LIFE AND SCIENTIFIC WORK

OF

PETER GUTHRIE TAIT

CAMBRIDGE UNIVERSITY PRESS

aonDon: FETTER LANE, E.G.

C. F. CLAY, MANAGER

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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; later ^e m
Edinburgh, 16-52; Tail as lecturer, 17-22; contributions to Chambers1 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,
39—41; 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., 114—116; 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,
149—150; 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, 186—187; description of scope and contents
of "T and T'," 187—194; Maxwell's criticisms, 195; German translation, 195-197; early
pamphlets, 197—200; "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 " Geits1 " 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 Glasgow1. 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 parson1 — 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 story1 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 M°Cosh, 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 accommodation1.
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 theory1.

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

4—2

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

5—2

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 carefully1. 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 Jufy> 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

6—2

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 autumn1. 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

7—2

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 (6—8 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.

8—2

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 printing1, 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 Exn Papers just at the time of the Assn 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 lecture1 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

9—2

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 Wanklyn1 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.
Henderson1 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 Smith1, 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 k<x (A#)s? A. is as nearly as possible the same in both.

I am working now with a platinum spiral heated by a current. I measure its
radiation by a pile and galvanometer, then suddenly1 for an instant shunt it into the
bridge and find its resistance. I am getting very steady results with different battery
power.

One of my students has attained great skill in finding specific heats ; and has
found that of best conducting copper to be slightly above that of bad, but to rise
more slowly with increase of temperature.

I have asked Tyndall whether he couldn't induce the Shoeburyness people to
fire a few stone bullets at a stone wall and get a party with spectroscopes to examine
the resulting flash. I think comets might be thus elucidated.

I sent a copy of my article to Lady Thomson last week.

Yours T'.

PS. Are you remembering poor Balfour and the Vortices ?

PS. [ Written across the top of the first page of the letter^ Your sets of tenths
of a unit not o.k. I get different values when I use 100 and 1000 as the next sides
of the quadrilateral. For instance I find 1775 to 1000 and 179 to 100 for the same
pair of wires.

In 1870 Tail began to communicate to the Royal Society his brief Notes
from the Physical Laboratory, the first set including J. W. Nichol's3 experi-

1 A marginal note by Thomson reads "March 28/71 Why suddenly? Rather keep it
always in the bridge under a constant El. M. F."

1 J. W. Nichol, F.R.A.S., accompanied the Transit of Venus Expedition to the Hawaian
Islands, and published in the Proc. R. S. E. (Vol. ix, 1875) a graphic account of a visit to
Mauna Loa and Killauea, the remarkable volcanos with their lava lakes only 15 miles apart
but differing in level by 10,000 feet. He died young; and his mother founded in his memory
the Nichol Foundation in the Physical Laboratory of Edinburgh University.

10 — 2

76 PETER GUTHRIE TAIT

ments on Radiation at various pressures of the surrounding gas, Brebner's work
on electrolysis, and Meik and Murray's investigations on the effect of load
on the resistance of copper wires.

Robertson Smith also found time for an exposure of Hegel's attack upon
the principles of the calculus as laid down in Newton's Principia, a kind of
criticism for which Smith, by virtue of his profound knowledge of both
mathematics and metaphysics, was singularly well equipped.

During the early years of his professoriate, Tait was on intimate terms
of friendship with W. H. Fox Talbot, best known for his discoveries in
photography and his deciphering of the cuneiform inscriptions. Fox Talbot
was a mathematician of distinct originality and was keenly interested in
experimental physics. He lived a good deal in Edinburgh during the sixties
and early seventies ; and on Saturday forenoons he often paid Tait a visit at the
College to experiment in light and magnetism. On May 15, 1871, Fox Talbot
communicated three short papers to the Royal Society of Edinburgh, the first
of which, "Note on the early History of Spectrum Analysis," was probably
suggested by Tait's address on that subject delivered the same evening before
the Society. The second, " On a New Mode of observing certain Spectra,"
ends with the remark that "all these experiments were made in the Physical
Laboratory of the University of Edinburgh by the kind permission and
assistance of Professor Tait." The third, " On the Nicol Prism," recalls some
of his earlier investigations and contains the description of a modified form of
polarising prism, which is made half of calc spar and half of glass. I have
often heard Tait express the very high opinion he held in regard to Fox
Talbot, whose discovery of anomalous dispersion was kept back from the
world by his own modesty and the too great caution of Sir David Brewster,
and had to be rediscovered many years afterwards by Le Roux and Christian-
sen.

The following letter touches on several points of interest.

17 DRUMMOND PLACE,

EDINBURGH, 11/1/71.
My dear Andrews,

We all heartily join in wishing you and yours many happy new years.
We are all well, but very busy — I at Physics, the rest at skating\ Even my wife has
become an enthusiast. 23 years ago I was wild about it, but I feel no inclination
to waste time on it now....

I am delighted to hear that you are getting on so well with your high pressures.
I often wish I were back again in Belfast. True I had more lecturing to do, and

THERMOELECTRICITY 77

less pay, but I had a great deal more leisure for private work. In fact I have barely
time for any private work during the winter session now-a-days.

However, I have got some students who are able and willing to work and I have
handed over my apparatus to them to make the best of it. At present I am entirely
engaged with " 1'effet Thomson " if you know what that is — the so-called specific heat
of electricityjin different conductors, which I think I have proved both experimentally and
theoretically to be proportional to the absolute temperature. This has led me to
construct a thermometer depending on two separate thermoelectric circuits working
against one another, so as to give galvanometric deflections rigorously proportional to
differences of absolute temperature through all ranges till the wires melt. I hope to
get the specific heats and melting points of various igneous rocks, &c., &c., true to a very
few degrees.

My Holtz machine — perhaps about the last thing that Ruhmkorff sent out of
Paris1 — is a splendid success ; 2-inch sparks from a jar with \ square yard of
coated surface at intervals of 4 seconds.

Tait was now in the heart of his thermoelectric investigations, which
for several years dominated the work of the Physical Laboratory. The
difficulties encountered and the methods by which they were overcome are
discussed in a series of short papers communicated to the Royal Society of
Edinburgh, afterwards worked up into the great Transactions paper of 1873.
In the earlier pioneer work Tait was helped by May and Straker, and a little
later by John Murray and R. M. Morrison. In the summer of 1873 he
instructed C. E. Greig and myself, who had spent one winter in the Laboratory,
to investigate by one and the same method the thermoelectric properties
of some twenty different metals paired in a sufficient number of ways ; and
these experiments which were made in the Natural Philosophy class room
formed the basis of the " First Approximation to the Thermoelectric Diagram."
The hot junctions were heated in oil up to a temperature of nearly 300° C.
Meanwhile Tait himself had been working with iron at still higher temperatures,
and making the first of what proved to be the most novel of his discoveries
in thermoelectricity, namely, the remarkable changes at certain temperatures
in the thermoelectric properties of iron and nickel.

Nearly all pairs of metals up to the temperatures of their melting points
have the thermoelectromotive force a parabolic function of the difference of
the temperatures of the junctions. When, however, iron or nickel is one of the
metals forming the thermoelectric couple this rule breaks down. Nevertheless
between particular limits of temperature the parabolic law is satisfied, so that
the relation between electromotive force and temperature can be fairly well

1 That is, before its investment by the German troops.

78 PETER GUTHRIE TAIT

represented by a succession of three parabolas with quite different parameters.
In the case of iron these peculiarities occur at high temperatures, which Tait
was able to measure by means of two alloys of platinum and iridium whose
thermoelectromotive force was very approximately proportional to the tem-
perature difference. These were known as M and N. Tait hoped to get a
series of such alloys having the same properties ; but though many specimens
of various percentage compositions were supplied him by Johnson and
Matthey, never again did he obtain a pair possessing the same simple
proportionality. The final experiments on iron at high temperatures were
entrusted to C. Michie Smith and myself in the winter of 1873. The three
wires M, N, and the particular specimen of iron under investigation had their
ends bound together to form one triple junction, while the other ends were
arranged so that the circuit M-N or the circuit N-Iron could be alternately
thrown into the galvanometer circuit. The triple junction was then inserted
within the hollow of a white-hot iron cylinder ; and as this cylinder cooled
to lower temperatures, the two circuits were thrown in rapid alternation into
the galvanometer circuit, and practically simultaneous measurements were
obtained of the N-Iron and M-N currents.

Nickel and cobalt were not easily obtained in the early seventies ; and
the first piece of nickel experimented with was a narrow ribbon not more than
two feet long, supplied by F. Lecoq de Boisbaudran. The following letter
to Andrews touches upon the work with these magnetic metals.

38 GEORGE SQUARE,
EDINBURGH, 13/12/75.

My dear Andrews,

Many thanks for your letter. I have been extremely remiss in not long
ago thanking you for the Nickel and Cobalt you kindly sent me. I know you will
be glad to learn what they have told me. Here it is : —

1. The new specimen of nickel gives almost exactly the same results as those
in my Thermoelectric Diagram. So that very curious result is verified.

2. The Cobalt specimen was not coherent enough for any but qualitative
results: — but it has shown me that cobalt lies (in the diagram) between Iron and
Nickel (at moderate temperatures), cutting copper, platinum, lead, zinc, cadmium, &c.,
so that the observations of a few neutral points will tell me all about it — except
(of course) the sinuosities which I have reason to think its line will show somewhere
about a white heat But I may be altogether wrong in this. Meanwhile with Crum
Brown's assistance I am preparing to deposit electrolytically films or foil of pure
cobalt.

THERMOELECTRICITY 79

The cobalt supplied by Andrews was probably far from pure ; for with the
rod of pure cobalt obtained by electrolytic deposition on aluminium, the
aluminium being afterwards dissolved away, J. G. MacGregor and C. M. Smith
found that the cobalt thermoelectric line lay below the nickel line and there-
fore further away from the iron line1.

Some of the difficulties encountered in these early days are not described
either in Tail's Transactions paper, or in the short laboratory notes which Tait
communicated from time to time to the Royal Society of Edinburgh.

Particularly interesting were the experiments on sodium and potassium,
the carrying out of which was entrusted to C. Michie Smith and myself.
The metals were prepared for Tait by (Sir) James Dewar, who sucked them
in the molten state up glass tubes under the surface of melted paraffin and
then allowed the whole to solidify. Each of the sodium and potassium bars
was thus enclosed in a glass tube, with solid paraffin ends protecting it from the
air. The ends were then slightly melted and platinum wires pushed through
the paraffin into the sodium or potassium. Sodium-platinum and potassium-
platinum circuits were thus constructed. Each bar was only a few inches long,
and as the one end had to be kept cool in running water while the other
was gently heated in an oil bath, the manipulation of the experiments was
not easy. There was moreover some risk of accident to the eye of the
operator who attended to the warmer junction.

Tait seems to have been led into his thermoelectric work in the hope of
testing a theoretic result he had obtained with reference to the " Thomson
Effect." Experimentally the work was a following up of much earlier investiga-
tions made by Thomson himself, to whom indeed the idea of the thermo-
electric diagram was due. What Tait did was (i) to establish for most metals
and through a considerable range of temperature the parabolic law for electro-
motive force, or the linear law for thermoelectric power, in virtue of which
each metal was represented by a straight line on the diagram ; (2)10 show how
the " specific heat of electricity " was indicated by the inclination of the
thermoelectric line and how the Peltier Effect and the Thomson Effect were
represented by areas on the diagram ; and (3) to discover the remarkable
changes of sign in the Thomson Effect for iron and nickel. His attempts to
measure the Thomson Effect directly were not successful, although he made
repeated attacks on the problem. For example, by passing a current first in

1 Working with a fairly pure specimen of rolled cobalt in 1891, I found that its thermo-
electric line lay above the nickel line at temperatures below 100° C. but below it at higher
temperatures.

8o PETER GUTHRIE TAIT

one direction and then in the other along a piece of thin platinum foil which
was cut away towards the centre until it became very narrow, he hoped to be
able to witness the shift of the glow at this narrowest part. When he got the
Gramme Dynamo about 1877, one of the first experiments he tried was to pass
the current from the Gramme machine along an iron bar when it had been
brought to a steady gradient of temperature along its length, after the
manner of Forbes' experiment in thermal conductivity. He hoped to detect
a change in the gradient of temperature ; but here again there was no success,
the current density not being great enough.

Another line of experiments on related effects, at which A. Macfarlane,
C. M. Smith, and I worked, was the coordination of the striking phenomena
which occur in iron about the dull red heat, namely, the loss of magnetic
susceptibility, the reglow as the iron wire cooled, the change of sign of
the Thomson Effect, and the change in the law of alteration of electrical
resistance with temperature, all of which Tait proved to be in the neighbour-
hood of the same temperature. In one of these experiments iron and platinum
wires were led through a white-hot iron cylinder side by side, while to the
middle of the iron were attached the M and N platinum-iridium wires. As
the whole gradually cooled, observations were taken in rapid succession of the
resistances of the iron and platinum wires and the thermoelectric currents in
the N-iron and N-M circuits. The method was no doubt rough and ready
and not susceptible of great accuracy, but it was effective enough to establish
conclusions which more carefully designed experiments of later date have
fully corroborated.

Among Maxwell's letters to Tait about this time the following quaint
remark was found written in three lines on a long strip of paper.

" If your straight lines, parabolas, &c. have no resemblance at all to those things
which men call by those names, I would as soon be J. Stuart Mill as call them so.
But if they differ very slightly, then T' is enrolled among the Boyle and Charles
of 0H1 who remain unhurt by Regnault, &c. But in Physics we must equally
avoid confounding the properties and dividing the substance. In the one case we
fall into the sin of rectification (Eccl. i. 15) and in the other we see in every zigzag
a proof of transubstantiation."

Although himself greatly taken up with the thermoelectric experiments, Tait
never lost sight of the investigation into the thermal conductivity of metal bars,
which was the first serious piece of experimental work he tackled in Edinburgh.
This following up of Forbes' important researches was begun under the

1 The Greek initials of Thermo-Electricity.

CROOKES' RADIOMETER 81

auspices of the British Association ; and Tait sent in two short Reports in
1869 and 1871. Most of the ' veteran ' students had a turn at the bars during
the seventies and eighties ; and Tail's paper on the application of Angstrom's
method of sending waves of heat along the bar (Proc. R. S. E, Vol. vni),
was based on observations made by A. L. MacLeish and C. E. Greig1 in the
early part of the year 1873. The harmonic analysis is fully worked out so as
to give the amplitudes and phases of the temperature oscillations at each
chosen point ; but the final calculation of the conductivities is not given. In
fact the simple and solvable form of the equation of conduction did not apply
even to a rough approximation. Tait therefore fell back upon Forbes' method,
and in 1878 he published a detailed account of his investigations, the main
purpose of which was to extend to other metals what Forbes had done for iron.
An important supplement to this memoir appeared in 1887 by (Professor)
Crichton Mitchell, who as an advanced student went over the whole ground
again, the one difference being that all the bars were now nickel plated.
Their surface conditions were thus rendered more nearly identical than in the
first set of experiments. One of the final conclusions come to was that

"We cannot yet state positively that there is any metal whose conductivity
becomes less as its temperature rises ; and thus the long sought analogy between
thermal and electric conductivity is not likely to be realised."

Early in 1875 Tait and Dewar made together a series of well planned
experiments on the phenomena of Crookes' radiometer. They gave a
demonstration of these before the Royal Society of Edinburgh on July 5, 1875 ;
but unfortunately no authoritative account of them was ever published. In
Nature of July 15, 1875, a report of the communication was given under the
title " Charcoal Vacua " which does not bring out clearly the real significance
of certain parts of Tait and Dewar's investigations. The following quotation
from Lord Kelvin's obituary notice read before the Royal Society of Edinburgh
puts the question in a clearer light :

" In a communication on ' Charcoal Vacua ' to the Royal Society of Edinburgh
of July 5, 1875, imperfectly reported in Nature of July 15 of that year, the true
dynamical explanation of one of the most interesting and suggestive of all the
scientific wonders of the nineteenth century, Crookes' radiometer, was clearly given.
The phenomenon to be explained is that in highly rarefied air a disc of pith or
cork or other substance of small thermal conductivity, blackened on one side, and
illuminated by light on all sides, even the cool light of a wholly clouded sky,

1 Dr A. L. MacLeish is now a physician resident in Los Angeles : the Rev. C. E. Greig
is a pastor in Paris.

T. II

82 PETER GUTHRIE TAIT

experiences a steady measurable pressure on the blackened side. Many naturalists, I
believe, had truly attributed this fact to the blackened side being rendered somewhat
warmer by the light; but none before Tait and Dewar had ever imagined the
dynamical cause — the largeness of the free path of the molecule of the highly
rarefied air, and the greater average velocity of rebound of the molecules from the
warmer side. Long free path was the open sesame to the mystery."

I had the good fortune to be present in the Laboratory when some of
the experiments were being made. One especially struck me as being of
peculiar significance. I cannot remember if this was shown before the
Royal Society of Edinburgh ; but it is not referred to in the published report.
A transparent light vane of rock salt was suspended under an ordinary air-
pump receiver and placed in front of and fairly close to a fixed blackened
surface. The energy rays were directed through the transparent vane on to
the blackened surface. At very moderate exhaustions repulsion was set up,
whereas for the ordinary form of Crookes' radiometer a very high vacuum is
needed. The whole question was thus proved to be one of the relation
between the free path and the distance between the repelling surfaces.

The following among other experiments are described in the Nature
Report. Two equal disks, one of glass and the other of rock salt, were
attached to the ends of a delicately suspended glass fibre. When the
radiation fell on the glass disk there was repulsion due to the heating of the
disk ; but when the radiation fell on the diathermanous rock salt there was
no repulsion — the heat was not absorbed sufficiently to produce the necessary
rise of temperature. The back of the rock salt disk was next coated with
lamp black, and after sufficient exhaustion was produced in the enclosing vessel,
the radiation was thrown through the rock salt on to the blackened surface. At
first one might expect an apparent attraction due to the repulsive action on
the far-away side ; but the disk was repelled exactly like the glass disk. This
was due to the bad conducting power of the lamp black, so that the rock salt
on the near side became heated by conduction more quickly than the outside
parts of the lamp black layer on the further side. In these experiments it was
necessary to use a very thin-walled enclosing vessel within which the vacuum
was formed, otherwise the glass vessel would itself absorb so much of the
low heat rays that the differential action of the glass and rock salt disks would
not be great enough to make itself apparent.

The next engrossing piece of experimental work was in connection with
the "Challenger" Reports. On the return of the "Challenger" Expedition

THE "CHALLENGER" THERMOMETERS 83

in 1876, Sir Wyville Thomson consulted Tait as to the corrections to be
applied to the readings of the deep sea temperatures given by the self-record-
ing thermometers which had been used. Experiments made by Captain
J. E. Davis, R.N., before the Expedition started on its four years' voyage,
indicated that a correction of at least half a degree Fahrenheit for every mile
depth under the sea had to be applied. A careful examination of the thermo-
meters with their protected bulbs convinced Tait that only very slight corrections
would be required ; and the necessity arose for retesting the thermometers.
In these laboratory experiments, as conducted first by Captain Davis and then
by Professor Tait, the conditions are very different from those under which
the thermometers record the temperatures of the ocean deeps. For example,
under the increasing pressure in the hydraulic press the temperature of the
surrounding water will be raised. Captain Davis and Professor Miller tried
to determine this rise of temperature by direct experiment ; and after taking
it into account they found a correction still to be applied, and this they referred
to the direct effect of pressure. This pressure correction accordingly was to
be applied to the readings obtained in deep sea observations. Tail's acuter
physical instinct saw no necessity for such a correction ; and after a prolonged
investigation into all the possible causes of temperature change he found that
the vulcanite mounting of the thermometers was the principal source of the
change which Davis and Miller failed to account for. The heating of the
vulcanite mounting due to compression would be of no consequence in the
deep sea experiments ; consequently no correction was needed. Or, to put
it quite accurately, the correction due to pressure was of an order distinctly
smaller than the errors of observation and therefore negligible. See Tail's
Report, "Challenger" Narrative, Vol. u, Appendix A; Sci. Pap. Vol. i,

P- 457-

The beautiful hydraulic apparatus designed by Professor Tait and supplied

by the Admiralty for making these tests was utilised by him in making further
investigations in the realm of high pressures. Some of these investigations
form the substance of a second " Challenger" Report (Physics and Chemistry,
Vol. ii, Part iv, 1888), bearing upon certain physical properties of fresh
water and of sea-water (Sci. Pap. Vol. n, p. i). The wide scope of this
enquiry may be best indicated by a few quotations from his own summary of
results. The compressibility of the glass of the piezometers was measured
by means of J. Y. Buchanan's apparatus, and found to be O'ooooo26 per atmo-
sphere. By a modified form of piezometer the compressibility of mercury was

ii — a

84 PETER GUTHRIE TAIT

determined, the value being 0-0000036 per atmosphere. These data were
necessary for the accurate determination of the compressibilities of the various
kinds of water and solutions. Within a range of temperature o° to 1 5° C. and
a range of pressures from 1 50 to nearly 460 atmospheres, the compressibility
of fresh water was approximately represented by the empirical formula

0-00186 / _ 31

36 + / \ 400 10,000.
The corresponding formula for sea-water was

0-OOI79/ / f \

38 + / \ 150 lo.ooo/'

In these t is Centigrade temperature and p is pressure in tons weight per
sq. inch. The point of minimum compressibility of fresh water is about 60° C.
at atmospheric pressure, and that of sea- water at about 56° C. ; both are
lowered by increase of pressure.

The average compressibility of solutions of NaCl for the first p tons
of additional pressure, at o° C., s being the amount of NaCl in 100 parts
by weight of water, could be represented very accurately by the formula
o-oo 1 86/(36 +/» + .$•).

The maximum density point of water was found to be lowered about 3° C.
by 1 50 atmospheres of pressure ; and from the heat developed by compression
of water Tait calculated that this lowering of the maximum density point
should be 3° per ton weight per square inch, (i ton weight per sq. inch =152-3
atmospheres.)

In most of his experimental work Tait did not apply his mind specially
to the invention of elaborate apparatus ; but that he could when the necessity
arose devise useful and ingenious forms appears very clearly in his compression
work.

Consider for example his high pressure gauge, constructed of a steel
cylinder, the measured change of volume of which under hydrostatic pressure
gives by a simple elastic formula the value of the pressure. In this instrument
the pressure is applied to the outside of the cylinder, and the change of
volume is measured by the alteration in level of mercury which fills the
cylinder and the narrow glass tube fitted to it above. This glass tube which
is in continuous connection with the interior of the steel cylinder is open above
and is not itself exposed to pressure. It projects through the top of the
outer vessel which surrounds the steel cylinder and within which the pressure

THE TAIT PRESSURE GAUGE 85

is applied. By a very simple but ingenious device Tait practically got rid of
the disturbing effects of temperature changes in the mercury filling the steel
cylinder. He placed within the cylinder a glass tube closed at both ends
which all but filled the cylinder. This left the action perfect as a pressure
gauge, and rendered negligible its action as a thermometer. Professor Carl
Barus in his memoir on the volume Thermodynamics of Liquids1 found his
modified form of the Tait Gauge highly efficient. One great merit was the
complete absence of cyclic quality so that the same pressure readings were
obtained whether they formed a series of ascending or descending pressures.

Another example of Tail's ingenuity is his electric contact device for
indicating when a definite compression has been produced in a piezometer
which is enclosed in an opaque hydraulic press and cannot therefore be seen
by the eye. His own description is in these words (second "Challenger"
Report, Appendix A).

" We have, therefore, only to fuse a number of platinum wires, at intervals, into
the compression tube, and very carefully calibrate it with a column of mercury which
is brought into contact with each of the wires successively. Then if thin wires, each
resisting say about one ohm, be interposed between the pairs of successive platinum
wires, we have a series whose resistance is diminished by one ohm each time the
mercury, forced in by the pump, comes in contact with another of the wires.
Connect the mercury with one pole of a cell, the highest of the platinum wires with
the other, leading the wires out between two stout leather washers; interpose a
galvanometer in the circuit, and the arrangement is complete. The observer himself
works the pump, keeping an eye on the pressure gauge, and on the spot of light
reflected by the galvanometer. The moment he sees a change of deflection he reads
the gauge...."

Amagat, between whom and Tait much correspondence passed at one
time with reference to pressure measurements, adopted this method with great
success in his later experiments. Regarding its efficacy he writes

"Sur la recommandation de 1'eminent physicien, je 1'ai essayd tout de suite et
n'en ai plus employe" d'autre, non seulement pour les liquides, mais encore pour le
gaz, dans les series allant jusqu'aux plus fortes pressions et pour les temperatures ne
depassant pas 50."

One general conclusion of great interest in these experiments is the
representation of the compressibility by an expression of the form Aj(B+fl),
where/ is the pressure and A and B depend only on the temperature. In
several subsequent papers Tait tested the applicability of this empirical
formula to experimental results obtained by other experimenters, notably
1 Bulletin of the United States Geological Survey, No. 96 (1892).

86 PETER GUTHRIE TAIT

Amagat. He also projected a series of investigations upon solutions of
varying concentration, so as to test the applicability of the formula Aj(B + s + /),
where s represents the percentage amount by weight of the solute. In 1893
he published results of a preliminary character on three solutions of the
substances Potassium Iodide, Potassium Ferrocyanide, Ammonium Sulphate,
Magnesium Sulphate and Barium Chloride, and found that the formula
applied fairly well. Five years later he published a preliminary note on the
compressibility of solutions of sugar based upon experiments which were
carried out by A. Shand (Nichol Foundationer). The results were not
very concordant ; but they indicated that the effect of sugar was, weight for
weight, barely one-third that of common salt in reducing the compressibility.
Mr Shand was planning a continuation of the experiments, when his early
death deprived the Edinburgh University of an experimenter of real ability
and resourcefulness.

The new compression apparatus, familiarly known as the " Big Gun,"
was not received till 1 879 ; and it was first set up in a small cellar on the
basement of the north side of the College. Here all the experiments dealing
with the testing of the " Challenger " thermometers were carried out. The
accommodation was very limited, and the light was poor ; but in a few years
the apparatus was transferred to a much larger basement cellar, in the north-
west corner ; and here all the later experiments on compression were made.

This change was part of a general expansion of the Physical Laboratory
consequent on the removal of the Anatomical Department in 1880 to the
New Buildings which were to be wholly devoted to medical studies. Till
that date the Dissecting Rooms occupied the top story of the north side of
the College with the exception of the small room which had served for a
physical laboratory under the care of Tait's successive assistants, W. Robertson
Smith, D. H. Marshall, and P. R. Scott Lang. During my first year of the
assistantship (1879-80) the whole suite of four rooms became transformed
into the physical laboratory. There was ample accommodation, so far as
mere area of floor space was concerned ; and it was possible to arrange a
junior laboratory and rooms for special magnetic and optical work. On the
basement Tait secured the large cellar already mentioned, in which were installed
the compression apparatus, the dynamo, the gas engine for driving the dynamo
and for working up to high pressures, and latterly the " guillotine " for the
impact experiments to be afterwards described. In a neighbouring cellar
fifty secondary cells were in due course installed ; and there was also a third

PROFESSOR CHRYSTAL 87

cellar which was used originally as a store room but which after 1892 was
equipped as a research room for my work on magnetic strains.

The old anatomical theatre was adapted to the purposes of the Mathe-
matical Department under Professor Chrystal, who began his Edinburgh
professoriate in the same transition year 1879-80. These changes brought the
Mathematical and Natural Philosophy Departments into closer contiguity; but,
what was of still greater importance, Professor Tait found in his new colleague
an enthusiastic experimentalist, who from 1880 to about 1886 passed the
summer sessions in the Physical Laboratory, exercising a stimulating influence
upon many of the students who were devoting themselves to practical physics.
Chrystal had just written the articles on Electricity and Magnetism for the
Encyclopaedia Britannica and was thoroughly posted on all the recent work
in these rapidly developing branches of physics. In carrying out his important
researches on the differential telephone and the measurements of inductances,
he had all the facilities of the laboratory placed at his disposal ; and both
directly and indirectly he gave many a hint to the students who were able
to take advantage of their opportunities. When I left for Japan in 1883
Chrystal was almost as strong an influence in the Laboratory as Tait himself ;
but after a few years the increasing duties of his own chair and the fact that
he found himself to be appropriating more and more of the really serviceable
apparatus for his own experiments obliged him to relinquish experimenting
for some time.

When, mainly through the exertions of Dr Buchan, the Ben Nevis
Observatory was started in 1883, attention was drawn to the difficulty of
measuring humidities of the atmosphere under the conditions which frequently
existed on the top of the mountain. Both Chrystal and Tait suggested forms
of instrument for the purpose. Chrystal's was on the principle of Dine's
hygrometer, the nickel plated copper box, into which the thermometer bulb
was inserted, being supplied by means of a double tap arrangement with warm
or cold water at will. The temperature was adjusted until a film began to
form on the box. Tail's instrument was constructed on a totally different
principle, that of the atmometer. The following is Tail's description from
his paper of February 16, 1885 (Proc, R. S. E. Vol. xin, p. 116).

" The atmometer is merely a hollow ball of unglazed clay, to which a glass tube
is luted. The whole is filled with boiled water and inverted so that the open end
of the tube stands in a dish of mercury. The water evaporates from the outer
surface of the clay (at a rate depending partly on the temperature, partly on the

88 PETER GUTHRIE TAIT

dryness of the air) and in consequence the mercury rises in the tube. In recent
experiments this rise of mercury has been carried to nearly 25 inches during dry
weather. But it can be carried much farther by artificially drying the air round the
bulb... I found, by inverting over the bulb of the instrument a large beaker lined with
moist filter paper, that the arrangement can be made extremely sensitive. The mercury
surface is seen to become flattened the moment the beaker is applied, and a few
minutes suffice to give a large descent, provided the section of the tube is small
compared with the surface of the ball.

" I propose to employ the instrument in this peculiarly sensitive state for the
purpose of estimating the amount of moisture in the air, when there is considerable
humidity ; but in its old form when the air is dry. For this purpose the end of the
tube of the atmometer is to be connected, by a flexible tube, with a cylindrical glass
vessel, both containing mercury. When a determination is to be made in moist air
the cylindrical vessel is to be lowered till the difference of levels of the mercury
amounts to (say) 25 inches, and the diminution of this difference in a definite time
is to be carefully measured, the atmospheric temperature being observed. On the
other hand, if the air be dry, the difference of levels is to be made nil, or even
negative at starting, in order to promote evaporation."

Experiments were made to test the applicability of the method ; but the
manipulation demanded more care and attention than could be expected from
a busy observer at a meteorological station.

In 1887 Tait who had been for many years a keen devotee of the game
of golf was led to consider various physical problems suggested by the flight
of a golf ball, from the moment of impact of the club to its final fall to earth.
The first consideration is the manner in which the momentum of the club is
communicated to the ball. Given the club moving with a certain speed, with
what speed will the ball be projected ? This is the one stage over which the
player has any control. After the ball has left the club its further progress is
conditioned by the initial conditions of flight and the continuous subsequent
interplay between the moving ball and the surrounding air.

Accordingly to study the laws of impact of various materials Tait set up a
simple but very effective form of apparatus which he humorously called the
"guillotine1." The name occurs early in the third paragraph of the first paper
on Impact (Sci. Pap. No. LXXXVIII). The block whose impact on the
material was to be studied " slid freely between guide rails, precisely like the
axe of a guillotine." As this block fell and rose again after several rebounds,
a pointer attached to it bore with sufficient pressure upon the blackened
surface of a revolving plate-glass wheel. The curve traced out in this manner

1 The name is preserved historically in the new Physical Laboratory of Edinburgh
University, the room in which the apparatus is now installed being called the Guillotine Room-

EXPERIMENTS ON IMPACT 89

contained a complete record of the whole motion of the impinging block ; and
from this record all the numerical data of the experiment could be obtained,
such as the successive heights of rebound, the time of duration of impact, and
the amount of compression of the substance on which the block fell.

To be able to measure time intervals it was necessary to have a definite
time record impressed on the revolving plate. This was simply effected by
means of a tuning fork delicately adjusted so that a tracing point attached to
one end traced out a sinuous curve on the plate concentric with the curves
traced out by the impinging block.

Although originally undertaken with the aim of determining the resilience
of rubber and guttapercha, the experiments were not confined to these golf
ball materials. The first series of experiments dealt with the impact proper-
ties of plane tree, cork, vulcanite, and vulcanised indiarubber ; and in the
second series lead, steel, glass, new native indiarubber, and various kinds of
golf balls were added to the list of substances experimented with.

The results embodied in these papers are of the highest physical interest.
Among the practical applications we may mention Tait's estimates of the
duration of impact of a hammer and nail (croooa sec.) and of the time-average
force (300 lb.-wt.). As regards the golf ball problem which suggested the
experiments, the very much smaller speeds of approach attainable in the
experiments render the data not very directly applicable. But it was estimated
that the time-average of the force during the collision (which may have lasted
about O'OO5 sec.) of the golf club and ball must be reckoned in tons' weight.

Closely connected with the golf ball enquiry were the ballistic pendulum
experiments, described in the second paper on the path of a spherical projectile
(1896, Set. Pap. Vol. ii, p. 371). The final type of pendulum used was a
bifilar suspension with a bob formed by a long horizontal bar oscillating in
the plane of the bifilar. The one end of the bar was faced with clay, and
into this soft material a golf ball was driven from a "tee" a few feet away.
The momentum of the ball at impact was transferred to the pendulum and
ball together, and could be easily estimated in terms of the distance through
which the bob was driven to the extremity of its range. This was observed
directly by an observer who was protected from being hit by the ball (should
that by any chance miss the clay) by the half-closed door past which the
pendulum swung. Tait's son Freddie and other powerful players visited the
laboratory and experimented with this form of apparatus. The general con-
clusion was that under ordinary conditions a well driven golf ball left the " tee "

T. 12

90 PETER GUTHRIE TAIT

with a speed of not more than 300 feet per second. Assuming 2/3 as the
coefficient of restitution, Tait found that the head of the club must have been
travelling at the rate of about 200 feet per second at the instant of impact.
Other questions relating to the flight of the golf ball have been already
discussed in Chapter i. ; and Tait's own final views will be found below in
his article on Long Driving, which appeared in the Badminton Magazine.

With a mind always on the alert for scientific problems, it is not surprising
that Tait occasionally failed to find what he was in search of. His attempts
to obtain distinct evidence of the Thomson Effect in thermo-electricity have
already been noted ; and I remember him spending the better part of a summer
session in the experimental study of electrification due to sudden evaporation
or condensation. Morning after morning he would come with a new arrange-
ment to try, meeting my enquiry with the remark, " Now, at last, I have got
the crucial experiment." He devised for this research large flat metallic
dishes, which we facetiously dubbed " frying pans " ; but nothing came of
it. Tait's conclusion was that his surfaces were not big enough.

Another enquiry which occupied his mind at intervals from his Belfast
days was the possibility of doubling an absorption line through the influence
of magnetism. That such an effect should take place was an inference he
made from Faraday's discovery of the rotation of the plane of polarization in
a strong magnetic field. In a short paper read before the Royal Society of
Edinburgh in 1876 he gave briefly the grounds for his belief (Set. Pap.
Vol. i, p. 255). With more powerful magnetic fields than were at that time
available there is little doubt that he would have observed an effect of the
kind looked for, and thus anticipated Zeeman's closely allied discovery of 1896.

"In consequence of the severe lightning stroke with which Skerry vore
Lighthouse was visited on 2nd February 1876, occasioning considerable
disturbance to the internal fittings of the lighthouse and the destruction of the
entrance door," D. and T. Stevenson, Engineers to the Board of Com-
missioners of the Northern Lights, suggested "the propriety of consulting
Professor Tait on the general question of protecting the lighthouse towers
against the effects of lightning." Professor Tait accordingly accompanied
David Stevenson and others of the Commissioners on their annual visitation
during the ensuing summer, and his opinions and advice are given in the
Report, of which the opening sentences have just been quoted. During this
trip of inspection Tait's attention was drawn to the methods of producing
fog signals, and experiments were afterwards made in Edinburgh to test the

FOG-SIGNALLING 91

applicability of a method of alternating out-blast and suction for producing
economically sounds of high intensity. The original idea did not develop
satisfactorily ; but some experiments in conjunction with Crum Brown led
him in 1878 to the construction of a new form of siren suitable for fog-
signalling. In these experiments an organ note was made discontinuous by
being sounded through a partition and a revolving disk cut into separate
sectors. Unfortunately the siren effect superposed on the effects which were
being studied disturbed somewhat the quality of tone. Tait found that when
there was no organ note being sounded the mere rotation of the perforated
disk produced a sound whose intensity could be greatly increased by soldering
plates perpendicularly to the revolving disk so as to increase the thickness of
the back edges of the apertures. When rapid rotation was set up the sound
emitted was almost terrifying in its intensity. It shrieked out through the
open window of the Natural Philosophy Class Room into the quadrangle to
such a degree as to interfere with the lectures in the neighbouring class rooms.
Tait was accordingly obliged to conduct his experiments at hours when no
classes met.

In February, 1880, Tait communicated a short note to the Royal Society
of Edinburgh describing his unsuccessful attempt to measure the velocity of
the particles which constitute the cathode rays in a Crookes' tube, by means
of observations of the spectrum made in directions perpendicular and parallel
to the lines of motion of the charged particles. One cause of the failure
was the loss of light by multiplied reflections when a powerful spectroscope
was used. This led him to construct a rotatory polariscope whose principle
depended upon the rotation by quartz of the plane of polarisation, combined with
sufficient prism dispersion just to separate the various bright lines of the source
from one another. The final form of the apparatus was described in Nature,
Vol. xxii (Sci. Pap. Vol. i, p. 423). When a plane polarised ray of
light is subjected to rotatory dispersion by transmission through quartz,
and, after further transmission through a double image prism, is dispersed
prismatically by means of a direct vision spectroscope, there appear side by
side two spectra of the original ray crossed by one or more dark bands
according to the thickness of the quartz plate used. The dark bands in the
one spectrum correspond in position with the bright bands on the other.
When the polarising nicol is rotated the bands move along the spectra.
Tail's idea was to use this form of apparatus for studying the bright line
spectra of faintly luminous objects, such as nebulae and comets. By employing

12 — a

92 PETER GUTHRIE TAIT

first a thin piece of quartz and then a much thicker piece, he showed how the
wave length of the light examined could be determined. There were how-
ever several practical difficulties in the way, and the method on trial did not
turn out to be so sensitive as Tait had expected.

The following brief note to Thomson shows that Tait was thinking over
the still debated question of the relative motion of the earth and the aether.

THE CLUB HOUSE,
ST ANDREWS,

26/4/82.

D' T.

Srcuf says No ! He says that in such a case period is everything.
But I have set Piazzi1 on to try his magnificent Gitter. If the ether be in
motion relatively to the earth, the absolute deviations of lines in the diffraction
spectrum should be different in different azimuths : unless (of course) the relative
motion of earth and ether be vertical. Anser. Yrs.

I am not aware however that either Piazzi Smyth or Tait ever tried
experiments of the kind indicated. Tait had clearly taken Stokes' opinion,
but was not convinced.

Another problem to which Tait again and again recurred was the question
of the diathermancy of water vapour. He strongly doubted the accuracy of
Tyndall's well-known experiments on this subject — see, for example, his letter
to Andrews given above, p. 68. In 1882 he described in a letter to Thomson,
who communicated it to the B. A. Meeting at Southampton, a new form of
apparatus for investigating absorption of radiant heat by gases. The letter
was published in Nature, Oct. 26, 1882. (See Set. Pap. Vol. n, p. 71.)
The general idea was to measure the absorption by the increase of pressure
in the gas due to the heating. The apparatus was simply a double walled
cylinder. While cold water was kept circulating in the jacket, steam could be
blown into the double top. The changes of pressure in the gaseous contents
were measured by a manometer U tube placed at the bottom. Several series
of experiments were carried out by J. G. MacGregor and T. Lindsay
(Proc. R. S. E. Vol. xn, 1882, p. 24), the conclusion being "that the
absorption of air containing 1-3 per cent, of water vapour is between that of

1 i.e., Piazzi Smyth, Astronomer Royal for Scotland and Professor of Astronomy in
Edinburgh University, a well-known worker in spectroscopy. The signature is a compact
monogram giving all three initials P. G. T. The phonetic spelling of "answer" is of course
intentional, just as in the case of " Stokes " ; these contractions were frequent, especially between
Tait and Maxwell.

RHYMING CORRESPONDENCE 93

air containing O'o6 per cent, and that of air containing 0*2 per cent, of
olefiant gas." This result was in agreement with what Tait himself obtained
from a first rough experiment, and indicated that damp air was less absorbent
of low heat rays than air mixed with a small quantity of olefiant gas.

These are a few of the investigations which did not yield all that Tait
hoped for ; in some instances indeed they were wholly abortive, or did not
reach even the stage of being tried.

When any promising idea occurred to his mind, Tait was in the habit of
referring it to the judgment of Stokes or Maxwell ; and although unfortunately
Tail's letters to Maxwell were not preserved, there has survived an amusing
rhyming correspondence bearing upon the nature of electricity. In June 1877
Tait thus described the experiment he was thinking of making :

Will mounted ebonite disk

On smooth unyielding bearing
When turned about with motion brisk

(Nor excitation sparing)
Affect the primitive repose

Of + or — in the wire,
So that while either downward flows

The other upwards shall aspire?
Describe the form and size of coil

And other things that we may need,
Think not about increase of toil

Involved in work at double speed.
I can no more, my pen is bad,

It catches in the roughened page —
But answer us and make us glad,

THOU ANTI-DISTANCE-ACTION SAGE !
Yet have I still a thousand things to say

But work of other kinds is pressing,
So your petitioner will ever pray

That your defence be triple messing\

This last Anglo-German pun on the well-known Horatian text is a good
example of one of Tail's forms of humour.

The following is Maxwell's reply as preserved in the original letter pasted
into Tail's Scrap Book. The annotations are Maxwell's even to the references
lo Art. 770 at verse i and Art. 577 at verse 7. These were pencilled in and
refer to Maxwell's Electricity and Magnetism, First Edilion.

94 PETER GUTHRIE TAIT

GLENLAIR,

DALBEATTIE,

25 June 1877.
Art. 770

The mounted disk of ebonite

Has whirled before nor whirled in vain,
Rowland of Troy, that doughty knight,

Convection currents did obtain1
In such a disk, of power to wheedle
From her loved north, the pensile needle.

'Twas when Sir Rowland, as a stage

From Troy to Baltimore, took rest
In Berlin, there old Archimage

Armed him to follow up this quest
Right glad to find himself possessor
Of the "irrepressible professor8."

But would'st thou twirl that disk once more?

Then follow in Sir Rowland's train
To where in busy Baltimore

He rears the bantlings of his brain ;
As he may do who still prefers
One Rowland to two Olivers8.

But Rowland, — no, nor Oliver, —

Could get electromotive force
[Which fact and reason both aver,

Has change of some kind as its source]
Out of a disk in swift rotation
Without the least acceleration.

But with your splendid roundabout

Of mighty power, new hung and greasy,

With galvanometer so stout,

Some new research would be as easy —

Some test which might perchance disclose

Which way the electric current flows.

Take, then, a coil of copper pure

And fix it on your whirling table,
Place the electrodes firm and sure

As near the axis as you're able
And strive to learn the way to work it
With galvanometer in circuit.

1 Berlin Monatsberichte.

' Sylvester's address to Johns Hopkins.

8 Heaviside and Lodge.

HIS PHYSICAL INTUITION 95

Art. 577

Not while the coil in spinning sleeps

On her smooth axle swift and steady,
But when against the stops she sweeps,

To watch the light spot then be ready,
That you may learn from its deflexion
The electric current's true direction.

It may be that it does not move,

Or moves, but for some other reason,
Then let it be your boast to prove

(Though some may think it out of season
And worthy of a fossil Druid)
That there is no electric fluid.

There is no evidence that Tait ever began on the line of work here
indicated.

Taking a general view of Tait's experimental work we find it characterised
by a true physical insight into the essential nature of each problem. Super-
fine accuracy was never his aim ; and perhaps from this point of view some
of his investigations lack finish. His methods were in many cases rough
and ready, but they were always under complete mathematical control.
Having laid down the broad lines of attack on any question he put together
his apparatus with little apparent attention to detail ; but his intuitions generally
led him right. In many cases the first rough arrangement was committed to
the care of two of his "veteran" students, in whose hands the final form of
apparatus gradually evolved itself as difficulty after difficulty was surmounted.
In this way the resourcefulness of the master and the enthusiasm and
patient skill of the disciples worked together towards the perfected end. In
his published accounts Tait never failed to give full credit to those who
helped him in carrying his ideas to fruition.

The most laborious experiments undertaken by Tait were those on the
conduction of heat in bars, on the errors of the " Challenger " thermometers,
on the compression of liquids and on the laws of impact. In all of these
Thomas Lindsay was his righthand man ; and the successive bands of students
who helped in the work consisted, in a sense, of picked men, for in those
days only real enthusiasts ever thought of continuing their laboratory work
so as to rank as "veterans." Already I have incidentally named a few of the

96 PETER GUTHRIE TAIT

students of whom Tait made use. Others, however, are equally worthy of
mention. Thus the compression work with the " Big Gun " owed much of
its success to the labours of R. T. Omond, afterwards superintendent of the
Ben Nevis Observatory, and H. N. Dickson, at present Lecturer on Geography
in University College, Reading. Again it was largely through the exertions
of A. J. Herbertson, now Reader in Geography, Oxford University, and
R. Turnbull, now Inspector, Department of Agriculture, Dublin, that the
impact apparatus evolved itself from the first rude form to the final perfected
arrangement ; and the later set of experiments and their reductions were
practically carried out by Alexander Shand.

It is interesting to note that many of Tail's students who helped him in
research work did not become professional physicists. Under the old system,
which present day pigeon-hole organisers rather despise, men had time to put
in valuable work which lay outside their official course of study. Tait and
his assistants soon saw who were the more resourceful among the laboratory
students, and these were quickly enrolled in the unofficial squadron of workers.
Under the present system of detailed courses of obligatory work, carefully
scheduled for the benefit of the average student, such a method as Tait
commanded in his day could hardly be applied. Many more students are
trained now than formerly to make physical measurements ; and the training
is more systematic and thorough ; but, with the exception of those who expect
to follow out physics in their life career, very few ever come in touch with the
stimulus which real research work gives. The day apparently is past for
fruitful physical work to be effected in their student days by men who
afterwards become clergymen, physicians, geographers, botanists, zoologists,
or even engineers.

This account of Tait as an experimental philosopher would not be com-
plete without some reference to the encouragement he gave to any of his
students following out researches of their own, which had not been directly
suggested by him. In particular, to those of us who desired to prepare an
experimental thesis for the Doctorate in science, he gave every facility in the
way of accommodation and apparatus. Among the more extended investiga-
tions of this independent nature I might mention Ewing and MacGregor's
measurement of the electric conductivity of saline solutions, Macfarlane's
experiments on the electric discharge through air and other dielectrics, my
own work on contact electricity, Crichton Mitchell's study of the rate of

ENCOURAGEMENT TO RESEARCH 97

cooling of bodies in steady currents of air, and C. M. Smith's experiments
on conduction of heat in insulating material.

These and other similar pieces of research by his successive assistants
and demonstrators could never have been carried out, had Tait not generously
given us unrestricted access to his laboratory. Once we gained his confidence,
we could roam at will through the whole department, and appropriate for our
own purposes any apparatus which for the moment was not being used.
There could be no truer way of encouraging research.

T.

CHAPTER III
MATHEMATICAL WORK

IN the preceding chapter Tail's experimental work has been dealt with
apart from the other scientific activities of his mind. At no time however
did he limit his attention to one problem exclusively ; and while with the aid
of his company of voluntary workers he was for the last thirty years of his
life busy with experiments in the laboratory, at home in his study he was
using his mathematical powers with great effect in all kinds of enquiries.
This mathematical and theoretical work may be conveniently classified under
three headings : namely, quaternions, mathematics and mathematical physics
outside the quaternion method, and the labours incidental to the writing of
his more mathematical treatises. The quaternion work will be considered in
an appropriate chapter ; another chapter will be devoted to the preparation of
Thomson and Tail's Natural Philosophy ; and Tail's other lilerary conlribu-
lions in book form will have a similar separale Irealmenl. Here, in a some-
whal disconlinuous manner, I propose lo give a general accounl of ihe more
malhemalical of his scienlific papers and noles, iracing as far as possible iheir
genesis and iheir conneclion wilh olher lines of research.

Passing over his early qualernionic papers in the Quarterly Journal of
Mathematics, the Messenger of Mathematics, and the Proceedings of the Royal
Society of Edinburgh, we come in 1865 to a purely mathemalical paper on
ihe Law of Frequency of Error ( Trans. R. S. E. Vol. xxiv ; Sci. Pap, Vol. i,
p. 47). He was led lo enquire inlo ihe foundalions of ihe iheory of errors
when he was wriling ihe article Probabililies for ihe first edilion of Chambers'
Encyclopaedia, his aim being lo eslablish the ordinary law of errors by a
" natural process " free from ihe malhemalical complicalions which characlerise
ihe work of aulhorilies like Laplace and Poisson. Slarling from a simple case
of drawing while and black balls from a bag, he deduced ihe well-known
exponenlial expression, and ihen generalised ihe demonslralion. If we excepl
his much laler papers on ihe kinelic iheory of gases Tail does nol seem to
have relurned lo queslions involving ihe iheory of probabililies.

BRACHISTOCHRONES 99

The preparation of the great Treatise on Natural Philosophy led his
mind into various dynamical problems, such as central forces, the hodograph,
and the theory of Action. On these he communicated short notes to the
Messenger of Mathematics and to the Proc. R. S. E. His greatest effort
at this time was, however, his paper " On the Application of Hamilton's
Characteristic Function to special Cases of Constraint." In it he showed how
brachistochrones or paths of shortest time were to be discussed by the same
general method which Hamilton had applied to the theory of Action. Most
of the investigation was embodied in the second edition of Tait and Steele's
Dynamics of a Particle. Before publishing the paper Tait took the precaution
of asking Cayley if he had been forestalled. Cayley replied :

" I have only attended to the direct problem of Dynamics, to find the motion
of a system under given circumstances, — whereas the question of brachistochrones
belongs of course to the inverse one... — and I really hardly know anything about
it. My impression is that the subject is new."

In his address to Section A of the British Association in 1870 Clerk
Maxwell, when referring to the rival theories of light, said

" To understand the true relation of these theories in that part of the field where
they seem equally applicable we must look at them in the light which Hamilton has
thrown upon them by his discovery that to every brachistochrone problem there
corresponds a problem of free motion, involving different velocities and times, but
resulting in the same geometrical path. Professor Tait has written a very interesting
paper on this subject1."

Now this discovery which Maxwell ascribes to Hamilton was really made
by Tait in the paper under discussion. Maxwell was usually very accurate in
his history, and we can imagine the glee with which Tait found his friend
tripping. He would by some merry joke make fun of Maxwell's momentary
deviation from the lines of historic truth. Accordingly on July 14, 1871,
Maxwell apologised in quaint fashion on an unsigned post card as follows :

" O T' Total ignorance of H and imperfect remembrance of T' in Trans. R. S. E.
caused — to suppose that H in his optical studies had made the statement in the

form of a germ which T' hatched. I now perceive that T' sat on his own egg, but
as his cackle about it was very subdued compared with some other incubators, I was
not aware of its origin when I spoke to B. A. When I examined hastily H on
Rays I expected to find far more than was there. But the good of H is not in
what he has done but in the work (not nearly half done) which he makes other
people do. But to understand him you should look him up, and go through all
1 See Maxwell's Scientific Papers, Vol. n, p. 228.

13—2

ioo PETER GUTHRIE TAIT

kinds of sciences, then you go back to him, and he tells you a wrinkle. I have
done lines of force and = potls. of double tangent galvanometer in a diagram,
showing the large uniform field. Is T still in London ? "

It was in this paper also that Tait proved his neat theorem " that a planet
moving about a centre of force in the focus of its elliptic orbit is describing
a brachistochrone (for the same law of speed as regards position) about the
other focus," or in other words, " while time in an elliptic orbit is measured by
the area described about one focus, action is measured by that described about
the other." These statements are intimately bound up with the general
theorem connecting brachistochrones and free paths already referred to.

In December 1871 Tait communicated to the Royal Society of Edinburgh
a mathematical note on the theory of spherical harmonics (see Proc. R. S. E.
Vol. vn, pp. 589-596). The interest of the note lies entirely in the simple
manner in which certain fundamental relations are deduced. The article.
seems to have taken form to some extent under the influence of Maxwell,
who, on a post card of date Sept. 5, 1871, wrote

" Spherical Harmonics first written in 1867 but worked up from T and T' when
that work appeared. Have you a short and good way to find //(^iw)2^5? If so
make it known at ice that I may bag it lawfully as T' 4nion path to harmonic
analysis."

Tait seems to have replied by sending a sketch of a new method, for
Maxwell on October 23 wrote (again on a post card)

" O. T' ! R. U. AT 'OME ? //Spharc'^/S was done in the most general form in 1867.
I have now bagged f and 17 from T and T' and done the numerical value of
//(Ff'")2^ in 4 lines, thus verifying T+T"s value of tf^MJdS. Your plan seems
indept. of T and T' or of me. PUBLISH ! "

This was followed up ten days later by a fairly long letter bearing upon
Tail's notes, the one quaternionic and the other in ordinary analysis. Tait
must now have sent his analytical note very much in the form in which it
was finally published. Regarding it Maxwell wrote :

it SCROOPE TERRACE,
CAMBRIDGE,

2 Nov. 1871.
O T'

Your notes have ravished me. An interest in 2<£apf being revived this is
exactly what is wanted for a quantitative or computative discussion of the symmetrical
system considered as depending only on certain symbols * and s.

MAXWELL ON SPHERICAL HARMONICS 101

It seems to have little or nothing to do with your 4nionic reduction which
is of course indept. of a selected axis1.

My method is also indept. of a selected axis, but does not seem to be
equivalent to your 4nion reduction which goes by steps.

Murphy is not at all bad in his way and affords a very good specimen of a
Caius man working a calculation.

How is it that 1,<f>ap^ 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<£ap£ are very rich and a judicious man might get up a
capital book for Cambridge, in which the wranglers would lade themselves with
thick clay till they became blind to the concrete.

But try and do the 4nions. The unbelievers are rampant. They say "show
me something done by 4nions which has not been done by old plans. At the
best it must rank with abbreviated notations."

You should reply to this, no doubt you will.

But the virtue of the 4nions lies not so much as yet in solving hard
questions, as in enabling us to see the meaning of the question and of its solution,
instead of setting up the question in x y z, sending it to the analytical engine, and
when the solution is sent home translating it back from x y z so that it may
appear as A, B, C to the vulgar.

There appears to be a desire for thermodynamics in these regions more than
I expected, but there are some very good men to be found.

You will observe a tendency to bosch in this letter which pray xqs as I
have been reading an ill assorted lot of books till I cannot correct prooves.

yours truly

d±>
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

0

i

i

0

o

0

O ...X

0

o

0

0

0

I

2

i

i

o

0

o

0

o

o

0

I

3

4

4

i

i

0

o

0

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

0

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

14—2

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.IO20 seconds is the same as that
of 3.IO20 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 pv be then
the mean free path; and if C be the number of collisions in 3.IO20 seconds, we have

nvC

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 (io13 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 pvjv. 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 tube1, 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<f> + Ctr* cos 4</> + etc.,
where x = ar cos 0 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 dl (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 H8 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 4nionss 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.

15—2

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 ither1 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/<?."
Tait's reply to this long letter was as follows :

Q. C. BELFAST,

October 19^/58.
My dear Sir William Hamilton

Plunged as I now am in the middle of the entrance & Scholarship
examinations for this session, I shall not have for some days the amount of time
requisite for a careful reading of your excellent No. V ____

I am tolerably familiar with the works of Moigno — and I quite agree with you
in your estimate of him. Did you ever see his ' Repertoire d'Optique Moderne ' ?
It is the strangest mixture of valuable matter and utter trash I ever came across.
I should like very much to know your opinion of Cauchy's investigations in the
Undulatory Theory — for I have found it possible by apparently legitimate uses of

WARREN'S ANTICIPATION OF HAMILTON 123

his methods to prove almost anything. But I have given up these speculations for
the present till I see whether I cannot get the requisite command of Quaternions,
as I feel that they must inevitably much simplify the investigations....

All that I have to say on the subject of my School (though I fancy myself
rather a cosmopolitan) as regards Differentials, &c., I must beg you to let me
reserve for some days till I have comparative leisure again —

Yours very truly

PETER G. TAIT.

While Letter v was in process of construction, Hamilton sent two shorter
letters, Nos. vi and vn, relating to other quaternion questions. In the
former he discussed the surface of revolution in the form

p = at(j>ua.~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 equation1

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 differentials1
— 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 Motion1." 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
equation1 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 <j>~* ( ) 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

<fji~l8p = •or
(where each member represents the whole elastic force called into play),

Your symbol has over mine the great advantage of being separable from the
subject, so that you can write

o = Sfj,~l ($-* - /*-V/*-1.

Having thus (as I hope) sufficiently allowed the superiority of your notation,
I may be permitted to remark that I think mine has one advantage as I have
applied it, namely, that of introducing directly the half of your operator <f>~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 <f>, 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

i3o 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 <f> 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 (P3 - r1)-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 termination1: 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 <j> 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

<f> 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 + <f>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 <j> 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's1 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 Theorems1,
which I have lately proved (as I think) anew, and which are intimately connected
with the question. —

THEOREM I. If $p be a distributive and vector and real function of a real
vector p, such that Sa<pp = Sptjxr, (a), then the eqn Vp$p = o, (/9), is satisfied by
(at least) one real direction of p.

THEOREM n. Whatever be the given and real dirns of p, (at least) two real and
rectangular directions, p and p", can be assigned, for a vector ra-, which shall satisfy
the two eqns Spvr = o, (7), and Sp-nfasi — o, (S).

THEOREM III. If p and TO- satisfy the system of the three eqns, (/3) (7) (&),
then w satisfies (/9), or more fully FUJ-^CT = o, (e).

THEOREM IV. (Extension of Theorem I.) The equation (/3) is always satisfied
by at least one system of three real and rectangular directions, plt p.,, p,, of p.

Proof obvious, from what precedes.

THEOREM V. The functional symbol <f> 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 eqM,
(4>+gi)Pi = 0, (<£+#,) Ps = 0, (<t>+g,)ps = 0, (97),

are satisfied by the 3 rectangular directions plt pt, 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 <f>p = - (g^xlpl +g*x*pt +g,x,ps\ (t),

and Sp<f>p = - (g&p? +g*x?p? +gtxfpt)> (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^;g3, and 3 real and rectangular
unit vectors, plt pt, 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 =£? + 2S<tpSpp, (ft),

without any sign of summation in the 2nd number ; and g~, a, $\ can always be
made real.

Having written so far, and even had the first sheet of this letter copied (into A.
1859), I think that I may now indulge myself in opening your letter received this
morning.... For I have been apprehensive of your anticipating me, or hitting on my
old train of thought, before I had (as above) recovered it for myself.

Tait, on April 21, replied:

I was greatly pleased with the transformations in XIX. I can easily prove all
your theorems with the exception of the first, i.e. that " Vp<f>p = 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 letter1 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 <f> 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 <^>~1i/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 <f> 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 <f>~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 <f> 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 problem1. 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 <j> and in letter 23 gave the wave-surface equation in

the new form T (p* + <f?)~ p = i ; a. form whose elegance Hamilton at once
recognised and continued thereafter to use.

(3) The theory of the linear vector function. This is chiefly contained
in Hamilton's xix, xxiu, xxv, and in Tail's 32 and 33. The essential parts
are reproduced in Hamilton's Elements and in Tail's Elementary Treatise.

(4) The theory of envelopes. This was begun by Tail's problem
of the paraboloid cylinder which forms section 321 of his Treatise (3rd edilion).
The problem greally look Hamillon's fancy. He began ihe discussion in
letter vm, and developed it in elaborate detail by quaternion processes in
letters xi and xm.

(5) The planning of the new ireatises on the calculus. Early in 1859
Hamilton began to write his " Manual," which finally appeared in 1866 as the
Elements, unfortunately incomplete in consequence of ihe dealh of the author
in 1865. Tail's own trealise was projected during the summer of 1859, bul
was wilhheld from publicalion until Hamillon's work should appear. It was
finally published in 1867.

In connection with the preparation of Tail's Quaternions the following
letter to Sir John Herschel is of considerable interest. Tait had senl Herschel
copies of some of his qualernion contributions to ihe Quarterly Journal of
Mathematics and, in reply lo Herschel's acknowledgement, wrote on Dec. 14,
1 864, as follows :

My Dear Sir

I am much obliged by your very kind note just received....

Five years ago, Messrs Macmillan & Co. advertized for speedy publication
an " Elementary Treatise on Quaternions " by me ; but, as my good friend
Sir W. R. Hamilton thought that it might possibly interfere with his forthcoming
" Elements of Quaternions " I withdrew it — and have published only the few articles
I recently sent you — all of them with his approval.

I had no idea that you had been engaged in preparing such a work ; and I

142 PETER GUTHRIE TAIT

merely write to say that I shall be most happy if you will persevere in your intention
of publishing an elementary volume on the subject. In fact the papers I have sent
you contain nearly the whole of my researches in the elementary part of the theory.
I have an immense store of work in MSS relating to its higher applications — but
unfit for an elementary treatise.

Since I projected the treatise I have ceased to be a Professor of Mathematics ;
and with private experiments and the ordinary preparation for the work of my class,
I feel that I have barely time enough to contribute my fair share to the " Treatise
on Natural Philosophy " which Thomson and I have undertaken. And, as this Treatise
is certain to extend to three volumes at least, of which (after two years work) not
even one is yet published, I feel that it may be years before I shall be in a position
to write on Quaternions in a carefully considered popular style. I am sure that
my old friend Macmillan would be delighted to have the chance of substituting your
name for mine in the advertisement, which he has been hopelessly repeating for
some years.

But the consent of Sir W. R. Hamilton is absolutely necessary to anyone
undertaking the work.

Believe me, my dear Sir,

Yours very truly

P. GUTHRIE TAIT.
Sir J. F. W. Herschel, Bart.

It is certainly remarkable that Herschel at the age of 72 should have
thought of such a project.

