 'to play,' in how many ways may the game finish ? " The paper in which Tait considered the problem is called a question of arrangement and probabilities. He first solved the simpler question as to the number of ways the player who is x up and y to play may win. Let this number of ways of winning be represented by P (x, y). Then starting with P (x + i, y+ i), we see that at the first stage the player may win, halve, or lose the next hole, and the number of possible ways of winning will then be represented by P (x+2, y}, P (x+i, y}, and P (x, y) respectively ; hence follows Tail's fundamental equation If then we construct a coordinate scheme with x measured horizontally and y vertically downwards, and place in the position xy the number P (x, y), we can at once pass by simple addition of three consecutive values of x for any one value of y to the values for the next higher value of y. The following is the scheme as far as jx=5- o o o o o i i o O ...X o I 2 i i o o o o I 3 4 4 i i o o o I 4 8 II 9 6 i I o o o I 5 13 23 28 26 16 8 I I o I 6 i9 4i 64 77 70 50 25 IO I etc. etc. The zero positions are enclosed in the double lines ; and the meaning of the entries to the left of the vertical lines is the number of ways in which the player may lose. The unit values on the right and left flanks are determined by the limiting conditions, which show that when x is greater than y, the game is won, so that P (x, y} = i . Similarly, when x is not less than y, the player cannot lose. Hence P(~x, y} = o. These considerations also explain why the fundamental equation given above does not apply to the second last unit on the right of each row. As an example, let a player be 2 up and 4 to play ; he may win in 26 different ways. His opponent who is 2 down and 4 to play may of course lose in the same number of ways. But the number of ways in which the player who is 2 up may lose is only 5. These numbers 26 and 5 PETER GUTHRIE TAIT give an estimate of the respective probabilities of either player winning. The number of possible draws is obtained from the same fundamental equation, the limiting conditions being P(x, y) = i when x=y, P(x,y)=o when x>y, whether x is positive or negative. The values are represented by the following scheme. o i o O I i I O I 2 3 2 I O O I 3 6 7 6310 014 10 16 19 16 10 4 i o etc. | etc. Thus when the one player is 2 up and 4 to play, the game may be drawn in 10 different ways, and hence the number of distinct ways in which such a game may end is 26 + 5 + 10 = 41 . These schemes were expressed by Tait in a formula based upon the expansion of the expression (a + i + i/a) raised to the power y. In a brief paper on a Fundamental Principle in Statics, communicated to the Royal Society on Dec. 21, 1874, Tait compared in a remarkably simple manner the gravitational attraction between the two hemispheres of the earth and the tendency to split across the diametral plane separating these in con- sequence of the earth's rotation. He thus proved that it was gravitation and not cohesion which kept the material of the earth together. A planet of the earth's mean density and of tensile strength equal to that of steel would be held together as much by cohesion as by gravitation if its radius were 409 miles. I believe this must be the result referred to by Kelvin in a short letter to Tait, which was written from White's workshop in Glasgow, but of which the date unfortunately had been torn off. It runs Dear T' I thought as much. It is not the thing I object to but your PFian way of doing it. However enough of that. I still think your planet the greatest step in dynamics made in the second half of the i Qth century I am up to see new electrometers but find them too unfinished. Yours T. VORTICES AND KNOTS 105 Not able to understand the reference to the planet I sent the note to Kelvin himself, who, writing on Oct. 3, 1907, said " I return my old pencilled letter to Tait, which has come to me enclosed with yours of yesterday. I have no recollection of the wonderful planet. " PFian meant Pecksniffian. Pecksniff was a great hero of Tait's in respect to his almost superhuman selfishness, cunning, and hypocrisy, splendidly depicted by Dickens." The only other planetary theorem with which Tait's name is associated is the one already referred to in connection with Action and Brachistochrones ; but this comparison between the effects of cohesion and gravitation when first made was just the kind of thing to appeal to Thomson. Tait's excursions into the field of pure mathematics were not frequent ; and his paper on the Linear Differential Equation of the Second Order (Jan. 3, 1876) practically stands alone. It contains some curious results and suggests several lines of further research. The general idea of the paper is to compare the results of various processes employed to reduce the general linear differential equation of the second order to a non-linear equation of the first order. The properties of the operators of the form ( x } are \8;tr ar/ incidentally considered, and the question is asked as to the evaluation at one step of the integral At the British Association Meeting of 1876, Tait communicated a note on some elementary properties of closed plane curves, especially with regard to the double points, crossings, or intersections. He pointed out the connection of the subject with the theory of knots, on which he was now about to begin a long and fruitful discussion. He was attracted to a study of knots by the problem of the stability of knotted vortex rings such as one might imagine to constitute different types of vortex atoms. Some of these were figured in Kelvin's great paper, which itself was the outcome of Tait's own experimental illustrations of Helmholtz's theorems of vortex motion. The conception of the vortex atom gave an extraordinary impulse to the study of vortex motion, and the following early letter of Maxwell indicates some of the lines of research ultimately prosecuted by Thomson and Tait. T. 14 106 PETER GUTHRIE TAIT GLENLAIR, DALBEATTIE, Nov. 13, 1867. Dear Tait If you have any spare copies of your translation of Helmholtz on " Water Twists " I should be obliged if you could send me one. I set the Helmholtz dogma to the Senate House in '66, and got it very nearly done by some men, completely as to the calculation, nearly as to the interpretation. Thomson has set himself to spin the chains of destiny out of a fluid plenum as M. Scott set an eminent person to spin ropes from the sea sand, and I saw you had put your calculus in it too. May you both prosper and disentangle your formulae in proportion as you entangle your worbles. But I fear that the simplest indivisible whirl is either two embracing worbles or a worble embracing itself. For a simple closed worble may be easily split and the parts separated but two embracing worbles preserve each other's solidarity thus though each may split into many, every one of the one set must embrace every one of the other. So does a knotted one. yours truly J. CLERK MAXWELL. Here Maxwell expressed very clearly one of the ideas which Tait finally made the starting point of his discussion of knots. The trefoil knot, the simplest of all knots, was chosen by Balfour Stewart and Tait as a symbolic monogram on the title page of the Unseen Universe ; and some of the speculations put forward in that work must have been closely connected with the line of thought which found a scientific development in Tail's later papers. It may have been while thinking out the attributes of vortex atoms in an almost frictionless fluid that Tait came to see there was a mathematical problem to attack in regard to the forms of knotted vortex rings. If we take a cord or, better still, a long piece of rubber tubing, twist it round itself in and out in any kind of arbitrary fashion, then join its ends so as to make a closed loop with a number of interlacings on it, we get a vortex ORDERS OF KNOTTINESS 107 knot. We may suppose it drawn out and flattened until the crossings have been well separated and reduced to the lowest possible number. Projected on the plane this will appear as a closed curve with a certain number of double points. Hence the fundamental mathematical problem may be thus stated : Given the number of its double points, find all the essentially different forms which a closed continuous curve can assume. Beginning at any point of the curve and going round it continuously we pass in succession through all the double points in a certain order. Every point of intersection must be gone through twice, the one crossing (in the case of the knot) being along the branch which passes above, the other along the branch which passes below. If we lay down a haphazard set of points and try to pass through them continuously in the way described, we shall soon find that only certain modes are possible. The problem is to find those modes for any given number of crossings. Let us begin to pass the point A by the over-crossing branch. We shall evidently pass the second point by an under-crossing branch, the third by an over- crossing again, and so on. Calling the first, third, fifth, etc., by the letters A, B, C, etc., we find that after we have exhausted all the intersections the even number crossings will be represented by the same letters interpolated in a certain order. To fulfil the conditions of a real knot, it is clear that neither A nor B can occupy the second place, neither B nor C the fourth, and so on. This at once suggests the purely mathematical problem : How many arrangements are there of n letters when a particular one cannot be in the first or second place, nor another particular one in the third or fourth, nor a third particular one in the fifth or sixth, and so on. Cayley and Thomas Muir both supplied Tait with a purely mathematical solution of this problem ; but even when that is done, there still remain many arrangements which will not form knots, and others which while forming knots are repetitions of forms already obtained. These remarks will give an idea of the difficulties attending the taking of a census of the knots, say, of nine or ten intersections what Tait called knots of nine-fold and ten-fold knottiness. If we take a piece of rubber tubing plaited and then closed in the way suggested above, we shall be surprised at the many apparently different forms a given knot may take by simple deformations. Conversely, what appear to the eye to be different arrangements, become on closer inspection Proteus-like forms of the same. While engaged in this research, Tait came into touch with the Rev. T. P. Kirkman, a mathematician of marked originality, and one of the pioneers in the theory of Groups. Kirkman's intimate knowledge of the properties of polyhedra 142 io8 PETER GUTHRIE TAIT suggested to him a mode of attack on knots quite distinct from that developed by Tait. Taking advantage of Kirkman's extension of the census to knots of eight-fold and nine-fold knottiness, Tait was able to give in his second paper (1884) all the forms of knots of the first seven orders of knottiness, the numbers being as follows : Order of knottiness 34567 8 9 Number of forms i i 2 4 8 21 47 A year later in his third paper Tait, basing his enumeration on Kirkman's polyhedral method of taking the census, figured the 1 23 different forms of ten-fold knottiness. Higher orders have been treated by Kirkman and Little (Trans. R. S. E. Vols. xxxn, xxxv, xxxvi, xxxix). In his second paper Tait pointed out that with the first seven orders of knottiness we have forms enough to supply all the elements with appropriate vortex atoms. A curious problem in arrangements suggested by the investigations in the properties of knots was thus enunciated by Tait : " A Schoolmaster went mad, and amused himself by arranging the boys. He turned the dux boy down one place, the new dux two places, the next three, and so on until every boy's place had been altered at least once. Then he began again, and so on ; till, after 306 turnings down all the boys got back to their original places. This disgusted him, and he kicked one boy out. Then he was amazed to find that he had to operate 1120 times before all got back to their original places. How many boys were in the class?" The answer is 18 (see Proc. R. S. E. Jan. 5, 1880; Sci. Pap. Vol. i, p. 402). In his discussion of knots Tait established a new vocabulary and gave precise meanings to such terms as knottiness, beknottedness, plait, link, lock, etc. He introduced with effect the old Scottish word "flype" which has no equivalent in southern English speech, the nearest being " turn-out- side-in." Clerk Maxwell has described some of Tail's processes in the following rhymes : (CATS) CRADLE SONG. By a Babe in Knots. Peter the Repeater Platted round a platter Slips of silvered paper Basting them with batter. MIRAGE 109 Flype 'em, slit 'em, twist 'em, Lop-looped laps of paper ; Setting out the system By the bones of Neper. Clear your coil of kinkings Into perfect plaiting, Locking loops and linkings I nterpenetrating. Why should a man benighted, Beduped, befooled, besotted, Call knotful knittings plighted, Not knotty but beknotted? It's monstrous, horrid, shocking, Beyond the power of thinking, Not to know, interlocking Is no mere form of linking. But little Jacky Homer Will teach you what is proper, So pitch him, in his corner, Your silver and your copper. One of Tait's most beautiful self-contained papers is his paper on Mirage (1881), published in the Transactions of the R. S. E. (Sci. Pap. Vol. i, No. LVIII). It is worked out as an example of Hamilton's general method in optics. Not only is it an elegant piece of mathematics, but it shows to advantage the clearness of Tait's physical intuition in his assumption of a practically possible vertical distribution of temperature and density capable of explaining all the observed phenomena. A less technical account of the paper on Mirage was published in Nature (Vol. xxvm, May 24, 1883) under the title " State of the Atmosphere which produces the forms of Mirage observed by Vince and by Scoresby." This article is printed below. In 1886 Tait's attention was strongly drawn to the foundations of the Kinetic Theory of Gases, on which subject he communicated four memoirs to the Transactions of the Royal Society of Edinburgh and a fifth (in abstract) to the Proceedings within the six succeeding years. His first aim, as indicated in the title, was to establish sure and strong the fundamental statistical propositions in the distribution of speeds and energy among a great many small smooth spheres subject only to their mutual collisions ; and the one initial point aimed at was a rigorous proof of Maxwell's theorem of the equal partition of energy. An interesting question carefully considered by no PETER GUTHRIE TAIT Tait was how to define the Mean Free Path, in regard to which he differed from Maxwell. He also laid stress on the principle that throughout the investigation each step of the process of averaging should not be performed before the expressions were ripe for it. Some of his views are put very succinctly in a letter to Thomson in 1888, just about the time he was printing the third paper of the series. We may regard it as containing Tail's last statement on the question. 38 GEORGE SQUARE, EDINBURGH, 27/2/88. O. T. Ponder every word of this and report. , Since there is absolute social equality in the community called a simple gas, the average behaviour of any one particle during 3.IO 20 seconds is the same as that of 3.IO 20 particles (the content of a cubic inch) for one second. Hence if be the chance that the speed is from v to v + dv, and if p v be then the mean free path; and if C be the number of collisions in 3.IO 20 seconds, we have n v C as the number of collisions in which the speed is v to v + dv, and the path /. Thus the whole space travelled over in 3.10" seconds (io 13 years nearly) is C2 (/). This consists of C separate pieces. The average of these, i.e. the Mean Free Path, is therefore 2 (a. A) .......................................... (0- Also the interval between two collisions, when the speed is v, is p v jv. Hence the whole time spent on C collisions is C ( ) . This is 3.10" seconds. Thus the average number of collisions per particle per second is Both of these results differ from those now universally accepted. Instead of (i) they, Maxwell, Meyer, Boltzmann etc., give 2 ( v) * A) and instead of (2) Both are, I think, obviously wrong. Yrs. KINETIC THEORY OF GASES m There is no record what reply Thomson made to this very clear statement. Having established the fundamental propositions in the first paper Tait proceeded in his later papers to develop the subject in its application to viscosity, thermal conduction, diffusion, the virial, and the isothermal equations. Certain strictures which Tait in his fourth paper applied to Van der Waals' method of evolving his well-known isothermal equation led to a discussion with Lord Rayleigh and Professor Korteweg (see Nature, Vols. XLIV, XLV, 1891-92). While accepting their explanations of Van der Waals' process he was not convinced that the process was valid in the sense of being a logical following out of the virial equation. On November 23, 1893, Tait reviewed in Nature (Vol. XLIX) the second edition of Dr Watson's Treatise on the Kinetic Theory of Gases : and the following paragraphs give very clearly his own view of the significance and aim of his papers on the subject : "I believe that I gave, in 1886 (Trans. /?. 6". E. Vol. XXXIII), the first (and possibly even now the sole) thoroughly legitimate, and at least approximately complete, demonstration of what is known as Clerk-Maxwell's Theorem, relating to the ultimate partition of energy between or among two or more sets of hard, smooth, and perfectly elastic spherical particles. And I then pointed out, in considerable detail, the logical deficiencies or contradictions which vitiated Maxwell's own proof of 1859, as well as those involved in the mode of demonstration which he subse- quently adopted from Boltzmann. Dr Boltzmann entered, at the time, on an elaborate defence of his position ; but he did not, in my opinion, satisfactorily dispose of the objections I had raised. Of course I am fully aware how very much easier it is for one to discover flaws in another man's logic than in his own, and how unprepared he usually is to acknowledge his own defects of logic even when they are pointed out to him. But the only attacks which, so far as I know, have been made on my investigation, were easily shown to be due to misconception of some of the terms or processes employed " From the experimental point of view, the first great objection to Boltzmann's Theorem is furnished by the measured specific heats of gases ; and Dr Watson's concluding paragraphs are devoted to an attempt to explain away the formidable apparent inconsistency between theory and experiment. In particular he refers to a little calculation, which I made in 1886 to show the grounds for our confidence in the elementary principles of the theory. This was subsequently verified by Natanson (Wied. Ann. 1888) and Burbury (Phil. Trans. 1892). Its main feature is its pointing out the absolutely astounding rapidity with which the average amounts of energy per particle in each of two sets of spheres in a uniform mixture approach to equality in consequence of mutual impacts. Thus it placed in a very clear light the difficulty of accepting Boltzmann's Theorem, if the degrees of freedom of a complex molecule at all resemble those of an ordinary dynamical system." ii2 PETER GUTHRIE TAIT The calculation referred to here was given in the first paper as Part v, the earlier parts being concerned with the mean free path, the number of collisions, and the general proof of Maxwell's theorem. Part vi is devoted to the discussion of some definite integrals, and the remaining three parts of the first paper take up the question of the mean free path in a mixture of two systems, the pressure in a system of colliding spheres, and the effect of external potential. In the second paper Tait proceeded to apply the results of the first paper " to the question of the transference of momentum, of energy, and of matter, in a gas or gaseous mixture ; still, however, on the hypothesis of hard spherical particles, exerting no mutual forces except those of impact." Before entering on this line of investigation, Tait took occasion to answer certain criticisms which had been made^>f his methods in the first paper, especially in regard to the number of assumptions necessary for the proof of Maxwell's theorem concerning the distribution of energy in a mixture of a gas. Tait contended however that all he demanded was " that there is free access for collision between each pair of particles, whether of the same kind or of different systems ; and that the number of particles of one kind is not overwhelmingly greater than that of the other." In the third paper, a special case of molecular attraction is dealt with. The particles which are under molecular force are assumed to have a greater average kinetic energy than the rest. In terms of this assumption the expression for the virial is developed in the fourth paper, leading finally to Tail's form of the isothermal equation C A-eE v + y v+a where C, A, e, y, a are constants, and E is a quantity which in the case of vapour or gas of small density has the value ^2,mu*, where u is the speed of the particle of mass m. This average kinetic energy is generally assumed to be proportional to the absolute temperature ; but Tait had grave reasons for not accepting this view. He said : " It appears to me that only if E above (with a constant added when required, as will presently be shown) is regarded as proportional to the absolute temperature, can the above equation be in any sense adequately considered as that of an Iso- thermal. If the whole kinetic energy of the particles is treated as proportional to the absolute temperature, the various stages of the gas as its volume changes with E constant correspond to changes of temperature without direct loss or gain of heat, and belong rather to a species of Adiabatic than to an Isothermal. Neither Van der Waals nor Clausius, so far as I can see, calls attention to the fact that when ISOTHERMAL EQUATION 113 there are molecular forces the mean-square speed of the particles necessarily increases with diminution of volume, even when the mean-square speed of a free particle is maintained unaltered ; and this simply because the time during which each particle is free is a smaller fraction of the whole time. But when the whole kinetic energy is treated as constant (as it must be in an Isothermal, when that energy is taken as measuring the absolute temperature), it is clear that isothermal compression must reduce the value of E ____ "For the isothermal formation of liquid, heat must in all cases be taken from the group M. This must have the effect of diminishing the value of E. Hence, in a liquid, the temperature is no longer measured by E, but by E + c, where c is a quantity whose value steadily increases, as the temperature is lowered, from the value zero at the critical point..." Fritting then E = Rt, where / is the absolute temperature, Tait intro- duced the pressure temperature and volume at the critical point, and threw his equation into the form where the barred letters refer to the critical values. He compared this with the corresponding equations of Van der Waals and Clausius and pointed out that, although they all three agreed in form for the critical isothermal, they could not do so for any other. He then found, by direct calculations from Amagat's results for Carbon Dioxide, that the pressures obtained by his formula for given volumes at the critical temperature agree almost perfectly with the measured pressures, between a range of volume from i to 0*003 5. This practically finishes the series of papers on the Foundations of the Kinetic Theory of Gases ; for the fifth instalment was printed only in abstract and indicates lines of investigation which were never completed. For five full years Tait occupied his mind with these researches ; and if we except his quaternion work there is no other line of investigation which made such serious demands upon both his mathematical powers and his physical intuitions. Throughout the whole series he is essentially the natural philosopher, using mathematics for the elucidation of what might be called the metaphysics of molecular actions. No writer on the subject has put more clearly the assumptions on which the statistical investigation is based ; and apparently he was the first to calculate the rate at which under given conditions the "special state" is restored when disturbed. His abhorrence of long and intricate mathematical operations is strongly expressed more than once. He was convinced of the general accuracy of Maxwell's T. 15 U4 PETER GUTHRIE TAIT conclusions ; but he could not admit the validity of all his demonstrations. If we may judge from a letter written to him by Maxwell as early as August 1873, Tait had been seeking enlightenment years before he himself thought of tackling the problem. Maxwell's letter consists of a set of numbered paragraphs, i, 3, 7, 5, evidently in answer to a set of corresponding questions put by Tait. Paragraph (5) runs thus : " By the study of Boltzmann I have been unable to understand him. He could not understand me on account of my shortness, and his length was and is an equal stumbling-block to me. Hence I am very much inclined to join the glorious company of supplanters and to put the whole business in about six lines." Maxwell then gave the conclusion of his paper on the Final State of a System of Molecules in motion subject to forces of any kind (Nature, Vol. vin, 1873: Scientific Papers, Vol. n, pp. 351-4) and continued: " In thermal language Temperature uniform in spite of crowding to one side by forces. Molecular volume of all gases equal. Equilibrium of mixed gases follows Dalton's Law of each gas acting as vacuum to the rest (in fact it acts as vacuum to itself also). In my former treatise I got these results only by way of conclusions. Now they come out before any assumption is made as to the law of action between molecules." A few months later (Dec. i, 1873) Maxwell returned to the subject evidently in reply again to Tait. This letter of Maxwell's touches upon a great variety of points, all in reference to Tail's varied activities at the time ; and it seems better to give the letter here as a whole with footnote eluci- dations than to break it up into bits distributed throughout the volume. Natural Sciences Tripos, i Dec. 1873. O T'. For the flow of a liquid in a tube 1 , axis z dp = Surface condition fj,-^- = \w .................................... (2), where v is the normal drawn towards the liquid. When the curvature is small, (2) is equivalent to supposing the walls to be removed back by /t/X and then X made oo or w = o. For glass and water by Helmholtz and Pietrowski /*/X = o. If so, and if the value of w is C(i x^/a? ~l + 7i)+p =o > which gives C. 1 See Tait's Laboratory Notes (Proc. J?. S. E. vui, p. 208) : On the Flow of Water through fine Tubes. The experiments were made by C. Michie Smith and myself with tubes of circular and elliptic bore. Tait had asked Maxwell to give him the theory of the phenomenon as a problem in viscosity. LETTER FROM MAXWELL 115 If not, you may write w = A + Br* + C*r* cos 2 + C t r* cos 4 + etc., where x = ar cos and y = br sin 6 and then and you satisfy (2) the best way you can when r= i. As to Ampere of course you may lay on d l (anything) where d^ is with respect to the element of a circuit. Have you studied H" on the potential ' of two elements? or Bertrand? who, with original bosh of his own rushes against the thicker bosches of H*'s buckler and says that H 8 believes in a force which does not diminish, with the distance, so that the reason why Ampere or H" or Bertrand observe* peculiar effects is because some philosopher in a Centauri happens to be completing a circuit. XQq D [tails]' as I am surrounded by Naturals and cannot give references. In introducing 4nions s do so by blast of trumpet and tuck of drum. Why should V. a/3y come in sneaking without having his style and titles proclaimed by a fugleman ? Why even . should be treated with due respect and we should be informed whether he is attractive or repulsive. What do you think of " Space-variation " as the name for Nabla ? It is only lately under the conduct of Professor Willard Gibbs that I have been led to recant an error which I had imbibed from your 6k.cs, namely that the entropy of Clausius is unavailable energy, while that of T' is available energy*. The entropy of Clausius is neither one nor the other. It is only Rankine's Thermo- dynamic Function.... I have also a great respect for the elder of those celebrated acrobats, Virial and Ergal, the Bounding Brothers of Bonn. Virial came out in my paper on Frames, R. S. E. 1870 in the form 2Rr = o, when there is no motion. When there is motion the time average of $2,Rr = time average of ^Mv\ where R is positive for attraction. But it is rare sport to see those learned Germans contending for the priority of the discovery that the 2nd law of O&cs is the Hamiltonsche Princip, when all the time they assume that the temperature of a body is but another name for the vis viva of one of its molecules, a thing which was suggested by the labours of Gay 1 The reference is to H(ermann) H(elmholtz)'s electrodynamic investigation which supplied the true criterion in place of the hasty generalisation of 385 in the first edition of Thomson and Tait. * The [tails] are drawn as arrow-headed wiggles of various lengths and forms. 1 See the chapter on Quaternions for other remarks by Maxwell on Tail's quaternion work. Maxwell was reading Kelland and Tail's Introduction to Quaternions which he reviewed in Nature shortly after. 4 Tait suggested in the first edition of his Thermodynamics (contracted into Obcs by Maxwell) that the word Entropy should be used in this sense. In the second edition he went back to the original meaning as given by Clausius. 152 n6 PETER GUTHRIE TAIT Lussac, Dulong, etc., but first deduced from dynamical statistical considerations by H?. The Hamiltonsche Princip, the while, soars along in a region unvexed by statistical considerations, while the German Icari flap their waxen wings in nephelo- coccygia amid those cloudy forms which the ignorance and finitude of human science have invested with the incommunicable attributes of the invisible Queen of Heaven.... General [quaternion] exercise. Interpret every 4nion expression in literary geo- r\ metrical language, e.g., express in neat set terms the result of - . 7. 