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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% The Project Gutenberg EBook of Ten British Mathematicians of the 19th   %
+% Century, by Alexander Macfarlane                                        %
+%                                                                         %
+% This eBook is for the use of anyone anywhere in the United States and most
+% other parts of the world at no cost and with almost no restrictions     %
+% whatsoever.  You may copy it, give it away or re-use it under the terms of
+% the Project Gutenberg License included with this eBook or online at     %
+% www.gutenberg.org.  If you are not located in the United States, you'll have
+% to check the laws of the country where you are located before using this ebook.
+%                                                                         %
+%                                                                         %
+%                                                                         %
+% Title: Ten British Mathematicians of the 19th Century                   %
+%                                                                         %
+% Author: Alexander Macfarlane                                            %
+%                                                                         %
+% Release Date: April 24, 2015 [EBook #9942]                              %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ASCII                                           %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK TEN BRITISH MATHEMATICIANS ***%
+%                                                                         %
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+\small
+\begin{PGtext}
+The Project Gutenberg EBook of Ten British Mathematicians of the 19th
+Century, by Alexander Macfarlane
+
+This eBook is for the use of anyone anywhere in the United States and most
+other parts of the world at no cost and with almost no restrictions
+whatsoever.  You may copy it, give it away or re-use it under the terms of
+the Project Gutenberg License included with this eBook or online at
+www.gutenberg.org.  If you are not located in the United States, you'll have
+to check the laws of the country where you are located before using this ebook.
+
+
+
+Title: Ten British Mathematicians of the 19th Century
+
+Author: Alexander Macfarlane
+
+Release Date: April 24, 2015 [EBook #9942]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK TEN BRITISH MATHEMATICIANS ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+\clearpage
+
+%%%% Credits and transcriber's note %%%%
+\begin{center}
+\begin{minipage}{\textwidth}
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+Produced by David Starner, John Hagerson, and the Online
+Distributed Proofreading Team
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+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
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+
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+
+\documentclass[oneside]{book}
+\usepackage[polutonikogreek,english]{babel}
+\selectlanguage{english}
+\usepackage{amsmath,amssymb,enumerate,graphicx,verse}
+\begin{document}
+
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+\small
+\begin{verbatim}
+
+\end{verbatim}
+\normalsize
+\fi
+%%%%% End of original header %%%%
+
+\newpage
+
+
+\frontmatter
+
+\begin{center}
+\noindent \Large MATHEMATICAL MONOGRAPHS
+
+\bigskip
+\footnotesize\textsc{EDITED BY} \\
+\normalsize \textsc{MANSFIELD MERRIMAN and ROBERT S. WOODWARD}
+
+\bigskip\bigskip\huge
+No. 17
+
+\bigskip
+\LARGE \textsc{lectures on} \\
+\huge TEN BRITISH MATHEMATICIANS \\
+\LARGE \textsc{of the Nineteenth Century}
+
+\bigskip \bigskip \normalsize BY
+
+\bigskip \large ALEXANDER MACFARLANE,
+
+\bigskip\footnotesize\textsc{
+Late President for the International Association for Promoting \\
+the Study of Quaternions}
+
+1916
+\end{center}
+
+\newpage
+
+\noindent\fbox{\parbox{\columnwidth}{
+\textbf{MATHEMATICAL MONOGRAPHS.} \\
+\small\textsc{edited by}\normalsize \\
+\textbf{Mansfield Merriman and Robert S. Woodward.}
+
+\bigskip
+\textbf{No. 1. History of Modern Mathematics.} \\
+By \textsc{David Eugene Smith.}
+
+\smallskip
+\textbf{No. 2. Synthetic Projective Geometry.} \\
+By \textsc{George Bruce Halsted.}
+
+\smallskip
+\textbf{No. 3. Determinants.} \\
+By \textsc{Laenas Gifford Weld.}
+
+\smallskip
+\textbf{No. 4. Hyperbolic Functions.} \\
+By \textsc{James McMahon.}
+
+\smallskip
+\textbf{No. 5. Harmonic Functions.} \\
+By \textsc{William E. Byerly.}
+
+\smallskip
+\textbf{No. 6. Grassmann's Space Analysis.} \\
+By \textsc{Edward W. Hyde.}
+
+\smallskip
+\textbf{No. 7. Probability and Theory of Errors.} \\
+By \textsc{Robert S. Woodward.}
+
+\smallskip
+\textbf{No. 8. Vector Analysis and Quaternions.} \\
+By \textsc{Alexander Macfarlane.}
+
+\smallskip
+\textbf{No. 9. Differential Equations.} \\
+By \textsc{William Woolsey Johnson.}
+
+\smallskip
+\textbf{No. 10. The Solution of Equations.} \\
+By \textsc{Mansfield Merriman.}
+
+\smallskip
+\textbf{No. 11. Functions of a Complex Variable.} \\
+By \textsc{Thomas S. Fiske.}
+
+\smallskip
+\textbf{No. 12. The Theory of Relativity.} \\
+By \textsc{Robert D. Carmichael.}
+
+\smallskip
+\textbf{No. 13. The Theory of Numbers.} \\
+By \textsc{Robert D. Carmichael.}
+
+\smallskip
+\textbf{No. 14. Algebraic Invariants.} \\
+By \textsc{Leonard E. Dickson.}
+
+\smallskip
+\textbf{No. 15. Mortality Laws and Statistics.} \\
+By \textsc{Robert Henderson.}
+
+\smallskip
+\textbf{No. 16. Diophantine Analysis.} \\
+By \textsc{Robert D. Carmichael.}
+
+\smallskip
+\textbf{No. 17. Ten British Mathematicians.} \\
+By \textsc{Alexander Macfarlane.} \normalsize }}
+
+\newpage
+
+\chapter{PREFACE}
+
+During the years 1901-1904 Dr. Alexander Macfarlane delivered, at
+Lehigh University, lectures on twenty-five British mathematicians
+of the nineteenth century. The manuscripts of twenty of these
+lectures have been found to be almost ready for the printer,
+although some marginal notes by the author indicate that he had
+certain additions in view. The editors have felt free to disregard
+such notes, and they here present ten lectures on ten pure
+mathematicians in essentially the same form as delivered. In a
+future volume it is hoped to issue lectures on ten mathematicians
+whose main work was in physics and astronomy.
+
+These lectures were given to audiences composed of students,
+instructors and townspeople, and each occupied less than an hour
+in delivery. It should hence not be expected that a lecture can
+fully treat of all the activities of a mathematician, much less
+give critical analyses of his work and careful estimates of his
+influence. It is felt by the editors, however, that the lectures
+will prove interesting and inspiring to a wide circle of readers
+who have no acquaintance at first hand with the works of the men
+who are discussed, while they cannot fail to be of special
+interest to older readers who have such acquaintance.
+
+It should be borne in mind that expressions such as ``now,''
+``recently,'' ``ten years ago,'' etc., belong to the year when a
+lecture was delivered. On the first page of each lecture will be
+found the date of its delivery.
+
+For six of the portraits given in the frontispiece the editors are
+indebted to the kindness of Dr.\ David Eugene Smith, of Teachers
+College, Columbia University.
+
+Alexander Macfarlane was born April 21, 1851, at Blairgowrie,
+Scotland. From 1871 to 1884 he was a student, instructor and
+examiner in physics at the University of Edinburgh, from 1885 to
+1894 professor of physics in the University of Texas, and from
+1895 to 1908 lecturer in electrical engineering and mathematical
+physics in Lehigh University. He was the author of papers on
+algebra of logic, vector analysis and quaternions, and of
+Monograph No.\ 8 of this series. He was twice secretary of the
+section of physics of the American Association for the Advancement
+of Science, and twice vice-president of the section of mathematics
+and astronomy. He was one of the founders of the International
+Association for Promoting the Study of Quaternions, and its
+president at the time of his death, which occured at Chatham,
+Ontario, August 28, 1913. His personal acquaintance with British
+mathematicians of the nineteenth century imparts to many of these
+lectures a personal touch which greatly adds to their general
+interest.
+
+\begin{center}
+\includegraphics[width=25mm]{images/AMpic.png} \\
+\textsc{Alexander Macfarlane}\\
+From a photograph of 1898
+\end{center}
+
+\tableofcontents
+
+%%PORTRAITS of MATHEMATICIANS
+
+%% GEORGE PEACOCK (1791-1858)
+%% A Lecture delivered April 12, 1901.
+
+%% AUGUSTUS DE MORGAN (1806-1871)
+%% A Lecture delivered April 13, 1901.
+
+%% SIR WILLIAM ROWAN HAMILTON (1805-1865)
+%% A Lecture delivered April 16, 1901.
+
+%% GEORGE BOOLE (1815-1864)
+%% A Lecture delivered April 19, 1901.
+
+%% ARTHUR CAYLEY (1821-1895)
+%% A Lecture delivered April 20, 1901.
+
+%% WILLIAM KINGDON CLIFFORD (1845-1879)
+%% A Lecture delivered April 23, 1901.
+
+%% HENRY JOHN STEPHEN SMITH (1826-1883)
+%% A Lecture delivered March 15, 1902.
+
+%% JAMES JOSEPH SYLVESTER (1814-1897)
+%% A Lecture delivered March 21, 1902.
+
+%% THOMAS PENYNGTON KIRKMAN (1806-1895)
+%% A Lecture delivered April 20, 1903.
+
+%% ISAAC TODHUNTER (1820-1884)
+%% A Lecture delivered April 13, 1904.
+
+%% INDEX
+
+\MainMatter
+
+\chapter [George Peacock (1791-1858)]
+{GEORGE PEACOCK\footnote{This Lecture was delivered April 12,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1791-1858)}\end{center}\normalsize
+
+George Peacock was born on April 9, 1791, at Denton in the north
+of England, 14 miles from Richmond in Yorkshire. His father, the
+Rev.\ Thomas Peacock, was a clergyman of the Church of England,
+incumbent and for 50 years curate of the parish of Denton, where
+he also kept a school. In early life Peacock did not show any
+precocity of genius, and was more remarkable for daring feats of
+climbing than for any special attachment to study. He received his
+elementary education from his father, and at 17 years of age, was
+sent to Richmond, to a school taught by a graduate of Cambridge
+University to receive instruction preparatory to entering that
+University. At this school he distinguished himself greatly both
+in classics and in the rather elementary mathematics then required
+for entrance at Cambridge. In 1809 he became a student of Trinity
+College, Cambridge.
+
+Here it may be well to give a brief account of that University, as
+it was the alma mater of four out of the six mathematicians
+discussed in this course of lectures\footnote{Dr.\ Macfarlane's
+first course included the first six lectures given in this
+volume.---\textsc{Editors.}}.
+
+At that time the University of Cambridge consisted of seventeen
+colleges, each of which had an independent endowment, buildings,
+master, fellows and scholars. The endowments, generally in the
+shape of lands, have come down from ancient times; for example,
+Trinity College was founded by Henry VIII in 1546, and at the
+beginning of the 19th century it consisted of a master, 60 fellows
+and 72 scholars. Each college was provided with residence halls, a
+dining hall, and a chapel. Each college had its own staff of
+instructors called tutors or lecturers, and the function of the
+University apart from the colleges was mainly to examine for
+degrees. Examinations for degrees consisted of a pass examination
+and an honors examination, the latter called a tripos. Thus, the
+mathematical tripos meant the examinations of candidates for the
+degree of Bachelor of Arts who had made a special study of
+mathematics. The examination was spread over a week, and those who
+obtained honors were divided into three classes, the highest class
+being called \emph{wranglers}, and the highest man among the
+wranglers, \emph{senior wrangler}. In more recent times this
+examination developed into what De~Morgan called a ``great writing
+race;'' the questions being of the nature of short problems. A
+candidate put himself under the training of a coach, that is, a
+mathematician who made it a business to study the kind of problems
+likely to be set, and to train men to solve and write out the
+solution of as many as possible per hour. As a consequence the
+lectures of the University professors and the instruction of the
+college tutors were neglected, and nothing was studied except what
+would pay in the tripos examination. Modifications have been
+introduced to counteract these evils, and the conditions have been
+so changed that there are now no senior wranglers. The tripos
+examination used to be followed almost immediately by another
+examination in higher mathematics to determine the award of two
+prizes named the Smith's prizes. ``Senior wrangler'' was
+considered the greatest academic distinction in England.
+
+In 1812 Peacock took the rank of second wrangler, and the second
+Smith's prize, the senior wrangler being John Herschel. Two years
+later he became a candidate for a fellowship in his college and
+won it immediately, partly by means of his extensive and accurate
+knowledge of the classics. A fellowship then meant about
+\pounds200 a year, tenable for seven years provided the Fellow did
+not marry meanwhile, and capable of being extended after the seven
+years provided the Fellow took clerical Orders. The limitation to
+seven years, although the Fellow devoted himself exclusively to
+science, cut short and prevented by anticipation the career of
+many a laborer for the advancement of science. Sir Isaac Newton
+was a Fellow of Trinity College, and its limited terms nearly
+deprived the world of the \emph{Principia}.
+
+The year after taking a Fellowship, Peacock was appointed a tutor
+and lecturer of his college, which position he continued to hold
+for many years. At that time the state of mathematical learning at
+Cambridge was discreditable. How could that be? you may ask; was
+not Newton a professor of mathematics in that University? did he
+not write the \emph{Principia} in Trinity College? had his
+influence died out so soon? The true reason was he was worshipped
+too much as an authority; the University had settled down to the
+study of Newton instead of Nature, and they had followed him in
+one grand mistake---the ignoring of the differential notation in
+the calculus. Students of the differential calculus are more or
+less familiar with the controversy which raged over the respective
+claims of Newton and Leibnitz to the invention of the calculus;
+rather over the question whether Leibnitz was an independent
+inventor, or appropriated the fundamental ideas from Newton's
+writings and correspondence, merely giving them a new clothing in
+the form of the differential notation. Anyhow, Newton's countrymen
+adopted the latter alternative; they clung to the fluxional
+notation of Newton; and following Newton, they ignored the
+notation of Leibnitz and everything written in that notation. The
+Newtonian notation is as follows: If $y$ denotes a fluent, then
+$\dot{y}$ denotes its fluxion, and $\ddot{y}$ the fluxion of
+$\dot{y}$; if $y$ itself be considered a fluxion, then $y^\prime$
+denotes its fluent, and $y^{\prime\prime}$ the fluent of
+$y^\prime$ and so on; a differential is denoted by \textsc{o}. In
+the notation of Leibnitz $\dot{y}$ is written $\frac{dy}{dx}$,
+$\ddot{y}$ is written $\frac{d^2 y}{dx^2}$, $y^\prime$ is
+$\int\!ydx$, and so on. The result of this Chauvinism on the part
+of the British mathematicians of the eighteenth century was that
+the developments of the calculus were made by the contemporary
+mathematicians of the Continent, namely, the Bernoullis, Euler,
+Clairault, Delambre, Lagrange, Laplace, Legendre. At the beginning
+of the 19th century, there was only one mathematician in Great
+Britain (namely Ivory, a Scotsman) who was familiar with the
+achievements of the Continental mathematicians. Cambridge
+University in particular was wholly given over not merely to the
+use of the fluxional notation but to ignoring the differential
+notation. The celebrated saying of Jacobi was then literally true,
+although it had ceased to be true when he gave it utterance. He
+visited Cambridge about 1842. When dining as a guest at the high
+table of one of the colleges he was asked who in his opinion was
+the greatest of the living mathematicians of England; his reply
+was ``There is none.''
+
+Peacock, in common with many other students of his own standing,
+was profoundly impressed with the need of reform, and while still
+an undergraduate formed a league with Babbage and Herschel to
+adopt measures to bring it about. In 1815 they formed what they
+called the \emph{Analytical Society}, the object of which was
+stated to be to advocate the \emph{d}'ism of the Continent versus
+the \emph{dot}-age of the University. Evidently the members of the
+new society were armed with wit as well as mathematics. Of these
+three reformers, Babbage afterwards became celebrated as the
+inventor of an analytical engine, which could not only perform the
+ordinary processes of arithmetic, but, when set with the proper
+data, could tabulate the values of any function and print the
+results. A part of the machine was constructed, but the inventor
+and the Government (which was supplying the funds) quarrelled, in
+consequence of which the complete machine exists only in the form
+of drawings. These are now in the possession of the British
+Government, and a scientific commission appointed to examine them
+has reported that the engine could be constructed. The third
+reformer---Herschel---was a son of Sir William Herschel, the
+astronomer who discovered Uranus, and afterwards as Sir John
+Herschel became famous as an astronomer and scientific
+philosopher.
+
+The first movement on the part of the Analytical Society was to
+translate from the French the smaller work of Lacroix on the
+differential and integral calculus; it was published in 1816. At
+that time the best manuals, as well as the greatest works on
+mathematics, existed in the French language. Peacock followed up
+the translation with a volume containing a copious
+\emph{Collection of Examples of the Application of the
+Differential and Integral Calculus}, which was published in 1820.
+The sale of both books was rapid, and contributed materially to
+further the object of the Society. Then high wranglers of one year
+became the examiners of the mathematical tripos three or four
+years afterwards. Peacock was appointed an examiner in 1817, and
+he did not fail to make use of the position as a powerful lever to
+advance the cause of reform. In his questions set for the
+examination the differential notation was for the first time
+officially employed in Cambridge. The innovation did not escape
+censure, but he wrote to a friend as follows: ``I assure you that
+I shall never cease to exert myself to the utmost in the cause of
+reform, and that I will never decline any office which may
+increase my power to effect it. I am nearly certain of being
+nominated to the office of Moderator in the year 1818-1819, and as
+I am an examiner in virtue of my office, for the next year I shall
+pursue a course even more decided than hitherto, since I shall
+feel that men have been prepared for the change, and will then be
+enabled to have acquired a better system by the publication of
+improved elementary books. I have considerable influence as a
+lecturer, and I will not neglect it. It is by silent perseverance
+only, that we can hope to reduce the many-headed monster of
+prejudice and make the University answer her character as the
+loving mother of good learning and science.'' These few sentences
+give an insight into the character of Peacock: he was an ardent
+reformer and a few years brought success to the cause of the
+Analytical Society.
+
+Another reform at which Peacock labored was the teaching of
+algebra. In 1830 he published a \emph{Treatise on Algebra} which
+had for its object the placing of algebra on a true scientific
+basis, adequate for the development which it had received at the
+hands of the Continental mathematicians. As to the state of the
+science of algebra in Great Britain, it may be judged of by the
+following facts. Baron Maseres, a Fellow of Clare College,
+Cambridge, and William Frend, a second wrangler, had both written
+books protesting against the use of the negative quantity. Frend
+published his \emph{Principles of Algebra} in 1796, and the
+preface reads as follows: ``The ideas of number are the clearest
+and most distinct of the human mind; the acts of the mind upon
+them are equally simple and clear. There cannot be confusion in
+them, unless numbers too great for the comprehension of the
+learner are employed, or some arts are used which are not
+justifiable. The first error in teaching the first principles of
+algebra is obvious on perusing a few pages only of the first part
+of Maclaurin's \emph{Algebra}. Numbers are there divided into two
+sorts, positive and negative; and an attempt is made to explain
+the nature of negative numbers by allusion to book debts and other
+arts. Now when a person cannot explain the principles of a science
+without reference to a metaphor, the probability is, that he has
+never thought accurately upon the subject. A number may be greater
+or less than another number; it may be added to, taken from,
+multiplied into, or divided by, another number; but in other
+respects it is very intractable; though the whole world should be
+destroyed, one will be one, and three will be three, and no art
+whatever can change their nature. You may put a mark before one,
+which it will obey; it submits to be taken away from a number
+greater than itself, but to attempt to take it away from a number
+less than itself is ridiculous. Yet this is attempted by
+algebraists who talk of a number less than nothing; of multiplying
+a negative number into a negative number and thus producing a
+positive number; of a number being imaginary. Hence they talk of
+two roots to every equation of the second order, and the learner
+is to try which will succeed in a given equation; they talk of
+solving an equation which requires two impossible roots to make it
+soluble; they can find out some impossible numbers which being
+multiplied together produce unity. This is all jargon, at which
+common sense recoils; but from its having been once adopted, like
+many other figments, it finds the most strenuous supporters among
+those who love to take things upon trust and hate the colour of a
+serious thought.'' So far, Frend. Peacock knew that Argand,
+Fran\c{c}ais and Warren had given what seemed to be an explanation
+not only of the negative quantity but of the imaginary, and his
+object was to reform the teaching of algebra so as to give it a
+true scientific basis.
+
+At that time every part of exact science was languishing in Great
+Britain. Here is the description given by Sir John Herschel: ``The
+end of the 18th and the beginning of the 19th century were
+remarkable for the small amount of scientific movement going on in
+Great Britain, especially in its more exact departments.
+Mathematics were at the last gasp, and Astronomy nearly so---I
+mean in those members of its frame which depend upon precise
+measurement and systematic calculation. The chilling torpor of
+routine had begun to spread itself over all those branches of
+Science which wanted the excitement of experimental research.'' To
+elevate astronomical science the Astronomical Society of London
+was founded, and our three reformers Peacock, Babbage and Herschel
+were prime movers in the undertaking. Peacock was one of the most
+zealous promoters of an astronomical observatory at Cambridge, and
+one of the founders of the Philosophical Society of Cambridge.
+
+The year 1831 saw the beginning of one of the greatest scientific
+organizations of modern times. That year the British Association
+for the Advancement of Science (prototype of the American, French
+and Australasian Associations) held its first meeting in the
+ancient city of York. Its objects were stated to be: first, to
+give a stronger impulse and a more systematic direction to
+scientific enquiry; second, to promote the intercourse of those
+who cultivate science in different parts of the British Empire
+with one another and with foreign philosophers; third, to obtain a
+more general attention to the objects of science, and the removal
+of any disadvantages of a public kind which impede its progress.