Only a careful comparison of the pages of Hamilton's and Tail's works
could establish to what extent Tail's contribulions were essenlially original.
Their melhods were markedly different Hamillon revelled in geomelrical
development of all kinds, ihe fertility of his malhemalical imagination
lending al limes lo make him discursive and almosl prolix. Tail's endeavour
in all his really original quaternion work was lo grapple wilh physical and
dynamical problems. Compare for example ihe Hamillonian development of
ihe properlies of ihe linear vector funclion wilh ihe chapler on slrains which
Tail conlribuled lo Kelland and Tail's Introduction to Quaternions — each
mode of irealmenl admirable in ils way.

The linear veclor funclion conlinued lo absorb much of Tail's allenlion
up to ihe very lasl day of his life. He made important contribulions lo
ihe iheory as well as many inleresling applications of ils power. See for
example papers xv, xxi, xxvi, cxiv, cxx, cxxi, cxxn, cxxiv in ihe Scientific
Papers, Vols. i and n — especially ihe firsl-named, lhal on ihe Rolalion of a
Rigid Solid.

Unqueslionably, however, Tail's greal work was his developmenl of

THE OPERATOR "NABLA" 143

the powerful operator V. Hamilton introduced this differential operator in
its semi-Cartesian trinomial form on page 610 of his Lectures and pointed
out its effects on both a scalar and a vector quantity. This, it will be
remembered, was one of the points especially brought forward by Tait when
he began the correspondence with Hamilton. Neither in the Lectures nor
in the Elements, however, is the theory developed. This was done by Tait
in the second edition of his book (V is little more than mentioned in the
first edition) and much more fully in the third and last edition.

From the resemblance of this inverted delta to an Assyrian harp Robertson
Smith suggested the name Nabla. The name was used in playful intercourse
between Tait and Clerk Maxwell, who in a letter of uncertain date
finished a brief sketch of a particular problem in orthogonal surfaces by
the remark "It is neater and perhaps wiser to compose a nablody on this
theme which is well suited for this species of composition."

In 1870, when engaged in writing his Treatise on Electricity and
Magnetism, Maxwell sent Tait the following suggestions as to names for
the results of V acting on scalar and vector functions :

GLENLAIR, DALBEATTIE,

Nov. 7, 1870.
Dear Tait

n • d , -d , id

V = z -r + J-r + k-T-.
dx J dy dz

What do you call this? Atled?

I want to get a name or names for the result of it on scalar or vector functions
of the vector of a point.

Here are some rough hewn names. Will you like a good Divinity shape their
ends properly so as to make them stick?

(1) The result of V applied to a scalar function might be called the slope of
the function. Lam<§ would call it the differential parameter, but the thing itself is a
vector, now slope is a vector word, whereas parameter has, to say the least, a scalar
sound.

(2) If the original function is a vector then V applied to it may give two
parts. The scalar part I would call the Convergence of the vector function, and
the vector part I would call the Twist of the vector function. Here the word twist
has nothing to do with a screw or helix. If the word turn or -version would do
they would be better than twist, for twist suggests a screw. Twirl is free from
the screw notion and is sufficiently racy. Perhaps it is too dynamical for pure
mathematicians, so for Cayley's sake I might say Curl (after the fashion of
Scroll). Hence the effect of V on a scalar function is to give the slope of that
scalar, and its effect on a vector function is to give the convergence and the twirl

144 PETER GUTHRIE TAIT

of that function. The result of V' applied to any function may be called the
concentration of that function because it indicates the mode in which the value
of the function at a point exceeds (in the Hamiltonian sense) the average value of
the function in a little spherical surface drawn round it.

Now if a- be a vector function of p and F a scalar function of p

VF is the slope of F

. VF is the twirl of the slope which is necessarily zero
V*F is the convergence of the slope, which is the concentration of F.
Also SV <r is the convergence of <r

Wo- is the twirl of a:

Now, the convergence being a scalar if we operate on it with V, we find that
it has a slope but no twirl.

The twirl of <r is a vector function which has no convergence but only a twirl.
Hence W, the concentration of <r, is the slope of the convergence of <r together
with the twirl of the twirl of <7, the sum of two vectors.

What I want is to ascertain from you if there are any better names for these
things, or if these names are inconsistent with anything in Quaternions, for I am
unlearned in quaternion idioms and may make solecisms. I want phrases of this
kind to make statements in electromagnetism and I do not wish to expose either
myself to the contempt of the initiated, or Quaternions to the scorn of the profane.

Yours truly

J. CLERK MAXWELL.

A week later (Nov. 14, 1870) Maxwell, when returning Robertson Smith's
letter in which the philology of Nabla was discussed in detail, wrote :

" I return you Smith's letter. If Cadmus had required to use V and had
consulted the Phoenician Professors about a name for it there can be no question
that Nabla would have been chosen on the JO 3 principle. It is plain that Hamilton's
V derives itself with all its congeners from Leibnitz' d, which has become consecrated
along with D d & etc., and a name derived from its shape is hardly the thing.

" With regard to my dabbling in Hamilton I want to leaven my book with
Hamiltonian ideas without casting the operations into Hamiltonian form for which
neither I nor I think the public are ripe. Now the value of Hamilton's idea of
a vector is unspeakable, and so are those of the addition and multiplication of
vectors. I consider the form into which he put these ideas, such as the names
Tensor, Versor, Quaternion, etc., important and useful, but subject to the approval
of the mathematical world....

" The names which I sent you were not for V but for the results of V. I shall
send you presently what I have written, which though it is in the form of a
chapter of my book is not to be put in but to assist in leavening the rest. I shall
take the learned Auctor1 and the grim Tortor1 into my serious consideration,
though Tortor has a helical smack which is distasteful to me but poison to T."

' These seem to have been suggestions made by Tail himself, probably more in joke
than in serious mood.

THE OPERATOR "NABLA" 145

It was probably this reluctance on the part of Maxwell to use the term
Nabla in serious writings which prevented Tait from introducing the word
earlier than he did. The one published use of the word by Maxwell is in
the title to his humorous Tyndallic Ode1, which is dedicated to the " Chief
Musician upon Nabla," that is, Tait.

The following letter from Maxwell shows how clearly he had grasped
the significance of the quaternion notation.

ARDHALLON,

DUNOON,

Jan. 23, 1871.
Dr T'

Still harping on that Nabla ?

You will find in Stokes on the Dynamical Theory of Diffraction something of
what you want, this at least which I quote from memory.
I. For all space — your eqn

Vo- = V"(T + 7>)

where <r is given and T and v are to vanish at oo gives but one solution for r and
one for v, the first derived by integration from FVo- and the second from >SV<r by
the potential method, and we then get the result in the form

(because, as Helmholtz has shown (Wirbelbewegung) 5Vr = o). All this is as old
as 1850 at least. See Stokes.

Now we leave all space and consider a region 2 within which VlP = o and therefore
V/* has no convergence. Now if a vector function has no convergence it ought to
be capable of being represented as the curl of a vector function, or there ought to

be a vector <r such that

FVo- =

The simplest case to begin with is of course the potential due to unit of mass
at the origin. Find <r and T for that case ! The difficulty arises from the fact that
the region in which V2/3 = o is here periphractic and surrounds completely the origin
where this is not true. If we draw a closed surface including the origin then

Si

whereas

1 5 Uv Wads = O, necessarily*.

Hence to make it impossible for the region to include the origin we must get
rid of periphraxy by drawing a line from the origin to oo and defining the region 2 so
as not to interfere with this line.

1 Reproduced partly in facsimile at the end of this Chapter.
' Because \\Sdv VV<r = \\ \ dvSV W<r = o for V'<r is a vector.

T. 19

146 PETER GUTHRIE TAIT

We may then write/ for i/r and

= I
J o

0

If we suppose the line to be in the axis of x, this gives

... xz xy

<7 = / (O) -f- 7 K r

T(y^ ~f- z^j T (y^ -\- Zj

an exceedingly ugly form for a thing derived from so symmetrical a beginning.
But this cannot be avoided if the algebraic sum of the masses is finite.
If it is o, we may treat it as magnetic matter.
If, in a region I,' in which there is magnetization, the intensity of magnetization be

3 = iA +JB + kC
and if /= ijr, where r is the distance between xyz and x'y'z', then

P

••• '*A

=jjjA (x - *) + B(j-y) + C(z'- M) dx,dy,^

Also
where F= jfj C^~^ ^^'^dxdy'dz', &c.

or

All this occurs in passing from the old theory of magnetism to the electro-
magnetic.

I have put down a lot of imitations of your jargon mainly that you may check
me in any solecism. I think if you are making a new edition of 4nions you should
give prominence to the rules defining the extent of the application of symbols such
as V, S, T, U, K, &c., which are consecrated letters, not to be used for profane
purposes ____

What do you make of this ?

You say that the constituents of T are potentials with densities — — &c. Well,

47T dx

then, take P=i/r and dP/dx=-x/r* &c., then the constituents of T will be xfiirr
&c. or

and VT = — . - = P, a scalar.

?r r

POTENTIALS AND STRESS FUNCTIONS 147

In fact of whatever scalar form P be, if V*T = V/>, 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 <r as determined above is such that SV<r = o so that we may
truly say V<r = V/>.

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 (Edinh) 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 Ld 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 it1.

" Search for a physical basis for

5 . V'aVo-

as a term of the energy developed in a medium by a variable displacement <r.
When found, make a note of, and apply to oil of turpentine, eau sucre"e, &c., for it
brings out the right sort of action on light of all colours. But the mischief is FV<r,
which it is manifest, can be produced without making any physical change inside
a body. The very rotation of 0 produces it. Now VV is a vector. Turn it
alternately in the direction of FV<r and oppositely and you have increase & diminu-

1 See Tait's Note on the Strain Function, Proc. Jt. S. E. 1872 ; Sci. Pap. Vol. i, pp. 196-7.

19—2

148 PETER GUTHRIE TAIT

tion of energy, & therefore a tendency to set like a magnet. The comfort is that
V'o- cannot subsist of itself.

" Of course the resultant force on an element is of the form FVV, and if <r is
a function of z only, and Ska- = o,

"This is the only explanation of terms of this form in an isotropic or fluid
medium, and since the rotation of plane of polarization is roughly proportional to
the inverse square of the wave length, terms of this form must exist.

dp,,
df

Thus, on the one hand, we have Tait submitting his quaternionic
theorems to Maxwell's critical judgment, and Maxwell recognising the power
of the quaternion calculus as handled by Tait in getting at the heart of
a physical problem.

Unfortunately Tail's letters to Maxwell have not been preserved ; and
we can only infer as to the general nature of Tail's replies to Maxwell's
constant enquiries regarding quaternion terms and principles. There can be
no doubt however that, in introducing the operator V and the Hamiltonian
notation associated with it, Maxwell was strongly influenced not only by
Tail's masterly paper on Green's and other Allied Theorems but also by
his intimale correspondence.

The fundamental properties of V as a differential operator may be
expressed very simply in dynamical language. When it acts on a scalar
function of the position of a point it gives in direction and magnilude the
maximum space rate of change of this funclion. For example, if u is a
polenlial, V« is the corresponding force. Its effect upon a vector quantity
is, in general, to produce a quaternion, with its scalar and vector parts.
Suppose the vector quantity to be the velocity of flow of a fluid, symbolised
by cr ; then Vcr consists of two parts, the scalar and vector parts. The former,
SVcr, represents what Maxwell called the Convergence, indicating a change
of density in the fluid at the point where cr is the velocity ; and for the
latter, symbolised by FVcr, and measuring in the present case the vorticity,
Maxwell's name of Curl has been generally accepted.

It is instructive to read Tail's early papers discussing the properties of V,
and to follow the growth of his power in dealing wilh it. At first he was
contenl lo begin wilh Hamillon's trinomial definition, as in the paper of 1862

ROTATION OF A RIGID BODY 149

on the Continuous Displacements of the Particles of a Medium (Scientific
Papers, Vol. I, p. 37). But ere long he discovered a less artificial definition,
free from Cartesian symbolism. This mode of establishing the theory of
V is given in the appendix to his great paper already mentioned, that on
Green's and other Allied Theorems (1870, Sci. Pap. Vol. r, p. 136). Here
we find developed in an original manner the quaternion integrals through
volumes, over surfaces, and along edges, which include as special cases the
theorems of Green, Gauss, and Stokes.

In 1868 Tait published an elaborate memoir on the Rotation of a Rigid
Body about a Fixed Point (Sci. Pap. Vol. i, p. 86), concerning which, while it
was in preparation, he had a good deal of correspondence with Cayley. On
August 1 8, 1868, he sent Cayley the concise quaternion equations of §§ 15, 19,
21, and asked if the results are "merely a shortening of yours"; and on
October 17, the same year, he drew Cayley 's attention to the Cartesian formulae
in §§ 28, 29, and to the fact that, " without integrating Euler's equations at
all (and I think from your second Report that the problem has always been
solved by first finding /, q, r), I find the following equations for w, x, y, z
[equation 24 in Paper]." On October 21, Cayley replied :

" The rotation formulae are deducible by an easy transformation from formulae
in my paper (Cam. and Dub. Math. Journ. Vol. I, 1846).. ..But the actual form you
have given to the formulae is, so far as I am aware, new ; and a very decided improve-
ment as reducing the denominator to be of the third order."

For these two quaternion papers Tait was awarded the Keith Prize by
the Royal Society of Edinburgh. The Secretary asked Clerk Maxwell to
draw up a statement to be read when the prize was formally awarded by the
President ; and Maxwell responded with a playful humour which considerably
mystified Professor J. H. Balfour. A copy was preserved in Tait's Scrap
Book.

(Balfour, having asked Maxwell to write something which could be read at a
meeting of the R. S. E. when I was to get the Keith medal, was mystified as
follows. P. G. T.)

GLENLAIR, DALBEATTIE, 28/11/70.

Dear Professor Balfour,

I do not presume to inform an officer of the Society with respect to
its recent awards. I saw that Tait had got the Keith Prize which is or ought to
be known to the public. I have not yet got a copy of the reasons for which it
was awarded, so if I coincide with them it does not arise from imitation.

The question seems to be, What is Tait good for ? Now I think him good,
first, for writing a book on Quaternions, and for being himself a living example of

ISO PETER GUTHRIE TAIT

a man who has got the Quaternion mind directly from Hamilton. I am unable
to predict the whole consequences of this fact, because, besides knowing Quaternions,
Tait has a most vigorous mind, and is well able to express himself especially in
writing, and no one can tell whether he may not yet be able to cause the Quaternion
ideas to overflow all their mathematical symbols and to become embodied in ordinary
language so as to give their form to the thoughts of all mankind.

I look forward to the time when the idea of the relation of two vectors will
be as familiar to the popular mind as the rule of three, and when the fact that
ij=—ji will be introduced into hustings' speeches as a telling illustration. Why
not? We have had arithmetical and geometrical series and lots of odd scraps of
mathematics used in speeches.

Nevertheless I do not recommend some of Tail's mathematical papers to be
read as an address to the Society, ore rotunda. That on Rotation is very powerful,
but the last one on Green's and other allied Theorems is really great.

The work of mathematicians is of two kinds, one is counting, the other is
thinking. Now these two operations help each other very much, but in a great
many investigations the counting is such long and such hard work, that the
mathematician girds himself to it as if he had contracted for a heavy job, and
thinks no more that day. Now Tait is the man to enable him to do it by thinking,
a nobler though more expensive occupation, and in a way by which he will not
make so many mistakes as if he had pages of equations to work out

I have said nothing of his book on Heat, because, although it is the clearest
thing of the sort, it is not so thoroughly imbued with his personality as his Quaternion
works. In this however I am probably entirely mistaken, so I advise you to ask
Tait himself who I have no doubt could hit off the thing much better than any
one.

I remain

yours truly

J. CLERK MAXWELL.

It seems appropriate here to reproduce from Maxwell's letters to Tait
some extracts bearing upon the quaternion calculus, for which it is clear
Maxwell had a profound admiration. The letters of Nov. 2, 1871, and
Dec. i, 1873, have been already given in extenso (see above, pp. 101,
115); the following are culled from other letters:

"(Dec. 21, 1871.) Impress on T. that (-^\ + (4rY + (-j-Y = - V1 and not + V1 as

he vainly asserts is now commonly believed among us. Also how much better and
easier he would have done his solenoidal and lamellar business if in addition to what
we know is in his head he had had say, 20 years ago, Qns. to hunt for Cartesians
instead of vice versa. The one is a flaming sword which turns every way ; the
other is a ram, pushing westward and northward and (downward ?). What we want
a Council to determine is the true doctrine of brackets and dots and the limits of
the jurisdiction of operators.

MAXWELL ON QUATERNIONS 151

"(Oct. 4, 1872.) How about electromagnetic 4nions as in proof slip 106, 107?
I suspect I am not sufficiently free with the use of the Tensor in devectorizing such
things as r (distance between two points). The great need of the day is a grammar
of 4nions in the form of dry rules as to notation and interpretation not only of
S T U V but of . ( ) and the proper position of da, etc. Contents, Notation, Syntax,
Prosody, Nablody.

"(Oct. 9, 1872.) I think I had better consecrate p [in the Treatise] to its pre-
scriptive office of denoting indicating or reaching forth unto the point of attention
(xyz) — Has p a name? It is no ordinary vector carrying a point. It is rather a
tentacle or feeler which reaches from the subject to the object. Is he the scrutator?
I am glad to hear of the second edition of 4nions. I am going to try, as I have already
tried, to sow 4nion seed at Cambridge. I hope and trust nothing I have yet done may
produce tares. But the interaction of many is necessary for the full development
of a new notation, for every new absurdity discovered by a beginner is a lesson.
Algebra is very far from o. k. after now some centuries, and diff. calc. is in a mess
and fff is equivocal at Cambridge with respect to sign. We put down everything,
payments, debts, receipts, cash credit, in a row or column, and trust to good sense
in totting up.

"(March 5, 1873.) O T'. If, in your surface integrals, ds is an element of surface,
is not ds a vector? and does not multiplication by Uv scalarize it? In your next
edition tell us if you consider an element of surface otherwise than as Vdadft
where a and ft are vectors from the origin to a point in the surface defined by
parameters a, b. Here the element of surface is a vector whose tensor is the area
and whose versor is Uv. These things I have written that our geometrical notions
may in Quaternions run perpetual circle, multiform, and mix and nourish all things.
Such ideas are slowly percolating through the strata of Cartesianism, trilinearity,
and determinism that overlie what we are pleased to call our minds.

"(Sep. 7, 1878.) Here is another question. May one plough with an ox and an
ass together? The like of you may write everything and prove everything in pure
4nions, but in the transition period the bilingual method may help to introduce and
explain the more perfect.

" But even when that which is perfect is come that which builds over three axes
will be useful for purposes of calculation by the Cassios1 of the future.

" Now in a bilingual treatise it is troublesome, to say the least, to find that the
square of AB is always positive in Cartesians and always negative in 4nions, and that
when the thing is mentioned incidentally you do not know which language is being
spoken.

" Are the Cartesians to be denied the idea of a vector as a sensible thing in real
life till they can recognise in a metre scale one of a peculiar system of square roots
of- I?

" It is also awkward when you are discussing, say, kinetic energy to find that to

1 " And what was he ?

Forsooth, a great arithmetician" (Othello, Act i, Scene i).
A neat example of Maxwell's ingenuity in literary suggestion.

152 PETER GUTHRIE TAIT

ensure its being + ve you must stick a — sign to it, and that when you are proving
a minimum in certain cases the whole appearance of the proof should be trending
towards a maximum.

" What do you recommend for El. and Mag. to say in such cases ?

" Do you know Grassmann's Ausdehnungslehre ? Spottiswoode spoke of it in
Dublin as something above and beyond 4nions. I have not seen it, but Sir W.
Hamilton of Edinburgh used to say that the greater the extension the smaller the
intention."

We have not the record of Tait's reply to the question of the sign,
a question which many later users of vector notations have attempted to
answer by simply ignoring one of the distinctive features of Hamilton's
calculus. So long as it is a question merely of a concise notation no harm
is done ; and Maxwell without seriously affecting the symbolic presentation
of his theory of electromagnetism might have adopted this method. But he
had too great a regard for the founder of Quaternions, and too deep an
insight into the inwardness of the quaternion calculus, to allow mere
expediency to play havoc with far-reaching fundamental principles.

Meanwhile Tait's activity in developing quaternion applications continued
throughout the seventies. In a Note on Linear Differential Equations in
Quaternions (Proc. R. S. E. 1870; Set. Pap. Vol. i, p. 153) he struck out on
new paths. Here he gave an extremely simple solution of the problem
of extracting the square root of a strain or linear vector function.

In a letter to Cayley of date Feb. 28, 1872, Tait gave the Cartesian
statement of the problem and continued, —

My quaternion investigation, which is very simple, leads to the biquadratic

4m,

where m, ««„ and mt are known functions of [the elements of the strain]; and from 6
the values of [the elements of the square root of the strain] can easily be found.

Thomson and I wish to introduce this into the new edition of our first volume
on Natural Philosophy — but he objects utterly to Quaternions, and neither of us
can profess to more than a very slight acquaintance with modern algebra — so that
we are afraid of publishing something which you and Sylvester would smile at as
utterly antiquated if we gave our laborious solutions of these nine quadratic equations.

As I said before the question is of interest in another way (for my Report on
Quaternions), for if <f> 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" ( ) = <f>'<f> ( ), 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 <f>.

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~ff( )(gr-1)-*.

Of course it may also be put in the form (g~lry ( ) (g-1r)~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

q1 r™ q1 ( ) 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^ ylt 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( )fl 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 expression1

^{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 ft1 + k cos 7')},
that is of the sum of any two quaternions

w + ix +jy + kz, v/ + ix" +jy' + kz1.

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, <f>, 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 Sp<j>p = —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

X1 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<pp = — i is strictly kinematical, and defines an
ellipsoid (or other central conicoid) with reference to a strain — in this case a pure
strain — the conception of which is vividly realistic.

gives no physical suggestion at all.

I should have said, in my paper, that we have to thank Cartesian processes
for the idea of an Invariant. In pure quaternions you have them always, so that
they present no feature for remark, p itself is an Invariant just as much as V is.
But what do you say to my little three term formula (on p. 95) which is equivalent
to 189 Cartesian terms?

Cayley seems to have made no immediate reply to this letter ; but
on June 6, 1894, in a short note on other matters he threw in the
remark :

" I wish you would tell me in what sense you consider Quaternions to be a method:
I do not see that they are so, in the sense in which coordinates are a method ; and
I consider them rather as a theory."

Tait replied on June 10, 1894:

As to your question about Quaternions, I fear that I do not quite catch your
meaning, so far at least as regards the technical distinction between a " Theory " and
a " Method." From my point of view, Quaternions are a mode of representing
geometrical or physical facts in such a clear way that one can see their mutual
relations and their consequences. They assist me in these in the same sort of way
as a figure or a model does, in the case of a knot or a complex surface : — or as an
experiment of a crucial kind does. In fact they help one to think. I look upon
them as contrasted with, rather than as related to, numerical work whether by
logarithms or by definite integrals. These in themselves do not help you to think,
though they are vitally important when you wish to measure ; and though the working
of them out may require very much thought. I fear this, in its turn, will not be very
comprehensible to you : — but I have not been in the habit of dealing with such
classes of questions ; or at all events of trying to express, in language, my notions
about them.

The discussion now entered upon a more definite phase, and on
June 1 8, Cayley wrote :

" Considering coordinates and quaternions each as a method I should formulate
the relation between them as follows : We seek to determine the position of a
variable point P in space in regard to a fixed point O. Thro" 0 draw the rectangular
axes Ox, Oy, Oz.

" Then

(coordinates) the position is determined by the coordinates x, y, z.
(quaternions) the position is determined by means of the vector a ( = OP).

CORRESPONDENCE WITH CAYLEY 163

" But then what do you mean by the vector a ? I mean

so that the knowledge of a implies that of the coordinates....

" But your claim for the superiority of quaternions rests, as I understand it, on
the non-necessity of any explicit use of the equation in question a=*ix + jy + kz, or
a = — iSia —jSja — kSka.. . .

" As to the modus operand!, if in regard to the points (x, y, z) and (x1, y1, z1) one
has to consider the combinations yz'—y'z, zx' — z'x, xy' — x'y, I consider these directly
as the minors of the matrix

x, y, z

x, y', z"

— whereas you represent them (in what seems to me an artificial manner) as the
components of Fa/3.... "

Tail's reply I give in full, since it presents in the briefest possible form
the fundamental principles of quaternions as Tait regarded them.

38 GEORGE SQUARE, EDINBURGH,

19/6/94.
My dear Cayley

In the very first paper I published on Qns. (Mess. Math. 1862) I said
"the method is independent of axes... and takes its reference lines solely from the
problem it is applied to." Unless under compulsion, I keep to a, and do not write
either ix+jy + kz or — iSia — &c.

Hamilton said (Lectures, p. 522) " I regard it as an inelegance, or imperfection,
in quaternions, or rather in the state to which it has hitherto been unfolded, whenever
it becomes or seems to become necessary to have recourse to " x, y, z, &c.

Unfortunately like all who have been brought up on Cartesian food I now
and then think of a as —iSia — &c. (Hamilton himself was a terrible offender in
this way: his i, j, k, was almost a fatal blot on his system). But I know that
I ought not to do so, because a better way is before me. Thus :

(P.S. What follows is, I see, Prosy. But it is necessary.)

Position is essentially relative (though in physics direction may be regarded as,
in a sense, absolute) so we must have an origin, p then, or OP, I look on as P — O,
the displacer which takes a point from O to P. Should it subsequently be displaced
to Q we have

Hence all the COMPOSITION laws of Vectors. And of course the notion
of repetition of any one displacer, so that we get the idea of the tensor, and of the
unit vector.

To COMPARE vectors, we may seek their quotient, or the factor which will
change one into the other. There are two obvious ways of looking at this.

(a) The first is mathematical rather than physical. Here we introduce the
idea of a factor, such that

a/p X p = <r.

21 - 2

1 64 PETER GUTHRIE TAIT

And we see at once that it consists (or may be regarded as consisting) of two
independent and commutative factors, its Tensor and its Versor.

But then comes the life of the whole; the recognition of the fact that when
POQ is a right angle, the versor of a-jp may be treated as in all respects lawfully
equivalent to the unit vector drawn perpendicular to p and to a. Thus every unit
vector is a quadrantal versor, and conversely. And, further, every versor is a power
of a unit vector. Thus, if the angle QOP be, in circular measure, A and if T be
the unit vector above defined, we have

U - = T**l' = cosA + r sin A.
P

Thus the separation of <r/p into the sum of its scalar and vector parts.

(b) The second is physical rather than mathematical. We think of vectors in
a homogeneously strained solid, and if OP be strained into OQ we write a- = tpp.

We recognise the conjugate strain <f>' and see the criterion of the pure strain
in <f> = <f>', 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' dy1 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.

33—3

i72 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 enclosed1.

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.

0 where are those high feasts of science ?

0 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 equations1. 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

Ha + 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 matter1. 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

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"#£:

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182 PETER GUTHRIE TAIT

done so much small print (per cent.) in what is already printed, that we must defer
further violent manifestations of such a tendency to a later stage — excepting always
what must come in Hamiltonianism and birds of a like feather....

As to the R. S. E. I shall give notice of two papers — one by you on certain
kinematical and dynamical theorems, another by myself on a certain most important
quaternion transformation.... The quat. theorem in question is this — and is really of
the greatest value (as witness all your, Maxwell's, Neumann's, etc., attempts at potential
function expressions for distributions of magnetism, electricity static or kinetic, etc., etc.),

which contains all about potential and will go into the Book in splendid style.

P.S. Send the Sph. Hares, soon and let me see Digitalis1 on Newton as
connected with Energy before a fortnight is out as I shall be arranging my lecture
on Energy for the R. S. E.

Two days later Tait continued :

6 G. G. E.

1/4/63.
Dear T.,

You have (no doubt) got the R. S. E. billet, and you have seen that I
am responsible for some " Kinematical & Dynamical Theorems " supposed to be
sent by you.

I shall give them Twist — but you ought to send me, by Monday morning at
latest, a few sentences upon Greed and Laziness which will make the affair more
complete.

Kirchhoff pitches into you & Stokes in the last Pogg. & also in the April
Phil. Mag., while in the latter T" [i.e. Tyndall] lets me off without comment ! ! !

Yrs. Ever,

T'.

P.S. Answer by instant return of post. Is there to be a small volume by
October or not? Our Coll. Calendar is printing, and my enunciation is wanted on
Monday — so say at once & I shall put T & T' (small) instead of Goodwin.

The kinematical and dynamical theorems referred to here were duly
published in very brief form in the Proceedings R. S. £., Vol. v. They
constitute the nuclei of several paragraphs in the Treatise. The few
sentences on Twist become §§ 120-123; and the dynamical theorems, which

1 Possibly a complex pun : Digitalis = poison = Poisson. The lecture referred to was an
address before the Royal Society of Edinburgh on the Conservation of Energy. An out-
standing feature was the discussion of Newton's Second Interpretation of Lex III (see below
p. 191) ; and it is possible that Thomson may have remembered a reference to it in one of
Poisson's many memoirs.

LETTER TO THOMSON 183

are simply enunciated in the R. S. E. paper, become §§311 and 312.
Tait nicknamed these theorems Greed and Laziness, namely,

(1) Greed: Given any material system at rest, and subjected to an
impulse of any given magnitude and in any specified direction, it will move
off so as to take the greatest amount of kinetic energy which the specified
impulse can give it.

(2) Laziness: Given any material system at rest. Let any parts of
it be set in motion suddenly with given velocities, the other parts being
influenced only by their connection with those which are set in motion,
the whole system will move so as to have less kinetic energy than belongs
to any other motion fulfilling the given velocity conditions.

The postscript indicates that there was still some hope of publishing
the more elementary book before the other.

The following letter touches on a variety of subjects more or less con-
nected with the book :

6 G. G. E.

24/4/63-
Dear T.,

I have been working up my Quat. article for Proc. R. S. E. I find that
in your " Mechanical Representation of Electric etc. Forces " you need never have
taken the rotations of the solid — you can always get displacements suiting any of
the forms of force, and they are in fact simpler than yours as given in the Math.
Journal. I will send you a proof to-morrow so that you may advise on it before it
goes to press.

The whole mystery of Electromagnetism lies in the operator 5 + \ V, as I
pointed out to Sir W. R. H. four years ago — but I cannot get a closer insight, at
least at present....

Also Stewart and I have simultaneously struck on an idea of which "Conserva-
tion of Energy" is a particular case — and we intend to develop from it some
tremendous consequences — numbers of which have already been booked and talked
over. A third volume of the book will be required for it, if it do not in fact
destroy the necessity for the first two. It is a gushing, gasping, idea. But, more
anon. As it will at once obliterate everybody who pretends to science but is not
acquainted with it, we propose shortly to initiate you, in order that you may not
be lost in the stramash that will follow its publication.

Gray promised me Sheet 7 (with the cuts) to-night — but it has not come yet,
and of course won't to-night. Look out for it on Monday.

Also look sharp to Chap. II for I have no doubt you see the propriety of
putting Sph. Hares. & Quatns. in an appendix at the end of the Vol.

Ponder this.

In Statics and Kinetics we should in the large volume keep to general theorems

184 PETER GUTHRIE TAIT

and leave Balance and Screws and Atwood's Machines, etc., for the small book —
I mean in the -Second Division of the Book — Abstract Dynamics....

These extracts together with the letters which have been published in
Prof. Sylvanus Thompson's Life of Lord Kelvin (Vol. I, pp. 453-464), show
that the final plan of the book must have been agreed upon late in 1862 or
early in 1863. Tait had already written a first draft of large parts of the
book ; but Thomson could not be got to apply himself steadily to the task.
Like a sleuth-hound Tait was always on his colleague's tracks begging for
copy or for proofs. When in May 1864 Thomson visited Creuznach for the
sake of his wife's health, he received the following letter, whose immediate
purpose was to bring him to his desk and pen.

6 G. G. E.

20/5/64.
Dear T.,

Send something for $. I have only written the examples twice yet, and
the Kinetics of a particle but once....

Rather a curious thing to-day. Thunderstorm. Divided-ring (very low charge)
going, and discharging every two minutes, when a thunder-cloud was close, by a
spark from a fine wire i mm. from the electrode. Abscissae G. M. T. (as supplied
by Her M. A. Royal for Scotland). Ordinates — potential at class-room window —
I inch = 500 zinc-copper pairs.

ih 1m 2m

Note the effect of the gun1 — ist on cloud north of the Coll.; then (after more
than 30 s.) on one to the south....

When you ARE at Xnach, I hope you won't spend ALL your time with
Dellmann, but will devote at least 4 h. per day to Kinetics of Rigid (about point)
and Elastic Bodies (including Fluids). If you do, we can easily appear in August,
and have the S. B. [ = small book] ready for November.

I send you a proof of my Energy2 signed for press — the publishers want me to
make a little volume of it and its predecessor (with mathematical developments).
I said I would be guided by you and E. They said Engineers had urged them to
get it done. Ponder O. T. and C. E. on this, as well as on other subjects'.

1 The Edinburgh Time Gun is fired every day, except Sunday, at i p.m. Greenwich
Mean Time.

* One of the articles in the North British Review, which finally became the second
chapter of Tail's Sketch of Thermodynamics.

8 The initials are difficult of interpretation. They may refer to Rankine the well-known
Professor of Civil Engineering in Glasgow University. Possibly C.E. means see Engineers.

QUATERNIONS EXCLUDED 185

It is now clear that there is no hope of the small book appearing
first.

The references in these letters to Quaternions show that there was in the
early days an intention to have an appendix on quaternionic treatment of
physical problems. There were obvious practical difficulties in the way of
introducing a mathematical method which in spirit and notation was so unlike
that in ordinary use. It could not have been done without a preliminary
statement of the purely mathematical laws of combination and transforma-
tion. Then again Thomson was not merely not interested but emphatically
antagonistic. In a letter to Cayley in 1872, quoted above (p. 152), Tait
mentioned incidentally that Thomson " objects utterly to Quaternions " ; and
Kelvin in 1901 in a letter to Chrystal wrote regarding Tait :

" We have had a thirty-eight year war over quaternions. He had been captured
by the originality and extraordinary beauty of Hamilton's genius in this respect ;
and had accepted I believe definitely from Hamilton to take charge of quaternions
after his death, which he has most loyally executed. Times without number I offered
to let quaternions into Thomson and Tait if he could only show that in any case our
work would be helped by their use. You will see that from beginning to end they
were never introduced."

The implication here is that Tait was unable to show that the use of
quaternions would be of any advantage. It should be remembered, however,
that Tail's own most important applications and developments were of later
date than the years preceding 1867 when "T and T"' was published; and
that, although Tait was himself fully convinced of the value of quaternions as
an instrument of research, it was a very different matter to get Thomson to
look at the matter from his point of view. Rather than risk the collapse
of the whole enterprise Tait relinquished what must have been at one
time a keen desire.

Meanwhile the book grew like an organism. With the exception of
appendices and parts introduced bodily from Thomson's published papers,
the manuscript first supplied to the printer was largely contributed by Tait.
Then, in proof, certain paragraphs, whose subject matter seized the mind of
Thomson, developed in a wonderful way. New sections and extensions of
old sections were added, and were altered, pruned and expanded by both
writers, until it was difficult to say how much was due to each.

When the Treatise finally appeared in 1867, after six years' preparation,
it was at once seen to be a great work ; and within two years the edition was

T. 24

186 PETER GUTHRIE TAIT

sold out. The reviews were all highly complimentary, partly perhaps for the
reason that the ordinary reviewer very soon found himself out of his depth.
The Athenaeum of Oct. 5, 1867, thus chronicled its advent :

"A professor of Glasgow and one of Edinburgh, both well known to the world
of science, have produced 750 pages as a first volume only. We defer description
until we see the whole. The mathematical part is of a formidable character, and
of the most modern type. The authors are thoroughly up to their subject, and
have strong physical as well as mathematical tastes. The size of the volume is the
fault of the subject and not of the authors, who have, so far as we have looked
closely, kept down details. If anything they have not sufficiently diluted the
mathematical part with expanded demonstration. But what of that? The higher
class students for whom this work is intended are rats who can gnaw through any-
thing: though even their teeth will be tried here and there, we can tell them."

The last sentence suggests the touch of De Morgan.

The Engineer of Nov. i, and the Medical Times and Gazette of Nov. 16,
1867, both journals of scientific standing, described at some length the contents
of the book, the latter ending with the sentences :

"Should the three succeeding volumes at all come up in value to the present
one, Thomson and Tait's Natural Philosophy will deserve to take place with
Newton's Principia and Laplace's Mtcanique Celeste. This is strong language, but
not too strong."

A long and somewhat discursive review appeared in the Scotsman of
Nov. 6, 1868, fully a year after the publication of the Treatise, The reviewer
was, however, keenly alive to the real merits of the contribution made by
Thomson and Tait to the scientific literature of our time. Referring to the
authors he wrote :

" They are to a certain extent a happy complement of each other — the one being
deeply speculative but slightly nebulous in the utterance of his original thoughts,
as often happens with profound thinkers ; the other, though not deficient in
originality, being clear, dashing, direct and practical.... What they exactly know they
state in a plain intelligible manner.... What they do not know they do not pretend
to explain....

"The authors zealously adhere to Newton, and they restore his methods and
doctrines where they can — not without reason. They do not much pretend to
originality — indeed they do not pretend to it at all. They quite openly and frankly
lay hold of every mathematician or philosopher who has anything useful to them,
and pillage him outright — always, of course, with the most handsome acknowledge-
ments. Lagrange, Laplace, Fourier, Euler, Gauss, Joule, d'Alembert, Liouville, and
the Irish (at least not the Scotch) Sir W. Hamilton are all laid under contribution,
and the hard heavy scientific slab which each other had dug for himself out of the
big bottomless quarry of Nature, as his personal title to immortality, is looked at

FRESHNESS OF TREATMENT 187

carefully, found to be suitable, seized hold of, and at once, after a few chips and
modifications (if necessary) laid into this big pyramid, in the climbing of which the
youth of this country will hereafter have free scope for trying their strength, and
looking down from the sides and the top of it upon this puzzling intricate physical
world....

"The world of which they give the natural philosophy is real and not ideal. It
is not the abstract world of Cambridge examination papers — in which matter is
perfectly homogeneous, pulleys perfectly smooth, strings perfectly elastic, fluids
perfectly incompressible.... — but it is the concrete world of the senses, which approxi-
mates to but always falls short alike of the ideal of the mathematical as of the
poetic imagination."

The review finishes with a reference to the doctrine of the dissipation
of energy, and a mild expression of wonder as to what the " natural theologian "
will make of it.

To a real student of physical science conversant with the text-books
of the middle of last century Thomson and Tail's Treatise on Natural
Philosophy must have come as a revelation. Instead of the usual
approach through Statics, with Duchayla's proof of the parallelogram of
forces, he found himself introduced to what the authors called Preliminary
Notions, arranged in four chapters under the titles Kinematics, Dynamical
Laws and Principles, Experience, Measures and Instruments. In opening
with the subject of Kinematics or the Geometry of Motion, Thomson and
Tait followed to some extent the example of certain recent French writers,
such as Delaunay and Duhamel ; but, in thus discussing motion and
displacement apart from dynamical relations, they carried out the idea
much more thoroughly than had ever been done by their predecessors.
In other English books of the day there was absolutely nothing like it.
In the first few paragraphs the geometrical conceptions of curvature and
tortuosity are treated in a novel and elegant manner as illustrative of
motion along a curve or line of changing direction. Even the familiar
quantities, velocity and acceleration, are discussed in a fresh way ; but it is
in the later sections of the first chapter that originality of treatment rivets
the attention of the careful reader. Simple Harmonic Motion, which is
the foundation of all kinds of wave-motion, is the avenue by which Fourier's
Theorem is approached. Composition of angular velocities, the rolling of
curves and surfaces on one another, curvature of surfaces, etc., etc., fall
into line as the subject develops. An altogether unique part is to be
found in §§135 to 138 inclusive (somewhat extended in the second edition).

24 — 2

i88 PETER GUTHRIE TAIT

Once, in conversation with Kelvin, I learned incidentally that these sections
had been inspired by Bolzani, professor of Mathematics in Kasan. Writing
subsequently for fuller information, I received from him the following letter
of date February u, 1904:

"I learned a good deal from Bolzani in the course of a few days that he stayed
with us in Arran, some time about i8601. He gave me a very clear statement
regarding Gauss' curvatura Integra in relation to a normal through the boundary,
which I had never before seen in print or learned otherwise. He did not give the
name Horograph, but with the aid of my colleague Lushington I devised it and
put it into T and 1". I have never seen it elsewhere since that time."

The discussion of the curvature of surfaces leads naturally to the
question of flexibility and developability of surfaces ; and then follows an
important series of paragraphs on strain or change of configuration. A
comparison of the sections in "T and T"' with the corresponding sections
in Tait's Quaternions, which was published in the same year, is very
instructive in the light of what has just been said regarding Thomson's
attitude to Quaternions. In a few lines the quaternion author gives what
in the larger book requires paragraphs and even pages. Tait had a great
liking for the theory of strains, and usually discussed it in considerable
detail in his Advanced Class lectures.

The last sections of the first chapter deal with degrees of freedom,
and conditions of restraint — a subject for which Kelvin had a particular
fondness. Probably no instrument of any importance was ever constructed
by him which did not contain some neat example of geometrical constraint.
The illustrations given in the Treatise are fundamental and far-reaching.

True to their general plan the authors finish the chapter on Kinematics
with a brief discussion of generalised components of position, velocity, and
acceleration, preparing the way for the great dynamical developments of
the next chapter.

In an important appendix, Thomson and Tait give (a) an Extension
of Green's Theorem, (b) Spherical Harmonic Analysis. In the latter of
these, which covers 20 pages in the first edition (extended to 48 pages in

1 The date was probably 1862 ; for among Tait's correspondence is a letter from Bolzani
of date November a of that year, written from Hull on board the "Volga" as he was
leaving for home. In this letter he thanked Tait for various kindnesses and for a copy of a
quaternion paper. He also asked Tait to get for him as early as possible a copy of
Hamilton's new work on quaternions.

SPHERICAL HARMONICS 189

the second edition) new ground is broken in the treatment of what had
till then been called " Laplace's Coefficients."

In a letter to Cayley of date February 4, 1886, Tait makes a curious
remark which touches on the genesis of this remarkable appendix. He
had been poking fun at Sylvester and Thomson for their raptures over
new ideas, and concluded :

Thomson, about 1860, used to lay down the law that the smallest experimental
novelty was of more value than the whole of mathematical physics. Then he met
with an accident which prevented his experimenting for a whole winter. During
this period he became acquainted with Spherical Harmonics, and then his fundamental
dictum was wellnigh reversed !

Tait of course is having his joke ; but the facts implied are probable
enough. Tait's intimacy with Thomson began in 1860 before his appoint-
ment to the Edinburgh Chair. Thomson was at the time very busy with
electrical work largely the outcome of his labours in furthering the develop-
ment of ocean telegraphy ; and Tait must have been impressed with the
eagerness of his new friend in all kinds of experimental work. Then came
the accident which lamed Thomson for life ; and necessity forced him for
a time to devote his energies to mathematical investigations in the famous
green-backed note books which were ever afterwards his inseparable com-
panions. It is not improbable that the new treatment of spherical harmonics
was begun and to some extent developed during these months of 1 860-61.

In Maxwell's letters to Tait there are several passing references to
this section of the " Archiepiscopal Treatise" as Maxwell playfully called
it, in humorous reference to the fact that the Archbishops of York and
Canterbury were at the time also Thomson and Tait. Thus on Dec. n,
1867, he wrote :

" I am glad people are buying T and T'. May it sink into their bones ! I shall
not see it till I go to London. I believe you call Laplace's Coeffts. Spherical
Harmonics. Good. Do you know that every Sp. Harm, of degree n has n axes ?
I did not till recently. When you know the directions of the axes (or their poles
on the sphere) you have got your harmonic all but its strength. For one of the
second degree they are the poles of the two circular equipotential lines of the sphere.
I have a picture of them."

Again, on 18 July, 1868, he wrote:

" How do T and T' divide the Harmonics between them ? I had before getting
hold of T and T' done mine for electricity, but I should be delighted to get rid of

190 PETER GUTHRIE TAIT

the subject out of the book except in the way of reference to T and T'. My
method is to treat them as the neighbourhood of singular points in potential systems,
those of positive degree being points of equilibrium, and those of negative degree
being infinite points."

And then followed a brief outline of the method subsequently developed
in Electricity and Magnetism, in which the harmonic of the «th degree is
determined by its n axes and a constant fixing its strength.

Thomson and Tail's treatment of Spherical Harmonics is essentially
physical, their object being — to quote their own phrase — " the expression
of an arbitrary periodic function of two independent variables in the proper
form for a large class of physical problems involving arbitrary data over a
spherical surface, and the deduction of solutions for every point of space."
It is this object which guides them in their analytical work ; and through
it all it is abundantly clear that the theory of the potential is ever present
to their minds. In no true sense can the appendix be regarded as a
sustained piece of mathematical reasoning. The convergency of the series
is practically assumed, or, let us say, left to be proved by the reader.
But the combined mathematical power and physical intuition are shown at
every stage ; and the use of the imaginary linear transformation, a distinct
novelty in 1867, leads to an elegant and simple deduction of useful forms.
Further on in the book in the sections on statics the authors give other
useful developments ; and it is then that they introduce their names, Zonal,
Sectorial, and Tesseral Harmonics, according to the character of the nodal
circles on the spherical surface.

Passing on to Chapter n of Division i we find it devoted to the
discussion of Dynamical Laws and Principles. One feature of the early
sections of this chapter is specially emphasised by the authors : it is a
return to Newton. This means in the main two things. In the first place
Newton's Laws of Motion are given in Newton's own words and the whole
fabric of dynamics is raised on them as the sufficient foundation. In the
second place, by adopting the Newtonian definition of force as being measured
by the change of motion produced, Thomson and Tait get rid of the wearisome
proof of the parallelogram of forces which was one of the marked features of
the text- books of the middle of last century. In fact, as Newton showed,
the composition and resolution of concurrent forces follow immediately from
the second law of motion. Tait frequently remarked that Thomson and
he "rediscovered Newton for the world." They also seem to have been

LAG RANGE AND HAMILTON 191

the first to point out clearly the significance of Newton's second interpretation
of his Third Law. I have heard Tait tell the story of the search after this
interpretation. "The Conservation of Energy," he said to Thomson one
day, " must be in Newton somewhere if we can only find it." They set
themselves to re-read carefully the Principia in the original Latin, and ere
long discovered the treasure in the finishing sentences of the Scholium
to Lex in.

Considerable portions of the earlier sections of the dynamical chapter
are, as they themselves point out, simply paraphrased from Newton. But
a marked feature of the discussion is the introduction of a new terminology
at once precise and suggestive. Moreover, old words are used with clearly
defined meanings, and never used except with these meanings. The
Conservation of Energy occupies the first place, and the terms Kinetic
Energy and Potential Energy give a new unity to the whole treatment.
The Moment of Inertia is defined, in the first instance, in terms of kinetic
energy. The conditions of equilibrium are established on their true kinetic
basis. The principle of virtual velocities and d'Alembert's principle, which
the older writers regarded with such reverence, are shown to be special
enunciations of the great laws of energy. Gradually, by almost imperceptible
advances along several lines of comparatively simple dynamical reasoning,
the way is paved for the entrance into the shrine of Lagrange's generalised
coordinates ; and thence into the spacious temple of Hamiltonian Dynamics.
At the time the book was being planned Thomson seems not to have studied
the more recent developments of Lagrange's dynamical method ; and his own
mathematical methods were based largely on those of Fourier. As Tait
once epigrammatically put it, " Fourier made Thomson." Tait used to tell
how, when the chapter on Dynamical Principles was being sketched, he
remarked, " Of course we must bring in Hamilton's dynamics." Thomson
having expressed unfamiliarity with Hamilton's theory, Tait rapidly sketched
it on a sheet of foolscap. Thomson was enraptured, took the sheet off with
him to Glasgow, and in a short time had the sections written out very
much as they appeared in the first edition of the Treatise.

The typical examples chosen to illustrate the power of Lagrange's
generalised equations are of great variety and interest. They include the
gyroscopic pendulum and the hydrokinetic problems of solids moving through
perfect fluids. This latter class of problem had been imagined by Thomson
as early as 1858; but it was in the pages of "T and T'" that the

192 PETER GUTHRIE TAIT

demonstrations were first given. From §331 to §336 of the first edition1
these hydrokinetic problems are brought into touch with physical questions
of far-reaching import. The discussion concludes with the remark that
" it must be remembered that the real circumstances differ greatly, because
of fluid friction, from those of the abstract problem, of which we take leave
for the present." At a much later date Kelvin published a few papers on
closely connected problems. In these as in other papers, as well as in the
Treatise, constant use is made of Thomson's general theorem of " Laziness "
as Tait called it (see above p. 182).

Chapters in and iv of Division i, on Experience and on Measures
and Instruments respectively, are brief, touching only on the more
fundamental aspects of the subject.

Division n (Abstract Dynamics) opens with a short introductory chapter
of a few pages, followed by Chapter vi on Statics of a Particle. The
academic problem familiar to all students of statics is quite ignored. Four
short paragraphs give the solution of the general problem of a particle
in equilibrium, under the action of given forces, some of which may be
forces of constraint ; and the remaining part of the chapter is occupied
with the theory of attractions according to the Newtonian Law of the
inverse square. By the easy steps of Kelvin's geometrical extensions of
Newton's original demonstrations of the attraction of a spherical shell, the
reader is led by simple physical reasoning, with the minimum of mathematics,
to the enunciation and proof of Green's problem of the unique surface
distribution of matter satisfying the assigned potential values at the surface.
Attractions of ellipsoids and centrobaric bodies generally are discussed in
considerable detail ; and this leads to the simpler applications of spherical
harmonics to problems of attraction. It may be mentioned in passing
that some of the demonstrations are simplified in the second edition.

Chapter vn is devoted to the equilibrium of solids and fluids, flexible
cords being included as constituting a kind of intermediate case between
the equilibrium of perfectly rigid solids and the deformation of perfectly
elastic solids. The transition from the flexible cord to the elastic wire
may seem at first sight abrupt, but it is not really so when regard is had
to the fundamental kinematical considerations involved. The whole mode

1 These correspond to §§ 320-325 of the second edition, the sections on Action which
originally preceded the hydrokinetic applications having been placed after them in the new
edition.

PROBLEM OF THE EARTH'S RIGIDITY 193

of treatment is novel and original in the highest degree. It is instructive
to see how by means of suitable and sufficiently obvious assumptions the
elastic wire and elastic thin plate can be discussed without the preliminary
laying down of the fundamental elastic principles. These are introduced,
in due course, as the authors pass on to the general theory of the equilibrium
and deformation of elastic solids. At the same time, a student reading
these sections for the first time will find them very hard to understand,
unless he has already some knowledge of the general theory of elasticity.

In the sections on Elasticity, the important practical problems of twist
and bending of bars are completely solved, and then the solutions are
indicated of more general cases, such as the elastic deformation of spherical
shells and solid spheres subject to given surface tractions and body forces.
These problems afford interesting examples of the use of the spherical
harmonic analysis developed in the Appendix to the First Chapter. The
discussion ends with a promise of further illustrations under " Properties
of Matter," a promise which curiously enough still appears in the second
edition, in spite of the prefatory remark that the book was not to be
carried to completion.

An important application of the laws of deformation of an elastic sphere
is made to our earth under the deforming stresses due to sun and moon ;
but, before this problem can be attacked, it is necessary to investigate the
deformation in a spheroid of incompressible fluid. This consideration forms
the transition from elasticity to hydrodynamics, and, after a brief discussion of
the more simple questions of equilibrium and flotation, Thomson and Tait
enter upon the final great series of problems connected with the equilibrium
of spheroidal rotating masses of fluid, the theory of the tides and tidal
stresses, and the closely related question of the rigidity of the earth. " To
promote an intelligent comprehension of the subject," all the polar harmonic
elements of the 6th and 7th orders are worked out, tabulated, and represented
graphically. The various kinds of tidal influence are then investigated ; and
it is shown by combined use of hydrostatic and elastic principles that if the
earth were only as rigid as steel it would as a whole yield to the tidal action
of sun and moon by an amount equal to about one-third of the yield in the
case of no rigidity. This concluding, highly original, part is greatly extended
by Sir George Darwin in the second edition of the book.

The distinctive feature of the Treatise when compared with the earlier
great works on the same branches of Natural Philosophy, such as Lagrange's
T. 25

i94 PETER GUTHRIE TAIT

Mdcanique Analytique, is the prominence given to motion rather than to
equilibrium. Dynamical Laws and Principles are regarded as Preliminary ;
and then, as the first part of Abstract Dynamics, we are introduced to Statics
of particles, of solids, and of fluids. In a letter of 1863 (quoted above, page
181) Tail protested against Attractions being brought in before Properties
of Matter, of which they formed a natural part. It had become customary —
and the custom still persists — to treat Attractions and the theory of the
Potential as part of Statics of Particles. And yet, physically, attraction
means motion ; and equilibrium is maintained only by the introduction of
balancing forces of another type, cohesion, adhesion, surface constraint, etc.
Tail's suggestion to treat Gravitation as a property of matter was certainly
more in line with a truly logical arrangement than was the plan ultimately
adopted. But Thomson had already published a series of beautiful geo-
metrical demonstrations on the subject ; and no doubt he saw a splendid
chance of utilising this material and of working out far-reaching problems
of terrestrial dynamics. In other words, the problem of the earth's rigidity
dominated to a marked degree the composition of the whole section of
Abstract Dynamics which constituted Division n. Only in this way can
we explain the unusual scope of Chapter vn. No doubt Lagrange, in his
classical work, had included under Statics general discussions of the equi-
librium of flexible elastic wires and thin plates ; but Thomson and Tail
were the first to bring within the limits of one chapter the laws of equi-
librium of perfect fluids at rest or in steady rotation, and of solids ideally
rigid or deformable. Much of this chapter indeed belongs as truly to the
prospective section on the Properties of Matter as to the more abstract
branch of dynamics. Profound though the influence of " Thomson and Tait "
has been on the teaching and coordination of the principles of Natural
Philosophy, no later writers have followed their example in devoting a single
chapter to ordinary Statics of extended bodies, strings, and flexible wires,
to Hydrostatics, and to Elasticity.

As regards the division of labour in the production of the book, I think
there are strong indications that there is more of Tail's initial work in the
earlier lhan in ihe laler portions of ihe volume, and lhal Thomson's hand is
particularly in evidence in the last chapter. Each was the other's severe
critic, and many a senlence must have undergone great internal changes under
ihe chiselling pen of each in lurn.

It would be easy lo find, especially in ihe more elemeniary parts, some

HELMHOLTZ ON "T AND T'" 195

faults of logical presentation. For example, very soon after the publication
of the work Maxwell in a letter to Tait made the following criticism of
some of the statements in paragraphs 207 and 208.

"207. Matter is never perceived by the senses. According to Torricelli quoted
by Berkeley ' Matter is nothing but an enchanted vase of Circe, fitted to receive
Impulse and Energy, essences so subtle that nothing but the inmost nature of
material bodies is able to contain them.'...

" 208. Newton's statement is meant to distinguish matter from space or volume,
not to explain either matter or density.

" Def. The mass of a body is that factor by which we must multiply the velocity
to get the momentum, and by which we must multiply the half square of the velocity
to get its energy.

" Hence if we take the exchequer pound as unit of mass (which is made of
platinum) and if we find a piece of copper such that when it and the exchequer
pound move with equal velocity they have the same momentum (describe experiment)
then the copper has a mass of one pound.

"You may place the two masses in a common balance (which proves their
weights equal), you may then cause the whole machine to move up or down. If
the arm of the balance moves parallel to itself the masses must also be equal.

"Some illustration of this sort (what you please) is good against heresy in the
doctrine of the mass. Next show examples of things which are not matter, though
they may be moved and acted on by forces, (i) The path of a body, (2) Its axis
of rotation, (3) The form of a steady motion, (4) An undulation (sound or light),
(5) Boscovich's centres of force. Next things which are matter such as the luminiferous
aether, and if there be anything capable of momentum and kinetic energy."

But faults of the kind indicated were like spots in the sun. The greatness
of the book became more evident the closer it was studied. Since the days
of Newton's Principia, no work on Natural Philosophy of anything like the
same originality had been produced in England. Thomson and Tail's
Treatise must ever rank with the classical works of Lagrange and Laplace.

It was not long before Helmholtz took steps for the preparation of a
German translation; and in 1871, after some delay on account of the
Franco-German War, this translation was finally published under the combined
authority of Helmholtz and Wertheim. A few sentences from Helmholtz's
Preface will indicate his own view of the value of the Treatise.

"The present volume will introduce to the physical and mathematical German
public the beginning of a work of high scientific significance, which will, in the most
excellent fashion, fill in a very perceptible gap in the literature of the subject...

"One of the authors, Sir William Thomson, has long been known in Germany
as one of the most penetrating and ingenious of thinkers who have applied them-
selves to our Science. When such a one undertakes to lead us, as it were, into the

i96 PETER GUTHRIE TAIT

workshop of his thoughts and to reveal the way in which he looks at things, to
disentangle the guiding threads which have helped him in his bold intellectual
combinations to master and coordinate the intractable and tangled material, we can
but feel towards him the liveliest gratitude. For this work which would indeed
overstrain the powers of a single much occupied man, he has found in P. G. Tait,
Professor of Natural Philosophy in Edinburgh, a highly fit and gifted collaborator.
Only perhaps by such a happy union could the task as a whole have been completed."

Helmholtz also speaks of the difficulty of finding German equivalents for
some of the new scientific terms invented by Thomson and Tait. That
verbal difficulties other than scientific troubled the minds of the German
translators is clear from the following letter from Tait to Helmholtz :

17, DRUMMOND PLACE, EDIN.

20/5/71.
Dear Professor Helmholtz,

I have postponed my answer to your letter till I could catch Thomson,
so that the answer to your queries should come from him as well as from me.
Now that he has got a yacht he goes off for three weeks at a time ; and is now
on his way to Lisbon, and perhaps Gibraltar. However, he quite agrees with the
following answers.

1. To "scull a boat" has two meanings1. In a sea-boat it means to work a
single oar at the stern like the tail of a fish. In a light racing boat on a river or
lake it means to work two oars in the rowlocks, one with either hand. This is
what we mean in the text. Thomson won the "sculls" in this sense when he was
a student in Cambridge.

2. To "run up on the wind" means to turn the ship's head to the wind.

3. To "carry a weather helm" means to put the tiller (Helmholtz?) to wind-
ward so that the rudder goes to the other side, tending to turn the ship's head from
the wind, providing she is moving fast enough to make the rudder act.

As to your proposed changes on the new matter, I have no doubt whatever
that they will be improvements, and we have reason to rejoice that you are kind
enough to make them.

Thomson desires me to say that, while we all regret you cannot come to the
B. A. Meeting, he will not forego your company on a cruise if it is possible to get
hold of you, and will make arrangements to start at any time that may suit you,
from August to November inclusive. But the best plan would be for you to try
to get to Edinburgh in time for the last day or two of the Meeting, which lasts
a week usually. This year we hold in Edinburgh the Centenary of Sir Walter Scott
just at the end of the Association Meeting ; and it is probable that, if you care at
all about his writings, you would be interested in seeing the collection of things
connected with him which will be exhibited then. If Sir William be not ready to
start, you might come with me to St Andrews, where my wife will be delighted

1 The reference is to § 336 in the first edition, considerably extended in § 325 in the
second edition.

VISIT OF HELMHOLTZ 197

to see you, and where you may learn (at its head-quarters) the mysteries of GOLF !
I have secured a house there, and so has my brother-in-law, Crum Brown, for the
months of August and September. It appears that Huxley also has done the same,
so that we may take to scientific discussions in the intervals of exercise.

I forgot to say that, with us, there are but rarely masts in canal-boats, and
therefore the point of attachment is not usually in the axis of the boat. Perhaps
you might put into the German translation the qualifying clause "provided the
rope be not attached to a point in the axis of the boat," which is not necessary
for an English reader.

All your other surmises are correct. I have told Vieweg to publish by instal-
ments if he likes. I hope you will write a Preface to the first instalment, so that
the weight of your authority may be brought to bear upon the reception of the
new terms introduced1.

While the first volume of the Treatise was shaping itself to a finish, the
authors had also in mind the more elementary book for the use of the ordinary
student attending the Natural Philosophy classes in Edinburgh and Glasgow.
The intention was to make large use of the same material for the two kinds
of book, the less mathematical portions being common to both.

There is fairly strong evidence that Division i, including Kinematics,
Dynamical Laws and Principles, Experience, and Measures and Instruments
was to be in the first instance Tail's domain ; while Thomson was to be
mainly responsible for Division n, namely, Abstract Dynamics, including
Statics, Attractions and Elasticity. Tait very soon put together sufficient
material to form a small pamphlet which he printed for the sake of his
students in 1863. In the Edinburgh Calendar for the session 1862-3, he
had already inserted the following optimistic reference, — " as a text-book on
the general subject of the lectures, in case the forthcoming volume by Professors
Thomson and Tait did not appear before Christmas, one of the following may
be named, etc." Next year's Calendar announced that " In October 1863 there
will be published the first volume of a Treatise on Natural Philosophy by

1 As indicated in this letter, Helmholtz did not reach Scotland in time to be present
at the British Association Meeting. He spent some days with Tait in St Andrews, but,
unlike Huxley, did not yield to the fascinations of golf. In a letter to his wife of date
August 20, 1871, Helmholtz gave his impressions of St Andrews life: "St Andrews hat
eine prachtige Bai, feine Sandflache, die dann mil einer scharfen Kante in grime Grasflachen
iibergeht....Es ist grosses Leben von Badegasten, eleganten Damen und Kindern, Gentlemen
in sporting Costiimen, welche golfing spielen. . . . Mr Tait kennt hier nichts anderes als
golfing. Ich musste gleich mit, die ersten Schlage gelangen mir, nachher traf ich entweder
nur die Erde Oder die Luft....Tait ist eine eigenthiimliche Art von Wildem Mann, lebt
hier, wie er sagt, nur fur seine Muskeln, und erst heute am Sonntag, wo er nicht spielen
durfte, aber auch nicht in die Kirche ging, war er zu verniinftigen Gegenstanden zu bringen."

198

PETER GUTHRIE TAIT

Comparison of the paragraphs of the Sketch of Elementary Dynamics (1863), the Treatise
on Natural Philosophy (1867), and the Elements of Natural Philosophy (1873).