8 dp df There is a close association between these remarks by Maxwell in 1873 and some of Tail's own comments in his Kinetic Theory papers published thirteen years later. In 1896, in a note on Clerk Maxwell's Law of Distribution of Velocity in a Group of equal colliding Spheres (Proc. R. S. E. Vol. xxi), Tait published his last views on the subject. He repelled certain criticisms of Maxwell's solution brought forward by Bertrand in the Comptes Rendus of that year. Bertrand's enunciation of what he conceived to be the problem attacked by Maxwell, and the enunciation of the problem really attacked, were set side by side ; and Bertrand was condemned out of his own mouth. At the same time Tait strengthened the experimental foundations of the argument that the solution of the problem is unique and cannot be destroyed by collisions, by an application of Doppler's principle to the radiations of a gas. The results of Tait's investigations into the flight of a golf ball have already been detailed (Chap, i, p. 27). A brief sketch of the mathematical method by which he deduced his results is appropriately given here. Tait published two papers on the Path of a Rotating Spherical Projectile, the first in 1893, the second in 1896 (Trans. R. S. E. Vols. xxxvn, xxxix). The foundation of the theory was the assumption that, in virtue of the combination of a linear speed v and a rotation &) about a given axis, the ball is acted on by a force proportional to the product of the speed and the rotation, and perpendicular both to the line of flight and to the axis of rotation. This transverse force acts in addition to the retarding force due to the resistance of the air ; and the first problem solved by Tait was the case in which no other than these two forces act. It is easy to show that under the influence of such forces the sphere will move in a spiral whose curvature will be inversely as the speed of translation and whose tangent will rotate with a constant angular velocity. The projection on the horizontal plane of the GOLF BALL TRAJECTORY, ETC. 117 path of a pulled or sliced golf ball will be very approximately portions of this spiral. The introduction of gravity acting constantly in one direction greatly complicates the problem, which cannot be solved, even to a first approximation, except on the supposition that the path nowhere deviates greatly from the horizontal. To obtain forms of paths at all like those observed, somewhat lengthy numerical calculations require to be made. The method by which Tait builds up the curve is very instructive and is a good example of his insight into the essence of a physical problem and of his capacity in working out a sufficient solution. The practical details will be found in the article on Long Driving reprinted below. In addition to the greater efforts of his mathematical powers, Tait contributed to the Messenger of Mathematics, to the Proceedings of the Royal Society of Edinburgh, and latterly to the Proceedings of the Edinburgh Mathematical Society, a variety of small notes, many of which he incorporated in the successive editions of his books. These notes were always interesting in themselves and frequently presented old truths from new points of view. In not a few of them his skill as a geometrician comes strongly into evidence. Tait, in fact, was no juggler with symbols ; and when taking up a new subject he invariably tried to make of it a geometrically tangible creation ; otherwise he would have none of it. Maxwell expressed this view of Tail's mental habitude in a letter in which, replying evidently to a demand of Tail's to consider a problem in conduction of heat, he wrote : " O T' If a man will not read Lam6 how should he know whether a given thing is v? Again, if a man throws in several triads of symbols and jumbles them up, pretending all the while that he has never heard of geometry, will not the broth be thick and slab ? If the problem is to be solved in this way by mere heckling of equations through ither 1 I doubt if you are the man for it as I observe that you always get on best when you let yourself and the public know what you are about." Of those casual things which Tait threw off largely as mathematical recreations, about a dozen were communicated to the Edinburgh Mathematical Society. The subjects treated of are nearly as numerous as the papers, including plane strains, summations of series, orthogonal systems of curves, 1 " Through ither," an expressive Scottish phrase, meaning lack of method so that things get tangled up one with the other higgledy-piggledy comes near it. It is often used with reference to a thriftless housewife who has no method but drives through her work anyhow. "Heckling of equations through ither" means assorting the equations in a random manner in the hope that they will be disentangled and simplified. n8 PETER GUTHRIE TAIT circles of curvature, attractions, centrobaric distributions, logarithms, etc. The note on centrobaric distributions he afterwards simplified and extended in his booklet on Newton's Laws of Motion, and gave a remarkably simple geometrical proof that the potential of a uniform spherical shell is constant throughout the interior, and varies for external points inversely as the distance from the centre. The last published paper not connected with quaternions was on a generalization of Josephus' problem (1898, Proc, R, S. E. Vol. xxn). The original problem stated simply is to arrange 41 persons in a circle in such a way that when every third person beginning at a particular position is counted out, a certain named one will be left. What position relatively to the first one counted will he occupy ? It is said that by this means Josephus saved his life and that of a companion out of a company who had resolved to kill themselves so as not to fall into the hands of the enemy. Josephus is said to have put himself in the 3131 place and his friend in the i6th place. Tail's generalization consists in pointing out that, if we know the position of "safety" for any one number, we can without going through the labour of the obvious sifting-out process at once say where the position of " safety " will be if the number is increased by one. This position is simply pushed forward by as many places as there are in the grouping by which the successive individuals are picked out. By successive application of the process, Tait quickly found that if every third man is picked out of a ring of 1,771,653 men, the one who is left last is the occupier of place 2 in the original arrangement. Hence if there were 2,000,000 in the circle the place to be assigned to the last one left after the knocking out by threes is evidently 2 + 3 x (2,000,000 - 1,77 1,653) = 2 + 3 x 228,347 = 2 + 685,041 = 685,043. When the number reaches 2,657,479 a new cycle will begin with the place of safety in position i. The general rule given by Tait is: " Let n men be arranged in a ring which closes up its ranks as each individual is picked out. Beginning anywhere, go continuously round, picking out each ftith man until r only are left. Let one of these be the man who originally occupied the /th place. Then if we had begun with + i men one of the r left would have been originally the (p + *)th, or (if p + m>n + i) the (/> + -- i)th." CHAPTER IV QUATERNIONS TAIT'S quaternion work was unique ; and his influence in the development of the calculus was second only to that of the great originator himself. He alone of all Hamilton's contemporaries seems to have been able to grasp the real significance of the method by direct perusal of Hamilton's Lectures. The extraordinary seventh " Lecture " bristled with novelties and difficulties. In grappling with these in his later Cambridge days Tait saw the value of quaternions as an instrument of research. But it was not till he was settled in Belfast that he began to make headway. On August ii, 1858, Dr Andrews wrote Hamilton a note introducing his young mathematical colleague as one who " had been directing his attention of late to Quaternions, and is anxious to be allowed to correspond with you on that subject" In a cordial response to this letter Hamilton speaks of having recently turned his attention to " differential equations and definite integrals in connection with old but revived researches of my own (I do not mean, just now, those which Jacobi has enriched by his comments)." He enclosed, no doubt to test the powers of his would-be correspondent, a number of questions, some of which Tait answered in his second letter of August 20. The first letter, of date August 19, must ever be regarded as of great historic importance. It began a remarkable correspondence, which brought Hamilton himself back to the study and further development of the subject, culminating finally in the production of both Hamilton's Elements and Tail's Elementary Treatise, After thanking Hamilton for the very kind manner in which he had responded to Andrews' request, Tait continued : I attacked your volume on Quaternions immediately on its appearance, and easily mastered the first 6 lectures but the portions I was most desirous of under- standing, viz. the physical applications of the method, have given me very considerable trouble; and, but for your offered assistance, I am afraid I should have had to relinquish all hopes of using Quaternions as an instrument in investigation, on 120 PETER GUTHRIE TAIT account of the time I should have had to spend in acquiring a sufficient knowledge of them. I have all along preferred mixed, to pure, mathematics, and since I left Cambridge, where the former are little attended to, have been busy at the Theories of Heat, Electricity, etc. Your remarkable formula for = \- = \- ^- as the square da? dy* dz* of a vector form, and various analogous ones with quaternion operators, appear to me to offer the very instrument I see*, for some general investigations in Potentials, and it is therefore almost entirely on the subject of Differentials of Quaternions that I shall trespass on your kindness.... The correspondence thus begun continued week by week with wonderful continuity until July 1859, when Hamilton began to print the Elements. The successive letters were numbered (Hamilton's in Roman, and Tail's in Indian, numerals) and copies kept by the writers themselves, so that there might be no difficulty in referring to questions raised by either at all stages of the correspondence. In his letter of August 20, 1858, Tait mentioned particularly certain difficulties : Perhaps it is only due to the novelty of the subject, but I have felt at several points that the otherwise known result was (perhaps not necessary but at all events) very desirable, in suggesting the transformation suitable for its proof. As instances I may mention * found in Art. 474 of your Lectures for the value of p* + 4 (t )* Sip Step, and the transformation of the Tractor function for the 2nd integration of the equation of motion of a planet.... Again in Art. 591 I cannot see how you infer that v is a normal vector when the equation to a surface is put in the form Svdp = o, Tdp not being indefinitely small, because it seems to me that in such a case v is a vector perpendicular to the chord dp. It was in reply to Tail's difficulties regarding the notion of finite differentials that Hamilton wrote the long letter v, which might have been a chapter in a treatise on the fundamental conception of the fluxion or differential method. Hamilton subsequently gave the argument clearly in his second treatise, the Elements of Quaternions, developing the whole discussion from the definition : Simultaneous Differentials (or Corresponding Fluxions) are limits of equimultiples of simultaneous and decreasing Differences. In this remarkable letter (dated October n to October 16, 1858) which occupies 45 closely written pages of large-sized note paper, and is subdivided HAMILTON ON DIFFERENTIALS 121 into 32 paragraphs, Hamilton began by comparing himself to the fox in Chaucer's story, The Nonne Prest, his Tale, and quoted : " But, Sire, I did it in no wick(ed) entent : Com doun, and I schal telle you what I ment." " But," continued Hamilton, " it is time to make a prodigious, a mortal leap, and to pass from Chaucer to Moigno. By the way did you ever meet the Abbd ? ' a little, round, fat, oily man of God ' who has however been sometimes called, in Paris, 'le diable de M. Cauchy.' "(2) Your name was familiar to me, before Dr Andrews was so good as to propose that we should have some personal acquaintance with each other. But I regret (and perhaps ought to be ashamed) to say, that as yet I have not had an opportunity of reading any of your works. However from the specimen sheet which you sent me, along with your first letter, of a book of yours on analytical mechanics, & in which you did me the honour to introduce the subject of the Hodograph, I collect that you consider it judicious, at least (if not absolutely necessary) in instruction, to use differential coefficients only & to exclude differentials themselves. And perhaps you may have adopted, even publicly as Airy has done, using the (to me) uncouth notation / fl ( ) for /( )d6 the system which rejects differentials. If so, I can only plead that I am not intentionally, nor knowingly, controverting anything which you have published. And if I now quote Moigno, it is merely to show that / am not wishing to be singular." Moigno's book from which Hamilton quoted with criticisms and comments was published in 1840; but before the letter was finished Hamilton's copy of Cauchy's Lefons sur le Calcul diffdrentiel (1829) was discovered "buried under masses of papers " in a corner of his library. There (as he expected) he found the inspiration of Moigno's views without Moigno's mistakes. Cauchy is then quoted and shown to treat throughout of differentials, and only in a secondary sense of differential coefficients ; and not only so, but Cauchy's differentials may have any arbitrary values and are not essentially infinitesimal. Then followed what must have delighted the heart of Tait. "(29) Although it was, perhaps, allowed to suppose that you might not have access to Cauchy's Leqons sur le Calcul diffe'rentiel (1829), which may be out of print, and even that Moigno (1840) might not be in your hands, I must not presume to imagine that a Cambridge man can possibly be unacquainted with the Principia. It may, however, be just permitted to remind you, that in the Lemmas VII, VIII IX of the ist Book, Newton's ' intelligantur (or intelligatur) semper ad puncta longinqua produci,' as also his ' recta semper finita ' in Lemma vil, and his 'triangula tria semper finita' of Lemma vm, are conceptions, to which the process of construction proposed in paragraph (16) of the present Letter appears to have much analogy. And in that famous Second Lemma of the Second Book, which is stated by himself, in his appended Scholium, to contain the foundation of his Method of T. 16 122 PETER GUTHRIE TAIT Fluxions (' methodi hujus generalis fundamentum continetur in lemmate praecedenti ') Newton expressly says...' Neque enim spectator in hoc lemmate magnitude momen- torum, sed prima nascentium proportio. Eodem recidit si loco momentorum usurpentur vel velocitates incrementorum ac decrementorum (quas etiam motus, mutationes et fluxiones quantitatum ^ominare licet) vel finitae quaevis quantitates velocitatibus hisce proportionales.' The finite differentials of Cauchy & myself, & doubtless of other moderns, are therefore really the fluxions of Newton in disguise ; and I ought to talk, or at least might talk, of fluxions of quaternions, and of their functions. "(30) Before I was 17 years old, I had diligently studied at least the three first sections of the ist Book of the Principia..,.'B\A. I think it was about that age, that I was carried away by the attractions of the French School, & specially by that of Lagrange. The Calcul des Fonctions charmed me, & for several years I supposed it to be, not merely an elegant and original production of a genius, whose mathematics almost sublimed themselves into poetry, but a sound and sufficient basis for the superstructure of the Differential Calculus.... But you may possibly be aware that it is now a long time since I pointed out a fatal defect in the foundation of Lagrange's theory, as set forth in the Calcul des Fonctions. ...... I suppose that no one now contests the necessity of founding the differential calculus on the notion of limits; at least, if it be desired that the structure should be a weather-proof and habitable house: or, in short, good for anything. In that respect, at least, though certainly not in the notation of fluxions, we are all glad to go back to Newton. "(31) To connect my definition more closely still with Newton's views, we have only to conceive that, if r = dq = A^, the quaternion function, fq, of the quaternion variable q, GROWS,... and passes, GRADUALLY, by such GROWTH, through the n I intermediate stages (of state, rather than of quantity) where n is a large positive whole number, until it ATTAINS at last the state /(*+?) =/<* + r > =/ (* + **> =fl + A / u a.~ e , where < is any vector function of the scalar u, a is the vector parallel to the axis of revolution and t is a second scalar variable. Letter vn contained an interesting historic note with reference to the dp quantity -^ : " I have lately observed that Mr Warren, of Cambridge, as long ago as 1828, in his Treatise on the Geom. Representation of the Square Roots of Negative Quantities,... gives, in his page 119, that very symbol -J- to represent a line which in length, and direction measures the velocity of a moving point My p has no necessary dependence on any sq. root of I, so long as we are merely using it to form such expressions as pi or -j- for the vector of velocity, ...or p" or -^ for the vector of acceleration; where p', p" are fairly entitled to be called ' derived functions' of t, of the 1st and 2nd orders, the primitive function being p." In letter 7 of date October 25, 1858, Tait wrote: I do not intend even today to enter upon the subject of differentials though I may state that I have re-read with great care your letter No. V, and have quite understood, and agreed with, it while at the same time I must confess that a good deal of it besides that referring more particularly to Quaternions was new to me. Towards the end of this letter Tait propounded the problem to find the envelope of the surface S*ap + 2Sa(3p = d" when Ta= i. He had given it incorrectly in a postscript to a previous letter, and Hamilton at once saw there must be some mistake. Tait, after making the correction, continued thus: The first equation represents I suppose a paraboloid and the second was intended (though I presume it is not explicit enough) to mean that a might be any 16 2 124 PETER GUTHRIE TAIT unit vector. The question had reference to the finding the locus of ultimate inter- sections of the series of paraboloids, a problem which arose out of an investigation I was lately making and which I felt was too much for me at the time but if you will permit me to withdraw it again for a little, I think I may perhaps manage it now. ^ The problem will be found solved in Tail's Quaternions, 321 (3rd edition), very much as Tait solved it in his letter 8. Hamilton was greatly taken with the question and discussed the geometry of the envelope at great length in his letters xi and xui. The envelope is a surface of revolution of the fourth degree having the quaternion equation 1 and this Hamilton proposed to call Tail's Surface. It is curious to note that the first solution sent by Hamilton to Tait did not agree with Tail's. By his first method of elimination, in fact, Hamilton introduced a " foreign factor" in the form of a sphere. In the very short letter xn he writes : " Your investigation would look much better in print than my own ; for you see that I take no pains, in this correspondence, to put any check on a natural tendency to diffuseness & scarcely ever copy from a draught, although the style of the composition would thereby be greatly improved, especially in the way of condensa- tion. "It takes, you know, -more pains to write a short than a long letter, or essay, on any subject : not that I pretend to have taken any pains with this short note ! but I must tell you, some time or other, of its once costing me half a quire of paper to write a note of one page to a lady who wanted my opinion on an astronomical manuscript of her own." Meanwhile along with the prolonged discussion of Tail's Surface in letter xui Hamilton was continuing his elucidation of the theory of differentials in letter x. After acknowledging receipt of parts of these letters on November 13, 1858, Tait continued in letter 10 in these words : For a week I have been hard at work trying to deduce the equation to Fresnel's wave-surface by a process purely quaternionic starting from the data employed by Archibald Smith in the Cam. Math. Journal. As yet I have only deduced the directions of the planes of polarization for any wave-front, and the law connecting the velocities of the two rays, and these come out with admirable simplicity. In attempting to find the equation to the surface I have come upon a terrible array of Versors. Of the latter I have still a sort of horror arising principally I suppose from my having avoided the use of them on any occasion on which it was possible. 1 The Cartesian equation is <*"(*"+/) = (*'+/ + THE WAVE SURFACE 125 Hamilton acknowledged the receipt of this letter by sending the first instalment of letter xiv. OBSERVATORY, Nov. \Tth. 1858. My dear Mr Tait Although X and XIII are still unfinished, not to mention IX, which is little more than begun, I am in a mood to commence now a new letter, of a perfectly miscellaneous nature, and free from the tyranny of any fixed idea. You tell me that you have been making progress with treatment of Fresnel's wave by Quaternions, but that you have not (or had not at the time of writing) completed the investigation. Whenever you have quite satisfied yourself with a result, or set of results, upon that subject, I should prefer you not immediately communicating such result, or results, to me ; because I should like to try, either to re- investigate the equation of the wave, or perhaps to hunt out an old investigation of it, in one of my manuscript books. The fairest, or at least the pleasantest course for both of us may therefore be, that we should agree upon some day and each of us on that day post a letter containing some of our separate results. This suggestion was warmly welcomed by Tait ; and in his letter 1 2 of date Nov. 29, 1858, the following reference was made to the agreement : You mentioned no day in particular for our exchanging results on the Wave Surface. I have (in a sense) completed my investigations but they are far from simple and I suspect strongly that there is some very elementary theorem of Trans- formation with which I am not acquainted which would immensely simplify them at once. I would therefore, to avoid knocking my head longer against eliminations which at present I find impracticable though I know they must be possible, request you to name as early a day as may be consistent with your perfect convenience, as you then may be able to tell me in a moment the reason of my imperfect success. At the close of letter x, Hamilton, writing on December i, fixed December 4 as the day for exchanging confidences on the Wave Surface. On that date accordingly Tait sent Hamilton his investigation along with the following letter : Q. C. BELFAST, $th Dec. 1858. My dear Sir William Hamilton I have to acknowledge the receipt of the rest of X with PS on two separate occasions, also of pp. 17 28 of XIV....I shall take an early opportunity of expressing my ideas with respect to V and X on the subject of finite differentials 1 meanwhile, as it is now late, I must explain as I best can the enclosed, which ' This expression of ideas seems never to have been given. Other and more important Quaternion developments had to be considered. 126 PETER GUTHRIE TAIT with all its deficiencies is the best I could make out of the subject before today when a new idea suggested itself that of avoiding the fearful eliminations which my method would seem to require in obtaining the equation of Fresnel's Wave Surface. The idea, which I have easjjy satisfied myself is correct, is to show that surfaces derived from reciprocal ellipsoids are themselves reciprocal. Meanwhile on December 3 Hamilton began his letter xv on the Wave Surface and dispatched the early sheets of it along with some pages of letter xiv, in which he acknowledged receipt of Tail's "...note No. 13 together with its very valuable enclosure of two sheets entitled ' Quaternion Proofs of some Theorems connected with the Wave Surface in Biaxal Crystals.'... I have read the first sheet of your Quaternion Proofs, and must say that they appear to me to be wonderfully elegant and to exhibit a very remarkable degree of mastery (so far) over the calculus of Quaternions, used as an instrument of expression and of investigation. " It would interest me much to know, whether (previous to our present cor- respondence) you had received ANY assistance from any other student of that calculus. Or did you learn all that you had acquired from the BOOK itself, combined (no doubt) with your own private exercises of various sorts? If the ' Lectures on Quaternions ' have been your ONLY teacher, I must consider the result of such a state of things to be not merely creditable to your own talents and diligence, but also complimentary to, and evidence of, some (scarcely hoped for) didactic capabilities of my volume ; which ought to tend to console me, under my artistic consciousness (as an author) of so many faults of execution, that if I could afford the expense of bringing out a New Edition I should be more likely to make it a New Work... My old friend John T. Graves called my attention about a year ago to a highly favourable, and very eloquent, article in the North American Review for July, 1857, on the subject of the Quaternions, and of my Book. But a conscientious Author wishes rather to be read, than to be praised, and therefore I should like to be informed, what drew your attention to my Book, and whether you had any personal assistance in studying it." To this request Tait replied in his letter 14 of date December 7, 1858 : With regard to my study of Quaternions I may affirm with some certainty that when I ordered your book, on account of an advertisement in the Athenaeum, I had NO IDEA what it was about. The startling title caught my eye in August '53, and as I was just going off to shooting quarters I took it and some scribbling paper with me to beguile the time However, as I told you in my first letter I got easily enough through the first six Lectures and I have still a good many notes I made at that time from which it now seems to me that I had not fully appreciated the simplicity of the method but had used quaternions generally in the shape EARLY EFFORTS IN QUATERNIONS 127 and treated i, j, k as imaginaries (like *J i) though of course according to their proper laws of combination. For fun I extract this Much of course could not have been made of this, and accordingly on my return to Cambridge I set to read other things, and to write my recently published Treatise on Particle Dynamics. The Theories of Heat, Electricity and Light have since occupied much of my spare time, and it was only in August last that I suddenly bethought me of certain formulae I had admired years ago at p. 610 of your Lectures and which I thought (and still think) likely to serve my purpose exactly. [The matter which more immediately suggested this to me was a paper of Helm- holtz's in Crelle's Journal (Vol. LV) which I was reading in July last as soon as we received it, and which put the subject of Potentials before me in a very clear light. The title (in German) I forget but an MS translation of my own which I have now beside me is headed "Vortex Motion 1 ." It refers to the integration of the general equations in Hydrodynamics, when udx + vdy + wdz is not a perfect differential.]... So far from having any assistance, save what you have so kindly given me, I am not even acquainted with any one who knows aught about quaternions (except Boole of Cork with whom however I have not exchanged a remark on the subject, and who, I suspect, looks on them in their analytical capacity only). So you see that, if there is any credit in my progress, it is entirely to your Lectures and Letters that it is due. Hamilton's letter xiv, which was begun on Nov. 17, and continued at fairly short but irregular intervals till Feb. 5, 1859, when it reached 88 closely written pages, ran on till April 3, in the form of eight postscripts. There seems to be no later reference to Tail's confession of how he began the study of Quaternions ; but various sections call for quotation because of the bearing they have on the subsequent history. In his letter 19, of date Jan. 3, 1859, Tait wrote as follows of Quaternions in general : About quaternions in general I may remark (as indeed I very frequently feel) that the processes are sometimes perplexingly easy by which I mean that one is often led in a step or two and without (at once) knowing it to the solution of what would be by ordinary methods a work not so much of difficulty as of labour. This however I take it must form one of its great excellencies in the hands of a person very well acquainted with it. A drawback to a beginner, but (as I am gradually being led to perceive) an immense advantage to one well skilled in the analysis, is the enormous variety of transformations of which even the simplest formulae are susceptible ; a variety fully justifying a remark of yours (Lectures Art. 504) which not many months ago used somewhat to puzzle me. If I had gained nothing more 1 The translation was published in Phil. Mag. July 1867. iz8 PETER GUTHRIE TAIT by reading this subject than the facility of making problems and transformations for Examination papers (especially in Trigonometry) and so saving an immense amount of time and trouble, I sh(61d have considered myself amply rewarded, but I hope in time to be able to apply it to perfectly original work (if anything can be quite original in these days).... In the portion of letter xiv which containing his reply to this letter from Tait Hamilton suggested publishing in the Philosophical Magazine his own investigations on the Wave Surface, and referred in particular to certain sections of his letter xv which might form the substance of this note. He said : "(54) It seems to me that some such sketch..., instead of forestalling your own communication, which appears likely to be of weight enough to deserve ampler space than the pages of a Magazine could afford, might, on the contrary, serve as a not ungraceful introduction to whatever you were disposed to publish afterwards. But let me know. ..what your FEELINGS in the matter are. I am quite aware that I can implicitly rely on your allowing me at least as much credit as you may be of opinion that I deserve ; and I think that you have really made the subject your own by your laborious and (so far as I yet know) successful investigations." To this Tait replied : Q. C. BELFAST, 7/1/59- My dear Sir William Hamilton Many thanks for your very kind letter containing XIV pp. 57-60 & xv pp. 93, 94, which I received this morning... I had been casting about as to how I should ask you to do the very thing you have just proposed as I have, as you will see when you look at the recent sheets of my Quat. Proofs, found one or two things which I believe were given by you for the first time but which I had either not received from you or not read until my own investigations were advanced beyond that point. For instance, I consider that I am not directly indebted to you for the quaternion form of the equation 1 to the wave in i, K, though of course you had it years before I knew of such a thing as quaternions at all. But then, knowing as I do the date of your discovery of that formula, I could not have published my own investigation without specially mentioning that you had communicated it to me, and the latter course it was impossible to follow, as I consider your letters private. You see then that I was in a difficulty and I should probably have tried at some other matter for a paper to publish, but for your last. I am delighted at the idea of being introduced to the Phil. Mag. (in which I have never written) in connection with "quaternions by you, especially when the subject as well as the 1 This is equation 13 in Tail's paper published in the Quarterly Journal of Mathematics, May 1859 (Set. Pap. Vol. i, page 7), namely, (K" - i*) = \S (f - K') p}' + ( Wp + TV'pf. COMPARISON OF NOTATIONS 129 method owes so much to you. But before venturing to publish under such auspices I must wait for your own opinion on my investigation itself which I think you may find