+One of the first resolutions adopted was to procure reports on the
+state and progress of particular sciences, to be drawn up from
+time to time by competent persons for the information of the
+annual meetings, and the first to be placed on the list was a
+report on the progress of mathematical science. Dr.\ Whewell, the
+mathematician and philosopher, was a Vice-president of the
+meeting: he was instructed to select the reporter. He first asked
+Sir W.~R.\ Hamilton, who declined; he then asked Peacock, who
+accepted. Peacock had his report ready for the third meeting of
+the Association, which was held in Cambridge in 1833; although
+limited to Algebra, Trigonometry, and the Arithmetic of Sines, it
+is one of the best of the long series of valuable reports which
+have been prepared for and printed by the Association.
+
+In 1837 he was appointed Lowndean professor of astronomy in the
+University of Cambridge, the chair afterwards occupied by Adams,
+the co-discoverer of Neptune, and now occupied by Sir Robert Ball,
+celebrated for his \emph{Theory of Screws}. In 1839 he was
+appointed Dean of Ely, the diocese of Cambridge. While holding
+this position he wrote a text book on algebra in two volumes, the
+one called \emph{Arithmetical Algebra}, and the other
+\emph{Symbolical Algebra}. Another object of reform was the
+statutes of the University; he worked hard at it and was made a
+member of a commission appointed by the Government for the
+purpose; but he died on November 8, 1858, in the 68th year of his
+age. His last public act was to attend a meeting of the
+Commission.
+
+Peacock's main contribution to mathematical analysis is his
+attempt to place algebra on a strictly logical basis. He founded
+what has been called the philological or symbolical school of
+mathematicians; to which Gregory, De~Morgan and Boole belonged.
+His answer to Maseres and Frend was that the science of algebra
+consisted of two parts---arithmetical algebra and symbolical
+algebra---and that they erred in restricting the science to the
+arithmetical part. His view of arithmetical algebra is as follows:
+``In arithmetical algebra we consider symbols as representing
+numbers, and the operations to which they are submitted as
+included in the same definitions as in common arithmetic; the
+signs $+$ and $-$ denote the operations of addition and
+subtraction in their ordinary meaning only, and those operations
+are considered as impossible in all cases where the symbols
+subjected to them possess values which would render them so in
+case they were replaced by digital numbers; thus in expressions
+such as $a + b$ we must suppose $a$ and $b$ to be quantities of
+the same kind; in others, like $a - b$, we must suppose $a$
+greater than $b$ and therefore homogeneous with it; in products
+and quotients, like $ab$ and $\frac{a}{b}$ we must suppose the
+multiplier and divisor to be abstract numbers; all results
+whatsoever, including negative quantities, which are not strictly
+deducible as legitimate conclusions from the definitions of the
+several operations must be rejected as impossible, or as foreign
+to the science.''
+
+Peacock's principle may be stated thus: the elementary symbol of
+arithmetical algebra denotes a digital, i.e., an integer number;
+and every combination of elementary symbols must reduce to a
+digital number, otherwise it is impossible or foreign to the
+science. If $a$ and $b$ are numbers, then $a + b$ is always a
+number; but $a - b$ is a number only when $b$ is less than $a$.
+Again, under the same conditions, $ab$ is always a number, but
+$\frac{a}{b}$ is really a number only when $b$ is an exact divisor
+of $a$. Hence we are reduced to the following dilemma: Either
+$\frac{a}{b}$ must be held to be an impossible expression in
+general, or else the meaning of the fundamental symbol of algebra
+must be extended so as to include rational fractions. If the
+former horn of the dilemma is chosen, arithmetical algebra becomes
+a mere shadow; if the latter horn is chosen, the operations of
+algebra cannot be defined on the supposition that the elementary
+symbol is an integer number. Peacock attempts to get out of the
+difficulty by supposing that a symbol which is used as a
+multiplier is always an integer number, but that a symbol in the
+place of the multiplicand may be a fraction. For instance, in
+$ab$, $a$ can denote only an integer number, but $b$ may denote a
+rational fraction. Now there is no more fundamental principle in
+arithmetical algebra than that $ab = ba$; which would be
+illegitimate on Peacock's principle.
+
+One of the earliest English writers on arithmetic is Robert
+Record, who dedicated his work to King Edward the Sixth. The
+author gives his treatise the form of a dialogue between master
+and scholar. The scholar battles long over this difficulty,---that
+multiplying a thing could make it less. The master attempts to
+explain the anomaly by reference to proportion; that the product
+due to a fraction bears the same proportion to the thing
+multiplied that the fraction bears to unity. But the scholar is
+not satisfied and the master goes on to say: ``If I multiply by
+more than one, the thing is increased; if I take it but once, it
+is not changed, and if I take it less than once, it cannot be so
+much as it was before. Then seeing that a fraction is less than
+one, if I multiply by a fraction, it follows that I do take it
+less than once.'' Whereupon the scholar replies, ``Sir, I do thank
+you much for this reason,---and I trust that I do perceive the
+thing.''
+
+The fact is that even in arithmetic the two processes of
+multiplication and division are generalized into a common
+multiplication; and the difficulty consists in passing from the
+original idea of multiplication to the generalized idea of a
+\emph{tensor}, which idea includes compressing the magnitude as
+well as stretching it. Let $m$ denote an integer number; the next
+step is to gain the idea of the reciprocal of $m$, not as
+$\frac{1}{m}$ but simply as $/m$. When $m$ and $/n$ are compounded
+we get the idea of a rational fraction; for in general $m/n$ will
+not reduce to a number nor to the reciprocal of a number.
+
+Suppose, however, that we pass over this objection; how does
+Peacock lay the foundation for general algebra? He calls it
+symbolical algebra, and he passes from arithmetical algebra to
+symbolical algebra in the following manner: ``Symbolical algebra
+adopts the rules of arithmetical algebra but removes altogether
+their restrictions; thus symbolical subtraction differs from the
+same operation in arithmetical algebra in being possible for all
+relations of value of the symbols or expressions employed. All the
+results of arithmetical algebra which are deduced by the
+application of its rules, and which are general in form though
+particular in value, are results likewise of symbolical algebra
+where they are general in value as well as in form; thus the
+product of $a^{m}$ and $a^{n}$ which is $a^{m+n}$ when $m$ and $n$
+are whole numbers and therefore general in form though particular
+in value, will be their product likewise when $m$ and $n$ are
+general in value as well as in form; the series for $(a+b)^{n}$
+determined by the principles of arithmetical algebra when $n$ is
+any whole number, \emph{if it be exhibited in a general form,
+without reference to a final term}, may be shown upon the same
+principle to the equivalent series for $(a+b)^n$ when $n$ is
+general both in form and value.''
+
+The principle here indicated by means of examples was named by
+Peacock the ``principle of the permanence of equivalent forms,''
+and at page 59 of the \emph{Symbolical Algebra} it is thus
+enunciated: ``Whatever algebraical forms are equivalent when the
+symbols are general in form, but specific in value, will be
+equivalent likewise when the symbols are general in value as well
+as in form.''
+
+For example, let $a$, $b$, $c$, $d$ denote any integer numbers,
+but subject to the restrictions that $b$ is less than $a$, and $d$
+less than $c$; it may then be shown arithmetically that
+\begin{displaymath}
+(a - b)(c - d)=ac + bd - ad - bc.
+\end{displaymath}
+Peacock's principle says that the form on the left side is
+equivalent to the form on the right side, not only when the said
+restrictions of being less are removed, but when $a$, $b$, $c$,
+$d$ denote the most general algebraical symbol. It means that $a$,
+$b$, $c$, $d$ may be rational fractions, or surds, or imaginary
+quantities, or indeed operators such as $\frac{d}{dx}$. The
+equivalence is not established by means of the nature of the
+quantity denoted; the equivalence is assumed to be true, and then
+it is attempted to find the different interpretations which may be
+put on the symbol.
+
+It is not difficult to see that the problem before us involves the
+fundamental problem of a rational logic or theory of knowledge;
+namely, how are we able to ascend from particular truths to more
+general truths. If $a$, $b$, $c$, $d$ denote integer numbers, of
+which $b$ is less than $a$ and $d$ less than $c$, then
+\begin{displaymath}
+(a - b)(c - d)=ac + bd - ad - bc.
+\end{displaymath}
+It is first seen that the above restrictions may be removed, and
+still the above equation hold. But the antecedent is still too
+narrow; the true scientific problem consists in specifying the
+meaning of the symbols, which, and only which, will admit of the
+forms being equal. It is not to find \emph{some meanings}, but the
+\emph{most general meaning}, which allows the equivalence to be
+true. Let us examine some other cases; we shall find that
+Peacock's principle is not a solution of the difficulty; the great
+logical process of generalization cannot be reduced to any such
+easy and arbitrary procedure. When $a$, $m$, $n$ denote integer
+numbers, it can be shown that
+\begin{displaymath}
+a^ma^n = a^{m+n}.
+\end{displaymath}
+According to Peacock the form on the left is always to be equal to
+the form on the right, and the meanings of $a$, $m$, $n$ are to be
+found by interpretation. Suppose that $a$ takes the form of the
+incommensurate quantity $e$, the base of the natural system of
+logarithms. A number is a degraded form of a complex quantity
+$p+q^{\sqrt{-1}}$ and a complex quantity is a degraded form of a
+quaternion; consequently one meaning which may be assigned to $m$
+and $n$ is that of quaternion. Peacock's principle would lead us
+to suppose that $e^me^n = e^{m+n}$, $m$ and $n$ denoting
+quaternions; but that is just what Hamilton, the inventor of the
+quaternion generalization, denies. There are reasons for believing
+that he was mistaken, and that the forms remain equivalent even
+under that extreme generalization of $m$ and $n$; but the point is
+this: it is not a question of conventional definition and formal
+truth; it is a question of objective definition and real truth.
+Let the symbols have the prescribed meaning, does or does not the
+equivalence still hold? And if it does not hold, what is the
+higher or more complex form which the equivalence assumes?
+
+
+\chapter [Augustus De~Morgan (1806-1871)]{AUGUSTUS
+DE~MORGAN\footnote{This Lecture was delivered April 13,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1806-1871)}\end{center}\normalsize
+
+Augustus De~Morgan was born in the month of June at Madura in the
+presidency of Madras, India; and the year of his birth may be
+found by solving a conundrum proposed by himself, ``I was $x$
+years of age in the year $x^2$.'' The problem is indeterminate,
+but it is made strictly determinate by the century of its
+utterance and the limit to a man's life. His father was Col.\
+De~Morgan, who held various appointments in the service of the
+East India Company. His mother was descended from James Dodson,
+who computed a table of anti-logarithms, that is, the numbers
+corresponding to exact logarithms. It was the time of the Sepoy
+rebellion in India, and Col.\ De~Morgan removed his family to
+England when Augustus was seven months old. As his father and
+grandfather had both been born in India, De~Morgan used to say
+that he was neither English, nor Scottish, nor Irish, but a Briton
+``unattached,'' using the technical term applied to an
+undergraduate of Oxford or Cambridge who is not a member of any
+one of the Colleges.
+
+When De~Morgan was ten years old, his father died. Mrs.\ De~Morgan
+resided at various places in the southwest of England, and her son
+received his elementary education at various schools of no great
+account. His mathematical talents were unnoticed till he had
+reached the age of fourteen. A friend of the family accidentally
+discovered him making an elaborate drawing of a figure in Euclid
+with ruler and compasses, and explained to him the aim of Euclid,
+and gave him an initiation into demonstration.
+
+De~Morgan suffered from a physical defect---one of his eyes was
+rudimentary and useless. As a consequence, he did not join in the
+sports of the other boys, and he was even made the victim of cruel
+practical jokes by some schoolfellows. Some psychologists have
+held that the perception of distance and of solidity depends on
+the action of two eyes, but De~Morgan testified that so far as he
+could make out he perceived with his one eye distance and solidity
+just like other people.
+
+He received his secondary education from Mr.\ Parsons, a Fellow of
+Oriel College, Oxford, who could appreciate classics much better
+than mathematics. His mother was an active and ardent member of
+the Church of England, and desired that her son should become a
+clergyman; but by this time De~Morgan had begun to show his
+non-grooving disposition, due no doubt to some extent to his
+physical infirmity. At the age of sixteen he was entered at
+Trinity College, Cambridge, where he immediately came under the
+tutorial influence of Peacock and Whewell. They became his
+life-long friends; from the former he derived an interest in the
+renovation of algebra, and from the latter an interest in the
+renovation of logic---the two subjects of his future life work.
+
+At college the flute, on which he played exquisitely, was his
+recreation. He took no part in athletics but was prominent in the
+musical clubs. His love of knowledge for its own sake interfered
+with training for the great mathematical race; as a consequence he
+came out fourth wrangler. This entitled him to the degree of
+Bachelor of Arts; but to take the higher degree of Master of Arts
+and thereby become eligible for a fellowship it was then necessary
+to pass a theological test. To the signing of any such test
+De~Morgan felt a strong objection, although he had been brought up
+in the Church of England. About 1875 theological tests for
+academic degrees were abolished in the Universities of Oxford and
+Cambridge.
+
+As no career was open to him at his own university, he decided to
+go to the Bar, and took up residence in London; but he much
+preferred teaching mathematics to reading law. About this time the
+movement for founding the London University took shape. The two
+ancient universities were so guarded by theological tests that no
+Jew or Dissenter from the Church of England could enter as a
+student; still less be appointed to any office. A body of
+liberal-minded men resolved to meet the difficulty by establishing
+in London a University on the principle of religious neutrality.
+De~Morgan, then 22 years of age, was appointed Professor of
+Mathematics. His introductory lecture ``On the study of
+mathematics'' is a discourse upon mental education of permanent
+value which has been recently reprinted in the United States.
+
+The London University was a new institution, and the relations of
+the Council of management, the Senate of professors and the body
+of students were not well defined. A dispute arose between the
+professor of anatomy and his students, and in consequence of the
+action taken by the Council, several of the professors resigned,
+headed by De~Morgan. Another professor of mathematics was
+appointed, who was accidentally drowned a few years later.
+De~Morgan had shown himself a prince of teachers: he was invited
+to return to his chair, which thereafter became the continuous
+center of his labors for thirty years.
+
+The same body of reformers---headed by Lord Brougham, a Scotsman
+eminent both in science and politics---who had instituted the
+London University, founded about the same time a Society for the
+Diffusion of Useful Knowledge. Its object was to spread scientific
+and other knowledge by means of cheap and clearly written
+treatises by the best writers of the time. One of its most
+voluminous and effective writers was De~Morgan. He wrote a great
+work on \emph{The Differential and Integral Calculus} which was
+published by the Society; and he wrote one-sixth of the articles
+in the \emph{Penny Cyclopedia}, published by the Society, and
+issued in penny numbers. When De~Morgan came to reside in London
+he found a congenial friend in William Frend, notwithstanding his
+mathematical heresy about negative quantities. Both were
+arithmeticians and actuaries, and their religious views were
+somewhat similar. Frend lived in what was then a suburb of London,
+in a country-house formerly occupied by Daniel Defoe and Isaac
+Watts. De~Morgan with his flute was a welcome visitor; and in 1837
+he married Sophia Elizabeth, one of Frend's daughters.
+
+The London University of which De~Morgan was a professor was a
+different institution from the University of London. The
+University of London was founded about ten years later by the
+Government for the purpose of granting degrees after examination,
+without any qualification as to residence. The London University
+was affiliated as a teaching college with the University of
+London, and its name was changed to University College. The
+University of London was not a success as an examining body; a
+teaching University was demanded. De~Morgan was a highly
+successful teacher of mathematics. It was his plan to lecture for
+an hour, and at the close of each lecture to give out a number of
+problems and examples illustrative of the subject lectured on; his
+students were required to sit down to them and bring him the
+results, which he looked over and returned revised before the next
+lecture. In De~Morgan's opinion, a thorough comprehension and
+mental assimilation of great principles far outweighed in
+importance any merely analytical dexterity in the application of
+half-understood principles to particular cases.
+
+De~Morgan had a son George, who acquired great distinction in
+mathematics both at University College and the University of
+London. He and another like-minded alumnus conceived the idea of
+founding a Mathematical Society in London, where mathematical
+papers would be not only received (as by the Royal Society) but
+actually read and discussed. The first meeting was held in
+University College; De~Morgan was the first president, his son the
+first secretary. It was the beginning of the London Mathematical
+Society. In the year 1866 the chair of mental philosophy in
+University College fell vacant. Dr.\ Martineau, a Unitarian
+clergyman and professor of mental philosophy, was recommended
+formally by the Senate to the Council; but in the Council there
+were some who objected to a Unitarian clergyman, and others who
+objected to theistic philosophy. A layman of the school of Bain
+and Spencer was appointed. De~Morgan considered that the old
+standard of religious neutrality had been hauled down, and
+forthwith resigned. He was now 60 years of age. His pupils secured
+a pension of \$500 for him, but misfortunes followed. Two years
+later his son George---the younger Bernoulli, as he loved to hear
+him called, in allusion to the two eminent mathematicians of that
+name, related as father and son---died. This blow was followed by
+the death of a daughter. Five years after his resignation from
+University College De~Morgan died of nervous prostration on March
+18, 1871, in the 65th year of his age.
+
+De~Morgan was a brilliant and witty writer, whether as a
+controversialist or as a correspondent. In his time there
+flourished two Sir William Hamiltons who have often been
+confounded. The one Sir William was a baronet (that is, inherited
+the title), a Scotsman, professor of logic and metaphysics in the
+University of Edinburgh; the other was a knight (that is, won the
+title), an Irishman, professor of astronomy in the University of
+Dublin. The baronet contributed to logic the doctrine of the
+quantification of the predicate; the knight, whose full name was
+William Rowan Hamilton, contributed to mathematics the geometric
+algebra called Quaternions. De~Morgan was interested in the work
+of both, and corresponded with both; but the correspondence with
+the Scotsman ended in a public controversy, whereas that with the
+Irishman was marked by friendship and terminated only by death. In
+one of his letters to Rowan, De~Morgan says, ``Be it known unto
+you that I have discovered that you and the other Sir W.~H.\ are
+reciprocal polars with respect to me (intellectually and morally,
+for the Scottish baronet is a polar bear, and you, I was going to
+say, are a polar gentleman). When I send a bit of investigation to
+Edinburgh, the W.~H.\ of that ilk says I took it from him. When I
+send you one, you take it from me, generalize it at a glance,
+bestow it thus generalized upon society at large, and make me the
+second discoverer of a known theorem.''
+
+The correspondence of De~Morgan with Hamilton the mathematician
+extended over twenty-four years; it contains discussions not only
+of mathematical matters, but also of subjects of general interest.
+It is marked by geniality on the part of Hamilton and by wit on
+the part of De~Morgan. The following is a specimen: Hamilton
+wrote, ``My copy of Berkeley's work is not mine; like Berkeley,
+you know, I am an Irishman.'' De~Morgan replied, ``Your phrase `my
+copy is not mine' is not a bull. It is perfectly good English to
+use the same word in two different senses in one sentence,
+particularly when there is usage. Incongruity of language is no
+bull, for it expresses meaning. But incongruity of ideas (as in
+the case of the Irishman who was pulling up the rope, and finding
+it did not finish, cried out that somebody had cut off the other
+end of it) is the genuine bull.''
+
+De~Morgan was full of personal peculiarities. We have noticed his
+almost morbid attitude towards religion, and the readiness with
+which he would resign an office. On the occasion of the
+installation of his friend, Lord Brougham, as Rector of the
+University of Edinburgh, the Senate offered to confer on him the
+honorary degree of LL.D.; he declined the honor as a misnomer. He
+once printed his name: Augustus De~Morgan,
+\begin{displaymath}
+\mbox{H}\cdot\mbox{O}\cdot\mbox{M}\cdot\mbox{O}\,\cdot\,
+\mbox{P}\cdot\mbox{A}\cdot\mbox{U}\cdot\mbox{C}\cdot\mbox{A}
+\cdot\mbox{R}\cdot\mbox{U}\cdot\mbox{M}\,\cdot\,\mbox{L}\cdot\mbox{I}
+\cdot\mbox{T}\cdot\mbox{E}\cdot\mbox{R}\cdot\mbox{A}\cdot\mbox{R}
+\cdot\mbox{U}\cdot\mbox{M.}
+\end{displaymath}
+\noindent He disliked the country, and while his family enjoyed
+the seaside, and men of science were having a good time at a
+meeting of the British Association in the country he remained in
+the hot and dusty libraries of the metropolis. He said that he
+felt like Socrates, who declared that the farther he got from
+Athens the farther was he from happiness. He never sought to
+become a Fellow of the Royal Society, and he never attended a
+meeting of the Society; he said that he had no ideas or sympathies
+in common with the physical philosopher. His attitude was
+doubtless due to his physical infirmity, which prevented him from
+being either an observer or an experimenter. He never voted at an
+election, and he never visited the House of Commons, or the Tower,
+or Westminster Abbey.
+
+Were the writings of De~Morgan published in the form of collected
+works, they would form a small library. We have noticed his
+writings for the Useful Knowledge Society. Mainly through the
+efforts of Peacock and Whewell, a Philosophical Society had been
+inaugurated at Cambridge; and to its Transactions De~Morgan
+contributed four memoirs on the foundations of algebra, and an
+equal number on formal logic. The best presentation of his view of
+algebra is found in a volume, entitled \emph{Trigonometry and
+Double Algebra}, published in 1849; and his earlier view of formal
+logic is found in a volume published in 1847. His most unique work
+is styled a \emph{Budget of Paradoxes}; it originally appeared as
+letters in the columns of the \emph{Athen\ae{}um} journal; it was
+revised and extended by De~Morgan in the last years of his life,
+and was published posthumously by his widow. ``If you wish to read
+something entertaining,'' said Professor Tait to me, ``get
+De~Morgan's \emph{Budget of Paradoxes} out of the library.'' We
+shall consider more at length his theory of algebra, his
+contribution to exact logic, and his Budget of Paradoxes.
+
+In my last lecture I explained Peacock's theory of algebra. It was
+much improved by D.~F.\ Gregory, a younger member of the Cambridge
+School, who laid stress not on the permanence of equivalent forms,
+but on the permanence of certain formal laws. This new theory of
+algebra as the science of symbols and of their laws of combination
+was carried to its logical issue by De~Morgan; and his doctrine on
+the subject is still followed by English algebraists in general.