Sketch

Treatise

Elements

Dynamics

i-3

A
Kinematics

4-47

Chap, i
Kinematics

i, 2

3-5. '4, 15. '7. 19-29. 3i, 32.
35. 36, (235), 41-43, 45, 53-58,
79-81, 83, 86, 90, 92, 93, 95, 96,
99, 102, 105

Chap, i
Kinematics

7-9, 16, 18, 20-25, 28-36, 38, 39,
43-48, 54-58, 63, 70-75, 91-93, 95,
97, 98, ioo, 103, 104, 106-108, no,
113, 116

B

Dynamical Laws
and Principles

48-95

96
97

Chap, n
Dynamical Laws and Principles

208, 210-214, 216, 217, 220, 223-

230, 281, 234-241, 244-247, 251,

253-256, 258-267, 268-273, 278,

281

289 and 292

451 (in Chap, v)

Chap, ii
Dynamical Laws and Principles

174, 176-180, 182, 183, 185, 188-
195, J98, 201-207, 210-213, 217,
219-222, 224-233, 240-245, 250,

235

254 and 257
404 (in Chap, v)

C

Statics

98

99

ioo

IOI

Chap, vii
Statics of Solids and Fluids

572, Example n
572, Example in
586

Chap, vn
Statics of Solids and Fluids

592, Example n
592, Example in
592, Final Sentences

D

Kinetics
102-108

109

no

unrepresented

Appendix
Kinetics
(*) to (g)
(52 in Kinematics)
(h)

E

Hydrodynamics

111-118
119

120
121, 122

Chap, vii
Statics of Solids and Liquids

742-745, 748, 750-752, 762

unrepresented

753

Chap, vn
Statics of Solids and Liquids

684-687, 690, 692-694, 703

unrepresented

694

(/, k, in Appendix)

THE EDINBURGH AND GLASGOW PAMPHLETS 199

[etc., etc.] ; and also an elementary book on the same subject for less advanced
students will soon appear (a portion having been specially printed with the
title Elementary Dynamics]" Year after year the same announcement was
printed in the Calendar, the date of appearing of the forthcoming volume
being simply pushed on one more year! At last in the 1868-9 Calendar,
" will be published " was changed to " was published " ; but the book on
The Elements of Natural Philosophy was still far from its final form.

Now there is not the least doubt that this small pamphlet of 44 pages,
which was published in 1863 by Maclachlan and Stewart, an old Edinburgh
firm whose shop faced the College, was almost entirely the work of Tait. With
a few exceptions the paragraphs still exist in their earliest draft in his MS
note book ; and are reproduced practically verbatim in the large type of the
Treatise. Of the few which are not represented in the Treatise, the majority
treat of parts of the subject which lay outside the scope of Volume i. The
pamphlet was called Sketch of Elementary Dynamics, and was issued under
the joint names of W. Thomson and P. G. Tait. Its contents are indicated
in the first column of the Table on the opposite page.

Simultaneously with the publication of this pamphlet in Edinburgh,
Thomson brought out a pamphlet in Glasgow, under the title "Elements of
Dynamics, Part i, edited, with permission, by John Ferguson1, M. A., from notes
of lectures delivered by William Thomson." With the exception of three short
introductory paragraphs and four later paragraphs upon Gauss' absolute unit
of force and Kater's measure of gravity at Leith Fort, this pamphlet differs
in toto from the one published in Edinburgh. It is concerned almost wholly
with Statics ; and the treatment is that of the old days before the " return
to Newton." Paragraphs 22 to 70, covering Chapters i and n and half of
Chapter in are unrepresented in " T and T'," and are indeed quite out of
sympathy with the whole mode of treatment of the Treatise, giving in
familiar fashion the now superseded Duchayla's proof of the parallelogram
of forces. In the Edinburgh Sketch the whole thing is disposed of in one
paragraph in the true Thomson-and-Tait style. Paragraphs 71 to 124 and
128 and 1 66 (covering Chapters in to vm) in the Glasgow pamphlet are
reproduced verbatim in Chapters vi and vn of the Elements of Natural
Philosophy as published in 1873, but have no place whatever in the large
Treatise. Serious students of the Elements were never greatly attracted
by these sections. They did not seem to fit in well with the rest of the
1 Now Professor of Chemistry in Glasgow University.

200 PETER GUTHRIE TAIT

treatise. They constituted a careful logical setting forth of what might be
called the analytical geometry of forces and couples, necessary perhaps
for solving problems, but not demanding much thought on the part
of the man familiar with elementary Cartesian methods. This Glasgow
pamphlet then may be considered to be Thomson's earliest attempt to
contribute his share to the elementary book as originally planned. Its
first, second and third paragraphs are identical in language with the corre-
sponding parts of the Edinburgh Sketch; paragraph 16 on the formula for
gravity is the same as paragraph 222 in the Treatise; and paragraphs 17
to 21 (the last being on Kater's pendulum measurements) correspond with
57 and 58 in the Edinburgh Sketch, and 223 and 226 in the Treatise.
With the exception of these few paragraphs the Glasgow pamphlet forms
no essential part of the Treatise on Natural Philosophy. The fact that
the Edinburgh Sketch, though bearing the names of both authors, was quite
unknown among Glasgow students, and is not mentioned in Professor
Sylvanus Thompson's bibliography at the end of Lord Kelvin's Life, proves
that it was practically the work of Tail.

It is this Edinburgh Sketch, accordingly, which must be considered to
be the earliest published form of the real " Thomson and Tait." I can best
exhibit this by comparing in tabular form (see page 198) the corresponding parts
of the three distinct publications, the Sketch, the Treatise, and the Elements
of Natural Philosophy. In many cases the original paragraphs of the
Sketch are simply reproduced ; in other cases they are expanded, but in
the expanded form the original sentences exist practically unchanged. Of
course in the Treatise the expansion takes in addition an analytical form.

Thus we see that with the exception of the part on Kinetics and the
last two paragraphs on Hydrokinetics, which lay quite outside the scope of
Volume i, the Edinburgh Sketch passed bodily into the Treatise, forming
indeed the nucleus about which the first and second chapters of the work
crystallised. The great work having been completed, the large type parts
of the first four chapters, along with the Kinetic and Hydrodynamic parts
of the Edinburgh Sketch of 1863, were pieced together so as to form a
pamphlet of 120 pages. This was issued in 1867 by the Clarendon Press
for the use of the students both in Edinburgh and Glasgow, and was followed
in 1868 by a second edition ("not published") of 138 pages. The Glasgow
pamphlet was then partly incorporated in the way already indicated ; and
with the addition of large type parts from Division n of the Treatise, the

THE ELEMENTS OF NATURAL PHILOSOPHY 201

Elements of Natural Philosophy took its final form and was issued in 1873.
In the Appendix a few typical kinetic problems appear exactly as they
were originally written down in Tail's Edinburgh Sketch of 1863. With
the exception of 43 pages devoted to the composition and resolution of
forces and couples, and essentially reproduced from the Glasgow pamphlet,
the Elements of Natural Philosophy of 279 pages is simply an abridgement
of the Treatise on Natural Philosophy.

To the earnest capable student it was and still is a mine of wealth ; but
what hours of misery it caused to many a hapless youth! I remember a
student from abroad coming in the summer session and asking Tail what
he should read so as to prepare . himself for the Natural Philosophy class
in the succeeding winter. Tait with a smile took up " Little T and T'," said
that this was the text-book in use, and recommended the youth to read the
first ten pages and come back to report progress. In a few weeks he
returned in the direst distress. He had read and re-read the introductory
sections, and his mind was an absolute blank !

Indeed, except for the senior students and for men looking forward to
Honours, the Elements was to a large extent a sealed book and was indeed
in certain parts more difficult than the larger work. It was too concise for
the ordinary average student, who never got deep enough into the subject
to appreciate its aim and scope. The authors wrote with their eye on
what was to come ; the average student was content with what little
knowledge he could gain now. J. M. Barrie, in his brightly written An
Edinburgh Eleven, calls the book "The Student's first glimpse of Hades."
The sentence in the Preface of the Elements which speaks of it as being
" designed more especially for use in schools and junior classes in Universities "
must have been penned with a chuckle on the part of Tait at any rate.

Clerk Maxwell reviewed the Elements in Nature, March 27, 1873
(Vol. vn), and expressed his

" sympathy with the efforts of men, thoroughly conversant with all that mathematicians
have achieved, to divest scientific truths of that symbolic language in which
mathematicians have left them, and to clothe them in words, developed by legitimate
methods from our mother tongue but rendered precise by clear definitions, and
familiar by well-rounded statements."

He did not however appraise the work from the point of view of
the tiro.

The first edition of the large Treatise was very soon sold out ; and the
T. 26

202 PETER GUTHRIE TAIT

authors had to consider the preparation of a new edition of Volume i, before
any progress had been made in getting ready the other promised volumes.
It was not, however, till 1875 that by mutual agreement the original contract
with the Clarendon Press was cancelled, and a new arrangement was made
with the Cambridge University Press for the publication of a second edition.
In this edition many sections were considerably expanded ; but on the whole
the large type portions remained unaltered. The addition of new matter
compelled the issue of the book in two volumes known as Part I and Part n.
The First Part was published in 1879, and the Second in 1883.

In the foregoing analysis of the contents of the first edition, incidental
references have been made to the corresponding parts of the second edition.
Without in any way exhausting the list of important changes, I might
mention the following subjects as having received new or improved treatment ;
Spherical Harmonics, Lagrange's generalised coordinates, the ignoration of
coordinates, Hamilton's general dynamic theory, gyrostatic action, attraction
of ellipsoids and the tides, tidal stresses and strains, due to the influence of
the sun and moon.

As the preparation of the new edition proceeded, it gradually became
clear to both authors that, judging from past experience, they could hardly
hope to accomplish the task on which they had entered with so much
enthusiasm in 1861.

Kelvin thus explained the situation in his obituary notice of Tait
communicated to the Royal Society of Edinburgh :

"The making of the first part of 'T and T" was treated as a perpetual joke,
in respect of the irksome details, of the interchange of drafts for 'copy,' amend-
ments in type, and final corrections in proof. It was lightened by interchange of
visits between Greenhill 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.... About 1878 we got to the end of our Division II on Abstract
Dynamics ; and according to our initial programme should then have gone on to
Properties of Matter, Heat, Light, Electricity, Magnetism. Instead of this we agreed
that for the future we could each work more conveniently and on more varied
subjects, without the constraint of joint effort to produce as much as we could of
an all-comprehensive text-book of Natural Philosophy. Thus our book came to
an end with only a foundation laid for our originally intended structure."

That they even completed the first volume of the projected treatise
was largely due to the indefatigable zeal of Tait in keeping Thomson to his

THE ELEMENTS OF NATURAL PHILOSOPHY 203

share of the task. In the revision of the proof sheets for the second edition,
Thomson's whole method of working seriously exercised the printers.

Any new aspect which opened up to his mind as he read the pages of
the first edition led at once to expansion and interpolation, sometimes of the
most alarming dimensions. The great sheet on which the original page was
pasted became covered with the new matter ; bits were pasted on like wings
bearing precious symbols ; while, not unfrequently, the discussion overflowed
into extra sheets, subsection after subsection being piled on regardless of
space and proof correction charges. Difficult indeed was the proof reading
in such circumstances ; and both Kelvin and Tait felt always deeply grateful
to Professors Burnside and Chrystal and Sir George Darwin for the aid they
gave in the final correction of the sheets. The last-named, in fact, added
several sections to the second volume on the problems of tidal action.

Richer and fuller and more complete in many respects though the second
edition was, it could not excel in beauty of printing the original first volume,
which though finally published in Oxford was printed by Constable of
Edinburgh. The authors refer to this very pointedly in the preface, as well
as to the great care with which the diagrams were made.

No finer tribute to the remarkable influence of " T and T'" could be
penned than that with which Clerk Maxwell enriched the pages of Nature
(Vol. xx), when he reviewed the new edition, Vol. i, Part i, shortly before his
death. A few quotations form a fitting conclusion to this sketch of the
genesis and growth of one of the most important scientific publications of the
Nineteenth Century.

" What," asked Maxwell, " is the most general specification of a material system
consistent with the condition that the motions of those parts of the system which
we can observe are what we find them to be? It is to Lagrange in the first place
that we owe the method which enables us to answer this question without asserting
either more or less than all that can be legitimately deduced from the observed facts.
But though the method has been in the hands of mathematicians since 1788, when
the Me"canique Analytique was published, and though a few great mathematicians,
such as Sir W. R. Hamilton, Jacobi, etc., have made important contributions to the
general theory of dynamics, it is remarkable how slow natural philosophers at large
have been to make use of these methods.

" Now however we have only to open any memoir on a physical subject in order
to see that these dynamical theorems have been dragged out of the sanctuary of
profound mathematics in which they lay so long enshrined, and have been set to
do all kinds of work, easy as well as difficult, throughout the whole range of
physical science.

26 — 2

204 PETER GUTHRIE TAIT

" The credit of breaking up the monopoly of the great masters of the spell, and
making all their charms familiar to our ears as household words, belongs in great
measure to Thomson and Tait. The two northern wizards were the first who, with-
out compunction or dread, uttered in their mother tongue the true and proper
names of those dynamical concepts which the magicians of old were wont to invoke
only by the aid of muttered symbols and inarticulate equations. And now the
feeblest among us can repeat the words of power and take part in dynamical
discussions which but a few years ago we should have left for our betters.

" In the present edition we have for the first time an exposition of the general
theory of a very potent form of incantation, called by our authors the Ignoration
of Coordinates. We must remember that the coordinates of Thomson and Tait are
not the mere scaffolding erected over space by Descartes, but the variables which
determine the whole motion...."

There then followed a remarkably clear statement of the conditions under
which certain kinds of coordinates not only may be ignored but ought to be
ignored, and the final illustration was in these words :

"There are other cases, however, in which the conditions for the ignoration of
coordinates strictly apply. For instance, if an opaque and apparently rigid body
contains in a cavity within it an accurately balanced body, mounted on frictionless
pivots, and previously set in rapid rotation, the coordinate which expresses the angular
position of this body is one which we are compelled to ignore, because we have
no means of ascertaining it. An unscientific person on receiving this body into his
hands would immediately conclude that it was bewitched. A disciple of the
northern wizards would prefer to say that the body was subject to gyrostatic domi-
nation."

CHAPTER VI
OTHER BOOKS

As a writer of scientific books Tait was eminently successful. His
books were the outcome of his daily work, and are stamped throughout
with the vigour and clear-mindedness of their author. The special
characteristics of each will appear as we proceed.

His first venture as an author was in conjunction with W. J. Steele,
his college friend and competitor in the great Tripos contest. Tait and
Steele's Dynamics of a Particle was begun soon after their graduation.
It was published in 1856 and at once caught on as a useful Cambridge
text-book. Steele's early death compelled Tait to write by far the greater
part of the book, Steele's contribution being only about one-tenth of the
whole. In a truly chivalrous spirit Tait continued to the end to bring out
the successive editions under the joint names, leaving intact the portions
contributed by his lamented friend.

The second edition appeared in 1865. Meanwhile a remarkable
revolution in the whole dynamical outlook had been effected by Thomson
and Tait working together in view of the publication of their Natural
Philosophy; and to bring "Tait and Steele" into harmony with the new
conceptions the second chapter had to be completely recast. This second
edition was, therefore, the first book published in which the "return to
Newton" was fully effected. The contrast between the two editions is
well brought out in the remark made by Chrystal (Nature, July 25, 1901)
that

"the first edition of Tait and Steele's ' Dynamics '.. .does not. . .contain either of the
words work or energy. In, its, original form it was founded on Pratt's ' Philosophy/
and written on old-fashioned Cambridge lines, which knew not of Lagrange or
Hamilton."

Already in 1856 Tait knew enough of Hamilton's work to introduce
the Hodograph, but as he confesses in the preface to the second edition

206 PETER GUTHRIE TAIT

he had not himself in 1855 "read Newton's admirable introduction to
the Principia" When exactly Tait made full acquaintance with Newton's
dynamical foundation we cannot tell ; but, if we may judge from the following
sentences in his inaugural lecture at Edinburgh, it must have been during
1860 at the latest:

That godlike mortal, as Halley does not scruple to call him, who, finding the very
laws of motion imperfectly understood, in a few years not only gave them fully and
accurately, and devised a mathematical method of almost unlimited power for their
application, but explained most of the phenomena of the Solar System including
Tides, Precession and Perturbations (though this is but one part of his contributions
to Natural Philosophy) — and who was only, after repeated solicitations, persuaded
that he had anything worthy to offer to the world, will remain to all time the beau-
ideal of magnificent genius and devoted application, alike unstained by vanity and
unwarped by prejudice.

In this inaugural lecture we find also an absolutely clear account
of the meaning of the Conservation of Energy. A few quotations will
make this clear.

When we talk of the Conservation of Force as a principle in Nature, it is to be
carefully noted that we do not mean force in the ordinary acceptation of the word —
and, indeed, the principle is now better known as the Conservation of Energy. As
this is a matter of very considerable moment I shall treat of it with a little detail.

Energy may be Actual* or Potential. Actual Energy belongs to moving bodies

Potential Energy belongs to a mass or a particle in virtue of its position, and is in

general work which can be got out of it on account of that position Supposing that

you have now an idea as to the meaning of these two terms, I give the principle of
the Conservation of Energy as it has been put by Professor Rankine to whom these
terms are due —

"In any system of bodies, the sum of the potential and actual energies of the
bodies is never altered by their mutual action."

It is abundantly evident that before Tait entered on his Edinburgh
career he had already thought deeply on the new doctrine of energy. When
the time came for the second edition of Dynamics of a Particle, the founda-
tions of the subject were at once brought into line with these modern views,
although in most other respects the book did not materially change its
character. It is amusing to read Tail's own pencilled annotation on the
first page of the preface to the second edition, "Very poor book — others
poorer."

As edition after edition was called for, Tait often felt a strong inclination
to recast the whole work. Lack of the necessary leisure combined with
1 The term Kinetic was not invented till two years later.

"TAIT AND STEELE" 207

the desire of preserving more or less intact Steele's original contribution to
the treatise no doubt stood in the way. It is certain, however, that Tait
would never in his later days have put together a book of the type of
" Tait and Steele." Written by him when he had just emerged triumphantly
from the Tripos examination it was meant for students with like ambitions.
Important examples were fully worked out and numerous exercises appended
to the chapters for the eager student to sharpen his weapons on. The
ground covered was considerable ; and when compared with other similar
books Dynamics of a Particle in its later editions maintained its position
both for accurate treatment of important classes of problems and for
the great amount of original matter it contained. The treatment was
analytical throughout ; but here and there, as edition succeeded edition, Tait
inserted some of his neat geometrical demonstrations. The contents are
fairly well indicated by the titles of the chapters : Kinematics, Laws of
Motion, Rectilinear Motion, Parabolic Motion, Central Orbits, Elliptic
Motion, Resisted Motion, Constrained Motion, General Theorems (Action,
Brachistochrones, etc.), Impact. The chapter on Central Orbits, especially
in the numerous examples of extraordinary laws of attraction adjusted to
give integrable solutions, reflects the Cambridge School of Wrangler trainers
of half a century ago. For these and other artificial problems invented
for examination purposes Tait had during the last forty years of his life a
genuine horror.

It is curious to note that, as Dynamics of a Particle was the first
book Tait gave to the world, so the last piece of composition he penned
was the preface to the seventh edition (Nov. 7, 1900). It finishes with
the sentence,

Meanwhile I once more despatch the Veteran on a campaign, with a few necessary
patches on his battered harness.

In little more than a month ill-health compelled him to lay aside his
own harness and weapons, and before many months had passed his battle
of life had closed.

The years immediately preceding 1867 were remarkably productive
from the literary point of view. Thomson and Tait's Natural Philosophy
was launched on a wondering world ; and as if this did not give scope
enough for his busy pen Tait found time to instruct the mathematicians
and natural philosophers in the use of Quaternions and the history of

ao8 PETER GUTHRIE TAIT

Thermodynamics. The Natural Philosophy and Tail's Quaternions are
discussed in appropriate chapters. Here we shall consider more carefully
his contributions to the history of the doctrine of Heat and Energy.
For work of this kind Tail was admirably fitted. Having no claim to
be regarded as one of the founders of the modern theory of energy, he
was early in closest touch with four of the great pioneers, Joule, Helmholtz,
Thomson, and Rankine. He witnessed the striking development of the
fruitful ideas of last century, helped in no small way to forge and fix
appropriate nomenclature, and probably did more than any other single
man to spread a knowledge of the true meaning of the first and second
laws of Thermodynamics. Year after year he led his two hundred students
round the Carnot cycle, and impressed upon them his weighty reflections
on energy and matter ; but to a still larger audience he appealed through
his writings, and one of the most characteristic of his elementary books
is his Sketch of Thermodynamics. This book was the direct outcome of
a controversy with Tyndall as to the historic development of the theory
of heat.

To write complete history when it is in the making is probably a
human impossibility. No man, however talented and well informed, can
see at one and the same time all the influences at work. Nor can he
trace accurately the manner of the working. The personal equation
necessarily enters in. The contemporary historian is apt to be biassed,
though it may be unconsciously. The mental picture will depend upon
the observer, just as witnesses of the same scene do not always tell the
same story. Such general considerations must be borne in mind when
we consider Tail's contributions to the history of modern science.

At the same time it must be remembered that Tait diligently read
the literature of any subject in which he was specially interested ; and that
his knowledge and appreciation of the real significance of the far-reaching
work of the early half of last century were probably unsurpassed. His
great intimacy with Thomson and Maxwell, two of the geniuses of our
time, brought him into immediate contact with the springs of physical
thought.

Bearing this in mind, no later historian can pass by Tait's attitude on
the history of the development of the doctrine of energy without a careful
consideration of the reasons for this attitude. These reasons Tait himself
gave frankly and fully.

ARTICLES IN NORTH BRITISH REVIEW 209

Of the many facts in the history of energy three only became matter
of serious controversy, namely, the true place to be assigned to Mayer
as one of the founders of the theory of heat, the sufficiency of Clausius'
axiom as a basis for the Second Law of Thermodynamics, and the claim
of Clausius to the Entropy integral.

The occasion which led to Tait's incursion into historic fields was a
lecture on "Force" delivered by Tyndall in 1862 in the Royal Institution.
The subject was what we now call " Transformations of Energy," and in
several particulars corresponded curiously with Tait's own inaugural lecture
of 1860. In developing the subject Tyndall, however, proclaimed that
"the striking generalisations " laid before his audience were "taken from
the labours of a German physician named Mayer." This no doubt was
true ; but the manner in which the work of Mayer was lauded seemed to
imply that the examples given of energy transformations were peculiarly
Mayer's and had been imagined by no other natural philosopher, and that
Mayer's priority claims had been hitherto altogether overlooked. This implica-
tion was certainly not true, and controversy was naturally roused. Joule, who
was himself one of the first to bring Mayer's work to the notice of scientific
men, sent a dignified protest to the Philosophical Magazine • and, as a
corrective to what they regarded as erroneous history, Thomson and Tait
communicated an article on Energy to Good Words for October 1862.
This article is historically important as the occasion on which the term
Kinetic Energy saw the light. Tyndall replied in the February and
June numbers of the Philosophical Magazine for 1863, and Tait's rejoinders
appeared in April and August.

Following up this controversy there appeared two unsigned articles
in the North British Review for 1864 — one in February on the Dynamical
Theory of Heat, the other in May on Energy. In the preface to his
Thermodynamics (1867) Tait refers to these reviews as from his pen —
indeed they constitute with necessary changes a large part of the book.
The earlier article traces the modern theory of heat from the independent
experiments of Rumford and Davy in 1798 through the remarkable reasoning
of Carnot (1824) and the epoch-making experiments of Joule (1840-48)
to the final establishment of thermodynamics at the hands of Clausius,
Rankine, and Thomson ; and then, being avowedly a review of Verdet's
Expost de la Thdorie Mtcanique de la Chaleur (1863) and Tyndall's Heat
considered as a Mode of Motion (1863), closes with appreciations and criticisms

T. 27

210 PETER GUTHRIE TAIT

of these books, the criticisms being mainly in regard to the history of the
subject. The second article on Energy is a review of some of the writings
of Joule, Mayer, Helmholtz, and Verdet.

Shortly after the appearance of these reviews, Verdet wrote Tait a
letter of date August 6, 1866, in which he referred to some of Tail's
criticisms :

"En demandant a 1'^diteur de le North British Review de me faire connaitre
1'auteur des articles sur la Theorie me"canique de la Chaleur je n'avais d'autre d£sir
que de savoir qui je devais remercier des e'loges donnds a mon expos6 de cette
th^orie. Je vous prie d'etre persuade que j'y ai e"t6 fort sensible et que je le suis
davantage depuis que j'en connais 1'auteur. Je ne fais aucune difficult^ de recon-
naitre plusieurs inexactitudes historiques que j'avais commises en attribuant a M. Favre
une experience de Joule et en oubliant que Joule avait des 1843 indiqu^ le principe
des relations entre la physiologie animale et la throne me"canique de la chaleur.
Je m'explique difficilement cette derniere omission, car des 1852, lorsque j'avais
insert dans les Annales de Chimie et de Physique un extrait du premier travail de
Joule, j'avais terming cet extrait par une traduction du passage dont il s'agit. Au
reste une lettre que M. Joule voulut bien m'e'crire lorsque je lui envoyai il y a deux
ans un tirage a part de mes Lemons me fait croire qu'en somme il a 6td satisfait de
la maniere dont j'ai parl£ de ses travaux. Personne ne les admire plus que moi ;
et je n'Wsiterais pas a signer tout ce que vous en dites de ses articles.

" Permettez moi apres cette declaration de conserver mon opinion sur Mayer et
de croire qu'une aussi grande ddcouverte que celle de la Conservation de 1'Energie
peut suffir a la gloire de deux et meme de trois inventeurs (car il ne me parait
pas possible de passer Colding sous silence)....

" J'espere, Monsieur, que j'aurai quelque jour le plaisir de vous remercier de vive
voix de tout ce que vous avez dit de flatteur pour moi, etc."

Tait had a high opinion of Verdet as an eminent physicist, who, he was
wont to remark, had just fallen short of the level of genius.

With fuller knowledge afterward gained of the early work of Mohr,
S^guin and Colding, Tait in his second edition of Thermodynamics, as well
as in Recent Advances and in Heat, somewhat modified his original sketch ;
but his contention to the end was that, however ingenious the views advanced
by these other pioneers, their work was not to be put on the same plane with
that of Joule as regarded either the soundness of the theory or the accuracy
of the experimenting.

Mayer's earliest pamphlet of 1842 was a discourse based on the mediaeval
dogma " Causa aequat effectum," whence he deduced the prime property
of all Causes, Indestructibility. He defined Kraft to mean very much
what we now call potential energy, illustrated its transformability into motion

MOHR, COLDING, AND MAYER 211

by a discussion of falling bodies, defined heat as Kraft and not as motion,
drew an analogy between compression of air and fall of bodies, and deduced
a value for the mechanical equivalent of heat on the assumption that the
heat generated in suddenly condensing a gas was equal to the work done
in condensation. Tait maintained that such a mingling of truth and error
could not be accepted as a sound basis for the true doctrine of thermodynamics ;
that it was doubtful if in any particular which could be accepted as sound
physics Mayer had anticipated others ; that his argument in regard to the
experiment of heating air by compression involved a gratuitous assumption
which might or might not be true ; and that already Joule, in his remarkable
experiments on the production of heat by electricity and friction (1840 to
1843), had in an irreproachable scientific manner elucidated the true nature
of heat.

Mayer's later pamphlets of 1845 an^ 1848 contained, as Tait pointed
out in the second chapter of his Thermodynamics, many beautiful examples
of the law of transformations. So also did Mohr's papers of 1837, Grove's
Correlation of the Physical Forces (1842), Joule's papers and lectures between
1840 and 1847, Colding's publications (1840) and Helmholtz's great memoir
of 1847. Indeed, as early as 1834, Mrs Somerville in her Connection of the
Physical Sciences, had called attention to the generality of such transformations.
In fact the notions of transformability and of the equivalence of heat and
work were in the air. The time at last was ripe for the full comprehension,
appreciation, and development of the much earlier experiments of Davy and
of Rumford. It is not surprising that several minds of the first order were
pondering over the significance of these and related phenomena, each
investigator approaching the subject in his own way and to a large extent
independent of the others. Yet by the great majority of their contem-
poraries the early work of these true philosophers was not fully appreciated.
Mohr's first paper, " Ueber die Natur der Warme," appeared in 1837 'm
Liebig's Annalen (Vol. xxiv) ; but Poggendorf declined an expansion of
this paper, which was however accepted for publication in Baumgartner's
and v. Holger's Zeitschrift fur Physik, a publication of comparatively
limited circulation. Mayer's 1842 paper, " Bemerkungen iiber die Krafte
der unbelebten Natur," made its ddbut in Liebig's Annalen, just five years
later than Mohr's similar paper. Neither of them seems to have had any
traceable influence in moulding contemporary scientific thought. Helmholtz
apparently knew nothing of them when he wrote his tract in 1847; and

27—2

212 PETER GUTHRIE TAIT

Joule does not seem to have become acquainted with Mayer's essay till
about 1850.

When in 1876 Tail had his attention drawn to Mohr's paper of 1837,
he sent a translation of it to the Philosophical Magazine (Vol. n, 5th Series,
p. no) and added the following statement :

About the time when Colding and Joule took up the experimental investigation
of Energy at the point where it had been left by Rumford and Davy, there were
published a great many speculations as to the nature of Heat and its relation to
work. Several of these speculations, especially those of Mayer and S6guin, have
been discussed, and at least in part reprinted, in the Philosophical Magazine. It is
right, therefore, that the same journal should recall attention to the above paper,
which was recently pointed out to me by Professor Crum Brown, and contains
what are in some respects the most remarkable of all these speculations.

Singularly enough, it is not even referred to by Mayer, though his much
belauded earliest paper appeared only five years later and in the very same journal.
It contains, in a considerably superior form, almost all that is correct in Mayer's
paper ; and, though it contains many mistakes, it avoids some of the worst of
those made by Mayer, especially his false analogy and his a priori reasoning.

Polarisation of Heat is ascribed to Melloni instead of Forbes ; the calculation
from the compressibility and expansibility of water is meaningless ; and the con-
fusion between the two perfectly distinct meanings of the word Kraft is nearly as
great as that which some modern British authors are attempting to introduce into
their own language by ascribing a second and quite indefensible meaning to the
word Force.

On the other hand, several of the necessary consequences of the establishment
of the Undulatory nature of Radiant Heat are well stated ; and the very process
(for determining the mechanical equivalent of heat by the two specific heats of air)
for which Mayer has received in some quarters such extraordinary praise — though
it is in principle, albeit not in practice, utterly erroneous — is here stated1 much
more clearly than it was stated five years later by Mayer.

As regards the experimental determination of the dynamical equivalent
of heat, Tait's position is practically upheld in the calm judgment given by

1 Mohr's argument is : " If any species of gas is heated more strongly it strives not only
to increase the number of its vibrations, but also to enlarge their amplitudes. If one prevents
this expansion, it appears as increased tension. One would require therefore a smaller quantity
of heat to warm a gas shut in by firm walls than a gas contained in yielding walls, since, if
heat be the cause of the expansion, just as much heat must become latent as there would
be cold developed if the gas were allowed to expand as much as before but without supply
of heat." In an earlier section he has already stated that " Heat is an oscillatory motion
of the smallest parts of bodies.... Heat appears as 'Kraft '...the expansion of bodies by
heat is a force-phenomenon of the highest kind."

"SKETCH OF THERMODYNAMICS" 213

Sir George Stokes when Mayer was awarded the Copley Medal by the Royal
Society of London. Commenting on this determination, Stokes wrote :

" This was undoubtedly a bold idea, and the numerical value of the mechanical
equivalent of heat obtained by Mayer's method is, as we know, very nearly correct.
Nevertheless it must be observed that one essential condition in a trustworthy
determination is wanting in Mayer's method ; the portion of matter operated on does
not go through a cycle of changes. Mayer reasons as if the production of heat were the
sole effect of the work done in compressing air. But the volume of the air is changed
at the same time, and it is quite impossible to say a priori whether this change may
not involve what is analogous to the statical compression of a spring, in which a
portion or even a large portion of the work done in compression may have been
expended. In that case the numerical result given by Mayer's method would have
been erroneous, and might have been even widely erroneous. Hence the practical
correctness of the equivalent got by Mayer's method must not lead us to shut our
eyes to the merit of our countryman Joule in being the first to determine the
mechanical equivalent of heat by methods which are unexceptionable, as fulfilling
the essential condition that no ultimate change of state is produced in the matter
operated upon."

Whatever view may be taken of the question, one thing is clear. It
was Tyndall's eulogy of Mayer which led to the writing of Tail's Sketch
of Thermodynamics. The first and second chapters, to a large extent
reproductions of the articles in the North British Review, were printed
privately for class use in 1867 under the title Historical Sketch of the
Dynamical Theory of Heat; and the book complete in three chapters
appeared in 1868.

When Tait was preparing his Thermodynamics for the press he asked
Maxwell for some hints. Maxwell's reply of date Dec. u, 1867, was very
characteristic and of great interest as being probably the first occasion on
which he put in writing his conception of those fine intelligences — Maxwell's
Demons as Kelvin nicknamed them — who operating on the individual
molecules of a gas could render nugatory the second law of thermodynamics.
He wrote :

" I do not know in a controversial manner the history of thermodynamics, that
is, I could make no assertions about the priority of authors without referring to
their actual works.. ..Any contributions I could make to that study are in the
way of altering the point of view here and there for clearness or variety, and
picking holes here and there to ensure strength and stability.

" As for instance I think that you might make something of the theory of the
absolute scale of temperature by reasoning pretty loud about it and paying it due
honour at its entrance. To pick a hole — say in the 2nd law of ©Acs., that if two

2i4 PETER GUTHRIE TAIT

things are in contact the hotter cannot take heat from the colder without external
agency.

" Now let A and B be two vessels divided by a diaphragm and let them contain
elastic molecules in a state of agitation which strike each other and the sides.

" Let the number of particles be equal in A and B but let those in A have the
greatest energy of motion. Then even if all the molecules in A have equal velocities,
if oblique collisions occur between them their velocities will become unequal, and I
have shown that there will be velocities of all magnitudes in A and the same in
B, only the sum of the squares of the velocities is greater in A than in B.

"When a molecule is reflected from the fixed diaphragm CD no work is lost or
gained.

" If the molecule instead of being reflected were allowed to go through a hole
in CD no work would be lost or gained, only its energy would be transferred from
the one vessel to the other.

" Now conceive a finite being who knows the paths and velocities of all the
molecules by simple inspection but who can do no work except open and close a
hole in the diaphragm by means of a slide without mass.

" Let him first observe the molecules in A and when he sees one coming the
square of whose velocity is less than the mean sq. vel. of the molecules in B let
him open the hole and let it go into B. Next let him watch for a molecule of B,
the square of whose velocity is greater than the mean sq. vel. in A, and when it
comes to the hole let him draw the slide and let it go into A, keeping the slide
shut for all other molecules.

"Then the number of molecules in A and B are the same as at first, but the
energy in A is increased and that in B diminished, that is, the hot system has got
hotter and the cold colder and yet no work has been done, only the intelligence of
a very observant and neat-fingered being has been employed.

" Or in short if heat is the motion of finite portions of matter and if we can
apply tools to such portions of matter so as to deal with them separately, then we
can take advantage of the different motion of different proportions to restore a
uniformly hot system to unequal temperatures or to motions of large masses.

" Only we can't, not being clever enough."

To this is appended a pencilled annotation by Thomson :

"Very good. Another way is to reverse the motion of every particle of the
Universe and to preside over the unstable motion thus produced."

In an undated letter, which must have been written about this time,
Maxwell constructed the following Catechism :

" Concerning Demons.

"i. Who gave them this name? Thomson.

"2. What were they by nature? Very small BUT lively beings incapable of
doing work but able to open and shut valves which move without friction or
inertia.

MAXWELL RECEIVES THE "HISTORIC SKETCH" 215

" 3. What was their chief end ? To show that the and Law of Thermodynamics
has only a statistical certainty.

"4. Is the production of an inequality of temperature their only occupation?
No, for less intelligent demons can produce a difference in pressure as well as
temperature by merely allowing all particles going in one direction while stopping
all those going the other way. This reduces the demon to a valve. As such value
him. Call him no more a demon but a valve like that of the hydraulic ram,
suppose."

Maxwell, writing on Dec. 23, 1867, acknowledged receipt of the
two-chaptered pamphlet in these words :

" I have received your histories of Thermodynamics and Energetics, and will
examine them, along with Robertson on the Unconditioned who holds that our
ultimate hope of sanity lies in sticking to metaphysics and letting physics go down
the wind. I have read some metaphysics of various kinds and find it more or less
ignorant discussion of mathematical and physical principles, jumbled with a little
physiology of the senses. The value of the metaphysics is equal to the mathematical
and physical knowledge of the author divided by his confidence in reasoning from
the names of things.

" You have also some remarks on the sensational system of philosophising (sensation
in the American not the psychological sense). Beware also of the hierophantic or
mystagogic style. The sensationalist says, ' I am now going to grapple with the Forces
of the Universe, and if I succeed in this extremely delicate experiment you will see
for yourselves exactly how the world is kept going.' The hierophant says, ' I do not
expect to make you or the like of you understand a word of what I say, but you may
see for yourselves in what a mass of absurdity the subject is involved.'

"Your statement however seems tolerably complete considering the number of
pages. One or two ideas should be brought in with greater pomp of entry perhaps

" There is a difference between a vortex theory ascribed to Maxwell at page 57,
and a dynamical theory of magnetism by the same author in Phil. Trans. 1865.
The former is built up to show that the phenomena are such as can be explained
by mechanism. The nature of the mechanism is to the true mechanism what an
orrery is to the Solar System. The latter is built on Lagrange's Dynamical Equations
and is not wise about vortices. Examine the first part which treats of the mutual
action of currents before you decide that Weber's is the only hypothesis on the
subject....

"You wrote me about experiments in the Laboratory. There is one which is
of a high order but yet I think within the means and powers of students, namely,
the determination of Joule's Coefft. by means of mercury. Mercury is (i3'S7/O'O33)
times better than water so that about 9 feet would give 10° F....[Plan described for
obtaining a vertical fall of mercury and measuring temperatures above and below]...
I think it a plan free from many mechanical difficulties, and in a lofty room with
plenty of mercury and strong ironwork, and a cherub aloft to read the level and
the thermometer and a monkey to carry up mercury to him (called Quicksilver Jack),

216 PETER GUTHRIE TAIT

the thing might go on for hours, the coefficient meanwhile converging to a value
to be appreciated only by the naturalist"

Tait also sent copies of the two-chaptered pamphlet to Helmholtz and
Clausius for their criticisms before the publication of the complete work. In
his letter to Helmholtz of date Feb. 2, 1867, he said :

Herewith I send copies of the first two chapters of a little work which I intend
soon to publish. Its main object is to serve as a text-book for students till
Thomson and I complete our work on Natural Philosophy.... My object in sending
this to you at present is to ask you and through you Prof. Kirchhoff, whether
in attempting to do justice to Joule and Thomson I have done injustice to you or
your colleague.

Helmholtz replied at considerable length on Feb. 23, 1867. The greater
part of this letter was quoted by Tait in the preface to the book, and was
also reproduced in Helmholtz's collected papers. In another portion of the
letter, not quoted by Tait, Helmholtz said that he did not think it quite fair
to Kirchhoff to be mentioned simply in one line of print with his predecessors
in the field of radiation and absorption. On March i, Tait, after thanking
Helmholtz for his "frank and friendly letter" continued :

With regard to Kirchhoff my object was to ascertain whether his paper on what
is called, I think, Wirkungsfunction and which had reference to the solution of
gases in liquids, should have been referred to.... The spectrum analysis question is
referred to very briefly in my pamphlet which accounts for my not having given
his remarkable researches more prominence. But, with reference to your letter, I
was under the impression that Stewart had established his priority in giving a
complete proof of the equality of Radiation and Absorption.... What I recollect is
that Stewart answered KirchhofTs paper in the Philosophical Magazine, and that
Kirchhoff did not reply to that answer in which Stewart gave the details of his
(supposed) prior proof....

As to Mayer I had no idea that his illness was due to the cold way in which
his papers were received ; nor, had I known this, would I have written so strongly

against his claims to the establishment of the Conservation of Energy I have

always given him full credit for the developments and consequences which he drew
from his premises, but at the same time I have held that his premises (though now
known to be true) had no basis better than a piece of bad Latin....

In his letter of March 19, 1867, Helmholtz made the following remark
regarding Kirchhoff:

" He enters very unwillingly into controversies, and he told me that he had
regarded as sufficient what stood in his paper (in the Phil. Mag. 4, XXV, 259), and

HELMHOLTZ ON PRIORITY QUARRELS 217

had not found that any substantially new arguments had been brought forward by
his opponents1."

At the end of a long letter of date 27 March, 1867, mainly occupied
with a discussion of his and Stewart's experiments on the rotating disk in
vacuo, Tait asked Helmholtz :

Is it fair to ask you whether you think with Clausius that my little pamphlet
will only do me harm — or with Thomson and Joule (who, of course, are interested
parties) as well as Stewart, who have reported favourably on it ? I wish to avoid
strife and to produce a useful little text-book ; but, if Clausius is right, I had better
burn it at once.

In his reply of April 30, 1867, Helmholtz considered some of the
difficulties in the proposed experiments on the rotating disk, then thanked
Tait for his offer to publish his translation of the paper on " Vortex Motion,"
and finished with the following wise words :

" In regard to the question of the publication of your sketch of the history of
the mechanical theory of heat it is very difficult to give advice. For my part I
must say that I have a great aversion to all priority quarrels and have indeed never
myself protested against the greatest misappropriation (eingriffe) ; and I find that I
have in this way really come well off, and that in the end my Own has been
adjudged again. But I know that my best friends think differently on this matter,
and that I stand pretty much alone in my opinion. Further as regards the services
of Joule and Thomson in the matter under discussion they appear to me to be so
completely and generally recognised by all intelligent people with whom I have
spoken that a polemic in their interests is hardly needed.

" If then you divest your writing of its polemical garb it will in my opinion be
thankfully received and will have more influence than with this polemic.

"This is my opinion, since you have wished to hear it; naturally I shall not
take it ill if you do not follow it, since I do not know enough of the personal
conditions which may be moving you."

Tait however thought otherwise, and in the interests of his friends and
for what he regarded as the truth he sent forth his book in all its individuality.

The general character of the three chapters may be inferred from their
titles, namely, (i) Historical Sketch of the Dynamical Theory of Heat,
(ii) Historical Sketch of the Science of Energy, (iii) Sketch of the Funda-
mental Principles of Thermodynamics. The book opens with the arresting
question, What is Heat ? The gradual undermining of the old Caloric Theory

1 It may be remarked here that Lord Rayleigh, in his paper "On Balfour Stewart's
Theory of the Connexion between Radiation and Absorption" (Phil. Mag., Jan. 1901), agrees
with Tait that Stewart had made out his case.

T. '8

2i8 PETER GUTHRIE TAIT

by the discoveries of Black, Davy, and Rumford, and the ultimate triumph
of the Energy Theory are sketched in a racy interesting manner ; and for
the first time in an elementary text-book Carnot's Cycle of Operations and
his notion of the Perfect Reversible Engine are expounded in detail. The
early and practically contemporaneous researches in the dynamical theory
of heat of Clausius, Rankine, and Thomson are explained, and the whole
subject brought into relation with the laws of Radiation.

In the second chapter Tait takes a much wider sweep, and passes in
review a series of striking examples of the transformations and the conserva-
tion of energy. These are taken from the recognised branches of physics,
dynamics, sound, electricity and magnetism, in all their aspects, solar radiation,
gravitational energy, physiological activity, and tidal retardation and other
illustrations of the dissipation of energy.

The third chapter is an exposition of the dynamical theory of heat as
it was developed by the pioneer workers on the subject. When engaged in
putting it together Tait wrote the following three letters to Thomson :

6 G. G., E., 4/1/68.
Dear T,

The compts. and best wishes of the season from all here to you and Lady
Thomson, of whom we hope to hear good accounts....

I don't understand Macmillan at all.... He says everything is arranged, and
doesn't tell me how1. But I'll find out before I take any further step. Meanwhile,
how about the remaining sheets of the pamphlet which you took with you?

I have been writing at the third chapter of my D. T.' and have undergone

TT

some very laborious reading at Clausius' Abhandlungen. I find that he calls — (he

calls it T, not having begun by defining tempre properly) the equivalence-value
of a quantity H of heat at tempre /. Then the second law becomes the assertion
of equivalences

where H, is negative. Then he goes to your

for reversible processes (in your proofs of which, by the way, some lines and steps

1 This has reference to the 2nd edition of " T and T," which however was not seriously
entered upon till 1875.

1 Dynamical Theory of Heat

LETTERS TO THOMSON 219

are altogether left out, so that it is no proof at all, a fact I only discovered on copying
it out into my MSS), and he generalises it into

tdH

tdH

J~'

which he calls the " Entropy." This is the equivalence-value of the actual heat and
of the potential of internal work — the latter of which he calls " Disgregation." All
the rest is mere playing with diff1 =ns. Have you done no more about dissipation
than is to be found in the Phil. Mag. ? The entropy and equivalence-values, etc.,
are merely dissipation enunciated in other language. I have my third Chap, nearly
finished, and shall probably have both ready for your perusal when you return, as
I wish your "polishing" to be done before I go to press. I have kept a few of
Rankine's geometrical things at the commencement, and have then bagged freely
from you. In the 4th Chap. I shall do some geometrical things about Leyden jars
and then bag freely from H1 [i.e., H. Helmholtz's paper, " Ueber die Erhaltung der
Kraft"].

Macmillan has sent me an interleaved copy of Vol. I1, stitched up in 4 equal
parts. This looks like speedy work for the new edition.

I got hold of Maxwell lately, and have managed to extract from him a promise
to send constant contributions to the R. S. E. He has also undertaken to write
accounts of his own, and some other people's work for my D. T.

Snow, sleet, and hail alternately — while you are among the oranges and
myrtles t ! 1

Yrs. T'.

6 G. G., E.

13/1/68.
Dear T,

I have written to Bertrand, and sent him papers as you counselled — no
answer.

My laboratory is getting on capitally but it will take some time to dry before
we can safely work in it.

When two soap-bubbles unite into one, how much work is done? Note, that
the common surface diminishes, but the whole bulk increases.

I have finished Chap. Ill of my Sketch of Thermodynamics, and as you are
not available I have sent it to Maxwell to look over. You will be rather surprised
when you see the quiet way in which I have bagged from you and Rankine....

Yrs. T'.

6 G. G., E.

18/1/68.

Dr T,

Have you got a note I sent you a week or two ago — giving you Forbes'
address?...

1 That is, of the Natural Philosophy; compare footnote (i) on previous page.

28—2

220 PETER GUTHRIE TAIT

My third chapter is ready, and I had thought of sending it to press — but on
second thoughts, I refrained — knowing that you would make far more serious
alterations on printed pages (especially if formed into sheets) than you would care to
do on a mere MS. Once you have looked over it in MSS you will not have the
face to protest against it in type.

You are getting imbued with a little of Pecksniff — rather as regard motives and
actions than as regards style ; still you have caught some of the style also.

But it is simply true, as I told you, that your printed proof of

is no proof at all — not even a chain of reasoning, merely a set of detached links !
How you let it be printed in such a state I can't imagine. Everybody sees you
had the proof in your eye, but whether you or the printers omitted a leading step
I can't of course tell....

When do you return ? U must come to I, or I 2 U, as soon as possible, — for
there is very much pressing work.

Yrs. T.

In July, 1868, Maxwell acknowledged receipt of the complete work in
these words:

"I will write you about your treatise at earliest but (i) I, personally, am satisfied
with the book as a development of T' and as an account of a subject when the ideas
are new and as I well know almost unknown to the most eminent scientific men.
It is a great thing to get this expressed anyhow and I think you have done it
intelligibly as well as accurately. But with respect to the bits of matter I sent you,
do you not think there are breaches of continuity between some, e.g., the statement
about dynamical theories and the context, if they do not actually contradict the
context, at least the N. B. Review part of it. If you disagree with anything of mine,
out with it, for it is better to go into print having one opinion than with two
opinions to throw the reader into perplexity.

"(2) I shall see what case Clausius has.

"(3) Who is Charles that I might believe on him?"

In a review of the second edition of Tail's Thermodynamics in Nature,
Vol. xvn (1877), Maxwell wrote:

"In the popular treatise, whatever threads of science are allowed to appear, are
exhibited in an exceedingly diffuse and attenuated form, apparently with the hope that
the mental faculties of the reader, though they would reject any stronger food, may
insensibly become saturated with scientific phraseology, provided it is diluted with
a sufficient quantity of more familiar language.... In this way by simple reading the
student may become possessed of the phrases of the science without having been
put to the trouble of thinking a single thought about it...

"The technical treatises do less harm, for no one ever reads them except under
compulsion....

MAXWELL'S REVIEW OF "THERMODYNAMICS" 221

"Prof. Tait has not adopted either of these methods. He serves up his strong
meat for grown men at the beginning of the book, without thinking it necessary to
employ the language either of the nursery or of the school ; while for younger
students he has carefully boiled down the mathematical elements into the most
concentrated form, and has placed the result at the end as a bonne boucJu, so that
the beginner may take it in all at once and ruminate upon it at his leisure.

"A considerable part of the book is devoted to the history of thermodynamics,
and here it is evident that with Prof. Tait the names of the founders of his science
call up the ideas, not so much of the scientific documents they have left behind
them in our libraries, as of the men themselves, whether he recommends them to
our reverence as masters in science, or bids us beware of them as tainted with
error. There is no need of a garnish of anecdotes to enliven the dryness of science,
for science has enough to do to restrain the strong human nature of the author,
who is at no pains to conceal his own idiosyncrasies, or to smooth down the
obtrusive antinomies of a vigorous mind into the featureless consistency of a con-
ventional philosopher."

The succeeding paragraphs contained a masterly account of the scientific
methods of Rankine, Clausius, and Thomson.

In this bonne bouche of a third chapter, as Maxwell humorously
called it, Tait gives an extremely compact and instructive sketch of the
mathematical elements of the subject. Beginning with Watt's energy diagram
he developes Carnot's cycle in its modern form, and then, possibly following
Maxwell's advice quoted above, discusses with great clearness Thomson's
scale of absolute temperature. With this in hand and with the further
assumption based on experiment that Carnot's function is inversely as the
absolute temperature, Tait is able to present Thomson's original treatment

in a simplified form. When, however, Tait explicitly referred to , -j- as

" Thomson's expression for the amount of heat dissipated during the cycle "
Clausius found cause of complaint, claiming the above integral as his. In
the preface to the second edition Tait showed very clearly that Thomson
had the whole thing formulated as early as 1851 ; but not until he and Joule
had experimental evidence of the value of Carnot's function would Thomson
use any other than the unintegrated form with the symbol p for Carnot's
function.

In a postcard of date Feb. 12, 1872, Maxwell remarked:

"As for C., though I imbibed my ©Acs from other sources, I know that he is
a prime source, and have in my work for Longman been unconsciously acted on
by the motive not to speak about what I don't know. In my spare moments
I mean to take such draughts of Clausiustical Ergon as to place me in that state

222 PETER GUTHRIE TAIT

of disgregation in which one becomes conscious of the increase of the general sum
of Entropy. Meanwhile till

Ergal and Virial from their throne be cast
And end their strife with suicidal yell

I remain, Yrs. -£."
at

In a letter of date Oct. 13, 1876, Maxwell made clearer references to
the point at issue between Tail and Clausius, and gave at the same time
some interesting confessions as to his own knowledge or, rather, ignorance
of the subject. He wrote :

" When you wrote the Sketch your knowledge of Clausius was somewhat defective.
Mine is still, though I have spent much labour upon him and have occasionally been
rewarded, e.g., earlier papers, molecular slotting1, electrolysis, entropy, and concentra-
tion of rays....

" N.B. In the latter paper the name of Hamilton does not occur. When you are
a-trouncing him, trounce him for that. Only perhaps Kirchhoff ignored Hamilton first
and Clausius followed him unwittingly not being a constant reader of the R. I. A.
Transactions, and knowing nothing of H. except (lately) his Princip., which he and
others try to degrade into the 2nd law of ®A as if any pure dynamical statement
would submit to such an indignity.

"With respect to your citation of Thomson it would need to be more explicit.
The likest thing I find to what you give is in the ist paper on D. T. of H.
(17 March, 1851, p. 272 & 273), but I do not find dq divided by anything like t.

" I think Rankine, by introducing his thermodynamic function <f> which is Jdg/t, made
a great hit, because <f> 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 expression1

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 /0 / ~ 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 championship1, 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, Zt 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 :

<f>v<n,y, BiaBo^aiy a-)(T)fjLdr<av rpi<r/j,vploi<i,
d\\d<ra-erai Tuirtopa, TLptareax; Biierjv,
irdvrcov o<r' eaTi 7rotKt\a>Ta,TOV repay
T?}? 8' aJn' 'A.vdy/cr)<{ ear' dxivijTov aOevot,
fj,6vr] y dirdvTuv ravro Siafi.evovff' del
/3poT<£v re /cat 0e£v TTUVT' aTrorpvei yew).

238 PETER GUTHRIE TAIT

A footnote states :
Thus paraphrased for us:

Nature, bewildering in diversity,

Of marvels Marvel most inscrutable,

Like Proteus, altereth her shape and mould ;

But Fate remaineth ever immovable,

And, changeless in persistency, outwears

The Time of men, the gods' Eternity.

Recalling that Professor D'Arcy W. Thompson had once remarked to
me that he believed his father Professor D'Arcy Thompson of Galway had
supplied Tait with some Greek verses, I drew his attention to the lines, and
obtained the following reply of date June 4, 1908 :

"Many thanks indeed for your letter of the 3ist May, which, with its enclosures,
interests me very much indeed.

" I cannot of course absolutely testify that the verses are my father's, but everything
points that way :

(1) I know that my father did some verses for The Unseen Universe, and, as
far as my recollection goes, they were on just such lines as these you send ;

(2) The Greek is extremely like Euripides, the author whom my father told
me he had imitated ;

(3) The English paraphrase strikes me as being exactly in my father's style.

" My father certainly told me that Tait had asked him to make those verses for the
book, so that little piece of waggery of inserting them for the admiration of the reader

and the mystification of the scholar was Tail's doing

(Signed) D'ARCY W. THOMPSON."

This is an example of the way in which Tait prevailed upon his friends to
help in adding interest to the pages of The Unseen.

Robertson Smith, the eminent Semitic scholar and theologian, seems to
have given valuable hints throughout, as may be inferred from the following
letter written by Tait on June 5, 1875.

My dear Smith,

Macmillan gives me private information that in a few weeks a second
edition of the U. U. will be wanted. He deprecates any material change, partly on
its own merits, mainly on the inevitable delay it would involve.

Now, while I still most strongly hold to your kind promise to (some day soon)
rewrite the first chapter for us, I think Mac. is right that there should be no material
change in the second edition — especially as but few of the great critics have yet
spoken out, and we must not at once abandon our first essay as if afraid of what may

RECEPTION OF "THE UNSEEN UNIVERSE" 239

be ultimately said of it. We must be at first a Lucretian Atom not a vortex ring,
strong in solid singleness, not wriggling meanly away from the knife ! Will you
therefore, by little instalments, as it suits you, give me soon all the more vital
improvements which occur to you as possible without much altering the pages, etc.
(the type having been kept up, so as to save expense) ?

You have of course seen Clifford's painful essay in the Fortnightly

An advanced ritualist, MacColl, has cracked us up in a letter to the Guardian last
week. This week the other ritualist paper The Church Herald says our book is infidel.
Last week the Spiritualist said that with a few slight changes the book would be an
excellent text-book for its clients. The Edinburgh Daily Review says we are subtle
and dangerous materialists. Hanna (late of Free St John's here) says the work is the
most important defence of religion that has appeared for a long time ! Which of these
is nearest the truth ?

The Church Herald is down on us for your suggestion about "for a little while
lower than the angels."

Truly the reviews and critiques of The Unseen Universe were as varied as
the religious and irreligious views of the critics who wrote them. To one it
was a " masterly treatise," to another it was full of " the most hardened and
impenitent nonsense that ever called itself original speculation." Some sneered at
the authors for their ignorance of philosophical thought and phraseology ; others
were captivated by the "acute analytical faculty," the "broad logical candid
turn of mind " displayed. Many of the early appreciations of the book were
certainly crude, hastily conceived, and hurriedly presented before their readers.
On various sides the intention of the argument was not clearly apprehended.
There was a novelty in the mode of presenting it, with an appeal to the
profoundest truths of modern physics, which rather confused the mind of the
ordinary critic unskilled in Carnot cycles and reversible engines. One critic
there was, the versatile and brilliant Clifford, who knowing these truths in all
their purely physical significance, gave the authors a terrible trouncing in the
Fortnightly Review (June i, 1877). The critique is reprinted in his Lectures
and Essays, but with some of the liveliest passages deleted. The most
important omission is the opening paragraph, which in its original form
presented Clifford in the guise of a clever debater, who burlesques the
argument he intends to demolish. The final paragraph sufficiently shows
Clifford's point of view and is of interest here from its incidental description
of Tait as a "wide-eyed hero," between whom and Clifford there existed
indeed a warm affection, divergent though their views were on questions
of religion. Scoffing at the attempt to preserve the Christian faith in an
enlightened scientific age, Clifford wrote :

240 PETER GUTHRIE TAIT

" ' Only for another half-century let us keep our hells and heavens and gods.' It is
a piteous plea ; and it has soiled the hearts of these prophets, great ones and blessed,
giving light to their generation, and dear in particular to our mind and heart. These
sickly dreams of hysterical women and half-starved men, what have they to do with
the sturdy strength of a wide-eyed hero who fears no foe with pen or club ? This
sleepless vengeance of fire upon them that have not seen and have not believed, what
has it to do with the gentle patience of the investigator that shines through every page
of this book, that will ask only consideration and not belief for anything that has not
with infinite pains been wholly established ? That which you keep in your hearts, my
brothers, is the slender remnant of a system which has made its red mark in history,
and still lives to threaten mankind. The grotesque forms of its intellectual belief have
survived the discredit of its moral teaching. Of this what the kings could bear with,
the nations have cut down ; and what the nations left, the right hand of man by man
revolts against day by day. You have stretched out your hands to save the dregs of
the sifted sediment of a residuum. Take heed lest you have given soil and shelter to
the seed of that awful plague which has destroyed two civilisations, and but barely
failed to slay such promise of good as is now struggling to live among men."

Racy and instructive though it was, Clifford's review did not really touch
the central doctrines of Stewart and Tail's speculations. One of their aims was
to show that there was nothing in physical science which denied the possibility
of our intelligences existing after death in another universe. They also argued
that certain aspects of the modern theory of energy suggested, if they did not
demonstrate, the probability of such an Unseen Universe. The reasonings
could not satisfy either the extreme right or the extreme left. It was little
wonder then that the prophet and the agnostic alike fell foul of the book, —
the prophet, because the authors strove to bring under the Law of Continuity
certain mysteries of his faith, the agnostic, because starting from the known
they endeavoured to cross the fringe of the unknown.

Many of the ideas and speculations put forward by Stewart and Tait
were novelties to the vast majority of their readers. These ideas are now
familiar as the sunshine. It would be impossible to say, however, to what
extent the authors of The Unseen Universe impressed some of their views
upon the world, or to what degree they were simply the earliest exponents of
thoughts which were gradually taking shape in the human mind.

The tenth edition of The Unseen Universe was translated into French
by a naval Lieutenant A.-B., with a preface to French readers by Professor
D. de St-P. (Paris Libraire Germer Bailliere et Cie, 1883).

In 1878 Stewart and Tait published a sequel to The Unseen Universe
under the name of Paradoxical Philosophy. The book was cast into the form

MAXWELL REVIEWS "PARADOXICAL PHILOSOPHY" 241

of a dialogue, the purpose of it being to convert Dr Hermann Stoffkraft, a
German materialist, to a belief in the doctrines of The Unseen Universe.
The deed is done ; but of course, as in the orthodox novel, the end is obvious
from the beginning. A delicately humorous and yet scientifically critical
review was written for Nature (Dec. 19, 1878) by Clerk Maxwell. Certain
paragraphs from that review hit off with such remarkable clearness the whole
bearing of the two books that no apology is needed for their reproduction
here.

" We cannot accuse the authors of leading us through the mazy paths of science
only to entrap us into some peculiar form of theological belief. On the contrary, they
avail themselves of the general interest in theological dogma to imbue their readers
unawares with the newest doctrines of energy. There must be many who would never
have heard of Carnot's reversible engine, if they had not been led through its cycle of
operations while endeavouring to explore the Unseen Universe. No book containing
so much thoroughly scientific matter would have passed through seven editions in so
short a time without the allurement of some more human interest

" The words on the title-page : ' In te, Domine, speravi, non confundar in aeternum '
may recall to an ordinary reader the aspiration of the Hebrew Psalmist, the closing
prayer of the ' Te Deum ' or the dying words of Francis Xavier ; and men of science,
as such, are not to be supposed incapable either of the nobler hopes or of the nobler
fears to which their fellow men have attained. Here, however, we find these venerable
words employed to express a conviction of the perpetual validity of the ' Principle of
Continuity,' enforced by the tremendous sanction, that if at any place or at any time
a single exception to that principle were to occur, a general collapse of every intellect
in the universe would be the inevitable result.

" There are other well known words in which St Paul contrasts things seen with
things unseen. These also are put in a prominent place by the authors of The Unseen
Universe. What, then, is the Unseen to which they raise their thoughts ?

" In the first place the luminiferous aether, the tremors of which are the dynamical
equivalent of all the energy which has been lost by radiation from the various systems of
grosser matter which it surrounds. In the second place a still more subtle medium,
imagined by Sir William Thomson as possibly capable of furnishing an explanation of
the properties of sensible bodies ; on the hypothesis that they are built up of ring
vortices set in motion by some supernatural power in a frictionless liquid : beyond which
we are to suppose an indefinite succession of media, not hitherto imagined by anyone,
each manifoldly more subtle than any of those preceding it. To exercise the mind in
speculations on such media may be a most delightful employment for those who are
intellectually fitted to indulge in it, though we cannot see why they should on that
account appropriate the words of St Paul."