interesting (though cumbrous) as I see on comparing the two it differs so much from yours ...... I am delighted that you intend to publish soon, and as I have already said you may make any mention you choose of our correspondence. The next day, Jan. 8, 1859, Tait continued in a letter which he called PS. to 20 : Having posted 20 this morning, and having a respite of a couple of hours while 3 men are at work preparing our ozone with an electrical machine, I have compared our methods of deducing the equation to the wave. Your ~* ( ) is the same as my ( _ ), or, as your 8/3 is my CT, and your u my = , all our equations can be at once compared by putting ~ 1 , or what might be written $~*( ) which will be what I denote by (__) or -aiSi( )-bjSj( )-&c. I have not time to examine the point, but I fancy that the introduction of <~i into your process would make it even simpler than it is. As to the real question at issue I consider myself not to have used your function , as though my notation can be interpreted into something of the same kind it wants the peculiar advantage of concentration which yours possesses, and which forms one distinctive feature of your XV. Tait developed this new notation in his letters 22 and 23. Hamilton did not immediately reply to this suggestion, other questions which will be referred to in due course having absorbed his attention. On February 5, however, he remarked in [76] of Letter xiv : " But let me first get off my hands a remark about the new Form which you suggest for the equation of the Wave Surface. I read it as T. 17 i 3 o PETER GUTHRIE TAIT and on just now glancing at your No. 22 received yesterday or the day before, but quite unexamined hitherto...! see that the symbol occurs several times. You have therefore probably introduced some new definition of the functional symbol and I am not entitled tc>6ay that your formula requires any correction. Of course we cannot afford to part with a certain liberty, of notation. But with my meaning of as developed in my Lectures and Letters, I found, a few minutes ago the hint (as I admit) having been taken from your last letter that the formula, { 7X*- 1 - p')-* p}' = - Sp (P 3 - r 1 )- 1 P, is an identity ; and therefore that one of my symbolical forms of the equation of the wave, namely, the equation may be immediately transformed to the following a result which I confess that I had not expected, but which (I suppose) agrees substantially with yours ____ You deserve I think great credit for having percei 'ved this transformation...." Thus we owe to Tait the discovery that the square root of a linear vector function or matrix of the third order enters symbolically into certain expressions exactly like an ordinary algebraic quantity. He was led to this discovery by a comparison of his own special notation with the notation used by Hamilton, who, on his own confession, had never thought of treating the linear vector function in this way. It is not a little curious that, at the time, neither Hamilton nor Tait seemed to have considered the analytical significance of the square root of a linear vector function. This was done in 1870 by Tait whose results, based on kinematic considerations, led to an interesting correspondence with Cayley and a further development of the properties of the matrix (see below, p. 152). After a good deal of further correspondence on the subject of the Wave Surface, Hamilton communicated his method to the Royal Irish Academy, and Tait published his investigation in the Quarterly Journal of Mathematics. Meanwhile, in Hamilton's mind a new project had been forming itself, which was first referred to in paragraph 71 of letter xiv, written on January 21, 1859. Here Hamilton wrote: " [71] I must tell you however of a quite different project of mine, which may occupy a good part of the present year if a fair share of health is spared me. I want to prepare for 1860 though I do not forget a passage in St James either a new edition of my Lectures, or what may be better, an entirely new work, which HAMILTON'S APPRECIATION OF THECORRESPONDENCE 131 might perhaps be called a ' Manual of Quaternions.' In it I suppress (decidedly) more than half of the existing Book ; not that I am ashamed of it, but because I conceive that it has served its purpose : and that what we may call a working volume is wanted now. " I fear that No. XVI of the series of MS will never be completed, or will be brought abruptly to a termination 1 : but I don't think that you require my word, for you have perhaps already indications enough, that I possess a number of uncommunicated results, respecting the function for instance, which will yet throw additional light on the treatment by quaternions of surfaces of the second order "[72] January 31, 1859. I see that the enclosed sheet, though not yet sent off, was written ten days ago. I have not even thought about the Wave Surface since, much less written a line about it ; but I by no means abandon the project of publishing some such short paper as I described to you in a former sheet ; leaving it to you to develope, in whatever form you choose, your own independent investiga- tions and results. It really seems to me that there would be some impertinence in my having the air of examining whether your formulae on that subject are correct. You are quite as well able as myself to decide any such point : especially since you have got into the way of making transformations and of multiplying them. I trust however that it is not an impertinence in me to confess that I think (or at all events, hope) that this correspondence has been useful to you, in some degree ; chiefly by causing you to feel a greater degree of confidence in your own powers ; as applied to a new subject ; and as evincing that whatever obscurity may have been allowed to remain in parts of my printed Lectures, from want of skill of an artistic kind in the author, it has not been fatal to a comprehension of the Book, by such a Reader as yourself; although the particular obscurity (about dp), which led to our correspondence, has not (in my opinion) been at all sufficiently yet removed, by my Letters V and X. " [73] As to myself I cheerfully confess, that I consider myself to have, in several respects, derived advantage, as well as pleasure, from the Correspondence. It was useful to me, for example, to have had my attention recalled to the whole subject of the Quaternions, which I had been almost trying to forget; partly under the impression that nobody cared, or would soon care, about them. The result seems likely to be, that I shall go on to write some such ' Manual,' not necessarily a very short one, as that alluded to in a recent paragraph. "[74] In fact, after pretty nearly filling two books, A. 1858 and T. 1858 with matters relating to the 'Tait Correspondence' [for 'A' had happened to be reserved, although 'B,' ' C,' 'D,' and 'E' (at least) had been stuffed with things connected with De Morgan, and with Definite Integrals &c. and after a few more letters of the alphabet having been pressed into the service, I used ' Alliteration's artful aid' and made a sudden bound, in honour of you, to 'T'] I have lately 1 No. xvi was begun on Dec. 14, 1858, but the greater part was written on Jan. u, 1859. It was abruptly finished off on Feb. 4, 1859, after a few paragraphs on surfaces of the second order had been put together. 17 2 132 PETER GUTHRIE TAIT taken possession of a very large book, which book I call A. 1859, and which is to relate entirely to quaternions. As yet, in it, I have confined myself to a new discussion of FIRST PRINCIPLES." Tail's reply to this constituted the greater part of his letter 23. He said : Many thanks for your kind and flattering letter....! applaud your purpose of publishing a practical " Manual of Quaternions." I may mention to you that I had been thinking of attempting something of the kind (but of course a very elementary work) if the idea met with your approval but that was of course before I heard that you intended doing anything of the kind yourself. There was one feature of my dawning idea which might suit you that was to get it printed as one of Macmillan's Cambridge series of which my Treatise on Dynamics forms a portion. It would thus be directly introduced to the largest body of mathematicians in this country.... Another feature would have been (and without this no book takes in Cambridge) numerous examples of the great simplicity of the new method....! merely mention my own half-developed scheme to show you that I think your present proposal an excellent one, and perhaps to give you a useful hint or two with the object of Quaternionizing my own University. In letter xvi of date April 10, 1859, Hamilton referred in a remarkably prescient manner to the part which Tail was destined to play in the development of quaternions. He wrote : " Let me be permitted to congratulate YOU (as well as myself most sincerely do I add this last objective case) on your having taken up the Quaternions. They will owe MUCH to you ; but I think that you will owe something to them. This may be only the natural vanity of an author ; but I believe that an early appreciation of genius wins a corresponding appreciation, in its turn, from mankind, for itself; even if not accompanied, as in your case it is, and will be, by independent acts of discovery'.' These extracts show unmistakably that the mathematical world owes more to Tait than has yet been revealed. It was he who fired Hamilton with the ambition to write his second great Treatise on Quaternions. As we read the correspondence, and especially Hamilton's long chapter-like letters, we see some of the leading features of the Elements taking shape. Had Hamilton lived to write the Preface to the unfinished Elements he probably would have mentioned explicitly the value of the Tait Correspondence. All we have, however, in published form is a footnote towards the close of the unfinished work, where Tait is spoken of as one "eminently fitted to carry on, happily and usefully, this new branch of mathematical science ; and likely to become in it, if the expression may be allowed, one of the chief successors to its inventor." WAVE SURFACE TRANSFORMATIONS 133 The following extracts from Tail's letters in March and April of 1859 show how thoroughly he was becoming saturated with the quaternion ideas and methods. [March 2.] I have added a good many new theorems to the wave investigations, but I fear their importance is nothing particular. The problem of the wave-front for which there is the greatest angular separa- tion of the rays has only led me to some complicated and almost intractable equations. I have been led in connection with the wave surface to the study of the curve p = $*.a, where p (the vector of any point) is a function of the scalar x a being a given vector and ( *) aiSi{ )bjSj( ) &c. From this I have got some curious results, but have been stopped short by a difficulty of a kind new to me in Quaternions, while trying to find x from having the same meaning as before.... Here again a new difficulty presented itself the elimination of m (an arbitrary scalar) between two equations of the form (where & = nf + 3 ) You may see that I have my hands pretty full of work even if the matters in question be of no importance. [March 18.] I have been working farther at the wave of late and I think am in a fair way to find the equation to the central surface of the second order concentric with the wave which has the closest contact with it at a given point. The difficulty consists in the solution of a functional equation or rather in determining the general value of a certain i/r-^o), where ^ is a linear and vector function. I have at last attacked the subject of Potentials which was the cause of my recent (and, this time, successful so far) attempt at the study of Quaternions, and I think I have got the method of applying the calculus to the matter. I have also been working at some illustrative problems. I met with this in a Cambridge Examination Paper, 'Find the locus of the centre of a sphere which touches two given lines in space.' I modify it into ' Find the locus of the centre of a surface of the second order, whose axes are given in ratio and direction, and which touches two given lines.' The required locus is given in the form where fi and 7 are the unit vectors along the given lines, 2a is the common perpendicular and is the function of the surface. 134 PETER GUTHRIE TAIT In letter xvm, dated April 12, 1859, Hamilton returned to the wave surface, and after deducing afresh its equation remarked : " Could anything be simpler or more satisfactory ? Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? Never mind when ____ " De Morgan and I have long corresponded unofficially and said odd things to each other. He was the very first person to notice the quaternions in print, namely, in a paper on Triple Algebra in the Camb. Phil. Trans, of 1844. It was, I think, about that time, or not long afterwards, that he wrote to me, nearly as follows : 'I suspect, Hamilton, that you have caught ttte right sow by the earl' Between us, dear Mr Tait, I think that we shall begin the SHEARING of it." Tait replied in letter 31 of date April 13, 1859: I have just received XVII and XVIII, the latter an hour or two ago. Your deduction of Fresnel's construction from the symbolic form of the equation to the wave is very elegant. I have given (in a paper which I suppose is now being printed, for it has been sent off ten days or more) a proof of the same, which is a mere interpretation of some of the equations which I have written down in deducing that to the wave. I have recently (as I mentioned in letter 26) come to a seemingly formidable difficulty in Quaternions. It is to find the most general form of linear and vector function i/r from the equation where a- = (< a + p*)~V and where the scalar and vector constants of the required function i/r involve p, a- and the operation <.... In the third PS. to your VIII you mentioned a result of Maccullagh's 1 which I have since found in the Trans. R. I. A. I was lately trying the problem in an extended form. I find for instance the following amongst a host of other results. (1) If the two lines which move in the planes are not at right angles, let the cosine of their inclination be e, and let the third line be perpendicular to them ; it traces a cone of the 4th order.... (2) If one of the moving lines be a generating line of a cone of the second order, the second lying in a plane which passes through the vertex thereof, and the third perpendicular to the other two, the locus is in general a cone of the 8th order.... While this letter was being penned, Hamilton was beginning his letter xix, the importance of which demands a full transcription. 1 As given by Hamilton, the problem is, If three rectangular lines so issue from a common origin that two of them move in fixed planes, the third will describe a cone of the 2nd order, whose circular sections are parallel to the two planes. THE LINEAR VECTOR FUNCTION 135 SV, April itfk, 1859. My dear Mr Tait Although what I am about to write must be very short, and might be marked as PS. to No. XVII, yet, on the whole, I choose to number it as above, partly with a view to encourage myself to write short letters. [i.] There is, as you know, a very important problem of transformation, to which you have alluded, both in early and in recent letters, and of which I by no means deny that those letters may contain a sufficient solution or solutions : for I have hitherto avoided to examine them, in connexion with that problem, which I certainly conceived myself to have resolved, about ten years ago, and to which (as solved) I alluded at the end of art. 567, in page 569 of the Lectures.... [4.] The problem... haunted me, as it happened, yesterday, while I was walking from the Provost's house to that of the Academy, &c. ; and _though I wrote nothing down that day I resumed it this morning : and arrived at what you might call, in the language of your No. 19, a ' perplexingly easy' solution (in the sense of being very UNLABORIOUS, for I do not pretend that the reasoning does not require a close attention) ; not in any way introducing ij k, nor a /3 7 (of an ellipsoid) nor t, K, but depending entirely on the properties of the function (f>. So simple does this solution appear, that I hesitate as yet to place entire confidence in it ; and therefore, till I have fully written it out for at present it is partly mental and have given it a complete and thorough re-examination, I hesitate to communicate it to you. Meantime, however, I must say, that I am not conscious of having taken any hint, in this investigation, from any of your letters.... [5.] April isth I shall just jot down here the enunciation of a few Theorems 1 , which I have lately proved (as I think) anew, and which are intimately connected with the question. THEOREM I. If $p be a distributive and vector and real function of a real vector p, such that Sa satisfies a cubic equation, whereof the three roots are always real. 1 This is probably what Tait referred to in his paper on the intrinsic nature of the quaternion method (1844; Sd. Pap. Vol. n, p. 396), where he states that "one of his many letters to me gave, in a few dazzling lines, the whole substance of what afterwards became a Chapter in the Elements" 136 PETER GUTHRIE TAIT THEOREM VI. If these roots be also all unequal, then the eq M , (4>+gi)Pi = 0, (<+#,) Ps = 0, (+g,)p s = 0, (97), are satisfied by the 3 rectangular directions p lt p t , p, of Theorem iv, and by those directions (or their opposites) only. THEOREM vn. For any other vector, p=Xipi+Xip,+x0t, (6), we have p = - (g^x lpl +g*x*p t +g,x,p s \ (t), and Spp = - (g&p? +g*x?p? +g t xfpt) > (K). THEOREM VIII. Whatever the real scalar, g, and the real vectors, a, a',... and /3, /S', ... may be, it is possible to find 3 real scalars, gi,g^;g 3 , and 3 real and rectangular unit vectors, p lt p t , p,, such that the following shall be an identical transformation : THEOREM IX. The data, g, a, , a, ft, ... being still real we have finally this other transformation : gf + USapSpp =? + 2Sp = o admits of one real solution at least." It is certainly a very elegant mode of attacking the question, and I had never thought of so simple a point of view as the making the normal coincide with the radius vector. But when I try to prove your theorem, I fall back again into the cubic of my letter 1 30, or at all events a simple case of it, so that I do not see how you manage to avoid a reference to something or other equivalent to i,j, k. In a PS. to letter XXH, dated Easter Tuesday, 1859, Hamilton indicated the proof which Tait longed for : " My Theorem I, of Letter xix, was proved by showing, on the plan of Lecture vn, Art. 567, that the equation could be satisfied without our having also p = o, provided that g was a root of a certain cubic equation. It is not at all necessary, for this purpose, that < should satisfy the functional condition 1 In regard to letter 30 Hamilton had remarked that he liked the look of it. Unfortu- nately a copy of this particular letter does not seem to have been preserved by Tait. THE LINEAR VECTOR FUNCTION 137 but as I assumed that this condition was satisfied in most, if not in all, of the subsequent theorems, I believe that I thought it convenient to enunciate it at starting. Besides I wrote in some haste." Hamilton's letter xxm contains a systematic investigation of the linear vector function, which differs markedly in the details of development from the investigation given in his subsequent book The Elements of Quaternions. In its initial stages it resembles Tail's mode of presentation, which Tait himself calls "Hamilton's admirable investigation" (see Tait's Quaternions, 3rd edition, 156-159). Writing on May n, 1859, Tait in letter 33 remarked : Your No. XXIII (which I received yesterday) was indeed a treat. Nothing could be more beautiful than your method of attacking the equation of the second degree. I have been trying to supply for myself the demonstrations you suppressed and have succeeded completely, though perhaps not elegantly. Thus as assume r" 1 \^ = m and if m = m', your theorem about the interchange of and -fy is proved. The above equations are evidently equivalent to <~ l Fi/r-'X/t = m and m'^ V$\n Multiply together, and equate scalars, and we have at once m' (-S^V-S/tn/r-'X - \> 2 ) = m or m' = m since Stj>\fj, = and therefore also S^~ 1 \;i = Another curious property of these functions resulting from this last equation is that <^>~ 1 i/r is the conjugate of $-4f~*. I came upon the following (which seems neat). Generally, whether n be + or or even = o, which is true (of course) of also. What I was most puzzled with was the proof that m (in your notation) is a constant. I saw at once that it could not contain the tensors of X and ft, but I did not feel so sure about the versors. I have satisfied myself on that point by making use of the distributive property of ~ 1 . Six days later in letter 34, Tait made a further reference to the same investigation. T. 18 138 PETER GUTHRIE TAIT When I came to your equation (31) of xxin I tried to prove it for myself and was so successful that I was just about to send you a note on the subject when I luckily read on and found that your luminous thought had completely anticipated me. Here is my work as it stands in an MSS book. Change into $+g, &c. and multiply by M, or No letter from Hamilton of date later than July 19, 1859, has been preserved, although there are copies of eight of Tail's own letters to Hamilton ranging from Sept. 7, 1859, to January 14, 1861. From these we gather that Hamilton was absorbed in the preparation of his new book and was keeping Tait steadily supplied with the proof sheets of the earlier chapters. Meanwhile Tait was strengthening himself in the use of the calculus, and in letter 41 of date Sept. 26 gave, very much as it afterwards appeared in his Treatise, his quaternion investigation of Ampere's electro- dynamic theory. This investigation, especially in the more generalised form in which it was presented in his paper of 1873 on the various possible expressions for mutual forces of elements of linear conductors (Proc. R. S. E. vin ; Set. Pap. Vol. i, p. 237), is a good example of the directness with which the quaternion method deals with a general problem 1 . Beginning with a general form of function, involving the relative position and the directions of two current elements, Tait developed the form of this function by a skilful use of Ampere's fundamental experimental laws. In letters 42 and 43 of date Nov. 3, 1859, and March 22, 1860, Tait continued the development of his electrodynamic investigations, pointing out the importance of the vector Vaa! fdUa in all investigations connected with the action of a circuit, where a! is the element at the point a of the circuit. A few months later Tait commenced his Edinburgh career, having been helped thereto by the following testimonial from Hamilton : Understanding that Professor Peter Guthrie Tait, now of the Queen's College, Belfast, but formerly of St Peter's, Cambridge, is likely to become a candidate for 1 See also Clerk Maxwell's Electricity and Magnetism, Vol. n, Chap. n. MISUNDERSTANDINGS 139 the Professorship of Natural Philosophy in the University of Edinburgh, in the event of that office becoming vacant, I consider it to be only just to Mr Tait to attest that, in consequence of a rather copious correspondence between him and me, which has been carried on for somewhat more than a year, on mathematical and physical subjects, including Quaternions, and the Wave-surface of Fresnel, my opinion of the energy and other capabilities of Professor Tait for any such appointment is very favourable indeed. WILLIAM ROWAN HAMILTON. OBSERVATORY OF TRINITY COLLEGE, DUBLIN, Dec. loth, 1859. Tail's return to Edinburgh and his assumption of new duties meant a considerable break in the line of his mental activities ; and it was not till Dec. 4, 1860, that he wrote letter 44 of the quaternion series to Hamilton. A few days earlier he had sent Hamilton a copy of his inaugural address, in which he had referred in glowing terms to the " powers " of Hamilton's " tremendous engine," to the great secret of quaternion applications, which "seems to be the utter absence of artifice, and the perfect simplicity and naturalness of the original conceptions." EDINBURGH, Dec. 4tA, 1860. My dear Sir William Hamilton, I received your letter this morning and am glad you are pleased with my introductory lecture. Its treatment by others has not been in all cases so lenient, in fact I am now doing battle with at least two opponents, who have vigorously attacked different parts of it. I am sure I am not violating confidence in telling you that one of these attacks is directed against the mention of Quaternions (towards the end of the lecture) as "likely to aid us to a degree yet unsuspected in the interrogation of Nature." The writer, I daresay, is a personal friend of your own that I do not know but, at all events while speaking of you with admiration and due courtesy, he protests in the interests of Science against my having published such a sentence as that above quoted... I was sorry to see from your letter that we must have been completely misunderstanding each other for some time as to my projected publication on Quaternions. In the first place, to prevent all misconception, let me say that when Dr Andrews wrote a note introducing me to you as a correspondent, I had not the slightest idea of ever being the author of a Volume on the subject. So he could know nothing whatever about the matter. And I think you will acknowledge that the whole is a mistake when I tell you that it never entered into my head to write a Book on Quaternions till I was asked by some Cambridge friends to do so, that I at once wrote to you about it, and asked how far it might be consistent with your wishes or plans that I should undertake such a work. In my letter to you, No. 38, I proposed two forms of publication, one a dry practical treatise, very short, assuming most of the fundamental laws of Quaternion multiplication, but stuffed 1 8 2 140 PETER GUTHRIE TAIT with examples the other, the examples alone. I went on to say that even the first of these " could not in the least interfere with your (then projected) new work, as it would treat only of the practice of the method, and not at all of the principles." And I added, " I have not the least intention of publishing a volume on the subject without your approval." When (in XXVIl) you wrote in answer to the above " I should prefer the establishment of PRINCIPLES being left, at least for some time longer, say even 2 or 3 years in my own hands ; and I think you may be content to deduce the Associative Law from the rules of i, j, k, etc." I fancied that you meant me to give these deductions in print beginning from i"-=j* = k* = ijk = i as something established in your Lectures and Manual. When some months or so later, I wrote to you that I had asked Macmillan to advertize for me " An Elementary Treatise on Quaternions, with numerous examples" I had no idea whatever that I was giving you any annoyance But (as I have already quoted from 38) I am most desirous to avoid the slightest suspicion of interference with your intentions and I therefore particularly request you to give me a perfectly distinct idea of your desire in the matter and my advertisement and form of treatment shall be at once adapted to it. But I regret you did not tell me of this, at once, more than a year ago, when I enclosed a printed copy of Macmillan's advertisement Hamilton's reply to this was evidently very satisfactory, for on December n, 1860, Tait wrote: I am glad to find that my explanation has been sufficient, for I assure you that I had attributed the slackness of our correspondence of the last year to your having been bored and tired with my continued questions about various old and new points in Quaternions, and had no idea whatever that I had annoyed you in any way by the publication of my unlucky advertisement. In letter 46, January 14, 1861, Tait acknowledged receipt of proof sheets of the Elements, and made further references to his electrodynamic work. Here the correspondence practically ended. We learn from Tail's preface to his Treatise that Hamilton shortly before his death in 1865 urged Tait to push on with his book, as his own was almost ready for publica- tion. It is pleasing to know that the misconception of the situation which had fretted the mind of the master was entirely removed by the straight- forward honest dealing of the disciple. Broadly speaking the subject-matter of the Hamilton-Tait correspondence may be grouped under five heads. (i) Quaternion differentials. These are discussed at length in Hamilton's letters v and x, the former of 45 pages having been written between the dates of Oct. 1 1 and 1 6, and the latter of 48 pages between ANALYSIS OF CORRESPONDENCE 141 the dates of Oct. 25 and Dec. 2, 1859. The discussion is reproduced in essence in the Elements, although much more briefly. (2) Transformations connected with Fresnel's wave-surface. Tait began the discussion in letter 10 and continued it in many of the subsequent letters down to letter 34. Hamilton took up the theme in letter xiv and elaborated it in letters xv, xv', xv", which ran on consecutively for 96 pages. Here also the essential parts of the investigations both of Hamilton and Tait will be found in their works. In letter 20 Tait suggested the use of the form and in letter 23 gave the wave-surface equation in the new form T (p* + ) where SV, VT P, a pure scalar. Multiply this by dp (a pure vector) and you get a pure vector dpVr = dpP. Hence your expression sl V (dpV)T = SJdpP = o because if it is anything at all it is the integral of a vector multiplied by a scalar and that is a pure vector and the scalar part of it is o. I suppose this is nonsense arising from our being barbarians to one another. Will you therefore be so kind as to give me a code by which I may interpret the symbol VdpV, that is to say, tell me what these symbols, thus arranged, ask me to do.... Note the Vector . In electromagnetism P is the magnetic potential and VP is the magnetic force outside the magnet or inside it in a hollow tube whose sides are parallel to the magnetization. V<7 = V/ > outside but inside Vcr = V/> + 47r3 where 3 is the magnetization. Va- is the magnetic force in a crevasse x 3>. I have not been able to make much of your T. I coloured some diagrams of lines of force Blue and red but I must study the astronomer to define the magnetic tints and softness. Sir W. Hamilton (Edin h ) was partial to redintegration, an operation you should get a symbol for. Among other scientific expressions I would direct your attention to the salutary influence of Demon-stration and Deter-mination, and to two acids recently studied, Periodic and Gallery Thronic acids. The 1st you will find use for. The 2nd is for the L d High Commissioner. Yours J C M In another letter, of which the opening paragraph has already been given (page 1 1 7 above), Maxwell refers to Tail's quaternionic investigations in the stress function. The letter is on a half sheet of note paper and is undated, but was probably written towards the end of 1872. The continuation is as follows: " I return your speculations on the (f> ( Uv) ds. Observe that in a magnet placed in a magnetic field the stress function is not in general self-conjugate, for the elements are acted upon by couples. But the = n of = m is very properly got as you get it 1 . " Search for a physical basis for 5 . V'aVo- as a term of the energy developed in a medium by a variable displacement be the strain function and <' its conjugate, and if we try to resolve, the strain into a pure strain followed by a rotation, so that <( ) = $&( )f~ l , I find vt" ( ) = ' ( ), so that the pure strain is the square root of the given strain followed by its conjugate. ORTHOGONAL ISOTHERMAL SURFACES 153 Cayley replied, March 2, 1872: " I find that your question may be solved very simply by means of a theorem in my memoir on Matrices, Phil. Trans. 