+Thus Chrystal founds his \emph{Textbook of Algebra} on De~Morgan's
+theory; although an attentive reader may remark that he
+practically abandons it when he takes up the subject of infinite
+series. De~Morgan's theory is stated in his volume on
+\emph{Trigonometry and Double Algebra}. In the chapter (of the
+book) headed ``On symbolic algebra'' he writes: ``In abandoning
+the meaning of symbols, we also abandon those of the words which
+describe them. Thus addition is to be, for the present, a sound
+void of sense. It is a mode of combination represented by $+$;
+when $+$ receives its meaning, so also will the word addition. It
+is most important that the student should bear in mind that, with
+one exception, no word nor sign of arithmetic or algebra has one
+atom of meaning throughout this chapter, the object of which is
+symbols, and their laws of combination, giving a symbolic algebra
+which may hereafter become the grammar of a hundred distinct
+significant algebras. If any one were to assert that $+$ and $-$
+might mean reward and punishment, and $A$, $B$, $C$, etc., might
+stand for virtues and vices, the reader might believe him, or
+contradict him, as he pleases, but not out of this chapter. The
+one exception above noted, which has some share of meaning, is the
+sign $=$ placed between two symbols as in $A = B$. It indicates
+that the two symbols have the same resulting meaning, by whatever
+steps attained. That $A$ and $B$, if quantities, are the same
+amount of quantity; that if operations, they are of the same
+effect, etc.''
+
+Here, it may be asked, why does the symbol $=$ prove refractory to
+the symbolic theory? De~Morgan admits that there is one exception;
+but an exception proves the rule, not in the usual but illogical
+sense of establishing it, but in the old and logical sense of
+testing its validity. If an exception can be established, the rule
+must fall, or at least must be modified. Here I am talking not of
+grammatical rules, but of the rules of science or nature.
+
+De~Morgan proceeds to give an inventory of the fundamental symbols
+of algebra, and also an inventory of the laws of algebra. The
+symbols are $0$, $1$, $+$, $-$, $\times$, $\div$, $(\,)^{(\,)}$, and
+letters; these only, all others are derived. His inventory of the
+fundamental laws is expressed under fourteen heads, but some of
+them are merely definitions. The laws proper may be reduced to the
+following, which, as he admits, are not all independent of one
+another:
+\begin{enumerate}[I.]
+\item Law of signs. $+ + = +$, $+ - = -$, $- + = -$, $- - = +$,
+$\times \times = \times$, $\times \div = \div$, $\div \times =
+\div$, $\div \div = \times$.
+\item Commutative law. $a+b = b+a$, $ab=ba$.
+\item Distributive law. $a(b+c) = ab+ac$.
+\item Index laws. $a^b \times a^c = a^{b+c}$, $(a^b)^c = a^{bc}$,
+$(ab)^c = a^c b^c$.
+\item $a- a= 0$, $a \div a = 1$.
+\end{enumerate}
+\noindent The last two may be called the rules of reduction.
+De~Morgan professes to give a complete inventory of the laws which
+the symbols of algebra must obey, for he says, ``Any system of
+symbols which obeys these laws and no others, except they be
+formed by combination of these laws, and which uses the preceding
+symbols and no others, except they be new symbols invented in
+abbreviation of combinations of these symbols, is symbolic
+algebra.'' From his point of view, none of the above principles
+are rules; they are formal laws, that is, arbitrarily chosen
+relations to which the algebraic symbols must be subject. He does
+not mention the law, which had already been pointed out by
+Gregory, namely, $(a+b)+c = a+(b+c), (ab)c = a(bc)$ and to which
+was afterwards given the name of the \emph{law of association}. If
+the commutative law fails, the associative may hold good; but not
+\emph{vice versa}. It is an unfortunate thing for the symbolist or
+formalist that in universal arithmetic $m^n$ is not equal to
+$n^m$; for then the commutative law would have full scope. Why
+does he not give it full scope? Because the foundations of algebra
+are, after all, real not formal, material not symbolic. To the
+formalists the index operations are exceedingly refractory, in
+consequence of which some take no account of them, but relegate
+them to applied mathematics. To give an inventory of the laws
+which the symbols of algebra must obey is an impossible task, and
+reminds one not a little of the task of those philosophers who
+attempt to give an inventory of the \emph{a priori} knowledge of
+the mind.
+
+De~Morgan's work entitled \emph{Trigonometry and Double Algebra}
+consists of two parts; the former of which is a treatise on
+Trigonometry, and the latter a treatise on generalized algebra
+which he calls Double Algebra. But what is meant by Double as
+applied to algebra? and why should Trigonometry be also treated in
+the same textbook? The first stage in the development of algebra
+is \emph{arithmetic}, where numbers only appear and symbols of
+operations such as $+$, $\times$, etc. The next stage is
+\emph{universal arithmetic}, where letters appear instead of
+numbers, so as to denote numbers universally, and the processes
+are conducted without knowing the values of the symbols. Let $a$
+and $b$ denote any numbers; then such an expression as $a-b$ may
+be impossible; so that in universal arithmetic there is always a
+proviso, \emph{provided the operation is possible}. The third
+stage is \emph{single algebra}, where the symbol may denote a
+quantity forwards or a quantity backwards, and is adequately
+represented by segments on a straight line passing through an
+origin. Negative quantities are then no longer impossible; they
+are represented by the backward segment. But an impossibility
+still remains in the latter part of such an expression as
+$a+b\sqrt{-1}$ which arises in the solution of the quadratic
+equation. The fourth stage is \emph{double algebra}; the algebraic
+symbol denotes in general a segment of a line in a given plane; it
+is a double symbol because it involves two specifications, namely,
+length and direction; and $\sqrt{-1}$ is interpreted as denoting a
+quadrant. The expression $a+b\sqrt{-1}$ then represents a line in
+the plane having an abscissa $a$ and an ordinate $b$. Argand and
+Warren carried double algebra so far; but they were unable to
+interpret on this theory such an expression as $e^{a\sqrt{-1}}$.
+De~Morgan attempted it by \emph{reducing} such an expression to
+the form $b+q\sqrt{-1}$, and he considered that he had shown that
+it could be always so reduced. The remarkable fact is that this
+double algebra satisfies all the fundamental laws above
+enumerated, and as every apparently impossible combination of
+symbols has been interpreted it looks like the complete form of
+algebra.
+
+If the above theory is true, the next stage of development ought
+to be \emph{triple} algebra and if $a+b\sqrt{-1}$ truly represents
+a line in a given plane, it ought to be possible to find a third
+term which added to the above would represent a line in space.
+Argand and some others guessed that it was $a + b\sqrt{-1} +
+c\sqrt{-1}\,^{\sqrt{-1}}$ although this contradicts the truth
+established by Euler that $\sqrt{-1}\,^{\sqrt{-1}}=e^{-
+\frac{1}{2} \pi}$. De~Morgan and many others worked hard at the
+problem, but nothing came of it until the problem was taken up by
+Hamilton. We now see the reason clearly: the symbol of double
+algebra denotes not a length and a direction; but a multiplier and
+\emph{an angle}. In it the angles are confined to one plane; hence
+the next stage will be a \emph{quadruple algebra}, when the axis
+of the plane is made variable. And this gives the answer to the
+first question; double algebra is nothing but analytical plane
+trigonometry, and this is the reason why it has been found to be
+the natural analysis for alternating currents. But De~Morgan never
+got this far; he died with the belief ``that double algebra must
+remain as the full development of the conceptions of arithmetic,
+so far as those symbols are concerned which arithmetic immediately
+suggests.''
+
+When the study of mathematics revived at the University of
+Cambridge, so also did the study of logic. The moving spirit was
+Whewell, the Master of Trinity College, whose principal writings
+were a \emph{History of the Inductive Sciences}, and
+\emph{Philosophy of the Inductive Sciences}. Doubtless De~Morgan
+was influenced in his logical investigations by Whewell; but other
+contemporaries of influence were Sir W.\ Hamilton of Edinburgh, and
+Professor Boole of Cork. De~Morgan's work on \emph{Formal Logic},
+published in 1847, is principally remarkable for his development
+of the numerically definite syllogism. The followers of Aristotle
+say and say truly that from two particular propositions such as
+\emph{Some} $M$'s \emph{are} $A$'s, and \emph{Some} $M$'s
+\emph{are} $B$'s nothing follows of necessity about the relation
+of the $A$'s and $B$'s. But they go further and say in order that
+any relation about the $A$'s and $B$'s may follow of necessity,
+the middle term must be taken universally in one of the premises.
+De~Morgan pointed out that from \emph{Most} $M$'s \emph{are} $A$'s
+and \emph{Most} $M$'s \emph{are} $B$'s it follows of necessity
+that some $A$'s are $B$'s and he formulated the numerically
+definite syllogism which puts this principle in exact quantitative
+form. Suppose that the number of the $M$'s is $m$, of the $M$'s
+that are $A$'s is $a$, and of the $M$'s that are $B$'s is $b$;
+then there are at least $(a+b-m)$ $A$'s that are $B$'s. Suppose
+that the number of souls on board a steamer was $1000$, that $500$
+were in the saloon, and $700$ were lost; it follows of necessity,
+that at least $700+500-1000$, that is, $200$, saloon passengers
+were lost. This single principle suffices to prove the validity of
+all the Aristotelian moods; it is therefore a fundamental
+principle in necessary reasoning.
+
+Here then De~Morgan had made a great advance by introducing
+\emph{quantification of the terms}. At that time Sir W.\ Hamilton
+was teaching at Edinburgh a doctrine of the quantification of the
+predicate, and a correspondence sprang up. However, De~Morgan soon
+perceived that Hamilton's quantification was of a different
+character; that it meant for example, substituting the two forms
+\emph{The whole of} $A$ \emph{is the whole of} $B$, and \emph{The
+whole of} $A$ \emph{is a part of} $B$ for the Aristotelian form
+All $A$'s are $B$'s. Philosophers generally have a large share of
+intolerance; they are too apt to think that they have got hold of
+the whole truth, and that everything outside of their system is
+error. Hamilton thought that he had placed the keystone in the
+Aristotelian arch, as he phrased it; although it must have been a
+curious arch which could stand 2000 years without a keystone. As a
+consequence he had no room for De~Morgan's innovations. He accused
+De~Morgan of plagiarism, and the controversy raged for years in
+the columns of the \emph{Athen\ae{}um}, and in the publications of
+the two writers.
+
+The memoirs on logic which De~Morgan contributed to the
+Transactions of the Cambridge Philosophical Society subsequent to
+the publication of his book on \emph{Formal Logic} are by far the
+most important contributions which he made to the science,
+especially his fourth memoir, in which he begins work in the broad
+field of the \emph{logic of relatives}. This is the true field for
+the logician of the twentieth century, in which work of the
+greatest importance is to be done towards improving language and
+facilitating thinking processes which occur all the time in
+practical life. Identity and difference are the two relations
+which have been considered by the logician; but there are many
+others equally deserving of study, such as equality, equivalence,
+consanguinity, affinity, etc.
+
+In the introduction to the \emph{Budget of Paradoxes} De~Morgan
+explains what he means by the word. ``A great many individuals,
+ever since the rise of the mathematical method, have, each for
+himself, attacked its direct and indirect consequences. I shall
+call each of these persons a \emph{paradoxer}, and his system a
+\emph{paradox}. I use the word in the old sense: a paradox is
+something which is apart from general opinion, either in subject
+matter, method, or conclusion. Many of the things brought forward
+would now be called \emph{crotchets}, which is the nearest word we
+have to old \emph{paradox}. But there is this difference, that by
+calling a thing a crotchet we mean to speak lightly of it; which
+was not the necessary sense of paradox. Thus in the 16th century
+many spoke of the earth's motion as the \emph{paradox of
+Copernicus} and held the ingenuity of that theory in very high
+esteem, and some I think who even inclined towards it. In the
+seventeenth century the depravation of meaning took place, in
+England at least.''
+
+How can the sound paradoxer be distinguished from the false
+paradoxer? De~Morgan supplies the following test: ``The manner in
+which a paradoxer will show himself, as to sense or nonsense, will
+not depend upon what he maintains, but upon whether he has or has
+not made a sufficient knowledge of what has been done by others,
+especially as to the mode of doing it, a preliminary to inventing
+knowledge for himself\ldots. New knowledge, when to any purpose,
+must come by contemplation of old knowledge, in every matter which
+concerns thought; mechanical contrivance sometimes, not very
+often, escapes this rule. All the men who are now called
+discoverers, in every matter ruled by thought, have been men
+versed in the minds of their predecessors and learned in what had
+been before them. There is not one exception.''
+
+I remember that just before the American Association met at
+Indianapolis in 1890, the local newspapers heralded a great
+discovery which was to be laid before the assembled savants---a
+young man living somewhere in the country had squared the circle.
+While the meeting was in progress I observed a young man going
+about with a roll of paper in his hand. He spoke to me and
+complained that the paper containing his discovery had not been
+received. I asked him whether his object in presenting the paper
+was not to get it read, printed and published so that everyone
+might inform himself of the result; to all of which he assented
+readily. But, said I, many men have worked at this question, and
+their results have been tested fully, and they are printed for the
+benefit of anyone who can read; have you informed yourself of
+their results? To this there was no assent, but the sickly smile
+of the false paradoxer.
+
+The \emph{Budget} consists of a review of a large collection of
+paradoxical books which De~Morgan had accumulated in his own
+library, partly by purchase at bookstands, partly from books sent
+to him for review, partly from books sent to him by the authors.
+He gives the following classification: squarers of the circle,
+trisectors of the angle, duplicators of the cube, constructors of
+perpetual motion, subverters of gravitation, stagnators of the
+earth, builders of the universe. You will still find specimens of
+all these classes in the New World and in the new century.
+
+De~Morgan gives his personal knowledge of paradoxers. ``I suspect
+that I know more of the English class than any man in Britain. I
+never kept any reckoning: but I know that one year with
+another?---and less of late years than in earlier time?---I have
+talked to more than five in each year, giving more than a hundred
+and fifty specimens. Of this I am sure, that it is my own fault if
+they have not been a thousand. Nobody knows how they swarm, except
+those to whom they naturally resort. They are in all ranks and
+occupations, of all ages and characters. They are very earnest
+people, and their purpose is bona fide, the dissemination of their
+paradoxes. A great many---the mass, indeed---are illiterate, and a
+great many waste their means, and are in or approaching penury.
+These discoverers despise one another.''
+
+A paradoxer to whom De~Morgan paid the compliment which Achilles
+paid Hector---to drag him round the walls again and again---was
+James Smith, a successful merchant of Liverpool. He found $\pi = 3
+\frac{1}{8}$. His mode of reasoning was a curious caricature of
+the \emph{reductio ad absurdum} of Euclid. He said let $\pi = 3
+\frac{1}{8}$, and then showed that on that supposition, every
+other value of $\pi$ must be absurd; consequently $\pi =
+3\frac{1}{8}$ is the true value. The following is a specimen of De
+Morgan's dragging round the walls of Troy: ``Mr.\ Smith continues
+to write me long letters, to which he hints that I am to answer.
+In his last of 31 closely written sides of note paper, he informs
+me, with reference to my obstinate silence, that though I think
+myself and am thought by others to be a mathematical Goliath, I
+have resolved to play the mathematical snail, and keep within my
+shell. A mathematical \emph{snail}! This cannot be the thing so
+called which regulates the striking of a clock; for it would mean
+that I am to make Mr.\ Smith sound the true time of day, which I
+would by no means undertake upon a clock that gains 19 seconds odd
+in every hour by false quadrative value of $\pi$. But he ventures
+to tell me that pebbles from the sling of simple truth and common
+sense will ultimately crack my shell, and put me \emph{hors de
+combat}. The confusion of images is amusing: Goliath turning
+himself into a snail to avoid $\pi = 3\frac{1}{8}$ and James
+Smith, Esq., of the Mersey Dock Board: and put \emph{hors de
+combat} by pebbles from a sling. If Goliath had crept into a snail
+shell, David would have cracked the Philistine with his foot.
+There is something like modesty in the implication that the
+crack-shell pebble has not yet taken effect; it might have been
+thought that the slinger would by this time have been
+singing---And thrice [and one-eighth] I routed all my foes, And
+thrice [and one-eighth] I slew the slain.''
+
+In the region of pure mathematics De~Morgan could detect easily
+the false from the true paradox; but he was not so proficient in
+the field of physics. His father-in-law was a paradoxer, and his
+wife a paradoxer; and in the opinion of the physical philosophers
+De~Morgan himself scarcely escaped. His wife wrote a book
+describing the phenomena of spiritualism, table-rapping,
+table-turning, etc.; and De~Morgan wrote a preface in which he
+said that he knew some of the asserted facts, believed others on
+testimony, but did not pretend to know \emph{whether} they were
+caused by spirits, or had some unknown and unimagined origin. From
+this alternative he left out ordinary material causes. Faraday
+delivered a lecture on \emph{Spiritualism}, in which he laid it
+down that in the investigation we ought to set out with the idea
+of what is physically possible, or impossible; De~Morgan could not
+understand this.
+
+\chapter [Sir William Rowan Hamilton (1805-1865)]{SIR WILLIAM
+ROWAN~HAMILTON\footnote{This Lecture was delivered April 16,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1805-1865)}\end{center}\normalsize
+
+William Rowan Hamilton was born in Dublin, Ireland, on the 3d of
+August, 1805. His father, Archibald Hamilton, was a solicitor in
+the city of Dublin; his mother, Sarah Hutton, belonged to an
+intellectual family, but she did not live to exercise much
+influence on the education of her son. There has been some dispute
+as to how far Ireland can claim Hamilton; Professor Tait of
+Edinburgh in the Encyclopaedia Brittanica claims him as a
+Scotsman, while his biographer, the Rev.\ Charles Graves, claims
+him as essentially Irish. The facts appear to be as follows: His
+father's mother was a Scotch woman; his father's father was a
+citizen of Dublin. But the name ``Hamilton'' points to Scottish
+origin, and Hamilton himself said that his family claimed to have
+come over from Scotland in the time of James I\@. Hamilton always
+considered himself an Irishman; and as Burns very early had an
+ambition to achieve something for the renown of Scotland, so
+Hamilton in his early years had a powerful ambition to do
+something for the renown of Ireland. In later life he used to say
+that at the beginning of the century people read French
+mathematics, but that at the end of it they would be reading Irish
+mathematics.
+
+Hamilton, when three years of age, was placed in the charge of his
+uncle, the Rev.\ James Hamilton, who was the curate of Trim, a
+country town, about twenty miles from Dublin, and who was also the
+master of the Church of England school. From his uncle he received
+all his primary and secondary education and also instruction in
+Oriental languages. As a child Hamilton was a prodigy; at three
+years of age he was a superior reader of English and considerably
+advanced in arithmetic; at four a good geographer; at five able to
+read and translate Latin, Greek, and Hebrew, and liked to recite
+Dryden, Collins, Milton and Homer; at eight a reader of Italian
+and French and giving vent to his feelings in extemporized Latin;
+at ten a student of Arabic and Sanscrit. When twelve years old he
+met Zerah Colburn, the American calculating boy, and engaged with
+him in trials of arithmetical skill, in which trials Hamilton came
+off with honor, although Colburn was generally the victor. These
+encounters gave Hamilton a decided taste for arithmetical
+computation, and for many years afterwards he loved to perform
+long operations in arithmetic in his mind, extracting the square
+and cube root, and solving problems that related to the properties
+of numbers. When thirteen he received his initiation into algebra
+from Clairault's \emph{Algebra} in the French, and he made an
+epitome, which he ambitiously entitled ``A Compendious Treatise on
+Algebra by William Hamilton.''
+
+When Hamilton was fourteen years old, his father died and left his
+children slenderly provided for. Henceforth, as the elder brother
+of three sisters, Hamilton had to act as a man. This year he
+addressed a letter of welcome, written in the Persian language, to
+the Persian Ambassador, then on a visit to Dublin; and he met
+again Zerah Colburn. In the interval Zerah had attended one of the
+great public schools of England. Hamilton had been at a country
+school in Ireland, and was now able to make a successful
+investigation of the methods by which Zerah made his lightning
+calculations. When sixteen, Hamilton studied the Differential
+Calculus by the help of a French textbook, and began the study of
+the \emph{M\'ecanique c\'eleste} of Laplace, and he was able at
+the beginning of this study to detect a flaw in the reasoning by
+which Laplace demonstrates the theorem of the parallelogram of
+forces. This criticism brought him to the notice of Dr.\ Brinkley,
+who was then the professor of astronomy in the University of
+Dublin, and resided at Dunkirk, about five miles from the centre
+of the city. He also began an investigation for himself of
+equations which represent systems of straight lines in a plane,
+and in so doing hit upon ideas which he afterwards developed into
+his first mathematical memoir to the Royal Irish Academy. Dr.\
+Brinkley is said to have remarked of him at this time: ``This
+young man, I do not say \emph{will be}, but \emph{is}, the first
+mathematician of his age.''
+
+At the age of eighteen Hamilton entered Trinity College, Dublin,
+the University of Dublin founded by Queen Elizabeth, and differing
+from the Universities of Oxford and Cambridge in having only one
+college. Unlike Oxford, which has always given prominence to
+classics, and Cambridge, which has always given prominence to
+mathematics, Dublin at that time gave equal prominence to classics
+and to mathematics. In his first year Hamilton won the very rare
+honor of \emph{optime} at his examination in Homer. In the old
+Universities marks used to be and in some cases still are
+published, descending not in percentages but by means of the scale
+of Latin adjectives: \emph{optime, valdebene, bene, satis,
+mediocriter, vix medi, non}; \emph{optime} means passed with the
+very highest distinction; \emph{vix} means passed but with great
+difficulty. This scale is still in use in the medical examinations
+of the University of Edinburgh. Before entering college Hamilton
+had been accustomed to translate Homer into blank verse, comparing
+his result with the translations of Pope and Cowper; and he had
+already produced some original poems. In this, his first year he
+wrote a poem ``On college ambition'' which is a fair specimen of
+his poetical attainments.
+
+\begin{verse}
+  Oh! Ambition hath its hour \\
+  Of deep and spirit-stirring power; \\
+  Not in the tented field alone, \\
+  Nor peer-engirded court and throne; \\
+  Nor the intrigues of busy life; \\
+  But ardent Boyhood's generous strife, \\
+  While yet the Enthusiast spirit turns \\
+  Where'er the light of Glory burns, \\
+  Thinks not how transient is the blaze, \\
+  But longs to barter Life for Praise.