After a playful discussion of some of the theories of the origin of
consciousness and of the meaning of personality, Clerk Maxwell summed
up thus :

T. 31

242 PETER GUTHRIE TAIT

" The progress of science, therefore, so far as we have been able to follow it, has
added nothing of importance to what has always been known about the physical
consequences of death, but has rather tended to deepen the distinction between the
visible part, which perished before our eyes, and that which we are ourselves, and to
show that this personality, with respect to its nature as well as to its destiny, lies quite
beyond the range of science."

In his letters to Tait, Maxwell let his humour play round the curious
speculations and metaphysics of the authors of The Unseen Universe and of
The Paradoxical Philosophy. For example, at the end of the letter of
Sept. 7, 1878, from which quotations have already been given {pp. 151-2),
Maxwell remarked :

"It is said in Nature that U.U. is germinating into some higher form. If you
think of extending the collection of hymns given in the original work, do not forget
to insert 'How happy could I be with Ether.'"

After the publication of Paradoxical Philosophy, he sent Tait " A Para-
doxical Ode" consisting of three stanzas, said to be "after Shelley."
The movement of the verses, the rhythm and the rhyming, strongly suggest
portions of Prometheus Unbound, although the imitation is not quite accurate
as to form. With marvellous ingenuity has Maxwell woven into his verses
much that characterised not only the speculations of The Unseen Universe
but also certain features of Tail's scientific work in relation to the classifica-
tion of knots. The verses are given here as they appear in the original
draft which was pasted into Tail's Scrap Book. It differs in slight details
from ihe version published in Maxwell's Life.

TO HERMANN STOFFKRAFT, PH.D.

A PARADOXICAL ODE.

(After Shelley)

I.

My soul's an amphicheiral knot1

Upon a liquid vortex wrought
By Intellect in the Unseen residing,

While thou dost like a convict sit

With marlinspike untwisting it
Only to find my knottiness abiding,

1 An amphicheiral knot is a knot which can be changed into its own mirror reflexion —
amphicheiral similarity is the similarity between a right and a left hand. See Tait " On Knots "
(Trans. R.S.E., 1876-7; Sci. Pap., Vol. i, pp. 288, 314, etc.).

A PARADOXICAL ODE 243

Since all the tools for my untying

In four-dimensioned space are lying1,

Where playful fancy intersperses

Whole avenues of universes2,

Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.

II.

But when thy Science lifts her pinions

In Speculation's wild dominions
I treasure every dictum thou emittest;

While down the stream of Evolution

We drift, and look for no solution
But that of the survival of the fittest.

Till in that twilight of the gods

When earth and sun are frozen clods,

When, all its energy degraded,

Matter in aether shall have faded,

We, that is, all the work we've done-
As waves in aether, shall for ever run
In swift expanding spheres, through heavens beyond the sun'.

III.

Great Principle of all we see

Thou endless Continuity*
By thee are all our angles gently rounded ;

Our misfits are by thee adjusted,

And as I still in thee have trusted,
So let my methods never be confounded !

O never may direct Creation

Break in upon my contemplation

Still may the causal chain ascending,

Appear unbroken and unending,

And where that chain is lost to sight,
Let viewless fancies guide my darkling flight
Through Aeon-haunted worlds, in order infinite.

1 A tri-dimensional knot cannot exist in four-dimensional space.

2 See The Unseen Universe, 4th edit., § 220.

3 See The Unseen Universe, § 196.

4 See The Unseen Universe, Chapter 11.

244 PETER GUTHRIE TAIT

In a later letter Maxwell wrote :
" Last three lines of Ode to Stoffkraft should be as follows :

While Residents in the Unseen —
Aeons or Emanations — intervene,
And from my shrinking soul the Unconditioned screen."

On Aug. 28, 1879, ten weeks before his death, Maxwell sent Tait
a curious composition purporting to be a soliloquy or self-communion by
Tait himself. In spite of the rapid advance of the fatal illness to which he
succumbed Maxwell's quaint humour still found expression to his life-long
friend. In this last of many letters, the speculations in The Unseen Universe
and the quaternion operator Nabla which Tait used with so great effect are
mingled together in a fashion most strange and fanciful. The jest lurks in
the closing sentence, pathetic though this is in its confession of physical
weakness.

"HEADSTONE IN SEARCH OF A NEW SENSATION."

" While meditating, as is my wont on a Saturday afternoon, on the enjoyments
and employments which might serve to occupy one or two of the aeonian aetherial
phases of existence to which I am looking forward, I began to be painfully conscious
of the essentially finite variety of the sensations which can be elicited by the combined
action of a finite number of nerves, whether these nerves are of protoplasmic or
eschatoplasmic structure. When all the changes have been rung in the triple bob
major of experience, must the same chime be repeated with intolerable iteration
through the dreary eternities of paradoxical existence ? The horror of a somewhat
similar consideration had as I well knew driven the late J. S. Mill to the very verge
of despair till he discovered a remedy for his woes in the perusal of Wordsworth's
Poems.

" But it was not to Wordsworth that my mind now turned, but to the noble
Viscount the founder of the inductive philosophy and to the Roman city whence he
was proud to draw his title, consecrated as it is to the memory of the Protomartyr
of Britain.

"Might not I, too, under the invocation of the holy ALBAN become inspired
with some germinating idea, some age-making notion by which I might burst the
shell of circumstance and hatch myself something for which we have not even a
name, freed for ever from the sickening round of possible activities and exulting in
a life every action of which would be a practical refutation of the arithmetic of the
present world.

" Hastily turning the page on which I had inscribed these meditations, I noticed

MAXWELL'S LAST LETTER 245

just opposite the name of the saint another name which I did not recollect having
written. Here it is

MA8 JA

" Here then was the indication, impressed by the saint himself, of the way out
of all my troubles. But what could the symbol mean? I had heard that the harp
from which Heman or Ethan drew those modulations from the plaintive to the
triumphant which modern music with its fetters of tonality may ignore but can never
equal — I had heard that this harp had been called by a name like this. But not in
all Wales could such a harp be found, nor yet the lordly music which has not been
able to come down through the illimitable years.

" Here I was interrupted by a visitor from Dresden who had come all the way
with his Erkenntniss-Theorie under his arm, showing that space must have three
dimensions, and that there's not a villain living in all Denmark but he's an arrant
knave. Peruse his last epistle and see whether he could be transformed from a blower
of his own trumpet into a Nabladist.

" I have been so seedy that I could not read anything however profound without
going to sleep over it.

dp,,
~df

CHAPTER VII
ADDRESSES, REVIEWS, AND CORRESPONDENCE

IN general Tait did not, like some of his colleagues, begin each session
at Edinburgh University with a special introductory lecture upon some
chosen subject. Occasionally, however, recent discoveries or new ways of
looking at physical problems attracted his attention, and gave an unusual
character to the opening lecture. The following two examples will illustrate
what is meant.

In the opening lecture of November 1869 he gave an account of the
first great results in solar spectroscopy, and discussed to some extent the
nature of nebulae and comets. This was the occasion on which for the first
time he gave an explanation of some of the phenomena of comets' tails.
Tail's " beautiful sea-bird analogy " — as Kelvin called it — was also given in
a series of interesting articles on Cosmical Astronomy which appeared in
Good Words in 1875. The following quotation is taken from a full abstract
of the opening lecture just referred to, which was published in Nature, Dec. 16,
1869, and was translated in La Revue des Cours Scientifiques, 1870.

Finally let us consider what we have recently learned about comets — bodies
which have hitherto puzzled the astronomer quite as much as the nebulae.... There
seems to be good grounds for imagining that a comet is a mere shower of stones
(meteorites and fragments of iron). This at least is certain that such a shower
would behave, in its revolution about the sun, very much as comets are seen to
do....

Such small comets as have been observed have given continuous spectra from
their tails, so far as could be judged with regard to an object so feebly illuminated.
This, then, it would appear, is simply reflected solar light. The heads, however,
give spectra somewhat resembling those of the nebulae I have just mentioned — the
spectra of incandescent gases. This is quite consistent with the descriptions given
by Hevelius and others of some of the grander comets ; which presented no peculiari-
ties of colour in the tail, but where the head was blueish or greenish. Now these
appearances are easily reconciled with the shower-of-stones hypothesis. The nucleus,
or head, of a comet is that portion of the shower where the stones are most

COMETS' TAILS 247

numerous, where their relative velocities are greatest, and where, therefore, mutual
impacts, giving off incandescent gases, are the most frequent and the most violent.
This simple hypothesis explains easily many very striking facts about comets, such
as their sometimes appearing to send off in a few hours a tail many hundreds of
millions of miles in length. Wild notions of repulsive forces vastly more powerful
than the sun's gravity have been entertained ; bold speculations as to decomposition
(by solar light) of gaseous matter left behind it in space by the comet have also
been propounded ; but it would seem that the shower-of-stones hypothesis accounts
very simply for such an appearance. For, just as a distant flock of seabirds comes
suddenly into view as a dark line when the eye is brought by their evolutions into
the plane in which they fly, so the scattered masses which have lost velocity by
impact, while they formed part of the head, or those which have been quickened
by the same action, as well as those which lag behind the others in virtue of the
somewhat larger orbits which they describe, show themselves by reflected solar light
as a long bright streak whenever the earth moves into any tangent plane to the
surface in which they are for the time mainly gathered.

A year later Tait found occasion, after the usual exposition of the
significance of the Conservation of Energy, to warn his students against
looseness of language. He illustrated his point by quoting from Bain's logic.
It would seem that he was put on the track of this book by W. Robertson
Smith, who had been carrying on his theological studies at the Free Church
College, Edinburgh, during 1868-70, and at the same time acting as Tail's
Assistant. In an undated letter, the main subject matter of which fixes the
date as April of 1870, Robertson Smith asked " Have you seen Bain's
Logic ? Full of rubbish about conservation of force, by which he means
momentum ! ! !" Tait's own lively criticisms of Bain's inaccuracies in scientific
statement will be found in Nature, Dec. i, 1870.

It is the custom in the Scottish Universities for the Arts Professors in
rotation to address the graduates at the annual graduation ceremonial. In
the old days when the Arts Chairs were limited by statute to seven, each
Professor was called upon at intervals of about seven years to act as " Promoter"
and give an address. Tait was Promotor in 1866, 1874, 1881, and 1888.
Before his turn came round again, the new regulations had come into force,
and the Arts Faculty had been widened out to embrace nearly a dozen other
chairs in literature, history, education, science and art.

Tait's first address on the value of the Edinburgh degree of M.A. was
published by the Senatus as a pamphlet. It contained a strong protest
against the proposition to amalgamate the Scottish Universities as one grand
National University with a central Examining Board. A quotation from

248 PETER GUTHRIE TAIT

this address was given above (p. n) in reference to cramming or coaching
for examinations. The question of central boards of examiners and their
necessary concomitant cram received many a hard hit from Tait in his
graduation addresses, which present in somewhat whimsical guise his horror
of the examiner who is not at the same time a teacher. The same views are
expressed in an article on " Artificial Selection " which appeared in Macmillans
Magazine for 1872 and which contains some racy illustrations of how not to
examine. The following quotation from this article indicates the ideal
University which Tait at that time pictured to himself:

A combination of the Scottish and English University systems, to the exclusion
of what is manifestly bad in each, is the thing really wanted. England's superiority
consists in very great measure of money and lands — that of Scotland in making
the University Professors the actual teachers. Let us have in the great English
Universities Professors teaching the many, to take the place of the all-pervading
Coach — in addition of course to the almost unequalled body of Professors they now
possess.... In Scottish Universities let many of the chairs be doubled or even trebled ;
let there be, for instance, a Professor of Experimental Physics in each, and a
Professor of Applied Mathematics, in the place of the present solitary Professor of
the enormous subject Natural Philosophy ; let us have a Professor of Chemistry
and Medicine, and a Professor of the Theory of Chemistry, etc.. ..Let the multi-
farious duties now discharged by one over-burdened man be distributed among two,
three or four ; let their salaries not depend for so much as half the whole amount on
the numbers attending their classes, so that there shall be no possible incitement
to lower their standard to attract more listeners. But also let us take every care
that they be kept rigorously to their work, and at once laid aside whenever they
have ceased to be working teachers.

This unfortunately is not likely to be done. The extreme poverty of the
Scottish Universities, more especially of the Metropolitan one, prevents their doing
much. And Scotland's share of the Imperial Revenue has always been insignificant
compared with her contributions to it. Still it is surely possible that a few annual
thousands might be obtained from Parliament to furnish her universities properly
with laboratories ; and the overworked and underpaid professors with adequate
remuneration and with additional assistants, from whom in turn their successors
might be chosen. Then the country, having done something to deserve success,
cannot fail to attain it.

Recent developments have in some respects, although not in all, been
along the very lines here sketched by one who, because of his conservative
political sympathies, was believed by many to be averse to progress of any
kind. In one particular, however, we have not worked towards the ideal
imagined by Tait. The exaltation of the examination still continues ; and
some of his strong characteristic statements are quite to the point in these

THE EVILS OF CRAM 249

days. Take the following extracts from the address to the graduates of 1874
(see Nature, April 30, 1874), and from the address of 1881.

It is a mere common-place to say that examination, or, as I have elsewhere
called it, artificial selection is, as too often conducted, about the most imperfect of
human institutions; and that in too many cases it is not only misleading, but
directly destructive, especially when proper precautions are not taken to annihilate
absolutely the chances of a candidate who is merely crammed, not in any sense
educated. Not long ago I saw an advertisement to the effect : — " History in an
hour, by a Cambridge Coach" How much must this author have thought of the
ability of the examiners before whom his readers were to appear ? There is one, but
so far as I can see, only one, way of entirely extirpating cram as a system, it may be
costly — well, let the candidates bear the expense, if the country (which will be
ultimately the gainer) should refuse. Take your candidates, when fully primed for
examination, and send them off to sea — without books, without even pen and ink ;
attend assiduously to their physical health, but let their minds lie fallow. Continue
this treatment for a few months, and then turn them suddenly into the Examination
Hall. Even six months would not be wasted in such a process if it really enabled
us to cure the grand inherent defect of all modern examinations. It is amusing to
think what an outcry would be everywhere raised if there were a possibility of such
a scheme being actually tried — say in Civil Service Examinations. But the certainty
of such an outcry, under the conditions supposed, is of itself a complete proof of
the utter abomination of the cramming system. I shall probably be told, by
upholders of the present methods, that I know nothing about them, that I am
prejudiced, bigoted, and what not. That, of course, is the natural cry of those
whose " craft is in danger " — and it is preserved for all time in the historic words,
" Thou wert altogether born in sin, and dost thou teach us ? " I venture now to
state, without the least fear of contradiction, a proposition which (whether new or
not) I consider to be of inestimable value to the country at large : — Wherever the
examiners are not in great part the teachers also, there will cram to a great extent
supersede education

This Chinese passion for excessive examination threatens to become as great a
nuisance here as the Celestials themselves have proved in San Francisco. I had better
not tell you any more of my own experiences ; I have already said quite enough about
myself. But I can tell you what occurred to a friend of my own, a professor in
a neighbouring State. My friend belongs to the well-known kingdom of Yvetot —
that happy land in which (as Thackeray tells us) the good king

Each year called out his fighting men,
And marched a league from home ; and then —
Marched back again.

The Professor had once a favourite student, of much more than ordinary
promise, who sought a post under government — a post for which he was in all
respects singularly fitted. Now, none but the very highest of such posts can be
obtained in Yvetot except after a strict examination ; so the youth had to submit

T. 32

250 PETER GUTHRIE TAIT

to the ordeal. One of the chief subjects on which he was examined was the Mandingo
language. The attempt proved unsuccessful; the student was "remitted" — that is
the correct phrase; they are very dainty and delicate in their phrases in Yvetot.
Yet, as it happened, that youth had been for some years in an office in the capital
of Mandingo itself; and had acquired, not, indeed, a pedantic knowledge, but, what
is far better, a thorough working knowledge of the language ! He was remitted
because his examiner was unable to test that kind of knowledge.

Again, the Professor had an old serving-man who desired Government employ-
ment as a door-porter and messenger. He had done that kind of work extremely
well for years in private houses. But even posts like these are never given by
Government (in Yvetot) except after examination. No one has yet found out how
to examine in the art of door-keeping, so the would-be porter had to be examined
in physical geography and continued fractions ! Of course he also was remitted, and
he died of shame a few days later. But (in Yvetot) all Government door-keepers
must know physical geography !

My friend the Professor has an almost morbid hatred of cram, and his Rfyetiteur
thinks with him. But one day there came to him an unfortunate man who had
been remitted in a Government examination, and who desired to try his fate once
more. There were but a few days before the awful tribunal of examiners was again
to meet. To teach the victim in the available time was impossible ; so my friend
thought that for once he would try whether the art of cramming comes by nature,
or has to be painfully acquired. The Rfye'titeur entered into the scheme with
hearty goodwill, and conducted the process. When the examination was over the
lately remitted one was almost at the head of the list. The Professor's only remark
was, — " I should like to examine these examiners ! "

To return to our own land. While the present state of things continues, the
universities have no option. They must give degrees, and in consequence they must
examine. With human beings, as with guns and girders, testing is a very delicate
process. You may double-shot your gun, or load your girder far beyond what it
will thenceforth be required to bear, and both may stand the test ; yet the very-
testing may have produced a flaw in the metal, some day to finish its career by
what will then be called (euphemistically, of course) a terrible accident.

In presence of these painful realities, it is your duty, alike to the University
and to your fellow-men, to endeavour, as far as you have opportunity, to extend the
inestimable blessings and privileges of education. But, in doing so, never for a
moment forget what education really is. It is not Latin, nor Greek, nor philosophy,
whether natural or unnatural1. It is not even the three R's. These are indispensable
preliminaries to education, but preliminaries only. He who possesses them has been
taught; it does not at all follow that he has been educated. The confusion of
teaching with education is a common but monstrous fallacy. You may know Liddell
and Scott by heart without becoming Greek scholars. You may be able to differentiate,

1 I remember that when Tail uttered these words with characteristic emphasis he turned
with triumphant glee toward his colleague, Campbell Fraser, professor of Logic and Metaphysics,
as much as to say, " I've got you this time — there is no reply ! "

THE REDE LECTURE 251

and integrate, and solve all manner of regulation problems in mathematics, and
yet be no mathematicians. Machinery can be made to do all that sort of thing
better than any human being can. You must have risen far above the mere efforts
of memory, and of " rule of thumb," before you can consider yourselves educated.
Your minds must be able to do something which no machine can do for them ;
otherwise they sink below the level of a machine, for it is absolutely free from human
liabilities to error.

But there is no cause for dread less extended education should promote same-
ness in opinions or pursuits — a thing in itself undesirable. Unless human knowledge
were complete, there could not but be serious differences of opinion, even amongst
the most highly educated, and even on subjects of the gravest importance. Yet,
though extensive differences of opinion cannot but exist, you must not on that
account think it of little moment what opinions you hold. Every human being who
has received the priceless gift of reason is righteously responsible for its employment
to the uttermost in all the varied circumstances of life. Opinions held lightly or
on insufficient grounds will never be of much use in inciting to, or in directing, action.
" Let every man be fully persuaded in his own mind,"

On May 24, 1873, Tait delivered the Rede Lecture in the Senate House,
Cambridge, choosing for his subject the Thermoelectric Diagram. A fairly
full abstract was published in Nature, Vol. vin, and is reprinted as No. xxvui
in the Scientific Papers. An enlarged diagram showing the positions of the
lines of the more interesting metals was specially prepared, being drawn in
such a way as to give at a glance the Centimetre-gramme-second values of the
various quantities involved — electromotive power, Peltier Effect, Thomson
Effect, etc. This was subsequently shown at the South Kensington Loan
Exhibition of Scientific Apparatus.

In the preparations for the Rede Lecture, Maxwell gave valuable help,
as may be gathered from the following letter, written in reply to questions
from Tait.

ii SCROOPE TER.

10 March, 1873.

O T,

</>o/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."

32—2

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

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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 m£l£e 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

34—2

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 <r = in +jv + kw, we have the cumbersome form

instead of the vastly simpler and more expressive

272 PETER GUTHRIE TAIT

It may be, however, that this apparent inconsistency is in reality dictated by skill
and prudence. The suspicious reader, already put on his guard by Clerk Maxwell's
first cautious introduction of the evil thing, has to be treated with anxious care and
nicety of handling : lest he should refuse altogether to bite again. If he rises to the
present cast we shall probably find that Prof. Clifford has V, in the form as it were of a
gaff, ready to fix him irrevocably. That he will profit by the process, in the long run,
admits of no doubt : so the sooner he is operated on the better. What is now urgently
wanted, for the progress of some of the most important branches of mathematical
physics, is a " coming " race of intelligent students brought up, as it were, at the feet of
Hamilton ; and with as little as may be of their freshness wasted on the artificialities
of x, y, z. Till this is procured, quaternions cannot have fair play. Nut-cracking,
though occasionally successful for a moment, is the most wasteful and destructive
of all methods of sharpening the teeth.

What we have at some length discussed is the most prominent feature of the
present work, but by no means its only distinctive one. No writer, who has any claim
upon his readers at all, can treat even the most hackneyed subject without giving
a new and useful turn to many a long-known truth. Many of Prof. Clifford's proofs
are exceedingly neat, and several useful novelties (e.g. Three-bar Motion) are introduced.
We have to complain, however, of a great deal of unnecessary new and very strange
nomenclature : for a large part of which the author is not responsible, his error
(for such we cannot help considering it) consisting in giving this stuff a place of
honour in his book. One does not require to be very violently conservative to
feel dismayed at an apparently endless array of such new-fangled terms as Pedals,
Rotors, Cylindroids, Centrodes, Kites, Whirls, and Squirts ! Yet these are but a
few gleaned at random from the book. Something, it seems, must be hard in a
text-book — simplify the Mathematic, and the Anglic (i.e. the English) immediately
becomes perplexing.

In Nature of June u, 1885 (Vol. xxxn), Tait reviewed Clifford's
The Common Sense of the Exact Sciences, a book which was published
six years after the author's death. The following sentences seem of sufficient
permanent interest to deserve quotation :

Once more a characteristic record of the work of a most remarkable, but too brief,
life lies before us. In rapidity of accurate thinking, even on abstruse matters, Clifford
had few equals ; in clearness of exposition, on subjects which suited the peculiar bent
of his genius and on which he could be persuaded to bestow sufficient attention, still
fewer. But the ease with which he mastered the more prominent features of a subject
often led him to dispense with important steps which had been taken by some of his
less agile concurrents. These steps, however, he was obliged to take when he was
engaged in exposition ; and he consequently gave them (of course in perfect good
faith) without indicating that they were not his own. Thus, especially in matters
connected with the development of recent mathematical and kinematical methods,
his statements were by no means satisfactory (from the historical point of view) to

poiNCARE's "THERMODYNAMIQUE" 273

those who recognised, as their own, some of the best " nuggets " that shine here and
there in his pages. His Kinematic was, throughout, specially open to this objection :
and it applies, though by no means to the same extent, to the present work. On the
other hand, the specially important and distinctive features of this work, viz. the homely,
yet apt and often complete, illustrations of matters intrinsically difficult, are entirely
due to the author himself....

The chief good of this book, and in many respects it is very good, lies in the fact
that the versatility of its gifted author has enabled him to present to his readers many
trite things, simple as well as complex, from so novel a point of view that they acquire
a perfectly fresh and unexpected interest in the eyes of those to whom they had
become commonplace. Surely this was an object worthy of attainment ! But it is
altogether thrown away on the non-mathematical, to whom neither new nor old points of
view are accessible.

Tail's review of Poincare^s Thermodynamique appeared in Nature,
Jan. 14, 1892 (Vol. XLV). It is a good example of honesty in criticism; for
in spite of the great and deserved fame of the author, Tait could only
condemn the book as a physical treatise.

The great expectations with which, on account of the well-won fame of its author,
we took up this book have unfortunately not been realised. The main reason is not far
to seek, and requires no lengthened exposition. Its nature will be obvious from the
following examples.. . .

Some forty years ago, in a certain mathematical circle at Cambridge, men were
wont to deplore the necessity of introducing words at all in a physico-mathematical
text-book : the unattainable, though closely approachable, Ideal being regarded as
a work devoid of aught but formulae !

But one learns something in forty years, and accordingly the surviving members
of that circle now take a very different view of the matter. They have been taught,
alike by experience and by example, to regard mathematics, so far at least as physical
enquiries are concerned, as a mere auxiliary to thought.... This is one of the great truths
which were enforced by Faraday's splendid career.

And the consequence, in this country at least, has been that we find in the
majority of the higher class of physical text-books few except the absolutely
indispensable formulae. Take, for instance, that profound yet homely and unpreten-
tious work, Clerk Maxwell's Theory of Heat. Even his great work, Electricity, though
it seems to bristle with formulae, contains but few which are altogether unnecessary.
Compare it, in this respect, with the best of more recent works on the same advanced
portions of the subject.

In M. Poincard's work, however, all this is changed. Over and over again, in the
frankest manner (see, for instance, pp. xvi, 176), he confesses that he lays himself open
to the charge of introducing unnecessary mathematics: and there are many other
places where, probably thinking such a confession would be too palpably superfluous,
he does not feel constrained to make it.

T. 35

274 PETER GUTHRIE TAIT

M. Poincare' not only ranks very high indeed among pure mathematicians but has
done much excellent and singularly original work in applied mathematics : all the more
therefore should he be warned to bear in mind the words of Shakespeare :

"Oh! it is excellent

To have a giant's strength ; but it is tyrannous
To use it like a giant."

From the physical point of view, however, there is much more than this to be said.
For mathematical analysis, like arithmetic, should never be appealed to in a physical
enquiry till unaided thought has done its utmost. Then, and not till then, is the
investigator in a position rightly to embody his difficulty in the language of symbols,
with a clear understanding of what is demanded from their potent assistance. The
violation of this rule is very frequent in M. Poincare's work, and is one main cause of its
quite unnecessary bulk. Solutions of important problems, which are avowedly imperfect
because based on untenable hypotheses (see, for instance, §§ 284 — 286), are not useful to a
student, even as a warning : they are much more likely to create confusion, especially
when a complete solution, based upon full experimental data and careful thought, can
be immediately afterwards placed before him. If something is really desired, in
addition to the complete solution of any problem, the proper course is to prefix to the
complete treatment one or more exact solutions of simple cases. This course is almost
certain to be useful to the student. The whole of M. Poincare's work savours of the
consciousness of mathematical power : and exhibits a lavish, almost a reckless, use of it.

One test of the soundness of an author, writing on Thermodynamics, is his
treatment of temperature, and his introduction of absolute temperature. M. Poincare
gets over this part of his work very expeditiously. In §§ 15 — 17 temperature, t, is
conventionally defined as in the Centigrade thermometer by means of the volume
of a given quantity of mercury ; or by any continuous function of that volume
which increases along with it. Next (§ 22) absolute temperature, T, is defined,
provisionally and with a caution, as 273 + t; from the (so-called) laws of Marriotte
and Gay-Lussac. Then, finally (§ 118), absolute temperature is virtually defined afresh
as the reciprocal of Carnot's function. (We say virtually, as we use the term in the
sense defined by Thomson. M. Poincare"'s Fonction de Carnot is a different thing.)
But there seems to be no hint given as to the results of experiments made expressly
to compare these two definitions. Nothing, for instance, in this connection at all
events, is said about the long-continued early experimental work of Joule and Thomson,
which justified them in basing the measurement of absolute temperature on Carnot's
function.

In saying this, however, we must most explicitly disclaim any intention of
charging M. Poincare" with even a trace of that sometimes merely invidious, sometimes
purely Chauvinistic, spirit which has done so much to embitter discussions of the
history of the subject. On the contrary, we consider that he gives far too little
prominence to the really extraordinary merits of his own countryman Sadi Carnot.
He writes not as a partisan but rather as one to whom the history of the subject is
a matter of all but complete indifference. So far, in fact, does he carry this that the
name of Mayer, which frequently occurs, seems to be spelled incorrectly on by far the

POINCARE'S "THERMODYNAMIQUE" 275

greater number of these occasions ! He makes, however, one very striking historical
statement (§ 95) :

"Clausius...lui donna le nom de Principe de Carnot, bien qu'il 1'eut e'noncd sans
avoir connaissance des travaux de Sadi Carnot."

Still, one naturally expects to find, in a Treatise such as this, some little allusion
at least to Thermodynamic Motivity ; to its waste, the Dissipation of Energy ; and
to the rest of those important early results of Sir W. Thomson, which have had such
immense influence on the development of the subject. We look in vain for any
mention of Rankine or of his Thermodynamic Function ; though we have enough,
and to spare, of it under its later alias of Entropy. The word dissipation does indeed
occur, for we are told in the Introduction that the Principe de Carnot is " la dissipation
de tentropie','

We find Bunsen and Mousson cited, with regard to the effect of pressure upon
melting points, almost before a word is said of James Thomson ; and, when that word
does come, it wholly fails to exhibit the real nature or value of the great advance he made.

Andrews again, a propos of the critical point, and his splendid work on the
isothermals of carbonic acid, comes in for the barest mention only after a long
discussion of those very curves, and of the equations suggested for them by Van der
Waals, Clausius, and Sarrau : though his work was the acknowledged origin of their
attempts.

The reason for all this is, as before hinted, that M. Poincar6 has, in this work,
chosen to play almost exclusively the part of the pure technical analyst ; instead of
that of the profound thinker, though he is perfectly competent to do that also when
he pleases. And, in his assumed capacity, he quite naturally looks with indifference,
if not with absolute contempt, on the work of the lowly experimenter. Yet, in strange
contradiction to this, and still more in contradiction to his ascription of the Conservation
of Energy to Mayer, he says of that principle: "personne n'ignore que c'est un fait
experimental."

But the most unsatisfactory part of the whole work is, it seems to us, the entire
ignoration of the true (i.e. the statistical) basis of the second Law of Thermodynamics.
According to Clerk Maxwell (Nature, XVII, 278)

"The touch-stone of a treatise on Thermodynamics is what is called the
second law."

We need not quote the very clear statement which follows this, as it is probably
accessible to all our readers. It certainly has not much resemblance to what will be
found on the point in M. Poincare's work : so little, indeed, that if we were to judge
by these two writings alone it would appear that, with the exception of the portion
treated in the recent investigations of v. Helmholtz, the science had been retrograding,
certainly not advancing, for the last twenty years.

In his reply {Nature, March 3, 1892), Poincare' practically confined his
attention to the discussion of the Thomson Effect, offering fuller explanations
of his meaning. This, however, did not touch on Tail's chief objections
which were epitomised as follows (March 10, 1892) :

35— *

276 PETER GUTHRIE TAIT

1. The work is far too much a mere display of mathematical skill. It soars above
such trifles as historical details, while overlooking in great measure the experimental
bases of the theory ; and it leaves absolutely unnoticed some of the most important
branches of the subject.

(Thus, for instance, Sadi Carnot gets far less than his due, Rankine is not alluded
to, and neither Thermodynamic Motivity nor the Dissipation of Energy is even
mentioned !)

2. It gives an altogether imperfect notion of the true foundation for the reckoning
of absolute temperature.

3. It completely ignores the real (i.e. the statistical) basis of the Second Law of
Thermodynamics.

On March 24, Poincar£ gave his reasons why, in the cases cited, he did
not discuss the questions in the way desired by Tail. A final letter from
Tait on April 7 ended the discussion. As was customary in such discussions,
Tait sent a proof of this letter to Kelvin. The proof has been preserved,
and is interesting inasmuch as it shows that Kelvin was in entire agreement
with Tait. In his first letter Poincair£ had referred to the distinction between
true and apparent electromotive forces ; and Tait replied :

It is necessary to add that I made no reference whatever to M. Poincar£'s
distinctions between " true " and " apparent " electromotive force ; — simply because
I regard these, along with many other celebrated terms such as " disgregation " etc.,
as mere empty names employed to conceal our present ignorance.

Kelvin underlined the words " regard these " and wrote in the margin
"So do I, K." Other annotations were even more definite in their expression,
and Kelvin's general acquiescence in the position taken by Tait was indicated
by the brief note appended to the sheet, "Netherhall, Mar. 28/92 OT OK K."

Tait's review of A. McAulay's Utility of Quaternions in Physics
appeared in Nature, Dec. 28, 1893 (Vol. XLIX) under the title "Quaternions
as an Instrument in Physical Research." The following are some of the
characteristic passages :

Just as "one shove of the bayonet" was truly said to be more effective than any
number of learned discussions on the art of war: this really practical work, giving
genuine quaternion solutions of new problems as well as largely extended developments
of old ones, is of incomparably greater interest and usefulness than the recently
renewed, but necessarily futile, attempts to prove that a unit vector cannot possibly be
a quadrantal versor....

Here, at last, we exclaim, is a man who has caught the full spirit of the quaternion
system : " the real aestus, the awen of the Welsh bards, the divinus afflatus that
transports the poet beyond the limits of sublunary things !"... Intuitively recognising

McAULAY'S "UTILITY OF QUATERNIONS" 277

its power, he snatches up the magnificent weapon which Hamilton tenders to all, and
at once dashes off to the jungle on the quest of big game. Others, more cautious or
perhaps more captious, meanwhile sit pondering gravely on the fancied imperfections
of the arm ; and endeavour to convince a bewildered public (if they cannot convince
themselves) that, like the Highlander's musket, it requires to be treated to a brand-new
stock, lock, and barrel, of their own devising, before it can be safely regarded as fit
for service. "Non his juventas orta parentibus...." What could be looked for from
the pupils of a school like that?

Mr McAulay himself has introduced one or two rather startling innovations. But
he retains intact all the exquisitely designed Hamiltonian machinery, while sedulously
oiling it, and here and there substituting a rolling for a sliding contact, or introducing
a lignum vitae bearing....

The " startling innovations," however, as we called them above, are unquestionably
Mr McAulay's own — and he has certainly gone far to justify their introduction. He
has employed the sure tests of ready applicability and extreme utility, and these have
been well borne. Objections based upon mere unwontedness or even awkwardness
of appearance must of course yield when such important advantages as these (if they
be otherwise unattainable) are secured ; but it certainly requires a considerable
mental wrench to accustom ourselves to the use of

as an equivalent for the familiar expression

dX

dx'

If this be conceded, however, it is virtually all that Mr McAulay demands of
us, and we are free to adopt his system ....... A single example, of a very simple

character, must suffice. Thus in the strain of a homogeneous isotropic solid, due to
external potential u, we have for the strain-function <f> (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

<£V + V« = o,

which is the vector equation required. Here it is obvious that, in the usual order of
writing,

This simple example shows the nature of the gain which Mr McAulay's method

278 PETER GUTHRIE TAIT

secures. Those who wish to know its extent must read the work itself. They will
soon be introduced to novel forms of concentrated operators, with regard to which,
as I have not yet formed a very definite opinion, I shall content myself by hazarding
the remark that, while they are certainly powerful and eminently useful, they must
at present be regarded as singularly uncouth.

Some of Tail's contributions to Nature were signed G. H., i.e. Guthrie
Headstone = Guthrie T£te Peter. These were always short notes, at times
quizzical, bearing on some passing interest. One of serious import was the
reference to the Tay Bridge disaster of December 28, 1879. The letter to
Nature (July 22, 1880) is worth quoting from.

There are two interesting scientific questions, apart from engineering proper, which
are suggested by the late enquiry, although no reference seems to have been made
to them in the reports.

The first is the origin of the extraordinary flash seen at the moment of the
downfall of the bridge by many spectators several miles away. It is scarcely doubtful
that an impact was the only possible cause.

The second is the important question of the amount of wind-pressure which would
suffice to force a train bodily off from the top of the bridge

The flash seems to prove that the train had been blown off the rails, and had come
into violent contact with the sides of the high girders. Then, and not sooner, the piers
were subjected to a strain they were unable to bear.

Tail's strong objeclion lo acling as a scienlific witness prevented him
from appearing at the Enquiry, but this in no way interfered with his making
suggestions lo his old school friend Allan D. Slewart, M.Inst.C.E., who
had been consulled by ihe Engineer, Sir Thomas Bouch, in regard to the
design of the continuous girders, though nol of ihe columns whose failure
caused ihe calaslrophe. Several years later Tail's allenlion was called by
his son (ihen an Engineering Apprenlice)1 to an official diagram showing
how the engine, carriages, etc. and the girders were found relalive lo ihe
proper cenlre line of ihe bridge, and he agreed lhal ihe upselling or derailing
of a second-class carriage near ihe rear of ihe train was ihe immediale cause
of ihe collapse of ihe bridge. This view seemed lo be subslanliated by ihe
marks found on the girders, the smashing of ihe second-class carriage, and
ihe dislance separaling ihis carriage from ihe fronl portion of ihe train.

In Oclober 1873 ihere appeared in ihe British Quarterly Review
a irenchanl crilicism of Herbert Spencer's First Principles (Second Edilion).

1 W. A. Tail, of Leslie and Reid, Engineers to the Edinburgh and District Water Trust.

MOULTON ON SPENCER'S "FIRST PRINCIPLES" 279

The writer of the Review was the Senior Wrangler of 1 868, who is now the
Right Honourable Sir John Fletcher Moulton, one of the Lord Justices of
England. His strictures were of course replied to by Spencer, who, as is
usual in such controversies, passed by without remark the most condemnatory
parts. It is interesting to note, however, that in the next edition of the First
Principles readers will look in vain for certain of the most striking of the
quotations introduced by the Reviewer in his exposure of Spencer's
ignorance of physical principles. These sentences were silently removed
without a word of explanation, presumably on the principle elsewhere
enunciated by Spencer when in reply to other criticisms he remarked1
" Though still regarding the statement I had actually made as valid, I
concluded it would be best to remove the stumbling block out of the way of
future readers."

At the close of his " Replies to Criticisms " in the Fortnightly Review,
Herbert Spencer endeavoured to turn the keen edge of the British Quarterly
Reviewer's damaging attack; and in February 1874 this portion with
additions was issued as a pamphlet entitled " Mr Herbert Spencer and the
British Quarterly Review." To anyone acquainted with physical principles
the reply is a revelation of how a man with a marvellous power of assimilating
knowledge and a unique gift of exposition can fail in understanding not
merely the criticism but also the facts on which the criticism is based.
For example Herbert Spencer's conception of what is meant by the adiabatic
condensation and rarefaction of air through which sound is passing is not
merely crude and incomplete — it is profoundly erroneous.

One of Spencer's main positions was that "our cognition of the Per-
sistence of Force is a priori," against which the Reviewer quoted from Tail's
Thermodynamics to the effect that

" Natural Philosophy is an experimental, and not an intuitive science. No a priori
reasoning can conduct us demonstratively to a single physical truth."

Spencer believed he found a discrepancy between this statement and the
following sentence from Thomson and Tail's Treatise on Natural Philosophy :

"As we shall show in our chapter on Experience, physical axioms are axiomatic
to those only who have sufficient knowledge of the action of physical causes to enable
them to see at once their necessary truth."

1 See Appendix to First Principles dealing with Criticisms, p. 586, issued as a separate
pamphlet in July 1880.

28o PETER GUTHRIE TAIT

From this he argued that these physical axioms were & priori and main-
tained that Newton's Laws of Motion were also in this sense axiomatic.
He wrote (page 316 in Replies to Criticisms):

" Not a little remarkable, indeed, is the oversight made by Professor Tait, in
asserting that 'no d priori reasoning can conduct us demonstratively to a single
physical truth,' when he has before him the fact that the system of physical truths
constituting Newton's Principia, which he has joined1 Sir William Thomson in editing,
is established by & priori reasoning."

Unfortunately for Herbert Spencer's argument his quotation from Thomson
and Tail's Treatise was incomplete, and the British Quarterly Reviewer
shattered the support, which Spencer imagined he had found in the sentence
quoted, by simply continuing the quotation. Referring to the passage quoted
by Spencer, the Reviewer (in a note to The British Quarterly, January,
1874, pp. 215-8) remarked:

" Had Mr Spencer, however, read the sentence that follows it, we doubt whether
we should have heard aught of this quotation. It is : ' Without further remark we shall
give Newton's Three Laws; it being remembered that, as the properties of matter
might have been such as to render a totally different set of laws axiomatic, these laws
must be considered as resting on convictions drawn from observation and experiment,
not on intuitive perception.' This not only shows that the term 'axiomatic' is used
in the previous sentence in a sense that does not exclude an inductive origin, but it
leaves us indebted to Mr Spencer for the discovery of the clearest and most authoritative
expression of disapproval of his views respecting the nature of the Laws of Motion."

This awkward accusation of ignorance of ipsissima verba of the authors he
was quoting Spencer did not condescend to answer. Deprived of their
support, he turned his battery of words upon the position taken by Thomson
and Tait, and proceeded to propound this dilemma :

Consider, he says, what is implied by framing the thought that " the properties of
matter might have been such as to render a totally different set of laws axiomatic"...
Does it express an experimentally ascertained truth? If so, I invite Professor Tait
to describe the experiments. Is it an intuition? If so, then along with doubt of an
intuitive belief concerning things as they are, there goes confidence in an intuitive
belief concerning things as they are not. Is it an hypothesis? If so, the implication is
that a cognition of which the negation is inconceivable (for an axiom is such) may
be discredited by inference from that which is not a cognition at all, but simply a
supposition.

1 This is inaccurate : it was Professor Blackburn who was joined with Thomson in editing
the Principia.

TAIT'S FIRST REPLY TO SPENCER 281

This argument cleverly evades the real question by attacking a some-
what infelicitous way of stating the important principle that, apart from
experiment and observation, the human mind cannot formulate the laws of
nature. In its assumption of the meaning of the word axiom, it disregards
the obvious meaning of Thomson and Tail's statements in the very paragraph
quoted ("T and TV § 243). Spencer seemed to have recognised this, for
he no longer contended that Thomson and Tait gave any support to his
view; but he still continued to assert that Newton must himself have regarded
the Laws of Motion as a priori principles (Replies to Criticisms, p. 326).
In response to Spencer's challenge Tait now entered the field and penned
the following short letter to Nature, March 26, 1874.

HERBERT SPENCER -versus THOMSON AND TAIT.

A friend has lent me a copy of a pamphlet recently published by Mr Herbert
Spencer, in which certain statements of mine are most unsparingly dealt with,
especially in the way of attempted contrast with others made by Sir W. Thomson
and myself. I am too busy at the present season to do more than request you to
reprint one of the passages objected to (leaving it to your readers to divine to what
possible objections it is open), and to illustrate by a brief record of my college days
something closely akin to the mental attitude of the objector.

" Natural Philosophy is an experimental, and not an intuitive science. No a
priori reasoning can conduct us demonstratively to a single physical truth " (Tait,
Thermodynamics, p. i).

One of my most intimate friends in Cambridge, who had been an ardent
disciple of the late Sir W. Hamilton, Bart, and had adopted the preposterous notions
about mathematics inculcated by that master, was consequently in great danger of
being plucked. His college tutor took much interest in him, and for a long time
gave him private instruction in elementary algebra in addition to the college lectures.
After hard labour on the part of each, some success seemed to have been obtained,
as my friend had at last for once been enabled to follow the steps of the solution
of a question involving a simple equation. A flush of joy mantled his cheek, he
felt his degree assured, and he warmly thanked his devoted instructor. Alas, this
happy phase had but a brief duration ; my friend's early mental bias too soon
recovered its sway, and he cried in an agony of doubt and despair, " But what if x
should turn out, after all, not to be the unknown quantity?"

Compare this with the following extract from Mr Spencer's pamphlet:

"... if I examine the nature of this proposition that 'the properties of matter
might have been ' other than they are. Does it express an experimentally-ascertained
truth? If so, I invite Prof. Tait to describe the experiments!"

P. G. TAIT.

In his reply published in Nature, April 2, Spencer complained that
T. 36

282 PETER GUTHRIE TAIT

Tait had torn a sentence from its context (no doubt in imitation of Spencer's
own method of quotation), and maintained that the unknown quantity was
the application of Tail's story. It should be remembered of course that
Spencer had been arguing in his First Principles and in his Replies to
Criticisms that Newton's Laws of Motion were known it priori, whereas
Tait regarded the Properties of Matter, including the Laws of Motion, as
unknown until they were discovered by the legitimate methods of experience,
and by these alone. (See also Tail's reply, p. 285 below.)

This passage at arms excited a great deal of interest among the students
of the Physical Laboratory. One of our number was R. B. Haldane1,
second to none in knowledge of philosophy and in power of debate. We
fought the Spencer-Tait controversy over and over again. I remember that
W. K. Clifford visited Edinburgh about that time, and in the Tea Room at
one of the April Meetings of the Royal Society of Edinburgh much lively
talk went on regarding the controversy. Tait was in great spirits and said
to Clifford, " There is not a man in England I suppose, other than Herbert
Spencer, who does not see the point of my story." Clifford responded
with a hearty laugh, "No doubt, it is a very good story."

In the same reply of April 2, Spencer practically reproduced the argument
as given in his pamphlet, and made the remark that Tait himself,

" by saying of physical axioms that the appropriately-cultivated intelligence sees
at once their necessary truth, tacitly classes them with mathematical axioms, of which
this self-evidence is also the recognised character."

Writing to the same number of Nature the British Quarterly Reviewer
disposed of Spencer's claim that he knew what Newton thought by quoting
from two letters in which Newton wrote to Cotes in these words :

" In experimental philosophy it (i.e. hypothesis) is not to be taken in so large a
sense as to include the first Principles or Axiomes which I call the Laws of Motion.
These Principles are deduced from phenomena and made general by Induction,
%vhich is the highest evidence that a Proposition can have in this Philosophy"...

" On Saturday last I wrote you representing that Experimental philosophy
proceeds only upon phenomena and deduces general Propositions from them only by
Induction. And such is the proof of mutual attraction. And the arguments for
the impenetrability, mobility, and force of all bodies, and for the laws of motion are
no better."

On April 1 6, Herbert Spencer so far admitted his imperfect knowledge,
and withdrew his contention that Newton regarded the Laws of Motion as
axioms in the limited sense for which he had been all along arguing ; but he

1 Secretary of State for War since 1905.

SPENCER'S DEFENCE 283

went on to maintain that, in the sense in which he understood it, d priori
intuition preceded experimental verification.

He illustrated his meaning by deducing the second law of motion from
the d priori assumption that definite quantitative relations exist between cause
and effect. He evidently thought that " definite quantitative relations" meant
proportionality1. But whatever final meaning Spencer attached to the phrase
"& priori intuition," there was no getting away from the obvious meaning of
one of the passages quoted by the British Quarterly Reviewer : —

" Deeper than demonstration — deeper even than definite cognition — deep as the
very nature of mind is the postulate at which we have arrived (i.e. the Persistence
of Force). Its authority transcends all other whatever; for not only is it given in
the constitution of our own consciousness, but it is impossible to imagine a con-
sciousness so constituted as not to give it " (First Principles, p. 192).

This was one of the stumbling blocks which Spencer out of consideration
for the future reader removed from his later editions.

In reply to the criticism that the phrase Persistence of Force was used in
various quite distinct senses, Spencer remarked (Replies to Criticisms, p. 311)
that had " he (the Reviewer) not been in so great a hurry to find incon-
sistencies, he would have seen why, for the purposes of my argument, I
intentionally use the word Force : Force being the generic word, including
both that species known as Energy, and that species by which Matter
occupies space and maintains its integrity."

This recalls Maxwell's metrical Report on Tait's Lecture on Force.

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.

But those whose statements baffle all attacks,

Safe by evasion, —
Whose definitions, like a nose of wax,

Suit each occasion, etc.

(See above, p. 254.)

1 E.g., there is a definite quantitative relation between speed of projection and height reached,
between strength of electric current and heat generated ; but there is no simple proportionality.
The whole discussion in Nature (Vols. ix and x) is well worth reading. See especially the
British Quarterly Reviewer's letters on April 2, April 16, and June n. It is not an exaggeration
to say that in each succeeding letter Spencer takes up a different position, having been driven
from one after another of his fancied strongholds. The discussion was important as showing
to what extent Herbert Spencer's First Principles could be relied on as an exposition of physical
fact and theory.

36—2

284 PETER GUTHRIE TAIT

Towards the close of his letter to Tait of date 27 August, 1874, repro-
duced in full in the Appendix to Chapter IV, pp. 171-5 above, Maxwell
indulged in some exquisite fooling, in which Spencer's utterances are
humorously parodied.

Another sly hit at the synthetic philosopher was given on a post card of
date July 27, 1876, when Maxwell asked Tait

" Have you (read) Willard Gibbs on Equilibrium of Heterogeneous Substances ?
If not, read him. Refreshing after H. Spencer on the Instability of the Homo-
geneous."

In Nature, July 17, 1879, Tait reviewed Sir Edmund Becket's book On
the Origin of the Laws of Nature. It opened with these words :

This is a very clever little book and deserves to be widely read. Its subject,
however, is scarcely one for our columns. For it is essentially " apologetic," and its
strong point is not so much accurate science as keen and searching logic. It
dissects with merciless rigour some of the more sweeping assertions of the modern
materialistic schools, reducing them (when that is possible) to plain English so as to
make patent their shallow assumptions.... He follows out in fact, in his own way, the
hint given by a great mathematician (Kirkman) who made the following exquisite
translation of a well-known definition : —

"Evolution is a change from an indefinite, incoherent, homogeneity, to a definite,
coherent, heterogeneity, through continuous differentiations and integrations."

(Translation into plain English.)

"Evolution is a change from a no-howish, untalkaboutable, all-alikeness, to a
some-howish and in-general-talkaboutable not-all-alikeness by continuous something-
elsifications and sticktogetherations."

Some quotations were then given of the method of Sir Edmund Becket
in dealing with certain modes of argument, and Tait concluded

When the purposely vague statements of the materialists and agnostics are thus
stripped of the tinsel of high-flown and unintelligible language, the eyes of the
thoughtless who have accepted them on authority (!) are at last opened, and they
are ready to exclaim with Titania

" Methinks I was enamoured of an ass."

As the touch of Ithuriel's spear at once happily revealed the deceiver, these
frank and clear exposures of the pretensions of pseudo-science cannot fail of
producing great ultimate good.

In his appendix to First Principles, dealing with Criticisms, Spencer
replied at considerable length to the criticism which seemed to be implied in
these quotations and statements. He pointed out (p. 566) that a " formula
expressing all orders of change in their general course... could not possibly

TAIT'S SECOND REPLY 285

be framed in any other than words of the highest abstractness " ; and by way
of a general enquiry into mental idiosyncrasies proceeded to put together in
one group the two mathematicians Kirkman and Tail and two literary
men, a North American Reviewer and Matthew Arnold. We are told
(p. 570) that

" men of letters, dieted in their early days on grammars and lexicons and in
their later days occupied with belles lettres, Biography and a History made up
mainly of personalities, are by their education and course of life left almost without

scientific ideas of a definite kind The mathematician too and the mathematical

physicist, occupied exclusively with the phenomena of number space and time, or, in
dealing with forces, dealing with them in the abstract, carry on their researches in
such ways as may, and often do, leave them quite unconscious of the traits
exhibited by the general transformations which things, individually and in their
totality, undergo."

These exhibit "certain defects of judgment... to which the analytical
habit, unqualified by the synthetical habit, leads." Much of which seems to
be beside the mark. Tait certainly was not occupied exclusively with the
phenomena of number, space and time. His was a mind intuitively physical.
He was in experimental touch with things as they are in a way to which
Spencer was an absolute stranger. As for lack of scientific ideas of a definite
kind, no better examples can be found than in First Principles, especially
in the chapter on the Instability of the Homogeneous. Enough has been
given of Herbert Spencer's own words to enable us to appreciate Tail's
rejoinder. This was given as part of his opening lecture in 1880, and
appeared in Nature, Nov. 25, the same year; but it will suffice to reproduce
only those parts which are not a repetition of previous letters.

Mr Spencer has quite recently published a species of analytical enquiry into
my " mental peculiarities," " idiosyncrasies of thought," " habits of mind," " mental
traits," and what not. From his illustrative quotations it appears that some or all
of these are manifested wherever there are differences between myself and my critic
in the points of view from which we regard the elements of science. Hence they
are not properly personal questions at all, but questions specially fitted for discussion
here and now. I may, therefore, commence by enquiring what species of "mental
peculiarity" my critic himself exhibited when he seriously asked me whether I had
proved by experiment that a thing might have been what it is not ! !

The title of Mr Spencer's pamphlet informs us that it deals with Criticisms ;
and I am the first of the subjects brought up in it for vivisection, albeit I have been
guilty (on Mr Spencer's own showing) only of " tacitly " expressing an opinion ! Surely
my vivisector exhibits here also some kind of " mental peculiarity." Does a man

286 PETER GUTHRIE TAIT

become a critic because he quotes, with commendation if you like, a clever piece
of analysis or exposition published by another?

Mr Spencer complains that an American critic (whose estimate is "tacitly"
agreed in by Mr Matthew Arnold) says of the "Formula of Evolution": — "This
may be all true, but it seems at best rather the blank form for a universe than anything
corresponding to the actual world about us." On which I remark, with Mr Kirkman,
" Most just, and most merciful ! " But mark what Mr Spencer says :

" On which the comment may be that one who had studied celestial mechanics
as much as the reviewer has studied the general course of transformations, might
similarly have remarked that the formula — 'bodies attract one another directly as
their masses and inversely as the squares of their distances,' was at best but a
blank form for solar systems and sidereal clusters."

We now see why Mr Spencer calls his form of words a Formula, and why he
is indignant at its being called a Definition. He puts his Formula of Evolution
alongside of the Law of Gravitation ! Yet I think you will very easily see that it
is a definition, and nothing more. By the help of the Law of Gravitation (not very
accurately quoted by Mr Spencer) astronomers are enabled to predict the positions
of known celestial bodies four years beforehand, in the Nautical Almanac, with an
amount of exactness practically depending merely upon the accuracy of the observations
which are constantly being made : — and, with the same limitation, the prediction
could be made for 1900 A.D., or 2000 A.D., if necessary. If now Mr Spencer's form
of words be a formula, in the sense in which he uses the term as applied to the Law
of Gravitation, it ought to enable us to predict, say four years beforehand, the history
of Europe, with at least its main political and social changes ! For Mr Spencer says
that his "formula" expresses "all orders of changes in their general course, — astronomic,
geologic, biologic, psychologic, sociologic"; and therefore "could not possibly be framed
in any other than words of the highest abstractness."

Of Mr Spencer's further remarks there are but three which are directed specially
against myself. (Mr Kirkman is quite able to fight his own battles.) He finds evidence
of "idiosyncrasies" and what not, in the fact that, after proclaiming that nothing
could be known about the physical world except by observation and experiment,
I yet took part in writing the " Unseen Universe " ; in which arguments as to the
Unseen are based upon supposed analogies with the seen. He says: — "clearly, the
relation between the seen and the unseen universes cannot be the subject of any
observation or experiment ; since, by the definition of it, one term of the relation
is absent." I do not know exactly what " mental peculiarity " Mr Spencer exhibits
in this statement. But it is a curious one. Am not I, the thinker, a part of the
Unseen ; no object of sense to myself or to others ; and is not that term of relationship
between the seen and the Unseen always present ? But besides this, Mr Spencer
mistakes the object of the book in question. The theory there developed was not
put forward as probable, its purpose was attained when it was shown to be conceivable
and not inconsistent with any part of our present knowledge.

Mr Spencer's second fault-finding is a propos of a Review of Thomson and Tarts
Nat. Phil. (Nature, July 3, 1879) by Clerk Maxwell. Maxwell, knowing of course
perfectly well that the authors were literally quoting Newton, and that they had

DEFINITIONS OF FORCE 287

expressly said so, jocularly remarked " Is it a fact that ' matter ' has any power,
either innate or acquired, of resisting external influences ? " Mr Spencer says : —
"And to Prof. Clerk Maxwell's question thus put, the answer of one not having
a like mental peculiarity with Prof. Tait, must surely be — No." Mr Spencer, not
being aware that the passage is Newton's, and not recognising Maxwell's joke,
thinks that Maxwell is at variance with the authors of the book !

Finally, Mr Spencer attacks me for inconsistency etc. in my lecture on Force
(Nature, September 21, 1876). I do not know how often I may have to answer
the perfectly groundless charge of having, in that Lecture, given two incompatible
definitions of the same term. At any rate, as the subject is much more important
than my estimates of Mr Spencer's accuracy or than his estimates of my " mental
peculiarities," I may try to give him clear ideas about it, and to show him that there
is no inconsistency on the side of the mathematicians, however the idea of force
may have been muddled by the metaphysicians. For that purpose I shall avoid
all reference to " differentiations " and " integrations " ; either as they are known
to the mathematicians, or as they occur in Mr Spencer's " Formula." Of course
a single line would suffice, if the differential calculus were employed.

Take the very simplest case, a stone of mass M, and weight W, let fall. After
it has fallen through a height h, and has thus acquired a velocity v, the Conservation
of Energy gives the relation

MV-= Wh.

v*
Here both sides express real things; M— is the kinetic energy acquired, Wh the

work expended in producing it.

But if we choose to divide both sides of the equation by - (the average velocity
during the fall) we have (by a perfectly legitimate operation)

Mv= Wt,

where / is the time of falling. This is read : — the momentum acquired is the product
of the force into the time during which it has acted. Here, although the equation is
strictly correct, it is an equation between purely artificial or non-physical quantities,
each as unreal as is the product of a quart into an acre. It is often mathematically
convenient, but that is all. The introduction of these artificial quantities is, at least
largely, due to the strong (but wholly misleading) testimony of the "muscular" sense.

Each of these modes of expressing the same truth, of course gives its own
mode of measuring (and therefore of defining) force.

The second form of the equation gives

W- —

t •

Here, therefore, force appears as the time-rate at which momentum changes ; or, if
we please, as the time-rate at which momentum is produced by the force. In using
this latter phrase we adopt the convenient, and perfectly misleading, anthropomorphism
of the mathematicians. This is the gist of a part of Newton's second Law.

288 PETER GUTHRIE TAIT

The first form of the equation gives

so that the same force now appears as the space-rate at which kinetic energy changes ;
or, if we please, as the space-rate at which energy is produced by the force.

Here are some of Mr Spencer's comments: — "force is that which changes the
state of a body ; force is a rate, and a rate is a relation (as between time and distance,
interest and capital) ; therefore a relation changes the state of a body."

The contradiction which Mr Spencer detects here, and over which he waxes
eloquent and defiant, exists in his own mind only. The anthropomorphism which
has misled him is but a convenient and harmless relic of the old erroneous interpreta-
tions of the impressions of sense.

In his reply (Nature, Dec. 2, 1880) Herbert Spencer reproduced a good
deal of his pamphlet, got somewhat indignant over a side issue, and sneered
at Tail's mathematics. "If," he remarked, "his mathematics prove that
while force is an agent which does work, it is also the rate at which an
agent does work, then I say — so much the worse for mathematics." In a
brief rejoinder (Nature, Dec. 9, 1880) Tait wrote:

Mr Spencer has employed an old remark of Prof. Huxley as to what mathematics
can, and cannot, do ; but he has not employed it happily, for the question at issue is
really this : is it correct to speak, at one time, of force as an agent which changes
a body's state of rest or of motion, and again to speak of it as the time-rate at which
momentum changes or as the space-rate at which energy is transformed ?

I answer that there is not the slightest inconvenience here ; except, perhaps, in the
eyes of those metaphysicians (if there be any) who fancy they know what force is.
Such phrases as "the wind blows" or "the sun rises" though used by the most
accurate even of scientific writers, would otherwise (on account of their anthropomorphism)
have to be regarded as absolute nonsense.

Here the controversy ended, for Spencer's later letter of Dec. 16 took no
notice of the scientific questions which were supposed to be the subjects of a
discussion which he himself had originated. Tait regarded the controversy
as a joke ; for he knew it was hopeless to convince of any errancy a mind
which believed that the greatest physical generalization of modern times could
be established as an a priori intuition.

OBITUARY NOTICE OF BALFOUR STEWART 289

DR BALFOUR STEWART, F.R.S.
(From Nature, Dec. 29, 1887.)

In the genial Manchester Professor the scientific world has lost not only an
excellent teacher of physics but one of its ablest and most original investigators.
He was trained according to the best methods of the last generation of experimen-
talists, in which scrupulous accuracy was constantly associated with genuine scientific
honesty. Men such as he was are never numerous ; but they are the true leaders
of scientific progress : directly, by their own contributions ; indirectly, though (with
rare exceptions) even more substantially, by handing on to their students the choicest
traditions of a past age, mellowed by time and enriched from the experience of the
present. The name of Stewart will long be remembered for more than one striking
addition to our knowledge, but his patient and reverent spirit will continue to impress
for good the minds and the work of all who have come under its influence.

He was born in Edinburgh, on November ist, 1828, so that he had entered
his sixtieth year. He studied for a short time in each of the Universities of St
Andrews and Edinburgh, and began practical life in a mercantile office. In the
course of a business voyage to Australia his particular taste for physical science
developed itself, and his first published papers : " On the adaptation of the eye to
different rays," and " On the influence of gravity on the physical condition of the
Moon's surface," appeared in the Transactions of the Physical Society of Victoria
in 1855. On his return he gave up business for science, and resumed study under
Kelland and Forbes, to the latter of whom he soon became Assistant. In this
capacity he had much to do with the teaching of Natural Philosophy on occasions
when Forbes was temporarily disabled by his broken health. During this period, in
1858, Stewart was led to his well-known extension of PreVost's Law of Exchanges,
a most remarkable and important contribution to the theory of Radiation. He
seems to have been the first even to suggest, from a scientific stand-point, that
radiation is not a mere surface phenomenon. With the aid of Forbes' apparatus,
then perhaps unequalled in any British University, he fully demonstrated the truth
of the conclusions to which he had been led by theory ; and the award of the
Rumford Medal by the Royal Society, some years later, showed that his work had
been estimated at its true value, at least in the scientific world. In fact his proof
of the necessary equality between the radiating and the absorbing powers of every
substance (when divested of some of the unnecessary excrescences which often mask
the real merit of the earlier writings of a young author) remains to this day the
simplest, and therefore the most convincing, that has yet been given.