1858." He then proceeded to indicate the steps of a somewhat prolonged process by which the solution might be found ; but in a second letter written a few hours later he practically reproduced Tail's process by use of the symbolic cubic, the Matrix symbol M being written instead of the vector function . On March 5, Tail wrote: It is a most singular fact that you seem to have been working simultaneously with Hamilton in 1857-8, just as I found you had been in a very much earlier year ...... I have had but time for a hurried glance at your paper on Matrices and I see that it contains (of course in a very different form) many of Hamilton's properties of the linear and vector function....! send you a private copy of my little article, by which you will see how closely the adoption of Hamilton's method has led me to anticipate almost every line of your last note.... There is one point of Hamilton's theory to which I do not see anything analogous in your paper. Expressed in his notation it is that - 1 and are identical, if we have gh mSfT^^r^p. The Report referred to by Tait in his letter of Feb. 28 was a Report which, urged by Cayley, he had agreed to prepare for the British Associa- tion. Shortly afterwards he asked to be relieved of the task, as it would be of too personal a character, and suggested Clifford as eminently quali- fied to undertake it. Nothing further seems to have been done. The quaternion discussion of orthogonal isothermal surfaces was published in 1873 (Sci. Pap. Vol. i, p. 176). It is an interesting example of the use of Hamilton's rotational operator g( )^ -1 . The opening paragraphs of this paper are not quaternionic, and seem to have been introduced by Tait for the double purpose of showing how he originally began to attack the problem and how much more suggestive and concise the quaternion solution is. In a letter of date July 22, 1873, Maxwell referred in a deliciously humorous manner to the character of Tail's investigations in these words : " I beg leave to report that I consider the first two pages of Professor Tait's Paper on Orthogonal Isothermal Surfaces as deserving and requiring to be printed in the Transactions of the R. S. E. as a rare and valuable example of the manner of that Master in his Middle or Transition Period, previous to that remarkable condensation T. 20 154 PETER GUTHRIE TAIT not to say coagulation of his style, which has rendered it impenetrable to all but the piercing intellect of the author in his best moments." When this paper was passing through the press Tait had a brief correspondence with Cayley on the nature of his solution. After its publication, Cayley made some interesting comments in a letter of date March 25, 1874. He first reproduced one of his own results which shows that, in order that r = const, may represent a family of orthogonal surfaces, then r considered as a function of x y z must satisfy a somewhat complicated partial differential equation of the third order. Tail's equation da- = uqdpq' 1 , he then pointed out, must be the equivalent of this partial differential equation of the third order. He concluded in these words : "Do you know anything as to the solution when the limitations [imposed by Tait] are rejected, and imaginary solutions taken account of? Considering simply the equation of the third order and the equation a + 6 + c = o [that is W=o] it would seem probable that there must be a solution of greater generality than the confocal quadrics. I do not see my way to the discussion of the question. The condition a + & + c=O seems to make no appreciable simplification in the equation of the third order. I admire the equation da = uqdpq~ l extremely it is a grand example of the pocket map." This comparison of a quaternion formula to a pocket map was quite in accord with Cayley's attitude towards the quaternion calculus. He admitted the conciseness of its formulae, but maintained that they were like pocket maps : everything was there, but it had to be unfolded into Cartesian or quantic form before it could be made use of, or even understood. This view Tait combated with all the skill at his command ; and every now and again the two mathematicians had a friendly skirmish over the relative merits of quaternions and coordinates. Even when they exchanged views on quaternionic problems altogether apart from this central controversial question, their different mental attitude came clearly to the front in their correspondence. This is seen, for example, in the following series of letters. Dear Tait In the quaternion q = w + ix+jy + kz, assuming tan -/= + - y ' + ** , (r = V(* + j>' + *') and -, y -, - , = cos a, cos & cos 7) then the quaternion is q *=w + ix+jy + kz * = - i' > {cos \f+ sin \f(i cos a +j cos /3 + k cos 7)} sin ^T and we can interpret the quaternion in a twofold manner, viz., in the first form, CORRESPONDENCE WITH CAYLEY 155 disregarding the scalar part w, as the force represented by the lines x, y, z\ and in the second form, disregarding the tensor r/sin \f, as a rotation f about the axis (, ft, 7). Then sum of two quaternions, qua force, is the resultant force. Product of two quaternions, qua rotation, is the resultant rotation. But is there any interpretation for the sum qua rotation or for the product qua force} It would be very nice if there were. We enjoyed our American expedition very much. I was glad to hear from Thomson that he also was going to lecture at Johns Hopkins University.... Yours very sincerely A. CAYLEY. CAMBRIDGE, yd Nov. 1882. UNIVERSITY OF EDINBURGH, 4/11/82. My dear Cayley I was very glad to get your note, and to hear that you had enjoyed your venturous journey. Thomson's proposal was quite new to mel I have not seen him for months. I am absolutely overwhelmed with work just now ; as, besides my University work, and R.S.E. do., I have been virtually forced to give a course of lectures to ladies, and I am writing, against time, a very long article for the Encyc. Brit. Maxwell's death left the staff of the Encyc. in a state of great perplexity. He had drawn up a scheme for the scientific articles, and had done the greater part of the work himself. Had he lived, the article " Mechanics " would have been written by him, or entrusted to some competent writer, two years ago, at least. As it is, the acting editor discovered, only three months ago, how much had been referred forward to it; and I spent the greater part of my summer holiday in writing it Seeing it through the press is no joke! And the work of trying to boil down the whole of abstract dynamics into 60 pages has been very heavy. I fear I misunderstand your questions. Of course I know that Vq is a force and that V(q + r) = Vq + Vr; whatever quaternions q and r may be. But, as to rotation, I have always written (after Hamilton) 9( )r' where (of course) we need not trouble about the tensor. This gives qr( )r~ l q~ l as the result of r( )r~ l followed by q( )q~ l ; and may be written qr( }(qr}-\ Now, in asking about the interpretation of a sum qua rotation, do you mean the effect of (q + r) ( ) (q + r)~ l ? Also, as to the product qua force, do you refer to V.qrl I can easily answer these questions, but I fear I have not caught your mean- ing.... Before Cayley's reply to this was received, Tait wrote a second note. 156 PETER GUTHRIE TAIT UNIVERSITY OF EDINBURGH, 6/11/82. My dear Cayley Since I wrote you I have fancied that I ought to have sent you the answers, even if I have misunderstood you. I. When we deal with a sum of two quaternions, from the rotational point of view, the ratio of their tensor plays a prominent part. In fact <*+')( )(3 + rY* = (9r*Tr( )r->(qr-i)-* where x is a scalar, which is to be found from an equation of the form a sin A - -i - = tan xA. a cos A + I This seems an answer to your question " Is there any interpretation of the sum qu rotation?" It is the rotation r( )r~ l followed by (qr~ f f( )(gr- 1 )-*. Of course it may also be put in the form (g~ l ry ( ) (g- 1 r)~ v followed by g( ) q~ l where y is another scalar found from a transcendental equation. Compounding these it may also be expressed as which is more symmetrical. But it can also be expressed as q 1 r q 1 ( ) q~ l r^ q~ l . When / and m are found from two equations of the form 2 (a cos a + b cos = c sin 2/a sin m/3 + cos 2/a cos mft, sin 2/a cos mft, l> sin ft 2 a sin a itsinft all the quantities a, b, c, a, ft, being known scalars. Of course the number of such expressions is endless ; and I wait further light from you. 2. As to the product qua " force " (as you call it), we have V.qr^Sr. Vq + Vr.Sq+ V. VqVr so that the " force " of the product appears as the sum of three forces ; two of which are multiples of the separate forces ; the other is a force perpendicular to both. In great haste, yours truly P. G. TAIT. Cayley's letter of the same date which crossed this one was as follows : Dear Tait It is only a difference of expression : I say that q = cos ^/+ sin ^f{i cos a +_;' cos ft + k cos 7), is the symbol of a rotation because operating in a particular manner with q upon CORRESPONDENCE WITH CAYLEY 157 ix+jy + kz we obtain ix^+jy^ + kz^ the x^ y lt z^ being the new values of x, y, z produced by the rotation: viz. the particular operation is ixi +jyi + kzi = q (ix +jy + kz) g~ l and you say that q( )f l is the rotation. But of course q, r being the two quater- nions, qr in my mode of expression or qr( ) (qr)~ l in yours, belongs to the resultant rotation. In my mode of expression 1 ^{cos i/+ sin kf(* cos a +/cos yS + k cos 7)} is equally well with cos \f+ sin \f(i cos a +/cos /8 + k cos 7) the symbol of the rotation ; and my question was is there any interpretation, in connection with rotations, of the sum T {cos i/+ sin \f(i cos a +j cos /8 + k cos 7)} + T {cos \f + sin \f (i cos a' +/ cos ft 1 + k cos 7')}, that is of the sum of any two quaternions w + ix +jy + kz, v/ + ix" +jy' + kz 1 . I think therefore you have understood me quite rightly viz. in asking about the interpretation of a sum qua rotation, I do mean the effect of (q + r) ( ) (q + r)~\ and as to the product qua force I do refer to Vqr and shall be much obliged for the answer. Believe me, dear Tait, yours very sincerely A. CAYLEY. Nov. bth PS. I believe it was I who first gave in the Phil. Mag. the formula q(ix +jy + kz)q~ l , showing it was identical with that of Rodrigues for the effect of a rotation but Hamilton was doubtless acquainted with it. Tait replied to this the next day : 7/1 1/82. My dear Cayley The note I sent you yesterday, and which I hope you got, will now, I see, more than answer your question ; which (as I understand it) refers to the sum of two versors Uq+Ur 1 There is a strong resemblance here between Cayley's symbolism of the rotation involved in the quaternion and the discussion by Klein and Sommerfeld in their Ucber die Theorit des Kreisels of what they call "die Quaternionentheorie " (Chap, i, 7). See Tail's paper "On the claim recently made for Gauss to the Invention (not the Discovery) of Quaternions" (Proc. R. S- E- Vol. xxm, 1889); and "Professor Klein's View of Quaternions, a Criticism," by C. G. Knott (Proc. R. S. E. Vol. xxm, 1889). 158 PETER GUTHRIE TAIT (You write a T instead of a U\ but the form you adopt viz. cos a + (il+jm + kri) sin a is a versor, its tensor being i). Of course in this particular case, the formula I gave you yesterday is much simplified. For instance we have a = i and x=\. Thus (Vq+Vr)( ) ( Vq + Vr)~ l = (qr~^r ( ) r~* (qr*$. This and indeed the general cases of q + r, is easily seen by means of a diagram [proof given by use of a spherical triangle].... I send with this a copy of an old paper of mine bearing on the question raised in your last... The second of these gives the reference which shows that Hamilton anticipated you about the quaternion rotation. The third passage refers to what I thought was mine (i.e. putting Rodrigues' expressions in a simpler form) but your letter shows that you also use this versor form.... Cayley's reply was : Dear Tait Best thanks for the last two letters and the memoir. I am rather glad to find that the formula was first given by Hamilton. The (g + r)( )(q + r)~ l formulae are very curious, but I hardly see as yet what to make of them.... CAMBRIDGE, 8/A Nov. 1882. Cayley seems to have forgotten to some extent the contents of Tail's paper of 1868. Towards the end of 1884 an interesting correspondence arose between Tait on the one hand and Cayley and Sylvester on the other in regard to the solution of the quaternion equation aq = qb. Sylvester had just published his general solution of the linear matrix equation ; and taking a more general view of the quaternion q he obtained what seemed at a first glance to be a different solution from that given by Tait in his Quaternions. The analytical theory which admits the possibility of Tq vanishing a possibility never considered by Tait is given by Cayley in Chapter vi of the 3rd edition of Tail's Quaternions ; and parts of this contributed chapter are almost identical word for word with portions of Cayley's letters. On August 28, 1888, Tait in view of the preparation of this 3rd edition, .asked Cayley for suggestions in the way of improvements, especially on the analytical side. Cayley responded immediately with some notes which Tait gratefully accepted. Some weeks later Tait wrote: CORRESPONDENCE WITH CAYLEY 159 Since I returned to Edinburgh I have been considering more closely the question of the new edition of my Quaternions and looking up specially Sylvester's papers in the Comptes Rendus and the Phil. Mag. It seems to me from my point of view (which I think is that of Hamilton) that all these things, excellent and valuable as they are, are not Quaternions but developments of Matrices. As I understand Hamilton's quest, it was for a method which should supersede Cartesian methods, wherever it is possible to do so. Hence i, j, k, and their properties, though they were the stepping stones by which Hamilton got his method, are to be discarded in favour of a, q, , etc.: and no problem or subject is a fit one for the introduction of Quaternions if it necessitates the introduction of Cartesian Machinery.... The conclusion from this seems to be that I ought, instead of inserting your contributions in the text of my book as it stands, to make a new chapter " On the Analytical view of Quaternions" (or some such title) in which they will form the spinal column. Therein will naturally assemble all the disaffected or lob-sided members, which are not capable of pure quaternionic treatment but which are nevertheless valuable, like the occipital ribs and the anencephalous heads in an anatomical museum. Ten days later Cayley replied : "I... have not yet written out two further notes which I should like to send you for the new Chapter which (I take it kindly) you do not compare with the Chamber of Horrors at Madame Tussaud's....! need not say anything as to the difference between our points of view; we are irreconcileable and shall remain so: but is it necessary to express (in the book) all your feelings in regard to coordinates ? One remark : I think you do not give your symbol < a sufficiently formal introduction : it comes in incidentally through a particular case, without the full meaning of it being shown. The two notes will be on the equation aq + qb = o and on Sylvester's solution of af + bq + c = o." On Oct. 22, 1888, Tait wrote: I am very glad to know that you will give me two more of them [i.e. the notes]; especially as I found Sylvester's papers hard to assimilate. A considerable part of each paper seems to be devoted to correction of hasty generalizations in the preceding one ! I don't know that my point of view of coordinates is very different from yours, though my sight is vastly inferior. But I can see pretty clearly in the real world, with its simple Euclidean space, by means of the quaternion telescope. Witness a paper of Thomson's which I have just seen in type for the next Phil. Mag. ; where three pages of formulae can easily, and with immense increase of comprehensi- bility, be put into as many lines of quaternions. In his reply to this letter Cayley, after indicating his desire to see proofs of Tait's Preface to his coming new edition of his Quaternions, asked : i6o PETER GUTHRIE TAIT " Have you considered how far some of the geometrical proofs are independent of anything that is distinctively Quaternions, and depend only on the notion of ** +jy + kz, with i, j, k as incommensurable imaginaries not further defined ? " It was not till the summer of 1889 that the third edition began to be printed ; and this naturally led to a renewal of the correspondence on quaternionic subjects. Writing on June 15, 1889, Tait drew Cayley's attention to a new problem which had been interesting him. In looking over the Trans. R.S.E. for your notes for the Fortschritte d. Math. I suppose you saw Plarr's paper on the form of the spots which a blackened ellipsoid would make if it were made to slide about in the corner of the ceiling. I have been trying to simplify the analysis, and have reduced the question to one of mere elimination : but it is still very complex. With the view of studying what any point of the ellipsoid does, I had a very true ellipse cut out of thick sheet brass in my laboratory, and have traced the curves (of the 1 2th degree?) made by a pencil passed through various holes in it when it slides between two perpendicular guide-edges. This was the beginning of Tail's discussion of the glissettes of the ellipse and hyperbola. In reference to the problem Cayley remarked : " I abstracted Plarr's paper, but it did not seem to me that he had got out much of a result not that I saw my way to doing it better. It is a very good question, and a very difficult one. The plane question ought to be much easier tho' I fancy even that might be bad enough. I shall be very glad to see your curves." On November 21, 1889, Tait referring to the glissettes wrote: Connected with the curves I sent you in summer there is a very curious theorem which may, perhaps, be new to you. They can be traced by a point in the plane of a hyperbola which slides between rectangular axes. A month later, Dec. 21, Tait wrote: My dear Cayley Thanks for your second splendid volume, which has come just in time for my brief vacation, and contains in an accessible form the Quantics, which I have long wished to read properly. The same post brought me a specimen copy of Quaternions, with various colours of cloth to choose from. Brick red seems to be the most taking bait, so when you get it you will have something striking to look at if not into. You will see, in a few days, in the Phil. Mag. another plea for Quaternions as the physical calculus, par excellence. Perhaps it may lead to an increased sale of my volume. Have you ever considered the locus of intersection of two normals to an ellipse which are perpendicular to one another? CORRESPONDENCE WITH CAYLEY 161 I showed the R.S.E., on Monday last, an Ellipse and a Hyperbola separately tracing the same glissette. The uninitiated were much puzzled to see it, as the one curve merely oscillates while the other turns complete summersalts, and they could not conceive that the same curve could be traced by a point of each. But it comes merely to this: that [in the parallelogram linkage OABA' which was sketched in the letter] the ellipse describes B about O virtually by the two links OA, AB ; while the hyperbola does it by the other two sides of the parallelogram. The centre, A, of the ellipse has a to and fro motion through a limited angle, while A' (the centre of the hyperbola) goes completely round. A later letter from Tait gave a further investigation of this problem very much as it appeared in the published paper (Proc. R. S. E. Dec. 1889; Sci. Pap. Vol. ii, p. 309), which Cayley characterised as " very interesting." On January 24, 1890, after acknowledging the receipt of Tait's Quaternions and a copy of the Phil. Mag. paper on the Importance of Quaternions in Physics (Sci. Pap. Vol. n, p. 297), Cayley renewed the old discussion in these words : " Of course I receive under protest ALL your utterances in regard to coordinates. Really, I might as well say, in analytical geometry we represent the equation of a surface of the second order by 7=o; compare this with the cumbrous and highly artificial quaternion notation Spp = l. But you cannot contend that this last equation by itself contains the specification of the constants which determine the particular quadric surface ; and the fair parallel is between your quaternion equation and (*$*, y, z, i) J = o: and if you say yours is shortest, I should reply, mere shortness is no object, or again there is nothing easier than to use a single letter to denote (x, y, 2, i). Again, for a determinant y z X 1 y' z x" y" z there is here absolutely nothing superfluous, the determinant depends upon nine quantities which have to be specified : and these are not simply a set of nine, but they group themselves in two different ways into 3's as shown by the lines and columns." Tait replied as follows : 38 GEORGE SQUARE, EDINBURGH, 25/1/90. My dear Cayley I might say with a great rhetorist, " I am not careful to answer thee in this matter": but I think that most of your remarks seem to be based on ignoration of the Title of my little paper. It is the use in Physics that I am speaking of. i/=o is just as expressive in quaternions as in any other calculus, i.e. it is, in all, T. 31 162 PETER GUTHRIE TAIT a blank form to be filled up. But Sp' and see the criterion of the pure strain in = ', as well as the general relation where m is the factor by which volume is increased. Thus we have the linear and vector function, or (as you call it) the Matrix, with its fundamental characteristic. Finally we have Nabla, which is defined by the equation expressing total differentiation so far as the shift, or displacement, d is concerned. In all this there is no reference whatever to anything Cartesian : and no more need there be such in any application or development of these principles. And I have always not merely allowed but proclaimed that, in the eyes of the mathematician, Qns. have the fatal defect of being confined to Euclidian space. But this is one of their great recommendations to the physicist.... I should like to know at your convenience when and how the notion of the Matrix came to you : and whether Hamilton's simple case of it was an anticipation or an application of the general theory. In response to this, Cayley sent Tait an article he had written out for the Messenger of Mathematics on " Coordinates versus Quaternions," remarking in a covering letter, " I do not know what has made me write it just now, but it puts on record the views which I have held for many years past and which have not been before published." He also expressed his dissatisfaction with Tail's sarcastic reference to " Trilinear Coordinates " in the Preface to his Treatise on Quaternions, and added in a postscript to his letter : " I certainly did not get the notion of a matrix in any way through quaternions : it was either directly from that of a determinant; or as a convenient mode of expression of the equations x' = ax + by CORRESPONDENCE WITH CAYLEY 165 In his reply Tait suggested that Cayley should communicate his note to the Royal Society of Edinburgh, and then continued : Of course I do not agree with you, in fact we are as far as the poles asunder in regard to your main point. There we must continue to differ I scarcely think you do me justice in giving without its context the remark [in the Preface to Tait's Quaternions, ist, 2nd, and 3rd editions] " such elegant trifles as Trilinear Coordinates." I think that you will see that the context very considerably modifies the scope of the remark: so much so, in fact, that while I am still of the opinion I expressed I am not prepared to use the phrase " elegant trifles " even about Trilinear Coordinates (of Quadplanar Coords. I said nothing) without some such qualification, or setting, as it has always had in my Preface. Cayley replied : CAMBRIDGE, lyh Junt. Dear Tait Thanks for your letter. I am quite willing that the paper 'should be read at the R.S.E. did you mean also that it should be published in the Proceedings? if you did I am quite willing to let this be done instead of sending it to the Messenger. Please make the reference to the preface of the ist as well as the 2nd and 3rd editions and make any additions or explanations to show the context of the " elegant trifles." I was bound to refer to quadriplanar coordinates, because the comparison is between Quaternions, which refer to three-dimensional space, and the Cartesian coordinates x, y, z or in place thereof the quadriplanar coordinates, x, y, x, w. Of course you see my point. I regard the trilinear or quadriplanar coordinates as the appropriate forms including as particular cases the rectangular coordinates x, y or x, y, z and bringing the theory into connexion with that of homogeneous forms of quantics and as remarked in my last letter, it is only in regard to these that the notion of an invariant has its full significance ; so that trilinear coordinates very poor things, Invariants a grand theory, is to me a contradiction. In a long formula of Gauss which you quote for its length, you give the expanded form of a determinant the expression is the perfectly simple one x'-x, y'-y, z'-z -r (dist. P, Qf dx dy dz dx' dy 1 dz" Do you put any immediate interpretation on of = scalar, or consider it merely as a necessary consequence of the premises ? Believe me, dear Tait, yours very sincerely A. CAYLEY. Cayley's paper On Coordinates versus Quaternions and Tait's reply On the Intrinsic Nature of the Quaternion Method were published side by side in the R. S. E. Proceedings, Vol. xx, pp. 271-284. As each of them expressed it in the correspondence, they differed fundamentally. Cayley 166 PETER GUTHRIE TAIT thought in quantics and coordinates ; Tail laid hold of each physical quantity as an entity for which the quaternion notation supplied the complete mental image. To Cayley the quaternion of Hamilton was an algebraic complex which he and Sylvester regarded as a matrix of the second order. For Tait the quaternion was a quantity obeying certain laws, and yielding by its transformations endless physical interpretations. These interpretations were of little interest to Cayley ; just as the general solution of the linear matrix equation had small attractions for Tait. It is a misfortune that in this remarkable correspondence on things quaternion Tail's letters to Maxwell have not been preserved. Towards Hamilton Tait was the loyal disciple, eager to have the master's help at all stages, and always ready to give him the fullest credit as the prime source of every luminous thought. This deep loyalty no doubt prompted Tait to destroy certain of Hamilton's later letters, which did not show the great man at his best. To Cayley Tait turned as to the embodiment of mathematical wisdom and knowledge. In spite of their fundamental difference of outlook on quaternion fields a difference which gradually emerged as they corresponded on the subject Tait had the greatest confidence in Cayley's mathematical intuitions. Once when questioned as to his opinion of Cayley's discoveries in pure mathematics, he remarked: "Cayley is forging the weapons for future generations of physicists." But for Maxwell Tait had not only unstinted admiration as a man of science ; he had for him a deep strong love which had its roots in common school life and grew, strengthened and ripened with the years. He understood to the full Maxwell's intellectual oddities, his peculiar playful humour, his nobility of character, and the deeper thoughts which moved his mind but rarely found expression. Maxwell's letters, which Tait preserved with the greatest care, imply an equivalence of correspondence on Tail's side, the general nature of which may in certain cases be guessed, but the exact terms of which are no longer accessible. Just as Tait placed implicit confidence in Maxwell's physical intuitions, so Maxwell accepted the leadership of Tait in quaternion symbolism and interpretation. He once playfully remarked that he envied Tait the authorship of the quaternion paper on Green's Theorem ; and the extracts given above indicate how much he was influenced by Tait when preparing his great work on Electricity and Magnetism. Tail's quaternion work was indeed the necessary precursor of the qualernion symbolism and nomenclature which Maxwell inlroduced inlo his book. MAXWELL'S INDEBTEDNESS 167 Tait brought out the real physical significance of the quantities 5V a, FVcr, V#. Maxwell's expressive names, Convergence (or Divergence) and Curl, have sunk into the very heart of electromagnetic theory. His suggested word Slope has been replaced by Gradient or Grad, a word of more general etymological intelligibility. But the point is that Maxwell was led to see the far-reaching importance of these conceptions only after they had been presented by Tait in their simple direct quaternion guise. Lame, Green, Gauss, Stokes, Kelvin, and others had the ideas more or less disconnectedly in their minds and utilised them in analysis ; but it is through Hamilton's calculus alone as developed by Tait that the important space relations, Gradient, Divergence, and Curl, appear as parts of a whole. It was Tait who taught Maxwell this deep-lying truth ; and it was Maxwell who spread the good news by his epoch-making treatise on electricity. Most later workers have been content to take the names and the separate conceptions, and reject the central idea embodied in the quaternion operator V. It should not be forgotten, however, that these conceptions were first concisely symbolised and fully discussed in their physical significance by Tait, and remain as a rich legacy from him through Maxwell to the non-quaternionic world. Maxwell gave them names, " rough hewn" he called them in his letter to Tait, whom he invoked as a "good divinity" to "shape their ends properly so as to make them stick." He was their sponsor, but Tait was their parent. Probably very few now using these terms, or their equivalents, in electromagnetic literature have realised the debt they owe to Tait, who first polished the facets of the V diamond. Rough and uncut it passed to him from Hamilton ; and now all the scientific world more or less unconsciously benefit by its radiance. Here then is one outstanding result of Tail's quaternion labours. The many vector quantities which call for consideration in modern electrical theory demand some form of vector notation. This was first realised by Maxwell, who, guided by Tait, adopted Hamilton's vector symbolism. Later writers have in many cases followed Maxwell in the spirit but not in the letter. There have arisen in consequence some six or seven distinct systems of vector notations, which are also called systems of vector analysis. The common elements in these rival systems are, with one exception, also common to the quaternion system, which is demonstrably a real analysis and not simply a notation. So far as mere symbolism is concerned, there is little to choose among these various systems. But what- i68 PETER GUTHRIE TAIT ever be the principle of notation adopted, whether a modified Hamiltonian or Grassmannian, these systems are used as Maxwell used quaternions. Tait inspired Maxwell to use the quaternion vector symbolism. All vector analysts follow Maxwell in substituting for the diffuse Cartesian symbolism a more