+
+  Look round the arena, and ye spy \\
+  Pallid cheek and faded eye; \\
+  Among the bands of rivals, few \\
+  Keep their native healthy hue: \\
+  Night and thought have stolen away \\
+  Their once elastic spirit's play. \\
+  A few short hours and all is o'er, \\
+  Some shall win one triumph more; \\
+  Some from the place of contest go \\
+  Again defeated, sad and slow.
+
+  What shall reward the conqueror then \\
+  For all his toil, for all his pain, \\
+  For every midnight throb that stole \\
+  So often o'er his fevered soul? \\
+  Is it the applaudings loud \\
+  Or wond'ring gazes of the crowd; \\
+  Disappointed envy's shame, \\
+  Or hollow voice of fickle Fame? \\
+  These may extort the sudden smile, \\
+  May swell the heart a little while; \\
+  But they leave no joy behind, \\
+  Breathe no pure transport o'er the mind, \\
+  Nor will the thought of selfish gladness \\
+  Expand the brow of secret sadness.
+
+  Yet if Ambition hath its hour \\
+  Of deep and spirit-stirring power, \\
+  Some bright rewards are all its own, \\
+  And bless its votaries alone: \\
+  The anxious friend's approving eye; \\
+  The generous rivals' sympathy; \\
+  And that best and sweetest prize \\
+  Given by silent Beauty's eyes! \\
+  These are transports true and strong, \\
+  Deeply felt, remembered long: \\
+  Time and sorrow passing o'er \\
+  Endear their memory but the more.
+\end{verse}
+
+The ``silent Beauty'' was not an abstraction, but a young lady
+whose brothers were fellow-students of Trinity College. This led
+to much effusion of poetry; but unfortunately while Hamilton was
+writing poetry about her another young man was talking prose to
+her; with the result that Hamilton experienced a disappointment.
+On account of his self-consciousness, inseparable probably from
+his genius, he felt the disappointment keenly. He was then known
+to the professor of astronomy, and walking from the College to the
+Observatory along the Royal Canal, he was actually tempted to
+terminate his life in the water.
+
+In his second year he formed the plan of reading so as to compete
+for the highest honors both in classics and in mathematics. At
+graduation two gold medals were awarded, the one for distinction
+in classics, the other for distinction in mathematics. Hamilton
+aimed at carrying off both. In his junior year he received an
+\emph{optime} in mathematical physics; and, as the winner of two
+\emph{optimes}, the one in classics, the other in mathematics, he
+immediately became a celebrity in the intellectual circle of
+Dublin.
+
+In his senior year he presented to the Royal Irish Academy a
+memoir embodying his research on systems of lines. He now called
+it a ``Theory of Systems of Rays'' and it was printed in the
+\emph{Transactions}. About this time Dr.\ Brinkley was appointed
+to the bishopric of Cloyne, and in consequence resigned the
+professorship of astronomy. In the United Kingdom it is customary
+when a post becomes vacant for aspirants to lodge a formal
+application with the appointing board and to supplement their own
+application by testimonial letters from competent authorities. In
+the present case quite a number of candidates appeared, among them
+Airy, who afterwards became Astronomer Royal of England, and
+several Fellows of Trinity College, Dublin. Hamilton did not
+become a formal candidate, but he was invited to apply, with the
+result that he received the appointment while still an
+undergraduate, and not twenty-two years of age. Thus was his
+undergraduate career signalized much more than by the carrying off
+of the two gold medals. Before assuming the duties of his chair he
+made a tour through England and Scotland, and met for the first
+time the poet Wordsworth at his home at Rydal Mount, in
+Cumberland. They had a midnight walk, oscillating backwards and
+forwards between Rydal and Ambleside, absorbed in converse on high
+themes, and finding it almost impossible to part. Wordsworth
+afterwards said that Coleridge and Hamilton were the two most
+wonderful men, taking all their endowments together, that he had
+ever met.
+
+In October, 1827, he came to reside at the place which was
+destined to be the scene of his scientific labors. I had the
+pleasure of visiting it last summer as the guest of his successor.
+The Observatory is situated on the top of a hill, Dunsink, about
+five miles from Dublin. The house adjoins the observatory; to the
+east is an extensive lawn; to the west a garden with stone wall
+and shaded walks; to the south a terraced field; at the foot of
+the hill is the Royal Canal; to the southeast the city of Dublin;
+while the view is bounded by the sea and the Dublin and Wicklow
+Mountains; a fine home for a poet or a philosopher or a
+mathematician, and in Hamilton all three were combined.
+
+Settled at the Observatory he started out diligently as an
+observer, but he found it difficult to stand the low temperatures
+incident to the work. He never attained skill as an observer, and
+unfortunately he depended on a very poor assistant. Himself a
+brilliant computer, with a good observer for assistant, the work
+of the observatory ought to have flourished. One of the first
+distinguished visitors at the Observatory was the poet Wordsworth,
+in commemoration of which one of the shaded walks in the garden
+was named Wordsworth's walk. Wordsworth advised him to concentrate
+his powers on science; and, not long after, wrote him as follows:
+``You send me showers of verses which I receive with much
+pleasure, as do we all: yet have we fears that this employment may
+seduce you from the path of science which you seem destined to
+tread with so much honor to yourself and profit to others. Again
+and again I must repeat that the composition of verse is
+infinitely more of an art than men are prepared to believe, and
+absolute success in it depends upon innumerable \emph{minuti\ae{}}
+which it grieves me you should stoop to acquire a knowledge
+of\ldots Again I do venture to submit to your consideration,
+whether the poetical parts of your nature would not find a field
+more favorable to their exercise in the regions of prose; not
+because those regions are humbler, but because they may be
+gracefully and profitably trod, with footsteps less careful and in
+measures less elaborate.''
+
+Hamilton possessed the poetic imagination; what he was deficient
+in was the technique of the poet. The imagination of the poet is
+kin to the imagination of the mathematician; both extract the
+ideal from a mass of circumstances. In this connection De~Morgan
+wrote: ``The moving power of mathetical \emph{invention} is not
+reasoning but imagination. We no longer apply the homely term
+\emph{maker} in literal translation of \emph{poet}; but
+discoverers of all kinds, whatever may be their lines, are makers,
+or, as we mow say, have the creative genius.'' Hamilton spoke of
+the \emph{M\'ecanique analytique} of Lagrange as a ``scientific
+poem''; Hamilton himself was styled the Irish Lagrange. Engineers
+venerate Rankine, electricians venerate Maxwell; both were
+scientific discoverers and likewise poets, that is, amateur poets.
+The proximate cause of the shower of verses was that Hamilton had
+fallen in love for the second time. The young lady was Miss
+de~Vere, daughter of an accomplished Irish baronet, and who like
+Tennyson's Lady Clara Vere de~Vere could look back on a long and
+illustrious descent. Hamilton had a pupil in Lord Adare, the
+eldest son of the Earl of Dunraven, and it was while visiting
+Adare Manor that he was introduced to the De~Vere family, who
+lived near by at Curragh Chase. His suit was encouraged by the
+Countess of Dunraven, it was favorably received by both father and
+mother, he had written many sonnets of which Ellen de~Vere was the
+inspiration, he had discussed with her astronomy, poetry and
+philosophy; and was on the eve of proposing when he gave up
+because the young lady incidentally said to him that ``she could
+not live happily anywhere but at Curragh.'' His action shows the
+working of a too self-conscious mind, proud of his own
+intellectual achievements, and too much awed by her long descent.
+So he failed for the second time; but both of these ladies were
+friends of his to the last.
+
+At the age of 27 he contributed to the Irish Academy a
+supplementary paper on his Theory of Systems of Rays, in which he
+predicted the phenomenon of conical refraction; namely, that under
+certain conditions a single ray incident on a biaxial crystal
+would be broken up into a cone of rays, and likewise that under
+certain conditions a single emergent ray would appear as a cone of
+rays. The prediction was made by Hamilton on Oct.\ 22nd; it was
+experimentally verified by his colleague Prof.\ Lloyd on Dec.\
+14th. It is not experiment alone or mathematical reasoning alone
+which has built up the splendid temple of physical science, but
+the two working together; and of this we have a notable
+exemplification in the discovery of conical refraction.
+
+Twice Hamilton chose well but failed; now he made another choice
+and succeeded. The lady was a Miss Bayly, who visited at the home
+of her sister near Dunsink hill. The lady had serious misgivings
+about the state of her health; but the marriage took place. The
+kind of wife which Hamilton needed was one who could govern him
+and efficiently supervise all domestic matters; but the wife he
+chose was, from weakness of body and mind, incapable of doing it.
+As a consequence, Hamilton worked for the rest of his life under
+domestic difficulties of no ordinary kind.
+
+At the age of 28 he made a notable addition to the theory of
+Dynamics by extending to it the idea of a Characteristic Function,
+which he had previously applied with success to the science of
+Optics in his Theory of Systems of Rays. It was contributed to the
+Royal Society of London, and printed in their \emph{Philosophical
+Transactions}. The Royal Society of London is the great scientific
+society of England, founded in the reign of Charles II, and of
+which Newton was one of the early presidents; Hamilton was invited
+to become a fellow but did not accept, as he could not afford the
+expense.
+
+At the age of 29 he read a paper before the Royal Irish Academy,
+which set forth the result of long meditation and investigation on
+the nature of Algebra as a science; the paper is entitled
+``Algebra as the Science of Pure Time.'' The main idea is that as
+Geometry considered as a science is founded upon the pure
+intuition of space, so algebra as a science is founded upon the
+pure intuition of time. He was never satisfied with Peacock's
+theory of algebra as a ``System of Signs and their Combinations'';
+nor with De~Morgan's improvement of it; he demanded a more real
+foundation. In reading Kant's \emph{Critique of Pure Reason} he
+was struck by the following passage: ``Time and space are two
+sources of knowledge from which various \emph{a priori}
+synthetical cognitions can be derived. Of this, pure mathematics
+gives a splendid example in the case of our cognitions of space
+and its various relations. As they are both pure forms of sensuous
+intuition, they render synthetical propositions \emph{a priori}
+possible.'' Thus, according to Kant, space and time are forms of
+the intellect; and Hamilton reasoned that, as geometry is the
+science of the former, so algebra must be the science of the
+latter. When algebra is based on any unidimensional subject, such
+as time, or a straight line, a difficulty arises in explaining the
+roots of a quadratic equation when they are imaginary. To get over
+this difficulty Hamilton invented a theory of algebraic couplets,
+which has proved a conundrum in the mathematical world. Some 20
+years ago there nourished in Edinburgh a mathematician named Sang
+who had computed the most elaborate tables of logarithms in
+existence---which still exist in manuscript. On reading the theory
+in question he first judged that either Hamilton was crazy, or
+else that he (Sang) was crazy, but eventually reached the more
+comforting alternative. On the other hand, Prof.\ Tait believes in
+its soundness, and endeavors to bring it down to the ordinary
+comprehension.
+
+We have seen that the British Association for the Advancement of
+Science was founded in 1831, and that its first meeting was in the
+ancient city of York. It was a policy of the founders not to meet
+in London, but in the provincial cities, so that thereby greater
+interest in the advance of science might be produced over the
+whole land. The cities chosen for the place of meeting in
+following years were the University towns: Oxford, Cambridge,
+Edinburgh, Dublin. Hamilton was the only representative of Ireland
+present at the Oxford meeting; and at the Oxford, Cambridge, and
+Edinburgh meetings he not only contributed scientific papers, but
+he acquired renown as a scientific orator. In the case of the
+Dublin meeting he was chief organizer beforehand, and chief orator
+when it met. The week of science was closed by a grand dinner
+given in the library of Trinity College; and an incident took
+place which is thus described by an American scientist:
+
+``We assembled in the imposing hall of Trinity Library, two
+hundred and eighty feet long, at six o'clock. When the company was
+principally assembled, I observed a little stir near the place
+where I stood, which nobody could explain, and which, in fact, was
+not comprehended by more than two or three persons present. In a
+moment, however, I perceived myself standing near the Lord
+Lieutenant and his suite, in front of whom a space had been
+cleared, and by whom was Professor Hamilton, looking very much
+embarrassed. The Lord Lieutenant then called him by name, and he
+stepped into the vacant space. `I am,' said his Excellency, `about
+to exercise a prerogative of royalty, and it gives me great
+pleasure to do it, on this splendid public occasion, which has
+brought together so many distinguished men from all parts of the
+empire, and from all parts even of the world where science is held
+in honor. But, in exercising it, Professor Hamilton, I do not
+confer a distinction. I but set the royal, and therefore the
+national mark on a distinction already acquired by your genius and
+labors.' He went on in this way for three of four minutes, his
+voice very fine, rich and full; his manner as graceful and
+dignified as possible; and his language and allusions appropriate
+and combined into very ample flowing sentences. Then, receiving
+the State sword from one of his attendants, he said, `Kneel down,
+Professor Hamilton'; and laying the blade gracefully and gently
+first on one shoulder, and then on the other, he said, `Rise up,
+Sir William Rowan Hamilton.' The Knight rose, and the Lord
+Lieutenant then went up, and with an appearance of great tact in
+his manner, shook hands with him. No reply was made. The whole
+scene was imposing, rendered so, partly by the ceremony itself,
+but more by the place in which it passed, by the body of very
+distinguished men who were assembled there, and especially by the
+extraordinarily dignified and beautiful manner in which it was
+performed by the Lord Lieutenant. The effect at the time was
+great, and the general impression was that, as the honor was
+certainly merited by him who received it, so the words by which it
+was conferred were so graceful and appropriate that they
+constituted a distinction by themselves, greater than the
+distinction of knighthood. I was afterwards told that this was the
+first instance in which a person had been knighted by a Lord
+Lieutenant either for scientific or literary merit.''
+
+Two years after another great honor came to Hamilton---the
+presidency of the Royal Irish Academy. While holding this office,
+in the year 1843, when 38 years old, he made the discovery which
+will ever be considered his highest title to fame. The story of
+the discovery is told by Hamilton himself in a letter to his son:
+``On the 16th day of October, which happened to be a Monday, and
+Council day of the Royal Irish Academy, I was walking in to attend
+and preside, and your mother was walking with me along the Royal
+Canal, to which she had perhaps driven; and although she talked
+with me now and then, yet an undercurrent of thought was going on
+in my mind, which gave at last a result, whereof it is not too
+much to say that I felt at once the importance. An electric
+circuit seemed to close; and a spark flashed forth, the herald (as
+I foresaw immediately) of many long years to come of definitely
+directed thought and work, by myself if spared, and at all events
+on the part of others, if I should even be allowed to live long
+enough distinctly to communicate the discovery. Nor could I resist
+the impulse---unphilosophical as it may have been---to cut with a
+knife on a stone of Brougham Bridge, as we passed it, the
+fundamental formula with the symbols $i$,$j$,$k$; namely,
+\begin{displaymath}
+i^2 = j^2 = k^2 = ijk = -1,
+\end{displaymath}
+which contains the solution of the problem, but of course as an
+inscription has long since mouldered away. A more durable notice
+remains, however, in the Council Book of the Academy for that day,
+which records the fact that I then asked for and obtained leave to
+read a paper on Quaternions, at the first general meeting of the
+session, which reading took place accordingly on Monday the 13th
+of November following.''
+
+Last summer Prof.\ Joly and I took the walk here described. We
+started from the Observatory, walked down the terraced field, then
+along the path by the side of the Royal Canal towards Dublin until
+we came to the second bridge spanning the canal. The path of
+course goes under the Bridge, and the inner side of the Bridge
+presents a very convenient surface for an inscription. I have seen
+this incident quoted as an example of how a genius strikes on a
+discovery all of a sudden. No doubt a problem was solved then and
+there, but the problem had engaged Hamilton's thoughts and
+researches for fifteen years. It is rather an illustration of how
+genius is patience, or a faculty for infinite labor. What was
+Hamilton struggling to do all these years? To emerge from Flatland
+into Space; in other words, Algebra had been extended so as to
+apply to lines in a plane; but no one had been able to extend it
+so as to apply to lines in space. The greatness of the feat is
+made evident by the fact that most analysts are still crawling in
+Flatland. The same year in which he discovered Quaternions the
+Government granted him a pension of \pounds200 per annum for life,
+on account of his scientific work.
+
+We have seen how Hamilton gained two \emph{optimes}, one in
+classics, the other in physics, the highest possible distinction
+in his college course; how he was appointed professor of astronomy
+while yet an undergraduate; how he was a scientific chief in the
+British Association at 27; how he was knighted for his scientific
+achievements at 30; how he was appointed president of the Royal
+Irish Academy at 32; how he discovered Quaternions and received a
+Government pension at 38; can you imagine that this brilliant and
+successful genius would fall a victim to intemperance? About this
+time at a dinner of a scientific society in Dublin he lost control
+of himself, and was so mortified that, on the advice of friends he
+resolved to abstain totally. This resolution he kept for two
+years; when happening to be a member of a scientific party at the
+castle of Lord Rosse, an amateur astronomer then the possessor of
+the largest telescope in existence, he was taunted for sticking to
+water, particularly by Airy the Greenwich astronomer. He broke his
+good resolution, and from that time forward the craving for
+alcoholic stimulants clung to him. How could Hamilton with all his
+noble aspirations fall into such a vice? The explanation lay in
+the want of order which reigned in his home. He had no regular
+times for his meals; frequently had no regular meals at all, but
+resorted to the sideboard when hunger compelled him. What more
+natural in such condition than that he should refresh himself with
+a quaff of that beverage for which Dublin is famous---porter
+labelled $X^3$? After Hamilton's death the dining-room was found
+covered with huge piles of manuscript, with convenient walks
+between the piles; when these literary remains were wheeled out
+and examined, china plates with the relics of food upon them were
+found between the sheets of manuscript, plates sufficient in
+number to furnish a kitchen. He used to carry on, says his eldest
+son, long trains of algebraical and arithmetical calculations in
+his mind, during which he was unconscious of the earthly necessity
+of eating; ``we used to bring in a `snack' and leave it in his
+study, but a brief nod of recognition of the intrusion of the chop
+or cutlet was often the only result, and his thoughts went on
+soaring upwards.''
+
+In 1845 Hamilton attended the second Cambridge meeting of the
+British Association; and after the meeting he was lodged for a
+week in the rooms in Trinity College which tradition points out as
+those in which Sir Isaac Newton composed the \emph{Principia}.
+This incident was intended as a compliment and it seems to have
+impressed Hamilton powerfully. He came back to the Observatory
+with the fixed purpose of preparing a work on Quaternions which
+might not unworthily compare with the \emph{Principia} of Newton,
+and in order to obtain more leisure for this undertaking he
+resigned the office of president of the Royal Irish Academy. He
+first of all set himself to the preparation of a course of
+lectures on Quaternions, which were delivered in Trinity College,
+Dublin, in 1848, and were six in number. Among his hearers were
+George Salmon, now well known for his highly successful series of
+manuals on Analytical Geometry; and Arthur Cayley, then a Fellow
+of Trinity College, Cambridge. These lectures were afterward
+expanded and published in 1853, under the title of \emph{Lectures
+on Quaternions}, at the expense of Trinity College, Dublin.
+Hamilton had never had much experience as a teacher; the volume
+was criticised for diffuseness of style, and certainly Hamilton
+sometimes forgot the expositor in the orator. The book was a
+paradox---a sound paradox, and of his experience as a paradoxer
+Hamilton wrote: ``It required a certain capital of scientific
+reputation, amassed in former years, to make it other than
+dangerously imprudent to hazard the publication of a work which
+has, although at bottom quite conservative, a highly revolutionary
+air. It was part of the ordeal through which I had to pass, an
+episode in the battle of life, to know that even candid and
+friendly people secretly or, as it might happen, openly, censured
+or ridiculed me, for what appeared to them my monstrous
+innovations.'' One of these monstrous innovations was the
+principle that $ij$ is not $=ji$ but $=-ji$; the truth of which is
+evident from the diagram. Critics said that he held that $3 \times
+4$ is not $= 4 \times 3$; which proceeds on the assumption that
+only numbers can be represented by letter symbols.
+
+\begin{center}
+\includegraphics[width=25mm]{images/WRHfig1.png}
+\end{center}
+
+Soon after the publication of the Lectures, he became aware of its
+imperfection as a manual of instruction, and he set himself to
+prepare a second book on the model of Euclid's \emph{Elements}. He
+estimated that it would fill 400 pages and take two years to
+prepare; it amounted to nearly 800 closely printed pages and took
+seven years. At times he would work for twelve hours on a stretch;
+and he also suffered from anxiety as to the means of publication.
+Trinity College advanced \pounds200, he paid \pounds50 out of his
+own pocket, but when illness came upon him the expense of paper
+and printing had mounted up to \pounds400. He was seized by an
+acute attack of gout, from which, after several months of
+suffering, he died on Sept.\ 2, 1865, in the 61st year of his age.
+
+It is pleasant to know that this great mathematician received
+during his last illness an honor from the United States, which
+made him feel that he had realized the aim of his great labors.
+While the war between the North and South was in progress, the
+National Academy of Sciences was founded, and the news which came
+to Hamilton was that he had been elected one of ten foreign
+members, and that his name had been voted to occupy the specially
+honorable position of first on the list. Sir William Rowan
+Hamilton was thus the first foreign associate of the National
+Academy of Sciences of the United States.
+
+As regards religion Hamilton was deeply reverential in nature. He
+was born and brought up in the Church of England, which was then
+the established Church in Ireland. He lived in the time of the
+Oxford movement, and for some time he sympathized with it; but
+when several of his friends, among them the brother of Miss
+De~Vere, passed over into the Roman Catholic Church, he modified
+his opinion of the movement and remained Protestant to the end.
+
+The immense intellectual activity of Hamilton, especially during
+the years when he was engaged on the enormous labor of writing the
+\emph{Elements of Quaternions}, made him a recluse, and
+necessarily took away from his power of attending to the practical
+affairs of life. Some said that however great a master of pure
+time he might be he was not a master of sublunary time. His
+neighbors also took advantage of his goodness of heart.