Radiant Heat was, justly, one of Professor Forbes' pet subjects, and was there-
fore brought very prominently before his Assistant. Another was Meteorology, and
to this Stewart devoted himself with such enthusiasm and success that in 1859 he
was appointed Director of the Kew Observatory. How, for eleven years, he there
maintained and improved upon the memorable labours of Ronalds and Welsh needs

T- 37

29o PETER GUTHRIE TAIT

only to be mentioned here : it will be found in detail in the Reports of the British
Association. Every species of inquiry which had to be carried out at Kew : whether
it consisted in the testing of Thermometers, Sextants, Pendulums, Aneroids, or
Dipping-Needles, the recording of Atmospheric Electricity, the determination of the
Freezing-Point of Mercury or the Melting-Point of Paraffin, or the careful study of
the peculiarities of the Air-Thermometer: received the benefit of his valuable sug-
gestions and was carried out with his scrupulous accuracy.

About twenty years ago Stewart met with a frightful railway accident, from
the effects of which he did not fully recover. He was permanently lamed, and
sustained severe injury to his constitution. From the vigorous activity of the prime
of life he passed, in a few months, to grey-headed old age. But his characteristic
patience was unruffled, and his intellect unimpaired.

His career as Professor of Physics in the Owens College has been, since his
appointment in 1870, brilliantly successful. It has led to the production of an
excellent treatise on Practical Physics, in which every necessary detail is given
with masterly precision, and which contains (what is even more valuable, and could
only have been secured to the world by such a publication) the matured convictions
of a thorough experimenter as to the choice of methods for the attack of each
special Problem.

His Elementary Physics, and his Conservation of Energy, are popular works on
physics rather than scientific treatises : but his Treatise on Heat is one of the best
in any language, a thoroughly scientific work, specially characteristic of the bent
of mind of its Author.

Stewart published, in addition to his Kew Reports, a very large number of
scientific memoirs and short papers. Many of these (notably the article in the
Encyc. Brit., Qth ed.) deal with Terrestrial Magnetism, in itself as well as in its
relations to the Aurora and to solar disturbances. A valuable series of papers,
partly his own, partly written in conjunction with De la Rue and Loewy, deals
with Solar Physics. His paper on the "Occurrence of Flint Implements in the Drift"
(Phil. Mag. I, 1862) seems to have been ignored by the "advanced" geologists,
one of whose pet theories it tends to dethrone ; and to have been noticed only by
physicists, especially Sir W. Thomson, whose beautiful experiments have done so
much to confirm it. His paper on " Internal Radiation in Uniaxal Crystals," to which
Stokes alone seems to have paid any attention, shows what Stewart might have
done in Mathematical Physics, had he further developed the genuine mathematical
power which he exhibited while a student of Kelland's.

I made Stewart's acquaintance in 1861, when he was the first-appointed Ad-
ditional Examiner in Mathematics in the University of Edinburgh, a post which
he filled with great distinction for five years. A number of tentative investigations
ultimately based upon our ideas as to possible viscosity of the luminiferous medium,
effect of gravitation-potential, on the physical properties of matter, etc., led to the
publication of papers on " Rotation of a disc in vacuo, Observations with a rigid
spectroscope, Solar spots and planetary configurations," etc. These, as well as our
joint work called The Unseen Universe, have been very differently estimated by
different classes of critics. Of course I cannot myself discuss their value. There is,

OBITUARY NOTICE OF ROBERTSON SMITH 291

however, one of these speculations, so closely connected with Stewart's Radiation
work as to require particular mention, especially as it seems not yet to have
received proper consideration, viz. " Equilibrium of Temperature in an enclosure con-
taining matter in visible motion" (Nature, IV. p. 331, 1871). The speculations are all
of a somewhat transcendental character, and therefore very hard to reduce to forms
in which they can be experimentally tested ; but there can be no doubt that
Stewart had the full conviction that there is in them all an underlying reality, the
discovery of whose exact nature would at once largely increase our knowledge.

Of the man himself I cannot trust myself to speak. What I could say will
easily be divined by those who knew him intimately ; and to those who did not
know him I am unwilling to speak in terms which, to them, would certainly appear
exaggerated.

PROFESSOR ROBERTSON SMITH.
(From Nature, April 12, I894.)1

The death of Prof. Robertson Smith, on March 31, at a comparatively early
age, is a profound loss to the whole thinking world.

Unfortunately for Science, and (in too many respects) for himself, his splendid
intellectual power was diverted, early in his career, from Physics and Mathematics,
in which he had given sure earnest of success. He turned his attention to eastern
languages, and acquired a knowledge of Hebrew, Arabic, and other tongues, quite
exceptional in the case of a Briton.

Dr Smith was born at Keig, Aberdeenshire, in 1846, and educated at Aberdeen
University, the New College, Edinburgh, and the Universities of Bonn and Got-
tingen. In 1868 he became Assistant to the Professor of Physics in Edinburgh
University ; in 1 870, at the age of twenty-four, he was appointed to the chair of
Hebrew in the Free Church College of Aberdeen. A few years later he fell under
the suspicion of holding heterodox views concerning Biblical history. Orthodoxy
raised her voice against him in the newspapers, in the churches, in the Presbyteries,
and finally, in the General Assembly of the Free Church of Scotland, and the
clamour culminated in his dismissal from the Professorship at Aberdeen in 1881.
This was effected, not by a direct condemnation of his published opinions, but by
a monstrous (temporary) alliance between ignorant fanaticism and cultivated Jesuitry
which deplored the " unsettling tendency " of his articles !

He next became successor to Prof. Baynes in the Editorship of the last
edition of the Encyclopaedia Britannica ; and here his business qualities, as well

1 The Article is unsigned; but Tail initialled it as his in his own copy of Nature. It
seems necessary to mention this in view of the somewhat "hearsay" language of the sixth
paragraph. The strong language at the end of the third paragraph embodies Tail's often
expressed views of the great " Heresy Hunt."

37—2

292 PETER GUTHRIE TAIT

as his extraordinary range of learning, came prominently before the world. In
1883 Dr Smith was appointed Reader in Arabic at Cambridge, and three years
later he succeeded the late Mr Bradshaw as librarian to the University. He was
afterwards elected to a Fellowship at Christ's College, and to the Professorship of
Arabic.

What Smith might have done in science is shown by his masterly paper " On
the Flow of Electricity in Conducting Surfaces" (Proc. R. S. E., 1870), which was
rapidly written in the brief intervals of leisure afforded by his dual life as simul-
taneously a Student in the Free Church College, and Assistant to the Professor of
Natural Philosophy in Edinburgh University.

We understand that his engagement as Assistant to Prof. Tait had its origin
in the extremely remarkable appearance made by young Smith as a Candidate in
the Examination for the Ferguson Scholarships, an examination in which most of
the very best men in the four Scottish Universities are annually pitted against one
another.

In Edinburgh University he did splendid service in the work of initiating the
Physical Laboratory : and there can be no doubt that the esprit de corps, and the
genuine enthusiasm for scientific investigation, which he was so influential in ex-
citing there, have inaugurated and promoted many a successful career (not in this
country alone, but in far regions everywhere), and that, near and far, his death will
be heard of with heart-felt sorrow.

A light and playful feature of his too few years of scientific work consisted in
his exposures of the hollowness of the pretensions of certain " philosophers," when
they ventured to tread on scientific ground. Several of these will be found in
the Proceedings and Transactions of the Royal Society of Edinburgh (1869-71).
Smith treats his antagonist " tenderly " as if he loved him, but the exposure is
none the less complete.

A writer in The Times thus testifies to Dr Smith's remarkable powers : " In him
there has passed away a man who possessed not only one of the most learned but
also one of the most brilliant and striking minds in either of the great English
Universities, and who was held in the highest regard by the leading orientalists of
the continent. His extraordinary range of knowledge, the swiftness and acuteness
of his intellect, and his passionate love of truth combined to make an almost unique
personality. His talents for mathematics and physical science were scarcely less
remarkable than those for linguistic studies, and if he had not preferred the latter,
there is no question that he could have reached great eminence in the former."

"RELIGION AND SCIENCE" 293

RELIGION AND SCIENCE.
(From The Scots Observer, Dec. 8, 1888.)

" For not by the rays of the sun, nor the glittering shafts of the day,
Must the fear of the gods be dispelled ; but by words and their wonderful play."

Thus said Clerk Maxwell, one of the genuine scientific men of the generation
just past, in ridicule of the assumptions and pretensions of the hydra-headed pseudo-
science. And he said rightly, for true science is not, and cannot be, at variance
with religion.

That there is a conflict between Religion and Science, not a mere difference
between certain theologians and certain "scientists," is very frequently stated as a
matter of fact. It is occasionally spoken of in grandiloquent phrase as the Incom-
patibility of Religion and Science ! Of course, if there be even a fragment of truth
in this, it is matter of very serious import indeed ; but it is prudent in all such
cases to ask the simple question, " On what authority is the statement made ? "
The answer to this will either make careful investigation imperative, or render it
altogether unnecessary. To this question, then, and to this alone, the present
article is devoted.

There are not very many people who really know what science is, though
there are many who think they do, and who even pose successfully, so far as the
public is concerned, as scientific men. This may seem a somewhat bold statement,
but it is capable of easy proof. Let us take an analogy or two to begin with.

1. A is a " medical man." What do you understand by such a phrase ? Does
it necessarily imply that, if you were seriously ill, you could with confidence submit
yourself to A's professional care? Are there no quacks on the Register? If you
think there are none, don't read any further. I am not addressing you. I look for
some common-sense in my readers.

2. B is a " military man." Does this imply that he is capable of conducting a
campaign, of leading a company, or even of putting an awkward squad through
their facings ? By no means, and you know it quite well. Yet you persist in calling
B a "military man," though he may be merely a re-incarnation of Bardolph, Pistol,
or Parolles !

3. C is a "business man." The term includes the trusty family lawyer, as well
as the bankrupt speculator !

4. D is a " scientific man." We've got to him at last. Well ! the term was
applied to Lord Brougham, and to Dionysius Lardner ; it is still applied to Isaac
Newton.

It was for this reason that, in speaking of Clerk Maxwell, I had to make a
slight but important qualification of the term, and to call him a genuine scientific
man. But in that italicised word there is a whole world of meaning. It opens an
abyss of impassable width among the group of so-called "scientific men" — leaving

294 PETER GUTHRIE TAIT

the vast majority hopelessly on the wrong side. There we will leave them for a
little, in order to discuss a collateral issue.

What is science? Whatever it is, it is certainly not mere knowledge, even of
facts. If I had Babbage's Logarithms by heart, as a school-boy has the multiplica-
tion table, I should know more than any other human being, besides being the
most wonderful calculator that ever existed ; but my knowledge would not be science.
No more, indeed, than it would be scholarship, if it consisted in my being able to
repeat the whole of Liddell and Scott. Every decently-educated Greek knew all
the contents of that huge volume (its errors excepted) two thousand years ago,
better than did these learned pundits. But it would never occur to the least dis-
cerning among us to speak of that Greek's "scholarship." These things are what,
now-a-days, we call cram. Descriptive botany, natural history, volumes of astro-
nomical observations, etc., are collections of statements, often facts, from which
scientific truth may ultimately be extracted, but they are not science. Science
begins to dawn, but only to dawn, when a Copernicus, and after him a Kepler or a
Galilei, sets to work on these raw materials, and sifts from them their essence.
She bursts into full daylight only when a Newton extracts the quintessence. There
has been, as yet, but one Newton ; there have not been very many Keplers. Thus the
great mass of what is commonly called science is totally undeserving of the name.
But, the name being given (though in error) to the crude collection of undigested
facts, the worker at them gets the name of " scientific man " from an undiscerning
public. And it is to be particularly observed that statements as to the so-called
incompatibility between religion and science come, all but exclusively, from this
class of persons. Not from all — not even from the majority of them ; but from
their ranks alone. How trustworthy is their judgment, as a general rule, may be
seen from some of their recent statements. Nothing like these has been heard since
" all, with one voice, about the space of two hours, cried out, Great is Artemis of the
Ephesians." Take this one — unique, it is to be hoped, in its absurdity — " When the
boldness of (Darwin's) generalisations, and the great school which he has founded,
are taken into account, it is perhaps no exaggeration to bracket him with Newton,
Kepler, and Tycho Brahe " ! So careful and conscientious an observer as Darwin
may justly be bracketed with Tycho, who was distinguished for these very qualities ;
but Tycho himself was no more than the mere hodman of Kepler. Should the
hypothesis of Wallace and Darwin turn out to be correct, its authors may perhaps
claim the position of Copernicus; otherwise they fall back to that of Ptolemy. The
mention of Newton in such a connection is only an ingenuous confession of inability
to comprehend the very nature of Newton's work. There cannot be a Newton in
natural science until there has been at least one Kepler ; and he has not yet
appeared, nor do we see much promise of his coming.

Let us revert to Maxwell's lines, — "Words, and their wonderful play." Has any
one, in a single phrase, ever more fully, or more fittingly, characterised at once the
one talent, and the abundant but temporary success, of the pseudo-scientific ? Roar
me in King Cambyses' vein, and you have the multitude at your feet. This is their
distinctive mark.

I have purposely made the above remarks in a disconnected manner, insidi-

"RELIGION AND SCIENCE" 295

ously working towards the enemy's goal. Hitherto we have had narrative and
description only. We will now, gentlemen, have a little logic, if you please, and
take the place-kick.

First. The so-called incompatibility of religion and science is proclaimed solely
from the ranks of those whose subject has not yet reached the scientific stage, and
from the ranks of pseudo-science.

Second. In both of these ranks, "words and their wonderful play" are the chief
weapons ; and they are employed with a pertinacity truly amazing.

Third. There has not, in all, been any very persistent or even vociferous
assertion of the so-called incompatibility.

From these it indubitably follows : —

Fourth. That, even among those whose subject has not yet reached the scien-
tific stage, and among the pseudo-scientific, there can be but few who maintain the
so-called incompatibility.

Fifth. As this tenet is held only by a small minority of those who are on the
wrong side of the abyss above mentioned, which separates the genuine scientific men
from the rest of the species, it might be entirely disregarded were it not for its
pernicious effect upon those who do not even pretend to be scientific.

Thus the really serious question is, " How can this pernicious effect be neutra-
lised or remedied ? " Not, certainly, by anathematising as infidels those who are
rather to be pitied for their ignorance. And, most certainly, not by pulpit denun-
ciations, in which the most transparently absurd dicta of pseudo-science are com-
placently used as weapons against adversaries armed with the very same — a
Kilkenny-cat display without claws or teeth on either side. If the clergy — upon
whom, more than on any others, this task ought to fall — would lay much more
stress than they have hitherto thought of doing upon the humility which is charac-
teristic of all true knowledge, whether it be religious or scientific, and upon the
blatant boasting equally characteristic of ignorance, they would be able easily to
convince their flocks that the present outcry is the work of a small minority only,
and has no countenance whatever from those who really know science.

CHAPTER VIII
POPULAR SCIENTIFIC ARTICLES

THUNDERSTORMS

Lecture in the City Hall, Glasgow, January 2^tk, 1880.

WHEN I was asked to give this lecture I was also asked to give a short list
of subjects from which your directors might select what they thought most fit.
I named three — one being of purely scientific importance, the others being of
practical importance as well. Regarded from the scientific point of view, one of
them was to be considered as fully understood in principle, and requiring only
additional experimental data to make it complete. This was the Conduction of Heat
in Solids. Another was to a certain extent scientifically understood, but its theory
was, and still is, in need of extended mathematical development. This was the
popular scientific toy, the Radiometer. The third was, and remains, scarcely under-
stood at all. Of course it was at once selected for to-night. I might have foreseen
that it would be. I had incautiously forgotten the nursery rule that a slice of
bread always falls with the buttered side down, and the stern conviction of sailors
that a marlinspike never drops from aloft except point foremost.

You may well ask, then, why I am here to-night. What can I say about a
subject which I assert to be scarcely understood at all ? To such a question there
are many answers, but the most satisfactory are supplied by analogy.

Would interesting and even scientifically attractive matter have failed a lecturer,
do you think, if he had chosen Astronomy for his subject, in days before Newton,
before Kepler, before even Copernicus? Yet he could certainly not have properly
described even the arrangement of the planets in the solar system, far less the laws
of their motions, and least of all the dynamical basis of these laws. Still, you will
grant that he might have given an admirable lecture. But stay : would he now be
any better off? After all that Copernicus, Kepler, Newton, and their successors
have done, do we yet scientifically understand why the planets move as they do ?
Certainly not. The mechanism of gravitation is still to us, as it was to Newton,
an absolute mystery. Only one even plausible attempt to explain it has yet been
made ; and that, in spite of Sir W. Thomson's very ingenious attempts to improve
it, we cannot yet venture to call probable. But, for all that, a lecturer on gravita-
tion has a magnificent field before him. Though he knows nothing of its mechanism

ELECTRICITY AND LIGHT 297

he knows its universal law, and he might profitably occupy your attention for many
evenings in tracing the consequences of that grand but simple statement of physical
fact. Think, for instance, of the many apparently altogether incompatible results to
which it leads : planetary perturbations, combined with stability of the solar system ;
precession and nutation of the earth's axis, yet permanence of the seasons ; and
constant tidal agitation of the sea, yet no consequent submersion of continents.
He might next trace the growth of stars, with their attendant planets, from an
original chaotic distribution of matter — calculate even the temperature which each
would acquire during its growth — all by the aid of this recognised law, of whose
own explanation he yet remains absolutely ignorant.

So it is with the splendid phenomenon of optics. No experimenter has yet, to
his certain knowledge, verified by direct proof the existence of that wonderful
elastic jelly which we know must fill all space without offering perceptible resistance
to bodies moving through it — that jelly whose inconceivably rapid quivering is the
mechanism by which not only do we see all objects, from the sun to the most
minute of stars, but by which comes to us continually from the sun that supply
of energy without which vegetable life, and with it that of animals, must at once
cease from the earth. Yet a lecturer could never be at a loss on matters connected
with light and colour, or with radiation in any of its varied forms. And all the
time he is consciously in total ignorance of the nature of this extraordinary lumini-
ferous ether upon which they all entirely depend.

A few years ago no qualified physicist would have ventured an opinion as to
the nature of electricity. Magnetism had been (to a certain extent, at least) cleared
up by an assumption that it depended on electric currents ; and from Orsted and
Ampere to Faraday and Thomson, a host of brilliant experimenters and mathematicians
had grouped together in mutual interdependence the various branches of electro-
dynamics. But still the fundamental question remained unsolved, What is Electricity ?
I remember Sir W. Thomson, eighteen years ago, saying to me, " Tell me what
electricity is, and I'll tell you everything else." Well, strange as it may appear to
you, I may now call upon him to fulfil his promise. And for good reason, as you
shall see.

Science and Scotland have lately lost in Clerk Maxwell one of their greatest
sons. He was, however, much better known to science than to Scotland. In
scientific ranks true merit is almost always certain to be recognised — in popular
ranks modest merit scarcely ever is. His was both true and exceptionally modest
merit. One grand object which he kept before him through his whole scientific
life was to reduce electric and magnetic phenomena to mere stresses and motions
of the ethereal jelly. And there can be little doubt that he has securely laid the
foundation of an electric theory — like the undulatory theory of light admirably
simple in its fundamental assumptions, but, like it, requiring for its full development
the utmost resources of mathematical analysis. It cannot but seem strange to the
majority of you to be told that we know probably as much about the secret
mechanism of electricity as we do about that of light, and that it is more than
exceedingly probable that a ray of light is propagated by electric and electro-
magnetic disturbances. It can be but a small minority of you who have been at

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298 PETER GUTHRIE TAIT

college so recently as to have been taught this. Yet, from the purely scientific
point of view, it is one of the most remarkable advances made during this century.

But to know what electricity is, in the same sense as we may be said to know
what light is, does not necessarily guide us in the least degree to a notion of its
source in any particular instance. What, for example, causes the luminosity of
fireflies, glowworms, etc., among natural objects, and of phosphorescent watch-dials
among artificial ones ? The answer is not yet ready, though it may soon come.
So we might know quite well what is electricity and yet be, as I told you at
starting, we are, almost entirely uncertain of the exact source of atmospheric electricity.

To come to my special subject. I am not going to try to describe a thunder-
storm. First, because I am certain that I could not do it without running the
risk of overdoing it, and thus becoming sensational instead of scientific ; and
secondly, because the phenomenon must be quite familiar, except perhaps in some
of its more singular details, to every one of you. From the artistic point of view
of the poets, who claim the monopoly of the expressions of wonder and awe, you
have descriptions without end. Who does not know at least the finest of them,
from that of Lucretius of old to that of Byron in modern times ?

But science has to deal with magnitudes which are very much larger or smaller
than those which such words as huge, enormous, tiny, or minute are capable of
expressing. And though an electric spark, even from our most powerful artificial
sources, appears to the non-scientific trifling in comparison with a mile-long flash of
lightning, the difference (huge, if you like to call it) is as nothing to others with
which scientific men are constantly dealing. The nearest star is as much farther
from us than is the sun, as the sun is farther from us than is London. The sun's
distance is ninety-three millions of miles. If that distance be called enormous, and
it certainly is so, what adjective have you for the star's distance? The particles of
steam are as much less than a drop of water as an orange is less than the whole
earth. How can you fitly characterise their smallness? Ordinary human language,
and specially the more poetic forms of it, were devised to fit human feelings and
emotions, and not for scientific purposes : for the Imagination, not for the Reason.
As there is a limit alike to pleasure and to pain, so there is to wonder and
astonishment ; and with words expressive of something near these limits ordinary
language rests content. A thoroughly scientific account of a thunderstorm, if it were
possible to give one, would certainly be at once ridiculed as pedantic.

Let us therefore, instead of attempting to discuss the phenomenon as a whole,
consider separately some of its more prominent features. And first of all, what are
these features when we are in the thunderstorm ?

By far the most striking, at least if the thunderstorm come on during the day,
is the extraordinary darkness. Sometimes at midday in summer the darkness
becomes comparable with that at midnight, and in another sense it much resembles
midnight darkness, for it is very different in kind as well as intensity from that
produced by the densest fog. Objects are distinctly visible through it at distances
of many miles, whether when self-luminous or when instantaneously lit up by
lightning. The darkness then is simply intense shadow, produced by the great
thickness and great lateral extension of the cloud-masses overhead This altogether

ATMOSPHERIC ELECTRICITY 299

unusual amount of cloud always forms a prominent feature in a great thunderstorm.
We have thus obtained one very important clue to the origin of the phenomenon.
In cases where the darkness is not so great, though the storm is visibly raging
overhead, it is always observed either that we are not far from the edge of its
area or, in rarer cases, that the lower cloud strata lie much higher than usual.
These form, therefore, no exception to the general conclusion just given as to the
abnormal amount of cloud present. Seen from a distance, the mass of cloud
belonging to the storm usually presents a most peculiar appearance, quite unlike any
other form of cloud. It seems to boil up, as it were, from below, and to extend
through miles of vertical height. The estimated height of its lower surface above the
ground varies within very wide limits. Saussure has seen it as much as three miles;
and in one case noticed by De 1'Isle it may have been as much as five miles. On
the other hand, at Pondicherry and Manilla it is scarcely ever more than half a
mile. Haidinger gives the full details of an extraordinary case, in which the thunder-
loud formed a stratum of only 25 feet thick, raised 30 yards above the ground.
Yet two people were killed on this occasion. Other notable instances of a similar
extreme character are recorded.

Careful experiment shows us that the air is scarcely ever free from electricity,
even in the clearest weather. And even on specially fine days, when large separate
cumuli are floating along, each as it comes near produces a marked effect on the
electrometer. Andrews obtained by means of a kite, on a fine clear day, a steady
decomposition of water by the electricity collected by a fine wire twisted round the
string. Thanks to Sir W. Thomson, we can now observe atmospheric electricity in
a most satisfactory manner. I will test, to show you the mode of proceeding, the
air inside and outside the hall. [The experiment was shown, and the external air
gave negative indications.]

On several occasions I have found it almost impossible, even by giving extreme
directive force to the instrument by means of magnets, to measure the atmospheric
potential with such an electrometer, and had recourse to the old electroscope, with
specially long and thick gold-leaves. On February 26th, 1874, when the sleet and
hail, dashing against the cupola of my class-room, made so much noise as to
completely interrupt my lecture, I connected that instrument with the water-dropper,
and saw the gold-leaves discharge themselves against the sides every few seconds,
sometimes with positive, sometimes, often immediately afterwards, with negative
electricity. Such effects would have required for their production a battery of tens
of thousands of cells. Yet there was neither lightning nor thunder, and the water
was trickling from the can at the rate of only two and a half cubic inches per
minute. Probably had there not been such a violent fall of sleet steadily discharging
the clouds we should have had a severe thunderstorm. That this is no fancied
explanation is evident from the fact that falling rain-drops are often so strongly
charged with electricity as to give a spark just before they touch the ground. On
such occasions, when the fall occurs at night, the ground is feebly lit up. This
"luminous rain," as it has been called, is a phenomenon which has been over and
over again seen by competent and trustworthy observers. In the Comptes Rendus
for November last we read of the curious phenomenon of electrification of the

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300 PETER GUTHRIE TAIT

observer's umbrella by a slight fall of snow, to such an extent that he could draw
sparks from it with his ringer.

In calm clear weather the atmospheric charge is usually positive. This is very
commonly attributed to evaporation of water, and I see no reason to doubt that the
phenomena are closely connected. I will show you one of the experiments upon
which the idea is based. I can take no other form of experiment than a somewhat
violent one, as the effects of the more delicate ones could not easily or certainly be
made visible to so large an audience. [A few drops of water were sprinkled on a
heated crucible, insulated, and connected with the electrometer.]

There can be no doubt that, whatever be the hidden mechanism of this experi-
ment, the steam has carried with it a strong charge of positive electricity, for it has
left the rest of the apparatus with a strong negative charge. We might reverse the
subject of measurement by connecting the electrometer with the escaping steam, but
I omit it, to save time, and because we will now try that form of the experiment
in another way. [High-pressure steam escaping from a little boiler was made to play
upon an insulated conductor furnished with spikes, and connected with the electro-
meter, which then showed a strong positive charge.]

There are many substances which produce on evaporation far greater electric
developments than water does, some of positive, others of negative, electricity. By
far the most remarkable in this respect to which attention has yet been called is an
aqueous solution of sulphate of copper. (Proc. R.S.E., 1862.) The smallest drop of
this solution thrown on a hot dish gives an intense negative effect — so great, in fact,
that it may be occasionally employed to charge a small Leyden jar. But this, like
the smaller effect due to water under similar circumstances, is not yet completely
explained.

The next striking features are the flashes of lightning which at intervals light
up the landscape with an intensity which must in the majority of cases far exceed
that produced by the full moon. To the eye, indeed, the flash does not often
appear to furnish more than the equivalent of average moonlight, but it must be
remembered that it lasts for a period of time almost inconceivably short, and that
the full effect of light on the eye is not produced until after the lapse of a con-
siderable fraction of a second. Professor Swan has estimated this interval at about
one-tenth of a second ; and he has proved that the apparent intensity of illumina-
tion for shorter intervals is nearly proportional to the duration. (Trans. R.S.E., 1849.)
I can illustrate this in a very simple manner. [Two beams of light were thrown
upon the screen by reflection from mirrors, each of which was fixed nearly at right
angles to an axis. When matters were so adjusted that the brightness of the two
illuminated spots was the same, one mirror was made to rotate. The corresponding
light spot described a circle about the other, and its brightness became less the
larger the circle in which it was made to revolve.] The lightning flash itself on this
account, and for the further reason that its whole apparent surface is exceedingly
small, must be in some degree comparable with the sun in intrinsic brilliancy —
though, of course, it cannot appear so. The fact that its duration is excessively
short is easily verified in many ways, but most simply by observing a body in
rapid motion. The spokes of the wheels of the most rapidly-moving carriage appear

THE LIGHTNING FLASH 301

absolutely fixed when illuminated by its light alone. One can read by its light a
printed page stuck on a disc revolving at great speed. But the most severe test is
that of Sir Charles Wheatstone's revolving mirror. Seen by reflection in such a
mirror, however fast it may be rotating, a flash of lightning is not perceptibly
broadened, as it certainly would be if its duration were appreciable.

The apparatus which, in our laboratories, enables us to measure the time which
light, moving at nearly 200,000 miles per second, takes to pass over a few feet, is
required to prove to us that lightning is not absolutely instantaneous. Wheatstone
has shown that it certainly lasts less than a millionth part of a second. Take this,
along with Swan's datum, which I have just given you, and you see that the apparent
brightness of the landscape, as lit up by a lightning flash, is less than one hundred
thousandth part of what it would be were the lightning permanent. We have thus
rough materials for instituting a comparison between the intrinsic brightness of
lightning and of the sun.

Transient in the extreme as the phenomenon is, we can still, in virtue of the
duration of visual impressions, form a tolerably accurate conception of the form of
a flash ; and in recent times instantaneous processes of photography have given us
permanent records of it. These, when compared with photographic records of ordinary
electric sparks, bear out to the full the convictions at once forced by appearances on the
old electricians, that a flash of lightning is merely a very large electric spark. The
peculiar zig-zag form, sometimes apparently almost doubling back on itself, the
occasional bifurcations, and various other phenomena of a lightning flash, are all shown
by the powerful sparks from an electric machine. [These sparks were exhibited
directly ; and then photographs, of which one is represented in the woodcut, were
exhibited.]

But the spectroscope has recently given us still more convincing evidence of
their identity, if any such should be wanted. This is a point of great importance, but
I have not now time to discuss it. Thus, though on a very small scale comparatively,
we may study the phenomena of lightning at convenience in our laboratories. We
thus know by experiment that electricity chooses always the easiest route, the path
of least resistance. Hence the danger of taking the otherwise most natural course of
standing under a tree during a thunderstorm. The tree, especially when wet, is a
much better conductor than the air, and is consequently not unlikely to be chosen
by the discharge ; but the human body is a conductor much superior to the tree,
and therefore is chosen in preference so far as it reaches. The bifurcations of a flash
can puzzle no one who is experimentally acquainted with electricity, but the zig-zag

302 PETER GUTHRIE TAIT

form is not quite so easily explained. It is certainly destroyed, in the case of short
sparks, by heating the air. [Photographs of sparks in hot and in cold air were
exhibited. One of each kind is shown in the woodcut. The smoother is that which
passed through the hot air. The other passed through the cold air nearer the camera,
and is therefore not quite in focus.]

Now heating in a tube or flame not only gets rid of motes and other combustible
materials but it also removes all traces of electrification from air. It is possible,
then, that the zig-zag form of a lightning flash may, in certain cases at least, be due
to local electrification, which would have the same sort of effect as heat in rarefying
the air and making it a better conductor.

A remark is made very commonly in thunderstorms which, if correct, is
obviously inconsistent with what I have said as to the extremely short duration of
a flash.

Even if we supposed the flash to be caused by a luminous body moving along,
like the end of a burning stick whirled around in a dark room, it would pass with
such extraordinary rapidity that the eye could not possibly follow its movements.
Hence it is clear that when people say they saw a flash go upwards to the clouds
from the ground, or downwards from the clouds to the ground, they must be mistaken.

The origin of the mistake seems to be a subjective one, viz., that the central parts
of the retina are more sensitive, by practice, than the rest, and therefore that the
portion of the flash which is seen directly affects the brain sooner than the rest.
Hence a spectator looking towards either end of a flash very naturally fancies that
end to be its starting point.

Before I can go farther with this subject it is necessary that I should give
some simple facts and illustrations connected with ordinary machine electricity.
These will enable you to follow easily the slightly more difficult steps in this part
of our subject which remain to be taken.

Since we are dealing mainly with tnotion of electricity, it is necessary to
consider to what that motion is due. You all know that winds, i.e., motions of the
air, are due to differences of pressure.

If the pressure were everywhere the same at the same level we should have no
winds. Similarly the cause of the motion of heat in a body is difference of
temperature. When all parts of a body are at the same temperature there is no
change of distribution of heat. Now electricity presents a precisely analogous case.
It moves in consequence of difference of potential. Potential, in fact, plays, with

ELECTRICAL CAPACITY 303

regard to electricity, a part precisely analogous to the rdle of pressure, or of tempera-
ture, in the case of motions of fluids and of conducted heat. It would tax your
patience too much were I to give an exact definition of potential in a lecture like
this ; but you may get a sufficiently approximate notion of it in a very simple way
— again by analogy. Suppose I wished to specify the power of the pump used to
compress air in an air-gun receiver. I should say it can produce a limiting pressure
of 40 or 100 atmospheres, as the case may be. If you try to go any farther it
leaks. This would be considered quite definite information. But, mark you, nothing
is said as to the capacity of the receiver. When the pump has done its utmost,
the receiver, be it large or small, contains air at the definite pressure of the 40 or
100 atmospheres which measure the power of the pump.

A longer time will be required for a more capacious receiver, but the ultimate
pressure is the same in all. And when two receivers contain air at the same
pressure you may open a communication between them, but no air will pass, how-
ever much they may differ in capacity. Similarly, you may measure the power
of a flame or a furnace by the highest temperature it can produce. It will take a
longer time to effect this the greater is the thermal capacity of the body to be
heated ; but when two bodies are at the same temperature no heat passes from one
to the other. Similarly, the power of an electrical machine may be measured by
the utmost potential it can give to a conductor. The greater the capacity of the
conductor the longer time will be required for the machine to charge it ; but no
electricity passes between two conductors charged to the same potential. Hence
the power of a machine is to be measured by using the simplest form of a conductor,
a sphere, and finding the utmost potential the machine can give it. It is easily
shown that the potential of a solitary sphere is directly as the quantity of electricity,
and inversely as the radius. Hence electricity is in equilibrium on two spheres
connected by a long thin wire when the quantities of electricity on them are pro-
portional— not to their surfaces, nor to their volumes, as you might imagine — to
their radii. In other words, the capacity is proportional to the radius. This, how-
ever, is only true when there are no other conductors within a finite distance.
When a sphere is surrounded by another concentric sphere, which is kept in metallic
connection with the ground, its capacity is notably increased, and when the radii
of the spheres are nearly equal the capacity of the inner one is directly as its
surface, and inversely as the distance between the two spheres. Thus the capacity
is increased in the ratio of the radius of one sphere to the difference of the radii
of the two, and this ratio may easily be made very large. This is the principle
upon which the Leyden jar depends.

We may usefully carry the analogy of the pump a good deal further. Supposing
the piston to be fully pressed home every stroke, the amount of work spent, even
if the whole be kept cool, on each stroke continually increases, so that more than
double the amount of work is required to charge the receiver to 40 atmospheres
instead of 20. The same holds with electricity. Each successive unit of the charge
requires more work to force it in than did the preceding one, because the repulsion
of all already in has to be overcome. It is found, in fact, that the work required
to put in a charge is proportional to the square of the charge. Of course less work

304 PETER GUTHRIE TAIT

is required for a given charge the greater is the capacity of the receiver. Conversely,
the damage which can be done by the discharge, being equal to the work required
to produce the charge, is proportional to the square of the charge, and inversely to
the capacity of the receiver. Or, what comes to the same thing, it is proportional
to the square of the potential and to the capacity of the conductor directly. Thus
a given quantity of electricity gives a greater shock the smaller the capacity of the
conductor which contains it. And two conductors, charged to the same potential,
give shocks proportional to their capacities. But in every case, a doubling of the
charge, or a doubling of the potential, in any conductor, produces a fourfold shock.

The only other point I need notice is the nature of the distribution of electricity
on a conductor. I say on a conductor, because it is entirely confined to the surface.
The law is — it is always so arranged that its attractions or repulsions in various
directions exactly balance one another at every point in the substance of the conductor.
It is a most remarkable fact that this is always possible, and in every case in one
way only. When the conductor is a single sphere the distribution is uniform. When
it is elongated the quantity of electricity per square inch of its surface is greater
at the ends than in the middle ; and this disproportion is greater the greater is the
ratio of the length to the transverse diameter. Hence on a very elongated body,
terminating in a point, for instance, the electric density — that is, the quantity per
square inch of surface — may be exceedingly great at the point while small every-
where else. Now in proportion to the square of the electric density is the outward
pressure of the electricity tending to escape by forcing a passage through the
surrounding air. It appears from experiments on the small scale which we can
make with an electrical machine, that the electric density requisite to force a passage
through the air increases under given circumstances, at first approximately as the
square root of the distance which has to be traversed, but afterwards much more
slowly, so that it is probable that the potential required to give a mile-long flash
of lightning may not be of an order very much higher than that producible in our
laboratories.

But from what I have said you will see at once that under similar circumstances
an elongated body must have a great advantage over a rounded one in effecting
a discharge of electricity. This is easily proved by trial. [The electric machine
being in vigorous action, and giving a rapid series of sparks, a pointed rod connected
with the ground was brought into the neighbourhood, and the sparks ceased at
once.] In this simple experiment you see the whole theory and practical importance
of a lightning conductor. But, as a warning, and by no means an unnecessary one,
I shall vary the conditions a little and try again. [The pointed rod was now
insulated, and produced no observable effect.] Thus you see the difference between
a proper lightning-rod and one which is worse than useless, positively dangerous.
There is another simple way in which I can destroy its usefulness — namely, by
putting a little glass cap on the most important part of it, its point, and thus
rendering impossible all the benefits it was originally calculated to bestow. [The
pointed rod was again connected with the ground, but furnished with a little glass
cap. It produced no effect till it was brought within four or five inches of one of
the conductors of the machine, and then sparks passed to it.] You must be strangely

LIGHTNING RODS 305

well acquainted with the phases of human perversity if you can anticipate what I
am now going to tell you, namely, that this massive glass cap, or reseller, as it was
fondly called, was only a year or two ago taken off from the top of the lightning-
rod employed to protect an important public building. [The repeller was exhibited.
It resembled a very large soda-water bottle with a neck much wider than the usual
form.] From the experiments you have just seen it must be evident to you that
the two main requisites of an effective lightning-rod are that it should have a sharp
point (or, better, a number of such points, lest one should be injured), and that it
should be in excellent communication with the ground. When it possesses these,
it does not require to be made of exceptionally great section ; for its proper function
is not, as is too commonly supposed, to parry a dangerous flash of lightning: it
ought rather, by silent but continuous draining, to prevent any serious accumulation
of electricity in a cloud near it. That it may effectually do this it must be thoroughly
connected with the ground, or (if on a ship or lighthouse) with the sea. In towns
this is easily done by connecting it with the water mains, at sea by using the
copper sheathing of the ship, or a metal plate of large surface fully immersed. Not
long ago a protected tower was struck by lightning. No damage was done in the
interior, but some cottages near its base were seriously injured. From a report on
the subject of this accident it appears that the lower end of the lightning-rod was
"jumped" several feet into the solid rock! Thus we see, in the words of Arago,
how " False science is no less dangerous than complete ignorance, and that it
infallibly leads to consequences which there is nothing to justify."

That the lightning-rod acts as a constant drain upon the charge of neighbouring
clouds is at once proved when there is, accidentally or purposely, a slight gap in
its continuity. This sometimes happens in ships, where the rod consists of separate
strips of metal inlaid in each portion of the mast. If they are not accurately
fitted together, a perfect torrent of sparks, almost resembling a continuous arc of
light is seen to pass between them whenever a thunderstorm is in the neigh-
bourhood.

I cannot pass from this subject without a remark upon the public as well as
private duty of having lightning-rods in far greater abundance than we anywhere
see them in this country. When of proper conducting power, properly pointed,
properly connected with the ground and with every large mass of metal in a building
they afford absolute protection against ordinary lightning — every single case of
apparent failure I have met with having been immediately traceable to the absence
of one or other of these conditions. How great is their beneficial effect you may
gather at once from what is recorded of Pietermaritzburg, viz., that till lightning-
rods became common in that town it was constantly visited by thunderstorms at
certain seasons. They still come as frequently as ever, but they cease to give
lightning-flashes whenever they reach the town, and they begin again to do so as
soon as they have passed over it.

A knight of the olden time in full armour was probably as safe from the
effects of a thunderstorm as if he had a lightning-rod continually beside him ; and
one of the Roman emperors devised a perfectly secure retreat in a thunderstorm
in the form of a subterraneous vault of iron. He was probably led to this by

T. 39

306 PETER GUTHRIE TAIT

thinking of a mode of keeping out missiles, having no notion that a thin shell of
soft copper would have been quite as effective as massive iron. But those emperors
who, as Suetonius tells us, wore laurel crowns or sealskin robes, or descended into
underground caves or cellars on the appearance of a thunderstorm, were not protected
at all. Even in France, where special attention is paid to the protection of buildings
from lightning, dangerous accidents have occurred where all proper precautions seemed
to have been taken. But on more careful examination it was usually found that
some one essential element was wanting. The most common danger seems to lie
in fancying that a lightning-rod is necessarily properly connected with the earth if
it dips into a mass of water. Far from it. A well-constructed reservoir full of
water is not a good " earth " for a lightning-rod. The better the stone-work and
cement the less are they fitted for this special purpose, and great mischief has been
done by forgetting this.

A few years ago the internal fittings of the lighthouse at Skerryvore were
considerably damaged by lightning, although an excellent lightning-rod extended
along the whole height of the tower. But a long copper stove-pipe, rising through
the whole interior of the tower, and the massive metallic ladder rising from the
ground to the lowest chamber, though with a considerable gap between them, offered
less resistance than the rod, for the lower end of the ladder was nearer to the sea
than was the pool on the reef into which the lightning-rod plunged. Hence the
main disruptive discharge took place from the stove-pipe to the ladder, blowing
the intervening door to pieces. The real difficulty in these situations, exposed to
tremendous waves, lies in effecting a permanent communication between the lightning-
rod and the sea. But when this is done the sea makes far the best of " earths."

When a lightning-rod discharges its function imperfectly, either from insufficient
conducting power or because of some abnormally rapid production of electricity, a
luminous brush or glow is seen near its point. This is what the sailors call St Elmo's
Fire, or Castor and Pollux. In the records of mountain climbing there are many
instances of such discharges to the ends of the alpenstocks or other prominent
pointed objects. One very remarkable case was observed a few months ago in
Switzerland, where at dusk, during a thunderstorm, a whole forest was seen to
become luminous just before each flash of lightning, and to become dark again at
the instant of the discharge.

Perhaps the most striking of such narratives is one which I will read to you
from the memoirs of the Physical and Literary Society, from which sprang the
Royal Society of Edinburgh. These Essays are rare and curious, and the names of
Maclaurin, James and Matthew Stewart, Whytt, and Monro, appear among their
authors.

The following observations on Thunder and Electricity by Ebenezer McFait, M.D.,
show how, in the search for truth, men may unwittingly put themselves in the gravest
danger: —

"The experiment proposed by Mr Franklin, to prove that lightning and the electrical
fire are the same, has often been repeated with success both in England and abroad, so that
the most noted electrical experiments have been performed by fire drawn from the clouds.

"Mr Franklin also first discovered that sharp points attract and discharge the electrical

McFAIT'S EXPERIMENTS IN 1752 307

matter most copiously ; and from thence supposes, that a very sharp-pointed rod, fixed to the
extremity of the top-mast of a ship, with a wire conducted down from the foot of the rod
round one of the shrouds, and over the ship's side into the sea, would silently lead off the
electrical fire, and save the ship from thunder in hot countries ; and that, by a similar method,
buildings might be preserved.

"So useful a proposal deserves to be examined: variety of experiments may give hints
for new improvements. For this reason the following observations are communicated, though
not so complete as might be wished, being the result of one trial only.

"It seldom thunders in this northern clime. In June, 1752, there seemed to be some
thunder at a distance from Edinburgh; but from the beginning of July to the beginning of
October we had nothing almost but continual rains. The last summer was uncommonly warm
and dry; and yet we had only a few claps of thunder at Edinburgh, one evening, and my
attempts for making any of those experiments were entirely unsuccessful until Saturday night,
September 15, when we had a very great storm.

" I used a round iron rod, two-tenths of an inch diameter, about eleven feet long,
sharpened at one end ; the other end was inserted in a glass tube, and that tube stood in a
common glass bottle, which I held in my hand.

"I used also another rod about three feet long, sharpened in like manner at one end,
which stood with the other end in a glass tube, which was stuck in the ground. I began
upon the Gallon Hill. The lightning and fire in the air abounded greatly, and yet it was
some time before anything else appeared. At last some rain began to fall, and the air
turned moister; then fire appeared upon the extremities of each of the rods in a small pretty
blaze, very like the fire which is discharged from the point of a sword in the dark, when the
person that holds it is electrified, and stands upon glass or resin; or like that which appears
upon any sharp point, when presented to an electrified gun barrel, but in greater quantity.
I touched the long rod with my finger, but had no sparks from it. The short rod was
accidentally taken out of its tube, and yet continued to burn and blaze as formerly. In like
manner the flame continued upon the end of the long rod, though I took hold of it anywhere
at pleasure above the glasses, until I moved my hand or finger along, within a few inches
of the flame; then it was attracted by my hand, and vanished.

" I went from the Calton Hill to the Castle Hill, at the other end of the town ; and in
passing through the streets no fire appeared upon either of the rods; but almost immediately
when I got clear of the houses, upon the open hill, the point of the longer iron rod took
fire. In the dark I had lost the tube belonging to the shorter rod, and the point of it did
not catch fire when the longer one was kindled. Perhaps I did not wait long enough for a
proper trial, for I soon touched the flame upon the long rod with the sharp point of the
short one, and then it also took flame, and continued burning, as before, without any further
dependence upon the longer one.

" I held the shorter rod by the sharp end, and approached the blunt end of it to the
flame upon the point of the longer rod ; then this blunt end caught the fire, and the flame
upon the points of the two rods continued rather stronger than on the single one before, so
long as I kept them in contact, and the fires within three or four inches of one another ; but
when I drew them farther asunder the flame upon the extremity of the blunt rod vanished.
This happened as often as I tried it, and it is evident that in like manner I could have got
the fire to fix upon the points of a great many rods, and so have had them all flaming
together. Once or twice a flash of lightning seemed to dart directly against the point of
the rod; then the fire, as I thought, expanded itself and united with the lightning, but it
immediately began to shine again when the lightning was past.

308 PETER GUTHRIE TAIT

"Though it rained much in time of these observations, yet the fire upon the ends of
the rods did not go out until it became so heavy as if it were pouring down out of funnels.

"After this I went home for some time, resolving to come abroad again when the storm
was more tolerable ; but it continued to rain all night, so violently, that I was obliged, with
regret, to leave several experiments to the chance of some future opportunity. For example,
I suspected that the glass tubes had not been of great use on this occasion, and wanted to
have tried whether I should have had the same appearances by using the rods alone, without
any other apparatus. This is very probable, as also that the glasses, by being wet, allowed
the electrical fire to flow off as it was attracted.

" I beg leave to add a few remarks relative to this subject. It would seem that experiments
of this kind may be made without danger when the thunder is at a moderate distance. The
lightning expands itself, as it flies, and by expansion loses its vigour. Perhaps there is one
simple and easy way of protecting masts and spires from thunder, viz., to fix horizontally
upon the highest parts of them a flat round piece of wood, of a foot diameter or more, in
order to prevent those blazing fires from fixing upon them, and accumulating.

"This storm passed directly over Edinburgh, and came on from the south by west, as
nearly as could be estimated. There was a great deal of lightning that night, above sixty
miles to the westward, but no thunder heard. At Glasgow there was very much lightning,
and a few distant faint claps of thunder. On the road from Belford or Berwick it lightened
incessantly, but two claps of thunder only were heard, and those very faint, so that there is
reason to think that the fire of this storm spread over the breadth of 130 miles at least.
I wish I could also give some account where this thunder began, and how far it ran before
it was extinguished.

"On September 3rd there was a great deal of streamers, which rose nearly from the
same point that the thunder afterwards came from, and gradually worked north till they
descended below the horizon. The air had a thunder-like appearance for several days before
this storm ; and for some nights after it the streamery vapour appeared equally diffused, muddy,
inert, and languid, and not vibrating any variety of colours, as if the more volatile parts had
been consumed. It is highly probable that lightning and the aurora borealis are of the same
materials. In hot countries streamers are not seen, or but rarely, because they are kindled
into thunder and flashes of lightning. In cold countries streamers abound, and it seldom
thunders. The streamers have served to predict thunder to follow next day, in summer, and
they have been also seen to break out into flashes of lightning. Thunder disturbs the motion
of the magnetic needle, and it has been lately found in Sweden that streamers do the same.
Thus thunder, electricity, magnetism, and the aurora borealis, appear all wonderfully related ;
and many things remain undiscovered in this vast field, which is but just newly opened.

"As it is probable that the height which some philosophers have assigned for the streamers
in the atmosphere is by several hundreds of miles too much, it were to be wished that people
in various latitudes would carefully observe their altitude at different times of the night, that
by comparing simultaneous observations this matter may be determined with more certainty."

At first reading one is inclined to regard this as a joke, but there is nothing
whatever to justify the notion. That Dr McFait was not killed on this occasion
was certainly not due to any want of precautions on his part, well calculated to make
such an event all but certain. He wanted only a knob on the blunt end of his
short rod. We are reminded of the remark made by one of the seconds in a well-
known duel about his principal, " To come on horseback to a fight with pistols ! and
in a white waistcoat, too! Couldn't he have got a bull's eye painted on it, just

DESTRUCTIVE EFFECTS OF LIGHTNING 309

over his heart? It would have expedited matters, and made them still more simple."
Richmann, of St Petersburg, had just before been killed while apparently in far less
danger than Dr McFait, and other incautious experimenters have since similarly
suffered.

The destructive effects of lightning are familiar to all of you, so that I need
not spend time in illustrating them on a puny scale by the help of Leyden jars.

All the more ordinary effects can thus easily be reproduced on a small scale.
How small you may easily conceive, when I tell you that a three-foot spark is
considered a long one, even from our most powerful machines, while it is quite
certain that lightning flashes often exceed a mile in length, and sometimes extend
to four and five miles. One recorded observation, by a trustworthy observer, seems
to imply a discharge over a total length of nearly ten miles.

When a tree is struck by a violent discharge it is usually split up laterally into
mere fibres. A more moderate discharge may rupture the channels through which
the sap flows, and thus the tree may be killed without suffering any apparent external
damage. These results are usually assigned to the sudden vaporisation of moisture,
and the idea is probably accurate, for it is easy to burst a very strong glass tube,
if we fill it with water and discharge a jar by means of two wires whose extremities
are placed in the water at a short distance from one another. The tube bursts
even if one end be left open, thus showing that the extreme suddenness of the
explosion makes it act in all directions, and not solely in that of least resistance.
When we think of the danger of leaving even a few drops of water in a mould
into which melted iron is to be poured, we shall find no difficulty in thus accounting
for the violent disruptive effects produced by lightning.

Heated air is found to conduct better than cold air, probably on account of
the diminution of density only. Hence we can easily see how it is that animals are
often killed in great numbers by a single discharge, as they crowd together in a
storm, and a column of warm air rises from the group.

Inside a thundercloud the danger seems to be much less than outside. There
are several instances on record of travellers having passed through clouds from
which, both before and after their passage, fierce flashes were seen to escape. Many
remarkable instances are to be found in Alpine travel, and specially in the reports
of the officers engaged in the survey of the Pyrenees. Several times it is recorded
that such violent thunderstorms were seen to form round the mountain on which
they were encamped that the neighbouring inhabitants were surprised to see them
return alive.

Before the use of lightning-rods on ships became general great damage was
often done to them by lightning. The number of British ships of war thus wholly
destroyed or much injured during the long wars towards the end of the last and
the beginning of the present century is quite comparable with that of those lost
or injured by gales, or even in battle. In some of these cases, however, the damage
was only indirectly due to lightning, as the powder magazines were blown up. In
the powder magazine of Brescia, in 1769, lightning set fire to over two million
pounds of gunpowder, producing one of the most disastrous explosions on record.

A powerful discharge of lightning can fuse not only bell-wires, but even stout

310 PETER GUTHRIE TAIT

rods of iron. It often permanently magnetises steel, and in this way has been the
cause of the loss of many a good ship ; for the magnetism of the compass-needles
has been sometimes destroyed, sometimes reversed, sometimes so altered that the
compass pointed east and west. And by the magnetisation of their steel parts the
chronometers have had their rates seriously altered. Thus two of the sailor's most
important aids to navigation have been simultaneously rendered useless or, what is
worse, misleading ; and this, too, at a time when, because of clouds, astronomica
observations were generally impossible. All these dangers are now, however, easily
and all but completely avoidable.

Sheet-lead punctured by lightning.

A very singular effect of lightning sometimes observed is the piercing of a hole
in a conducting-plate of metal, such as the lead-covering of a roof. In such a case
it is invariably found that a good conductor well connected with the ground approaches
near to the metal sheet at the part perforated. We can easily repeat the experi-
ment on a small scale with tinfoil. [A thick piece of sheet-lead from the lower
buildings of Nelson's monument, Edinburgh, punctured by lightning, was exhibited.
It is figured in the woodcut, reprinted (by permission) from Proc, R.S.E., 1863.]

The name thunderbolt, which is still in use, even by good writers, seems to have
been introduced in consequence of the singular effects produced when lightning
strikes a sandhill or sandy soil. It bores a hole often many feet in length, which
is found lined throughout with vitrified sand. The old notion was that an intensely
hot, solid mass, whose path was the flash of lightning, had buried itself out of sight,
melting the sand as it went down. It is quite possible that this notion may have
been strengthened by the occasional observation of the fall of aerolites, which are
sometimes found, in the holes they have made, still exceedingly hot. And at least
many of the cases in which lightning is said to have been seen in a perfectly clear
sky are to be explained in the same way. Everyone knows Horace's lines : —

FORKED LIGHTNING 311

" Diespiter

Igni corusco nubila dividens
Plerumque, per purum tonantes
Egit equos volucremque currum."

But Virgil's remark is not so commonly known. He is speaking of prodigies of
various kinds, and goes on : —

" Non alias coelo ceciderunt plura sereno
Fulgura; nee diri toties arsere cometae."

It is very singular that he should thus have associated comets and meteorites
which quite recent astronomical discovery has shown to have a common origin.

Another remarkable peculiarity, long ago observed, is the characteristic smell
produced when lightning strikes a building or a ship. In old times it was supposed
to be sulphurous ; now-a-days we know it to be mainly due to ozone. In fact, all
the ready modes of forming ozone which are as yet at the disposal of the chemist
depend upon applications of electricity. But besides ozone, which is formed from
the oxygen of the air, there are often produced nitric acid, ammonia, and other
compounds derived from the constituents of air and of aqueous vapour. All these
results can be produced on a small scale in the laboratory.

Hitherto I have been speaking of lightning discharges similar in kind to the
ordinary electric spark, what is commonly called forked or zig-zag lightning. Our
nomenclature is very defective in this matter, and the same may be said of the
chief modern European languages. For, as Arago remarks, by far the most common
form of lightning flash observed in thunderstorms is what we have to particularise,
for want of a better term, as sheet-lightning. He asserts that it occurs thousand-
fold as often as forked lightning ; and that many people have never observed the
latter form at all ! It is not at all easy to conceive what can be the nature of
sheet-lightning, if it be not merely the lighting up of the clouds by a flash of forked
lightning not directly visible to the spectator. That this is, at least in many cases,
its origin is evident from the fact that its place of maximum brightness often takes
the form of the edge of a cloud, and that the same cloud-edge is occasionally lit up
several times in quick succession. You will remember that we are at present dealing
with the appearances observed in a thunderstorm, so that I do not refer to that
form of sheet-lightning which commonly goes by the name of summer-lightning, and
which is not, audibly, at least, followed by thunder.

The next remarkable feature of the storm is the thunder, corresponding, of
course, on the large scale to the snap of an electric spark. Here we are on com-
paratively sure ground, for sound is very much more thoroughly understood than is
electricity. We speak habitually and without exaggeration of the crash of thunder,
the rolling of thunder, and of a peal of thunder ; and various other terms will
suggest themselves to you as being aptly employed in different cases. All of these
are easily explained by known properties of sound. The origin of the sound is in
all cases to be looked for in the instantaneous and violent dilatation of the air along
the track of the lightning flash ; partly, no doubt, due to the disruptive effects of
electricity of which I have already spoken, but mainly due to the excessive rise of

312 PETER GUTHRIE TAIT

temperature which renders the air for a moment so brilliantly incandescent. There
is thus an extremely sudden compression of the air all round the track of the
spark ; and a less sudden, but still rapid, rush of the air into the partial vacuum
which it produces. Thus the sound wave produced must at first be of the nature
of a bore or breaker. But as such a state of motion is unstable, after proceeding
a moderate distance the sound becomes analogous to other loud but less violent
sounds, such as those of the discharge of guns. Were there few clouds, were the
air of nearly uniform density, and the flash a short one, this would completely
describe the phenomenon, and we should have a thunder crash or thunder clap
according to the greater or less proximity of the seat of discharge. But, as has
long been well known, not merely clouds but surfaces of separation of masses of air
of different density, such as constantly occur in thunderstorms, reflect vibrations in
the air; and thus we may have many successive echoes, prolonging the original
sound. But there is another cause, often more efficient than these. When the flash
is a long one, all its parts being nearly equidistant from the observer, he hears the
sound from all these parts simultaneously ; but if its parts be at very different
distances from him, he hears successively the sounds from portions farther and farther
distant from him. If the flash be much zig-zagged, long portions of its course may
run at one and the same distance from him, and the sounds from these arrive
simultaneously at his ear. Thus we have no difficulty in accounting for the rolling
znA pealing of thunder. It is, in fact, a mere consequence, sometimes of the reflection
of sound, sometimes of the finite velocity with which it is propagated. The usual
rough estimate of five seconds to a mile is near enough to the truth for all ordinary
calculations of the distance of a flash from the observer.

The extreme distance at which thunder is heard is not great, when we consider
the frequent great intensity of the sound. No trustworthy observation gives in
general more than about nine or ten miles, though there are cases in which it is
possible that it may have been heard 14 miles off. But the discharge of a single
cannon is often heard at 50 miles, and the noise of a siege or naval engagement
has certainly been heard at a distance of much more than 100 miles. There are
two reasons for this : the first depends upon the extreme suddenness of the production
of thunder ; the second, and perhaps the more effective, on the excessive variations
of density in the atmosphere, which are invariably associated with a thunderstorm.
In certain cases thunder has been propagated, for moderate distances from its apparent
source, with a velocity far exceeding that of ordinary sounds. This used to be
attributed to the extreme suddenness of its production ; but it is not easy, if we
adopt this hypothesis, to see why it should not occur in all cases. Sir W. Thomson
has supplied a very different explanation, which requires no unusual velocity of
sound, because it asserts the production of the sound simultaneously at all parts of
the air between the ground and the cloud from which the lightning is discharged.

We now come to an exceedingly strange and somewhat rare phenomenon, to
which the name of fire-ball or globe-lightning has been given. As we are as yet
unable to produce anything of this kind by means of our electrical machines, some
philosophers have tried to cut the Gordian knot of the difficulty by denying that
any such thing can exist. But, as Arago says, " Oil en serions nous, si nous nous

FIRE BALLS 313

mettions a nier tout ce qu'on ne sait pas expliquer ? " The amount of trustworthy and
independent evidence which we possess as to the occurrence of this phenomenon is
such as must convince every reasonable man who chooses to pay due attention to
the subject. No doubt there is a great deal of exaggeration, as well as much
imperfect and even erroneous observation, in almost all of these records. But the
existence of the main feature (the fire-ball) seems to be proved beyond all doubt.

The most marked peculiarities of this species of lightning-discharge are its com-
paratively long duration and its comparatively slow motion. While a spark, or
lightning flash, does not last longer than about a millionth part of a second, if so
long, globe-lightning lasts from one to ten seconds, sometimes even longer, so that
a sufficiently self-possessed spectator has time carefully to watch its behaviour. The
general appearance is that of a luminous ball, which must be approximately spherical,
because it always appears circular in outline, slowly and steadily descending from a
thundercloud to the ground. It bursts with a loud explosion, sometimes before
reaching the ground, sometimes as it impinges, and sometimes after actually rebound-
ing. Its size varies from that of a child's head to a sphere of little less than a yard
in diameter. On some occasions veritable flashes of lightning were seen to proceed
from large fire-balls as they burst. It is difficult to imagine what these balls can
be if they be not a species of natural Leyden jar very highly charged. If it be so,
no ordinary lightning-rod can possibly prevent danger from it ; and we may thus
be able to explain the very few cases in which damage has been done by lightning
to thoroughly protected buildings. To guard against this form nothing short of a
pretty close net-work of stout copper wires would suffice. Meanwhile I give a
brief sketch of two out of the long series of descriptions of such phenomena which
Arago has patiently collected. The first is given on the authority of Babinet, who
was deputed by the Academy of Sciences to make inquiries into the case.

Shortly, but not immediately, after a loud peal of thunder, a tailor who was
sitting at his dinner saw the paper ornament which covered his fire-place blown
down as if by a gentle breeze, and a globe of fire, about the size of a child's head,
came gently out and moved slowly about at a slight elevation above the floor. It
appeared bright rather than hot, and he felt no sensation of warmth. It approached
him like a little kitten which desired to rub itself in play against his legs ; but he
drew his feet away, and by slow and cautious movements avoided contact with it.
It remained several seconds near his feet, while he leaned forward, and carefully
examined it. At last it rose vertically to about the level of his head, so he threw
himself back in his chair and continued to watch it. It then became slightly
elongated, and moved obliquely towards a hole pierced to the chimney about a
yard above the mantel piece. This hole had been made for the chimney of a
stove which was used in winter. " But," as the tailor said, " the globe could not
see the hole, for paper had been pasted over it." The globe went straight for the
hole, tore off the paper, and went up the chimney. After the lapse of time which
at the rate at which he had seen it moving, it would have required to get to the
top of the chimney, a terrific explosion was heard, and a great deal of damage was
done to the chimney and the roofs around it.

The next is even more striking: In June, 1849, in the evening of one of the

T. 40

3i4 PETER GUTHRIE TAIT

days when cholera was raging most formidably in Paris, the heat was suffocating,
the sky appeared calm, but summer lightning was visible on all sides. Madame
Espert saw from her window something like a large red globe, exactly resembling
the moon, when it is seen through mist. It was descending slowly towards a tree.
She at first thought it was a balloon, but its colour undeceived her; and while she
was trying to make out what it was, she saw the lower part of it take fire ("fe vis
le feu prendre au has de ce globe"), while it was still some yards above the tree.
The flames were like those of paper burning slowly, with sparks and jets of fire.
When the opening became twice or thrice the size of one's hand, a sudden and
terrific explosion took place. The infernal machine was torn to pieces, and a dozen
flashes of zig-zag lightning escaped from it in all directions. The dtbris of the
globe burned with a brilliant white light, and revolved like a catherine-wheel. The
whole affair lasted for at least a minute. A hole was bored in the wall of a house,
three men were knocked down in the street, and a governess was wounded in a
neighbouring school, besides a good deal of other damage.