compact and graphic vector notation. In imitating Maxwell they become disciples of Tait : and once more we realise the close historic connection between Tait's quaternion labours and the developments of modern vector analyses applied to physical problems. It was indeed for the sake of physical applications that Tait made himself master of the quaternion calculus. The conditions under which his Elementary Treatise* was prepared have already been described ; and in judging of the merits of the book, especially in its first edition, we must bear in mind that Hamilton's expressed wishes considerably tied Tait's hands. It was necessary for the sake of the student that the foundations of the calculus should be established in one of the several ways which Hamilton himself had already indicated. But Tait's aim, as indicated in the Preface to his Treatise, was to bring out the value of Quaternions in physical investigations. In the earlier chapters (I refer at present only to the first edition of the Treatise) there was of course little scope for Tait to show any originality of treatment. The first chapter in which he began, as it were, to beat out his own path, was Chapter V, on the solution of equations. In discussing the properties of the linear vector function he followed a line suggested by Hamilton in one of his letters ; but he followed it out in his own way. In Tait's eyes the linear vector function was a strain ; and to a reader acquainted with the theory of strains it is abundantly evident that, even when explicitly confining himself to the purely mathematical side of the question, Tait had the strain conception vividly before his mind. In this early chapter he emphasised those properties which became all important in the later chapters on Kinematics and Physics. The linear vector function continued to occupy Tait's attention through- out the remaining years of his life ; and many interesting applications were added in the second and third editions of his book. These usually appeared, in the first instance, as notes to the Royal Society of Edinburgh. He never found time to put his investigations into the form of a complete memoir. All he was able to accomplish was a series of abstracts giving 1 First edition, 1867; and edition, 1873; 3rd edition, 1890. HELMHOLTZ'S APPRECIATION 169 the main results, and indicating various lines of investigation and gaps to be filled in by subsequent research. The last connected set of notes on the linear vector function began to appear in May 1896, six years after the publication of the third edition of the Treatise, and continued till May 1899 (Set. Pap. Vol. u, pp. 406, 410, 413, 424). During his last illness Tait spoke several times of the importance of the linear vector function, regarding which he felt that some great advance was still to be made. On July 2, 1901, two days before his death, he was able to put some jottings on a sheet of paper, which he handed to his son asking him to place it in a safe place as it contained the germ of an important development. These notes were published in facsimile by the Royal Society together with a commentary in which I indicate their relation to previous papers. The general aim of these later papers is to classify and analyse strains with special reference to related pure strains, those, namely, which are unaccompanied with rotation. One of the most interesting of the results is the theorem that any strain in which there are three directions unaltered can be decomposed into two pure strains ; and, conversely, two pure strains successively applied are equivalent to a strain in which there are three directions unchanged but not in general at right angles to one another. It is more particularly in reference to the development of the properties of V that the second and third editions of the Treatise show marked advance upon the first. As regards the extent and variety of the physical applications the second edition is indeed an entirely different book. This was the edition which was translated into German by Dr G. v. Scherff (Leipzig, Teubner, 1880) and into French by Gustave Plarr (Paris, Gauthier- Villars, 1884). There was a suggestion as early as 1871 to prepare a German translation to be published by Vieweg. But the project was not carried out. Helmholtz, in a letter to Vieweg of date November 19, 1871, spoke of Tait's Quaternions in these words : " As regards Tait's Quaternions, it is indeed an ingenious and interesting mathematical book. But it uses a method of mathematical research which, so far as I know, has hardly been taken up in Germany. When perhaps some enthusiast (Liebhaber) working his way into it, will undertake the translation, the book will, I feel almost certain, find a great sale. It is keen and penetrating, full of new ideas and conceptions, and one can speak of its scientific value only with the highest recognition. But it lies somewhat removed from the usual paths of mathematical T. 22 170 PETER GUTHRIE TAIT study, and the method has not as yet furnished new results of a kind to attract attention." These words were spoken of the first edition, which was strictly, in accordance with its title, an Elementary Treatise. But it is in the later editions that Tait displays his strength. The two chapters on Kinematics and Physical Applications abound in numerous illustrations of the power and flexibility of the calculus. The last chapter, which extends to 101 pages in the third edition, passes over nearly the whole range of mathematical physics, from statics and kinetics of bodies through optics and electrodynamics to the series of remarkable sections dealing with the operator Nabla, V. Here we find treated gravitational and magnetic potential, hydrodynamics, elasticity, varying action, brachistochrones, catenaries, etc. These later sections are not easy reading. They suffer from what Maxwell playfully called " the remarkable condensation, not to say coagulation, of his style," and cannot be fully appreciated by a student who has not already made some acquaintance with the subjects taken up. It should be remembered, however, that this was exactly what Tait had in mind. His aim was not to write a quaternion treatise on mathematical physics, but to show forth the power and conciseness of the quaternion method when applied to important physical problems. With descriptive letter-press interpolated after the manner of scientific treatises and the details of the symbolism worked out, the last chapter of Tail's Elementary Treatise on Quaternions would form a most admirable text book for advanced students in applied mathematics. The greater generality of the quaternion attack as compared with the usual methods introduces some striking novelties, which the ardent student would do well to follow up. MAXWELL'S TYNDALLIC ODE 171 APPENDIX TO CHAPTER IV Maxwell's "Tyndallic Ode" was dedicated to Tait as the chief musician upon Nabla. As with several of Maxwell's clever rhymes, it was no doubt suggested by some of Tait's own utterances. It is at any rate certain that there would have been no Ode if there had been no Tait. Maxwell could not indeed write to his old school mate without indulging in some quaint fancy or hidden joke; and Tait was wont to respond in similar fashion. In this appendix I reproduce the original Ode as it was handed to Tait in the first instance ; and then give a later letter, in which a new verse is added to the Ode, and in which other matters are touched on in Maxwell's inimitable way. The Ode is a humorous imitation of the style of the popular scientific lecturer. Two copies were preserved by Tait. The first rough draft, consisting of four verses, was written in pencil on paper which bears the printed inscription " British Association, Edinburgh." It was evidently dashed off by Maxwell in the B. A. Reception Room during the Edinburgh meeting of 1871. The heading is "To the Chief Musician upon Nabla A Tyndallic Ode Tune The Brook." This was the version which appeared at the time in Nature (Vol. IV, p. 261), where it is spoken of as a paper read before the " Red Lions " ! The other copy, also in Maxwell's hand-writing, is in ink, and seems to have been written the same evening. It must be regarded as the true complete original with its seven verses, the first four of which show several well-marked textual improvements upon the earlier pencilled draft. The peculiar interest of this copy lies in the heading which is elaborately written in Hebrew, in all probability by W. Robertson Smith, to whom the name of Nabla for the inverted Delta was due. This Hebrew title is after the manner of the Hebrew Psalter and is a literal translation of the title given in the first draft. The evidence for the personality of the Hebrew scribe is complete. It is recorded that T. M. Lindsay and W. R. Smith read a paper before Section A On Democritus and Lucretius, A Question of Priority in the Kinetical Theory of Matter (B. A. Reports 1871, Transactions of the Sections p. 30) ; and Principal Lindsay tells me that Robertson Smith was continually in the company of Tait and Maxwell during the Meeting of the British Association, that V was the source of many jokes, and that there is little doubt the Hebrew inscription is from the " reed " of Robertson Smith. The order of the verses is that indicated by pencil in the original, and differs from the order given in Lewis Campbell's Life of Maxwell. There are also some textual variants. 333 i 7 2 PETER GUTHRIE TAIT I come from empyrean fires, From microscopic spaces, Where molecules with fierce desires Shiver in hot embraces. The atoms clash, the spectra flash, Projected on a screen The double D, magnesian b And Thallium's living green. 3 We place our eye where these dark rays Unite in this dark focus; Right on the source of power we gaze Without a screen to cloak us. Next, where the eye was placed at first, We place a disk of platinum; It glows, then puckers like to burst : By Jove, I'll have to flatten him ! MAXWELL'S TYNDALLIC ODE 173 4 I light this sympathetic flame My slightest wish that answers. I sing, it sweetly sings the same, It dances with the dancers. I shout, I whistle, clap my hands, I stamp about the platform, The flame responds to my commands In this form and in that form. 5 This crystal tube, the electric ray Shows optically clean, No dust nor haze within, but stay, All has not yet been seen. What gleam is this of heavenly blue, What wondrous form appearing, What mystic fish, what whale, that through The ethereal void is steering ! Here let me pause, these passing facts These fugitive impressions Must be transformed by mental acts To permanent possessions. Then summon up your grasp of mind Your fancy scientific, That sights and sounds, with thoughts combined, May be of truth prolific. 7 Go to ! prepare your mental bricks, Bring them from every quarter, Firm on the sand your basement fix With best asphaltic mortar. The pile shall rise to heaven on high To such an elevation That the swift whirl with which we fly Shall conquer gravitation. w The following letter written to Tait immediately after the Belfast meeting of the B. A. in 1874, when Tyndall delivered the presidential address, gives an additional verse to the Ode as well as other quaint imaginings. 174 PETER GUTHRIE TAIT GLENLAIR, 27/A Aug. 1874. O. T'. B. A. Trans. 1874. In the expected presidential address the following has been inserted as an antipenultimate. On the atmosphere as a vehicle of sound. What means that thrilling drilling scream ! Protect me ! 'tis the Siren Her heart is fire ! her breath is steam ! Her larynx is of iron ! Sun ! dart thy beams. In tepid streams Rise, viewless exhalations ! And lap me round, that no rude sound May mar my meditations. Phil. Trans. 1874, p. 183. Notes of the actual address are enclosed 1 . The effect on the British Ass. is described in the following adaptation of H. Heine. Tune Loreley. I know not what this may betoken That I feel so wonderful wise, The dream of existence seems broken Since Science has opened mine eyes. At the British Association I heard the President's speech, And the methods and rules of creation Seemed suddenly placed in my reach. My life's undivided devotion To Science I solemnly vowed, I'd dredge up the bed of the ocean, I'd draw down the spark from the cloud ; To follow my thoughts as they go on Electrodes I'd plunge in my brain, Nay, I'd swallow a live entozoon New feelings of Life to obtain. where are those high feasts of science ? where are those words of the wise ! 1 hear but the roar of Red Lions, 1 eat what their Jackal supplies. I meant to be so scientific, But science seems turned into fun ; And this with their roaring terrific These old red lions have done. 1 This refers to the clever rhyming epitome of Tyndall's address, which appeared at the time in Iackwood's Magazine, and will be found in Lewis Campbell's Life of Maxwell. MAXWELL'S TYNDALLIC ODE 175 The following instance of domestic evolution was submitted to Mr Herbert Spencer who was present in Section A. "The ancients made enemies saved from the slaughter Into hewers of wood and drawers of water ; We moderns, reversing arrangements so rude, Prefer ewers of water and drawers of wood." Mr Spencer in the course of his remarks regretted that so many members of the Section were in the habit of employing the word Force in a sense too limited and definite to be of any use in a complete theory of evolution. He had himself always been careful to preserve that largeness of meaning which was too often lost sight of in elementary works. This was best done by using the word sometimes in one sense and sometimes in another and in this way he trusted that he had made the word occupy a sufficiently large field of thought. The operations of differentiation and integration which appeared, from the language of previous speakers, to be already in some degree familiar to members of the Section, were, he observed, essential steps in the normal progress of evolution. It gave him great pleasure to learn that members of Section A were now turning their attention to these processes. He was also glad to see how entirely the Section concurred with his view of nervous action as a wave of accumulation, and he hoped they would also direct their attention to the mode in which the exhausted nerve recuperates its energy by absorption of heat from the neighbouring tissues which form its environ- ment. In Professor Tait's new edition of his work on Thermodynamics he had no doubt this subject would be ably treated. Mr Spencer, whose speech was throughout one of the most didactive, exhaustive and automatic efforts ever exerted, then left the Section. In Section B, Prof. W. K. Clifford read a paper on Chemical equations 1 . The equation was the first selected. He observed that both the constituents of the left member were in the liquid state and that though the resultant might not be familiar to some members, he could warrant it 2XL. From an equation of similar form H a + a, = 2HCI he deduced by an easy transformation whence by extracting the square root H-Cl=o or H=Cl, a result even more remarkable than that obtained by Sir B. C. Brodie. dp -Jf 1 Clifford's paper On the General Equations of Chemical Decomposition was read before Section A; but the Title only is given in the B. A. Reports (1874). An abstract appeared in Nature, Sept. 24, 1874, and was reprinted in the Preface to the Mathematical Papers, p. xxv. Maxwell's amusing parody has a striking superficial resemblance to the original. CHAPTER V THOMSON AND TAIT "T AND T"' OR THOMSON AND TAIT'S NATURAL PHILOSOPHY. THE publication of Thomson and Tail's Natural Philosophy was an event of the first importance in the history of physical science. No more momentous work had been given to the world since the days of the brilliant French mathematicians, Laplace, Lagrange, and Fourier. Thoroughly familiar with the mathematical methods invented and developed by these great writers, Thomson and Tait conceived the project of an all-embracing treatise on Natural Philosophy, in which physical conceptions and mathematical analysis would be rationally blended in an harmonious interpretation of the phenomena of Nature. The intention, it is true, was realised only in part. The first volume appeared in 1867, and a second edition greatly enlarged was issued, Part I in 1879, Part II in 1883. But in the Preface to Part II the authors announced that "the intention of proceeding with the other volumes is now definitely abandoned." No reasons were given; simply the fact was stated. Fortunately there was no longer the same necessity for a continuation of the work on the extensive scale originally imagined. Since the appearance of the first edition, other important works had been published covering a large part of the domain of Natural Philosophy. Clerk Maxwell's Electricity and Magnetism and Lord Rayleigh's Theory of Sound were the most conspicuous of these ; and Thomson's own Reprint of Papers on Electrostatics and Magnetism supplemented in a striking manner the doctrines inculcated in the Natural Philosophy. In all treatises published since its appearance the impress of " Thomson and Tait " is clearly seen. Nevertheless, the world of science must ever lament that the two Scottish Professors did not put in type the sections on Properties of Matter, frequently mentioned in the first edition, and usually with reference to a particularly attractive part of the subject. BEGINNINGS OF T AND T' 177 Occasionally in conversation Tait would refer to the manner in which the great work, familiarly known as "T and T'," took form and grew, and to the amusing difficulties which frequently arose, especially when proof-sheets were mislaid. Some of the earlier reminiscences have been fortunately preserved as contemporaneous history in Tail's letters to Andrews. Those letters were all kept, and through the kindness of the Misses Andrews I am able to give in Tait's own language the genesis and early development of the Natural Philosophy. The first quotation is taken from a letter of date Dec. 18, 1861 : I told Slesser [Tait's successor at Belfast] to tell you that I had agreed to write a joint book on Physics with Thomson. In fact I had nearly arranged the matter with Macmillan, when Thomson, to my great delight, offered to join. We contemplate avoiding the extreme details of methods which embarrass the otherwise excellent French books (vide Jamin, Daguin, etc.) and which, though they may have led their authors to results, are not those that would generally be used in verification. Also we propose a volume, quite unique, on Mathematical Physics. I know of no such work in any language and in fact have acquired all my knowledge of the subject by hunting up papers (often contradictory, and more often unsatisfactory) in Journals, Transactions, Proceedings, etc. Such a book is one I would willingly have paid almost any price for during the last ten years but it does not yet exist. And I think that Thomson and I can do it. We shall commence printing as soon as we have made arrangements with the Publisher, for our first two volumes will contain simply the essence of the Glasgow and Edinburgh Experimental Lectures blended into (I hope) an harmonious whole. A little difficulty arises at the outset, Thomson is dead against the existence of atoms ; I though not a violent partisan yet find them useful in explanation but I suppose we can mix these views well enough... The incidental remark as to Thomson's disbelief in atoms reads strange in these days when we recall how much of Kelvin's later work had to do with the ultimate constitution of matter 1 . For example, before the decade was finished the Vortex Atom, which was suggested to Thomson by Tait's smoke-ring illustrations of Helmholtz's theory of vortex motion (see above, p. 69), had been launched on its chequered voyage in the sea of molecular speculation. Through the kindness of Lady Kelvin I am able to give the following extract from a letter from Tait to Thomson of date Jan. 6, 1862: I like your draft index to Vol. I very well. I have made a few insertions in it, and may perhaps make more before I send it back redrafted, which I will soon 1 See however Thomson's paper on "The Nature of Atoms" (Proc. Manchester Lit. and Phil. Society, 1862) quoted in Larmor's Aether and Matter, p. 319. T. 23 1 78 PETER GUTHRIE TAIT do. Meanwhile I think we may tell Macmillan that the illustrations will in the main be diagrams and not wood engravings (i.e. sketches) You see [from an enclosed letter] the information he wants as to advertising. I wish you would give me a hint or two of your likings and dislikes on such a delicate point. If we can settle on the nature of, and constitution of, our Preface, I think a sort of precis of it would do very well for Advertisement. I think also that we might begin even now to point out as looming in the future our Great work, which so far as I know will be unique ; of course I mean the Principia Mathematica, whatever be the title it is to bear. We may gain considerable credit, and perhaps profit, by the present undertaking ; but the other will go over Europe like a statical charge. Don't you think it would be prudent to warn the profane off such ground by a timely notice? Such as this In preparation, by the same authors, A MATHEMATICAL TREATISE ON NATURAL PHILOSOPHY, containing the elements of the mathematical treatment of Elasticity, Capillarity, Electricity, Heat etc., etc., or anything tending to such a purpose.... A fortnight later (Jan. 20, 1862) Tait detailed to his Belfast friend more concerning the coming book. In reply to a demand for information regarding Heats of Combination, Andrews had referred to the discrepancies between his measurements and those made a little later by Favre and Silbermann ; and Tait replied in his turn : My immediate occasion for information on Heat of Combination (for my lectures) is over, but I am sure Thomson and myself will have particular pleasure in putting you right with regard to Favre and Silbermann, etc. But as matters are at present arranged that will be in our second volume. I will give you here a short index of the proposed Vol. I, with which we are busily engaged. I may merely tell you that I don't feel any alarm on the point you mentioned in your first letter Thom- son has thought far longer, and far more deeply, about matter than I have. The major part of the writing will be done by me as Thomson feels a repugnance to it which is not common. I have already sent him two chapters, and a general abstract of a section; and he speaks of them in the very highest terms.... Here we are, Vol. I. Section I. Chap. I. Introductory. II. Matter, Motion, Mass, etc. III. Measures and Instruments of Precision. IV. Energy, Vis viva, Work, v. Kinematics. VI. Experience (Experiment and Observation). PLAN OF FIRST VOLUME 179 Section II. Abstract Mechanics (Perfect solids, fluids, etc.). Chap. I. Introductory (I have written this and will let you see it soon). II. Statics. III. Dynamics (Laws of Motion, NEWTON. Did you ever read his Latin? Do). IV. Hydrostatics and Dynamics. Section III. Properties of Matter, Elasticity, Capillarity, Cohesion, Gravity, Inertia, etc., etc. (This is to be mine.) Section IV. Sound. Section V. Light. This will give you as good an idea as I yet possess as to the contents of our first volume. All the other physical forces will be included in Vol. II, which will finish up with a great section on the one law of the Universe, the Conservation of Energy. No mathematics will be admitted (except in notes, and these will be more or less copious throughout the volume, being printed in the text but in smaller type). But we shall give very little in that way as my great object in joining Thomson in this work is to have him joined to me in the great work which is to follow, on the Mathematics of Nat. Phil., which I do not believe any living man could attempt alone, not even Helmholtz. On September 9, of the same year, when Thomson seems to have been holidaying in Ireland, Tait wrote to Andrews : Pray impress on Thomson that he should get home again as soon as possible and get into harness else we cannot begin printing in October as was arranged. I think the first chapter, at least my part of it (for I have not got Thomson's yet), will please you. It has greatly pleased myself. It is all about Motion, Actual and Relative, and such matters as Rotations, Displacements, etc., and I hope to make the large type part of it intelligible even to savages or gorillas. It thus appears that, during the few months intervening between the dates of these letters, the plan of the book had somewhat altered. Instead of being Chapter v, Kinematics is to form Chapter i. This was the arrangement finally adopted. Tait at once began to prepare his manuscript. On the fly-leaf of one of the large quarto volumes of blue tinted paper which he used in the early days for lecture notes and all kinds of scientific work we find the inscription "Mainly written in 1861-2 for T and T' (rough beginning)," and then below "Since used (1885) for K. T. 1 of Gases." Pasted to the fly-leaf are the two halves of a sheet of foolscap containing a table of contents similar to but differing in detail from the scheme sent in the 1 That is, "Kinetic Theory." 23 2 180 PETER GUTHRIE TAIT letter to Andrews quoted above. The manuscript proper begins, however, with Kinematics on page i ; and on assigned pages throughout the book introductory sentences on other branches of the subject are given in Tail's clear, strong hand-writing. The paragraphs on kinematics are the fullest and in many cases are the very paragraphs which appear almost verbatim in the published pages of "Thomson and Tait." Page 21 of this MS book is reproduced on the opposite page slightly reduced in size in the ratio of 23:30. It will be seen to correspond very nearly word for word with portions of paragraph 48 in the Treatise, and is given in illustration of the remarks just made, and also as an excellent example of the legibility of Tail's manuscript. In another similar volume marked with ihe same date 1862 there is a well planned series of paragraphs on the Properties of Matter, which no doubt were intended to be the large type portion of Division in of the second volume referred to so pointedly in ihe Preface lo ihe Firsl Edition. These seclions, although never printed as part of " T and T'," were afterwards utilised by Tait in his book, Properties of Matter. The same volume also contains the original draft of the sections on Experience and on Measures and Instruments. Here again many of the sentences are exactly as ihey appear in the Treatise, "T and T'." The name " T and T' " was applied by the authors themselves long before ihere was any hope of the book being published ; and from 1862 onwards till 1892, when " Kelvin" displaced "Thomson," T and T' were the usual forms of address and signature in their letters. The book progressed slowly. Thai il progressed at all was due to Tail's never flagging energies and delerminalion. The original plan of preparing a somewhal elemenlary work on Nalural Philosophy to be followed by a treatise on Mathemalical Physics was ullimalely given up, and ihe Treatise when il appeared in 1867 was a kind of combination of ihe iwo lypes of book at first conceived. Some notion of how the book took shape may be gathered from the following extracls from letters written by Tait to Thomson. Writing on March 30, 1863, Tait said : I think you are unwise in your suggested alteration of the Book. Attractions come naturally and nicely in Prop. Matt. but not sooner^. The fact is, we have 1 It is interesting to note that Tait, in his MS book on Properties of Matter, introduces sections on potential after the account of the Cavendish Experiment. FACSIMILE OF TAIT'S MANUSCRIPT 181 / /2 "#: which is Jdg/t, made a great hit, because is a real quantity whereas q is not, only dq = td$. There are many things in T which are equivalent to this because T has worked at the same subject and worked correctly and all mathematical truth is one, but you cannot expect Clausius to see this unless it stands very plain in print. In short Rankine's state- ments are identical with those of C, but T's are only equivalent... " With respect to our knowledge of the condition of energy within a body, both Rankine and Clausius pretend to know something about it. We certainly know how much goes in and comes out and we know whether at entrance or exit it is in the form of heat or work, but what disguise it assumes when in the privacy of bodies, or, as Torricelli says, " nell' intima corpulenza de' solide naturali," is known only to R., C. and Co." The paper mentioned by Maxwell was not, however, the paper referred to by Tail, as will appear immediately. Among Tail's correspondence an interesting letter from Thomson (Kelvin) 1 Stot, a Scottish word meaning to impinge and rebound, still in constant use among school children of all classes, e.g., to slot a ball. Compare German slossen. THE ENTROPY INTEGRAL 223 was found bearing on this controversy. It is valuable as showing that Tait's views were fully endorsed by his friend. The letter begins abruptly with a quotation from Thomson's 1852 paper "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy " (Proc. R. S. E. in ; also Phil. Mag. iv, Oct. 1852 ; Math, and Phys. Papers, Vol. i, No. LIX). To make the quotation quite intelligible, a preliminary sentence seems to be necessary (see Math, and Phys. Papers, Vol. I, p. 512). "'Let S denote the temperature of the steam...; T the temperature of the condenser; /* the value of Carnot's function, for any temperature t; and R the value of The letter begins with the integral, and then continues the quotation : " ' Then ( I R)w expresses the greatest amount of mechanical effect that can be economised in the circumstances from a quantity wjj of heat produced by the expendi- ture of a quantity w of work in friction, whether of the steam in the pipes and entrance ports, or of any solids or fluids in motion in any part of the engine ; and the remainder, Rw, is absolutely and irrecoverably wasted, unless some use is made of the heat dis- charged from the condenser." The whole thing is included in this illustration and the preceding ' universal ' generalisation of it, of which this is a particular illustration. I don't believe Clausius yet to this day understands as much of the fact of dissipation of energy as is stated in that first paper in which the theory is propounded and the name given, and it does not appear that he has ever made any acknowledgment what- ever of T in the matter. This must be because he does not understand it ; not because he would consciously appropriate what is not his own. " As for the very letters of the formula, T in the same article says, ' If the system of thermometry adopted be such that M= t+a' Accepting Clausius' statement that 'neither the expression (4>/-rJ nor anything of like meaning can be found in the article referred to by T',' the only conclusion is that he is ignorant of the fact that ' dt T+ a and so had his eyes closed to the fact that RwjJ means the same as or t,-j according to the notation of T'. 