+Surrounding the house there is an extensive lawn affording good
+pasture, and on it Hamilton pastured a cow. A neighbor advised
+Hamilton that his cow would be much better contented by having
+another cow for company and bargained with Hamilton to furnish the
+companion provided Hamilton paid something like a dollar per
+month.
+
+Here is Hamilton's own estimate of himself. ``I have very long
+admired Ptolemy's description of his great astronomical master,
+Hipparchus, as \selectlanguage{greek}>an'hr fil'oponos ka`i
+filal'hjhc\selectlanguage{english}; a labor-loving and
+truth-loving man. Be such my epitaph.''
+
+Hamilton's family consisted of two sons and one daughter. At the
+time of his death, the \emph{Elements of Quaternions} was all
+finished excepting one chapter. His eldest son, William Edwin
+Hamilton, wrote a preface, and the volume was published at the
+expense of Trinity College, Dublin. Only 500 copies were printed,
+and many of those were presented. In consequence it soon became a
+scarce book, and as much as \$35.00 has been paid for a copy. A
+new edition, in two volumes, is now being published by Prof.\
+Joly, his successor in Dunsink Observatory.
+
+\chapter [George Boole (1815-1864)]{GEORGE
+BOOLE\footnote{This Lecture was delivered April 19,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1815-1864)}\end{center}\normalsize
+
+George Boole was born at Lincoln, England, on the 2d of November,
+1815. His father, a tradesman of very limited means, was attached
+to the pursuit of science, particularly of mathematics, and was
+skilled in the construction of optical instruments. Boole received
+his elementary education at the National School of the city, and
+afterwards at a commercial school; but it was his father who
+instructed him in the elements of mathematics, and also gave him a
+taste for the construction and adaptation of optical instruments.
+However, his early ambition did not urge him to the further
+prosecution of mathematical studies, but rather to becoming
+proficient in the ancient classical languages. In this direction
+he could receive no help from his father, but to a friendly
+bookseller of the neighborhood he was indebted for instruction in
+the rudiments of the Latin Grammar. To the study of Latin he soon
+added that of Greek without any external assistance; and for some
+years he perused every Greek or Latin author that came within his
+reach. At the early age of twelve his proficiency in Latin made
+him the occasion of a literary controversy in his native city. He
+produced a metrical translation of an ode of Horace, which his
+father in the pride of his heart inserted in a local journal,
+stating the age of the translator. A neighboring school-master
+wrote a letter to the journal in which he denied, from internal
+evidence, that the version could have been the work of one so
+young. In his early thirst for knowledge of languages and ambition
+to excel in verse he was like Hamilton, but poor Boole was much
+more heavily oppressed by the \emph{res angusta domi}---the hard
+conditions of his home. Accident discovered to him certain defects
+in his methods of classical study, inseparable from the want of
+proper early training, and it cost him two years of incessant
+labor to correct them.
+
+Between the ages of sixteen and twenty he taught school as an
+assistant teacher, first at Doncaster in Yorkshire, afterwards at
+Waddington near Lincoln; and the leisure of these years he devoted
+mainly to the study of the principal modern languages, and of
+patristic literature with the view of studying to take orders in
+the Church. This design, however, was not carried out, owing to
+the financial circumstances of his parents and some other
+difficulties. In his twentieth year he decided on opening a school
+on his own account in his native city; thenceforth he devoted all
+the leisure he could command to the study of the higher
+mathematics, and solely with the aid of such books as he could
+procure. Without other assistance or guide he worked his way
+onward, and it was his own opinion that he had lost five years of
+educational progress by his imperfect methods of study, and the
+want of a helping hand to get him over difficulties. No doubt it
+cost him much time; but when he had finished studying he was
+already not only learned but an experienced investigator.
+
+We have seen that at this time (1835) the great masters of
+mathematical analysis wrote in the French language; and Boole was
+naturally led to the study of the \emph{M\'ecanique celeste} of
+Laplace, and the \emph{M\'ecanique analytique} of Lagrange. While
+studying the latter work he made notes from which there eventually
+emerged his first mathematical memoir, entitled, ``On certain
+theorems in the calculus of variations.'' By the same works his
+attention was attracted to the transformation of homogeneous
+functions by linear substitutions, and in the course of his
+subsequent investigations he was led to results which are now
+regarded as the foundation of the modern Higher Algebra. In the
+publication of his results he received friendly assistance from
+D.~F.\ Gregory, a younger member of the Cambridge school, and
+editor of the newly founded \emph{Cambridge Mathematical Journal}.
+Gregory and other friends suggested that Boole should take the
+regular mathematical course at Cambridge, but this he was unable
+to do; he continued to teach school for his own support and that
+of his aged parents, and to cultivate mathematical analysis in the
+leisure left by a laborious occupation.
+
+Duncan F.\ Gregory was one of a Scottish family already
+distinguished in the annals of science. His grandfather was James
+Gregory, the inventor of the refracting telescope and discoverer
+of a convergent series for $\pi$. A cousin of his father was David
+Gregory, a special friend and fellow worker of Sir Isaac Newton.
+D.~F.\ Gregory graduated at Cambridge, and after graduation he
+immediately turned his attention to the logical foundations of
+analysis. He had before him Peacock's theory of algebra, and he
+knew that in the analysis as developed by the French school there
+were many remarkable phenomena awaiting explanation; particularly
+theorems which involved what was called the separation of symbols.
+He embodied his results in a paper ``On the real Nature of
+symbolical Algebra'' which was printed in the \emph{Transactions}
+of the Royal Society of Edinburgh.
+
+Boole became a master of the method of separation of symbols, and
+by attempting to apply it to the solution of differential
+equations with variable coefficients was led to devise a general
+method in analysis. The account of it was printed in the
+\emph{Transactions} of the Royal Society of London, and brought
+its author a Royal medal. Boole's study of the separation of
+symbols naturally led him to a study of the foundations of
+analysis, and he had before him the writings of Peacock, Gregory
+and De~Morgan. He was led to entertain very wide views of the
+domain of mathematical analysis; in fact that it was coextensive
+with exact analysis, and so embraced formal logic. In 1848, as we
+have seen, the controversy arose between Hamilton and De~Morgan
+about the quantification of terms; the general interest which that
+controversy awoke in the relation of mathematics to logic induced
+Boole to prepare for publication his views on the subject, which
+he did that same year in a small volume entitled
+\emph{Mathematical Analysis of Logic}.
+
+About this time what are denominated the Queen's Colleges of
+Ireland were instituted at Belfast, Cork and Galway; and in 1849
+Boole was appointed to the chair of mathematics in the Queen's
+College at Cork. In this more suitable environment he set himself
+to the preparation of a more elaborate work on the mathematical
+analysis of logic. For this purpose he read extensively books on
+psychology and logic, and as a result published in 1854 the work
+on which his fame chiefly rests---``An Investigation of the Laws
+of Thought, on which are founded the mathematical theories of
+logic and probabilities.'' Subsequently he prepared textbooks on
+\emph{Differential Equations} and \emph{Finite Differences}; the
+former of which remained the best English textbook on its subject
+until the publication of Forsyth's \emph{Differential Equations}.
+
+Prefixed to the \emph{Laws of Thought} is a dedication to Dr.\
+Ryall, Vice-President and Professor of Greek in the same College.
+In the following year, perhaps as a result of the dedication, he
+married Miss Everest, the niece of that colleague. Honors came:
+Dublin University made him an LL.D., Oxford a D.C.L.; and the
+Royal Society of London elected him a Fellow. But Boole's career
+was cut short in the midst of his usefulness and scientific
+labors. One day in 1864 he walked from his residence to the
+College, a distance of two miles, in a drenching rain, and
+lectured in wet clothes. The result was a feverish cold which soon
+fell upon his lungs and terminated his career on December 8, 1864,
+in the 50th year of his age.
+
+De~Morgan was the man best qualified to judge of the value of
+Boole's work in the field of logic; and he gave it generous praise
+and help. In writing to the Dublin Hamilton he said, ``I shall be
+glad to see his work (\emph{Laws of Thought}) out, for he has, I
+think, got hold of the true connection of algebra and logic.'' At
+another time he wrote to the same as follows: ``All metaphysicians
+except you and I and Boole consider mathematics as four books of
+Euclid and algebra up to quadratic equations.'' We might infer
+that these three contemporary mathematicians who were likewise
+philosophers would form a triangle of friends. But it was not so;
+Hamilton was a friend of De~Morgan, and De~Morgan a friend of
+Boole; but the relation of \emph{friend}, although convertible, is
+not necessarily transitive. Hamilton met De~Morgan only once in
+his life, Boole on the other hand with comparative frequency; yet
+he had a voluminous correspondence with the former extending over
+20 years, but almost no correspondence with the latter.
+De~Morgan's investigations of double algebra and triple algebra
+prepared him to appreciate the quaternions, whereas Boole was too
+much given over to the symbolic theory to appreciate geometric
+algebra.
+
+Hamilton's biography has appeared in three volumes, prepared by
+his friend Rev.\ Charles Graves; De~Morgan's biography has
+appeared in one volume, prepared by his widow; of Boole no
+biography has appeared. A biographical notice of Boole was written
+for the \emph{Proceedings} of the Royal Society of London by his
+friend the Rev.\ Robert Harley, and it is to it that I am indebted
+for most of my biographical data. Last summer when in England I
+learned that the reason why no adequate biography of Boole had
+appeared was the unfortunate temper and lack of sound judgment of
+his widow. Since her husband's death Mrs.\ Boole has published a
+paradoxical book of the false kind worthy of a notice in
+De~Morgan's \emph{Budget}.
+
+The work done by Boole in applying mathematical analysis to logic
+necessarily led him to consider the general question of how
+reasoning is accomplished by means of symbols. The view which he
+adopted on this point is stated at page 68 of the \emph{Laws of
+Thought}. ``The conditions of valid reasoning by the aid of
+symbols, are: \emph{First}, that a fixed interpretation be
+assigned to the symbols employed in the expression of the data;
+and that the laws of the combination of those symbols be correctly
+determined from that interpretation; \emph{Second}, that the
+formal processes of solution or demonstration be conducted
+throughout in obedience to all the laws determined as above,
+without regard to the question of the interpretability of the
+particular results obtained; \emph{Third}, that the final result
+be interpretable in form, and that it be actually interpreted in
+accordance with that system of interpretation which has been
+employed in the expression of the data.'' As regards these
+conditions it may be observed that they are very different from
+the formalist view of Peacock and De~Morgan, and that they incline
+towards a realistic view of analysis, as held by Hamilton. True he
+speaks of interpretation instead of meaning, but it is a fixed
+interpretation; and the rules for the processes of solution are
+not to be chosen arbitrarily, but are to be found out from the
+particular system of interpretation of the symbols.
+
+It is Boole's second condition which chiefly calls for study and
+examination; respecting it he observes as follows: ``The principle
+in question may be considered as resting upon a general law of the
+mind, the knowledge of which is not given to us \emph{a priori},
+that is, antecedently to experience, but is derived, like the
+knowledge of the other laws of the mind, from the clear
+manifestation of the general principle in the particular instance.
+A single example of reasoning, in which symbols are employed in
+obedience to laws founded upon their interpretation, but without
+any sustained reference to that interpretation, the chain of
+demonstration conducting us through intermediate steps which are
+not interpretable to a final result which is interpretable, seems
+not only to establish the validity of the particular application,
+but to make known to us the general law manifested therein. No
+accumulation of instances can properly add weight to such
+evidence. It may furnish us with clearer conceptions of that
+common element of truth upon which the application of the
+principle depends, and so prepare the way for its reception. It
+may, where the immediate force of the evidence is not felt, serve
+as a verification, \emph{a posteriori}, of the practical validity
+of the principle in question. But this does not affect the
+position affirmed, viz., that the general principle must be seen
+in the particular instance---seen to be general in application as
+well as true in the special example. The employment of the
+uninterpretable symbol $\sqrt{-1}$ in the intermediate processes
+of trigonometry furnishes an illustration of what has been said. I
+apprehend that there is no mode of explaining that application
+which does not covertly assume the very principle in question. But
+that principle, though not, as I conceive, warranted by formal
+reasoning based upon other grounds, seems to deserve a place among
+those axiomatic truths which constitute in some sense the
+foundation of general knowledge, and which may properly be
+regarded as expressions of the mind's own laws and constitution.''
+
+We are all familiar with the fact that algebraic reasoning may be
+conducted through intermediate equations without requiring a
+sustained reference to the meaning of these equations; but it is
+paradoxical to say that these equations can, in any case, have no
+meaning or interpretation. It may not be necessary to consider
+their meaning, it may even be difficult to find their meaning, but
+that they have a meaning is a dictate of common sense. It is
+entirely paradoxical to say that, as a general process, we can
+start from equations having a meaning, and arrive at equations
+having a meaning by passing through equations which have no
+meaning. The particular instance in which Boole sees the truth of
+the paradoxical principle is the successful employment of the
+uninterpretable symbol $\sqrt{-1}$ in the intermediate processes
+of trigonometry. So soon then as this symbol is interpreted, or
+rather, so soon as its meaning is demonstrated, the evidence for
+the principle fails, and Boole's transcendental logic falls.
+
+In the algebra of quantity we start from elementary symbols
+denoting numbers, but are soon led to compound forms which do not
+reduce to numbers; so in the algebra of logic we start from
+elementary symbols denoting classes, but are soon introduced to
+compound expressions which cannot be reduced to simple classes.
+Most mathematical logicians say, Stop, we do not know what this
+combination means. Boole says, It may be meaningless, go ahead all
+the same. The design of the \emph{Laws of Thought} is stated by
+the author to be to investigate the fundamental laws of those
+operations of the mind by which reasoning is performed; to give
+expression to them in the symbolical language of a Calculus, and
+upon this foundation to establish the Science of Logic and
+construct its method; to make that method itself the basis of a
+general method for the application of the mathematical doctrine of
+Probabilities; and, finally to collect from the various elements
+of truth brought to view in the course of these inquiries some
+probable intimations concerning the nature and constitution of the
+human mind.
+
+Boole's inventory of the symbols required in the algebra of logic
+is as follows: \emph{first}, Literal symbols, as $x$, $y$, etc.,
+representing things as subjects of our conceptions; \emph{second},
+Signs of operation, as $+$, $-$, $\times$, standing for those
+operations of the mind by which the conceptions of things are
+combined or resolved so as to form new conceptions involving the
+same elements; \emph{third}, The sign of identity $=$; not
+equality merely, but identity which involves equality. The symbols
+$x$, $y$, etc., are used to denote classes; and it is one of
+Boole's maxims that substantives and adjectives alike denote
+classes. ``They may be regarded,'' he says, ``as differing only in
+this respect, that the former expresses the substantive existence
+of the individual thing or things to which it refers, the latter
+implies that existence. If we attach to the adjective the
+universally understood subject, `being' or `thing,' it becomes
+virtually a substantive, and may for all the essential purposes of
+reasoning be replaced by the substantive.'' Let us then agree to
+represent the class of individuals to which a particular name is
+applicable by a single letter as $x$. If the name is \emph{men}
+for instance, let $x$ represent \emph{all men}, or the class
+\emph{men}. Again, if an adjective, as \emph{good}, is employed as
+a term of description, let us represent by a letter, as $y$, all
+things to which the description \emph{good} is applicable, that
+is, \emph{all good things} or the class \emph{good things}. Then
+the combination $yx$ will represent \emph{good men}.
+
+Boole's symbolic logic was brought to my notice by Professor Tait,
+when I was a student in the physical laboratory of Edinburgh
+University. I studied the \emph{Laws of Thought} and I found that
+those who had written on it regarded the method as highly
+mysterious; the results wonderful, but the processes obscure. I
+reduced everything to diagram and model, and I ventured to publish
+my views on the subject in a small volume called \emph{Principles
+of the Algebra of Logic}; one of the chief points I made is the
+philological and analytical difference between the substantive and
+the adjective. What I said was that the word \emph{man} denotes a
+class, but the word \emph{white} does not; in the former a
+definite unit-object is specified, in the latter no unit-object is
+specified. We can exhibit a type of a \emph{man}, we cannot
+exhibit a type of a \emph{white}.
+
+The identification of the substantive and adjective on the one
+hand and their discrimination on the other hand, lead to different
+conceptions of what De~Morgan called the \emph{universe}. Boole's
+conception of the Universe is as follows (\emph{Laws of Thought},
+p. 42): ``In every discourse, whether of the mind conversing with
+its own thoughts, or of the individual in his intercourse with
+others, there is an assumed or expressed limit within which the
+subjects of its operation are confined. The most unfettered
+discourse is that in which the words we use are understood in the
+widest possible application, and for them the limits of discourse
+are coextensive with those of the universe itself. But more
+usually we confine ourselves to a less spacious field. Sometimes
+in discoursing of men we imply (without expressing the limitation)
+that it is of men only under certain circumstances and conditions
+that we speak, as of civilized men, or of men in the vigor of
+life, or of men under some other condition or relation. Now,
+whatever may be the extent of the field within which all the
+objects of our discourse are found, that field may properly be
+termed the universe of discourse.''
+
+Another view leads to the conception of the Universe as a
+collection of homogeneous units, which may be finite or infinite
+in number; and in a particular problem the mind considers the
+relation of identity between different groups of this collection.
+This \emph{universe} corresponds to \emph{the series of events},
+in the theory of Probability; and the characters correspond to the
+different ways in which the event may happen. The difference is
+that the Algebra of Logic considers necessary data and relations;
+while the theory of Probability considers probable data and
+relations. I will explain the elements of Boole's method on this
+theory.
+
+\begin{center}
+\includegraphics[width=25mm]{images/GBfig1.png} \\
+\footnotesize \textsc{Fig.\ 1.} \normalsize
+\end{center}
+
+The square is a collection of points: it may serve to represent
+any collection of homogeneous units, whether finite or infinite in
+number, that is, the universe of the problem. Let $x$ denote
+\emph{inside the left-hand circle}, and $y$ \emph{inside the
+right-hand circle}. $Uxy$ will denote the points inside both
+circles (Fig.\ 1). In arithmetical value $x$ may range from $1$ to
+$0$; so also $y$; while $xy$ cannot be greater than $x$ or $y$, or
+less than $0$ or $x+y-1$. This last is the principle of the
+syllogism. From the co-ordinate nature of the operations $x$ and
+$y$, it is evident that $Uxy = Uyx$; but this is a different thing
+from commuting, as Boole does, the relation of $U$ and $x$, which
+is not that of co-ordination, but of subordination of $x$ to $U$,
+and which is properly denoted by writing $U$ first.
+
+Suppose $y$ to be the same character as $x$; we will then always
+have $Uxx=Ux$; that is, an elementary selective symbol $x$ is
+always such that $x^2 = x$. These are but the symbols of ordinary
+algebra which satisfy this relation, namely $1$ and $0$; these are
+also the extreme selective symbols \emph{all} and \emph{none}. The
+law in question was considered Boole's paradox; it plays a very
+great part in the development of his method.
+
+\begin{center}
+\includegraphics[width=25mm]{images/GBfig2.png} \\
+\footnotesize \textsc{Fig.\ 2.} \normalsize
+\end{center}
+
+Let $Uxy = Uz$, where $=$ means \emph{identical with}, not
+\emph{equal to}; we may write $xy = z$, leaving the $U$ to be
+understood. It does not mean that the combination of characters
+$xy$ is identical with the character $z$; but that those points
+which have the characters $x$ and $y$ are identical with the
+points which have the character $z$ (Fig. 2). From $xy = z$, we
+derive $x = \frac{1}{y} z$; what is the meaning of this
+expression? We shall return to the question, after we have
+considered $+$ and $-$.
+
+\begin{center}
+\includegraphics[width=25mm]{images/GBfig3.png} \quad
+   \includegraphics[width=25mm]{images/GBfig4.png} \\
+\footnotesize \textsc{Fig.\ 3} \qquad \qquad \qquad
+   \textsc{Fig.\ 4} \normalsize
+\end{center}
+
+Let us now consider the expression $U(x+y)$. If the $x$ points and
+the $y$ points are outside of one another, it means the sum of the
+$x$ points and the $y$ points (Fig.\ 3). So far all are agreed. But
+suppose that the $x$ points and the $y$ points are partially
+identical (Fig.\ 4); then there arises difference of opinion. Boole
+held that the common points must be taken twice over, or in other
+words that the symbols $x$ and $y$ must be treated all the same as
+if they were independent of one another; otherwise, he held, no
+general analysis is possible. $U(x+y)$ will not in general denote
+a single class of points; it will involve in general a
+duplication.
+
+\begin{center}
+\includegraphics[width=25mm]{images/GBfig5.png} \\
+\footnotesize \textsc{Fig.\ 5.} \normalsize
+\end{center}
+
+Similarly, Boole held that the expression $U(x-y)$ does not
+involve the condition of the $Uy$ being wholly included in the
+$Ux$ (Fig.\ 5). If that condition is satisfied, $U(x-y)$ denotes a
+simple class; namely, the $Ux$'s \emph{without} the $Uy$'s. But
+when there is partial coincidence (as in Fig.\ 4), the common
+points will be cancelled, and the result will be the $Ux$'s which
+are not $y$ taken positively and the $Uy$'s which are not $x$
+taken negatively. In Boole's view $U(x-y)$ was in general an
+intermediate uninterpretable form, which might be used in
+reasoning the same way as analysts used $\sqrt{-1}$.
+
+Most of the mathematical logicians who have come after Boole are
+men who would have stuck at the impossible subtraction in ordinary
+algebra. They say virtually, ``How can you throw into a heap the
+same things twice over; and how can you take from a heap things
+that are not there.'' Their great principle is the impossibility of
+taking the pants from a Highlander. Their only conception of the
+analytical processes of addition and subtraction is throwing into
+a heap and taking out of a heap. It does not occur to them that
+the processes of algebra are \emph{ideal}, and not subject to
+gross material restrictions.
+
+If $x+y$ denotes a quality without duplication, it will satisfy
+the condition
+\begin{align*}
+                   (x+y)^2 &= x+y, \\
+               x^2+2xy+y^2 &= x+y, \\
+  \text{but } x^2 = x, y^2 &= y,   \\
+            \therefore 2xy &= 0.