As another instance, here is a description (taken from Dove) of one which fell
at Barbadoes, in 1831, during a terrific hurricane: At three o'clock in the morning
the lightning ceased for a few moments, and the darkness which enveloped the
town was indescribably terrible. Fiery meteors now fell from the clouds ; one in
particular, of spherical form, and of a deep red colour, fell perpendicularly from a
considerable height. This fire-ball fell quite obviously by its own weight, and not
under the influence of any other external force. As with accelerated velocity it
approached the earth it became dazzlingly white, and of elongated form. When it
touched the ground it splashed all about like melted metal, and instantly disappeared.
In form and size it resembled a lamp globe ; and the splashing about at impact
gave it the appearance of a drop of mercury of the same size.

I have never seen one myself, but I have received accounts of more than one
of them from competent and thoroughly credible eye-witnesses. In particular on a
stormy afternoon in November, 1868, when the sky was densely clouded over, and
the air in a highly electrical state, there was heard in Edinburgh one solitary short,
but very loud, clap of thunder. There can be no doubt whatever that this was due
to the explosion of a fire-ball, which was seen by many spectators in different parts
of the town, to descend towards the Calton Hill, and to burst whilst still about a
hundred feet or so above the ground. The various accounts tallied in most particulars,
and especially in the very close agreement of the positions assigned to the ball by
spectators viewing it from different sides, and in the intervals which were observed
to elapse between the explosion and the arrival of the sound.

The remaining phenomena of a thunderstorm are chiefly the copious fall of
rain and of hail, and the almost invariable lowering of the barometer. These are
closely connected with one another, as we shall presently see.

Almost all the facts to which I have now adverted point to water-substance, in
some of its many forms, as at least one of the chief agents in thunderstorms. And
when we think of other tremendous phenomena which are undoubtedly due to water,
we shall have the less difficulty in believing it to be capable of producing thunder-
storms also,

SOURCE OF HIGH POTENTIAL 315

First of all let us think of some of the more obvious physical consequences of
a fall of a mere tenth of an inch of rain. Suppose it to fall from the lowest mile
of the atmosphere. An inch of rain is 5 Ib. of water per square foot, and gives out
on being condensed from vapour approximately 3,000 units of heat on the centigrade
scale. The mass of the mile-high column of air a square foot in section is about
360 Ib., and its specific heat about a quarter. Thus its temperature throughout
would be raised by about 33° C., or 60° F. For one-tenth inch of rain, therefore,
we should have a rise of temperature of the lowest mile of the atmosphere amounting
to 3'3°C, quite enough to produce a very powerful ascending current. As the air
ascends and expands it cools, and more vapour is precipitated, so that the ascending
current is further accelerated. The heat developed over one square foot of the earth's
surface under these conditions is equivalent to work at the rate of a horse-power for
twelve minutes. Over a square mile this would be ten million horse-power for half
an hour. A fall of one-tenth of an inch of rain over the whole of Britain gives
heat equivalent to the work of a million millions of horses for half an hour ! Numbers
like these are altogether beyond the limits of our understanding. They enable us,
however, to see the full explanation of the energy of the most violent hurricanes
in the simplest physical concomitants of the mere condensation of aqueous vapour.

I have already told you that the source of atmospheric electricity is as yet
very uncertain. Yet it is so common and so prominent a phenomenon in many of
its manifestations that there can be little doubt that innumerable attempts have
been made to account for it. But when we consult the best treatises on meteor-
ology we find it either evaded altogether or passed over with exceedingly scant
references to evaporation or to vegetation. Not finding anything satisfactory in
books, I have consulted able physicists, and some of the ablest of meteorologists,
in all cases but one with the same negative result. I had, in fact, the feeling which
every one must experience who attempts to lecture on a somewhat unfamiliar subject,
that there might be much known about it which I had not been fortunate enough
to meet with. Some years ago I was experimentally led to infer that mere contact
of the particles of aqueous vapour with those of air, as they fly about and impinge
according to the modern kinetic theory of gases, produced a separation of the two
electricities, just as when zinc and copper are brought into contact the zinc becomes
positively electrified and the copper negatively. Thus the electrification was supposed
to be the result of chemical affinity. Let us suppose, then, that a particle of vapour,
after impact on a particle of air, becomes electrified positively (I shall presently
mention experiments in support of this supposition), and see what further consequences
will ensue when the vapour condenses. We do not know the mechanism of the pre-
cipitation of vapour as cloud, and we know only partially that of the agglomeration
of cloud-particles into rain-drops ; but of this we can be sure that, if the vapour-
particles were originally electrified to any finite potential the cloud-particles would
be each at a potential enormously higher, and the rain-drops considerably higher
still. For, as I have already told you, the potential of a free charged sphere is
proportional directly to the quantity of electricity on it and inversely to its radius ;
so when eight equal and equally charged spheres unite into one sphere of double
the radius, its potential is four times that of each of the separate spheres. The

40 — a

316 PETER GUTHRIE TAIT

potential in a large sphere, so built up, is in fact directly proportional to its surface
as compared with that of any one of the smaller equal spheres of which it is built.

Now, the number of particles of vapour which go to the formation of a single
average rain-drop is expressed in billions of billions; so that the potential of the
drop would be many thousands of billion times as great as that of a particle of
vapour. On the very lowest estimate this would be incomparably greater than any
potential we can hope to produce by means of electrical machines.

But this attempt at explanation of atmospheric electricity presents two for-
midable difficulties at the very outset.

1. How should the smaller cloud-particles ever unite if they be charged to
such high potentials, which of course must produce intense repulsions between them ?

2. Granting that, in spite of this, they do so unite, how are they separated
from the mass of negatively electrified air in which they took their origin?

I think it is probable that the second objection is more imaginary than real,
since there is no doubt that the diffusion of gases would speedily lead to a great
spreading about of the negatively electrified particles of air from among the precipi-
tated cloud-particles into the less highly electrified air surrounding the cloud. And
if the surrounding air were equally electrified with that mixed with the cloud, there
would be no electric force preventing gravity from doing its usual work. This
objection, in fact, holds only for the final separation of the whole moisture from the
oppositely electrified air ; and gravity may be trusted to accomplish this. That
gravity is an efficient agent in this separation is the opinion of Prof. Stokes. It
must be observed that as soon as the charge on each of the drops in a cloud rises
sufficiently, the electricity will pass by discharge to those which form the bounding
layer of the cloud.

The first objection is at least partially met by the remark that in a cloud-mass
when just formed, if it be at all uniform, the electric attractions and repulsions
would approximately balance one another at every point, so that the mutual repul-
sion of any two water-drops would be almost compensated except when they came
very close to one another.

But there is nothing in this explanation inconsistent with the possibility that
the particles of water may be caused to fly about repeatedly from cloud to cloud,
or from cloud to an electrified mass of air ; and in many of these regions the air,
already in great part deprived of its moisture, may have become much cooled by
expansion as it ascends, so that the usual explanation of the production of hail is
not, at least to any great extent, interfered with.

I may here refer to some phenomena which seem to offer, if closely investigated,
the opportunity for the large scale investigations which, as I shall presently show,
will probably be required to settle the source or sources of atmospheric electricity.

First, the important fact, well known nearly 2,000 years ago, that the column
of smoke and vapour discharged by an active volcano gives out flashes of veritable
lightning. In more modern times this has been repeatedly observed in the eruptions
of Vesuvius and other volcanos — such as, for instance, the island of Sabrina. It is
recorded by Sir W. Hamilton (British ambassador at Naples at the end of last
century) that in the eruption of 1794 these flashes were accompanied by violent

VOLCANIC ERUPTIONS 317

peals of thunder. They destroyed houses in the neighbourhood of the mountain,
and are said to have done considerable damage even at places 250 miles off, to
which the cloud of volcanic dust and vapour was carried by the wind. I will read
a couple of extracts from Hamilton's paper. It is in the Philosophical Transactions
for 1795 : —

"The electric fire in the year 1779, that played constantly within the enormous black
cloud over the crater of Vesuvius, and seldom quitted it, was exactly similar to that which is
produced on a very small scale by the conductor of an electrical machine communicating with an
insulated plate of glass, thinly spread over with metallic filings, etc., when the electric matter
continues to play over it in zig-zag lines without quitting it. I was not sensible of any noise
attending .that operation in 1779; whereas the discharge of the electrical matter from the
volcanic clouds during this eruption, and particularly the second and third days, caused
explosions like those of the loudest thunder ; and, indeed, the storms raised evidently by the
sole power of the volcano resembled in every respect all other thunder-storms, the lightning
falling and destroying everything in its course. The house of the Marquis of Berio at S. Jorio,
situated at the foot of Vesuvius, during one of these volcanic storms, was struck with lightning,
which, having shattered many doors and windows, and damaged the furniture, left for some
time a strong smell of sulphur in the rooms it passed through. Out of these gigantic and
volcanic clouds, besides the lightning, both during this eruption and that of 1779, I have,
with many others, seen balls of fire issue, and some of a considerable magnitude, which,
bursting in the air, produced nearly the same effect as that from the air-balloons in fire-works,
the electric fire that came out having the appearance of the serpents with which those fire-
work balloons are often filled. The day on which Naples was in the greatest danger from
the volcanic clouds, two small balls of fire, joined together by a small link like a chain-shot,
fell close to my casino at Posilipo. They separated, and one fell in the vineyard above the
house, and the other in the sea, so close to it that I heard a splash in the water; but, as I
was writing, I lost the sight of this phenomenon, which was seen by some of the company with
me, and related to me as above."

"The Archbishop of Taranto, in a letter to Naples, and dated from that city the i8th
of June, said, ' We are involved in a thick cloud of minute volcanic ashes, and we imagine
that there must be a great eruption either of Mount Etna or of Stromboli.' The bishop did
not dream of their having proceeded from Vesuvius, which is about 250 miles from Taranto.
We have had accounts also of the fall of the ashes during the late eruption at the very
extremity of the province of Leece, which is still farther off; and we have been assured
likewise that those clouds were replete with electrical matter. At Martino, near Taranto,
a house was struck and much damaged by the lightning from one of these clouds. In the
accounts of the great eruption at Vesuvius in 1631 mention is made of the extensive progress
of the ashes from Vesuvius, and of the damage done by the ferilli, or volcanic lightning,
which attended them in their course."

Sabine, while at anchor near Skye, remarked that the cloud-cap on one of the
higher hills was permanently luminous at night, and occasionally gave out flashes
resembling those of the aurora. I have not been able to obtain further information
as to this very important fact ; but I have recently received a description of a very
similar one from another easily accessible locality.

3i8 PETER GUTHRIE TAIT

My correspondent writes from Galway, to the following effect, on the 2nd of
the present month : —

"At the commencement of the present unprecedently long and severe storm the wind
blew from south-west and was very warm. After blowing for about two days it became,
without change of direction, exceedingly bitter and cold ; and the rain was, from time to time,
mixed with sleet and hail, and lightning was occasional. This special weather is common
for weeks together in March or early April. The air is (like what an east wind brings in
Edinburgh) cold, raw, dry, and in every way uncomfortable, especially to people accustomed
to the moist Atlantic winds. During these weeks a series of small clouds, whose shadows
would only cover a field of a few acres, seem to start at regular intervals from the peaks of
hills in Connemara and Mayo. They are all more or less charged with electricity. From
high ground, behind the city, I have at one time seen such a cloud break into lightning
over the spire of the Jesuits' church. At another, I have seen such a cloud pour down in a
thin line of fire, and fall into the bay in the shape of a small incandescent ball. On one
occasion I was walking with a friend, when I remarked, ' Let us turn and make a run for it.
We have walked unwittingly right underneath a little thundercloud.' I had scarcely spoken
when a something flashed on the stony ground at our very feet, a tremendous crash pealed over
our heads, and the smell of sulphur was unmistakable. I fancy that I have been struck with
these phenomena more than others, from the circumstance that they have always interfered
with my daily habits. My walks often extended to considerable distances and to very lonely
districts. Now these small local spurts of thunderstorms would hardly excite attention in the
middle of a town, all the less as the intervening weather is bright, though raw — these spurts
coming on every three or four quarters of an hour. Neither would they excite much atten-
tion in the country, as, while such a little storm was going on in one's immediate neighbourhood,
you would see at no great distance every sign of fine weather. In fact they always seem to
me like the small change of a big storm."

My correspondent, though a good observer and eloquent in description, is not
a scientific man1. But it is quite clear from what he says that a residence of a few
weeks in Galway, at the proper season, would enable a trained physicist to obtain,
with little trouble, the means of solving this extremely interesting question. He
would require to be furnished with an electrometer, a hygrometer, and a few other
simple pieces of apparatus, as well as with a light suit of plate armour, not of steel
but of the best conducting copper, to insure his personal safety. Thus armed he
might fearlessly invade the very nest or hatching-place of the phenomenon, on the
top of one of the Connemara hills. It is to be hoped that some of the rising genera-
tion of physicists may speedily make the attempt, in the spirit of the ancient chivalry,
but with the offensive and defensive weapons of modern science.

Another possible source of the electricity of thunderstorms has been pointed out
by Sir W. Thomson. It is based on the experimental fact that the lower air is
usually charged with negative electricity. If ascending currents carry up this lower
air the electricity formerly spread in a thin stratum over a large surface may, by
convection, be brought into a very much less diffused state, and thus be raised to a
potential sufficient to enable it to give a spark.

1 The correspondent was probably the late Professor D'Arcy Thompson, of Galway,
already referred to on p. 238.

INSTABILITY IN THE ATMOSPHERE 319

However the electrification of the precipitated vapour may ultimately be accounted
for, there can be no doubt of the fact that at least as soon as cloud is formed the
particles are electrified ; and what I have said as to the immense rise of potential
as the drops gradually increase in size remains unaffected. I have tried various
forms of experiment, with the view of discovering the electric state of vapour mixed
with air. For instance, I have tested the vapour which is suddenly condensed when
a receiver is partially exhausted ; the electrification of cooled bodies exposed to moist
air from a gas-holder ; and the deposition of hoar-frost from a current of moist air
upon two polished metal plates placed parallel to one another, artificially cooled, and
connected with the outer and inner coatings of a charged jar. All have given results,
but as yet too minute and uncertain to settle such a question. These experiments are
still in progress. It appears probable, so far, that the problem will not be finally solved
until experiments are made on a scale much larger than is usual in laboratories.

A great thunderstorm in summer is in the majority of cases preceded by very
calm sultry weather. The atmosphere is in a state of unstable equilibrium, the lower
strata are at an abnormally high temperature, and highly charged with aqueous
vapour. It is not easy, in a popular lecture like this, to give a full account of what
constitutes a state of stable equilibrium, or of unstable, especially when the effects
of precipitation of vapour are to be largely taken into account. It is sufficient for
my present purpose to say that in all cases of thoroughly stable equilibrium, a slight
displacement tends to right itself; while, in general, in unstable equilibrium, a slight
displacement tends to increase. Now, if two cubic feet of air at different levels
could be suddenly made to change places, without at first any other alteration, and
if, on being left to themselves, each would, under the change of pressure which it
would suddenly experience, and the consequent heating or cooling, with its associated
evaporation or precipitation of moisture, tend to regain its former level, the equilibrium
would be stable. This is not the case when the lower strata are very hot, and fully
charged with vapour. Any portion accidentally raised to a higher level tends to
rise higher, thus allowing others to descend. These, in consequence of their descent,
tend still farther to descend, and thus to force new portions up. Thus, when the
trigger is once pulled, as it were, we soon have powerful ascending currents of hot
moist air, precipitating their moisture as cloud as they ascend, cooling by expansion,
but warmed by the latent heat of the vapour condensed. This phenomenon of
ascending currents is strongly marked in almost every great thunderstorm, and is
precisely analogous to that observed in the centre of a West Indian tornado and of
a Chinese typhoon.

When any portion of the atmosphere is ascending it must be because a denser
portion is descending, and whenever such motions occur with acceleration the pressure
must necessarily be diminished, since the lower strata are not then supporting the
whole weight of the superincumbent strata. If their whole weight were supported
they would not descend. Thus even a smart shower of rain must directly tend to
lower the barometer. [A long glass tube, filled with water, was suspended in a
vertical position by a light spiral spring, reaching to the roof of the hall. A number
of bullets hung at the top of the water column, attached to the tube by a thread.
When the thread was burned, by applying a lamp, the bullets descended in the

32o PETER GUTHRIE TAIT

water, and during their descent the spring contracted so as to raise the whole tube
several inches.]

In what I have said to-night I have confined myself mainly to great thunder-
storms, and to what is seen and heard by those who are within their sphere of
operation. I have said nothing of what is commonly called summer-lightning, which
is probably, at least in a great many cases, merely the faint effect of a distant
thunderstorm, but which has also been observed when the sky appeared tolerably
clear, and when it was certain that no thunderstorm of the ordinary kind had occurred
within a hundred miles. In such cases it is probable that we see the lightning of a
storm which is taking place in the upper strata of the atmosphere, at such a height
that the thunder is inaudible, partly on account of the distance, partly on account
of the fact that it takes its origin in air of small density.

Nor have I spoken of the aurora, which is obviously connected with atmospheric
electricity, but in what precise way remains to be discovered. Various theories have
been suggested, but decisive data are wanting. Dr Balfour Stewart inclines to the
belief that great auroras, visible over nearly a whole terrestrial hemisphere, are due
to inductive effects of changes in the earth's magnetism. This is not necessarily
inconsistent with the opinion that, as ordinary auroras generally occur at times when
a considerable change of temperature takes place, they are phenomena due to the
condensation of aqueous vapour in far less quantity, but through far greater spaces,
than the quantities and spaces involved in ordinary thunderstorms.

In taking leave of you and of my subject I have two remarks to make. First,
to call your attention to the fact that the most obscure branches of physics often
present matter of interesting reflection for all, and, in consequence, ought not to be
left wholly in the hands of professedly scientific men. Secondly, that if the pre-
cautions which science points out as, at least in general, sufficient, were recognised
by the public as necessary, the element of danger, which in old days encouraged
the most debasing of superstitions, would be all but removed from a thunderstorm.
Thus the most timid would be able to join their more robust fellow-creatures in
watching fearlessly, but still of course with wonder and admiration, one of the most
exquisite of the magnificent spectacles which Nature from time to time so lavishly
provides.

MIRAGE 321

STATE OF THE ATMOSPHERE WHICH PRODUCES THE FORMS OF
MIRAGE OBSERVED BY VINCE AND BY SCORESBY.

(From Nature, Vol. xxviil, May 24, 1883.)

IN 1 88 1, when I wrote the article Light for the Encyc. Brit., I had not been
able to meet with any detailed calculations as to the probable state of the atmo-
sphere when multiple images are seen of objects situated near the horizon. I had
consulted many papers containing what are called "general" explanations of the
phenomena, but had found no proof that the requisite conditions could exist in
nature : except perhaps in the case of the ordinary mirage of the desert, where it is
obvious that very considerable temperature-differences may occur in the air within a
few feet of the ground. But this form of mirage is essentially unsteady, for it
involves an unstable state of equilibrium of the air. In many of Scoresby's obser-
vations, especially that of the solitary inverted image of his father's ship (then thirty
miles distant, and of course far below the horizon), the details of the image could be
clearly seen with a telescope, showing that the air must have been in equilibrium.
The problem seemed to be one well fitted for treatment as a simple example of
the application of Hamilton's General Method in Optics, and as such I discussed it.
The details of my investigation were communicated in the end of that year to the
Royal Society of Edinburgh, and will, I hope, soon be published. The paper itself
is too technical for the general reader, so that I shall here attempt to give a sketch
of its contents in a more popular form. But a curious little historical statement
must be premised.

It was not until my calculations were finished that I found a chance reference
to a great paper by Wollaston (Phil. Trans. 1800). I had till then known only of
Wollaston's well-known experiment with layers of different liquids in a small vessel.
But these, I saw, could not reproduce the proper mirage phenomena, as the rays
necessarily enter and emerge from the transition strata by their ends and not by their
lower sides. This experiment is by no means one of the best things in Wollaston's
paper, so far at least as the immediate object of the paper is concerned. That so much
has been written on the subject of mirage during the present century, with only a casual
reference or two to this paper, is most surprising. It may perhaps be accounted for
by the fact that Wollaston does not appear to have had sufficient confidence in his own
results to refrain from attempting, towards the end of his paper, a totally different
(and untenable) hypothesis, based on the effects of aqueous vapour. Be the cause
what it may, there can be no doubt that the following words of Gilbert were amply
justified when they were written, early in the present century: "In der That ist
Wollaston der Erste und Einzige, der die Spieglung aufwarts mit Gliick zu erklaren
unternommen hat." For his methods are, in principle, perfectly correct and sufficiently
comprehensive ; while some of his experiments imitate closely the state of the air
requisite for the production of Vince's phenomena. Had Wollaston only felt the
necessary confidence in his own theory, he could hardly have failed to recognise that
what he produced by the extreme rates of change of temperature in the small air-
T. 41

322 PETER GUTHRIE TAIT

space close to a red-hot bar of metal, could be produced by natural rates of change
in some ten or twenty miles of the atmosphere: and he would have deserved the
credit of having completely solved the problem.

Six months after my paper was read, another happy chance led me to seek for
a voluminous paper by Biot, of which I had seen no mention whatever in any of
the books I had previously consulted. The probable reason for the oblivion into
which this treatise seems to have fallen is a curious one. It forms a considerable
part of the volume for 1809 of the M£m. de I'Institut. But in the three first great
libraries which I consulted, I found this volume to be devoid of plates. In all
respects but this, each of the sets of this valuable series appeared to be complete.
Without the figures, which amount to no less than sixty-three, it is practically
impossible to understand the details of Biot's paper. The paper was, however,
issued as a separate volume, Recherclies sur les Refractions extraordinaires qui
ont lieu fres de Vhorizon (Paris, 1810), which contains the plates, and which I
obtained at last from the Cambridge University Library. I have since been able
to procure a copy for the Edinburgh University Library. Biot's work is an almost
exhaustive one, and I found in it a great number of the results which follow almost
intuitively from my methods : such as the possible occurrence of four images, under
the conditions usually assumed for the explanation of the ordinary mirage; the
effects of (unusual) refraction on the apparent form of the setting sun, etc. But it
seems to me that Biot's long-continued observations of the phenomena as produced
over extensive surfaces of level sand at Dunkirk have led him to take a somewhat
one-sided view of the general question. And, in particular, I think that his attempted
explanation of Vince's observations (so far as I am able to understand it; for it
is very long, and in parts extremely obscure and difficult, besides containing some
singular physical errors) is not satisfactory. His general treatment of the whole
question is based to a great extent upon the properties of caustics, though he
mentions (as the courbe des minima) the "locus of vertices" which I had employed
in my investigations, and which I think greatly preferable. There can be no doubt,
however, that Biot's paper comes at least next in importance to that of Wollaston :
though in his opinion Wollaston's work was complete only on the physical side of
the problem. " Sous le rapport de la physique son travail ne laisse rien a de'sirer."

But, if the chief theoretical papers on the subject have thus strangely been
allowed to drop out of notice, the case is quite different with several of those which
deal with the observed phenomena. Scoresby's Greenland, his Arctic Regions, and
his Voyage to the Northern Whale Fishery, are still standard works ; and in them, as
well as in Vols. IX and XI of the Trans. R.S.E., he has given numerous careful
drawings of these most singular appearances. The explanatory text is also
peculiarly full and clear, giving all that a careful observer could be expected to
record. It is otherwise with the descriptions and illustrations in Vince's paper {Phil.
Trans. 1799). In fact the latter are obviously not meant as drawings of what was
seen ; but as diagrams which exhibit merely the general features, such as the relative
position and magnitude of the images : the details being filled in at the option of
the engraver. That such was the view taken by Brewster, is obvious from the
illustrations in his Optics (Library of Useful Knowledge), for while one of Scoresby's

SCORESBY'S SKETCHES 323

drawings is there copied, one of Vince's is treated in a highly imaginative style by
the reproducer.

Scoresby's sketches are composite, as he takes care to tell the reader, so that in
the reproduction below (Fig. i), I have simply selected a few of the more remark -

m

Fig. i.

able portions which bear on the questions to be discussed. It is to be remarked
that the angular dimensions of these phenomena are always of telescopic magnitude :
the utmost elevation of an image rarely exceeding a quarter or a third of a degree.

Because the rays concerned are all so nearly horizontal, and (on the whole)
concave towards the earth; and because they must also have on the whole consider-
ably greater curvature than the corresponding part of the earth's surface, especially
if they happen to have points of contrary flexure ; it is clear that, for a preliminary
investigation, we may treat the problem as if the earth were a plane. This simplifies
matters very considerably, so that definite numerical results are easily obtained ; and
there is no difficulty in afterwards introducing the (comparatively slight) corrrections
due to the earth's curvature. But these will not be further alluded to here.

Of course I began, as almost every other person who has thought of the production
of the ordinary mirage of the desert must naturally have begun, by considering the
well-known problem of the paths of projectiles discharged from the same gun, with
the same speed but at different elevations of the piece. This corresponds, in the
optical problem, to the motion of light in a medium the square of whose refractive
index is proportional to the distance from a given horizontal plane. Instead, how-
ever, of thinking chiefly of the different elevations corresponding to a given range,
I sought for a simple criterion which should enable me to decide (in the optical
application) whether the image formed would, in any particular case, be a direct or
an inverted one. And this, I saw at once, could be obtained, along with the number
and positions of the images, by a study of the form of the locus on which lie the
vertices of all the rays issuing from a given point. Thus, in the ballistic problem,
the locus of the vertices of all the paths from a given point, with different elevations
but in the same vertical plane, is an ellipse.

Its minor axis is vertical, the lower end being at the gun ; and the major axis
(which is twice as long) is in the plane of projection. Now, while the inclination of
the piece to the horizon is less than 45°, the vertex of the path is in the lower half
of this ellipse, where the tangent leans forward from the gun ; and in this case a
small increase of elevation lengthens the range, so that the two paths do not inter-
sect again above the horizon. In the optical problem this corresponds to an erect

41—2

324

PETER GUTHRIE TAIT

image. But, when the elevation of the piece is greater than 45°, the vertex of the
path lies in the tipper half of the ellipse, where the tangent leans back over the
gun ; and a small increase of elevation shortens the range. Two contiguous paths,
therefore, intersect one another again above the horizon. And, in the optical
problem, this corresponds to an inverted image. In symbols, if the eye be taken as
origin and the axis of x horizontal, there will be a direct image for a ray at whose
vertex dyjdx and x (in the curve of vertices) have the same sign, an inverted image
when the signs are different.

Hence, whatever be the law of refractive index of the air, provided only it be
the same at the same distance from the earth's surface (i.e. the surfaces of equal
density parallel planes, and therefore the rays each symmetrical about a vertical
axis), all we have to do, in order to find the various possible images of an object
at the same level as the eye, is to draw the curve of vertices for all rays passing
through the eye, in tlte vertical plane containing the eye and the object, and find its
intersections with the vertical line midway between the eye and the object. As soon as
this simple idea occurred to me, I saw that it was the very kernel of the matter,
and that all the rest would be mere detail of calculation from particular hypotheses.
Each of the intersections in question is the vertex of a ray by which the object can
be seen, and the corresponding image will be erect or inverted, according as the
curve of vertices leans from or towards the eye at the intersection. Thus, in Fig. 2,

Fig. 2.

let E be the eye, and the dotted line the curve of vertices for all rays in the plane
of the paper, and passing through E. Let A be an object at the level of the eye,
A1A*A* the vertical line midway between E and A. Then A1, A*, A* are the
vertices of the various rays by which A can be seen. If we make the same con-
struction for a point B, near to A, we find that whereas the contiguous rays through
A1, B1 and through A*, B1 do not intersect, those through A', B3 do intersect. At
A1 and A* the curve of vertices leans from the eye, and we have erect images ; at
A* it leans back towards the eye, and we have an inverted image. And thus, if
this curve be continuous, the images will be alternately erect and inverted. The
sketch above is essentially the same as one given by Vince, only that he does not

CURVE OF VERTICES

325

employ the curve of vertices. If the object and eye be not at the same level, the
construction is not quite so simple. We must now draw a curve of vertices for rays
passing through the eye, and another for rays passing through the object. Their
intersections give all the possible vertices. (This construction of course gives the
same result as the former, when object and eye are at the same level.) But the
images are now by no means necessarily alternately erect and inverted, even though
the curve of vertices be continuous. However, I merely note this extension of the
rule, as we shall not require it in what follows.

I then investigated the form of the curve of vertices in a medium in which the
square of the refractive index increases by a quantity proportional to the square of
the distance from a plane in which it is a minimum, and found that (under special
circumstances, not however possible in air) three images could be produced in
such a medium. But the study of this case (which I could not easily explain
here without the aid of mathematics) led me on as follows.

As the curvature of a ray is given by the ratio of the rate of change of index
per unit of length perpendicular to the ray, to the index itself (a result which I find
was at least virtually enunciated by Wollaston) ; and as all the rays producing the
phenomena in question are very nearly horizontal : i.e. perpendicular to the direction
in which the refractive index changes most rapidly : their curvatures are all
practically the same at the same level. Hence if the rate of diminution of the
refractive index, per foot of ascent, were nearly constant, through the part of the
atmosphere in which the rays travel, the rays we need consider would all be
approximately arcs of equal circles ; and the curve of vertices would (so far as these
rays are concerned) lean wholly from the eye ; being, in fact, the inferior part of
another equal circle which has its lowest point at the eye. Hence but one image,
an erect one, would be formed; but it would be seen elevated above the true
direction of the object. This is practically the ordinary horizontal refraction, so far
as terrestrial objects on the horizon are concerned. The paths of the various rays

Fig. 3-

would be of the form in Fig. 3 (the drawing is, of course, immensely exaggerated)
and the locus of vertices, A, B, C, obviously leans from the eye. But now suppose
that, below a stratum of this kind, there were one of constant density, in which of

Fig. 4.

course the rays would be straight lines. Then our sketch takes the form Fig. 4
(again exaggerated) ; each of the portions of the ray in the upper medium being

3a6 PETER GUTHRIE TAIT

congruent to the corresponding one in the former figure (when the two figures are
drawn to the same scale), but pushed farther to the right as its extremities are less
inclined to the horizon. In its new form the curve of vertices ABC leans back
towards the eye, and we have an inverted image. The lower medium need not be
uniform as, for simplicity, we assumed above. All that is required is that the rate
of diminution of density upwards shall be less in it than in the upper medium.

Those who have followed me so far will at once see that, as a more rapid
increase of density, commencing at a certain elevation, makes the curve of vertices
lean back, so a less rapid decrease (tending to a " stationary state ") at a still higher
elevation will make the curve of vertices again lean forward from the eye. I need
not enlarge upon this.

Thus to repeat : the conditions requisite for the production of Vince's pheno-
menon, at least in the way conjectured by him, are, a stratum in which the refractive
index diminishes upwards to a nearly stationary state, and below it a stratum in
which the upward diminution is either less or vanishes altogether. The former con-
dition secures the upper erect image, the latter the inverted image and the lower
direct image.

In my paper read to the Royal Society of Edinburgh I have given the mathe-
matical details following from the above statement ; and have made full calculations
for the effect of a transition stratum, such as must occur between two uniform strata
of air of which the upper has the higher temperature. From Scoresby's remarks it
appears almost certain that something like this was the state of affairs when the
majority (at least) of his observations were made. When two masses of the same
fluid, at different temperatures, rest in contact ; or when two fluids of different
refractive index, as brine and pure water, diffuse into one another; the intervening
layer must have a practically " stationary " refractive index at each of its bounding
surfaces, and a stratum of greatest rate of change of index about midway between
them. The exact law of change in the stratum is a matter of comparatively little
consequence. I have assumed (after several trials) a simple harmonic law for the
change of the square of the refractive index within the stratum. This satisfies all
the above conditions, and thus cannot in any case be very far from the truth. But
its special merit, and for my purpose this was invaluable, is that it leads to results
which involve expressions easily calculated numerically by means of Legendre's
Tables of Elliptic Integrals. This numerical work can be done once for all, and
then we can introduce at leisure the most probable hypotheses as to the thickness of
the transition stratum, the height of its lower surface above the ground, and the
whole change of temperature in passing through it. I need not now give the
details for more than one case, and I shall therefore select that of a transition
stratum 50 feet thick, and commencing 50 feet above the ground. From the
physical properties of air, and the observed fact that the utmost angular elevation
of the observed images is not much more than a quarter of a degree, we find that
the upper uniform layer of air must under the conditions assigned be about 70° C.
warmer than the lower. Hence by the assumed law in the stratum, the maximum rise
of temperature per foot of ascent (about the middle of the transition stratum) must
be about o° "2 C. per foot. Such changes have actually been observed by Glaisher

TEMPERATURE GRADIENT IN AIR

327

in his balloon ascents, so that thus far the hypothesis is justified. But we have an
independent means of testing it. The form of the curve of vertices is now some-
what like the full lines in the following cut (Fig. 5), where 0 is the eye, and the
lines RS, TU represent the boundaries of the transition stratum. It is clear, that,
if PM be the vertical tangent, there can be but one image of an object unless
its distance from O is at least twice OM. This will therefore be called the " Critical
distance" If the distance be greater than this there are three images : — one erect,
seen directly through the lower uniform stratum — then an inverted one, due to the
diminution of refractive index above the lower boundary of the transition stratum —
and finally an erect image, due to the approximation to a stationary state towards
the upper boundary of that stratum. Now calculation from our assumed data gives
OM about six miles, so that the nearest objects affected should be about twelve
miles off. Scoresby says that the usual distance was from ten to fifteen miles. Thus
the hypothesis passes, with credit, this independent and severe test. A slight
reduction of the assumed thickness of the transition stratum, or of its height above
the ground, would make the agreement exact.

All the phenomena described in Vince's paper of 1799, as well as a great many
of those figured in Scoresby's works, can easily be explained by the above assump-
tions. Scoresby's remarkable observation of a single inverted image of his father's
ship (when thirty miles off, and of course far below the horizon) requires merely a
more rapid diminution of density at a definite height above the sea. His figure is
the second in Fig. i above. But Scoresby figures, as above shown, several cases
in which two or more inverted images, without corresponding erect ones, were seen
above the ordinary direct image. The natural explanation is, of course, a series of
horizontal layers of upward diminishing density and without a " stationary state "
towards their upper bounding planes. I find that, by roughly stirring (for a very
short time) a trough in which weak brine below is diffusing into pure water above, we
can reproduce this phenomenon with great ease. In fact, when temporary equilibrium
sets in, the fluids are arranged in a number of successive parallel strata with some-
what abrupt changes of density.

But the mathematical investigation, already spoken of, shows that it is quite
possible that there may be layers tending to a stationary state without any corre-
sponding visible images.

This depends on the fact that, while the inverted image (due to the lower part
of a stratum) is always taller than the object seen directly (though not much taller
unless the object is about the critical distance), the numerical calculation shows
that the erect image is in general extremely small, and can come into notice only

328 PETER GUTHRIE TAIT

when the object is not far beyond the critical distance. Thus there may have been,
in all of Scoresby's observations (though he has only occasionally noticed and depicted
them) an erect image above each inverted one, but so much reduced in vertical
height as to have been invisible in his telescope, or at least to have formed a mere
horizontal line so narrow that it did not attract his attention. The greatly superior
number of inverted images, compared with that of the direct ones, figured by
Scoresby, may thus be looked upon as another independent confirmation of the
approximate correctness of the hypothetical arrangement we have been considering.

To obtain an experimental repetition of the phenomena in the manner indicated
by the above hypothesis, a tank, with parallel glass ends, and about 4 feet long,
was half-filled with weak brine (carefully filtered). Pure water was then cautiously
introduced above it, till the tank was nearly filled. After a few hours the whole
had settled down into a state of slow and steady diffusion, and Vince's phenomenon
was beautifully shown. The object was a metal plate with a small hole in it, and
a lamp with a porcelain globe was placed behind it. The hole was triangular, with
one side horizontal (to allow of distinction between direct and inverted images)
and was placed near one end of the tank, a little below the surface-level of the
unaltered brine, the eye being in a corresponding position at the other end. A
little vertical adjustment of object and eye was required from time to time as the
diffusion progressed. The theoretical results that the upper erect image is usually
much less than the object, and that it is seen by slowly convergent rays, while the
inverted image is larger than the object and is seen by diverging rays, were easily
verified.

To contrast Wollaston's best-known experiment with this, a narrow tank with
parallel sides was half-filled with very strong brine, and then cautiously filled up
with pure water. (The strong brine was employed to make up, as far as possible,
for the shortened path of the rays in the transition stratum.) Phenomena somewhat
resembling the former were now seen, when object and eye were nearly at the
same distance apart as before, and the tank about half-way between them. But in
this case the disparity of size between the images was not so marked — the upper
erect image was always seen by diverging rays, the inverted image by rays
diverging or converging according as the eye was withdrawn from, or made to
approach, the tank. In this case, the curvature of each of the rays in the vessel
is practically constant, but is greatest for the rays which pass most nearly through
the stratum of most rapid change of refractive index. Hence, when a parallel beam
of light fell horizontally on the tank and was received on a sufficiently distant
screen, the lower boundary of the illuminated space was blue — and the progress of
the diffusion could be watched with great precision by the gradual displacement of
this blue band. I propose to employ this arrangement for the measurement of the
rate of diffusion, but for particulars I must refer to my forthcoming paper.

Wollaston's experiment with the red-hot poker was probably, his experiment
with the red-hot bar of iron almost certainly, similar to that above described with
the long tank, and the weak brine ; and not to that with the short tank, though
the latter is usually cited as Wollaston's contribution to the explanation of the
Vince phenomenon. We have seen how essentially different they are, and that the
latter does not correspond to the conditions presented in nature.

GOLF 329

LONG DRIVING
(From the Badminton Magazine*, March, 1896.)

Error ubique patet; falsa est doctrina periti :
Sola fides numeris intemerata manet.

In the great drama, familiarly known as a Round of Golf, there are many Acts,
each commonly but erroneously called a Stroke. Besides Acts of Driving, to which
this article is devoted, there are Acts of Approaching, Acts in (not always out of)
Hazards, and Acts of Holing-out. There is another class of Acts, inevitable as
human beings are constituted — Acts of Negligence, Timidity, or Temerity. Of these
we cannot complain, and they give much of its interest to the game. A philosophic
professional, after missing an easy putt, put this aspect of the game in words which
could scarcely be improved : " If we cud a' aye dae what we wantit, there wud be
nae fun in't." Besides these, there are, too frequently, other Acts wholly superfluous
and in general injurious, to the game : Acts of Gambling, Fraud, and Profanity.
These, however, belong to the domain of the moral, rather than of the natural,
philosopher.

Each Act of golf proper has several Scenes. An Act of Driving essentially
contains four; besides the mere preliminary work of the caddie, such as teeing the
ball (when that is permissible), handing his master the proper club, and clearing
loafers and nursery maids out of the way. These are : —

SCENE I. — Stance, Waggle, and Swing.

Here the only dramatis personae directly engaged are the player and his club.
This scene is rarely a brief one, even with the best of players, and is often absurdly
protracted.

SCENE II.— The Stroke Proper.

The club and the ball practically share this scene between them; but the
player's right hand, and the resistance of the air, take some little part in it. It is a
very brief one, lasting for an instant only, in the sense of something like one ten-
thousandth of a second. Yet in that short period most important events take place.
[Sometimes, it is true, this scene does not come off at all, the club passing, instead
of meeting, the ball. It is called a stroke for all that, and is sedulously noted on
the scorer's card.]

SCENE III.— The Carry.

Here the action is confined to the ball, gravity, and the atmosphere. The scene
may last for a second or two only, if the ball be topped, or if a poor player is at
work ; but with good drivers it usually takes six seconds at the very least.

1 The parts in square brackets were on a first proof preserved by Tait, but were deleted
in the final proof.

T. 4a

330 PETER GUTHRIE TAIT

SCENE IV.—TAe Run.

Here pure chance is the main actor. The scene has no measurable duration
when the ball lands in mud or soft sand. It may continue for two, three, or more
seconds if the ball be topped or get a running fall, and the links be hard and keen.
The ball's progress may be by mere rolling or by a series of leaps. This is usually
(at least on a "sporting" course) a most critical scene, and the player feels himself
breathing more freely when it is safely concluded.

Our chief concern is with the second and third of these scenes ; the fourth,
from its very nature, being of such a capriciously varying character that it would
be vain to attempt a discussion of it, and the first being of interest to our present
purpose only in so far as it is a necessary preliminary to the second. Its result, as
far as we are concerned, is merely to bring the club face into more or less orderly
and rapid contact with the ball. We say advisedly the club face (not merely the
club head), for operations such as hacking and sclaffing, however interesting in them-
selves, form no part of golf, properly so-called.

All the resources of the pen, the pencil, and the photographic camera have
been profusely employed on behalf of the public to convey to it some idea of the
humours of Scene I., so that we may omit the discussion of it also. [Yet, after all,
sketching and word painting, even when the most amazing vagaries of the most
fertile imagination have been freely taken advantage of, are sometimes far less effec-
tive in this matter than is the simple verity as recorded by the imperturbably
truthful camera. For there are many things, sad or laughable, unimportant or of
the most immense consequence to the discovery of peculiarities of style, which
escape the keenest vision, and yet are seized upon and preserved for leisurely after-
study in a single wink, as it were, of that terrible photographic Eye.]

Brief as is the duration of the second scene, the analysis of even its main
features requires considerable detail if it is to be made fully intelligible. I will
attempt to give this in as popular a form as possible. But before I do so, it may
be well to show its importance by a passing reference to some of its consequences,
as we shall thus have a general notion of what has to be explained. When the
ball parts company with the club this scene ends, and the third scene begins. Now,
at that instant, having by its elasticity just recovered from the flattening which it
suffered from the blow, the ball must be moving as a free rigid solid. It has a
definite speed, in a definite direction, and it may have also a definite amount of
rotation about some definite axis. The existence of rotation is manifested at once
by the strange effects it produces on the curvature of the path, so that the ball
may skew to right or left, soar upwards as if in defiance of gravity, or plunge head-
long downwards, instead of slowly and reluctantly yielding to that steady and
persistent pull. The most cursory observation shows that a ball is hardly ever
sent on its course without some spin, so that we may take the fact for granted, even
if we cannot fully explain the mode of its production. And the main object of this
article is to show that LONG CARRY ESSENTIALLY INVOLVES UNDERSPIN.

Now, if golf balls and the faces of clubs were both perfectly hard (i.e. not
deformable) the details of the effects of the blow would be a matter of .simple

CONDITIONS OF IMPACT 331

dynamical calculation, and Scene II. would be absolutely instantaneous. If, in ad-
dition, the ball were perfectly spherical, smooth, and of homogeneous material, no
blow could possibly set it in rotation ; if it were defective in any of these particulars,
we could easily calculate the direction of the axis of rotation and the amount of
spin produced in it by any assigned blow. But, unfortunately, neither balls nor
clubs can make an approach to perfect hardness. For there is never, even in a
gentle stroke, a mere point of contact between ball and club. In good drives the
surface of contact may often, as we see by an occasional trace from undried paint
or by the pattern impressed by the first drive on a new leather face, be as large
as a shilling. The exact mathematical treatment of so large a distortion is an
exceedingly complex and difficult matter. But fortunately we are not called upon
to attack it, for it is obvious, from the facts of common observation already cited,
that the final effect on the ball is of the same general character as if it had been
perfectly hard, though the speed of projection, and notably that of spin, will be
materially less. And it is with the character rather than with the amount of the
effect that we are mainly concerned, as will be seen farther on. Thus, in a great
part of what follows, we will argue as if both club and ball were hard, since we
seek to explain the character of the results and are not for the time concerned with
their magnitiides.

When we reflect on the very brief duration of the impact, during which the
average force exerted is about three tons' weight (while the player is, for an instant,
working at two or three horse-power at least), we see at once that we may practi-
cally ignore the effects of gravity, of the continued pushing forward of the club head,
and even of the resistance of the air (though amounting to, say, five-fold the weight
of the ball) during that short period ; so that we are concerned only with the
velocity and the orientation of the club face at the moment of impact.

The simplest case which we have to consider is that in which the club face,
at the instant of meeting the ball, is moving perpendicularly to itself. If the ball
be spherical and homogeneous, there can be no spin ; and thus we are concerned
only with the steps of the process by which the ball ultimately leaves the club in
the common direction of motion. The first effect is the impulsive pushing forward
of the part of the ball which is struck, the rest, by its inertia, being a little later in
starting. Thus the ball and club face are both distorted until they, for an instant,
form, as it were, one body, which has the whole momentum which the club head
originally possessed. As the club head is usually about five times more massive
than the ball, the common speed is five-sixths only of the original speed of the
head. [In the case of a more massive club head a correspondingly less fraction
of its speed would be lost, but a proportionately greater effort would be required
to give it a definite speed to begin with. A sort of compromise must thus be
made, and experience has led to the proportion cited above — so long, at least, as
we are dealing with balls of about ten or eleven to the pound.] But the ball and
club both tend to recover from their distortion, and experiment shows that they
exert on one another, during this recovery, an additional impulsive pressure which
is a definite fraction of that already exerted between them. This fraction is tech-
nically called the " co-efficient of restitution," and it is upon its magnitude that

42 — a

332 PETER GUTHRIE TAIT

the higher or lower quality of a ball, and of a club face, mainly depend. Its value,
when good materials are employed, is usually about O'6. Thus the club and ball
at last separate with a relative speed six-tenths of that with which the club approached
the ball. The ball, therefore, finally acquires a speed about one-third greater than
that which the club head originally had. Thus the head must have a pace of about
1 80 feet per second in order that it may drive the ball at the rate of 240 feet per
second. And, for various values of the co-efficient of restitution, the ultimate pace
of the ball can never be less than five-sixths, nor as much as five-thirds, of the
initial speed of the club head. We thus get at present about four-fifths of what
is (theoretically) attainable ; and we may perhaps, by means of greatly improved
materials, some day succeed in utilising a considerable part of this wasted fifth.

Where, as is almost invariably the case, the face of the club is not moving
perpendicularly to itself at impact there is always one perfectly definite plane which
passes through the centre of the ball and the point of first contact, and is parallel
to the direction of motion of the head. It is in this plane, or parallel to it, that
the motions of all parts of the ball and the club head (except, of course, some of
the small relative motions due to distortion) take place. Hence if the ball acquire
rotation it must be about an axis perpendicular to this plane. The whole circum-
stances of the motion can therefore be, in every case, represented diagram matically
by the section of the ball and club face made by this plane. The diagram may
take one or other of the two forms below, either of which may be derived from the
other by perversion and inversion.

[Thus, if the page be turned upside-down and held before a mirror, the result
will be simply to make the first figure into the second, and the second into the first
— merely, in fact, altering their order. Holding the page erect, before a mirror, we
get diagrams specially suited for a left-handed player.]

In each of the figures the velocity of the club head at impact is represented
by the line AB, and the dotted lines AC and CB represent its components parallel
and perpendicular to the club face respectively. By properly tilting the figures,
AB may be made to take any direction we please, i.e. — the club head may be
represented as moving in any direction whatever — but it is quite sufficient for our
purpose to treat it as moving horizontally. It is the existence of the component
velocity AC, in a direction parallel to the club face, which (alone) makes the differ-
ence between this case and the simple one which we have just treated. And if the
ball were perfectly smooth this component would lead to no consequences. But
because of friction this component produces a tangential force whose effect is partly
to give the ball as a whole a motion parallel to AC, partly to give it rotation in
the direction indicated by the curved arrow. The direction of motion of the ball

FAULTS OF STROKE 333

when free lies somewhere between the directions of AS and CB, say in the line DB.
[The reader must take this statement for granted, if it be not pretty obvious to
him ; for its proof, even in the simple case in which the ball is regarded as perfectly
hard, involves the consideration of moment of inertia, which I must not introduce.]

As already remarked, one or other of the diagrams above applies to any possible
case. But there are two special cases which are of paramount importance, and if
these be fully understood by him the reader can easily make for himself the appli-
cation to any other.

In the first of these special cases the plane of the diagram is to be regarded
as horizontal, and the club face (perpendicular to it by the conditions of the diagram)
consequently vertical, while the rotation given to the ball is about a vertical axis.
The spectator is, therefore, supposed to be looking down upon the club and ball
from a station high above them. The interpretation of the indicated result thus
depends upon the direction of the line joining the player's feet. If that line be
(as it ought to be) perpendicular to the face of the club, it is parallel to BC\ so
that the club (when it reaches the ball) is being pulkd in (first figure), or pushed
out (second figure), in addition to sweeping past in front of the player parallel to
the line joining his feet. The first of these is the very common fault called " slicing."
The second is not by any means so common, and I am not aware that it has ever
been dignified by a special name. If, on the other hand, the line of the feet be
parallel to AB, the sweep of the club head is in the correct line, but the face is
turned outwards (first figure), or inwards (second figure), and we have what is
called "heeling" or "toeing." These terms must not be taken literally, for heeling
may be produced by the toe of the club and toeing by the heel. Slicing and heeling
have thus precisely the same effect, so far as the rotation (and consequent " skewing ")
of the ball is concerned ; but the position of the line DB shows that, other things
being correct, a sliced ball starts a little to the left of the intended direction, while
the heeled ball commences its disastrous career from the outset by starting a little
to the right. It is most important to the player that he should be able to distin-
guish between these common faults, because, though their (ultimate) results are
identical, the modes of cure are entirely different. This, of course, is obvious from
what we have said above as to the intrinsic nature of each. Toeing, and the
innominate fault mentioned above, both give the opposite rotation to that produced
by heeling, and therefore the opposite skew. If slicing and toeing occur together,
each tends to mitigate the evil effects of the other ; so with heeling and the innomi-
nate. But slicing and heeling together will produce aggravation of each other's effects.

In the second special case the plane of the diagram is regarded as vertical,
and the spectator's line of sight passes horizontally between the player's feet from
a point behind him. The first diagram, therefore, corresponds to under-cutting, and
the second to topping, if AB be horizontal ; or to jerking, and bringing the club
upwards behind the ball1, respectively, if the face be vertical. The first diagram

1 This suggests a very favourite diagram employed by many professed instructors in the
game. It is usually embellished with a full circle, intended to show the proper path of the
head, and the ball is placed (on a high tee) a good way in front of the lowest point of the
circle. It will be seen from the text above that this virtual pulling in of the club head produces,
in a vertical plane, the same sort of result as does slicing in a horizontal plane.

334 PETER GUTHRIE TAIT

also represents the natural action of a spoon, or a "grassed" play club, AB being
horizontal. In all these cases the spin is about a horizontal axis, and therefore the
skewing is upwards or downwards. Thus, we have traced out generally, and also
specially for the most important cases, the processes of the second scene, which
usher the ball into the third with a definite speed and a definite rotation.

In the discussion of the third scene, in which the ball is left to its own
resources, to struggle as best it can against the persistent downward pull of gravity
and the ever-varying resistance of the air, we will treat fully of really good drives
— i.e., those in which the spin, if there be any, is about a horizontal axis perpen-
dicular to the plane of flight, and is such as to cause the ball to "soar," not to
"dook." Incidentally, however, we will notice (though with much less detail) the
causes which produce departures of various kinds from a high standard of driving.

We will treat, first, of the path as affected by gravity alone ; second, of the path
under gravity and resistance alone (the ball having no rotation) ; third, of the
path as it would be if the ball were spinning, but not affected by gravity ; fourth,
as it is when all these agents are simultaneously at work ; and, finally, the effects
of wind. The first and third of these, in each of which one of the most important
agents is left wholly out of account, though of less consequence than the others,
are necessary to the proper development of the subject, inasmuch as their prelimi-
nary treatment will enable us to avoid complications which might embarrass the
reader.

i. If there were no resistance, the path of a golf ball would be part of a
parabola, BAG, whose axis, AD, is vertical. The vertex, A, of the path would be
always midway along the range, BC; and the ball would reach the ground with
the speed given it from the tee. A golf ball would therefore be an exceedingly
dangerous missile. For fairish but high driving would easily make the range BC
something like a quarter of a mile ! And at that distance the ball would fall with
precisely the same speed as that with which it left the tee The range for any
definite " elevation " (i.e. angle at which the path was inclined to horizon at starting)
would be proportional to the square of the initial speed, so that double speed
would give quadruple range ; and for any given speed, it would increase with the
elevation up to 45°, and thence diminish with greater elevation. Any one can test
this last result by means of the jet from a garden engine. The speed of such a jet
is so small that the resistance is inconsiderable.

For comparison with some of the numerical results to be given below, we will
here give a few simple particulars.

SIMPLE THEORY OF PROJECTILES 335

Suppose the ball to have an initial speed of 200 feet per second ; we have

Elevation

Range

Height

Time

75°

631

579

12

45

1,242

3*0-5

8-8

IS

621

41 '6

3'2

7'5

321

io'6

1-6

The lengths are in feet and the times in seconds. Notice that for elevation 15"
we get a range of 207 yards, with a maximum height of 14 yards. These, so far,
are not very unlike what may sometimes occur in an actual drive. But we must
look to all the facts ; and this closer comparison shows the resemblance to be only
superficial. For, first, the vertex is midway along the path ; second, the ball comes
down, as it rose, at 200 feet per second. These are utterly contrary to experience.
But, third, this long journey is effected in little more than three seconds. A golfer
finds that it requires nearly seven seconds. The unresisted projectile theory is thus
completely at fault, so far as application to golf is concerned,

2. Let us next consider the effect of atmospheric resistance, the ball having
no spin. [This was, and unfortunately must continue to be, a matter of grave
concern to myself. For when I began to learn golf, my instructor (an elderly man,
but a very fair player for all that) urged me to bear constantly in mind that "all
spin is detrimental" This was, he told me, the definite result of his long experience.
It cost me much thought, and long practice, to carry out his recommendation, and
it is possible that I have more personal experience of the behaviour of balls almost
free from spin than has any other player. The more nearly I approached this
ideal the greater was the proportion of run to carry in my driving. I understand
it now — too late by thirty-five years at least.]

It has already been said that want of homogeneity in a spherical ball almost
certainly leads to its getting spin from the very tee. But, even should it be pro-
jected without rotation, it will soon acquire some as it moves through the air. The
spin so acquired will be of an uncertain and variable nature, and the flight of the
ball will be unsteady and erratic. [I have elsewhere explained how to test balls
for this defect, by merely making them oscillate while floating on mercury. Any
which oscillate quickly are absolutely useless.]

It seems to be pretty well established that, for the range of speed common in
golf, the resistance is as the square of the speed. In fact, the faster a ball moves
the more air does it displace in a given time and also the faster does it make that
air move. The most convenient mode of expressing the amount of resistance is
to assign the "terminal velocity" of the ball, i.e. its speed when the resistance is
just equal to its weight If a sack full of golf balls were emptied at a height of
three or four miles, the balls would reach the ground with their terminal velocity.

336

PETER GUTHRIE TAIT

Robins, long ago, gave data from which we can assign 114 feet per second as the
terminal velocity of a golf ball. His experiments were all made at moderate speeds.
The comparatively recent experiments of Bashforth, though mainly directed to very
high speeds, give for speeds under 800 feet per second the means of assigning 95
feet per second as the terminal velocity of a golf ball. When I began to make
calculations on this subject, I naturally took the more recent determination as the
correct one ; and was thus forced to assume at least 300 feet per second as the
initial speed of a golf ball in order to account for some of the simplest facts.
I have since found that this estimate is very considerably in excess of the truth,
and therefore that the terminal velocity, as assigned by Bashforth's data, is con-
siderably too small. I still, however, think that Robins' estimate is somewhat too
low, so that in what follows I shall assume 108 feet as the terminal velocity for an
ordinary golf ball. Trifling as may appear the differences among these numbers
(114, 95, and 108), experiments on initial speed seem to show that, if the first were
correct, we ought to drive somewhat further than we do ; while, if the second be
correct, it is quite certain that, on the open links, considerably greater initial speeds
are given than any which I have been fortunate enough to measure in my laboratory,
though I have had the kind assistance of some of the most slashing drivers of the
day1.

Assuming, then, 108 as the terminal velocity, it is easy to calculate by the help
of Bashforth's tables (which can be adapted to any amount of resistance) the
following sufficiently approximate results for different elevations, the initial speed
being assumed to be 240 feet per second.

Elevation

Range

Height

Speed at
Vertex

Time

Speed of
Descent

5°°

440

206

59

6'95

90

43

467

'75

68

6-5

88

37'5

474

144

76

6-16

84

34

475

119

83

5 '43

82

29

467

99

90

4 '9

82

23

445

70-6

IO2

4'i

82

18-5

412

5°

"3

3 '47

84

iS'3

389

38

1227

3'03

86

I3-3

341

28-5

I24-3

271

87-4

IO-2

324

20-5

I40-4

2'24

IOO

One feature, at least, in this table is much more consonant with experience
than that corresponding to the parabolic path. The ball reaches the ground in all
these courses with speed far inferior to that with which it started.

1 This paragraph and the paragraph and Table following were considerably abridged in
the article as published in the Badminton Magazine.

TRAJECTORIES FOR VARIOUS SPEEDS

337

I purposely assumed 240 feet per second for the initial speed, though it is
greater by ten per cent, than any I have yet actually measured1, in order to give
this form of the theory as fair play as could be equitably conceded. Yet the
utmost attainable range, as shown by the table, is short of 160 yards, while the
players whose pace I measured had habitually carried about 180 yards the previous
day, or were to do it the next. But I measured their habitual elevation as well as
their pace. It was always small, rarely more than i in 6, i.e. less than 10°. The
average was about I in 7, or little over 8°. Look again to the table, and we see
that the maximum range for a speed of 240 (though only about 160 yards at best)
involves an elevation of some 34°, altogether unheard of in long driving, while the
elevation of 10° gives a range of 108 yards only ! Thus this form of the theory
also breaks down completely. But, before we altogether dismiss it, let us test it in
two other and perfectly different ways.

It might be objected to the reasoning above that I have taken initial data
from experiments on a limited number of exceptionally good players ; and that I
have somewhat arbitrarily assumed a value for the resistance of the air greater than
that given by one recognised authority and considerably less than that given by
another. One good, though only partial, answer is to change my authority so far
as elevation is concerned ; and, keeping to the same co-efficient of resistance, find
the characteristics of drives with various initial speeds.

Some six years ago, Mr Hodge kindly measured for me, by means of a clino-
meter, the average elevation of drives made by a great number of good players
from the first tee at St Andrews. He estimated it at about i3'S°. Bashforth's
tables usually involve data for whole degrees only, so I shall assume 14° as the
standard elevation. Here are the results for various initial speeds, some wholly
unattainable :

Initial speed

66

"5

142

i74

«S

275

382

739

Final speed

S6'S

80

84

88

90

9i '3

91-2

907

Range ...

58

MS

197

257

329

421

SS4

833

Height

37

i°'3S

i47

20-3

27-6

44-6

56

101

Time

©•96

r6

1-9

2'2

2-6

3

3-6

47

Position of vertex

o'Si

o'S3

°-55

0'S6

o'S7

o«6

0-63

0-67

Note, first, that in all these paths the time is much too short ; second, that the
initial speed required for a carry of 184 yards is 382 feet per second, which must

1 In my laboratory experiments the players could not be expected to do full justice to
their powers. They had to try to strike, as nearly as possible in the centre, a ten-inch disc
of clay, the ball being teed about six feet in front of it. Besides this preoccupation, there
was always more or less concern about the possible consequences of rebound, should the
small target be altogether missed,

T. 43

338

PETER GUTHRIE TAIT

be regarded as totally unattainable; also that the position of the vertex in that
case is little more than three-fifths of the range from the tee.

Next, let us simply take 10° for elevation and 240 for initial speed, and find
the results for various amounts of resistance or terminal velocities. Here they are :

Terminal velocity

43'

245

176

'33

114

98

82

Range ...

566

506

440

363

328

287

240

Height

26

24-4

22-5

2O'6

18-9

17-2

I5'3

Time

2'S

2-46

2-36

2-23

2-14

2-05

2'OI

The only approach to the practical range of good driving is in the first column.
But the corresponding resistance is about one-sixteenth only of our estimate, and
the time is absurdly too small. Thus the ordinary resistance theory also fails to
explain the facts. When compared with them it breaks down almost as completely
as did the parabolic theory.

In what precedes I have endeavoured to make it perfectly clear that something
else besides mere speed and elevation is required, if all the ordinary facts of long
driving are to be simultaneously accounted for. Great initial speed is required if
the resistance is great, and the larger these are the further is the vertex from mid-
way, but then the time taken for a range of 180 yards will be much too short.
In order that 180 yards may be covered in six seconds the average horizontal speed
must be only 90 feet per second, and gravity would cut short the ball's flight long
before it had reached the goal ; unless, by way of preventing this, we gave it an
extravagant elevation at starting. And, in all cases, the path will be concave down-
wards throughout its whole extent. In many fine drives it is concave itpwards for
nearly half of the range. The sole additional consideration to which we can have
recourse to help us in reconciling these apparently inconsistent facts is rotation of
the ball — to which we are thus compelled to have recourse !

I have been very, perhaps even unnecessarily, cautious in leading up to this
conclusion ; and I have consequently been careful to fortify my position from time
to time by an appeal to the recognised maxim, Mundum regunt nutneri. I have a
vivid recollection of the " warm " reception which my heresies met with, some years
ago, from almost all of the good players to whom I mentioned them. The general
feeling seemed to be one in which incredulity was altogether overpowered by disgust.
To find that his magnificent drive was due merely to what is virtually a toeing
operation — performed, no doubt, in a vertical and not in a horizontal plane — is too
much for the self-exalting golfer !

The fact, however, is indisputable. When we fasten one end of a long un-
twisted tape to the ball and the other to the ground, and then induce a good
player to drive the ball (perpendicularly to the tape) into a stiff-clay face a yard or
two off, we find that the tape is always twisted ; no doubt to different amounts by

EFFECT OF ROTATION 339

different players, but in such wise as to show that the ball makes usually from
about i to 3 turns in six feet — say from 40 to 120 turns or so per second. This
is clearly a circumstance not to be overlooked.