224 PETER GUTHRIE TAIT " In that same article occurs the expression 1 which (considering that there is absolutely no limitation of the body to which the /// may be applied) supplies with tolerable completeness the / of the bone over which Clausius snarls, and triumphantly justifies T"s [Tait's] 178. "Lastly remark that the very formula for the 'part of it (the heat) rejected as waste into the refrigerator at the temperature T' in the other article' referred to by T ( 179) is _ dxdydecdt . e 7j T The dxdydzcdt here is T"s dq and e J] T is i/t when the thermodynamic thermo- metry is used. " Last lastly remark that while T was keeping the notation p he was working along with Joule (Phil. Mag. 1852, second half-year) to find whether #=///. a agreed approxi- mately enough with air thermometer ordinary reckoning to be a convenient assumption, and (Phil. Mag. 1853 first half-year) intimated that it was so (and set forth the same more fully afterwards with Joule, Trans. R. S.): and from 1851 (Dynamical Theory of H., Part I forward) T had the formula and kept putting forward in all his papers till he finally adopted [? JjT], leaving absolutely no room for Clausius' pretensions. Cl. in fact never showed any right whatever to and till this day has not put it on its right foundation. (Signed) T." The argument in this letter is practically identical with what Thomson himself allowed Tait to publish in the Philosophical Magazine for May 1879 as a note to Tait's own communication " On the Dissipation of Energy " ; but the tone of it is more personal. The statements in the last paragraph 1 [Marginal note by Thomson himself.] The misprints corrected in Phil. Mag., Jan. 1853, bemuddle the formula for final uniform temperature but not the meaning of the dissipation and the formula / / ~ for it. 1 "On the Restoration of Mechanical Energy from an unequally heated Space," Phil. Slag, v, 1853, Math, and Phys. Papers, VoL I, No. LXIII, p. 555. THE CHARGE OF CHAUVINISM 225 can easily be verified by referring to the papers mentioned, which are now conveniently collected together as Articles XLVIII, XLIX, LIX, and LXIII in Volume i of the Mathematical and Physical Papers. All of these except the later parts of XLVIII preceded the publication of Clausius' Fourth Memoir, which appeared in Poggendorf's Annalen in December 1854, and in which the Entropy integral is given by Clausius for the first time. The second edition of Tail's Thermodynamics was published in 1877. In it he makes more emphatic his criticism of the original form of the axiom which Clausius used as the basis of the Second Law of Thermodynamics, and is less eulogistic in his references to Clausius' thermodynamic work in general. The facts are all given in due order ; but Clausius was not satisfied with the manner in which his work was presented, and criticised strongly the general " Tendency " of Tait's historical sketch of the dynamical theory of heat. Tait has by some writers been accused of Chauvinism in his treatment of scientific history. It seems to me that the charge is ill-founded. His championship of Joule and Thomson as two of the real founders of Thermo- dynamics and of Balfour Stewart as having established, in relation to the laws of radiation, certain truths that were almost universally ascribed to Kirchhoff, is probably what is in the mind of those who make the charge. But all are agreed as to the eminence of Joule and Thomson, and nothing that Tait wrote could ever be interpreted as detraction of Kirchhoff. Nevertheless Balfour Stewart's work was not then appreciated at its true value. Even Lord Rayleigh's more recent championship 1 , which is quite as strong as Tait's, has not yet had its full impression on the scientific world. Probably the charge of Chauvinism against Tait may be attributed in some measure to the vigour of his onslaught on anything which he regarded as bad history, and to the glee with which he exposed it. Except in France, Boyle's Law is the name now universally given to what used to be even in this country called Marriotte's Law' ; but it needed Tait to discover evidence in Newton's Principia and in Marriotte's own writings that Marriotte had a skilful way of 1 See his paper " On Balfour Stewart's Theory of the Connexion between Radiation and Convection," Phil. Mag., i, January 1901, regarding which Lord Rayleigh says, "Kirchhoff's independent investigation of a year and a half later [Dec. 1859] is more formal and elaborate but scarcely more convincing." 2 The name occurs even in the First Edition of "T and T," 597 I T. 29 226 PETER GUTHRIE TAIT expounding other people's discoveries as if they were his own. Boyle, no doubt, was an Englishman ; but it cannot be claimed that Marriotte preceded him. There is no Chauvinism here on Tait's part. On the other hand it was Tait, who, accepting the statement of Gay-Lussac, secured for "le Citoyen Charles " the recognition of his rights in relation to the law of gases, named after Dalton in this country and after Gay-Lussac on the Continent. It was Tait more than any other individual writer who popularised Carnot's Cycle of Operations and the Perfect Engine, which are expounded not only in his purely scientific works but also in The Unseen Universe. Again, Tait, by translating Helmholtz's paper on Vortex Motion, gave a new direction to hydrodynamical study in this country. No doubt he felt warmly any attempt, conscious or unconscious, to credit to others discoveries made by any of his own countrymen, and in this he was not peculiar ; but I know of no case in which he claimed for a fellow countryman anything which could be demonstrably associated with the name of another at an earlier date. He used to say that, if laws are to be named after their first discoverers, then Ohm's Law should be called after Fourier, and Doppler's Principle after Romer. In these instances there is nothing Chauvinistic. There are now many books on Thermodynamics of various standards, each having its own merit. But as an account of the fundamental principles in their historic setting Tait's Sketch cannot be surpassed. The promi- nence given to Carnot's Principle, the simplicity and directness of the mathematical methods introduced into the third chapter, the beautiful illus- trations of the transformation of energy given in the second chapter, and the clear account of the manner in which Thomson seized hold of the original conception of absolute temperature, all give the book a character peculiarly its own. Abbe" Moigno, the well-known mathematician and editor of Les Mondes, saw at once the value of the work, and with the help of M. Alfred Le Cyre, published a French translation in 1870. The preface opens with these sentences : " Lorsque je lus pour la premiere fois 1'Esquisse historique de la the'orie dynamique de la chaleur, trois choses me frapperent vivement : i, 1'auteur resume rapidement et completement les travaux accomplis dans cette branche aujourd'hui si tendue de la physique mathematique ; 2, il rend parfaitement & chacun la justice qui lui est due ; 3, il etablit en quelques pages tres-nettes et tres-ele"gantes synthetiquement d'abord, analy- tiquement ensuite, les lois fondamentales de la dynamique de la chaleur." The next work by Tait which calls for notice is his Recent Advances in Physical Science (Macmillan & Co., 1876, 2nd Edition, 1876), the published "RECENT ADVANCES" 227 form of a course of lectures which Tait gave by request to a company of some ninety Edinburgh citizens, mostly professional men. The lectures were delivered in his usual style from the briefest notes, and the book was compiled from the verbatim shorthand report. Of all Tail's published works it gives the best idea of his method as a lecturer. One of its greatest merits to a real student of the subject is the exposition of Carnot's Principle. The name of Carnot was first introduced in Lecture IV on the Transformation of Energy, and occurred again and again throughout the succeeding chapter on The Transformation of Heat into Work. The story goes that when Tait began the Sixth Lecture with the words " I shall commence this afternoon by taking a few further consequences of the grand ideas of Carnot," an elderly pupil sitting towards the back was heard to protest vehemently against the name of Carnot. The published book contains thirteen lectures, but some of the lectures delivered were not published. I remember for example being one of a few undergraduates who were allowed to join the class on two of the occasions on which it met in the University. This change of meeting-place was for the sake of the experimental illustrations, which could not well be performed in an ordinary hall. These two lectures on the Polarisation of Light and Radiant Heat do not appear in the volume, probably because much of the subject matter could not be regarded as recent in the sense in which the doctrine of energy was recent. In addition to the clear exposition of the foundations of the modern theory of energy, Tait gave in these lectures an admirable account of the physical basis of spectrum analysis and the first great discoveries made by Kirchhoff and Bunsen, and by Huggins, Lockyer, Young and others. Astrophysics is now a branch of astronomy claiming its own specialists and possessing its own literature ; but, in the seventies, solar and stellar spectroscopy was but a particular illustration of the broad principle of spectrum analysis. Another important section of Recent Advances was devoted to the discussion of the atom and molecule, their magnitudes and masses, and even their ultimate constitution. The book was reviewed in all our best papers and journals at considerable length, in general with high commendation. The following quotation from an article in the Quarterly Review, Vol. 142, entitled " Modern Philosophers on the Probable Age of the Earth " may be taken as a good type of the appreci- ative notices which abounded. 29 2 228 PETER GUTHRIE TAIT " His lectures now before us, from their nature, belong to the class of composition for which we avow our predilection. They were delivered extempore to a scientific audience, and printed from short-hand notes. They lose nothing of their vigour, to use an expression of Lord Macaulay, by translation out of English into Johnsonese. We are allowed to seize the thought in the making, and if it loses anything in grace, the loss is more than counterbalanced by power. " Those who wish thoroughly to understand the subject of this paper should study Professor Tait's lectures on the sources of energy, and the transformation of one sort of energy into another. Matthew Arnold's phrase, ' let the mind play freely round ' any set of facts of which you may become possessed, often recurs to the mind on reading these papers. There is a rugged strength about Professor Tait's extempore addresses which, taken together with their encyclopaedic range, and the grim humour in which the professor delights, makes them very fascinating. They have another advantage. Men not professionally scientific find themselves constantly at a loss how to keep up with the rapid advance which has characterised recent years. One has hardly mastered a theory when it becomes obsolete. But in Professor Tait we have a reporter of the very newest and freshest additions to scientific thought in England and on the Continent, with the additional advantage of annotations and explanations by one of the most trustworthy guides of our time." The second edition of Recent Advances was translated into German by G. Wertheim (Braunschweig, F. Vieweg und Sohn, 1877); into French by Krouchkoll (Paris, Gauthier Villars, 1887); and into Italian by D' Angelo Emo (Fano, Tipografia Sonciniana, 1887). After the publication of Recent Advances Tait became occupied with the preparation of the second edition of " T and TV In the preface to the second volume which appeared in 1883 it is stated that the continuation of the great work had been abandoned. Tait accordingly turned his attention to the production of a series of elementary text-books, more in the line of what he originally intended before Thomson joined him in 1861. In 1884 and 1885 Tait brought out three books on Heat, Light, and Properties of Matter. What gives the book on Heat its distinguishing features are the introductory chapters, especially Chapter iv. After a rapid historic survey of the growth of the modern conception of heat, Tait introduces the First Law of Thermodynamics. Typical examples are given of the effects and production of heat, leading up to the great principle of Transformation and to the Second Law of Thermodynamics. Then follows Kelvin's definition of absolute temperature. By thus early introducing the true conception of temperature he is able to discuss all the familiar thermal changes in volume PROPERTIES OF MATTER 229 and state in terms of the absolute temperature. A German translation by Dr Ernst Lecher was published in 1885 (Wien, Toeplitz und Deuticke). The book on Light (second edition 1889 ; third edition 1900) was based on the article " Light " which he supplied to the ninth edition of the Encyclopaedia Britannica. Many paragraphs are identical in the two publications ; but the article contains a sketch of Hamilton's Characteristic Function which does not appear in the book ; while the book contains an able discussion of Radiation and Spectrum Analysis, which are done under separate headings in the Encyclopaedia. The mathematical discussions are of a higher order than in Heat, the geometrical theorem on which he finally builds the explanation of the rainbow being especially worthy of note. Particularly interesting are the quotations from Newton, Huyghens, and Laplace with reference to the undulatory and emission theories of light. Of these text-books written by Tait on different branches of natural philosophy perhaps the most characteristic is the Properties of Matter (1885, successive editions, 1890, 1894, 1899 and 1907, the last under the able editorship of Professor W. Peddie). A German translation by G. Siebert was published in 1888 (Wien, A. Pichler's Witwe und Sohn). The Properties of Matter is the book which will best recall to his former students the personality of Tait as a lecturer. It embodies much of the earlier half of the course of study through which Tait gave his many students a " common sense view of the world we live in." The headings of the chapters show the scope of the book, concerning which Lord Rayleigh in his review (see Nature, August 6, 1885, Vol. xxxn) remarked that it was not easy to give a reason why electric and thermal properties of matter should be excluded. The reason is undoubtedly historic, the phrase " Properties of Matter " dating from the time when the mechanical ponderable matter was distinguished from the imponderables heat, light, electricity and magnetism. The first three chapters are devoted to a dis- cussion of what matter is, and contains lively criticism of the metaphysicians. Then come Time and Space ; Impenetrability, Porosity, Divisibility ; Inertia, Mobility, Centrifugal Force ; Gravitation ; Deformability and Elasticity ; Compressibility of Gases and Vapours ; of Liquids ; Compressibility and Rigidity of Solids ; Cohesion and Capillarity ; Diffusion, Osmose, Transpira- tion, Viscosity ; Aggregation of Particles. Lord Rayleigh in his review specially referred to the treatment of elasticity, remarking that the Chapters on Deformation and Compression 230 PETER GUTHRIE TAIT "are perhaps the most valuable part of the work, and will convey a much needed precision of ideas to many students of physics whose want of mathematical training deters them from consulting the rather formidable writings of the original workers in this field. The connection of Young's modulus of elasticity... with the more fundamental elastic constants... is demonstrated in full.... In his treatment of the compression of solids and liquids the author is able to make valuable contributions derived from his own experimental work. "In the chapter on 'Gases' a long extract is given from Boyle's 'Defence of the Doctrine Touching the Spring and Weight of Air,' in order to show how completely the writer had established his case in 1662. As to this there can hardly be two opinions ; and Professor Tait is fully justified in insisting upon his objections to ' Marriotte's Law.' In Appendix IV a curious passage from Newton is discussed, in which the illustrious author appears to speak of Marriotte sarcastically. It is proper that these matters should be put right...." A paragraph from Balfour Stewart's review of Tail's Heat (Nature, June 26, 1884, Vol. xxx) seems to be worthy of quotation as an interesting description of Tail's melhod and style in all his books. " A treatise on heat by one so eminent, both as physicist and teacher of Physics, needs no apology, and yet no doubt the author is right in stating that his work is adapted to the lecture room rather than to the study or the laboratory. Freshness and vigour of treatment are its characteristics, and the intelligent student who reads it conscientiously will rise from it not merely with a knowledge of heat but of a good many other things beside. "'If science,' says our author, 'were all reduced to a matter of certainty, it could be embodied in one gigantic encyclopaedia, and too many of its parts would then have... little more than the comparatively tranquil or rather languid interest which we feel in looking up in a good gazetteer such places as Bangkok, Akhissar, or Tortuga. 1 Not a few text-books of science are precisely of the nature of such a guide without its completeness, and while they carry the student successfully to the end of his journey, the way before him is made so utterly deficient in human interest that he reaches his goal with a sigh of relief, and looks back upon his journey with anything but satisfaction as a task accomplished rather than a holiday enjoyed. Now the presence of such a human interest is the great charm of the work before us. It may be a fancy on our part, but we cannot help likening our author to the well-known guide of Christiana and her family. Both have been equally successful in the slaughter of those giants whom the older generation of pilgrims had to find out for themselves and encounter alone. But here the likeness ends, for it is quite certain that those who place themselves under the scientific guidance of our author will not be treated like women or children, but they will be taught to fight like men. And surely to combat error is an essential part of the education of the true man of science, for, if not trained up as a good soldier of the truth to defend the king's highway, he will be only too apt to turn freebooter and gain his livelihood by preying on the possessions of others." These text-books, especially the Heat and Properties of Matter, were of "DYNAMICS" 231 course very useful helps to the students of the general class of Natural Philosophy. In the earlier days " Little T and T'" and Tait's Thermodynamics were the only books which were serviceable in supplementing the lectures. The former was a sealed book to the majority of those studying for the ordinary M.A. degree ; and the latter in its first chapter covered a limited ground, while most of the second chapter was too condensed food for the ordinary mind to assimilate. We had, accordingly, to trust largely to the lectures, for the mode of treatment and the illustrations given were peculiarly Tait's own. The article "Mechanics" which Tait contributed to the Encyclopaedia Britannica in 1883 formed the foundation of an advanced text-book on Dynamics, which was published in 1895 (A. and C. Black). Having used paper-bound copies of the article as a text-book in his Honours Class for the twelve intermediate years Tait was able, when its publication in usual book form was determined on, to modify and improve along lines which experience had indicated. As explained by Tait in the letter to Cayley quoted above (p. 155), the Britannica article was originally planned by Maxwell; but the details had to be arranged by Robertson Smith, the editor, so as not to overlap other articles. The book accordingly, although largely a reprint, contains sections on Attraction, Hydrodynamics, and Waves which were not in the original article. If from the point of view of the student the book has a fault, it is that of brevity and conciseness. But there can be only one opinion as to its thoroughness and accuracy. The ground covered is greater than in any other book on the subject, for it includes not only what is ordinarily understood by Dynamics of particles and rigid bodies but also the more important parts of elasticity and motion of fluids. The foundations are Newton's Laws of Motion ; for although Tait had himself, in scientific papers and otherwise, tried to devise a system free from the explicit assumption of Force in the Newtonian sense, yet to the end he regarded Newton's Laws of Motion as the most practical way of introducing the student to a study of the subject. Naturally there are strong resemblances between Tait's Dynamics and " T and T'," especially in certain modes of proof; but in his own book Tait restrains himself from treating developments which make a great demand upon the mathematical knowledge of the reader. Occasionally the extreme brevity of a statement is such that the student on a first reading fails to see immediately all that is implied ; but a critical examination of such statements shows that they are complete without being 232 PETER GUTHRIE TAIT redundant. Among the parts which are particularly characteristic of Tail's methods the following may be mentioned : discussion of Fourier's series, of strains, of Attractions and Potential, of Action (under which is included the flow of electricity in a surface), of the strength of tubes under internal and external pressures, of the bending and vibration of rods, of vortex motion, and of surface waves on fluids. Perhaps the practical nature of the book is best indicated by the way in which Lagrange's generalised coordinates are introduced. Having established in ordinary Cartesian symbolism Hamilton's principle of Varying Action, Tait then uses this principle to deduce the usual Lagrangian equations of motion. The demonstration is not general or exhaustive, but it is sufficient for the kind of problems which most naturally present themselves to a student beginning the study of higher dynamics. Tail's demonstrations, whether geometrical or analytical, are characterised by neatness and elegance. He used to say that he could always improve a demonstration given by some one else. When reading a newly published paper he was able very rapidly to come to an opinion as to its originality and accuracy. Thus, as already noticed (p. 1 1 3), he was very critical of certain of the mathematical processes used in investigations regarding the kinetic theory of gases. If a theorem could not be proved without a prodigious array of symbols covering pages, he had a feeling that the theorem was not worth the proving. His attitude of mind towards much of mathe- matical literature is well brought out in the answer he gave to one of his sons about the year 1878. He was turning over the pages of a mathematical journal which had just come by post. When asked if he was going to read the journal right through, he remarked : "Certainly not. I am not such a flat as to read other people's mathematics. I look to see what result the beggar brings out, and then if he's right I can usually find a shorter cut." About 1892 Tait formed the project of printing a small pamphlet of concise paragraphs to take the place of lecture notes for his students, who would thus be able to pay undivided attention to the explanations and amplifications given in the lectures. Some twenty or thirty pages were put in type, but pressure of other work, more particularly the editing of the reprint of Scientific Papers, prevented the project being carried to completion. When reminded by the publishers that these pages had been lying in type for nearly six years Tait felt that he was not able to carry out fully the original intention, and compromised the matter by confining these notes to a highly condensed discussion of Newton's Laws of Motion, in other words, the "NEWTON'S LAWS OF MOTION" 233 foundations of dynamics. A small book of fifty-two pages, and entitled Newton s Laws of Motion, was finally published in 1899 by A. and C. Black. The book contained a brief introduction on Matter and Energy and then two chapters on Kinematics and Dynamics respectively. In a review by A. E. H. L. in the columns of Nature (Vol. LXI, January 18, 1900) the book was commended as being "for the most part excellent, the geometrical methods employed being especially elegant. Room is found for an elementary discussion of strain, of compounded simple harmonic motions, of attractions, including the distribution of electricity on a sphere under influence, and of the velocity of waves along a stretched cord, in addition to interesting and unhackneyed accounts of the matters which are the stock in trade of books on the elements of mechanics. The book on the whole is thoughtful, in many parts it is much better than the current text-books on the subject, and the parts that call for criticism are no worse than the corresponding parts of most other books on the subject; but they are the most important parts, and they might have been so much better. There was a great opportunity, but it has been missed." Part of the criticism virtually amounted to a complaint that certain sections were not sufficiently expanded. Tail's own preface may be regarded as an answer to this kind of objection ; for the book is explicitly stated not "to be a text-book " but " a short and pointed summary of the more important features of ... the basis of the subject." For example "to explain" (as was desired by the critic) " the mathematical notion of a limit " requires not " some space " but a good deal of space, if the explanation is to be complete. Nevertheless, the following brief paragraphs show that the conception of physical and dynamical continuity, on which fundamentally the notion of the limit rests, was explicitly recognised by Tait : 10. When we pass from the consideration of displacement to that of motion, the idea of time necessarily comes in. For motion essentially consists in continued displacement. In the kinematics of a point, all sorts of motion are conceivable : but we limit ourselves to such as are possible in the case of A f article of matter. 11. These limitations are simple, but very important. (a) The path of a material particle must be a continuous line. [A gap in it would imply that a particle could be annihilated at one place and reproduced at another.] (y9) There can be no instantaneous finite change in the direction, or in the speed, of the motion. \Inertia prevents these, unless we introduce the idea of finite transformations of energy for infinitely small displacements, or (in the Newtonian system) infinite forces.].... 14. If the speed be variable its value, during any period, must sometimes exceed T. 3 234 PETER GUTHRIE TAIT and sometimes fall short of the average value. But (by 1 1 (/9), above, and therefore solely in consequence of inertia) the shorter the period considered, the more closely will the actual speed of a material particle agree with the average value : and that without limit. Again, very early in the book, Tait warned the reader against the inevitable anthropomorphism which clings to our words and phrases ; yet he was attacked for using Newton's anthropomorphic definition of force as a cause and at the same time pointing out its true nature as simply a rate of change of a quantity in time. As regards the general criticism that Tait followed too slavishly Newton's presentation of the foundations of dynamics, there is a great deal to be said on both sides. Tail's experience had convinced him that for junior men Newton's method was the best, dealing as it did with immediate sensations and perceptions. For that reason he called the book Newton s Laws of Motion, But, although in this small pamphlet Tait felt himself compelled to adhere to Newton's method, every one interested in the subject knew that he had in one published paper attempted to establish the laws of motion on a wider basis free from the explicit use of the word Force. This paper " On the Laws of Motion, Part I," was printed (but only in Abstract) in the Proceedings of the Royal Society of Edinburgh, 1882 ; and a German translation appeared in a German mathematical journal. The Second Part was never written out in a form suitable for publication. When busy with the preparation of the 1882 paper, Tait wrote to Cayley on Nov. 20, 1882, in these words : Do you know of any attempt to construct the whole system of Mechanics (for it would, under the circumstances, be absurd to call it Dynamics') from general principles, such as Conservation and Transformation of Energy, Least Action, etc., without intro- ducing either Force, Momentum or Impulse? I have worked out a scheme of the kind having been led to it by writing a long article for the Encyc. Brit. Not that it goes in there, of course, but because in speaking of the anthropomorphic terms in which Newton's Laws are expressed (e.g., a body compelled by force to do so and so ; a body persevering in its state of etc. etc.) I tried to find out some simple mode of getting rid of what I find Maxwell has called Personation. Of course, Force constantly comes in, but not in any sense as an agent, merely as the space-rate of transformation of energy. It plays a part in some sense akin to that of temperature-gradient in heat-conduction. But I see, by the words I have doubly underlined, how very difficult it is to avoid anthropomorphism. I suppose it must always be so, unless scientific men protest effectively against " the sun rises," " the wind blows," etc. etc. If any such scheme has appeared, I should like to consider it before bringing my notions before the R.S.E. CAYLEY ON NEWTON'S LAWS 235 I have said in my article that no one who has ever rolled a pea on the table under the tips of his index and middle fingers, crossed, will afterwards believe anything whatever on the testimony of his "muscular sense" alone. Yet what other ground have we, for believing in the objectivity of force, than the impression on our muscular sense ? On January 20, 1883, Cayley replied : " Dear Tait, I ought to have written ever so long ago in answer to your question as to the construction of a system of mechanics from general principles without Force, Momentum, or Impulse but it could only have been to say that I did not know of any attempt at such a construction the idea was quite new to me, and I have not taken it in enough to see anything about it myself so that you will have lost nothing by the delay. I hope your proposed communication to the R.S.E. will be published " On February 26, 1883, Cayley acknowledged receipt of the Paper in these words : " Dear Tait, The whole discussion is beyond me I understand force I do not under- stand energy. I am willing to believe that Newton's Action = Reaction potentially includes d'Alembert's principle but I never saw my way with the former, and do see my way with the latter and I accept Virtual Velocities + d'Alembert's principle as the foundation of Mechanics. In this position of outer darkness, it would be quite useless to attempt any remark on your paper. " I send herewith a paper from the A.M.J.; please look at the statement pp. 2-4 of Abel's theorem in its most simple form " Tait replied as follows : 38 GEORGE SQUARE, EDINBURGH. 