+\end{align*}
+
+Similarly, if $x-y$ denote a simple quality, then
+\begin{align*}
+                 (x-y)^2 &= x-y, \\
+             x^2+y^2-2xy &= x-y, \\
+      x^2 = x, \quad y^2 &= y,   \\
+\text{therefore, } y-2xy &=-y,   \\
+            \therefore y &= xy.
+\end{align*}
+
+In other words, the $Uy$ must be included in the $Ux$ (Fig.\ 5).
+Here we have assumed that the law of signs is the same as in
+ordinary algebra, and the result comes out correct.
+
+Suppose $Uz$=$Uxy$; then $Ux=U\frac{1}{y}z$. How are the $Ux$'s
+related to the $Uy$'s, and the Uz's? From the diagram (in Fig.\ 2)
+we see that the $Ux$'s are identical with all the $Uyz$'s together
+with an indefinite portion of the $U$'s, which are neither $y$ nor
+$z$. Boole discovered a general method for finding the meaning of
+any function of elementary logical symbols, which applied to the
+above case, is as follows:
+
+When $y$ is an elementary symbol,
+\begin{align*}
+                  1&=y+(1-y). \\
+\text{Similarly } 1&=z+(1-z). \\
+       \therefore 1&=yz+y(1-z)+(1-y)z+(1-y)(1-z),
+\end{align*}
+\noindent which means that the $U$'s either have both qualities
+$y$ and $z$, or $y$ but not $z$, or $z$ but not $y$, or neither
+$y$ and $z$. Let
+\begin{equation*}
+\frac{1}{y}z = Ayz + By(1-z) + C(1-y)z + D(1-y)(1-z),
+\end{equation*}
+\noindent it is required to determine the coefficients $A$, $B$,
+$C$, $D$. Suppose $y=1$, $z=1$; then $1=A$. Suppose $y=1$, $z=0$,
+then $0=B$. Suppose $y=0$, $z=1$; then $\frac{1}{0}=C$, and $C$ is
+infinite; therefore $(1-y)z=0$; which we see to be true from the
+diagram. Suppose $y=0$, $z=0$; then $\frac{0}{0}=D$, or $D$ is
+indeterminate. Hence
+\begin{equation*}
+\frac{1}{y}z = yz+\text{an indefinite portion of }(1-y)(1-z).
+\end{equation*}
+
+\begin{center}
+*\hspace{1cm}*\hspace{1cm}*\hspace{1cm}*\hspace{1cm}*
+\end{center}
+
+Boole attached great importance to the index law $x^2=x$. He held
+that it expressed a law of thought, and formed the characteristic
+distinction of the operations of the mind in its ordinary
+discourse and reasoning, as compared with its operations when
+occupied with the general algebra of quantity. It makes possible,
+he said, the solution of a quintic or equation of higher degree,
+when the symbols are logical. He deduces from it the axiom of
+metaphysicians which is termed the principle of contradiction, and
+which affirms that it is impossible for any being to possess a
+quality, and at the same time not to possess it. Let $x$ denote an
+elementary quality applicable to the universe $U$; then $1 - x$
+denotes the absence of that quality. But if $x^{2} = x$, then $0 =
+x - x^{2}, 0 = x(1 - x)$, that is, from $Ux^{2}=Ux$ we deduce
+$Ux(1 - x) = 0$.
+
+He considers $x(1 - x) = 0$ as an expression of the principle of
+contradiction. He proceeds to remark: ``The above interpretation
+has been introduced not on account of its immediate value in the
+present system, but as an illustration of a significant fact in
+the philosophy of the intellectual powers, viz., that what has
+been commonly regarded as the fundamental axiom of metaphysics is
+but the consequence of a law of thought, mathematical in its form.
+I desire to direct attention also to the circumstance that the
+equation in which that fundamental law of thought is expressed is
+an equation of the second degree. Without speculating at all in
+this chapter upon the question whether that circumstance is
+necessary in its own nature, we may venture to assert that if it
+had not existed, the whole procedure of the understanding would
+have been different from what it is.''
+
+We have seen that De~Morgan investigated long and published much
+on mathematical logic. His logical writings are characterized by a
+display of many symbols, new alike to logic and to mathematics; in
+the words of Sir W.\ Hamilton of Edinburgh, they are ``horrent
+with mysterious spicul\ae{}.'' It was the great merit of Boole's
+work that he used the immense power of the ordinary algebraic
+notation as an exact language and proved its power for making
+ordinary language more exact. De~Morgan could well appreciate the
+magnitude of the feat, and he gave generous testimony to it as
+follows:
+
+``Boole's system of logic is but one of many proofs of genius and
+patience combined. I might legitimately have entered it among my
+\emph{paradoxes}, or things counter to general opinion: but it is
+a paradox which, like that of Copernicus, excited admiration from
+its first appearance. That the symbolic processes of algebra,
+invented as tools of numerical calculation, should be competent to
+express every act of thought, and to furnish the grammar and
+dictionary of an all-containing system of logic, would not have
+been believed until it was proved. When Hobbes, in the time of the
+Commonwealth, published his ``Computation or Logique'' he had a
+remote glimpse of some of the points which are placed in the light
+of day by Mr.\ Boole. The unity of the forms of thought in all the
+applications of reason, however remotely separated, will one day
+be matter of notoriety and common wonder: and Boole's name will be
+remembered in connection with one of the most important steps
+towards the attainment of this knowledge.''
+
+
+\chapter [Arthur Cayley (1821-1895)]{ARTHUR
+CAYLEY\footnote{This Lecture was delivered April 20,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1821-1895)}\end{center}\normalsize
+
+Arthur Cayley was born at Richmond in Surrey, England, on August
+16, 1821. His father, Henry Cayley, was descended from an ancient
+Yorkshire family, but had settled in St.\ Petersburg, Russia, as a
+merchant. His mother was Maria Antonia Doughty, a daughter of
+William Doughty; who, according to some writers, was a Russian;
+but her father's name indicates an English origin. Arthur spent
+the first eight years of his life in St.\ Petersburg. In 1829 his
+parents took up their permanent abode at Blackheath, near London;
+and Arthur was sent to a private school. He early showed great
+liking for, and aptitude in, numerical calculations. At the age of
+14 he was sent to King's College School, London; the master of
+which, having observed indications of mathematical genius, advised
+the father to educate his son, not for his own business, as he had
+at first intended, but to enter the University of Cambridge.
+
+At the unusually early age of 17 Cayley began residence at Trinity
+College, Cambridge. As an undergraduate he had generally the
+reputation of being a mere mathematician; his chief diversion was
+novel-reading. He was also fond of travelling and mountain
+climbing, and was a member of the Alpine Club. The cause of the
+Analytical Society had now triumphed, and the \emph{Cambridge
+Mathematical Journal} had been instituted by Gregory and Leslie
+Ellis. To this journal, at the age of twenty, Cayley contributed
+three papers, on subjects which had been suggested by reading the
+\emph{M\'ecanique analytique} of Lagrange and some of the works of
+Laplace. We have already noticed how the works of Lagrange and
+Laplace served to start investigation in Hamilton and Boole.
+Cayley finished his undergraduate course by winning the place of
+Senior Wrangler, and the first Smith's prize. His next step was to
+take the M.A.\ degree, and win a Fellowship by competitive
+examination. He continued to reside at Cambridge for four years;
+during which time he took some pupils, but his main work was the
+preparation of 28 memoirs to the \emph{Mathematical Journal}. On
+account of the limited tenure of his fellowship it was necessary
+to choose a profession; like De~Morgan, Cayley chose the law, and
+at 25 entered at Lincoln's Inn, London. He made a specialty of
+conveyancing and became very skilled at the work; but he regarded
+his legal occupation mainly as the means of providing a
+livelihood, and he reserved with jealous care a due portion of his
+time for mathematical research. It was while he was a pupil at the
+bar that he went over to Dublin for the express purpose of hearing
+Hamilton's lectures on Quaternions. He sat alongside of Salmon
+(now provost of Trinity College, Dublin) and the readers of
+Salmon's books on Analytical Geometry know how much their author
+was indebted to his correspondence with Cayley in the matter of
+bringing his textbooks up to date. His friend Sylvester, his
+senior by five years at Cambridge, was then an actuary, resident
+in London; they used to walk together round the courts of
+Lincoln's Inn, discussing the theory of invariants and covariants.
+During this period of his life, extending over fourteen years,
+Cayley produced between two and three hundred papers.
+
+At Cambridge University the ancient professorship of pure
+mathematics is denominated the Lucasian, and is the chair which
+was occupied by Sir Isaac Newton. About 1860 certain funds
+bequeathed by Lady Sadleir to the University, having become
+useless for their original purpose, were employed to establish
+another professorship of pure mathematicas, called the Sadlerian.
+The duties of the new professor were defined to be ``to explain
+and teach the principles of pure mathematics and to apply himself
+to the advancement of that science.'' To this chair Cayley was
+elected when 42 years old. He gave up a lucrative practice for a
+modest salary; but he never regretted the exchange, for the chair
+at Cambridge enabled him to end the divided allegiance between law
+and mathematics, and to devote his energies to the pursuit which
+he liked best. He at once married and settled down in Cambridge.
+More fortunate than Hamilton in his choice, his home life was one
+of great happiness. His friend and fellow investigator, Sylvester,
+once remarked that Cayley had been much more fortunate than
+himself; that they both lived as bachelors in London, but that
+Cayley had married and settled down to a quiet and peaceful life
+at Cambridge; whereas he had never married, and had been fighting
+the world all his days. The remark was only too true (as may be
+seen in the lecture on Sylvester).
+
+At first the teaching duty of the Sadlerian professorship was
+limited to a course of lectures extending over one of the terms of
+the academic year; but when the University was reformed about
+1886, and part of the college funds applied to the better
+endowment of the University professors, the lectures were extended
+over two terms. For many years the attendance was small, and came
+almost entirely from those who had finished their career of
+preparation for competitive examinations; after the reform the
+attendance numbered about fifteen. The subject lectured on was
+generally that of the memoir on which the professor was for the
+time engaged.
+
+The other duty of the chair---the advancement of mathematical
+science was---discharged in a handsome manner by the long series
+of memoirs which he published, ranging over every department of
+pure mathematics. But it was also discharged in a much less
+obtrusive way; he became the standing referee on the merits of
+mathematical papers to many societies both at home and abroad.
+Many mathematicians, of whom Sylvester was an example, find it
+irksome to study what others have written, unless, perchance, it
+is something dealing directly with their own line of work. Cayley
+was a man of more cosmopolitan spirit; he had a friendly sympathy
+with other workers, and especially with young men making their
+first adventure in the field of mathematical research. Of referee
+work he did an immense amount; and of his kindliness to young
+investigators I can speak from personal experience. Several papers
+which I read before the Royal Society of Edinburgh on the Analysis
+of Relationships were referred to him, and he recommended their
+publication. Soon after I was invited by the Anthropological
+Society of London to address them on the subject, and while there,
+I attended a meeting of the Mathematical Society of London. The
+room was small, and some twelve mathematicians were assembled
+round a table, among whom was Prof.\ Cayley, as became evident to
+me from the proceedings. At the close of the meeting Cayley gave
+me a cordial handshake and referred in the kindest terms to my
+papers which he had read. He was then about 60 years old,
+considerably bent, and not filling his clothes. What was most
+remarkable about him was the active glance of his gray eyes and
+his peculiar boyish smile.
+
+In 1876 he published a \emph{Treatise on Elliptic Functions},
+which was his only book. He took great interest in the movement
+for the University education of women. At Cambridge the women's
+colleges are Girton and Newnham. In the early days of Girton
+College he gave direct help in teaching, and for some years he was
+chairman of the council of Newnham College, in the progress of
+which he took the keenest interest to the last. His mathematical
+investigations did not make him a recluse; on the contrary he was
+of great practical usefulness, especially from his knowledge of
+law, in the administration of the University.
+
+In 1872 he was made an honorary fellow of Trinity College, and
+three years later an ordinary fellow, which meant stipend as well
+as honor. About this time his friends subscribed for a
+presentation portrait, which now hangs on the side wall of the
+dining hall of Trinity College, next to the portrait of James
+Clerk Maxwell, while on the end wall, behind the high table, hang
+the more ancient portraits of Sir Isaac Newton and Lord Bacon of
+Verulam. In the portrait Cayley is represented as seated at a
+desk, quill in hand, after the mode in which he used to write out
+his mathematical investigations. The investigation, however, was
+all thought out in his mind before he took up the quill.
+
+Maxwell was one of the greatest electricians of the nineteenth
+century. He was a man of philosophical insight and poetical power,
+not unlike Hamilton, but differing in this, that he was no orator.
+In that respect he was more like Goldsmith, who ``could write like
+an angel, but only talked like poor poll.'' Maxwell wrote an
+address to the committee of subscribers who had charge of the
+Cayley portrait fund, wherein the scientific poet with his pen
+does greater honor to the mathematician than the artist, named
+Dickenson, could do with his brush. Cayley had written on space of
+\emph{n} dimensions, and the main point in the address is derived
+from the artist's business of depicting on a plane what exists in
+space:
+
+\begin{verse}
+O wretched race of men, to space confined! \\
+What honor can ye pay to him whose mind \\
+To that which lies beyond hath penetrated? \\
+The symbols he hath formed shall sound his praise, \\
+And lead him on through unimagined ways \\
+To conquests new, in worlds not yet created.
+
+First, ye Determinants, in ordered row \\
+And massive column ranged, before him go, \\
+To form a phalanx for his safe protection. \\
+Ye powers of the $n$th root of $-1$! \\
+Around his head in endless cycles run, \\
+As unembodied spirits of direction.
+
+And you, ye undevelopable scrolls! \\
+Above the host where your emblazoned rolls, \\
+Ruled for the record of his bright inventions. \\
+Ye cubic surfaces! by threes and nines \\
+Draw round his camp your seven and twenty lines \\
+The seal of Solomon in three dimensions.
+
+March on, symbolic host! with step sublime, \\
+Up to the flaming bounds of Space and Time! \\
+There pause, until by Dickenson depicted \\
+In two dimensions, we the form may trace \\
+Of him whose soul, too large for vulgar space, \\
+In $n$ dimensions flourished unrestricted.
+\end{verse}
+
+The verses refer to the subjects investigated in several of
+Cayley's most elaborate memoirs; such as, Chapters on the
+Analytical Geometry of \emph{n} dimensions; On the theory of
+Determinants; Memoir on the theory of Matrices; Memoirs on skew
+surfaces, otherwise Scrolls; On the delineation of a Cubic Scroll,
+etc.
+
+In 1881 he received from the Johns Hopkins University, Baltimore,
+where Sylvester was then professor of mathematics, an invitation
+to deliver a course of lectures. He accepted the invitation, and
+lectured at Baltimore during the first five months of 1882 on the
+subject of the \emph{Abelian and Theta Functions}.
+
+The next year Cayley came prominently before the world, as
+President of the British Association for the Advancement of
+Science. The meeting was held at Southport, in the north of
+England. As the President's address is one of the great popular
+events of the meeting, and brings out an audience of general
+culture, it is usually made as little technical as possible.
+Hamilton was the kind of mathematician to suit such an occasion,
+but he never got the office, on account of his occasional breaks.
+Cayley had not the oratorical, the philosophical, or the poetical
+gifts of Hamilton, but then he was an eminently safe man. He took
+for his subject the Progress of Pure Mathematics; and he opened
+his address in the following na\"{\i}ve manner: ``I wish to speak
+to you to-night upon Mathematics. I am quite aware of the
+difficulty arising from the abstract nature of my subject; and if,
+as I fear, many or some of you, recalling the providential
+addresses at former meetings, should wish that you were now about
+to have from a different President a discourse on a different
+subject, I can very well sympathize with you in the feeling. But
+be that as it may, I think it is more respectful to you that I
+should speak to you upon and do my best to interest you in the
+subject which has occupied me, and in which I am myself most
+interested. And in another point of view, I think it is right that
+the address of a president should be on his own subject, and that
+different subjects should be thus brought in turn before the
+meetings. So much the worse, it may be, for a particular meeting:
+but the meeting is the individual, which on evolution principles,
+must be sacrificed for the development of the race.'' I daresay
+that after this introduction, all the evolution philosophers
+listened to him attentively, whether they understood him or not.
+But Cayley doubtless felt that he was addressing not only the
+popular audience then and there before him, but the mathematicians
+of distant places and future times; for the address is a valuable
+historical review of various mathematical theories, and is
+characterized by freshness, independence of view, suggestiveness,
+and learning.
+
+In 1889 the Cambridge University Press requested him to prepare
+his mathematical papers for publication in a collected form---a
+request which he appreciated very much. They are printed in
+magnificent quarto volumes, of which seven appeared under his own
+editorship. While editing these volumes, he was suffering from a
+painful internal malady, to which he succumbed on January 26,
+1895, in the 74th year of his age. When the funeral took place, a
+great assemblage met in Trinity Chapel, comprising members of the
+University, official representatives of Russia and America, and
+many of the most illustrious philosophers of Great Britain.
+
+The remainder of his papers were edited by Prof.\ Forsyth, his
+successor in the Sadlerian chair. The Collected Mathematical
+papers number thirteen quarto volumes, and contain 967 papers. His
+writings are his best monument, and certainly no mathematician has
+ever had his monument in grander style. De~Morgan's works would be
+more extensive, and much more useful, but he did not have behind
+him a University Press. As regards fads, Cayley retained to the
+last his fondness for novel-reading and for travelling. He also
+took special pleasure in paintings and architecture, and he
+practised water-color painting, which he found useful sometimes in
+making mathematical diagrams.
+
+To the third edition of Tait's \emph{Elementary Treatise on
+Quaternions}, Cayley contributed a chapter entitled ``Sketch of
+the analytical theory of quaternions.'' In it the $\sqrt{-1}$
+reappears in all its glory, and in entire, so it is said,
+independence of $i$, $j$, $k$. The remarkable thing is that
+Hamilton started with a quaternion theory of analysis, and that
+Cayley should present instead an analytical theory of quaternions.
+I daresay that Prof.\ Tait was sorry that he allowed the chapter
+to enter his book, for in 1894 there arose a brisk discussion
+between himself and Cayley on ``Coordinates versus Quaternions,''
+the record of which is printed in the Proceedings of the Royal
+Society of Edinburgh. Cayley maintained the position that while
+coordinates are applicable to the whole science of geometry and
+are the natural and appropriate basis and method in the science,
+quaternions seemed a particular and very artificial method for
+treating such parts of the science of three-dimensional geometry
+as are most naturally discussed by means of the rectangular
+coordinates $x$, $y$, $z$. In the course of his paper Cayley says:
+``I have the highest admiration for the notion of a quaternion;
+but, as I consider the full moon far more beautiful than any
+moonlit view, so I regard the notion of a quaternion as far more
+beautiful than any of its applications. As another illustration, I
+compare a quaternion formula to a pocket-map---a capital thing to
+put in one's pocket, but which for use must be unfolded: the
+formula, to be understood, must be translated into coordinates.''
+He goes on to say, ``I remark that the imaginary of ordinary
+algebra---for distinction call this $\theta$---has no relation
+whatever to the quaternion symbols $i$, $j$, $k$; in fact, in the
+general point of view, all the quantities which present
+themselves, are, or may be, complex values $a + \theta b$, or in
+other words, say that a scalar quantity is in general of the form
+$a + \theta b$. Thus quaternions do not properly present
+themselves in plane or two-dimensional geometry at all; but they
+belong essentially to solid or three-dimensional geometry, and
+they are most naturally applicable to the class of problems which
+in coordinates are dealt with by means of the three rectangular
+coordinates $x$, $y$, $z$."
+
+To the pocketbook illustration it may be replied that a set of
+coordinates is an immense wall map, which you cannot carry about,
+even though you should roll it up, and therefore is useless for
+many important purposes. In reply to the arguments, it may be
+said, \emph{first}, $\sqrt{-1}$ has a relation to the symbols $i$,
+$j$, $k$, for each of these can be analyzed into a unit axis
+multiplied by $\sqrt{-1}$; \emph{second}, as regards plane
+geometry, the ordinary form of complex quantity is a degraded form
+of the quaternion in which the constant axis of the plane is left
+unspecified. Cayley took his illustrations from his experience as
+a traveller. Tait brought forward an illustration from which you
+might imagine he had visited the Bethlehem Iron Works, and hunted
+tigers in India. He says, ``A much more natural and adequate
+comparison would, it seems to me, liken Coordinate Geometry to a
+steam-hammer, which an expert may employ on any destructive or
+constructive work of one general kind, say the cracking of an
+eggshell, or the welding of an anchor. But you must have your
+expert to manage it, for without him it is useless. He has to toil
+amid the heat, smoke, grime, grease, and perpetual din of the
+suffocating engine-room. The work has to be brought to the hammer,
+for it cannot usually be taken to its work. And it is not in
+general, transferable; for each expert, as a rule, knows, fully
+and confidently, the working details of his own weapon only.
+Quaternions, on the other hand, are like the elephant's trunk,
+ready at \emph{any} moment for \emph{anything}, be it to pick up a
+crumb or a field-gun, to strangle a tiger, or uproot a tree;
+portable in the extreme, applicable anywhere---alike in the
+trackless jungle and in the barrack square---directed by a little
+native who requires no special skill or training, and who can be
+transferred from one elephant to another without much hesitation.
+Surely this, which adapts itself to its work, is the grander
+instrument. But then, \emph{it} is the natural, the other, the
+artificial one.''