3. Some 230 years ago, Newton employed the analogy of the curved path
of a tennis ball " struck with an oblique racket " to aid him in explaining the separa-
tion of the various constituents of white light by a prism. And he says, in words
which apply aptly to the behaviour of a golf ball, " a circular as well as a progres-
sive motion being communicated to it by that stroke, its parts, on that side where
the motions conspire, must press and beat the contiguous air more violently than
on the other, and there excite a reluctancy and reaction of the air proportionally
greater." In other words, the pressure of the air is greater on the advancing than
on the retreating side of the ball, so that it is deflected from its course in the same
direction as that of the motion of its front part, due to the rotation. This explana-
tion has not since been improved upon, though the fact itself has been repeatedly
verified by many experimenters, including Robins and Magnus.

That the deflecting force thus called into play by the rotation of the ball may
be of considerable magnitude is obvious from the fact of the frequently observed
upward concavity of the earlier part of the path. For this shows that, at first,
the new force is greater than the weight of the ball. It is thus greater than one-
fifth of the direct resistance when the latter has its maximum value. Its magnitude
depends upon the rate of spin, and also upon the speed of the ball, and may be
regarded as directly proportional to their product. And we know, from the way in
which the ball behaves after falling, that the spin does not diminish very rapidly,
for a good deal of it remains at the end of the carry. It is probable that the spin
contributes to the direct resistance also ; and this was one of my reasons for assuming
a terminal velocity somewhat less than that deduced from the datum of Robins.
Two important effects of hammering, or otherwise roughening, the ball are now
obvious : it enables the club to " grip " the ball firmly, so as to secure as much spin
as possible, and it enables the ball, when free, to utilise its spin to the utmost.

Some of the effects due to resistance and spin alone are very curious. Thus a
top or " pearie " spinning with its axis vertical on a smooth horizontal plane is
practically free from the effect of gravity. If it receive a blow which tends to give
it horizontal motion only, it moves in an endless spiral, coming back, as it were,
to receive a second blow. The sense of the spiral motion is the same as that of the
rotation. If its spin were to fall off at the same rate as does its speed of transla-
tion, the spiral path would, as it were, uncoil itself into a circular one.

Closely related to this is Robins' experiment with a pendulum whose bob is
supported by two strings twisted together, so that they set it in rotation as they
untwist. The plane of the pendulum's vibration constantly turns round in the same
sense as does the bob.

If the bob be supported by a fine wire to whose upper end torsion can be
applied, it may be made to move as a conical pendulum. Then its path will shrink,
or open out, as the bob is made to rotate in, or against, the sense of the revolution.

When a narrow, rectangular, slip of paper is let fall, with its greater sides
horizontal, it usually begins to spin about its longer axis, and at a rate which is

43—*

340 PETER GUTHRIE TA1T

generally greater the narrower it is. Then it falls almost uniformly in a nearly
straight path, considerably inclined to the vertical. The deflection is always towards
the side to which the edge of the strip which is at any instant the lower, and
therefore the foremost, is being carried by the rotation. If the longer edges be
not quite horizontal, the path is usually a nearly perfect helix, the successive posi-
tions of the upper surface being arranged very much like the steps of a spiral
staircase. This is an exceedingly simple, as well as a beautiful and instructive
experiment ; and, besides, it has an intimate relation to our subject.

Finally, we need only refer to Robins' musket, which virtually solved the problem
of shooting round a corner. The barrel was slightly curved to the left near the
muzzle ; and the bullet (made purposely to fit loosely) rolled on the concave (right-
hand) side of the bore, and thus behaved precisely like a sliced golf ball, starting
a little to the left, and then skewing away to the right.

4. As the transverse force due to the spin is always in a direction perpen-
dicular to that of the ball's motion, it has no direct influence on the speed of the
ball. Its only effect is on the curvature of the path. Thus, so long as we are
dealing only with paths confined to one vertical plane, the axis of rotation must be
perpendicular to that plane, and the effect of the transverse force is merely, as it
were, an unbending of the path which would have been pursued had there been no
rotation. From this (very inadequate) point of view we see at once why, other
things being the same, even a moderate underspin greatly lengthens the carry, especi-
ally in the case of a low trajectory. But such analogies give us no hint as to the
actual amount of the lengthening in any particular case. They lead us, however,
to suspect that too great a spin may, in its turn, tend to shorten the carry ; and
that, if of sufficiently great amount, it might even bend the path over backwards
and thus lead to the formation of a kink. Nothing but direct calculation, however,
can give us definite information on these questions. And, unfortunately, we must
trust implicitly in the accuracy of the computer, for we have no independent means
of checking, from stage to stage of his work, the results of his calculations. In
dealing with the case of an unresisted projectile, such numerical work can be checked
at any stage by a simple and exact geometrical process. Even the more complex
conditions of a path in which the resistance is as the square of the speed admit of
exact analytical expression ; but when the transverse force due to rotation is taken
account of, the equations do not admit of integration in finite terms, so that the
computer has to work out the approximate details of the path by successive little
stages, say 6 feet at a time (or somewhere about 90 in all), and any errors of
approximation he may make at any stage will not only themselves be faithfully
represented in the final results, but will necessarily introduce other errors by furnish-
ing incorrect data for each succeeding stage of the calculation. As all of his work
was carried out to four places of figures at least, such errors are unlikely to have
any serious consequence. I have endeavoured to obtain a rough estimate of the
probable amount of error thus inevitably introduced by testing my computer (as
well as the formula which I gave him) upon examples in which I had the means
of independently calculating the exact result at each stage. His work bore this
severe test very well. Unfortunately for my present application, it was based through-

THE TRUE TRAJECTORY 341

out upon too high an estimate of the resistance, and the initial data also were
chosen with reference to this. Thus, when reduced to 108 feet as the terminal
velocity, the paths have been virtually worked out for intervals of about 9 feet
instead of 6, and the initial data appear of a rather haphazard character.

We see from what precedes that the full investigation of the path of a golf
ball, even when it is restricted to a vertical plane, would require voluminous tables
of at least triple entry ; for the form and dimensions of the path are now seen to
depend quite as essentially upon the amount of spin as upon the initial speed and
elevation. There are now no longer two, but innumerable, paths, which involve a
definite carry even when they are confined to a vertical plane, and the initial speed
is given. Of course no tables for their computation are in existence, and it would
prove a somewhat laborious, and therefore costly, work to prepare such tables, even
for a few judiciously selected values of elevation, speed, and initial diminution of
weight. In their absence it is impossible to make any statements more definite
than such very general ones as those above. But (as soon as trustworthy deter-
minations of the resistance of air and of the rate at which the spin of a golf ball
falls off are obtained) a couple of good computers, working in duplicate for a month
or two, would supply sufficient material for at least a rough approximation to any
path affected under the ordinary limits of the initial data of fairly good driving.
Let us hope that some wealthy club, or some enthusiastic patron of the game, may
be induced to further such an undertaking, at least so far as to enable us to give
a fairly approximate answer to such a question as, " Other things being the same,
what values of elevation and of initial diminution of weight will together secure the
maximum carry ? " When we are in a position to give an answer, the clearing up
of the whole subject may be regarded as at least fairly commenced.

30 C'O 90 120 150 180 yds

Meanwhile, as a specimen of what may be done in this direction, I give in
the annexed plate an approximate sketch of the path of a golf ball under the
following initial conditions, the spin being regarded as unaltered during the flight :

Initial speed 240 feet-seconds

Initial relief of weight ... ... 2 fold

Elevation 5-2"

This is fig. i on the plate, and it will be at once recognised as having at least
considerable resemblance to that class of really good, raking drives in which the
ball's path is concave upwards for more than a third of the range. Its one obvious

342 PETER GUTHRIE TAIT

defect, the too great obliquity of the descent, is due to the fact that, not knowing
the law according to which the spin falls off, I have assumed it to continue un-
changed throughout the path. The dotted curve, fig. 5, which gives a very close
approximation to the observed path, was obtained by rough calculations (little
more than estimates) from the same initial speed as fig. i, but with no elevation
to start with. The spin is initially about 50 per cent, greater than in fig. i, but it
has been assumed to fall off in geometric progression with the lapse of time. From
the mode in which this curve was obtained, I cannot insert on it, as I have done
on figs, i, 2 and 3, the points reached by the ball in each second of its flight;
but they will probably coincide pretty closely with those on fig. i. In the last-
mentioned figure, F is the point of contrary flexure and V the vertex. We have,
farther,

Range ... ... ... ... ... 186 yards

Time ... ... ... ... ... 6-2

Greatest height ... ... ... ... 60 feet

Position of vertex ... ... ... 0*71 of range

In fig. 2 the initial speed and rotation are the same as in fig. i, but the
elevation is increased to 12°. It will be seen that little additional carry is gained
in consequence. [Had there been no spin, the increase of elevation from 5° to 12°
would have made a very large increase in the range.] In fig. 3 the elevation is
96° only, but the initial diminution of weight is treble of the weight. In this
figure we see well shown the effect of supposing the spin to be constant throughout,
for it has two points of contrary flexure, /*", and Ft, and only between these is
it concave downwards.

For contrast with these I have inserted, as fig. 4, a path with the same initial
speed, but without spin. Though it has the advantage of 15° of elevation, it is
obviously far inferior to any of the others in the transcendently important matter
of range.

By comparing figs. 2 and 3, we see the effect of further increase of initial spin,
especially in the two points of contrary flexure in 3. Still further increasing the

spin, these points of flexure close in upon the
vertex of the path, and, when they meet it, the
vertex becomes a cusp as in the second of the cuts
shown. The tangent at the cusp is vertical, and
the ball has no speed at that point. This is a
specially interesting case, the path of a gravitating
projectile nowhere concave downwards. With still further spin, the path has a kink,
as in the first of these figures.

I have not yet been able to realise the kink (though I have reached the cusp
stage) with an ordinary golf ball. It would not be very difficult if we could get an
exceedingly hard ball, made hollow if necessary, and if we were to tee on a steepish
slope, and use a well-baffed cleek with a roughened face. I have, however, obtained
good kinks with other projectiles ; the first was one of the little French humming

INFLUENCE OF WIND 343

tops, made of very thin metal, the most recent being in the majestic flight of a
large balloon of very thin indiarubber. This is a very striking experiment ; eminently
safe, and thoroughly demonstrative.

As to the genesis of exceptionally long carries (in the absence of wind), it
will be seen from what precedes that I am not in a position to pronounce any very
definite opinion. How much may be due to an accidentally happy combination of
elevation and spin, how much to extravagant initial speed, can only be decided after
long and laborious calculation. That extra speed has a great deal to do with the matter
(always provided there be spin enough) is obvious from the numerical data given
above. Whatever the initial speed, it is cut down by resistance to half in 83 yards,
so that to increase the carry by 83 yards requires doubled speed. More particularly
to lengthen the carry by thirty yards in fig. I, the ball must have an initial speed
of 300 instead of 240. But such exceptional drives never occur in really careful
play:

Vis consili expers mole ruit sua,
Vim temperatam di quoque provehunt
In majus, etc.

would almost seem to have been written for golf. The vis temperata is the only
passport to a medal or a championship. Its congener, but also its opposite, usually
comes into play when two good drivers, playing for amusement and ready for a
" lark," find two other swipers ahead of them. Then the temptation is almost
irresistible to that " harmless pastime, sport fraternal," which consists in " tickling up "
the party in front as soon as they have " played their second." The law which
permits this furnishes in itself the strongest possible incentive to outrageously long
driving ; and thus, in one sense at least, tends to lower the standard of the game.

However this may be, a long drive is not essentially a long carry. In fact,
with luck and a hard, keen green, the veriest topper or skittles may occasionally
pass the best driver, provided he hits hard enough. But it is not golf, as rightly
understood, recklessly to defy hazards on the mere chance of being lucky enough
to escape them.

5. On the effects of wind little can be learned from calculation until we have
full data. For it is almost invariably the case that the speed of the wind varies
within very wide limits with the height above the ground. Even when the players
themselves feel none, there may be a powerful current sixty or a hundred feet
above them.

We will, therefore, simply in order to combat some current prejudices, treat
only of the case in which the wind is in the plane of the drive (i.e. a head wind

or a following wind) and is of the same speed at all levels within the usual rise of
the ball. Then the matter is easy enough. For the air, so far as we are concerned

344 PETER GUTHRIE TAIT

with it, is then moving as a whole, and in it the path of a golf ball depends only
upon the relative speed and elevation with which it was started. Find, then, with
these data, the path of the ball relatively to the air, and then compound with the
results the actual motion of the air, and we have the path of the ball as it appears
to a spectator. If, then, the ball be struck from 0 with velocity represented by
OA, and the reversed velocity of the air be represented by AB, the velocity of the
ball relatively to the air is given by OB1 or OB3 according as the wind is with the
ball or against it. Trace the successive positions of the ball in the moving air for
each of these, say at intervals of a second, and then displace these horizontally,
forward or backward, to the amount by which the air itself has advanced during
the time elapsed. The result is of course merely to compress or to lengthen each
portion of the path in proportion to the time which the ball took in traversing it.
There is no effect on the height of any part of the path, nor on the time of passing
through it. It is clear that the path, whose initial circumstances are shown by OBt
in the figure, will rise higher than that corresponding to OBt. Hence a ball
which has no spin rises higher when driven with a following wind than against an
equally strong head wind. This is in the teeth of the general belief, which is
probably based on the fact that the vertex of the path against a head wind is
brought closer to the spectator at the tee, and therefore its angular elevation is
increased. When the ball has spin, the conditions of this question become very
complex and no general statement can be made ; though a calculation can, of
course, be carried out for the data of each particular case.
I conclude, as I began, with the much-needed warning :

False views abound, the "cracks" are all mistaken;
In figures, only, rests our trust unshaken.

It seems appropriate to note, with regard to Tait's article on long
driving, how completely his theory has stood the test of later investi-
gations. In an interesting lecture on the dynamics of a golf ball delivered
at the Royal Institution on March 18, 1910, Sir J. J. Thomson1 went over
much of the ground covered by Tait in the preceding article and in his
papers on the path of a rotating spherical projectile, reproducing also by
way of illustration some of the experiments due to Robins and to Magnus.
The most novel feature of Sir J. J. Thomson's lecture was the practical
realisation of the possible paths calculated by Tait and shown on page 341.
The cusp and kink figured on page 342 were also demonstrated by
Sir J. J. Thomson by means of the same ingenious experiment. A stream
of negatively charged particles from a red-hot piece of platinum with a spot
of barium oxide upon it was caused to travel along the vacuum tube in
which the platinum strip was contained. This stream of particles was
1 See Nature of December 22, 1910, Vol. LXXXV, pp. 251-257.

THERMOELECTRICITY 345

luminous and visible. The stream passed between two plates which could
be brought at will to different electrical potentials. When the electric field
was established with the lines of force passing upwards, each negatively
charged particle was subject to a vertical force analogous to the force of
gravity in the case of the golf ball. The luminous stream showed the
path described by each particle. This corresponded to the golf ball path
when there was no underspin, as in fig. 4 on p. 341. A force analogous
to the upward force which acts on the properly spinning golf ball was then
applied to the moving electrified particles by introducing" a magnetic field,
with lines of magnetic force passing horizontally across the stream. The
moving particles were driven at right angles to their own motion and to
the line of magnetic force. By suitable adjustment of direction and strength
of magnetic field, the luminous stream could be made to assume forms
identical with those figured by Tait in the curves on page 341. By increasing
the strength of the field Sir J. J. Thomson obtained not only the kink figured
on page 342, which Tait had demonstrated with the light rubber balloon,
but he also obtained a succession of loops or kinks in the luminous stream.

The following paragraphs contain additional notes on three aspects of Tait's
life and work. The first note discusses some recent developments of thermoelectric
theory, the second touches on the social side of Tait's character, and the third
supplies further information regarding the production of one of Tait's most characteristic
books.

THERMOELECTRICITY.

Mr J. D. Hamilton Dickson, M.A., of Peterhouse, Cambridge, has
recently discussed with remarkable skill and ingenuity the data supplied
by Sir James Dewar and Professor Fleming's measurements1 of the thermo-
electric properties of various metals from — 20x3° C. to + 100° C. His con-
clusions, which are of great interest and form an important extension of
Tait's early results, were read before the Royal Society of Edinburgh on
Nov. 7, 1910, and are being published in full in the Transactions of that
Society.

As noted on page 79 above, one of Tait's main results was that through
a considerable range of temperature the electromotive force between a given
pair of metals was in general a parabolic function of the difference of

1 Phil. Mag., July 1895, Vol. XL, pp. 95-119.
T. 44

346 PETER GUTHRIE TAIT

temperature of the junctions. Was this law fulfilled through the range of
low temperatures which were the feature of Dewar and Fleming's experiments?
To settle this question Dickson first reduced the platinum thermometer
temperature readings to absolute scale, and then plotted with great care
on section paper the electromotive force of each metal lead couple against
the temperature of the junction whose temperature was varied. The curves
were parabolas with axis perpendicular to the temperature axis for gold,
silver, zinc and the alloy German silver, which metals therefore follow Tait's
law strictly.

It was obvious at a glance that in the other cases the electromotive-
force curve was not a parabola with axis perpendicular to the temperature
axis. By graphical construction of the loci of the points of bisection of
sets of parallel chords, Dickson proved that these curves must be curves
of the second degree ; for the loci mentioned were very approximately
straight lines. In the case of the antimony-lead curve the loci of different
sets of parallel chords passed approximately through one point, indicating
that the curve was a hyperbola. In every other case the loci of mid-points
of different sets of parallel chords were parallel straight lines, indicating
that the curve was a parabola with axis parallel to these loci but not necessarily
perpendicular to the temperature axis. If following Dickson we designate
by the " Tait-line " the graph which gives the relation between the temperature
and the rate of change of the electromotive force per unit change of
temperature, the results obtained by him may be thus summarised : — The
lead line being laid down horizontal, the Tait-lines for gold, silver, zinc
and German silver are straight, in accordance with Tait's theory ; for
platinum (two kinds, both of very great purity), copper, cadmium, nickel,
magnesium, palladium, and aluminium, the Tait-lines are cubical hyperbolas ;
and for antimony the Tait-line is a quartic curve. The two last statements
may be easily verified as properties of the parabola and hyperbola with axes
inclined to the temperature axis.

In one of his early notes on thermoelectricity, embodied in his " First
Approximation to the Thermoelectric Diagram" (see Set. Pap., Vol. i,
p. 220), Tait remarked that

When the temperatures were very high, the parabola was slightly steeper on
the hotter than on the colder side. This, however, was a deviation of very small
amount, and quite within the limits of error introduced by the altered resistance
of the circuit at the hotter parts, the deviations of the mercury thermometers from

THE EVENING CLUB 347

absolute temperature, and the non-correction of the indication of the thermometers
for the long column of mercury not immersed in the hot oil round the junction.

As mentioned on p. 77 above, Tait's " First Approximation to the
Thermoelectric Diagram" was based on the experiments made by C. E. Greig
and myself according to a method which Tait's earlier experiments had
shown to be convenient and sufficiently accurate. The thermoelectric curves
when drawn were all very approximately parabolic ; but generally when
a well-marked vertex was obtained a slight lack of perfect symmetry was
observable. This lack of symmetry Tait regarded as due to the mercurial
thermometers not giving a scale of temperature accurately in harmony with
the absolute scale ; and, since the deviation from symmetry in any particular
case was always very slight, the matter was passed over as of comparatively
small moment in obtaining what Tait called the First Approximation.
Dickson's results, however, show that this asymmetry may be an essential
feature of the curve itself, and not due to any errors involved in the measure-
ment either of the temperature or of the electromotive force, for he gives
similar examples of inclined axis from Tait's work, as well as from the
work of other experimenters.

THE EVENING CLUB.

As already noted (see above, pp. 33 and 49) Tait mingled very little
in general society during the last twenty years of his life. To those who
knew him only in these later years he appeared to be more or less of a
recluse, and (except during his holiday months in St Andrews) could not
in any sense be regarded as a Club-man. Yet he was splendid company
when occasion offered ; and it is well in this connection to recall that he
was one of the founders of the Evening Club, which was organised in 1869
on the model of the "Cosmopolitan" and "Century" Clubs in London.
The original prospectus was signed by fourteen well known citizens who
formed themselves into a provisional committee. Their names in order of
signature were : Dr John Muir, Professor P. G. Tait, Professor David
Masson, James Drummond, R.S.A.1, J. Matthews Duncan, M.D., Robert
Wallace, D.D.2, Robert Cox, W.S., James Donaldson, LL.D.', A. Findlater,

1 Curator of the National Gallery.

2 Afterwards M.P. for Edinburgh.

' At present Principal of St Andrews University.

44—2

348 PETER GUTHRIE TAIT

LL.D.1, D. Douglas, F.R.S.E., J. F. Maclennan8, Alex. Nicolson2, Aeneas
J. G. Mackay2, John Maitland2. These promoters of the Club, by personal
appeals among their friends, quickly gathered together more than a hundred
names of those willing to become members. Tail received the following
characteristic response from Macquorn Rankine, whose propensity for
forming Greek derived words was irrepressible :

59 ST VINCENT STREET,
GLASGOW,

2<)th October, 1869.

My dear Tait

I shall be very happy to join your Capnopneustic Club (as it may
appropriately be termed). I beg pardon for not having answered you sooner ; but
I have been greatly engrossed by business, both engineering and academic.

Believe me,

very truly yours,

W. J. MACQUORN RANKINE.

In 1870 the membership was 140, and in 1874, 150, including many
of the more prominent Edinburgh lawyers, artists, physicians, clergymen,
teachers both in college and school, bankers, commercial men, publishers,
engineers, etc. There was also a selection of members not resident in
Edinburgh, such as Professor Lewis Campbell, St Andrews, Sir M. E. Grant
Duff, Professor T. M. Lindsay, Glasgow, Sir William Stirling Maxwell,
Professor Nichol, Glasgow, Professor G. G. Ramsay, Glasgow, Sir George
Reid, R.S.A., then living in Aberdeen, Professor W. Robertson Smith,
Aberdeen, Sir William Thomson, Glasgow, Principal Tulloch, St Andrews.

The Club met every Saturday and Tuesday evening and on Monday
evenings immediately after the statutory fortnightly meetings of the Royal
Society, for purely social intercourse, cards and serious subjects of debate
being taboo. During the first twelve years of its existence Tait
frequently attended the gatherings. The roll of guests introduced by

1 Editor of Chambers' Encyclopaedia.

2 Advocate. Maclennan was the author of Primitive Marriage. Mackay was afterwards
Professor of History.

"RECENT ADVANCES" 349

members has been preserved, and we find Tait personally responsible for
introducing some seventy guests between 1870 and 1884. In August 1871,
when the British Association met in Edinburgh, the Club held several
meetings ; and among the guests introduced by Tait were Cayley, Clerk
Maxwell, Huxley, Bierens de Haan, Colding, Sylvester and Clifford.

After a vigorous existence for about twenty-five years, the Evening
Club began to lose vitality. The members attended irregularly and fitfully,
mainly on account of the growing lateness of the dinner hour ; and after
an effort to hold it together in a less formal manner the Club was disbanded
in 1897.

It formed an important episode in the life of Tait, bringing him into
close social touch with many men whom otherwise he would never have
met. Those who attended the meetings during the seventies recall Tait
as one of the great personalities, taking his full share in the talk, and
enjoying the relaxation from the hard thinking in which he usually passed
his evenings.

But the Club also had a direct bearing on scientific activity ; for it
was probably in the free and easy conversation of this Evening Club that
there germinated in the minds of George Barclay, the first treasurer of
the Club, and Thomas Stevenson, the well known engineer, the idea
which finally came to fruition in Tail's Lectures on some Recent Advances
in Physical Science.

RECENT ADVANCES IN PHYSICAL SCIENCE.

The recent death of Mr George Barclay at the advanced age of 91
has led to the discovery among his papers of further information regarding
the course of lectures just referred to (see also above, p. 327). On
February 14, 1874, Thomas Stevenson issued the circular to the subscribers
announcing that the lectures would be delivered in St George's Hall,
Randolph Place, on Tuesdays and Thursdays, at a quarter past four
o'clock, during the months of February, March, and April, the first lecture
to be given on Thursday, the igth February. This hall, although convenient
as regards situation, was found to be so unsuitable in other respects that
after a vain endeavour to find a better place of meeting in the New
Town, Thomas Stevenson and George Barclay, who acted respectively

350 PETER GUTHRIE TAIT

as secretary and treasurer, arranged for the delivery of the lectures in the
Natural Philosophy Class Room of the University. This was announced
in a circular issued by George Barclay on February 24th immediately
after the second lecture ; so that it appears that all save two of the
lectures were given in the University.

The lectures were not published in book form till nearly two years
after they were delivered. The reporter had lost his short-hand note of
one of them ; and Tait had to redeliver this lecture in his retiring room to
an audience of one. The book on Recent Advances is the only book of
Tail's which contains a dedication. It is in these words: "With this
work I desire to associate the names of George Barclay and Thomas
Stevenson, Fellows of the Royal Society of Edinburgh, by whose efforts
these Lectures were organised, and at whose wish they are published
as delivered. P. G. T."

BIBLIOGRAPHY

BEFORE his last illness Tail had begun to prepare material for the third volume of his
Scientific Papers. A special feature was to have been a complete bibliography of his contri-
butions to scientific journals, with short explanatory remarks after the titles of the shorter
papers which were not included either in the first or second volume of the collected papers.
In filling in these remarks on a prepared sheet Tail had got as far as the year 1870. I have
endeavoured to continue the work thus begun by Tail himself, giving in approximately
chronological order all his published papers and articles. Since by far the greater number of
these appear in the Proceedings of the Royal Society of Edinburgh, I have used the letters
Jt.S.E. to mean this publication. When a paper has appeared in the Transactions of that
Society the reference is to Trans. R.S.E. Any other journals are indicated by sufficiently
intelligible contractions. The two volumes of the republished Scientific Papers are referred to
simply as S.P. The first Sixty (LX) appear in Volume I and the rest from LXI to cxxxni in
Volume II. When any reference is made to Tail's books, the characteristic name of the book
is given in Italics.

1. Note on the density of Ozone. In conjunction with Thomas Andrews, 1857. Proc. Roy.
Soc., vii. Reprinted in Andrews' Scientific Papers.

2. Second Note on Ozone. In conjunction with Thomas Andrews, 1857-59. Proc. Roy.
Soc., ix. Reprinted in Andrews' Scientific Papers.

3. On the volumetric relations of Ozone and the action of the electric discharge on oxygen
and other gases. In conjunction with Thomas Andrews, 1859. Proc. Hoy. Soc., x. Phil.
Trans., 1860. Reprinted in Andrews' Scientific Papers.

4. Quaternion investigations connected with Fresnel's wave-surface. 1859. Quart. Journ.
Math. S.P., i. Tail's Quaternions, §§ 432-452.

5. Note on Ihe Carlesian equalion of Ihe wave-surface. 1859. Quart. Journ. Math., in.
S.P., n.

6. Qualernion investigations connected wilh eleclrodynamics and magnelism. 1860. Quart.
Journ. Math., in. S.P., in. Tail's Quaternions, §§ 458-472.

7. Malhemalical Noles: A spherical nebula consisls of concenlric shells of uniform density;
find the law of the latter that to a spectator at a greal dislance the nebula may appear
uniformly bright. The densily in shell of radius r is inversely as J(a'~f3). 1860.
Quart. Journ. Math., iv.

8. Qualernion invesligation of Ihe polenlial of a closed circuit. 1861. Quart. Journ. Math.,
IV. S.P., IV. Tail's Quaternions, §471.

9. Note on a modification of the apparatus employed for one of Ampere's fundamental
experiments in electrodynamics. 1861. Jt.S.E., iv. S.P., v.

10. Note on molecular arrangement in crystals. 1862. R.S.E., iv. Crystalline properties
illustrated by use of piles of marbles of equal size. A horizonlal layer may be arranged
in square or triangular order, and successive layers may be simply superposed or
inserted into interstices in the preceding ones. With interslilial arrangemenl Ihe square
and triangular order give same density.

352 BIBLIOGRAPHY

11. Formulae connected with continuous displacements of the particles of a medium. 1862.
X.S.E., iv. S.P., vi.

12. Note on electricity developed during evaporation and during effervescence from chemical
action. In conjunction with J. A. Wanklyn. 1862. fi.S.JZ., iv. Large charges were
produced by evaporation of a drop of bromine, etc., from a hot platinum dish.

13. On Determinants. 1862. Mess, of Math., I.

14. Quaternions. (A series of three papers.) 1862. Mess, of Math., i.

15. Energy. In conjunction with W. Thomson (Lord Kelvin). 1862. Good Words.

16. Note on central forces. 1862. Mess, of Math., \.

17. Note on a quaternion transformation. 1862. R.S.E., v. S.P., VH. Tait's Quaternions,

§§ MS. MS, 474-

18. Reply to Prof. Tyndall's remarks on a paper on "Energy" in Good Words. 1863.
Phil. Mag., xxv.

19. On the Conservation of Energy. 1863. Phil. Mag., xxvi. (Two letters, June and August.)

20. Note on the Hodograph for Newton's law of force. 1863. Mess, of Math., n.

21. Elementary physical applications of quaternions. 1863. Quart. Journ. Math., vi. Tait's
Quaternions, Chap. xn.

22. On the Conservation of Energy. 1863. Address before Royal Society of Edinburgh.
Second interpretation of Newton's Third Law given for first time in modern form. R.S.E., iv.

23. On the history of thermodynamics. 1864. Phil. Mag., xxvin.

24. The dynamical theory of heat. 1864. North British Review.

25. Energy. 1864. North British Review.

26. Note on Fermat's Theorem. 1864. R.S.E., v.

27. Note on the history of Energy. 1864. Phil. Mag., xxix (1865).

28. Note on action. 1865. R.S.E., v. Area about empty focus of planet's orbit represents
action.

29. On the law of frequency of error. 1865. Trans. R.S.E., xxiv. S.P., vm.

30. On the application of Hamilton's Characteristic Function to special cases of constraint.
1865. R.S.E., v. Trans. R.S.E., xxiv. S.P., ix.

31. Note on the behaviour of iron filings strewn on a vibrating plate, and exposed to the
action of a magnetic pole. 1865. R.S.E., v. The filings are kept on the ventral
segments by a pole placed above the plate, but instantly dispersed when it is below.

32. Preliminary note on the heating of a disk by rapid rotation in vacua. In conjunction
with Balfour Stewart. 1865. Proc. Roy. Soc., xiv. Phil. Mag., xxix.

33. On the heating of a disk by rapid rotation in vacua. In conjunction with Balfour
Stewart. 1865. Proc. Roy. Soc., xiv. Phil. Mag., xxx.

34. Note on the compression of air in an air bubble under water. 1865. R.S.E., v. A
propos of solution of air in water, and of the vesicular vapour of Clausius. Because
of surface tension, an air bubble of radius o'ooooi of an inch contains air at pressure
of ii atmospheres.

35- On some geometrical constructions connected with the elliptic motion of unresisted pro-
jectiles. 1866. R.S.E., v. Tail and Steele, § 121.

36. On the heating of a disk by rapid rotation in vacua. In conjunction with Balfour
Stewart. 1866. Proc. Roy. Soc., xv. Phil. Mag., xxxin.

37. On some capillary phenomena. 1866. R.S.E., v. Study of process of dividing soap
bubble into two, and of causing two to unite. Exhibition of motions in film by using
its posterior surface as a concave mirror reflecting selected portions of sunlight.

BIBLIOGRAPHY 353

38. On the Value of the Edinburgh Degree of M.A. An Address delivered to the Graduates
in Arts, April 24, 1866. Published by Maclachlan and Stewart, Edinburgh.

39. Sir William Rowan Hamilton. 1866. Biographical article in North British Review.
Chief source of article in Encyclopaedia Britannica, 1880. S.P., cxxvin.

40. Note on formulae representing the fecundity and fertility of women. 1867. Trans.
R.S.E., xxiv. Fecundity is found to depend linearly and fertility parabolically on age.
Reprinted with additions in Duncan's Fecundity and Fertility (Black, 1871). The
formulae are known to Statisticians as Tail's Laws.

41. Note on determinants of the third order. 1866. R.S.E., vi. Various quaternion
transformations leading to theorems in determinants.

42. Translation of Helmholtz's paper on "Vortex Motion." Phil. Mag., 1867.

43. Note on the reality of the roots of the symbolical cubic which expresses the properties
of a self-conjugate linear and vector function. 1867. R.S.E., vi. S.P., x.

44. Note on a celebrated geometrical problem proposed by Fermat. 1867. R.S.E., vi. S.P., xi.

45. Note on the radiant spectrum. 1867. R.S.E., vi.

46. Note on the hodograph. 1867. R.S.E., vi. S.P., xn.

47. Note on an inequality. 1868. R.S.E., vi. The conditions for the coalescence of two
soap bubbles lead to a mathematical inequality.

48. Mode of demonstrating equality of radiation and absorption. 1868. R.S.E., vi. Ink
letters on platinum strip.

49. Physical proof that the geometrical mean of any number of quantities is less than the
arithmetic mean. 1868. R.S.E., vi. S.P., xm.

50. On the Dissipation of Energy. 1868. R.S.E., vi. S.P., xiv.

51. Notice of Sir David Brewster. The Scotsman, n Feb. 1868.

52. On the rotation of a rigid body about a fixed point. 1868. Abstract in Proc. R.S.E.,
vi. Full Paper in Trans. R.S.E., xxv. S.P., xv.

53. On the motion of a pendulum affected by the rotation of the Earth and other disturbing
causes. 1869. JR.S.E., vi. Tail's Quaternions, §§ 427-430.

54. On the heating of a disk by rapid rotalion in vacua. In conjunclion with Balfour Stewart.
Reply to O. E. Meyer. 1869. Phil. Mag., xxxvn. Ann. d. Phys. u. Ch., 136, pp. 165-6.

55. Notice of Principal J. D. Forbes. The Scotsman, n Jan. 1869.

56. On comets. 1869. R.S.E., vi. British Assoc. Reports, xxxix. The sea bird Analogy.

57. Note on electrolytic polarisation. 1869. R.S.E., vi. S.P., xvi. Phil. Mag., xxxvni.

58. Provisional Report of a Commitlee consisting of Professor Tail, Professor Tyndall, and
Dr Balfour Stewart, appointed for the purpose of repeating Principal J. D. Forbes's Ex-
perimenls on Ihe Thermal Conductivity of Iron and of extending them to other Metals.
Brit. Ass. Rep., xxxix.

59. The Progress of Natural Philosophy. Nature, 1869.

60. On the Steady Motion of an incompressible perfect fluid in Iwo dimensions. 1870.
R.S.E., vn. S.P., xvii. In a letter to Cayley (6 Oct. 1877) Tail refers lo Ihis paper
as conlaining a "very curious hydrokinetic theorem which I remember greatly surprized
not only Rankine and Stokes bul even Thomson."

61. On the most general motion of an incompressible perfect fluid. 1870. R.S.E., vn.
S.P., xvin.

62. Mathematical Notes. 1870. R.S.E., vn. Every cube is difference of two squares, etc.

63. On Green's and other allied Theorems. 1870. Trans. R.S.E., xxvi. S.P., xix.
Abstract in Proc. R.S.E., vn.

T.

45

354 BIBLIOGRAPHY

64. Note on linear partial differential equations. 1870. R.S.E., vn. S.P., xx.

65. Notes from the Physical Laboratory of the University. 1870. R.S.E., vn. Account
of J. P. Nichol's experiments on radiation and convection of heat from blackened and
bright surfaces at various pressures of surrounding gas. At 10 mm. pressure the effect
mainly due to radiation. The effect of lowering the pressure in lessening the rate of
cooling was more marked in the case of the bright than of the blackened surface.

A. Brebner continued experiments of No. 57 above.

P. W. Meik and John Murray experimented on change of electric resistance of two
kinds of copper with increase of load. The change was proportional to the load and
was different in the two cases.

66. Address forming part of the Report of the Proceedings at the 32nd Annual General
Meeting of the Scottish Provident Institution on 3oth March 1870 (Professor Tait,
Convener).

67. Laboratory Notes. On thermoelectricity. 1870. R.S.E.,vu. Incorporated in Transactions
paper, No. 108 below.

68. Note on linear differential equations in Quaternions. 1870. R.S.E., vn. S.P., xxi.

69. Energy and Professor Bain's " Logic." 1870. Nature, HI.

70. On some Quaternion Integrals. 1870. R.S.E., vn. S.P., xxn.

71. Laboratory Notes. 1871. (i) On thermoelectricity. (2) On Phyllotaxis. Develops
formula for the fundamental divergence of the spiral. R.S.E., vn.

72. Review of Balfour Stewart's "Elementary Physics." 1870. Nature, HI."

73. Laboratory Notes. 1871. (i) On anomalous spectra, and on a simple Direct Vision
spectroscope. Suggests a combination of two liquids. (2) On a method of illustrating
to a large audience the composition of Simple Harmonic Motions under various con-
ditions. Instead of Lissajoux's apparatus suggests mirrors rotating about axes not quite
perpendicular to the planes of the mirrors. (3) On a simple mode of explaining the
optical effects of mirrors and lenses. Use of term divergence instead of reciprocal of
distance. R.S.E., vn.

74. Review of Tyndall's "Use and Limits of the Imagination in Science." 1871. Nature, in.

75. Review of Highton's " On the Relations between Chemical Change Heat and Fire."
1871. Nature, in.

76. Sensation and Science. 1871. Nature, iv.

77. Mathematical Notes, (i) On a quaternion integration. Given in Tail's Quaternions,
§ 431. (2) On the Ovals of Descartes. Results partly got by quaternions— according
to Cayley mostly to be found in Chasles' Aper^u Historique. 1871. R.S.E., vn.

78. On Spectrum Analysis. 1871. Abstract of an Address, important because of its historic
statements. 1871. R.S.E., vn.

79. Mathematical Notes. 1871. (i) On a property of self-conjugate linear and vector functions.
Tail's Quaternions, Chap, v, Question 18 at end. (2) Relation between corresponding
ordinates of two parabolas. Suggested by Thermoelectric Diagram. (3) On some qua-
ternion transformations. Tail's Quaternions, §§514-517. R.S.E., vn.

80. On an expression for the potential of a surface distribution and on the operator TV.
Tail's Quaternions, §§ 516, 517. 1871. R.S.E., vn.

81. Reply to Dr Ingleby. 1871. Nature, v.

82. Note on Spherical Harmonics. Suggested by a Quaternion mode of attack — referred to
by Maxwell as "ravishing." 1871. R.S.E., vn.

83. Review of " Physikalisches Repetitorium." 1871. Nature, v.

BIBLIOGRAPHY 355

84. Address to the Mathematical and Physical Section of the British Association at
Edinburgh. 1871. Brit. Ass. Reports, XLI. S.P., xxm.

85. On thermoelectricity. 1871. Brit. Ass. Reports, XLI.

86. Reply to Professor Clausius. 1872. Phil. Mag., XLIII.

87. On the history of the Second Law of thermodynamics in reply to Professor Clausius.
1872. Phil. Mag., XLIV.

88. Reply to Professor Clausius. 1872. Phil. Mag., XLIV. German translation in Annalen
d. Physik u. Chemie, CXLV.

89. Laboratory Notes. On thermoelectricity. 1872. R.S.E., vn. Incorporated in Trans-
actions paper, No. 108 below.

90. Note on a singular property of the retina. 1872. R.S.E., vn. S.P., xxiv.

91. On the operator <£ (v). 1872. R.S.E., vn. Tail's Quaternions, § 508.

92. Note on pendulum motion. 1872. R.S.E., vn. Tail's Dynamics, § 143.

93. Note on the Slrain Funclion. 1872. R.S.E., vn. S.P., xxvi.

94. On orthogonal isothermal surfaces. 1866-1872. Trans. R.S.E., xxvn. S.P., xxv.

95. Laboratory Notes. 1872. R.S.E., vn. (i) On thermoelectricity ; circuils with more
lhan one neutral point. Included in Transactions paper, No. 108 below. (2) On a
method of exhibiting the sympathy of pendulums. Magnets suspended horizontally
end on to one another in the same straight line. " Interesting illustration of the
linear propagation of dislurbances in a medium consisting of discrete massive particles
when only contiguous ones act on each other... It might be curious to enquire
whether a certain form of the equation of motion might not give some hints as to the
formation of a dynamical hypothesis of the action of transparent solids on the lumini-
ferous ether."

96. On some quaternion inlegrals, Part II. 1872. R.S.E., vn. S.P., xxn.

97. Scintillation (signed G. H.). 1872. Nature, vi, p. 181.

98. On a question of arrangement and probabilities. 1872. R.S.E., vin. S.P., xxvn.

99. Artificial Selection. 1872. Macmillan's Magazine, xxv.

100. On the stiffness of wires. 1872. R.S.E., vin.

101. Comets' tails (signed G. H.). 1872. Nature, vn, p. 105.

102. Sensation and Science. 1872. Nature, vi.

103. Laboratory Notes. 1872. (i) On the relation belween Ihermal and eleclric conductivity.
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remarkable fall off of conductivily (or increase of resislance) of iron al a red heat as
compared with platinum. (3) On the thermoelectric relations of pure iron; contained
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104. Notice of Professor Macquorn Rankine. The Glasgow Herald, Dec. 28, 1872.

105. Nole on Angslrom's Melhod for the conductivily of bars. 1873. Gives Ihe harmonic
analysis necessary wilh numerical details for certain bars deduced from experimental
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106. Additional Note on the Strain Function. 1873. R.S.E., vin. S.P., xxvi.

107. On the Ihermoeleclric properties of pure nickel. 1873. First time the nickel line was
obtained — showed same bend as iron but at lower temperalure. R.S.E., vin.

108. Firsl approximation lo a Thermoeleclric Diagram. 1873. Trans. R.S.E.t'xx.vu. S.P., xxix.

109. Thermoeleclricily. The Rede Lecture delivered at Cambridge in 1873. Abstracl in
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no. Laboratory Nole. On Ihe flow of water Ihrough fine lubes. 1873. Experiments by

45—2

356 BIBLIOGRAPHY

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112. Review of De Morgan's "Budget of Paradoxes." 1873. Nature, vn.

113. Tyndall and Forbes. 1873. Nature, vm.

114. Note on the transformation of double and triple integrals. 1873. R.S.E., vm. S.P., xxx.

115. Forbes and Tyndall. 1873. An Article intended originally for the Contemporary
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116. On the heating of a disk by rapid rotation in vacua. In conjunction with Balfour Stewart.

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117. Note on the various possible expressions for the force exerted by an element of one
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118. Note on Mr Bang's communication of 7th April 1873, on a singular property possessed
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for such an effect. Hence the apparent motion of the bubble. // is not the same
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119. Preliminary note on a new method of obtaining very perfect vacua. In conjunction
with Sir James Dewar. 1874. R.S.E., vm.

120. Todhunter's Experimental Illustrations. 1874. Nature, ix.

121. Laboratory Notes, (i) On Atmospheric electricity: observed alternations of positive and
negative electrifications during hail storm. (2) On the thermoelectric position of sodium.

1874. R.S.E., vm.

122. Herbert Spencer v. Thomson and Tail. 1874. Nature, ix.

123. "Cram." 1874. Nature, ix.

124. Double Rainbow. 1874. Nature, x.

125. Mathematical Notes, (i) On a singular theorem given by Abel. S.P., xxxn. (2) On
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126. On the thermoelectric positions of sodium and potassium. 1874. R.S.E., vm.

127. Review of Sedley Taylor's " Sound and Music." 1874. Nature, x.

128. Laboratory Notes, (i) Photographic records of sparks from the Holtz machine.

(2) Determination of surface tension of liquids by ripples produced by tuning fork.

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129. Further Researches in very perfect vacua. In conjunction with (Sir) James Dewar.

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130. Photographs of electric sparks in hot and cold air. 1875. Trans. R.S.E., xxvn.
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131. Cosmical Astronomy. A series of Lectures published in Good Words of 1875.

132. Laboratory Notes, (a) On the application of Sir W. Thomson's dead-beat arrangement
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BIBLIOGRAPHY 357

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134. On the linear differential equation of the second order. 1876. R.S.E., ix. S.P., xxxv.

135. Notice of Professor G. G. Stokes. 1876. Nature, xn.

136. Review of Miss Buckley's "Natural Science." 1876. Nature, xm.

137. Laboratory Notes. 1876. (a) On a possible influence of magnetism on the absorption
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138. Note on the origin of thunderstorms. 1876. Discusses production of vortex columns
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139. Translation of F. Mohr's "Views of the Nature of Heat." Phil. Mag. (The paper
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140. Force. Evening Lecture at British Association (Glasgow Meeting). 1876. Nature, xiv.
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141. Some elementary properties of plane closed curves. Brit. Ass. Reports, 1876. Messenger
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142. Applications of the theorem that two closed plane curves intersect an even number of
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143. Definition and Accuracy. 1876. Nature, xv.

144. Notice on some recent atmospheric phenomena. 1876. R.S.E., ix. Description of
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145. Article " Heat " in Handbook to the Loan Collection of Scientific Apparatus (at South
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146. Note on the measure of beknottedness. 1877. R.S.E., ix.

147. Notice of James Lindsay. The Scotsman, Jan. 5, 1877.

148. Note on the effect of heat on infusible impalpable powders. 1877. R.S.E., ix.
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149. On knots. With remarks by Listing. 1877. R.S.E., ix.

150. On links. 1877. R.S.E., ix.

151. On the relative percentages of the atmosphere and of the ocean which would flow
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152. Sevenfold knottiness. 1877. R.S.E., ix.

153. On amphicheiral forms and their relations. 1877. R.S.E., ix.

154. Preliminary Note on a new method of investigating the properties of knots. 1877.
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155. On knots. 1877. Trans. R.S.E., xxvm. S.P., xxxix.

156. Note on an identity. 1877. R.S.E., ix. S.P., XLIII.

157. Review of Maxwell's "Matter and Motion." 1877. Nature, xvi.

158. Review of Wormell's "Thermodynamics." 1877. Nature, xvn.

159. Review of Zollner's " Scientific Papers." 1877. Nature, xvn.

358 BIBLIOGRAPHY

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162. Note on a geometrical theorem as to the properties of three concentric circles, which
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163. Exhibited a double mouthpiece by means of which it is easy for two players to produce
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164. Note on the surface of a body in terms of a volume integral. 1878. Jt.S.E., ix.
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165. On the teaching of Natural Philosophy. The Contemporary Review, 1878.

166. Explanation of white rainbow observed by Sir Robert Christison. 1878. R.S.E., ix.
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167. Review of Clifford's "Dynamic." 1878. Nature, xvni.

168. On the strength of currents required to work a telephone. 1878. R.S.E., ix.

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169. Review of Freeman's "Fourier's Theory of Heat." 1878. Nature, xvni.

170. Thermal and electric conductivity. 1878. R.S.E., ix. Trans. R.S.E., xxvin.
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171. Review of Proctor's "Pleasant Ways in Science." 1878. Nature, xix.

172. On some definite integrals. 1878. R.S.E., ix. Suggested by electric image problems
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173. Note on electrolytic conduction. 1878. R.S.E., ix. S.P., XLIX.

174. Letter from Hermann Stoffkraft. 1878. Nature, xix.

175. Review of Bain's "Education." 1878. Nature, xix.

176. On certain effects of periodic variation of intensity of a musical note. In conjunc-
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177. Note on a mode of producing sounds of very great intensity. 1878. R.S.E., ix. S.P., L.

178. Does Humanity require a new Revelation? The International Review, 1878.

179. Quaternion Proof of Minding's Theorem. 1879. London Math. Soc. Proc., x.

180. On the Dissipation of Energy. (Letter to Sir W. Thomson.) Phil. Mag., vn, 1879.

181. On the Measurement of Beknottedness. 1879. R.S.E., x.

182. Review of Beckett's "On the Origin of the Laws of Nature." 1879. Nature, xx.

183. Laboratory Notes. 1879. (0 Measurements of the electromotive force of the Gramme
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184. Quaternion investigations connected with Minding's Theorem. 1879. R.S.E., x.

185. On methods in definite integrals. 1879. R.S.E., x.

186. Application of certain formulae to sum infinite series. 1879. R.S.E., x.

187. Obituary notice of Professor Kelland. In conjunction with Professor Chrystal. 1879.
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188. Obituary notice of Clerk Maxwell. 1879. R.S.E., x.

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193. Note on the velocity of gaseous particles at the negative pole of a vacuum tube.
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194. Scientific jokes (signed G. H.). 1880. Nature, xxi, pp. 349, 368, 396.

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196. On the colouring of maps. 1880. Various proofs that four colours suffice.

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207. Note on the temperature changes due to compression. 1881. R.S.E., xi.

208. On some effects of rotation in liquid jets. 1881. R.S.E., xi. A jet of mercury
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209. Note on thermal conductivity, and on the effects of temperature-change of specific
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210. Note on a simple method of showing the diminution of surface tension in water by
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211. Note on a singular problem in Kinetics. 1881. R.S.E., xi. S.P., LVII.

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217. Memoir of J. Macquorn Rankine. Prefixed to Scientific Papers. 1881.

218. Graduation Address. Edinburgh, April 20, 1881. Printed for private circulation.

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224. On a method of investigating experimentally the absorption of radiant Heat by gases.
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229. Notice of J. B. Listing. 1883. Nature, xxvm. S.P., LXV.

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231. Review of Jordan's "The New Principles of Natural Philosophy." 1883. Nature,

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232. Biographical Note on J.-A.-F. Plateau. 1883. Nature, xxvm.

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234. Professor Stokes' Works. 1883. Nature, xxix.

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243. On various suggestions as to the source of atmospheric electricity. 1884. Journal of
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245. On an improved method of measuring compressibility. 1884. Described above, p. 85.
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250. Note on the necessity for a condensation nucleus. 1885. Points out that the presence
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251. Review of Williamson's " Dynamics." 1885. Nature, xxxi.

252. On evaporation and condensation. 1885. Continues last note by considering possibility
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253. Hooke's anticipation of the Kinetic Theory and of synchronism. 1885. R.S.E., xm.
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254. Review of Clifford's "Exact Sciences." 1885. Nature, xxxti.

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259. Review of Stokes' "Light." 1885. Nature, xxxn.

260. On an application of the atmometer. 1885. Proposal to use the instrument in its
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265. On the foundations of the Kinetic Theory of Gases; with a continuation containing
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271. Weight and mass. 1887. Nature, xxxv.

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T. 46

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277. Notice of G. R. Kirchhoff. 1887. Nature, xxxvi. S.P., cxxvu.

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279. The unwritten chapter on golf. The Scotsman, Aug. 31, 1887. Nature, xxxvi.

280. An exercise on logarithmic tables. Proc. Edin. Math. Soc., v. 1887. S.P., LXXXV.

281. Notice of Balfour Stewart. 1887. Nature, xxxvu.

282. On glories. 1887. R.S.E., xiv. S.P., LXXXVI.

283. On the compressibility of water and of different solutions of common salt. 1887.
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284. On the thermoelectric properties of iron. 1888. Attention is called to Battelli's direct
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285. Review of Fleeming Jenkin's "Papers, Literary and Scientific," etc. 1888. Nature,

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290. Quaternion Notes. 1888. R.S.E., xv. S.P., xc.

291. Engineers v. Professors and College Men. 1888. Nature, xxxix.

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293. Address to the Graduates. 1888. (Printed for private circulation.)

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296. Report on some of the physical properties of fresh- water and of sea-water. 1888. From
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300. Obituary notice of Balfour Stewart. Proc. Roy. Soc. London, 1889. S.P., xci.

301. The relation among four vectors. 1889. R.S.E., xvi. S.P., xcn.

302. Note on Mr McAulay's paper on differentiation of a quaternion. 1888. R.S.E., xvi.

303. Additional remarks on the virial of molecular forces. 1887. R.S.E., xvi.

304. On the relation among the line surface and volume integrals (quaternionic). 1889.
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305. Quaternion note on a geometrical problem. 1889. R.S.E., xvi. S.P., xciv.

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S.P., c.

314. Review of Thurston's translation of Carnot's Essay, " Reflexions on the Motive Power
of Heat." 1890. Nature, XLII, p. 365.

315. Some points in the physics of golf. 1890. Nature, XLII.

316. The pace of a golf-ball. Golf, Dec. 1890.

317. On the foundations of the Kinetic Theory of Gases. Part IV. 1891. Trans. R.S.E.,
xxxvi. S.P., LXXX.

318. Quaternions and the Ausdehnungslehre. 1891. Nature, XLIV. S.P., cxxix (Appendix).

319. Note on the isothermals of ethyl oxide. 1891. R.S.E., xvn. S.P., ci.

320. Some points in the physics of golf. 11. 1891. Nature, XLIV.

321. Van der Waals' treatment of Laplace's pressure in the virial equation — two letters to
Lord Rayleigh. 1891. Nature, XLIV.

322. Note appended to Dr Sang's paper on Nicol's polarizing eyepiece. 1891. Mainly a
simplification of the mathematics as given by Sang in his early paper of 1837, which
was not published by the Royal Society of Edinburgh till 1891. The ingenious suggestion
made by Sang to construct a polarizer by placing a thin slice of Iceland Spar between
two glass plates was made independently much later by Bertrand in 1884. R.S.E.,
XVHI. S.P., en.

323. Hammering and driving. Golf, Feb. 1892.

324. On the virial equation for gases and vapours. 1892. Nature, XLV.

325. Review of Poincard's "Thermodynamics," with two letters in reply to Poincare". 1892.
Nature, XLV.

326. On the foundations of the Kinetic Theory of Gases. Part V. 1892. R.S.E., xix. S.P., LXXXI.

327. On impact, n. 1892. Trans. R.S.E., xxxvn. S.P., LXXXIX.

328. Note on Dr Muir's solution of Sylvester's elimination problem. 1892. R.S.E., xix.

329. Note on the thermal effect of pressure on water. 1892. R.S.E., xix. S.P., Civ.

330. Note on the division of space into infinitesimal cubes. 1892. R.S.E., xix. S.P., cv.

331. Note on attraction. Proc. Edin. Math. Soc., 1893. S.P., cvi.

332. Carry. Golf, Aug. 1893.

333- Carry and run. Golf, Sept. 1893.

334. Some points in the physics of golf. in. 1893. Nature, XLVIII.

335- On tne compressibility of liquids in connection with their molecular pressure. 1893.
R.S.E., xx. S.P., cvn.

336. Vector Analysis. 1893. Nature, XLIX.

337. Preliminary note on the compressibility of aqueous solutions in connection with molecular
pressure. 1893. R.S.E., xx. S.P., cvui.

338. Review of Watson's "Theory of Gases." 1893. Nature, XLIX.

339. On the path of a rotating spherical projectile. 1893. Trans. R.S.E., xxxvn. S.P., cxn.

46 — 2

364 BIBLIOGRAPHY

340. Quaternions as an instrument of research. 1893. Nature, XLIX.

341. Obituary notice of Professor Robertson Smith. 1894. Nature, XLIX.

342. On the compressibility of fluids. 1894. R.S.E., xx. S.P., cix.

343. Note on the antecedents of Clerk Maxwell's electrodynamical wave-equations. 1894.
R.S.E., xx. S.P., cxiv.

344. On the application of Van der Waals' equation to the compression of ordinary liquids.
1894. X.S.E., xx. S.P., ex.

345. The initial pace of a golf ball. Golf, July 1894.

346. On the electromagnetic wave surface. 1894. R.S.E., xxi. S.P., cxv.

347. On the intrinsic nature of the quaternion method. 1894. R.S.E., xx. S.P., cxvi.

348. Systems of plane curves whose orthogonals form a similar system. 1895. R.S.E., xx.
S.P., cxvu.

349. Note on the circles of curvature of a plane curve. Proc. Edin. Math. Soc., 1895.
S.P., cxvin.

350. On the path of a rotating spherical projectile. Part II. 1896. Jf.S.E., xxi. Abstract of
Transactions paper, No. 351 below.

351. On the path of a rotating spherical projectile. Part II. 1896. Trans. R.S.E., xxxix.
S.P., cxin.

352. Note on centrobaric shells. 1896. J?.S.E., xxi. S.P., cxix.

353. Long driving. 1896. Badminton Magazine.

354. Note on Clerk Maxwell's law of distribution of velocity in a group of equal colliding
spheres. 1896. R.S.E., xxi. S.P., cxxv.

355. On the linear and vector function. 1896. R.S.E., xxi. S.P., cxx.

356. On the linear and vector function. 1897. R.S.E., xxi. S.P., cxxi.

357. Note on the solution of equations in linear and vector functions. 1897. R.S.E., xxi.
S.P., cxxii.

358. On the directions which are most altered by a homogeneous strain. 1897. R.S.E., xxn.
S.P., cxxin.

359. On the generalization of Josephus' Problem. 1898. R.S.E., xxn. S.P., cxxvi.

360. Note on the compressibility of solutions of sugar. 1898. R.S.E., xxn. S.P., cxi.

361. Queries on the reduction of Andrews' measurements on Carbonic Acid. Short letter
appended to K. Tsuruta's letter in Nature, LIX, Feb. 2, 1899.

362. The experimental bases of Professor Andrews' paper on the continuity of the gaseous
and liquid states of matter (Phil. Trans., 1859). Title only in R.S.E. Proc. for Feb. 20,
1899; but the original MS prepared for this paper is included in "Andrews' measurements
of the compression of carbon dioxide and of mixtures of carbon dioxide and nitrogen,"
edited by C. G. Knott, who, by request of the Council of the Roy. Soc. Edin., completed
the work begun by Tail. R.S.E. , Vol. xxx, 1909—10.

363. On the linear and vector function. 1899. fi.S.E., xxn. S.P., cxxiv.

364. On the claim recently made for Gauss to the invention (not the discovery) of quater-
nions. 1900. R.S.E., xxin.

365. Quaternion Notes. 1902. R.S.E., xxiv. Published posthumously in facsimile from
a sheet of foolscap on which Tail had jotted down some quaternion formulae two
days before his death; with commentary by C. G. Knott.

BIBLIOGRAPHY 365

LIST OF BOOKS WHOLLY OR PARTLY WRITTEN BY P. G. TAIT.

1. A Treatise on Dynamics of a Particle. By P. G. Tail and W. J. Steele. Macmillan and
Co., London. 1856, 1865, 1871, 1878, 1882, 1889, 1900.

2. Sketch of Elementary Dynamics. By W. Thomson and P. G. Tail. Maclachlan and
Stewart, Edinburgh. 1863. A pamphlet of 44 pages, afterwards incorporated in the
large type part of the Treatise, No. 3 in this list, and also in Nos. 5 and 7.

3. A Treatise on Natural Philosophy. Vol. I. By Sir W. Thomson and P. G. Tail. The
Clarendon Press, Oxford. 1867. Second Edition in Two Parts, The University Press,
Cambridge. 1878, 1883.

4. An Elementary Treatise on Quaternions. By P. G. Tait. The Clarendon Press, Oxford.
1867. Later and enlarged editions, The University Press, Cambridge. 1873, 1890.

5. Elementary Dynamics. By Sir W. Thomson and P. G. Tait. The Clarendon Press,
Oxford. 1867. Second issue [not published], 1868. Afterwards enlarged to Elements
of Natural Philosophy.

6. Historical Sketch of the Dynamical Theory of Heat. [Not published.] Thomas Constable,
Edinburgh. 1867. The first and second chapters of the Sketch of Thermodynamics (1868).

7. Elements of Natural Philosophy. By Sir W. Thomson and P. G. Tait. The Clarendon
Press. 1873. Second Edition, Cambridge. 1879.

8. Sketch of Thermodynamics. By P. G. Tait. Edmonston and Douglas, Edinburgh. 1868.
Second Edition, David Douglas, Edinburgh. 1877.

g. Life and Letters of J. D. Forbes. By J. C. Shairp, P. G. Tait and A. Adams-Reilly.
Macmillan and Co., London. 1873.

10. Introduction to Quaternions. By Philip Kelland and P. G. Tait. Macmillan and Co.,
London. 1873, 1881, 1904 (the last edited by C. G. Knott).

11. The Unseen Universe. By B. Stewart and P. G. Tait. Macmillan and Co., London.
1875, 1875, 1875, 1876, 1876, 1878, 1879, 1881, 1882, 1885, 1886, 1888, 1890.

12. Recent Advances in Physical Science. By P. G. Tait. Macmillan and Co., London. 1876,
1876.

13. Paradoxical Philosophy. By B. Stewart and P. G. Tait. Macmillan and Co., London. 1878.

14. Miscellaneous Scientific Papers by W. J. Macquorn Rankine, with a memoir of the Author
by P. G. Tait. Edited by W. J. Millar. Charles Griffin and Co., London. 1881.

15. Heat. By P. G. Tait. Macmillan and Co., London. 1884.

1 6. Light. By P. G. Tait. Adam and Charles Black, London. 1884, 1889, 1900.

17. Properties of Matter. By P. G. Tait. Adam and Charles Black, London. 1885, 1890,
1894, 1899, 1907 (the last edited by W. Peddie).

18. The Scientific Papers of the late Thomas Andrews, with a Memoir by P. G. Tait and
A. Crum Brown. Macmillan and Co., London. 1889.

19. Dynamics. By P. G. Tait. Adam and Charles Black, London. 1895.

20. Newton's Laws of Motion. By P. G. Tait. Adam and Charles Black, London. 1899.

21. Scientific Papers. By P. G. Tait. Vol. I. The Cambridge University Press. 1898.

22. Scientific Papers. By P. G. Tait. Vol. II. The Cambridge University Press. 1900.

Of these books, Nos. 4, 8, n, and 12 were translated into French; Nos. 3, 4, 12, 15,
and 1 7 into German ; and No. 1 2 into Italian.

INDEX

Names of individuals are indexed in heavy type; names of books in italic.