28/2/83. My dear Cayley, Many thanks for your paper, which I have already looked at and will read. It seems to me that this work may, with a little trouble, be brought to bear on the very important and difficult question of Kinetic Stability. If so, I hope you will develop it largely. I suppose you know Boole's paper in Phil. Trans. It was from it that I first got a notion of what Abel's Theorem really means. Your disclaimer in reference to my Abstract is really a vote in my favour. For Virtual Velocities is merely the principle of Energy in a mathematical guise ; and d'Alembert's principle is either the first or second interpretation of Newton's Lex III ; and you say that you adhere to them. I say advisedly, either the first or second, for there are two quite distinct things which go by the name of d'Alembert's principle : 30 2 236 PETER GUTHRIE TAIT I. Some people say this is d'Alembert : Let m at x, y, z, be a particle the applied forces being X, Y, Z t and the internal forces f, 17, Then mx=X + ^, &c., whence 2(w^) = S^", &c., f &c. going out. And the same sort of thing when factors &e, &c. are used. II. Others say this is d'Alembert : Let the notation be as before. Then the statical conditions are 2(^r+f) = o, &c. whence, introducing the reversed effective forces, you get for the kinetical conditions 2(X-mx + Z) = o, &c. And the same sort of thing with any permissible displacements as factors. I is merely Lex III direct. II is amply met by Newton's second interpretation of Lex III, where he points out the Reactiones, "ex acceleratione oriundis," as forces to be taken into account. Which is your view of d'Alembert ? But there is a point in my paper which may interest you, where I show that the hitherto puzzling Least Action merely expresses the inertia condition, so far as the component motion parallel to an equipotential surface is concerned ...... In the winter of 1874, a few months after the delivery by Tyndall of his famous presidential address before the British Association at Belfast, it began to be whispered among the students of Edinburgh University that Tait was engaged on a book which was to overthrow materialism by a purely scientific argument. When, in the succeeding spring, The Unseen Universe* appeared it was at once accepted as the fulfilment of this rumour. The title page of the book contained the words, "THE UNSEEN UNIVERSE, or Physical Speculations on a Future State. The things which are seen are temporal, but the things which are not seen are eternal. London, Macmillan and Co. 1875"; and at the top was a trefoil knot, the symbol of the Vortex Atom imagined by Thomson and discussed at considerable length by the authors of the book. In spite of its anonymous publication it seemed to be known from the beginning that the work was written by Balfour Stewart and P. G. Tait. Anyone at all familiar with Tail's scientific style and with his views of the historic development of the modern theory of energy could not fail to see that his hand must have been mainly responsible 1 Tait greatly enjoyed Gustav Wiedemann's punning criticism that the book should be called the "Unsinn Univers." "THE UNSEEN UNIVERSE" 237 for Chapters in and iv, on the Present Physical Universe and on Matter and Ether. Whatever may be thought of the argument of the book, one merit was that, by means of these physical chapters, the great ideas associated with the names of Carnot and Joule were presented to the minds of vast numbers of readers who would never otherwise have come into touch with them. The book was heralded in a curious old-world fashion by means of an anagram, which was published in Nature, October 15, 1874, and signed West, that is, according to Tail's elucidation, We S(tewart) T(ait). This anagram spelled out the sentence " Thought conceived to affect the matter of another universe simultaneously with this may explain a future state." This sentence may therefore be regarded as one of the central doctrines of The Unseen Universe. It occurs at the end of paragraph 199 in Chapter VH. The book created a great sensation. It was at once recognised as the work of a scientific author or authors. The fourth edition, which was published in April 1876, exactly a year after the first publication, appeared with the authors' names on the title-page, and subsequent reprints did not differ materially from this edition. The one conspicuously new feature was an introduction setting forth succinctly the motive of the book, which had been strangely misunderstood by some of the earlier critics. Also a few important changes were made throughout, but on the whole the book was essentially the same through all the editions. One addition to the original form of the text is well worth attention, being a fine example of the kind of humour which Tait occasionally delighted in. The end of paragraph 103 originally ended with the sentence: " The one (i.e., matter) is like the eternal unchangeable Fate or Necessitas of the ancients ; the other (i.e., energy) is Proteus himself in the variety and rapidity of its transformations." In the later editions this sentence is followed by six lines of Greek verse, namely : vTa,TOV repay T?}? 8' aJn' 'A.vdy/cr)<{ ear' dxivijTov aOevot, fj,6vr] y dirdvTuv ravro Siafi.evovff' del /3poTo/3 AXXe?. (1) I have no Assistant. If I can do you any service, well and good, if not, why not ? (2) Prof. Liveing will lend you his bags, give you his gases, and furnish you with lime light. If you are particular about your lantern bring it yourself, like Guy Fawkes or the man in the Moon. The gases will go for half an hour. If you want them for longer, say so. Bring your own galvanometer. (3) Thermopylae exist, but Peltier only in the form of a repulsive electrometer, and the effet Thomson is an " effet defective." 322 252 PETER GUTHRIE TAIT (4) The Senate house is a place to write in, to graduate in, and to vote in. The Public Orator I believe can speak in it provided he employs the Latin tongue. What those venerable walls would say if the vernacular were sounded within them I dare not even think. If you have a good audience there will not be much echo from Geo. II or Pitt, and if you erect a lofty platform, the light spot on the screen and the under side of your table may be seen by all. (5) If you do your @H as you did your Quaternions to the British Asses you will do very well, always remembering that to speak familiarly of a 2nd Law, as of a thing known for some years, to men of culture who have never even heard of a ist Law, may arouse sentiments unfavourable to patient attention Both Moral and Intellectual Entropy are noble subjects, though the dictum of Pecksniff concerning the idea of Todgers be unknown to me and not easily verified. I do not know much about reversible operations in morals. The science or practice depends chiefly on the existence of singular points in the curve of existence at which influences, physically insensible, produce great results. The man of tact says the right word at the right time, and a word spoken in due season how good is it ? The man of no tact is like vinegar upon natron when he sings his songs to a heavy heart. The ill timed admonition only hardens the conscience, and the good resolution, made just when it is sure to be broken, becomes macadamized into pavement for the abyss. Yrs ^ df In the early seventies the Director of the Museum of Science and Art in Edinburgh, now the Royal Scottish Museum, arranged courses of scientific lectures to the Industrial Classes. Courses were given by Dr Buchan, Professor Tait, Professor Crum Brown, Dr (afterwards Professor) McKen- drick on special branches of their respective sciences. Tail's lectures on Cosmical Astronomy were delivered during January 1874, the titles of the successive lectures being (i) Our sources of information as to bodies non- terrestrial, (2) Their dimensions and distances, (3) Their masses and rates of motion, (4) Their composition and modes of aggregation, (5) Their mutual action, (6) Their ultimate state. When preparing these lectures, Tait took the opportunity of fulfilling a promise to Dr Norman Macleod, the editor of Good Words, and contributed a corresponding series of articles to that popular magazine. At the British Association Meeting in Glasgow in 1876, Professor Andrews was President ; and Tait out of a feeling of loyalty to his old friend and colleague agreed to give one of the evening lectures. The subject was " Force," and its main scientific features were a strong demand for accuracy in scientific language, and a demonstration that force in the strictly Newtonian LECTURE ON "FORCE" 253 sense of the word has no real objective existence but is a mere space-variation of energy. The lecture abounded in illustrations from all sides of human experience and was severely critical on laxity of thought and of expression on the part, not only of journalists essaying to speak of scientific things, but even of recognised writers of scientific books. The lecture was published in Nature and subsequently reprinted as an appendix to the second edition of Recent Advances in Physical Science. It appears as No. xxxvu in the Scientific Papers. The raciest and most critical passages were, however, omitted. In these Tait let himself go to the intense amusement of many of his audience and to the horror of some who did not quite appreciate the form Tait's humour occasionally assumed. Lord Brougham and Professor Tyndall, though not explicitly named, were singled out as having been guilty of carelessness of diction in the expression of scientific truth ; and the audience were startled when Tait capped his exposure of the recent President of the British Association by the question, "Are these thy gods, Oh Israel ?" Tait used to tell how he early noticed in the audience one alert listener who seemed almost to anticipate the points, so quickly did he respond to the humour and sarcasm of the lecturer. His expectant and eager expression was a delightful inspiration to Tait. The real fun of the lecture is well shown forth in the humorous verses which Maxwell sent to Tait a few days later, with the heading " For P. G. Tait but not for Ebony" meaning Black-woods Magazine. The following version is taken from the original draft, which was pasted into Tait's Scrap Book. REPORT OF TAIT'S LECTURE ON FORCE: B. A. 1876. Ye British Asses, who expect to hear Ever some new thing, I've nothing new to tell, but what, I fear, May be a true thing, For Tait comes with his plummet and his line Quick to detect your Old bosh new dressed, in what you call a fine Popular lecture. Whence comes that most peculiar smattering Heard in our section ? Pure nonsense, to a scientific swing Drilled to perfection ? 254 PETER GUTHRIE TAIT That small word "Force" is made a barber's block Ready to put on Meanings most strange and various, fit to shock Pupils of Newton. Ancient and foreign ignorance they throw Into the bargain ; The Sage of Leipzig mutters from below Horrible jargon. The phrases of last century in this Linger to play tricks Vis viva and Vis Mortua and Vis Accekratrix. These long-nebbed words that to our text-books still Cling by their titles, And from them creep, as entozoa will, Into our vitals. But see ! Tait writes in lucid symbols clear One small equation ; And Force becomes of Energy a mere Space-Variation. Force, then, is force, but mark you ! not a thing, Only a Vector; Thy barbed arrows now have lost their sting Impotent spectre! Thy reign, O Force ! is over. Now no more Heed we thine action ; Repulsion leaves us where we were before, So does attraction. Both Action and Reaction now are gone. Just ere they vanished, Stress joined their hands in peace, and made them one ; Then they were banished. The Universe is free from pole to pole Free from all forces. Rejoice! ye stars like blessed gods ye roll On in your courses. No more the arrows of the Wrangler race, Piercing, shall wound you. Forces no more, those symbols of disgrace Dare to surround you. But those whose statements baffle all attacks, Safe by evasion, Whose definitions, like a nose of wax, Suit each occasion, LECTURE ON THUNDERSTORMS 255 Whose unreflected rainbow far surpassed All our inventions, Whose very energy appears at last Scant of dimensions : Are these the gods in whom ye put your trust, Lordlings and ladies ? The " secret potency of cosmic dust " Drives them to Hades. While you, brave Tait ! who know so well the way Forces to scatter, Calmly await the slow but sure decay Even of matter. On January 29, 1880, in the City Hall, Glasgow, under the auspices of the Glasgow Science Lecture Association, Tait gave a lecture on Thunder- storms, for which he collected a vast amount of curious information. At one time he intended to include this lecture in the first volume of the Scientific Papers ; but gave up the idea on the ground no doubt that the lecture did not contain any distinct addition of his own to our scientific knowledge. Nevertheless it touches in an interesting way on many of the features of thunderstorms. It was reported in full in the columns of Nature and it has been thought well to reprint it in this volume as an admirable specimen of the popular scientific lecture. Had Tait devoted himself to popular lecturing, there is no doubt he would have impressed himself strongly on the community. He had a full command of terse vigorous language, a pleasant resonant voice, the power of speaking deliberately and emphatically, a clear utterance, and a strong personality behind it all. His humour could always be counted upon as adding a sparkle to the physical arguments and descriptions. Finally, his honesty of mind would never lead him to gloss over difficulties, or give a doubtful lead on the applications of some broad principle. Tait acted as Reviewer and Critic of many scientific works chiefly in the columns of Nature and occasionally in the Philosophical Magazine. It may be said emphatically that Tait never wrote for the mere sake of writing. His desire always was to bring out what he believed to be the truth, and this he did in many cases by exposing the errors. He had no patience with rhetorical writing in a book claiming to be scientific ; and it went hard with an author who indulged in such verbiage. Tait had also 256 PETER GUTHRIE TAIT a keen eye for faults of expression, for looseness of phrase, and for lack of precision in the ideas which it was intended to communicate. As examples of the severely critical vein we may refer to his two articles on Sensation and Science in Nature, Vol. iv, July 6, 1871, and Vol. vi, July 4, 1872. The first is devoted to an exposure of the extraordinary misconception on the part of Professor Haughton as to the physical significance of the Principle of Least Action. The criticism is deservedly severe. In a writer of Haughton's standing and reputation the misconception was inexcusable, for the simple reason that his words would carry weight and be accepted as scientific truth by very many of his hearers and readers. Haughton's aim was to apply to the animal kingdom this principle of least action, which appears sometimes in various more or less irreconcileable guises as " the minimum of effort," "the least quantity of material," "a wonderful economy of force," "a performing its allotted task (by a muscle) with the least amount of trouble to itself," " minimum amount of muscular tissue," and so on. "A very Proteus is this so-called principle," wrote Tait. "There is no knowing where to have it It is a minimum, an economy, a least quantity, and what not ; sometimes of effort, sometimes of material, then of trouble, and anon of muscular tissue, or of force of the same kind as that with which the bee constructs its cell ! But the most curious feature about it is that in none of its metamorphoses does it in the slightest degree resemble the Least Action of Maupertuis, with which it would seem throughout to be held as identical." The second article on Sensation and Science dealt with a book on Comets and things in general by Professor Zollner of Leipzig, an extraordinary man of brilliant but unequal parts. The work, as Tait described it, "deals not alone with the nature of Comets, the inferiority of British to German physicists, and the grave offence of which a German is guilty when he sees anything to admire except at home ; but also with the errors of Thomas Buckle, the relations of Science to Labour and Manufacture, and the analogies of development in Languages and Religion." Zollner was specially wrath with Helmholtz for sanctioning the German translation of Thomson and Tail's Natural Philo- sophy. Tait could not bring himself to take the man and his writings seriously ; but Helmholtz thought it necessary in his Preface to the Second Part of the German edition to reply at considerable length to Zollner's attacks. A translation (by Crum Brown) of this reply is given in Nature, Vol. x, 1874. DE MORGAN'S "BUDGET OF PARADOXES" 257 When reading Zollner's book Tail called Tyndall's attention to the terrible onslaught the author had made on Tyndall's theory of comets. In his reply Tyndall wrote : " I have glanced over it (Zollner's book) not read it, myself. I can see that he means to mangle me kill me first and chop me into mince- meat afterwards. But whether it is that the fire of my life has fallen to a cinder, the book has produced very little disturbance in my feelings.... Ten years ago I should have been at the throat of Zollner, but not now. I would rather see you and Clausius friends than Zollner and myself. Trust me C. is through and through an honest high-minded man." The reference to Clausius had to do with the controversy then going on between Tail and Clausius in regard to the second law of thermodynamics. In many of his reviews Tait found occasion not only to hit off the character of the writer but also to descant on the true way and the false in the teaching of science. A few examples may still be of interest. The following extracts are from a review which appeared in Nature on January 30, 1873, f De Morgan's inimitable Budget of Paradoxes. This work is absolutely unique. Nothing in the slightest degree approaching it in its wonderful combinations has ever, to our knowledge, been produced. True and false science, theological, logical, metaphysical, physical, mathematical, etc., are interwoven in its pages in the most fantastic manner : and the author himself mingles with his puppets, showing off their peculiarities, posing them, helping them when diffident, restraining them when noisy, and even occasionally presenting himself as one of their number. All is done in the most perfect good-humour, so that the only incongruities we are sensible of are the sometimes savage remarks which several of his pet bears make about their dancing master. De Morgan was a man of extraordinary information. We use the word advisedly as including all that is meant by the several terms knowledge, science, erudition, etc. Everywhere he was thoroughly at home. An old edition and its value-giving peculiarities or defects, a complex mathematical formula with its proof and its congeners, a debated point in theology or logic, a quotation from some almost- unheard-of author, all came naturally to him, and from him. With a lively and ready wit, and singularly happy style, and admirable temper, he was exactly fitted to write a work like this. And every page of it shows that he thoroughly enjoyed his task. De Morgan was a very dangerous antagonist. Ever ready, almost always thoroughly informed, gifted with admirable powers of sarcasm which varied their method according to the temperament of his adversary, he was ready for all comers, gaily tilted against many so-called celebrities ; and upset them. It is unfortunate that the issue of his grand contest with Sir William Hamilton (the great Scottish Oxford Philosopher) is but in part indicated in this volume it is softened down, T. 33 258 PETER GUTHRIE TAIT in fact, till one can hardly recognise the features of the extraordinary Athenaeum correspondence of 1847. There the ungovernable rage of the philosopher contrasts most strongly with the calm sarcasm of the mathematician, who was at every point his master, and who "played" him with the dexterity and the tenderness of old Isaak himself! But it is characteristic of De Morgan that, though he was grievously insulted throughout the greater part of this discussion, no trace of annoyance seems to have remained with him after the death of his antagonist ; for none would gather from the Budget more than the faintest inkling of the amount of provocation he received. Take again the following introductory paragraphs of a very full and instructive review of Clerk Maxwell's great work on Electricity and Mag- netism. The review appeared in Nature, April 24, 1873. In his deservedly celebrated treatise on " Sound," the late Sir John Herschel felt himself justified in saying, " It is vain to conceal the melancholy truth. We are fast dropping behind. In Mathematics we have long since drawn the rein and given over a hopeless race." Thanks to Herschel himself, and others, the reproach, if perhaps then just, did not long remain so. Even in pure mathematics, a subject which till lately has not been much attended to in Britain, except by a few scattered specialists, we stand at this moment at the very least on a par with the elite of the enormously disproportionate remainder of the world. The discoveries of Boole and Hamilton, of Cayley and Sylvester, extend into limitless regions of abstract thought, of which they are as yet the sole explorers. In applied mathematics no living men stand higher than Adams, Stokes, and W. Thomson. Any one of these names alone would assure our position in the face of the world as regards triumphs already won in the grandest struggles of the human intellect. But the men of the next generation the successors of these long-proved Knights are beginning to win their spurs, and among them there is none of greater promise than Clerk Maxwell. He has already, as the first holder of the new chair of Experimental Science in Cambridge, given the post a name which requires only the stamp of antiquity to raise it almost to the level of that of Newton. And among the numerous services he has done to science, even taking account of his exceedingly remarkable treatise on " Heat," the present volumes must be regarded as preeminent. We meet with three sharply-defined classes of writers on scientific subjects (and the classification extends to all such subjects, whether mathematical or not). There are, of course, various less-defined classes, occupying intermediate positions. First, and most easily disposed of, are the men of calm, serene, Olympian self- consciousness of power, those upon whom argument produces no effect, and whose grandeur cannot stoop to the degradation of experiment ! These are the d priori reasoners, the metaphysicians, and the Paradoxers of De Morgan. Then there is the large class, of comparatively modern growth, with a certain amount of knowledge and ability, diluted copiously with self-esteem haunted, how- ever, by a dim consciousness that they are only popularly famous and consequently straining every nerve to keep themselves in the focus of the public gaze. These, MAXWELL'S "ELECTRICITY AND MAGNETISM" 259 also, are usually, men of " paper " science, kid-gloved and black-coated with no speck but of ink. Finally, the man of real power, though (to all seeming) perfectly unconscious of it who goes straight to his mark with irresistible force, but neither fuss nor hurry reminding one of some gigantic but noiseless " crocodile," or punching engine, rather than of a mere human being. The treatise we have undertaken to review shows us, from the very first pages, that it is the work of a typical specimen of the third of these classes. Nothing is asserted without the reasons for its reception as truth being fully supplied there is no parade of the immense value of even the really great steps the author has made no attempt at sensational writing when a difficulty has to be met ; when necessary, there is a plain confession of ignorance without the too common ac- companiment of a sickening mock-modesty.... The main object of the work, besides teaching the experimental facts of electricity and magnetism, is everywhere clearly indicated it is simply to upset completely the notion of action at a distance. Everyone knows, or at least ought to know, that Newton considered that no one who was capable of reasoning at all on physical subjects could admit such an absurdity: and that he very vigorously expressed this opinion. The same negation appears prominently as the guiding consideration in the whole of Faraday's splendid electrical researches, to which Maxwell throughout his work expresses his great obligations. The ordinary form of statement of Newton's law of gravitation seems directly to imply this action at a distance ; and thus it was natural that Coulomb, in stating his experimental results as to the laws of electric and magnetic action which he discovered, as well as Ampere in describing those of his electrodynamic action, should state them in a form as nearly as possible analogous to that commonly employed for gravitation. The researches of Poisson, Gauss, etc., contributed to strengthen the tendency to such modes of representing the phenomena ; and this tendency may be said to have culminated with the exceedingly remarkable theory of electric action proposed by Weber. All these very splendid investigations were, however, rapidly leading philosophers away towards what we cannot possibly admit to be even a bare representation of the truth. It is mainly to Faraday and W. Thomson that we owe our recall to more physically sound, and mathematically more complex, at least, if not more beautiful, representations. The analogy pointed out by Thomson between a stationary distri- bution of temperature in a conducting solid, and a statical distribution of electric potential in a non-conductor, showed at once how results absolutely identical in law and in numerical relations, could be deduced alike from the assumed distance-action of electric particles, and from the contact-passage of heat from element to element of the same conductor. After quoting Maxwell's own frank and ample acknowledgement of his debt to these two men, Tait continued : It certainly appears, at least at first sight, and in comparison with the excessively 33 260 PETER GUTHRIE TAIT simple distance action, a very formidable problem indeed to investigate the laws of the propagation of electric or magnetic disturbance in a medium. And Maxwell did not soon, or easily, arrive at the solution he now gives us. It is well-nigh twenty years since he first gave to the Cambridge Philosophical Society his paper on Faraday's Lines of Force, in which he used (instead of Thomson's heat-analogy) the analogy of an imaginary incompressible liquid, without either inertia or internal friction, subject, however, to friction against space, and to creation and annihilation at certain sources and sinks. The velocity-potential in such an imaginary fluid is subject to exactly the same conditions as the temperature in a conducting solid, or the potential in space outside an electrified system. In fact the so-called equation of continuity coincides in form with what is usually called Laplace's equation. In this paper Maxwell gave, we believe for the first time, the mathematical expression of Faraday's Electro-tonic state, and greatly simplified the solution of many important electrical problems. Since that time he has been gradually developing a still firmer hold of the subject, and he now gives us, in a carefully methodised form, the results of his long-continued study.... It is quite impossible in such a brief notice as this to enumerate more than a very few of the many grand and valuable additions to our knowledge which these volumes contain. Their author has, as it were, flown at everything ; and, with immense spread of wing and power of beak, he has hunted down his victims in all quarters, and from each has extracted something new and invigorating for the intellectual nourishment of us, his readers. In his review of Maxwell's remarkable little book Matter and Motion (Nature, Vol. xvi, June 14, 1877) Tait was led into an interesting discussion of the necessity for accuracy and for paying attention to the things which count. He pointed his moral by quoting some sentences from recent text-books on Natural Philosophy (which it had been his intention to review along with Maxwell's book), and then proceeded to contrast them with Maxwell's unpretentious volume. Clerk Maxwell's book is not very easy reading. No genuine scientific book can be. But the peculiar characteristic of it is that (while anyone with ordinary abilities can read, understand, and profit by it) it is the more suggestive the more one already knows. We may boldly say that there is no one now living who would not feel his conceptions of physical science at once enlarged, and rendered more definite by the perusal of it... Clerk Maxwell's work, then, is simply Nature itself, so far as we understand it. The peaks, precipices, and crevasses are all there in their native majesty and beauty. Whoso wishes to view them more closely is free to roam where he pleases. When he comes to what he may fear will prove a dangerous or impassable place, he will find the requisite steps cut, or the needful rope attached, sufficiently but not obtrusively, by the skilful hand of one who has made his own roads in all directions, and has thus established a claim to show others how to follow. MAXWELL'S "MATTER AND MOTION" 261 In the rival elementary works the precipice and the crevasse are not to be seen : there are, however, many pools and ditches ; for the most part shallow, but very dirty. You are confined to the more easily accessible portions of the region. In the better class of such books these are trimly levelled the shrubs and trees are clipped into forms of geometrical (i.e. unnatural) symmetry like a Dutch hedge. Smooth straight walks are laid down leading to old well-known "points of view," and, as in Trinity of former days, undergraduates are warned against walking on the grass-plats. These " royal roads " to knowledge have ever been the main cause of the stagnation of science in a country. He would be a bold man indeed who would venture to assert that the country which, in times all but within the memory of many of us, produced such mighty master-minds as Lagrange, Fourier, Ampere, and Laplace, does not now contain many who might well have rivalled the achievements even of men like these. But they have no chance of doing so; they are taught, not by their own struggles against natural obstacles, with occasional slight assistance at a point of unexpected difficulty, but by being started off in groups, "eyes front" and in heavy marching order, at hours and at a pace determined for all alike by an Official of the Central Government, along those straight and level (though perhaps sometimes rough) roads which have been laid down for them ! Can we wonder that, whatever their natural fitness, they don't now become mountaineers? It seems appropriate at this point to reproduce parts of the account which Tail gave of the life and work of his life-long friend James Clerk Maxwell. Schoolboys at the same school, contemporaries at Cambridge, profoundly interested in the same great branch of science, and constant correspondents throughout their busy lives, they were the truest of friends knit heart to heart by bonds which only death could sever. Tait had an unstinted admiration for the genius of Maxwell, a deep love for the man, and a keen appreciation of his oddities and humour. In their correspondence they were always brimming over with fun and frolic, and puzzling each other with far-fetched puns, and literary allusions of the most extraordinary kind. I have been able throughout this memoir to give a good deal from Maxwell's letters to Tait. Unfortunately the other side of the correspondence has disappeared. Some lines written to Tait on a half sheet of note-paper whose contents referred to proof corrections are worth preserving as a neat example of Maxwell's power of moralising on physical truth : "The polar magnet in his heart of steel Earth's gentle influence appears to feel ; But trust him not ! he's biassed at the core Force will but complicate that bias more, No Power but that of all-dissolving Fire Can quite demagnetize the hardened wire." a62 PETER GUTHRIE TAIT The following extracts are from Tait's account of Maxwell's work in Nature, January 29, 1880: At the instance of Sir W. Thomson, Mr Lockyer, and others I proceed to give an account of Clerk Maxwell's work, necessarily brief, but I hope sufficient to let even the