+
+The reply which Tait makes, so far as it is an argument, is: There
+are two systems of quaternions, the $i$, $j$, $k$ one, and another
+one which Hamilton developed from it; Cayley knows the first only,
+he himself knows the second; the former is an intensely artificial
+system of imaginaries, the latter is the natural organ of
+expression for quantities in space. Should a fourth edition of his
+\emph{Elementary Treatise} be called for $i$, $j$, $k$ will
+disappear from it, excepting in Cayley's chapter, should it be
+retained. Tait thus describes the first system: ``Hamilton's
+extraordinary \emph{Preface} to his first great book shows how
+from Double Algebras, through Triplets, Triads, and Sets, he
+finally reached Quaternions. This was the genesis of the
+Quaternions of the forties, and the creature thus produced is
+still essentially the Quaternion of Prof.\ Cayley. It is a
+magnificent analytical conception; but it is nothing more than the
+full development of the system of imaginaries $i$, $j$, $k$;
+defined by the equations,
+\begin{equation*}
+i^{2} = j^{2} = k^{2} = ijk = -1,
+\end{equation*}
+\noindent with the associative, but \emph{not} the commutative,
+law for the factors. The novel and splendid points in it were the
+treatment of all directions in space as essentially alike in
+character, and the recognition of the unit vector's claim to rank
+also as a quadrantal versor. These were indeed inventions of the
+first magnitude, and of vast importance. And here I thoroughly
+agree with Prof.\ Cayley in his admiration. Considered as an
+analytical system, based throughout on pure imaginaries, the
+Quaternion method is elegant in the extreme. But, unless it had
+been also something more, something very different and much higher
+in the scale of development, I should have been content to admire
+it;---and to pass it by.''
+
+From ``the most intensely artificial of systems, arose, as if by
+magic, an absolutely natural one'' which Tait thus further
+describes. ``To me Quaternions are primarily a Mode of
+Representation:---immensely superior to, but of essentially the
+same kind of usefulness as, a diagram or a model. They are,
+virtually, the thing represented; and are thus antecedent to, and
+independent of, coordinates; giving, in general, all the main
+relations, in the problem to which they are applied, without the
+necessity of appealing to coordinates at all. Coordinates may,
+however, easily be read into them:---when anything (such as
+metrical or numerical detail) is to be gained thereby.
+Quaternions, in a word, exist in space, and we have only to
+recognize them:---but we have to invent or imagine coordinates of
+all kinds.''
+
+To meet the objection why Hamilton did not throw $i$, $j$, $k$
+overboard, and expound the developed system, Tait says: ``Most
+unfortunately, alike for himself and for his grand conception,
+Hamilton's nerve failed him in the composition of his first great
+volume. Had he then renounced, for ever, all dealings with $i$,
+$j$, $k$, his triumph would have been complete. He spared Agog,
+and the best of the sheep, and did not utterly destroy them. He
+had a paternal fondness for $i$, $j$, $k$; perhaps also a not
+unnatural liking for a meretricious title such as the mysterious
+word \emph{Quaternion}; and, above all, he had an earnest desire
+to make the utmost return in his power for the liberality shown
+him by the authorities of Trinity College, Dublin. He had fully
+recognized, and proved to others, that his $i$, $j$, $k$, were
+mere excrescences and blots on his improved method:---but he
+unfortunately considered that their continued (if only partial)
+recognition was indispensable to the reception of his method by a
+world steeped in---Cartesianism! Through the whole compass of each
+of his tremendous volumes one can find traces of his desire to
+avoid even an allusion to $i$, $j$, $k$, and along with them, his
+sorrowful conviction that, should he do so, he would be left
+without a single reader.''
+
+To Cayley's presidential address we are indebted for information
+about the view which he took of the foundations of exact science,
+and the philosophy which commended itself to his mind. He quoted
+Plato and Kant with approval, J.~S.\ Mill with faint praise.
+Although he threw a sop to the empirical philosophers at the
+beginning of his address, he gave them something to think of
+before he finished.
+
+He first of all remarks that the connection of arithmetic and
+algebra with the notion of time is far less obvious than that of
+geometry with the notion of space; in which he, of course, made a
+hit at Hamilton's theory of Algebra as the science of pure time.
+Further on he discusses the theory directly, and concludes as
+follows: ``Hamilton uses the term algebra in a very wide sense,
+but whatever else he includes under it, he includes all that in
+contradistinction to the Differential Calculus would be called
+algebra. Using the word in this restricted sense, I cannot myself
+recognize the connection of algebra with the notion of time;
+granting that the notion of continuous progression presents itself
+and is of importance, I do not see that it is in anywise the
+fundamental notion of the science. And still less can I appreciate
+the manner in which the author connects with the notion of time
+his algebraical couple, or imaginary magnitude, $a+b\sqrt{-1}$.''
+So you will observe that doctors differ---Tait and Cayley---about
+the soundness of Hamilton's theory of couples. But it can be shown
+that a couple may not only be represented on a straight line, but
+actually means a portion of a straight line; and as a line is
+unidimensional, this favors the truth of Hamilton's theory.
+
+As to the nature of mathematical science Cayley quoted with
+approval from an address of Hamilton's:
+
+``These purely mathematical sciences of algebra and geometry are
+sciences of the pure reason, deriving no weight and no assistance
+from experiment, and isolated or at least isolable from all
+outward and accidental phenomena. The idea of order with its
+subordinate ideas of number and figure, we must not call innate
+ideas, if that phrase be defined to imply that all men must
+possess them with equal clearness and fulness; they are, however,
+ideas which seem to be so far born with us that the possession of
+them in any conceivable degree is only the development of our
+original powers, the unfolding of our proper humanity.''
+
+It is the aim of the evolution philosopher to reduce all knowledge
+to the empirical status; the only intuition he grants is a kind of
+instinct formed by the experience of ancestors and transmitted
+cumulatively by heredity. Cayley first takes him up on the subject
+of arithmetic: ``Whatever difficulty be raisable as to geometry,
+it seems to me that no similar difficulty applies to arithmetic;
+mathematician, or not, we have each of us, in its most abstract
+form, the idea of number; we can each of us appreciate the truth
+of a proposition in numbers; and we cannot but see that a truth in
+regard to numbers is something different in kind from an
+experimental truth generalized from experience. Compare, for
+instance, the proposition, that the sun, having already risen so
+many times, will rise to-morrow, and the next day, and the day
+after that, and so on; and the proposition that even and odd
+numbers succeed each other alternately \emph{ad infinitum}; the
+latter at least seems to have the characters of universality and
+necessity. Or again, suppose a proposition observed to hold good
+for a long series of numbers, one thousand numbers, two thousand
+numbers, as the case may be: this is not only no proof, but it is
+absolutely no evidence, that the proposition is a true
+proposition, holding good for all numbers whatever; there are in
+the Theory of Numbers very remarkable instances of propositions
+observed to hold good for very long series of numbers which are
+nevertheless untrue.''
+
+Then he takes him up on the subject of geometry, where the
+empiricist rather boasts of his success. ``It is well known that
+Euclid's twelfth axiom, even in Playfair's form of it, has been
+considered as needing demonstration; and that Lobatschewsky
+constructed a perfectly consistent theory, wherein this axiom was
+assumed not to hold good, or say a system of non-Euclidean plane
+geometry. My own view is that Euclid's twelfth axiom in Playfair's
+form of it does not need demonstration, but is part of our notion
+of space, of the physical space of our experience---the space,
+that is, which we become acquainted with by experience, but which
+is the representation lying at the foundation of all external
+experience. Riemann's view before referred to may I think be said
+to be that, having \emph{in intellectu} a more general notion of
+space (in fact a notion of non-Euclidean space), we learn by
+experience that space (the physical space of our experience) is,
+if not exactly, at least to the highest degree of approximation,
+Euclidean space. But suppose the physical space of our experience
+to be thus only approximately Euclidean space, what is the
+consequence which follows? \emph{Not} that the propositions of
+geometry are only approximately true, but that they remain
+absolutely true in regard to that Euclidean space which has been
+so long regarded as being the physical space of our experience.''
+
+In his address he remarks that the fundamental notion which
+underlies and pervades the whole of modern analysis and geometry
+is that of imaginary magnitude in analysis and of imaginary space
+(or space as a \emph{locus in quo} of imaginary points and
+figures) in geometry. In the case of two given curves there are
+two equations satisfied by the coordinates ($x$, $y$) of the
+several points of intersection, and these give rise to an equation
+of a certain order for the coordinate $x$ or $y$ of a point of
+intersection. In the case of a straight line and a circle this is
+a quadratic equation; it has two roots real or imaginary. There
+are thus two values, say of $x$, and to each of these corresponds
+a single value of $y$. There are therefore two points of
+intersection, viz., a straight line and a circle intersect always
+in two points, real or imaginary. It is in this way we are led
+analytically to the notion of imaginary points in geometry. He
+asks, What is an imaginary point? Is there in a plane a point the
+coordinates of which have given imaginary values? He seems to say
+No, and to fall back on the notion of an imaginary space as the
+\emph{locus in quo} of the imaginary point.
+
+
+\chapter [William Kingdon Clifford (1845-1879)]{WILLIAM
+KINGDON~CLIFFORD\footnote{This Lecture was delivered April 23,
+1901.---\textsc{Editors.}}}
+
+\large\begin{center}{(1845-1879)}\end{center}\normalsize
+
+William Kingdon Clifford was born at Exeter, England, May 4, 1845.
+His father was a well-known and active citizen and filled the
+honorary office of justice of the peace; his mother died while he
+was still young. It is believed that Clifford inherited from his
+mother not only some of his genius, but a weakness in his physical
+constitution. He received his elementary education at a private
+school in Exeter, where examinations were annually held by the
+Board of Local Examinations of the Universities of Oxford and
+Cambridge; at these examinations Clifford gained numerous
+distinctions in widely different subjects. When fifteen years old
+he was sent to King's College, London, where he not only
+demonstrated his peculiar mathematical abilities, but also gained
+distinction in classics and English literature.
+
+When eighteen, he entered Trinity College, Cambridge; the college
+of Peacock, De~Morgan, and Cayley. He already had the reputation
+of possessing extraordinary mathematical powers; and he was
+eccentric in appearance, habits and opinions. He was reported to
+be an ardent High Churchman, which was then a more remarkable
+thing at Cambridge than it is now. His undergraduate career was
+distinguished by eminence in mathematics, English literature and
+gymnastics. One who was his companion in gymnastics wrote: ``His
+neatness and dexterity were unusually great, but the most
+remarkable thing was his great strength as compared with his
+weight, as shown in some exercises. At one time he would pull up
+on the bar with either hand, which is well known to be one of the
+greatest feats of strength. His nerve at dangerous heights was
+extraordinary.'' In his third year he won the prize awarded by
+Trinity College for declamation, his subject being Sir Walter
+Raleigh; as a consequence he was called on to deliver the annual
+oration at the next Commemoration of Benefactors of the College.
+He chose for his subject, Dr.\ Whewell, Master of the College,
+eminent for his philosophical and scientific attainments, whose
+death had occurred but recently. He treated it in an original and
+unexpected manner; Dr.\ Whewell's claim to admiration and
+emulation being put on the ground of his intellectual life
+exemplifying in an eminent degree the active and creating faculty.
+``Thought is powerless, except it make something outside of
+itself; the thought which conquers the world is not contemplative
+but active. And it is this that I am asking you to worship
+to-day.''
+
+To obtain high honors in the Mathematical Tripos, a student must
+put himself in special training under a mathematican, technically
+called a coach, who is not one of the regular college instructors,
+nor one of the University professors, but simply makes a private
+business of training men to pass that particular examination.
+Skill consists in the rate at which one can solve and more
+especially write out the solution of problems. It is excellent
+training of a kind, but there is no time for studying fundamental
+principles, still less for making any philosophical
+investigations. Mathematical insight is something higher than
+skill in solving problems; consequently the senior wrangler has
+not always turned out the most distinguished mathematician in
+after life. We have seen that De~Morgan was fourth wrangler.
+Clifford also could not be kept to the dust of the race-course;
+but such was his innate mathematical insight that he came out
+second wrangler. Other instances of the second wrangler turning
+out the better mathematician are Whewell, Sylvester, Kelvin,
+Maxwell.
+
+In 1868, when he was 23 years old, he was elected a Fellow of his
+College; and while a resident fellow, he took part in the eclipse
+expedition of 1870 to Italy, and passed through the experience of
+a shipwreck near Catania on the coast of the island of Sicily. In
+1871 he was appointed professor of Applied Mathematics and
+Mechanics in University College, London; De~Morgan's college, but
+not De~Morgan's chair. Henceforth University College was the
+centre of his labors.
+
+He was now urged by friends to seek admission into the Royal
+Society of London. This is the ancient scientific society of
+England, founded in the time of Charles II, and numbering among
+its first presidents Sir Isaac Newton. About the middle of the
+nineteenth century the admission of new members was restricted to
+fifteen each year; and from applications the Council recommends
+fifteen names which are posted up, and subsequently balloted for
+by the Fellows. Hamilton and De~Morgan never applied. Clifford did
+not apply immediately, but he became a Fellow a few years later.
+He joined the London Mathematical Society---for it met in
+University College---and he became one of its leading spirits.
+Another metropolitan Society in which he took much interest was
+the Metaphysical Society; like Hamilton, De~Morgan, and Boole,
+Clifford was a scientific philosopher.
+
+In 1875 Clifford married; the lady was Lucy, daughter of Mr.\ John
+Lane, formerly of Barbadoes. His home in London became the
+meeting-point of a numerous body of friends, in which almost every
+possible variety of taste and opinion was represented, and many of
+whom had nothing else in common. He took a special delight in
+amusing children, and for their entertainment wrote a collection
+of fairy tales called \emph{The Little People}. In this respect he
+was like a contemporary mathematician, Mr.\ Dodgson---``Lewis
+Carroll''---the author of \emph{Alice in Wonderland}. A children's
+party was one of Clifford's greatest pleasures. At one such party
+he kept a waxwork show, children doing duty for the figures; but I
+daresay he drew the line at walking on all fours, as Mr.\ Dodgson
+was accustomed to do. A children's party was to be held in a house
+in London and it happened that there was a party of adults held
+simultaneously in the neighboring house; to give the children a
+surprise Dodgson resolved to walk in on all fours; unfortunately
+he crawled into the parlor of the wrong house!
+
+Clifford possessed unsurpassed power as a teacher. Mr.\ Pollock, a
+fellow student, gives an instance of Clifford's theory of what
+teaching ought to be, and his constant way of carrying it out in
+his discourses and conversations on mathematical and scientific
+subjects. ``In the analytical treatment of statics there occurs a
+proposition called Ivory's Theorem concerning the attractions of
+an ellipsoid. The textbooks demonstrate it by a formidable
+apparatus of coordinates and integrals, such as we were wont to
+call a \emph{grind}. On a certain day in the Long Vacation of
+1866, which Clifford and I spent at Cambridge, I was not a little
+exercised by the theorem in question, as I suppose many students
+have been before and since. The chain of symbolic proof seemed
+artificial and dead; it compelled the understanding, but failed to
+satisfy the reason. After reading and learning the proposition one
+still failed to see what it was all about. Being out for a walk
+with Clifford, I opened my perplexities to him; I think that I can
+recall the very spot. What he said I do not remember in detail;
+which is not surprising, as I have had no occasion to remember
+anything about Ivory's Theorem these twelve years. But I know that
+as he spoke he appeared not to be working out a question, but
+simply telling what he saw. Without any diagram or symbolic aid he
+described the geometrical conditions on which the solution
+depended, and they seemed to stand out visibly in space. There
+were no longer consequences to be deduced, but real and evident
+facts which only required to be seen.''
+
+Clifford inherited a constitution in which nervous energy and
+physical stren\-gth were unequally balanced. It was in his case
+specially necessary to take good care of his health, but he did
+the opposite; he would frequently sit up most of the night working
+or talking. Like Hamilton he would work twelve hours on a stretch;
+but, unlike Hamilton, he had laborious professional duties
+demanding his personal attention at the same time. The consequence
+was that five years after his appointment to the chair in
+University College, his health broke down; indications of
+pulmonary disease appeared. To recruit his health he spent six
+months in Algeria and Spain, and came back to his professional
+duties again. A year and a half later his health broke down a
+second time, and he was obliged to leave again for the shores of
+the Mediterranean. In the fall of 1878 he returned to England for
+the last time, when the winter came he left for the Island of
+Madeira; all hope of recovery was gone; he died March 3, 1879 in
+the 34th year of his age.
+
+On the title page of the volume containing his collected
+mathematical papers I find a quotation, ``If he had lived we might
+have known something.'' Such is the feeling one has when one looks
+at his published works and thinks of the shortness of his life. In
+his lifetime there appeared \emph{Elements of Dynamic, Part I}.
+Posthumously there have appeared \emph{Elements of Dynamic, Part
+II; Collected Mathematical Papers; Lectures and Essays; Seeing and
+Thinking; Common Sense of the Exact Sciences}. The manuscript of
+the last book was left in a very incomplete state, but the design
+was filled up and completed by two other mathematicians.
+
+In a former lecture I had occasion to remark on the relation of
+Mathematics to Poetry---on the fact that in mathematical
+investigation there is needed a higher power of imagination akin
+to the creative instinct of the poet. The matter is discussed by
+Clifford in a discourse on ``Some of the conditions of mental
+development,'' which he delivered at the Royal Institution in 1868
+when he was 23 years of age. This institution was founded by Count
+Rumford, an American, and is located in London. There are
+Professorships of Chemistry, Physics, and Physiology; its
+professors have included Davey, Faraday, Young, Tyndall, Rayleigh,
+Dewar. Their duties are not to teach the elements of their science
+to regular students, but to make investigations, and to lecture to
+the members of the institution, who are in general wealthy and
+titled people.
+
+In this discourse Clifford said ``Men of science have to deal with
+extremely abstract and general conceptions. By constant use and
+familiarity, these, and the relations between them, become just as
+real and external as the ordinary objects of experience, and the
+perception of new relations among them is so rapid, the
+correspondence of the mind to external circumstances so great,
+that a real scientific sense is developed, by which things are
+perceived as immediately and truly as I see you now. Poets and
+painters and musicians also are so accustomed to put outside of
+them the idea of beauty, that it becomes a real external
+existence, a thing which they see with spiritual eyes and then
+describe to you, but by no means create, any more than we seem to
+create the ideas of table and forms and light, which we put
+together long ago. There is no scientific discoverer, no poet, no
+painter, no musician, who will not tell you that he found ready
+made his discovery or poem or picture---that it came to him from
+outside, and that he did not consciously create it from within.
+And there is reason to think that these senses or insights are
+things which actually increase among mankind. It is certain, at
+least, that the scientific sense is immensely more developed now
+than it was three hundred years ago; and though it may be
+impossible to find any absolute standard of art, yet it is
+acknowledged that a number of minds which are subject to artistic
+training will tend to arrange themselves under certain great
+groups and that the members of each group will give an independent
+and yet consentient testimony about artistic questions. And this
+arrangement into schools, and the definiteness of the conclusions
+reached in each, are on the increase, so that here, it would seem,
+are actually two new senses, the scientific and the artistic,
+which the mind is now in the process of forming for itself.''
+
+Clifford himself wrote a good many poems, but only a few have been
+published. The following verses were sent to George Eliot, the
+novelist, with a presentation copy of \emph{The Little People}:
+
+\begin{verse}
+Baby drew a little house, \\
+\vin  Drew it all askew; \\
+Mother saw the crooked door \\
+\vin  And the window too.
+
+Mother heart, whose wide embrace \\
+\vin  Holds the hearts of men, \\
+Grows with all our growing hopes, \\
+\vin  Gives them birth again,
+
+Listen to this baby-talk: \\
+\vin  'Tisn't wise or clear; \\
+But what baby-sense it has \\
+\vin  Is for you to hear.
+\end{verse}
+
+An amusement in which Clifford took pleasure even in his maturer
+years was the flying of kites. He made some mathematical
+investigations in the subject, anticipating, as it were, the
+interest which has been taken in more recent years in the subject
+of motion through the atmosphere. Clifford formed a project of
+writing a series of textbooks on Mathematics beginning at the very
+commencement of each subject and carrying it on rapidly to the
+most advanced stages. He began with the \emph{Elements of
+Dynamic}, of which three books were printed in his lifetime, and a
+fourth book, in a supplementary volume, after his death. The work
+is unique for the clear ideas given of the science; ideas and
+principles are more prominent than symbols and formulae. He takes
+such familiar words as \emph{spin, twist, squirt, whirl}, and
+gives them an exact meaning. The book is an example of what he
+meant by scientific insight, and from its excellence we can
+imagine what the complete series of textbooks would have been.
+
+In Clifford's lifetime it was said in England that he was the only
+mathematician who could discourse on mathematics to an audience
+composed of people of general culture and make them think that
+they understood the subject. In 1872 he was invited to deliver an
+evening lecture before the members of the British Association, at
+Brighton; he chose for his subject ``The aims and instruments of
+scientific thought.'' The main theses of the lecture are
+\emph{First}, that scientific thought is the application of past
+experience to new circumstances by means of an observed order of
+events. \emph{Second}, this order of events is not theoretically
+or absolutely exact, but only exact enough to correct experiments
+by. As an instance of what is, and what is not scientific thought,
+he takes the phenomenon of double refraction. ``A mineralogist, by
+measuring the angles of a crystal, can tell you whether or no it
+possesses the property of double refraction without looking
+through it. He requires no scientific thought to do that. But Sir
+William Rowan Hamilton, knowing these facts and also the
+explanation of them which Fresnel had given, thought about the
+subject, and he predicted that by looking through certain crystals
+in a particular direction we should see not two dots but a
+continuous circle. Mr.\ Lloyd made the experiment, and saw the
+circle, a result which had never been even suspected. This has
+always been considered one of the most signal instances of
+scientific thought in the domain of physics. It is most distinctly
+an application of experience gained under certain circumstances to
+entirely different circumstances.''
+
+In physical science there are two kinds of law---distinguished as
+``empirical'' and ``rational.'' The former expresses a relation
+which is sufficiently true for practical purposes and within
+certain limits; for example, many of the formulas used by
+engineers. But a rational law states a connection which is
+accurately true, without any modification of limit. In the
+theorems of geometry we have examples of scientific exactness; for
+example, in the theorem that the sum of the three interior angles
+of a plane triangle is equal to two right angles. The equality is
+one not of approximation, but of exactness. Now the philosopher
+Kant pointed to such a truth and said: We know that it is true not
+merely here and now, but everywhere and for all time; such
+knowledge cannot be gained by experience; there must be some other
+source of such knowledge. His solution was that space and time are
+forms of the sensibility; that truths about them are not obtained
+by empirical induction, but by means of intuition; and that the
+characters of necessity and universality distinguished these
+truths from other truths. This philosophy was accepted by Sir
+William Rowan Hamilton, and to him it was not a barren philosophy,
+for it served as the starting point of his discoveries in algebra
+which culminated in the discovery of quaternions.