Absolute Temperature, 226, 228

Absorption Line doubled by magnetism, 90

Academy, Edinburgh, 3

Action and time in planet's orbit, 100

Action, Theory of, 99

Address, illuminated, to Tait on his retirement, 39

Address to Graduates, 11, 247-251

Addresses, etc., 246-295

Adiathermancy of ozone, projected experiments, 68

Airy, G. B., on differential notation, criticised by

Hamilton, 121

Alban and Nabla, Maxwell on, 244-245
Alexander, Hugh and Pat, at St Andrews, 54
Amagat, E.-H., on Tail's electric contact, 85
Ampere, J. J. A., repulsion of current on itself,
66-67 ; electrodynamics by quaternions, 138 ;
electricity and magnetism, 259, 297
Anagram heralds The Unseen Universe, 237
Andrews, Thomas, experiments with Tait, 12 ;
influence on Tait, 13; letter to, 22; at St
Andrews, 56 ; letters to, on experimental
preparations, 64-66, about new laboratory, 70,
on thermoelectricity, 76-78 ; introduces Tait to
Hamilton, 119; letters to, about "T and T',"
177-179 ; president of Brit. Ass. in 1876, 252
Andrews, Mrs, Tail's letter to, 13
Andrews, Misses, supply letters to their Father,

22, 177

Angstrom, A. J., thermal conduction, Si
Arago, F. J. D., quoted, 305, 312 ; on fire-balls,

313

Archimedes, 52

Arnold, Matthew, classed by Herbert Spencer with
Tait as possessing mental idiosyncrasies, 285

Art of Golf, 59

Artificial Selection, article in Macmillan's Maga-
zine, 248

Artillery, Russian Review of, contains translation
of Kinetic Theory of Gases, 25

Assurance, Life, Tail's interest in, 24-25

Astronomy, HerschePs, used as texl-book, 20

Astronomy, Cosmical, articles in Good Words,

24, 246, 252 ; lectures in Museum to Industrial

Classes, 24, 252

Athenaeum reviews "T and T'," 186
Atmomeler, used as hygrometer, 87-88
Atmospheric Electricity, 184, 298-299 ; source of,

315 ; connection with Aurora, 320
Atmospheric Instability, 319
Atom, Vortex, of Thomson, 68-69, 105-106, 177
Aurora and atmospheric electricity, 320
Authors, Tail's favourile, 33

Babinet, J., on fire-balls, 313

Badminton Magazine, article on Long Driving,

28, 58, 90, 329-344
Bain, A., " Logic " reviewed, 247
Balfour, James, at St Andrews, 56
Balfour, J. H., secretary of Royal Society of

Edinburgh, 28, 149
Ballistic pendulum for measuring speed of golf

balls, 89, 337
Barclay, George, 349-350
Barrie, J. M., description of Tait lecturing, 17 ;

description of " Little T and T',1' 201
Bashforth, F., tables of air resistance, 26 ;

resistance of the air, 336
Baynes, T. S., editor of Encyclopaedia Britannica,

291
Becket, Sir Edmund, book on the origin of the

laws of nature reviewed, 284
Belfast, Tait in, 12-15
Bell, Beatson, schoolmate and secretary of the

Gumming Club, 9
Bennett, Hughes, experiments on eleclrical effects

in muscles, 65
Berkeley, G., objective and subjective, 19 ; referred

to by Maxwell in connection with Torricelli,

195
Bertrand, J., criticises Helmholtz, 115 ; criticises

Maxwell's Law, 1 16
Bethune at St Andrews, 56
Big Gun, The, 86, 96
Biot, J. B., on mirage, 322

368

INDEX

Black, Joseph, work on heat, 218

Blackie, John Stuari, stories concerning, 46

Blackwell, Dr, al Si Andrews, 55

Blaikie, W. B., designs The New Amphion, 73

Boisbaudran, F. Lecoq, supplied thermoelectric

material, 78

Boltzmann, L., Kinetic theory of gases, 1 1 1
Bolzani, T., of Kasan, helps "T and T'," 188
Books, Tail's published, 205-245
Boole, G., referred to by Tail, 127
Borthwick, Lord, al Si Andrews, 56
Bouch, Sir Thomas, engineer of Tay Bridge, 279
Boyle and Marriotte, 225-226, 230
Boys, C. V., lanlern leclure in Tail's drawing

room, 34

Brachistochrones, Tail's work on, 99
Brebner, A., experiments on eleclrolysis, 76
Brescia powder magazine exploded by lighlning,

Brewster, Sir David, 69

British Associalion, Tail presideni of Seclion A,
1871, 24 ; meelings of 1871, 1892, 33, 34 ; report
on Ihermal conduclion, 81 ; meeling of 1882, 92 ;
meeling of 1872, npie on plane curves, 105 ;
meeting of 1871, Maxwell writes Tyndallic ode,
171 ; meeling of 1874, Maxwell's description,
173, 174 ; proposed report on qualernions, 152 ;
meeting of 1874 at Belfast, 236; meeting of
1876 at Glasgow, lecture on Force, 252

British Quarterly Review, criticism of Herbert
Spencer's First Principles, 278-283

Brougham, Lord, referred to in lecture on Force,

2S3

Brown, Rev. Canon Cowley, 40
Brown, A. Crum, 41 ; phosphorescent golf balls,

57; preparation of pure cobaltfor ihermoeleclric

work, 78-79 ; discontinuities in organ notes,

91 ; at St Andrews, 197 ; tells Tail of Mohr,

212 ; lectures to Industrial Classes, 252 ;

Iranslates Helmholtz's reply to Zollner, 256
Bruntsfield Links, Golf al, 23, 54
Buchan, Alexander, consulls Tail, 51 ; Ben Nevis

Observatory, 87 ; leclures to Industrial Classes,

252

Buchanan, J. Y., compressibility apparatus, 83
Budget of Paradoxes, de Morgan's, reviewed,

257-258

" Bulger, The," verses by Tail, 61, 62
Bunsen, R. W., specirum analysis, 227
Burbury, S. H., on kinelic theory of gases,

in

Burnett Lectures by Stokes, reviewed, 266-269
Burnside, W., reads proofs of " T and T," second

edilion, 203
Byron, Lord, Tail's favourite poet, 33 ; description

of thunderstorm, 298

Calendar, Edinburgh University, reference lo "T
and T',1' 197

Gallon Hill, McFaii's experimenls on almospheric
eleclricily on, 307-308 ; fire-ball seen al, 314

Cambridge career, 8-n ; orchestra, 12

Campbell, Lewis, al Edinburgh Academy, 5 ; Life
of Maxwell, 171, 242; al ihe Evening Club,
348

Capacily, eleclrical, 303

Carnot, N. L. S., pioneer in iheory of heat and
energy, 208, 209 ; cycle of operalions and
perfect reversible engine, 218, 226; function
of, 221 ; his Principle expounded in Recent
Advances, in Unseen Universe, elc., 227, 237,
241

Cassio, "a greal arilhmelician," 151

Caslle Hill, McFaii's experiments in atmospheric
electricity on, 307-308

Castor and Pollux, electrical phenomenon, 306

"Cat's Cradle Song" by Maxwell, 108-109

Catenary models al Edinburgh University, 7

Cathode Rays, velocity of charged particles, 91

Cauchy, A. L., referred to by Hamilton and Tail,
121, 122

Cayley, A., 33 ; on brachistochrones, 99 ; on knot
problem, loo; on malrices, 130, 153; on
rolalion, 149 ; correspondence wilh Tail, 152-
166; orthogonal surfaces, 154; on quaternion
inlerpretations, 154-158; contributes chapter
lo Tail's Quaternions, 158-159; on coordinates
versus quaternions, 161-166 ; letter to, referring
to Kelvin's lameness, 189 ; on Laws of Motion,
234-235 ; at the Evening Club, 349

Centrobaric Bodies in "T and T'," 192

Cervantes, a favourile author, 33

Chair of Natural Philosophy at Edinburgh, election
lo, 16 ; retirement from, 39

Challenger Lodge, last days at, 40

Challenger Reports, thermomeler correclions, 24,
95 ; final report, 82-86

Chambers' Encyclopaedia, 23

Charcoal Vacua, experiments wilh Dewar, 81

Charles, J.-A.-C., and Ihe laws of gases, 226

Chaucer, quoted by Hamilton, 121

Chauvinism, 225

Chiene, John, at Si Andrews, 56

Christison, Sir Robert, portrail in R.S.E. rooms,
29

Chrystal, G., nolice of Tait in Nature, 42 ; works
in Physical Laboratory, 87 ; invenis hygrometer
for Ben Nevis Observatory, 87 ; letter from
Kelvin about quaternions and "T and T'," 185 ;
reads proofs of "T and T'," second edition, 203 ;
refers lo " Tail and Sleek," 205

Clark, Lord Rutherford, at St Andrews, 56

Clarke, C. B., of Queens', n

INDEX

369

Class mates at the Edinburgh Academy, 5, 9

Clausius, R., Second Law of Thermodynamics, 18 ;
isothermal equation, 112, 113; entropy, 115;
theory of heat, 209, 218, 221 ; Tail's pamphlet
on Thermodynamics, 216, 217; complains of
Tait's historic sketch, 221 ; referred to by
Maxwell, 222 ; Thomson on the entropy
integral, 223-224; Fourth Memoir discussed by
Tail, 225; "Tendency" of Tait's history
objected to, 225 ; referred to by Tyndall in
letter to Tail, 257

Cleek, Tait's furrowed, 28, 61

Clifford, W. K., 33 ; referred to in connection with
B. A. report on quaternions, 153 ; on chemical
equations, 1 75 ; reviews The Unseen Universe
in the Fortnightly Review, 239-240 ; Elements
of Dynamic reviewed, 270-272 ; The Common
Sense of the Exact Sciences reviewed, 272-273 ;
at the R.S.E., 282 ; at the Evening Club, 349

Clouds, electric condition in, 316

Club, the Evening, 347-349

Club, the Gumming, 3, 6, 9

Club House at St Andrews, 53, 62

Cockburn, General, a school mate, 5

Colding, L. A., work on heat, 210, 211, 212; at
the Evening Club, 349

Colleagues in Queen's College, Belfast, 12

Colleagues in Edinburgh University Senatus, 46, 47

College Memories by R. L. Stevenson, 73-74

Comet's Tails, Tait's views, 246-247

Compressibility of gases (Andrews), 66

Compressibility of various liquids, 83-86

Compression measured by electric contact, 85

Concentration, Maxwell's name for v2, 144

Conductivity, thermal, 80, 81, 296

Conductor, lightning, 305, 306

Conservation of Energy, discovered in Principia,
191 ; explained in inaugural lecture, 206

Constraint, cases of, and Hamilton's characteristic
function, 99

Constraint, geometrical, 188

Contemporary Review, Tyndall's defence in, 24

Continuity, Law of, 63, 240

Controversies, 18, 209, 281-288

Convergence, Maxwell's name for S.v, 143-144

Copernicus, 296

Correspondence, 32 ; rhyming, 93-95

Cosmical Astronomy, lectures on, 24, 252 ; articles
in Good Words, 24, 246, 252

Coulomb, C. A. de, 259

Cox, R., of the Evening Club, 347

Cram, denunciation of, 249-251

Cremona, L., 33

Critchley, L. O., photograph of Tail and Lindsay,
So

Critic, Tail as, 255

T.

Crookes' Radiometer, 81, 82

Camming, James, 4

Gumming Club, the, 3, 6, 9 ; banquet in honour

of Tail as Senior Wrangler, 10
Curl, Maxwell's name for V.v, 143-144
Cycle of Operations, Carnot's, 218, 226
Gyre, Alfred le, translates Tait's Thermodynamics,

226
Czar of Russia and Freddie Tait, 54

Daguerre, photography, 3

D'Alembert's Principle, described by Tait, 235-236

Dalkeith, Tait's birthplace, 3 ; Grammar School, 3

Darkness before thunder, 298-299

Darwin, Sir George, tidal stresses, 193 ; reads

proofs of " T and T','1 second edition, 203
Davis, Capt. J. E., tests deep-sea thermometers, 83
Davy, Sir Humphry, experiments on heat, 209,

211, 212, 218

Demons, Maxwell's, 213, 214
Density, electrical, 304

Dewar, Sir James, pupil and friend, 51 ; prepares
sodium and potassium for thermoelectric work,
79; experiments in high vacua, 81, 82 ; thermo-
electric investigations at low temperatures, 345
Diathermancy of water vapour, 92
Dickens, Charles, a favourite author, 33
Dickson, H. N., work on compression, 96
Dickson, J. D. Hamilton, story of Wranglership,
9 ; notice of Tait in Peterhouse Magazine, 42 ;
foundation of Tait Prize at Peterhouse, 50 ;
extension of thermoelectric diagram, 345-347
Differential Equations, linear, of second order, 105 ;

in quaternions, 152
Differential Telephone, Chrystal's, 87
Differentials in quaternions, 120-123, H1
Diffraction, Stokes' dynamical theory of, 145
Dissipation of energy, 24 ; Thomson's expression

for, during cycle, 221, 223, 224
Distance travelled by thunder, 312
Distinctions at school and college, 6, 7, 8, 9
Distribution of electricity, 304
Donaldson, James, and the Evening Club, 347
Doppler and Eomer, 226
Douglas, D., of the Evening Club, 348
Dove, H. W., account of fire-balls, 314
Driving, Long, article in Badminton Magazine, 28,

58, 90, 329-344

Dnunmond, James, and the Evening Club, 347
Duff, Sir M. E. Grant, and the Evening Club, 348
Dumas, Alexandra, a favourite author, 33
Duncan, J. Matthews, and the Evening Club, 347
Dux, permanent, at Edinburgh Academy, 4
Dynamic, Clifford's Elements of, reviewed, 270-272
Dynamical theory of heat, historic sketch of, 213-
225

47

370

INDEX

Dynamical Equivalent of heat, Maxwell's method

of experimenting, 215
Dynamics, Tail's, 21, 24, 231-232
Dynamics of a Particle, Tail and Steele's, 10, 205-

207 ; MS. preserved in Peterhouse, 10
Dynamo, Gramme, 80

Edinburgh, University student at, 6 ; Tail's career
at, 16-52 ; Malhematical Sociely of, 29 ; Royal
Society of, 28-31 ; Degree of M.A., address to
graduates, 247

Edinburgh pamphlet on Dynamics (1863), 197-200

Elastic wires and plates in "T and T'," 192-193

Elasticity, 21 ; in "T and T',n 193

Electricity, developed by muscular contraclion, 65 ;
repulsion of curreni on itself, Ampere and Tail,
66-67 > Tail's contact device in measuring com-
pression, 85 ; developed by evaporation, 68, 90,
300; atmospheric, 184, 298, 299, 315 ; nature of,
297 ; relation to light, 297 ; potential, 302-303 ;
capacily, 303 ; density and distribulion, 304 ;
eleclrical condition in ihunder cloud, 316 ;
eleclrical effects during volcanic eruplions, 317;
due lo condensalion, 319

Electricity and Magnetism, Maxwell's, 176; re-
viewed, 258-260

Eleclrified particles, projeclion of, 345

Electrostatics and Magnetism, Thomson's Reprint
of papers on, 1 76

Elements of Quaternions, 141

Elements of Dynamics, 199

Elements of Natural Philosophy, 197-201

Ellipse, glissettes of, 160

Elmo's Fire, St, 306

Emo, D'Angelo, translates Recent Advances into
Italian, 228

Encouragement to former students, 96-97

Encyclopaedia, Chambers', 23

Encyclopaedia Britannica, 24 ; articles by Chryslal,
87 ; article Mechanics, 155, 231 ; article Lighl,
229

Energeiics, i

Energy, article in Good Words, 23, 209 ; dissipalion
of, 24, 223-224 ; conservalion of, 179 ; discussed
in inaugural leclure, 206 ; kinelic, lerm first
used, 209; Wall's energy diagram, 221

Engineer, The, reviews " T and T'," 186

Enseignement mathtmatique, f, notice of Tail by
J. S. Mackay, 42

Entropy, Maxwell on Clausius, 115; Clausius" claim
lo ihe integral, 209, 224-225 ; moral and intel-
lectual, 252

Envelopes, Tail's Surface, 123-124

Equiangular Spiral, molion in, 102

Equilibrium of helerogeneous subslances (Willard
Gibbs), 284

Ergal, referred lo by Maxwell, 222

Error, frequency of, 98

Espert, Madame, accounl of fire-ball, 314

Euler's equalions in qualernions, 149

Evaporalion, electricily developed by, 68

Evening Club, ihe, 347-349

Evoluiion, Herbert Spencer's definilion or formula

of, 286

Ewing, J. A., resislance of solutions, 96
Examinalion, evils of, 249-250
Experiment to measure mass of earth projected,

15
Experimental work, 64-97 ; general character of

Tail's, 95
Experiments with Andrews, 12

Facsimile of Tail's manuscripl, 181

Faraday, Michael, 43, 52 ; views on conservalion
of force, 66 ; eleclricily and magnelism, 297

Famie, H. B., An of Golf, 59

" Father of Freddie Tail," 54

Faulls in golf strokes, 333

Favourite authors, Tail's, 33

Favre and Silbermann, heats of combination, 1 78

Ferguson, John, edils Thomson's Elements of
Dynamics (1863), 199

Fergusson, Ll.-Col. Alexander, aulhor of ihe
Chronicles of the Cummins? Club, 4

Findlater, Andrew, laughl Tail golf, 23 ; ihe Even-
ing Club, 347

Fire-ball, 312-314

First Principles, controversy over Herbert Spen-
cer's, 278-288

Fleming, A. B., thermoelectric investigations at
low temperalures, 345

Flexible cords in "T and T'," 192

Flint, Robert, appreciaiion of Tail, 44-46 ; Tail's
admiralion of, 46

Flule, Tail an adepi on ihe, 12

Fluxions and differenlials, 121-122

Fog signalling, 90, 91

Forbes, George, 24

Forbes, J. D., natural philosophy class, 6 ; relires
from chair, 16 ; glacier work, 18, 24 ; portrait
in R. S. E., 29 ; and James Lindsay, 66 ;
experiments on thermal conduction, 80 ; and
Balfour Stewart, 289

Force, lecture on, 252-255 ; lecture versified by
Maxwell, 253-255; persistence of, 279; and
energy, Spencer's views concerning, 283 ; defi-
nilions of, 287, 288

Forked lightning, 311

Formula of evolution, Herbert Spencer's, 286

Fortnightly Review, Herbert Spencer's replies lo
criticisms, 279

Foster, Carey, founds laboratory, 22

INDEX

Fourier, J. B. J., analysis, 22 ; influence on Thom-
son, 191 ; and Ohm, 226

Franklin, B., referred to by McFait, 306

Fraser, A. C, Tail's joke at, 250

Fraser, T. R., medical attendant, 41, 42

Freedom, degrees of, in "T and T'," 188

Frequency of error, law of, 98

Fresnel's wave surface, 124-130, 133-134, 141

Fronde, Anthony, Tail's reply to, 36

Fuller, F., candidate for Edinburgh chair, 16 ; as
teacher, 17

Future Siale, physical speculations on a, 236

Gases, Kinetic theory of, 25, 109-113; Russian
translation of Tail's papers on, 25 ; Watson's
treatise, in

Gases, Andrews' experiments on compressibility
of, 66

Gauge, high pressure, 84-85

Gauss, J. K. F., in relation to quaternions, 37 ;
theorem, 149 ; curvalura inlegra in " T and T',"
1 88; magnetism, 259

Gay-Lussac, J. L., and the laws of gases, 226

Geits al Edinburgh Academy, 4

Gerebiateffe, J., translates Tail's Kinetic Theory
of Gases into Russian in Russian Review of
Artillery, 25

Gibbs, Willard, 115 ; equilibrium of heterogeneous
substances, 284

Gibson, G. A., nolice of Tail in Edinburgh Medical
Journal, 42

Gifford Leclures at Edinburgh, Stokes, 35 ; Helm-
hollz, 34-35 ; al Glasgow, Tail asked lo give, 35

Gilbert, L. W., on mirage, 321

Glacier iheory, Forbes', 1 8, 24

Gladstone, W. E., as slalesman, dislrust of, 37

Glaisher, J., observations of temperature changes
in air, 326-327

Glasgow, pamphlet (1863), 199-200; lecture on
Force at meeting of Brit. Ass. in, 252, 253 ;
lecture on thunderstorms in, 255-296

Glissettes of ellipse and hyperbola, 160-161

Gloag, James, mathematical master at Edinburgh
Academy, 4, 5 ; elation on receiving news of
Tail's Wranglership, 9 ; at the Gumming Club
banquel, 10

Globe lighlning, 312-314

Goethe, Tail quotes, 48

Golf, at Bruntsfield Links, 23, 54 ; physical problem
in, 25, 27, 28, 88, 331-344 ; unwritten chapter
in, 26 ; laws of impact in relation lo, 28, 88, 89,
33'> 332 ; at Si Andrews, 52-63 ; at Mussel-
burgh, 54; "the Morning Round" at, 55;
contribulions to the magazine Golf, 58 ; articles
on, 58 ; art of, and science of, 59 ; article,
Long Driving, 329-344

Golf Ball, flight in air, 25, 26, 59, 60, 88, 116,
334-344 ; pulling, slicing, topping, etc., 26-27,
59-60, 117, 331-332; underspin in, 27, 59, 60,
337-344 ; " the Bulger," 61, 62 ; phosphore-
scent, 57 ; impact with club, 88, 330-331 ;
speed measured ballistically, 89, 90, 337 ; palh
imilated by Sir J. J. Thomson's streams of
negalively charged particles, 344-345

Golf Match problem, 103-104

Good Words, article on energy by Thomson and
Tail, 23, 209 ; articles by Tail on cosmical
aslronomy, 24, 246

Gordon, Rev. Cosmo Reid, candidale for ihe
Edinburgh chair (1860), 16

Graduates, addresses lo, u, 247, 249-251

Gramme Dynamo machine, 31, 80; experiments
wilh, by Sir W. Thomson, 32

Grant, Sir Alexander, Tail's regard for, 46

Grassmann's Ausdehnungslehre, referred lo by
Maxwell, 152

Gravilation, mechanism of, 296-297

Green's Theorem, 22, 149; and olher allied
theorems, quaternion discussion, 148, 149, 150;
extension of, in "T and T'," 188

Green's Problem in "T and T'," 192

Greig, C. E., experiments in thermoeleclricily, 77 ;
in ihermal conduclion, 8 1

Grove, W. R., Correlalion of ihe Physical Forces,
211

Guilloline, apparaius for sludying impacl, 86, 88 ;
guilloline room, 88

Gun, Ihe Big, 86, 96

Gunning Vicloria Jubilee Prize, 47

Gutnrie Headstone (G. H.), Tail's pseudonym, 278

Gyrostalic pendulum, 191 ; domination, 204

Haan, Bierens de, 33 ; at the Evening Club, 349

Hall, T. Wright, a class mate, 5

Halley, E., referred to in inaugural address, 206

Hamilton, Sir William Rowan, 12, 13, 14,43, '9i>
205 ; influence on Tail, 1 2 ; referred lo in
inaugural address, 24 ; Life by Tait in North
British Review and in Encyc. Brit., 24 ;
Characleristic Function referred to by Max-
well, 99, 229 ; Tail's correspondence with,
119-140; Hamiltonsche Princip, 115-116;
Lectures read by Tait, 119, 126-127; defini-
tion of differenlials, 120 ; quotes Chaucer,
Cauchy, Moigno, and Newton on fluxions,
121 ; criticises Airy, 121 ; discusses Tail's
envelope, 123-124; on wave surface, 124-
126, 128-130; publishes Elements of Quater-
nions, 132 ; teslimonial lo Tail, 138-139 ; ihe
hodograph, 205 ; referred lo by Maxwell as
having been ignored by Kirchhoffand Clausius,
222 ; general method in oplics, 321

47—2

372

INDEX

Hamilton, Sir W., ambassador at Naples, de-
scribes eruption of Vesuvius, 316-317

Hamilton, Sir W., of Edinburgh, referred to by
Tail, 281

Handwriting, facsimile of Tail's, 182

Hanna, Rev. W., speaks highly of The Unseen
Universe, 239

Harland, E., a school mate, 5

Harmonics, spherical, 100-102 ; in "T andT'," 188,
189, 190 ; Maxwell refers to, 100-102

Haughton, S., book reviewed by Tait, 256

Heat, Tail's book on, 24, 210, 228-231 ; reviewed
by Balfour Stewart, 230

Heat, dynamic theory of, 208 ; evolved during
condensation and precipitation, 315

Heats of combination, Andrews, Favre and Silber-
mann, 178

Hegel, G. W. F., criticism of Newton's fluxions
refuted by W. Robertson Smith, 76

Helmholtz, H. von, 32, 33 ; at Brit. Assoc. (1892),
34 ; asked to give Gifford Lectures, 34-35 ; at
St Andrews, 56, 197 ; paper on vortex motion
translated by Tait, 68, 105, 127 ; viscosity of
liquids, 114; electrodynamics, 115; on Tait
and quaternions, 169 ; vortex motion, 177, 226;
referred to in letter to Andrews, 1 79 ; German
translation of "T and T'," 195-196 ; letter to,
explaining nautical terms, 196, 197 ; describes
Tait at golf, 197 ; Tait in close touch with,
208 ; Ueber die Erhaltung der Kraft, reviewed
in North British Review, 210; on Tail's
Thermodynamics, 216-217 > advice to Tait lo
shun polemics, 217 ; reply lo Zollner, 256

Henderson, W., class room iransformed into
physical laboratory, 70

Herbertson, A. J., experiments on impact, 96

Hermite, C, 33

Herschel, Sir J. F. W., Astronomy used as text-
book, 20; letter to, from Tait about quaternions,
141-142

Hodge, Tom, at St Andrews, 56 ; measurement of
elevations, 337

Hodograph, 99, 102, 205

Hole, W., artisl of Quasi Cursores, 49

Homer, at School, 6

Homogeneous, instability of the, 284-285

Honours from Academies and Universities, 47

Hopkins, W., 8, H, 265

Horace, favourite author, 33 ; quoted, 311, 329

Horograph in "T and T'," 188

Huggins, Sir W., spectroscopy, 227

Huxley, T., at St Andrews, 56, 57, 197 ; at the
Evening Club, 349

Hydraulic apparatus for pressure experiments, 83

Hydrodynamics, 21

Hygrometers for Ben Nevis, Chrystal's and Tail's, 87

Hyperbola, glissettes of, 160-161

Ideal University, Tail's, 248

Illuminated address to Tait on retirement, 39

Impact, in relation to golf, 28, 88, 89 ; duration of,

88, 89, 331
Inaugural lecture in Edinburgh University, 22,

'39
Industrial classes, lectures to, 252

Inglis, Lord, at St Andrews, 54

Instability of the homogeneous, Herbert Spencer's,

284-285
International Review article on " Does Humanity

demand a new Revelation ? " 36
Intuitions, physical, 95
Isothermal equation, various forms, 112

Jackson, Sir J., Tait memorial fund, 50-5 1 ; student

in laboratory, 72

Jacobi, K. G. J., referred to by Hamillon, 1 19
Jenkin, Fleeming, a school male, 5 ; al R. S. E.,

3°

Josephus' problem, 118

Joule, J. P., 43, 47 ; Tail in close louch wilh, 208 ;
prolesls against Tyndall's history of energy,
209 ; experiments in heat, 209 ; wrilings re-
viewed in North British Review, 210 ; pioneer
work in ihermodynamics, 210-212 ; dynamical
equivalent, Maxwell's mode of measurement,
215-216 ; and Thomson, 224, 225

Jupiler, observalions on, 3

Keiih Prize of R. S. E. awarded iwice lo Tait, 47,

'49

Kelland, P., 6, 289

Kelland and Tait's Introduction to Quaternions,
149

Kelvin, Lord (see also W. Thomson), reference
lo Tait's marriage, 14 ; thermodynamical dis-
coveries, 19 ; prompts Tait to lake up ihe
kinetic theory of gases, 25 ; frequent visitor at
Tait's house and laboratory, 31-32 ; wild ex-
perimenting with Tait's Gramme machine, 32 ;
at Brit. Assoc. (1892), 34 ; at Stokes' Gifford
Lecture, 35 ; obituary notice of Tail, 43 ; un-
veils Tail's porlrail al Pelerhouse, 48 ; account
of Tait and Dewar's experimenls on high
vacua, 81, 82 ; lelter from, 104-105 ; Sylvanus
Thompson's Life of Kelvin, 1 84, 200 ; letter
to Chryslal regarding quaternions, 185 ; names
Maxwell's demons, 214 ; on Tait's sea-bird
analogy, 246 ; supported Tail in his crilique of
Poincard's Thermodynamique, 276

Kepler, J., 296

Kinelic Energy, term first used, 209

Kinelic Theory of Gases, 25, 109-115; Russian

INDEX

373

translation of Tail's papers, 25 ; Watson's
treatise on, in

Kingsburgh, Lord, at St Andrews, 56

Kirchhoff, G. R., referred to in letter to Helmholtz,
216 ; referred to by Maxwell, 222 ; laws of
radiation, 225 ; spectrum analysis, 227

Kirkman, Rev. T. P., on knots, 107-108 ; transla-
tion of Herbert Spencer's definition of evolu-
tion, 284 ; mental idiosyncrasies discussed by
Herbert Spencer, 285

Knocklayd Hill, proposed measurements of lati-
tude, 15

Knots, 24, 105, 106 ; Cayley, Muir, and Kirkman,
107 ; Maxwell's verses on, 108

Knott, C. G., assistant to Tait, 31, 86 ; experi-
ments on thermoelectricity, 77-80 ; contact
electricity, 96 ; flow of water in tubes, 114

Knottiness, orders of, 106

Korteweg, D. J., discussion on Van der Waals'
equation, in

Krouchkoll, M., translates Recent Advances into
French, 228

Laboratory, Tail's Physical, 22, 70, 71, 72 ; Thom-
son's and Carey Foster's, 22 ; expansion of
Tail's, 86 ; Chrystal works in, 87
Lagrange, J. L., Calcul des Fonctions, criticised

by Hamilton, 122 ; general dynamic methods,
176, 191 ; Mtcanique analytique, 194
Laidlay, J. E., at St Andrews, 60
Lam6, G., referred to by Maxwell, 117
Lang, Andrew, 59

Lang, P. R. S., assistant to Tait, 21, 86
Laplace, P. S., on Probabilities, 98 ; Laplace's

Coefficients or spherical harmonics, 102, 189 ;

referred to, 176

Larmor, Sir J., Aether and Matter, 177
Last illness, 38 ; last paper (on Gauss and qua-
ternions), 37 ; last printed statement, 38 ; last

written notes on quaternions, 40
Latin verses, skill in, 4
Laws of Nature, Origin of (Becket), 284
Lead punctured by lightning, 310
Least Action, principle of, 256
Lecker, E., translales Tail's Heat into German,

229

Lecture, Rede, on thermoelectric diagram, 251
Lecturer, Tail as, 17, 255
Lectures, to Edinburgh men, chiefly professional,

227. 349-35° ; to Induslrial Classes, 24, 252
Leslie, Sir John, 66
Life Assurance, 24
Lighl, article in Encyc. Brit., 24, 229 ; book on,

24, 228-229 ; and electricily, 297
Lighlhouse, Skerryvore, struck by lightning, 90

306

Lighlning, inlensily, 300; duralion, 300-301 ; zig-
zag characler, 301-302; rod, 305-306; de-
structive effects, 309-310 ; punctures sheet
lead, 310; forked, sheet, and summer, 311

Lindsay, James, mechanical assistanl lo Leslie,
Forbes and Tail, 66, 74 ; described in College
Memories by R. L. Slevenson, 73-74

Lindsay, Tom, succeeds his father as mechanical
assistanl lo Tait, 74 ; experiments in dialher-
mancy, 92 ; Tail's righl hand, 95

Lindsay, Rev. T. M., on W. Robertson Smith,
171 ; and the Evening Club, 348

Linear Vector Function, 130, 135-138, 141, 142,
1 68

Lister, Lord, at R. S. E., 30

"Lillle T and T'" as lexl-book, 21, 191-201, 231

Liveing, G. D., lends apparalus for Tail's Rede
Leclure, 251

Lockyer, Sir J. N., 24

Loewy, B., 290

Long Carry of golf ball, 60

Long Driving in Badminton Magazine, 28, 58, 90,

329-344
Low, J. L., author of F. G. Tait, a Record, 2, 40 ;

on Tait at Si Andrews, 52-63
Lucretius, on ihundersiorms, 298
Lushington, E. L., helps Thomson lo coin names,

1 88

McAulay, A., Utility of Quaternions reviewed,

276-278

McColl, Rev. Dr, praises The Unseen Universe, 239
McCosh, Rev. J., colleague at Belfast, 12
Maccullagh, theorem of, 134
McFait, E., experiments on atmospheric electricity,

306-309
Macfarlane, A., notice of Tait in Physical Review,

42 ; experimenls on eleclric properties of iron,

80 ; electrical discharge, 96
MacGregor, J. G., experiments on ihermoelectricity,

79 ; on dialhermancy, 92 ; on eleclrical resisl-

ance of solutions, 96
Mackay, Aeneas J. G., and ihe Evening Club, 348
Mackay, J. S., notice of Tait in I ' Enseignement

Mathtmatique, 42
McKendrick, J. G., lectures lo Industrial Classes,

252

Maclaurin, referred to, 306
MacLeish, A. L., experimenls on ihermal con-

duciion, 8 1

Maclennan, J. F., and ihe Evening Club, 348
Macleod, Rev. N., editor of Good Words, 252
Macmillan's Magazine, article on Artificial Selec-
tion, 248
Magnetic doubling of absorption line, 90 ; poten-

lial, 147

47—3

374

INDEX

Magnetism, nature of, 297

Magneto-optics, 147

Magnus, P., on deviation of rotating ball, 27, 60,

339

Maitland, John, and the Evening Club, 348
Manuscript of Dynamics of a Particle preserved

at Peterhouse, 10
Manuscript note-books, 179-180
Marriage, 14

Marriotte and Boyle, 225, 226, 230
Marshall, D. H., assistant to Tait, 72, 86
Masson, David, and the Evening Club, 347
Mathematical Society of Edinburgh, 29, 117
Mathematical Work, Tail's, 98-118
Matrix, square root of, 130, 152, 153 ; Cayley on,
153 ; Sylvester's solution of linear matrix equa-
tion, 158-159

Matter, Properties of, lectures, 19 ; book on, 24,
176, 228-230; book reviewed by Lord Ray-
leigh, 229-230

Matter and Motion, Maxwell's, reviewed, 260-261
Maximum density point of water affected by pres-
sure, 84

Maxwell, J. Clerk, at Edinburgh Academy, 5 ; at
Edinburgh University, 7 ; candidate for Edin-
burgh chair, 16 ; his creative genius, 17 ; at
Tail's, 33 ; referred to by Kelvin, 43 ; letter from,
about thermoelectricity, 80 ; rhyming corre-
spondence, 93-95 ; on brachistochrones, 99 ;
on Hamilton's work, 99 ; on spherical har-
monics, 100-102, 189, 190; quaternions, 101,
145, 147-150, 151, 252 ; ihermodynamic func-
tion, 101, 1 15 ; on vortex motion, 105 ; verses on
knots, 108-109 ; Theorem, 109 ; kinetic theory
of gases, 111-114; °n Boltzmann, 114 ; letter
to Tait, 114-115; on Helmholtz and Pie-
trowski, 114, 115 ; on Ampere, 115 ; on Ber-
trand, 115; on Nabla, 115; on Clausius,
Rankine, Willard Gibbs, enlropy, ihermo-
dynamic function, virial, ergal, Hamiltonsche
Princip, 115; distribution of velocities, 116;
on Tail's mathemalical methods, 117; on
Nabla and its effects, 143 ; on Hamiltonian
ideas, 144 ; on Stokes on diffraction, 145 ;
Tyndallic Ode, 145, 171-173 ; on eleclro-
magnetism, 146 ; on Hamilton of Edinburgh,
147 ; magnetic aclion on light, 147 ; magnetic
potential, 147 ; on Tait as Keith Prizeman, 149-
150 ; on Tail's "middle period," 153 ; Electri-
city and Magnetism, 1 70 ; criticises " T and T' "
in leller, 195 ; reviews Elements of Natural
Philosophy, 201 ; reviews "T and T','' 203-204 ;
inlimacy wilh Tait, 208 ; on Tail's Thermo-
dynamics, 213-215 ; calechism concerning
demons, 214-215 ; suggests measuring Joule's
equivalenl by fall of mercury, 215-216; reviews

Tail's Thermodynamics, 220 ; on Clausius,
Hamilton, Kirchhoff, Rankine and Thomson,
222 ; quoles Torricelli, 222 ; reviews Para-
doxical Philosophy, 241-242 ; a Paradoxical
Ode, 242-244 ; lasl leller lo Tait, 244-245 ;
arrangements for Tail's Rede Lecture, 251,
252 ; rhyming report of Tail's Leclure on
Force, 253-255 ; Electricity and Magnetism
reviewed, 258-260 ; Matter and Motion re-
viewed, 260-261 ; Tail's account of scientific
work of, 262-264 ; Tail's admiralion and love
for, 267 ; humorous hils al Herbert Spencer's
First Principles, 283-284 ; quoled in Scots
Observer article, 292 ; eleclricily and lighl,
297 ; al ihe Evening Club, 349

Maxwell, Sir W. Stirling, the Evening Club, 348

Mayer, J. R., eulogised by Tyndall, 209 ; writings
reviewed by Tait, 210 ; pioneer work in Heat,
210-212 ; receives Copley Medal, 213 ; referred
to by Tait in letler lo Helmhollz, 216

Mean Free Path, no

Mechanics, article in Encyc. Brit., 21, 24, 155, 231

Medical Times and Gazette, reviews " T and T," 186

Meik, P. W., experimenls on electrical resistance
of wires under load, 76

Meikle, John, the actuary, 25

Memorial to Tait at Edinburgh University, 50

Memories, Some College, by R. L. Stevenson, 73-74

Memory, Tail's accurale verbal, 6, 33

Miller, W. A., experimenls lo test deep-sea ther-
mometers, 83

Mirage, 24, 109, 321-328

Mitchell, A. Crichton, experimenls on ihermal
conduction, 8 1 ; on laws of cooling, 96

Mohr, F., experiments on Heat, 210-212 ; article
translated by Tait in Phil. Mag., 212

Moigno, Abbe", referred to by Hamilton, 121 ;
Repertoire d'Optique, 122 ; publishes French
translation of Tail's Thermodynamics, 226

Monro, Alex., 306

Morgan, A. de, 186 ; Budget of Paradoxes reviewed,
257

" Morning Round, The," verses by Tait, 55

Morris, Tom, at St Andrews, 56

Morrison, R. M., experimenls on ihermoeleclricily,

77

Moulton, Sir J.Flelcher, reviews Herbert Spencer's
First Principles in The British Quarterly,
279-283 ; replies lo Spencer, 282, 283

Muir, John, and ihe Evening Club, 347

Muir, T., on Knot Problem, 106

Murphy, R., on spherical harmonics, referred to by
Maxwell, 102

Murray, Sir John, gives Challenger Lodge to Tait
during last illness, 40 ; student in laboratory,
72 ; experimenls on electrical resislance of

INDEX

375

wires under load, 76 ; experiments on thermo-
electricity, 77

Muscular Contraction, electrical effects of, 65
Museum of Science and Art, Lectures in, 252
Musselburgh Golf Course, 54

Nabla, in quaternions, name suggested by W.
Robertson Smith, 143 ; name discussed by
Maxwell, 143-144 ; the Chief Musician upon,
145, 171 ; Tail's development of, 167; Max-
well's names for results of, 167 ; and Alban,
Headstone's soliloquy on, according to Maxwell,
244

Natanson, J., kinetic theory of gases, 1 1 1

Natural Philosophy Chair at Edinburgh, 16

Natural Philosophy, Thomson and Tail's, 21, 23,
142, 176-204, 208; reception of, 185-186;
second edition, 228 ; Elements of, 197-201

Nature, conlributions to, 24, 26, 27, 28, 58, 247,
249, 251, 253, 255, 256, 257, 258, 260, 262, 278,
281-295, 321-328 ; Reports of R. S. E. meet-
ings, 31 ; Tail's books reviewed in, 201, 203,
229, 230, 233, 241 ; Chrystal's notice of Tail in,
42, 204

Neaves, Lord, Presidenl R. S. E., 30

Nebulae, Tail's views of, 246

Nevis, Ben, Observatory, 87 ; superintendent of, 96

New Amphion, The, Book of Edinburgh University
Union Bazaar, 73

Newcomb, S., 33

Newton, Sir Isaac, Principia used as text-book,
21 ; swerving tennis ball in air, 27, 60, 339 ;
principles of fluxional calculus attacked by
Hegel, defended by W. Robertson Smith, 76 ;
Laws of Motion, 118, 232-236; quoted by
Hamilton, 121-122; rediscovered by Thomson
and Tail, 190 ; referred to in Tail's inaugural
lecture, 206 ; appealed lo by Herberi Spencer
in support of his a priori intuilions, 279, 280,
282 ; Spencer admits that he has no support
from, 282 ; referred to in lecture on thunder-
storms, 296

Nichol, J. N., and the Evening Club, 348

Nichol, J. W., student in Laboratory, 75 ; experi-
ments on radiation, 74; Foundation in
Laboratory in memory of, 75

Nicolson, Alex., and the Evening Club, 348

North British Review, 23, 209, 213

Northern Lights, Commissioners of, 90

Obituary notices, of Tail, 42-43 ; by Tait, of Balfour
Stewart, 289-291 ; of Robertson Smith, 291-292
Obligations to Andrews, 13
Observatory, Ben Nevis, 87

Ode, Tyndallic, 145, 171-173; Paradoxical, 242-244
Oersted, H. C., electricity and magnetism, 297

Ohm and Fourier, 226

Omond, R. T., pressure experiments, 96 ; superin-
tendent Ben Nevis Observatory, 96

Orchestra, Cambridge, 12

Orthogonal Isothermal Surfaces, 153

Ozone, experimenlal work on, 13 ; produced by
lighlning, 311

Paradoxical Philosophy, 240-245 ; reviewed by
Maxwell, 241-242 ; Ode to Dr Sloffkraft by
Maxwell, 242-244

Paris Exposition (1855), 64

Path of projectile in air, 334-344

Peddle, W., edits last edition of Tail's Properties
of Matter, 229

Pendulum, Ballistic, for measuring speed of golf
balls, 89, 337

Persistence of force, 279

Peterhouse, Cambridge, 8-n; Master of, 14;
History of, 14 ; Magazine, notice of Tait by
J. D. Hamilton Dickson, 42 ; portrail of Tait,
48 ; Tait Prize, 50

Pettigrew, J. B., experiments on electrical effects
of muscles, 65

Phosphorescent golf balls, 57

Physical and Literary Society of Edinburgh, 306

Physical Laboralory. See Laboratory.

Physical Review, notice of Tail by Macfarlane in, 42

Physics of golf, 25, 27, 28, 88, 331

Pietrowski, G. v., viscosity experiments, 114

Pitman, Frederick, a school mate, 5

Planet's orbit, Time and Action in, ico

Planet held together by gravitation, 104

Playfair, Lord, professor of chemistry in Edin-
burgh, 68

Poincare', J. H., Thermodynamiqtte reviewed, 273-
276 ; replies to Tait, 275-276

Poisson, S. D., on Probabilities, 98; electricity, 259

Polariscope, rotatory, 91-92

Porter, W. A., u, 14; James, 14; J. S., 14, 41

Portraits of Tait, in Natural Philosophy Library,
Edinburgh University, 48 ; in Royal Society
of Edinburgh Council Room, 29, 48 ; National
Portrait Gallery, Queen Street, Edinburgh,
48 ; Hall of Peterhouse, Cambridge, 48-50 ;
in Quasi Cursores, 49

Potential, Quaternion ireatment, 127, 133; mag-
netic, 147 ; electric, 302-304 ; increase due to
coalescence of rain drops, 315

Practical joking, 57

Pressure, experiments under high, 83 ; effect on maxi-
mum density point of water, 84 ; gauge, 84-85

Principia, Newton's, used as text-book, 21 ; princi-
ples of calculus attacked by Hegel, defended
by W. Robertson Smith, 76 ; conservation of
energy discovered in, 191

376

INDEX

Prizes, Keith and Gunning Victoria Jubilee, 47
Prize, Tail, at Peterhouse, Cambridge, 50
Probabilities, article on, 98
Projectile, rotating spherical, 28, 116, 340-342;

path imitated by Sir J. J. Thomson's stream

of negatively electrified particles, 344-345
Promoter at Graduation Ceremonial, 247
Properties of Matter, lectures on, 21 ; book on, 24,

228-231 ; book reviewed by Lord Rayleigh,

229-230

Pulling in golf, 26, 333
Pupil of Kelland and Forbes, 6
Pupil, story of Tail's one, 1 1

Quarterly Review, review of Tail's Recent Ad-
vances, 227

Quarterly Review, British. See British Quarterly
Review

Quasi Cursores, portrait in, 49

Quaternions, 12, 13, 14, 24, 119; article in Encye.
Brit., 24 ; and Gauss, 37 ; last notes on, 40,
169 ; first suggestion of book on, 65 ; Max-
well's views on, 101, 145, 147-150, 151, 252;
researches by Tail, 102 ; correspondence with
Hamilton, 119-141 ; differentials, 120-123, 141 ;
wave surface discussion, 1 24- 1 26, 1 28-1 30 ; Tail
describes his early studies, 126-127 > potentials,
127, 133, 182 ; Ampere's electrodynamic theory,
138 ; Hamilton's Elements published, 141 ;
Tait's Treatise, 141, 170, 208 ; Kelland and
Tail's Introduction to, 142 ; Integrals, Green's
Theorem, etc., 149 ; interpreiations discussed
by Cayley and Tail, 155-158; versus coordi-
nates, argument between Cayley and Tail,
161-166; translations of Tait's Treatise, 169;
Helmholtz on, 169 ; excluded from "T and T',''
183-185 ; as an instrument of research, 276-278;
review of McAulay's Utility of Quaternions in
Physics, 276-278

Queen's College, Belfast, 12

Radiant Spectrum, Brewster's observations, 69 ;

Tait's explanation, 70
Radiation, article in Encyc. Brit., 24 ; reference to

Balfour Stewart's work, 216, 217, 225
Radiometer, Crookes', 81, 82, 296
Ramsay, G. G., and the Evening Club, 348
ilankine, W. J. Macquorn, I, 209 ; thermodynamic

function, 115, 222 ; work in Heat, 208,218-221 ;

letter from, regarding Evening Club, 348
Rayleigh, Lord, discussion with Tait on Van der

Waals ' formula, in; Theory of Sound, 1 76 ;

on Balfour Stewart's theory of radiation and

absorption, 217, 225
Recent Advances in Physical Science, Tait's, 210,

226-228, 349-350

Rede Lecture on the thermoelectric diagram, 251

Reid, Sir George, portraits of Tait, 47-50 ; the
Evening Club, 348

Reid, Rev. H. S., son-in-law, 40

Religion, Tait on, 36, 293-295

Rendu's Glaciers of Savoy, 24

Repeller on lightning rod, 305

Reports, Challenger, 82-86

Reports of R. S. E. meetings to Nature, 31

Resilience of rubber, vulcanite, glass, steel, etc., 89

Resistance of air, 26, 335-342

Resistance, electrical, of wires under load, 76

Retirement from chair, 39

Reversible engine, Carnot's, 218, 226

Review of Artillery, Russian, 25

Review in British Quarterly Review, of Herbert
Spencer's First Principles by J. Fletcher
Moulton, 278-283

Reviewer, Tait as, 255

Reviews by Tait, de Morgan's Budget of Para-
doxes, 257-258 ; Maxwell's Electricity and
Magnetism, 258-260 ; Maxwell's Matter and
Motion, 260-261 ; Stokes' Mathematical and
Physical Papers, 265-266 ; Stokes' Burnett
Lectures, 266-269 > Clifford's Elements of
Dynamic, 270-272 ; Clifford's The Common
Sense of the Exact Sciences, 272-273 ; Poin-
card's Thermodynamique, 273-276 ; McAulay's
Utility of Quaternions, 276-278 ; Becket's On
the Origin of the Laws of Nature, 284

Rhyming correspondence, 93-95

Rigid Body, rotation of, 149

Rising club shot, in golf, 60

Robins, B., on resistance of the air, 336 ; effect of
rotation, 339, 340

Homer and Doppler, 226

Ronaldson, John, Tait's uncle, 3 ; letter to, regard-
ing Knocklayd Experiment, 15

Rotation of a rigid body, 149 ; Maxwell on, 150

Rotatory polarisation, 91, 92

Routh, E. J., candidate for chair, 16 ; trainer of
senior wranglers, 17

Rowland, H. A., 94

Royal Society of Edinburgh, 28-30; elected Fellow,
28 ; elected Secretary, 28 ; old Meeting Room,
29 ; removal to new premises, 29 ; portraits,
29 ; minutes of council in reference to Tait's
death, 42 ; award of prizes, 47

Royal Society of London, 49 ; award of Royal
Medal, 47

Rumford, Count, experiments on heat, 209-212,
218

Russia, Czar of, and Freddie Tait, 54

Russian Review of Artillery, 25

Sabine, Sir E., on luminous clouds, 317

INDEX

377

St Andrews, Tail at, 52-63 ; Club House, 53, 62 ;
Crum Brown, Helmholtz, and Huxley at, 56,

57, 197

St Elmo's Fire, 306

Sang, Edward, candidate for Edinburgh chair, 16

Schoolmaster, the Mad, Tail's problem of, 108

Science of golf, 59

Science, Religion and, article in The Scots Ob-
server, 292-295

Scientific articles, popular, 296-344

Scoresby, W., observations of mirage, 109,321-326

Scots Observer, The, verses in, 61 ; article on
Religion and Science, 292-295

Scotsman, The, the unwritten chapter on golf, 26 ;
Tail's description of laboratory, 70 ; review of
" T and TV' 186

Scott, Sir Walter, portrait in R. S. E., 29 ; a
favourite author, 33 ; centenary, 196

Scottish Provident Institulion, Director of, 25

Scrap Book, Tail's, 2, 3, 8, 242

Sea-bird analogy of comet's lails, 246-247

Se"guin, A., work in Heat, 210

Selection, Artificial, article in Macmillaris Maga-
zine, 248

Senate House examinations, preparation for, 8

Senatus, minute of, in reference to Tail's retire-
ment, 39 ; colleagues in, 47

Senior Wrangler, 9

Sensation and Science, 256

Stand, A., experiments on compression, 86; ex-
periments on impact, 96

Sharpey, W., University Commission of 1872, 71

Shaw, Sir Doyle Money, letter to, 9 ; president of
Gumming Club, 9, 10

Sheet lightning, 311

Sherriff, General, a school mate, 5

Siebert, G., translates Tail's Properties of Matter,
229

Silbernrann and Favre, 178

Sime, James, Forbes' medallist, 7

Simpson, Sir W., on golf, 59

Sinclair, Rt Hon. Thomas, pupil of Tail in Bel-
fast, 12

Siren for fog-signalling, 91

Skerryvore Lighthouse, struck by lightning, 90, 306

Sketch of Elementary Dynamics, 197-200

Sketch of Thermodynamics, 208-226

Slesser, G. M., Tail's successor in Belfast, 177

Slicing in golf, 26, 27, 333

Smith, Archibald, wave surface, 124

Smith, H. J. S., University Commission of 1872, 71

Smith, C. Michie, experimenis on thermoelectricity,
78-80, 97 ; flow of water in tubes, 1 14

Smith, W. Robertson, at St Andrews, 56 ; Ferguson
scholar, 71 ; assistanl lo Tail, 71, 86; experi-
ments on flow of electricity in plates, 72 ;

refutes Hegel's criticisms of Newton's method
of establishing the fluxional or differential
calculus, 76 ; suggests name Nabla, 143 ;
at Brit. Assoc. meeting of 1871, 171 ; writes
Hebrew superscription to Maxwell's Tyndallic
Ode, 171-172; editor of Encyc. Brit., 231;
letter to, about The Unseen Universe, 238 ;
refers to Bain's Logic, 247 ; obituary notice by
Tail, 291-292 ; the Evening Club, 348
Smythe, Piazzi, at R. S. E., 30; referred to by

Tail, 92

Solids moving in fluid, 191
Solulions, compressibilily of, 84-86
Somerset Cottage, Tail's early home, 3
Somerville, Mrs, Connection of the Physical

Sciences, 211

Sound, Lord Rayleigh's Theory of, 176
Spectrum, radiant, Brewster's observations, 69;

Tail's explanation, 70
Spectrum analysis and spectroscopy, 227
Spencer, Herbert, at Brit. Assoc. meeting in Belfast
(1874), 175 ; First Principles reviewed in
British Quarterly by J. Flelcher Moullon,
278-283 ; replies to criticisms in The Fort-
nightly, 279 ; appendix to First Principles
dealing with criticisms, 279 ; attacks Tail's ex-
perimenlal philosophy, 279 ; allacks "TandT',"
280 ; is replied to by Tail, 281, 284, 285-288 ;
replies to Tail, 281-282, 284-285, 288 ; on ihe
menial idiosyncrasies of Mallhew Arnold,
Kirkman, Tail, men of lelters and mathema-
ticians generally, 285 ; allacks Tail as aulhor
of The Unseen Universe, 286
Spherical Harmonics, 100-102, 182-183, 188-190
Spottiswoode, W., refers to Grassmann's Aus-

dehnungslehre, 152

Stas, J. S., of Brussels, visits Edinburgh, 68
Steele, W. J., college friend, 8, 9 ; second wrangler,
9 ; co-aulhor wilh Tail of Dynamics of a
Particle, 10, 205 ; early death, 10 ; portrait in
group wilh Tail, 1 1

Stevelly, Tail's colleague at Belfast, 12
Stevenson, D. and T., prelection of lighthouses

from lightning, 90

Stevenson, Robert Louis, studenl in ihe Labora-
tory, 72 ; paper on temperalure as influenced
by woods, 73 ; word piclure of Old Lindsay,
73-74 ; Some College Memories, 73-74
Stevenson, Thomas, lighihouse engineer, 73

90

Stewart, A. D., a school mate, 5 ; selected designs
for Forth Bridge, 5 ; at St Andrews, 54 ;
designer of girders of Tay Bridge, 278
Stewart, Balfour, at Edinburgh University, 7 ;
experiments along with Tail, 23, 68 ; co-author
of The Unseen Universe, 24, 36, 105, 236-237 ;

378

INDEX

on Radiation, 216, 217, 225 ; obituary notice
by Tait, 289-291 ; on Auroras, 320

Stewart, James and Matthew, 306

Stoffkraft, Hermann, hero of the Paradoxical
Philosophy, 241 ; Maxwell's Ode to, 242-244

Stokes, Sir G. G., 33, 34, 43, 92 ; Gifford Lecturer,
35; letter of condolence to Mrs Tait, 41 ; dynami-
cal theory of diffraction, 145 ; theorem, 149 ; on
Mayer's work, 213 ; Tail's account of Stokes'
work, 264-265 ; Mathematical and Physical
Papers reviewed, 265-266 ; Burnett Lectures
reviewed, 266-269

Strains, theory of, 22

Straker, J. H., experiments on thermoelectricity,

77

Straths, the, at St Andrews, 56
Student, The, Flint's appreciation of Tait in, 44-46
Study, Tail's, 33 ; Order of Merit on wall, 43
Swan, W., candidale for Edinburgh chair, 16 ;

intensily and duralion of light, 300-301
Sylvester, J. J., 33 ; at St Andrews, 56 ; linear

malrix equation, 158 ; at the Evening Club,

349

"T and T'," 21, 176, 179, 180, 188, 189, 202, 203,
228; "Little," 197-201

Tait, Mrs, presents MS. of "Tail and Steele" to
Peterhouse library, 10; marriage, 14; at home,
33, 34; letters of condolence to, 41, 42 ; presents
earliest portrait of Tait to the Natural Philo-
sophy department, 48 ; story of the stolen
umbrellas, 58

Tait, Alex., 57

Tait, Archibald Campbell, Archbishop of Canter-
bury, 9, 189

Tait, Freddie, or Lieut. F. G., 2, 25, 26, 54, 57-60,
63, 89; Record of Life, by J. L. Low, 2, 40;
wounded at Magersfontein, 37 ; killed al
Koodoosberg, 37 ; and ihe Czar of Russia, 54

Tait, Jack, al St Andrews, 56

Tait, W. A., 278

"Tait and Steele," 10, 205-207

Tail-line in ihermoeleclric diagram, 346-347

Tail Memorial al Edinburgh Universily, 50

Tail Memorial Fund, 50

Tail Prize at Peterhouse, 50

Tail's one pupil, slory of, 1 1

Talbot, W. Fox, pholography, 3 ; experimenls in
Tail's Laboralory, 76 ; discovery of anomalous
dispersion, 76

Tay Bridge disasler, 278

Telephone, Chrystal's differenlial, 87

Temperature, absolute, treated by Tait in his
Thermodynamics and Heat, 226, 228

Tercentenary Celebrations at Edinburgh Uni-
versity, 33

Terminal velocity, 335, 336

Testimonial to Tait from Hamilton, 138, 139

Thackeray, W. M., a favourite author, 33 ; quoted,
249

Thermal conduclivily, 80, 81

Thermodynamic Funclion, Rankine's, 222

Thermodynamics, Sketch of, 23, 184, 208, 226;
French Iranslation, 226

Therntodynamique, Poincard's, reviewed, 275-276

Thermoelectric Diagram, Rede Lecture, 251 ;
extension of, by J. D. H. Dickson, 345-347

Thermoelectricity, 77-80, 345~347

Thermometers, Challenger Deep Sea, 83

Thompson, D'Arcy, composes Greek verses for
The Unseen Universe, 237-238 ; on thunder
spurts, 318

Thompson, D'Arcy Wenlworlh, discusses his
father's Greek verses in The Unseen Universe,
238

Thompson, Sylvanus P., Life of Kelvin, 184, 200

Thomson, James, 12

Thomson, Sir J. J., on golf ball paths and electron
streams, 344-345

Thomson, Sir William (see also under Kelvin),
Laboratory in Glasgow, 22 ; at R. S. E., 30 ;
R. S. E. reports for Nature, 31 ; al Tail's
Laboralory, 31 ; experimenls wilh Tail's
Gramme machine, 31-32 ; al Slokes' Gifford
Leclure, 35 ; new forms of galvanomelers and
electrometers, 67, 68 ; the vortex alom, 68, 69 ;
describes Tail's experimenls on vorlex rings,
69; lellers lo, 75, 92, no, 180-184, 218-220;
lelters from, 104, 223-224 ; Reprint of Papers
on Electrostatics and Magnetism, 176; joins
Tail in book, 177 ; repugnance lo wriling
books, 178; pioneer work in Heal, 208,
218-221 ; wriles with Tait article Energy for
Good Words, 209 ; names Maxwell's demons,
213; on Clausius and Ihe Enlropy integral,
223-224 ; on thermodynamics, 225 ; referred
to as editor of Principia, 280 ; referred to in
connection with anthropological iheories, 290;
on gravilation, 296 ; on electricily and mag-
netism, 297; on atmospheric electricily, 299;
explains apparenl high velocity of thunder,
312; source of atmospheric electricity, 318;
the Evening Club, 348

Thomson, William, Archbishop of York, 189

Thomson, Sir Wyville, colleague in Belfast, 12 ;
story regarding, 15 ; consults Tait about
Challenger ihermomelers, 83

Thomson and Tait's Treatise on Natural Philo-
sophy, 21, 31, 176-204; beginnings of, 177;
reviewed in various journals, 186 ; general
plan described, 187 ; criiicised by Maxwell in
lelter, 195 ; reviewed by Maxwell, 203-204 ;

INDEX

379

German translation, 195-196, 256 ; as inter-
preted by Herbert Spencer, 279, 282

Thunder, bolt, 310, 311 ; crash, peal, roll, 311, 312 ;
cloud, 318

Thunderstorms, lecture on, 255, 296-320

Tides and tidal stresses, 193

Time and action in planet's orbit, 100

Todhunter, I., recommends book on quaternions,

65

Topping in golf, 26, 60, 333
Torricelli, E., quoted by Maxwell, 195, 222
Translations, " T and T," 195-196 ; Thermo-
dynamics, 226 ; Recent Advances, 228 ; Pro-
perties of Matter, 229 ; Heat, 229 ; The Unseen
Universe, 240
Treatise on Natural Philosophy. See Natural

Philosophy

Trefoil knot, symbolic of vortex atom, 105, 236
Trilinear coordinates, Cayley and Tail on, 164-165
Tulloch, Rev. J., and the Evening Club, 348
Turnbull, R., experiments on impact, 96
Turner, Sir William, at R. S. E., 30
Twist, Thomson's theorems on, 182
Tyndall, J., on Forbes, 18, 24 ; on history of
thermodynamics, 23 ; on diathermancy, 68,
92 ; controversy with Tail, 208 ; eulogy of
Mayer, 209, 213 ; book on heat reviewed, 209;
president of Brit. Assoc., 236 ; referred to in
Tail's lecture on Force, 253 ; letter to Tait on
Zollner and Clausius, 257
Tyndallic Ode, 145, 171-173

Underspin in golf ball, 26, 27, 59, 330-344

University, Tail's ideal, 248

University of Edinburgh, student at, 6 ; elected to
chair, 16 ; Tercentenary, 33

University Commissioners, 52, 71

University Union Bazaar, 73

Unseen Universe, The, 24, 36, 105, 236-245 ; re-
viewed in leading journals, 239-241 ; French
translation, 240

Unveiling of Peterhouse portrait, 48

Vacua, experiments on high, 81, 82
Vacuo, rotation of disk in, 23, 68

Vapour, diathermancy of water, 92

Vector methods, 14

Velocity of charged particles in cathode rays, 91

Verdet, E., letter to, 32, 210; book reviewed,

209, 210
Verses, by Tait, 55-56, 61-62, 93 ; by Maxwell,

94-95, 242-243, 253-255, 261 ; by D'Arcy

Thompson, 237, 238
Vertices, curve of, 324-326
Vince, S., observations of mirage, 109
Virgil, on thunderbolts and comets, 311
Viscosity in water flowing in tubes, 1 14
Vortex, rings, 68, 69 ; atom, 68, 69, 105, 177 ;

knotted, Maxwell on, 105 ; motion, 127
Vorticity and curl, 148

Waals, Van der, isothermal equation, m, 112

Walker, T. A., history of Peterhouse, 14

Walking stick, story of Tail's, 72-73

Wallace, Robert, and the Evening Club, 347

Walsh, General, at St Andrews, 57

Wanklyn, J. A., experiments with Tail on elec-

Iricily developed during evaporalion, 68
War, South African, 37
Waler Vapour, dialhermancy of, 92
Watson, W., treatise on Kinetic Theory of Gases,

ill
Watson, Sir Patrick Heron, school mate, 5 ;

description of Tait, 1 8
Wave surface, 124-126, 128-130, 141
Wertheim, G., German translalion of " T and T',"

195-196

Wheatstone's revolving mirror, 301
White, Rev. E., condilional immorlalily, 37
Whytt, R., 306
Williams, Archdean, 6
Wilson, Andrew, aulhor of the Abode of Snow,

school mate, 5

Wollaston, W. H., on mirage, 321, 322, 325, 328
Wrangler, Senior, 9

Yvelot, posls in, 249-250

Zeeman, P., 90

Zollner, F., book reviewed, 256-257

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