non-mathematical reader see how very great were his contributions to modern science. I have the less hesitation in undertaking this work that I have been intimately acquainted with him since we were schoolboys together. If the title of mathematician be restricted (as it too commonly is) to those who possess peculiarly ready mastery over symbols, whether they try to understand the significance of each step or no, Clerk Maxwell was not, and certainly never attempted to be, in the foremost rank of mathematicians. He was slow in "writing out," and avoided as far as he could the intricacies of analysis. He preferred always to have before him a geometrical or physical representation of the problem in which he was engaged, and to take all his steps with the aid of this : afterwards, when necessary, translating them into symbols. In the comparative paucity of symbols in many of his great papers, and in the way in which, when wanted, they seem to grow full-blown from pages of ordinary text, his writings resemble much those of Sir William Thomson, which in early life he had with great wisdom chosen as a model. There can be no doubt that in this habit, of constructing a mental representation of every problem, lay one of the chief secrets of his wonderful success as an investigator. To this were added an extraordinary power of penetration, and an altogether unusual amount of patient determination. The clearness of his mental vision was quite on a par with that of Faraday ; and in this (the true) sense of the word he was a mathematician of the highest order. But the rapidity of his thinking, which he could not control, was such as to destroy, except for the very highest class of students, the value of his lectures. His books and his written addresses (always gone over twice in MS) are models of clear and precise exposition ; but his extempore lectures exhibited in a manner most aggra- vating to the listener the extraordinary fertility of his imagination. Clerk Maxwell spent the years 1847-50 at the University of Edinburgh, without keeping the regular course for a degree. He was allowed to work during this period, without assistance or supervision, in the Laboratories of Natural Philosophy and of Chemistry : and he thus experimentally taught himself much which other men have to learn with great difficulty from lectures or books. His reading was very extensive. The records of the University Library show that he carried home for study, during these years, such books as Fourier's Thforie de la Clialeur, Monge's Gtome'trie Descriptive, Newton's Optics, Willis' Principles of Mechanism, Cauchy's Calcul Difftrentiel, Taylor's Scientific Memoirs, and others of a very high order. These were read through, not merely consulted. Unfortunately no list is kept of the books consulted in the Library. One result of this period of steady work consists in two elaborate papers, printed in the Transactions of the Royal Society of Edinburgh. The first (dated 1849), "On the Theory of Rolling Curves," is a purely mathematical treatise, supplied with an immense collection of very elegant particular examples. The second (1850) is "On the Equilibrium of Elastic Solids." Considering the age of the writer at the time, this MAXWELL'S SCIENTIFIC WORK 263 is one of the most remarkable of his investigations. Maxwell reproduces in it, by means of a special set of assumptions, the equations already given by Stokes. He applies them to a number of very interesting cases, such as the torsion of a cylinder, the formation of the large mirror of a reflecting telescope by means of a partial vacuum at the back of a glass plate, and the Theory of Orsted's apparatus for the compression of water. But he also applies his equations to the calculation of the strains produced in a transparent plate by applying couples to cylinders which pass through it at right angles, and the study (by polarised light) of the doubly-refracting structure thus produced. He expresses himself as unable to explain the permanence of this structure when once produced in isinglass, gutta percha, and other bodies. He recurred to the subject twenty years later, and in 1873 communicated to the Royal Society his very beautiful discovery of the temporary double refraction produced by shearing in viscous liquids. During his undergraduateship in Cambridge he developed the germs of his future great work on "Electricity and Magnetism" (1873) in the form of a paper "On Faraday's Lines of Force," which was ultimately printed in 1856 in the "Trans, of the Camb. Phil. Society." He showed me the MS of the greater part of it in 1853. It is a paper of great interest in itself, but extremely important as indicating the first steps to such a splendid result. His idea of a fluid, incompressible and without mass, but subject to a species of friction in space, was confessedly adopted from the analogy pointed out by Thomson in 1843 between the steady flow of heat and the phenomena of statical electricity. After a fairly exhaustive account of Maxwell's principal contributions to scientific literature, Tait continued : Maxwell has published in later years several additional papers on the Kinetic Theory, generally of a more abstruse character than the majority of those just described. His two latest papers (in the Phil. Trans, and Camb. Phil. Trans, of last year) are on this subject : one is an extension and simplification of some of Boltzmann's valuable additions to the Kinetic Theory. The other is devoted to the explanation of the motion of the radiometer by means of this theory. Several years ago (Nature, Vol. XII, p. 217), Prof. Dewar and the writer pointed out, and demonstrated experimentally, that the action of Mr Crookes' very beautiful instrument was to be explained by taking account of the increased length of the mean free path in rarefied gases, while the then received opinions ascribed it either to evaporation or to a quasi- corpuscular theory of radiation. Stokes extended the explanation to the behaviour of disks with concave and convex surfaces, but the subject was not at all fully investigated from the theoretical point of view till Maxwell took it up. During the last ten years of his life he had no rival to claim concurrence with him in the whole wide domain of molecular forces, and but two or three in the still more recondite subject of electricity. " Everyone must have observed that when a slip of paper falls through the air, its motion, though undecided and wavering at first, sometimes becomes regular. Its general path is not in the vertical direction, but inclined to it at an angle which remains nearly constant, and its fluttering appearance will be found to be due to 264 PETER GUTHRIE TAIT a rapid rotation round a horizontal axis. The direction of deviation from the vertical depends on the direction of rotation.... These effects are commonly attributed to some accidental peculiarity in the form of the paper..." So writes Maxwell in the Cam. and Dud. Math. Jour. (May, 1854) and proceeds to give an exceedingly simple and beautiful explanation of the phenomenon. The explanation is, of course, of a very general character, for the complete working out of such a problem appears to be, even yet, hopeless ; but it is thoroughly characteristic of the man, that his mind could never bear to pass by any phenomenon without satisfying itself of at least its general nature and causes. Similar in character to the quotations just given are the following culled from a series of articles and reviews which appeared in Nature between 1875 and 1887, all dealing with the life and work of Sir George Stokes. The earliest article formed the fifth of the Nature series of Scientific Worthies (Nature, July 15, 1875). GEORGE GABRIEL STOKES. A great experimental philosopher, of the age just past, is reported to have said, " Show me the scientific man who never made a mistake, and I will show you one who never made a discovery." The implied inference is all but universally correct, but now and then there occur splendid exceptions (such as are commonly said to be requisite to prove a rule), and among these there has been none more notable than the present holder of Newton's Chair in Cambridge, George Gabriel Stokes, Secretary of the Royal Society. To us, who were mere undergraduates when he was elected to the Lucasian Professorship, but who had with mysterious awe speculated on the relative merits of the men of European fame whom we expected to find competing for so high an honour, the election of a young and (to us) unknown candidate was a very startling phenomenon. But we were still more startled, a few months afterwards, when the new Professor gave public notice that he considered it part of the duties of his office to assist any member of the University in difficulties he might encounter in his mathematical studies. Here was, we thought (in the language which Scott puts into the mouth of Richard Cceur de Lion), " a single knight, fighting against the whole mle of the tournament." But we soon discovered our mistake, and felt that the undertaking was the effect of an earnest sense of duty on the conscience of a singularly modest, but exceptionally able, and learned man. And, as our own knowledge gradually increased, and we became able to understand his numerous original investigations, we saw more and more clearly that the electors had indeed consulted the best interests of the University ; and that the proffer of assistance was something whose benefits were as certain to be tangible and real as any that mere human power and knowledge could guarantee. And so it has proved. Prof. Stokes may justly be looked upon as in a sense one of the intellectual parents of the present splendid school of Natural Philosophers whom STOKES' MATHEMATICAL PAPERS 265 Cambridge has nurtured the school which numbers in its ranks Sir William Thomson and Prof. Clerk Maxwell. All of these, and Stokes also, undoubtedly owe much (more perhaps than they can tell) to the late William Hopkins. He was, indeed, one whose memory will ever be cherished with filial affection by all who were fortunate enough to be his pupils. But when they were able, as it were, to walk without assistance, they all (more or less wittingly) took Stokes as a model. And the model could not but be a good one : it is all but that of Newton himself. Newton's wonderful combination of mathematical power with experimental skill, without which the Natural Philosopher is but a fragment of what he should be, lives again in his successor. Stokes has attacked many questions of the gravest order of difficulty in pure mathematics, and has carried out delicate and complex experimental researches of the highest originality, alike with splendid success. But several of his greatest triumphs have been won in fields where progress demands that these distinct and rarely associated powers be brought simultaneously into action. For there the mathematician has not merely to save the experimenter from the fruitless labour of pushing his enquiries in directions where he can be sure that (by the processes employed) nothing new is to be learned ; he has also to guide him to the exact place at which new knowledge is felt to be necessary and attainable. It is on this account that few men have ever had so small a percentage of barren work, whether mathematical or experimental, as Stokes. The following review by Tait of Stokes' Mathematical and Physical Papers (Vols. i and n) appeared in Nature, December 13, 1883 : There can be but one opinion as to the value of the collection before us, and (sad to say) also as to the absolute necessity for it. The author, by common consent of all entitled to judge, takes front rank among living scientific men as experimenter as well as mathematician. But the greater part of his best work has hitherto been buried in the almost inaccessible volumes of the Cambridge Philosophical Transactions, in company with many other papers which deserve a much wider circulation than they have yet obtained. Stokes' well-deserved fame was thus practically secured by means of a mere fraction of his best work.... The present publication will effect a very remarkable amount of transference of credit to the real author, from those who (without the possibility of suspicion of mala fides) are at present all but universally regarded as having won it. Two or three years ago, only, the subject for a Prize Essay in a Continental scientific society was The nature of unpolarized, as distinguished from polarized, light. But all that science is even yet in a position to say, on this extremely curious subject, had been said by Stokes thirty years ago in the Cambridge Philosophical Transactions.... Prof. Stokes has wisely chosen the chronological order, in arranging the contents of the volumes. Such a course involves, now and then, a little inconvenience to the reader ; but this is much more than compensated for by the insight gained into the working of an original mind, which seems all along to have preferred a bold attack upon each more pressing scientific difficulty of the present, to attempts at smoothing the beginner's road into regions already well explored. When, however, Prof. Stokes does T. 34 266 PETER GUTHRIE TAIT write an elementary article, he does it admirably. Witness his Notes on Hydrodynamics, especially that entitled On Waves. Before that article appeared, an article as comprehensive as it is lucid, the subject was almost a forbidden one even to the best student, unless he were qualified to attack the formidable works of Laplace and Airy, or the still more formidable memoirs of Cauchy and Poisson. Here he finds at least the main points of this beautiful theory, disencumbered of all unnecessary complications, and put in a form intelligible to all who have acquired any right to meddle with it. It is quite impossible to tell how much real good may be done by even one article like this. Would there were more such ! There are few, even of the most gifted men, who do not occasionally require extraneous assistance after the earlier stages of their progress : all are the better for it, even in their maturer years. The contents of these two volumes consist mainly, almost exclusively, of papers connected with the Undiilatory Theory of Light or with Hydrodynamics. On the former subject at least, Stokes stands, without a living rival, the great authority. From the Aberration of Light, the Constitution of the Luminiferous Ether, the full explanation of the singular difficulties presented by Newton's Rings, to the grand theoretical and experimental treatise on the Dynamical Theory of Diffraction, we have a series of contributions to this branch of optics which, even allowing for improved modern surroundings, will bear comparison with the very best work of Newton, Huyghens, Young, or Fresnel in the same department. Specially remarkable among the Hydrodynamical papers is that on Oscillatory Waves, to which a very important addition has been made in the reprint. The investigation of the " profile " of such a wave is here carried to a degree of approximation never before attempted. Besides these classes of papers we have the very valuable treatise on Friction of Fluids in Motion, and on the Equilibrium and Motion of Elastic Solids. This was Stokes" early masterpiece, and it may truly be said to have revolutionized our knowledge on the subjects it treats. To mention only one point, though an exceedingly important one, it was here that for the first time was clearly shown the error of assuming any necessary relation between the rigidity and the compressibility of an elastic solid, such as had been arrived at from various points of view by the great Continental mathematicians of the earlier part of the present century. Of the few purely mathematical papers in the present volumes the most important is the well-known examination of the Critical Values of the Sums of Periodic Series, a subject constantly forced on the physicist whenever he has to treat a case of discontinuity.... Tait contributed to Nature three reviews on Stokes' Burnett Lectures, which were delivered in Aberdeen in three Courses and were published in three corresponding volumes about a year apart. The review of the First Course, On the Nature of Light, appeared on April 10, 1884 (Vol. xxix). The Second Course, On Light as a Means of Investigation, was reviewed on August 20, 1885 (Vol. xxxn). The following extracts are of special interest: STOKES' "BURNETT LECTURES" 267 The interest raised by the first series of these lectures is fully sustained by this second instalment, though the subject-matter is of a very different order. Then, the main question was the nature of light itself; now, we are led to deal chiefly with the uses of light as an instrument for indirect exploration. It is one of the most amazing results of modern science that the nature of mechanisms, too minute or too distant to be studied directly with the help of the microscope or the telescope, can be thus in part at least, revealed to reason. This depends on the fact that a ray of light, like a human being, bears about with it indications alike of its origin and of its history ; and can be made to tell whence it sprang and through what vicissitudes it has passed. The lecturer begins by pointing out that this indirect use of light already forms an extensive subject ; and he then specially selects for discussion half-a-dozen important branches of it... The first of these is Absorption. Here we have the explanation of the colours of bodies; the testing ray having gone in, and come out "shorn." This leads to the application of the prism in the immediate discrimination of various solutions which, to the unaided eye, appear to have the same colour. It is shown how, by a mere glance, the chemist may often be saved from fruitless toil, occasionally from grave error. From the study of what rays are absorbed, the transition is an easy and natural one to the study of what becomes of them when they are absorbed. Here we have heating, chemical changes, phosphorescence, etc. The remainder of the lecture is devoted to an exceedingly interesting treatment of the beautiful subject of fluorescence. The second lecture begins with Rotation of the Plane of Polarisation of light by various liquids, with its important application to saccharimetry. Then we have Faraday's discovery of the corresponding phenomenon produced in the magnetic field, with its application in the discrimination of various classes of isometric compounds Then comes the " still vexed " question of the history of Spectrum Analysis. The present view of it must, of course, be carefully read : it is much too long to be here extracted in full, and to condense would be to mutilate it. Of course the claims of the author himself are the only ones to which scant justice is done. But the President of the British Association of 1871 fortunately gave, in his opening address, the means of filling this lacuna. Just as the Gravitation-theory of an early Lucasian Professor was publicly taught in Edinburgh University before it became familiar among scientific men, so the present Lucasian Professor's suggestions for the analysis of the solar atmosphere, by means of the dark lines in the spectrum, were publicly explained in the University of Glasgow for eight successive years before the subject became generally known through the prompt and widespread publicity given to the papers of Bunsen and Kirchhoff! The following are Sir William Thomson's words of 1871: "It is much to be regretted that this great generalisation was not published to the world twenty years ago... because we might now be (sic) in possession of the inconceivable riches of astronomical results which we expect from the next ten years' investigation by 342 268 PETER GUTHRIE TAIT spectrum analysis, had Stokes given his theory to the world when it first occurred to him." The third lecture is devoted to the information which spectrum analysis affords as to the chemical composition of the sun's atmosphere, and its physical condition ; the classification of stars, the constitution of nebulae, and the nature of comets.... The remarks on the nebulae and on comets will be read with great avidity ; and, by the majority of readers, with some surprise. For it is stated that the planetary nebulae, " making abstraction of the stellar points, consist of glowing gas." And of comets we find : " There can no longer be any doubt that the nucleus consists, in its inner portions at least, of vapour of some kind, and we must add incandescent vapour..." An ingenious suggestion as to the source of this incan- descence is introduced as the "green-house theory." The nucleus is supposed to be surrounded by an envelope of some kind, transparent to the higher but opaque to the lower forms of radiation. Thus solar heat can get freely at the nucleus, but cannot escape until it has raised the nucleus (in part at least) to incandescence. The coma and tail are formed by the condensation of small quantities of this vapour, so that they are mere mists of excessive tenuity. Herschel's suggestion, that the development of the tail is due to electric repulsion exerted by a charge on the sun, is spoken of with approval ; and the production of the requisite charge of the mist- particles is regarded as a concomitant of condensation. Nothing, however, is said as to the opposite charge which the comet itself must receive, nor of the peculiar effects which would arise from this cause : whether in the form of a modification of the shape of the comet's head, or of a modification of its orbit and period due to a constantly increasing attraction exerted by the sun upon a constantly diminishing mass. Of course, if this novel theory can stand the test of a full comparison with facts, it will have established its claim to become part of science. But it is hard to take leave of the simple old ideal comet : the swarm of cosmical brickbats : some- thing imposing because formidable : and to see it replaced by what is, in comparison, a mere phantom, owing its singular appearance to the complexity of the physical properties it possesses and the recondite transformations perpetually taking place in its interior. The old idea of a comet's constitution was not only formidable, but was capable of explaining so much, and of effecting this by means so simple and so natural, that one almost felt it deserved to be well-founded ! The new idea makes it resemble the huge but barely palpable 'Efreet of the Arabian Nights, who could condense himself so as to enter the bottle of brass with the seal of Suleymdn the son of Daood ! The following sentences are from Tait's review of the Third Course of Stokes' Burnett Lectures, namely, On the Beneficial Effects of Light (see Nature, June 2, 1887, Vol. xxxvi) : This volume completes the course of the First Burnett Lecturer on the New Foundation. We have already (Vol. XXIX, p. 545, and Vol. XXXII, p. 361) noticed STOKES' "BURNETT LECTURES" 269 the first two volumes ; and we are now in a position to judge of the work as a whole. But we must first speak of the contents of the present volume. The author commences by extending the term "Light" to radiation in general.... Next comes a curious suggestion of analogy between the behaviour of fluo- rescent bodies (which always degrade the refrangibility of the light they give off) and the heat-radiation from bodies which have been exposed to sun-light. Sun-light, as it reaches us after passing through the atmosphere, is less rich in ultra-red rays than is the radiation from the majority of terrestrial sources; while the radiation from bodies which have been heated by direct sun-light is entirely ultra-red. Here we have, for the terrestrial atmosphere, the "green-house theory" which, in the second course, was applied to explain some of the singular phenomena exhibited by comets. This is followed by an extremely interesting discussion of the functions of the colouring matters of blood and of green leaves : with the contrasted effects, upon plants, of total deprivation of light, and of continuously maintained illumination. A particularly valuable speculation, as to the probable nature of the behaviour of chlorophyll, is unfortunately too long for extraction. So far, radiation has been treated without any special reference to vision. But the author proceeds to describe the physical functions and adaptations of the eye : with particular reference to the arrangements for obviating such of the theoretical defects as, while involved in its general plan, would also tend to diminish its practical usefulness. The introduction of this obviously natural proviso, one which we do not recollect having seen prominently put forward till now, exhibits in a quite new light the intrinsic value of those objections to the " argument from design " which have been based upon the alleged imperfection of the eye as an optical instrument. The analogy of fluorescence is once more introduced, but now for the purpose of suggesting a mechanical explanation of the mode in which the sense of vision is produced. This is brought forward after the modern photo-chemical theory of vision has been discussed The triplicity of the colour-sense, and the mechanism of single vision with two eyes, are treated at some length. But throughout this part of the work it is frankly confessed that there are many elementary questions, some of funda- mental importance, which we are still unable even approximately to answer No higher praise need be bestowed on the scientific part of this third volume than is involved in saying that it is a worthy successor to the other two. Together, they form a singularly instructive, and yet (in the best sense) popular, treatise on a fascinating branch of natural philosophy. Were this their only aim, no one could deny that it has been thoroughly attained. But their aim is of a loftier character. Here and there throughout the work there have been occasional references to the main purpose which has determined the author's mode of arranging his facts and his deductions from them. In the few closing pages this purpose is fully developed, and a brief but exceedingly clear statement shows at once how much in one sense, and yet how little in another, can be gathered as to the personality and the character of the Creator from a close and reverent study of His works. 270 PETER GUTHRIE TAIT Tait contributed important reviews on two works by W. K. Clifford. The first of these, which appeared in Nature (Vol. xvm) on May 23, 1878, referred to the Elements of Dynamic, Part i, Kinematic, which was particularly interesting to Tait because of the use the author made throughout of quaternion methods. I give the review in full : Though this preliminary volume contains only a small instalment of the subject, the mode of treatment to be adopted by Prof. Clifford is made quite obvious. It is a sign of these times of real advance, and will cause not only much fear and trembling among the crammers but also perhaps very legitimate trepidation among the august body of Mathematical Moderators and Examiners. For, although (so far as we have seen) the word quaternion is not once mentioned in the book, the analysis is in great part purely quaternionic, and it is not easy to see what arguments could now be brought forward to justify the rejection of examination-answers given in the language of quaternions especially since in Cambridge (which may claim to lay down the law on such matters) Trilinear Coordinates, Determinants, and other similar methods were long allowed to pass unchallenged before they obtained formal recognition from the Board of Mathematical Studies. Everyone who has even a slight knowledge of quaternions must allow their wonderful special fitness for application to Mathematical Physics (unfortunately we cannot yet say Mathematical Physic !) : but there is a long step from such semi-tacit admissions to the full triumph of public recognition in Text-Books. Perhaps the first attempt to obtain this step (in a book not ostensibly quaternionic) was made by Clerk Maxwell. In his great work on Electricity all the more important Electrodynamic expressions are given in their simple quaternion form though the quaternion analysis itself is not employed : and in his little tract on Matter and Motion {Nature, Vol. XVI, p. 119) the laws of composition of vectors are employed throughout. Prof. Clifford carries the good work a great deal farther, and (if for this reason alone) we hope his book will be widely welcomed. To show the general reader how much is gained by employing the calculus of Hamilton we may take a couple of very simple instances, selecting them not because they are specially favourable to quaternions but because they are familiar in their Cartesian form to most students. Every one who has read Dynamics of a Particle knows the equations of non-acceleration of moment of momentum of a particle, under the action of a single centre of force, in the form xy yx = o\ yz zy =o\ zx xz oj with their first integrals, which express the facts that the orbit is in a plane passing through the centre, and that the radius-vector describes equal areas in equal times. But how vastly simpler as well as more intelligible is it not to have these three equations written as one in the form CLIFFORD'S "DYNAMIC" 271 and the three first integrals above referred to as the immediate deduction from this in the form Vpp = a. Take again Gauss's expression for the work done in carrying a unit magnetic pole round any closed curve under the action of a unit current in any other closed circuit. As originally given, it was [a long unwieldy expression in x, y, z, x', y', d~\. With the aid of the quaternion symbols this unwieldy expression takes the compact form i . pdpdp' The meanings of the two expressions are identical, and the comparative simplicity of the second is due solely to the fact that it takes space of three dimensions as it finds it ; and does not introduce the cumbrous artificiality of the Cartesian coordinates in questions such as this where we can do much better without them. In most cases at all analogous to those we have just brought forward, Prof. Clifford avails himself fully of the simplification afforded by quaternions. It is to be regretted, therefore, that in somewhat higher cases, where even greater simplification is attainable by the help of quaternions, he has reproduced the old and cumbrous notations. Having gone so far, why not adopt the whole ? Perhaps the most valuable (so far at least as physics is concerned) of all the quaternion novelties of notation is the symbol .9 .9 .3 V = I + J~~ + K^~, dx J dy dz whose square is the negative of Laplace's operator : i.e. A glance at it is sufficient to show of what extraordinary value it cannot fail to be in the theories of Heat, Electricity, and Fluid Motion. Yet, though Prof. Clifford discusses Vortex Motion, the Equation of Continuity, etc., we have not observed in his book a single V. There seems to be a strange want of consistency here, in coming back to such " beggarly elements " as Sftt + ty + &,iv instead of - SVtr, especially when, throughout the investigation, we have a used for ui+vj+ wk, and when, in dealing with strains, the Linear and Vector Function is quite freely used. Again, for the vector axis of instantaneous rotation of the element at x, y, z (p), when the displacement at that point is (when there are no molecular couples) the equation = o which (in virtue of the property of a, already spoken of) is equivalent to three independent scalar conditions. Suppose we wish to express these, without the a, in the form of one vector condition. Mr McAulay boldly writes the first term as 5.00'jVj or rather as S.a^V, for in so simple a case the suffixes are not required, and the strain-function is self- conjugate under the restriction above. Then, at once, the property of a shows us that