+
+This philosophy was admired but not accepted by Clifford; he was,
+so long as he lived, too strongly influenced by the philosophy
+which has been built upon the theory of evolution. He admits that
+the only way of escape from Kant's conclusions is by denying the
+theoretical exactness of the proposition referred to. He says,
+``About the beginning of the present century the foundations of
+geometry were criticised independently by two mathematicians,
+Lobatchewsky and Gauss, whose results have been extended and
+generalized more recently by Riemann and Helmholtz. And the
+conclusion to which these investigations lead is that, although
+the assumptions which were very properly made by the ancient
+geometers are practically exact---that is to say, more exact than
+experiment can be---for such finite things as we have to deal
+with, and such portions of space as we can reach; yet the truth of
+them for very much larger things, or very much smaller things, or
+parts of space which are at present beyond our reach, is a matter
+to be decided by experiment, when its powers are considerably
+increased. I want to make as clear as possible the real state of
+this question at present, because it is often supposed to be a
+question of words or metaphysics, whereas it is a very distinct
+and simple question of fact. I am supposed to know that the three
+angles of a rectilinear triangle are exactly equal to two right
+angles. Now suppose that three points are taken in space, distant
+from one another as far as the Sun is from $\alpha$ Centauri, and
+that the shortest distances between these points are drawn so as
+to form a triangle. And suppose the angles of this triangle to be
+very accurately measured and added together; this can at present
+be done so accurately that the error shall certainly be less than
+one minute, less therefore than the five-thousandth part of a
+right angle. Then I do not know that this sum would differ at all
+from two right angles; but also I do not know that the difference
+would be less than ten degrees or the ninth part of a right
+angle.''
+
+You will observe that Clifford's philosophy depends on the
+validity of Lobatchewsky's ideas. Now it has been shown by an
+Italian mathematician, named Beltrami, that the plane geometry of
+Lobatchewsky corresponds to trigonometry on a surface called the
+\emph{pseudosphere}. Clifford and other followers of Lobatchewsky
+admit Beltrami's interpretation, an interpretation which does not
+involve any paradox about geometrical space, and which leaves the
+trigonometry of the plane alone as a different thing. If that
+interpretation is true, the Lobatchewskian plane triangle is after
+all a triangle on a special surface, and the \emph{straight} lines
+joining the points are not the shortest absolutely, but only the
+\emph{shortest} with respect to the surface, whatever that may
+mean. If so, then Clifford's argument for the empirical nature of
+the proposition referred to fails; and nothing prevents us from
+falling back on Kant's position, namely, that there is a body of
+knowledge characterized by absolute exactness and possessing
+universal application in time and space; and as a particular case
+thereof we believe that the sum of the three angles of Clifford's
+gigantic triangle is precisely two right angles.
+
+Trigonometry on a spherical surface is a generalized form of plane
+trigonometry, from the theorems of the former we can deduce the
+theorems of the latter by supposing the radius of the sphere to be
+infinite. The sum of the three angles of a spherical triangle is
+greater than two right angles; the sum of the angles of a plain
+triangle is equal to two right angles; we infer that there is
+another surface, complementary to the sphere, such that the angles
+of any triangle on it are less than two right angles. The
+complementary surface to which I refer is not the pseudosphere,
+but the equilateral hyperboloid. As the plane is the transition
+surface between the sphere and the equilateral hyperboloid, and a
+triangle on it is the transition triangle between the spherical
+triangle and the equilateral hyperboloidal triangle, the sum of
+the angles of the plane triangle must be exactly equal to two
+right angles.
+
+In 1873, the British Association met at Bradford; on this occasion
+the evening discourse was delivered by Maxwell, the celebrated
+physicist. He chose for his subject ``Molecules.'' The application
+of the method of spectrum-analysis assures the physicist that he
+can find out in his laboratory truths of universal validity in
+space and time. In fact, the chief maxim of physical science,
+according to Maxwell is, that physical changes are independent of
+the conditions of space and time, and depend only on conditions of
+configuration of bodies, temperature, pressure, etc. The address
+closed with a celebrated passage in striking contrast to
+Clifford's address: ``In the heavens we discover by their light,
+and by their light alone, stars so distant from each other that no
+material thing can ever have passed from one to another; and yet
+this light, which is to us the sole evidence of the existence of
+these distant worlds, tells us also that each of them is built up
+of molecules of the same kinds as those which are found on earth.
+A molecule of hydrogen, for example, whether in Sirius or in
+Arcturus, executes its vibrations in precisely the same time. No
+theory of evolution can be formed to account for the similarity of
+molecules, for evolution necessarily implies continuous change,
+and the molecule is incapable of growth or decay, of generation or
+destruction. None of the processes of Nature since the time when
+Nature began, have produced the slightest difference in the
+properties of any molecule. We are therefore unable to ascribe
+either the existence of the molecules or the identity of their
+properties to any of the causes which we call natural. On the
+other hand, the exact equality of each molecule to all others of
+the same kind gives it, as Sir John Herschel has well said, the
+essential character of a manufactured article, and precludes the
+idea of its being eternal and self-existent.''
+
+What reply could Clifford make to this? In a discourse on the
+``First and last catastrophe'' delivered soon afterwards, he said
+``If anyone not possessing the great authority of Maxwell, had put
+forward an argument, founded upon a scientific basis, in which
+there occurred assumptions about what things can and what things
+cannot have existed from eternity, and about the exact similarity
+of two or more things established by experiment, we would say:
+`Past eternity; absolute exactness; won't do'; and we should pass
+on to another book. The experience of all scientific culture for
+all ages during which it has been a light to men has shown us that
+we never do get at any conclusions of that sort. We do not get at
+conclusions about infinite time, or infinite exactness. We get at
+conclusions which are as nearly true as experiment can show, and
+sometimes which are a great deal more correct than direct
+experiment can be, so that we are able actually to correct one
+experiment by deductions from another, but we never get at
+conclusions which we have a right to say are absolutely exact.''
+
+Clifford had not faith in the exactness of mathematical science
+nor faith in that maxim of physical science which has built up the
+new astronomy, and extended all the bounds of physical science.
+Faith in an exact order of Nature was the characteristic of
+Faraday, and he was by unanimous consent the greatest electrician
+of the nineteenth century. What is the general direction of
+progress in science? Physics is becoming more and more
+mathematical; chemistry is becoming more and more physical, and I
+daresay the biological sciences are moving in the same direction.
+They are all moving towards exactness; consequently a true
+philosophy of science will depend on the principles of mathematics
+much more than upon the phenomena of biology. Clifford, I believe,
+had he lived longer, would have changed his philosophy for a more
+mathematical one. In 1874 there appeared in \emph{Nature} among
+the letters from correspondents one to the following effect:
+
+An anagram: The practice of enclosing discoveries in sealed
+packets and sending them to Academies seems so inferior to the old
+one of Huyghens, that the following is sent you for publication in
+the old conservated form:
+\begin{displaymath}
+A^{8}C^{3}DE^{12}F^{4}GH^{6}J^{6}L^{3}M^{3}N^{5}O^{6}PR^{4}S^{5}T^{14}U^{6}V^{2
+}WXY^{2}.
+\end{displaymath}
+
+This anagram was explained in a book entitled \emph{The Unseen
+Universe}, which was published anonymously in 1875; and is there
+translated, ``Thought conceived to affect the matter of another
+universe simultaneously with this may explain a future state.''
+The book was evidently a work of a physicist or physicists, and as
+physicists were not so numerous then as they are now, it was not
+difficult to determine the authorship from internal evidence. It
+was attributed to Tait, the professor of physics at Edinburgh
+University, and Balfour Stewart, the professor of physics at Owens
+College, Manchester. When the fourth edition appeared, their names
+were given on the title page.
+
+The kernel of the book is the above so-called discovery, first
+published in the form of an anagram. Preliminary chapters are
+devoted to a survey of the beliefs of ancient peoples on the
+subject of the immortality of the soul; to physical axioms; to the
+physical doctrine of energy, matter, and ether; and to the
+biological doctrine of development; in the last chapter we come to
+the unseen universe. What is meant by the \emph{unseen universe}?
+Matter is made up of molecules, which are supposed to be
+vortex-rings of an imperfect fluid, namely, the luminiferous
+ether; the luminous ether is made up of much smaller molecules,
+which are vortex-rings in a second ether. These smaller molecules
+with the ether in which they float are the unseen universe. The
+authors see reason to believe that the unseen universe absorbs
+energy from the visible universe and \emph{vice versa}. The soul
+is a frame which is made of the refined molecules and exists in
+the unseen universe. In life it is attached to the body. Every
+thought we think is accompanied by certain motions of the coarse
+molecules of the brain, these motions are propagated through the
+visible universe, but a part of each motion is absorbed by the
+fine molecules of the soul. Consequently the soul has an organ of
+memory as well as the body; at death the soul with its organ of
+memory is simply set free from association with the coarse
+molecules of the body. In this way the authors consider that they
+have shown the physical possibility of the immortality of the
+soul.
+
+The curious part of the book follows: the authors change their
+possibility into a theory and apply it to explain the main
+doctrines of Christianity; and it is certainly remarkable to find
+in the same book a discussion of Carnot's heat-engine and
+extensive quotations from the apostles and prophets. Clifford
+wrote an elaborate review which he finished in one sitting
+occupying twelve hours. He pointed out the difficulties to which
+the main speculation, which he admitted to be ingenious, is
+liable; but his wrath knew no bounds when he proceeded to consider
+the application to the doctrines of Christianity; for from being a
+High Churchman in youth he became an agnostic in later years; and
+he could not write on any religious question without using
+language which was offensive even to his friends.
+
+The \emph{Phaedo} of Plato is more satisfying to the mind than the
+\emph{Unseen Universe} of Tait and Stewart. In it, Socrates
+discusses with his friends the immortality of the soul, just
+before taking the draught of poison. One argument he advances is,
+How can the works of an artist be more enduring than the artist
+himself? This is a question which comes home in force when we
+peruse the works of Peacock, De~Morgan, Hamilton, Boole, Cayley
+and Clifford.
+
+
+\chapter [Henry John Stephen Smith (1826-1883)]{HENRY JOHN
+STEPHEN~SMITH\footnote{This Lecture was delivered March 15,
+1902.---\textsc{Editors.}}}
+
+\large\begin{center}{(1826-1883)}\end{center}\normalsize
+
+Henry John Stephen Smith was born in Dublin, Ireland, on November
+2, 1826. His father, John Smith, was an Irish barrister, who had
+graduated at Trinity College, Dublin, and had afterwards studied
+at the Temple, London, as a pupil of Henry John Stephen, the
+editor of Blackstone's \emph{Commentaries}; hence the given name
+of the future mathematician. His mother was Mary Murphy, an
+accomplished and clever Irishwoman, tall and beautiful. Henry was
+the youngest of four children, and was but two years old when his
+father died. His mother would have been left in straitened
+circumstances had she not been successful in claiming a bequest of
+\pounds10,000 which had been left to her husband but had been
+disputed. On receiving this money, she migrated to England, and
+finally settled in the Isle of Wight.
+
+Henry as a child was sickly and very near-sighted. When four years
+of age he displayed a genius for mastering languages. His first
+instructor was his mother, who had an accurate knowledge of the
+classics. When eleven years of age, he, along with his brother and
+sisters, was placed in the charge of a private tutor, who was
+strong in the classics; in one year he read a large portion of the
+Greek and Latin authors commonly studied. His tutor was impressed
+with his power of memory, quickness of perception, indefatigable
+diligence, and intuitive grasp of whatever he studied. In their
+leisure hours the children would improvise plays from Homer, or
+Robinson Crusoe; and they also became diligent students of animal
+and insect life. Next year a new tutor was strong in the
+mathematics, and with his aid Henry became acquainted with
+advanced arithmetic, and the elements of algebra and geometry. The
+year following, Mrs.\ Smith moved to Oxford, and placed Henry
+under the care of Rev.\ Mr.\ Highton, who was not only a sound
+scholar, but an exceptionally good mathematician. The year
+following Mr.\ Highton received a mastership at Rugby with a
+boardinghouse attached to it (which is important from a financial
+point of view) and he took Henry Smith with him as his first
+boarder. Thus at the age of fifteen Henry Smith was launched into
+the life of the English public school, and Rugby was then under
+the most famous headmaster of the day, Dr.\ Arnold. Schoolboy life
+as it was then at Rugby has been depicted by Hughes in ``Tom
+Brown's Schooldays.''
+
+Here he showed great and all-around ability. It became his
+ambition to crown his school career by carrying off an entrance
+scholarship at Balliol College, Oxford. But as a sister and
+brother had already died of consumption, his mother did not allow
+him to complete his third and final year at Rugby, but took him to
+Italy, where he continued his reading privately. Notwithstanding
+this manifest disadvantage, he was able to carry off the coveted
+scholarship; and at the age of nineteen he began residence as a
+student of Balliol College. The next long vacation was spent in
+Italy, and there his health broke down. By the following winter he
+had not recovered enough to warrant his return to Oxford; instead,
+he went to Paris, and took several of the courses at the Sorbonne
+and the Coll\`ege de France. These studies abroad had much
+influence on his future career as a mathematician. Thereafter he
+resumed his undergraduate studies at Oxford, carried off what is
+considered the highest classical honor, and in 1849, when 23 years
+old, finished his undergraduate career with a double-first; that
+is, in the honors examination for bachelor of arts he took
+first-class rank in the classics, and also first-class rank in the
+mathematics.
+
+It is not very pleasant to be a double first, for the outwardly
+envied and distinguished recipient is apt to find himself in the
+position of the ass between two equally inviting bundles of hay,
+unless indeed there is some external attraction superior to both.
+In the case of Smith, the external attraction was the bar, for
+which he was in many respects well suited; but the feebleness of
+his constitution led him to abandon that course. So he had a
+difficulty in deciding between classics and mathematics, and there
+is a story to the effect that he finally solved the difficulty by
+tossing up a penny. He certainly used the expression: but the
+reasons which determined his choice in favor of mathematics were
+first, his weak sight, which made thinking preferable to reading,
+and secondly, the opportunity which presented itself.
+
+At that time Oxford was recovering from the excitement which had
+been produced by the Tractarian movement, and which had ended in
+Newman going over to the Church of Rome. But a Parliamentary
+Commission had been appointed to inquire into the working of the
+University. The old system of close scholarships and fellowships
+was doomed, and the close preserves of the Colleges were being
+either extinguished or thrown open to public competition. Resident
+professors, married tutors or fellows were almost or quite
+unknown; the heads of the several colleges, then the governing
+body of the University, formed a little society by themselves.
+Balliol College (founded by John Balliol, the unfortunate King of
+Scotland who was willing to sell its independence) was then the
+most distinguished for intellectual eminence; the master was
+singular among his compeers for keeping steadily in view the true
+aim of a college, and he reformed the abuses of privilege and
+close endowment as far as he legally could. Smith was elected a
+fellow with the hope that he would consent to reside, and take the
+further office of tutor in mathematics, which he did. Soon after
+he became one of the mathematical tutors of Balliol he was asked
+by his college to deliver a course of lectures on chemistry. For
+this purpose he took up the study of chemical analysis, and
+exhibited skill in manipulation and accuracy in work. He had an
+idea of seeking numerical relations connecting the atomic weights
+of the elements, and some mathematical basis for their properties
+which might enable experiments to be predicted by the operation of
+the mind.
+
+About this time Whewell, the master of Trinity College, Cambridge,
+wrote \emph{The Plurality of Worlds}, which was at first published
+anonymously. Whewell pointed out what he called law of waste
+traceable in the Divine economy; and his argument was that the
+other planets were waste effects, the Earth the only oasis in the
+desert of our system, the only world inhabited by intelligent
+beings; Sir David Brewster, a Scottish physicist, inventor of the
+kaleidoscope, wrote a fiery answer entitled ``More worlds than
+one, the creed of the philosopher and the hope of the Christian.''
+In 1855 Smith wrote an essay on this subject for a volume of
+Oxford and Cambridge Essays in which the fallibility both of men
+of science and of theologians was impartially exposed. It was his
+first and only effort at popular writing.
+
+His two earliest mathematical papers were on geometrical subjects,
+but the third concerned that branch of mathematics in which he won
+fame---the theory of numbers. How he was led to take up this
+branch of mathematics is not stated on authority, but it was
+probably as follows: There was then no school of mathematics at
+Oxford; the symbolical school was flourishing at Cambridge; and
+Hamilton was lecturing on Quaternions at Dublin. Smith did not
+estimate either of these very highly; he had studied at Paris
+under some of the great French analysts; he had lived much on the
+Continent, and was familiar with the French, German and Italian
+languages. As a scholar he was drawn to the masterly disquisitions
+of Gauss, who had made the theory of numbers a principal subject
+of research. I may quote here his estimate of Gauss and of his
+work: ``If we except the great name of Newton (and the exception
+is one which Gauss himself would have been delighted to make) it
+is probable that no mathematician of any age or country has ever
+surpassed Gauss in the combination of an abundant fertility of
+invention with an absolute vigorousness in demonstration, which
+the ancient Greeks themselves might have envied. It may be
+admitted, without any disparagement to the eminence of such great
+mathematicians as Euler and Cauchy that they were so overwhelmed
+with the exuberant wealth of their own creations, and so
+fascinated by the interest attaching to the results at which they
+arrived, that they did not greatly care to expend their time in
+arranging their ideas in a strictly logical order, or even in
+establishing by irrefragable proof propositions which they
+instinctively felt, and could almost see to be true. With Gauss
+the case was otherwise. It may seem paradoxical, but it is
+probably nevertheless true that it is precisely the effort after a
+logical perfection of form which has rendered the writings of
+Gauss open to the charge of obscurity and unnecessary difficulty.
+The fact is that there is neither obscurity nor difficulty in his
+writings, as long as we read them in the submissive spirit in
+which an intelligent schoolboy is made to read his Euclid. Every
+assertion that is made is fully proved, and the assertions succeed
+one another in a perfectly just analogical order; there nothing so
+far of which we can complain. But when we have finished the
+perusal, we soon begin to feel that our work is but begun, that we
+are still standing on the threshold of the temple, and that there
+is a secret which lies behind the veil and is as yet concealed
+from us. No vestige appears of the process by which the result
+itself was obtained, perhaps not even a trace of the
+considerations which suggested the successive steps of the
+demonstration. Gauss says more than once that for brevity, he
+gives only the synthesis, and suppresses the analysis of his
+propositions. \emph{Pauca sed matura}---few but
+well-matured---were the words with which he delighted to describe
+the character which he endeavored to impress upon his mathematical
+writings. If, on the other hand, we turn to a memoir of Euler's,
+there is a sort of free and luxuriant gracefulness about the whole
+performance, which tells of the quiet pleasure which Euler must
+have taken in each step of his work; but we are conscious
+nevertheless that we are at an immense distance from the severe
+grandeur of design which is characteristic of all Gauss's greater
+efforts.''
+
+Following the example of Gauss, he wrote his first paper on the
+theory of numbers in Latin: ``De compositione numerorum primorum
+form\ae{} $4^n+1$ ex duobus quadratis.'' In it he proves in an
+original manner the theorem of Fermat---``That every prime number
+of the form $4^n+1$ ($n$ being an integer number) is the sum of
+two square numbers.'' In his second paper he gives an introduction
+to the theory of numbers. ``It is probable that the Pythagorean
+school was acquainted with the definition and nature of prime
+numbers; nevertheless the arithmetical books of the elements of
+Euclid contain the oldest extant investigations respecting them;
+and, in particular the celebrated yet simple demonstration that
+the number of the primes is infinite. To Eratosthenes of
+Alexandria, who is for so many other reasons entitled to a place
+in the history of the sciences, is attributed the invention of the
+method by which the primes may successively be determined in order
+of magnitude. It is termed, after him, `the sieve of
+Eratosthenes'; and is essentially a method of exclusion, by which
+all composite numbers are successively erased from the series of
+natural numbers, and the primes alone are left remaining. It
+requires only one kind of arithmetical operation; that is to say,
+the formation of the successive multiples of given numbers, or in
+other words, addition only. Indeed it may be said to require no
+arithmetical operation whatever, for if the natural series of
+numbers be represented by points set off at equal distances along
+a line, by using a geometrical compass we can determine without
+calculation the multiples of any given number. And in fact, it was
+by a mechanical contrivance of this nature that M. Burckhardt
+calculated his table of the least divisors of the first three
+millions of numbers.''
+
+In 1857 Mrs. Smith died; as the result of her cares and exertions
+she had seen her son enter Balliol College as a scholar, graduate
+a double-first, elected a fellow of his college, appointed tutor
+in mathematics, and enter on his career as an independent
+mathematician. The brother and sister that were left arranged to
+keep house in Oxford, the two spending the terms together, and
+each being allowed complete liberty of movement during the
+vacations. Thereafter this was the domestic arrangement in which
+Smith lived and worked; he never married. As the owner of a house,
+instead of living in rooms in college he was able to satisfy his
+fondness for pet animals, and also to extend Irish hospitality to
+visiting friends under his own roof. He had no household cares to
+destroy the needed serenity for scientific work, excepting that he
+was careless in money matters, and trusted more to speculation in
+mining shares than to economic management of his income. Though
+addicted to the theory of numbers, he was not in any sense a
+recluse; on the contrary he entered with zest into every form of
+social enjoyment in Oxford, from croquet parties and picnics to
+banquets. He had the rare power of utilizing stray hours of
+leisure, and it was in such odd times that he accomplished most of
+his scientific work. After attending a picnic in the afternoon, he
+could mount to those serene heights in the theory of numbers
+
+\begin{verse}
+``Where never creeps a cloud or moves a wind, \\
+Nor ever falls the least white star of snow, \\
+Nor ever lowest roll of thunder moans, \\
+Nor sound of human sorrow mounts, to mar \\
+Their sacred everlasting calm.''
+\end{verse}
+
+Then he could of a sudden come down from these heights to attend a
+dinner, and could conduct himself there, not as a mathematical
+genius lost in reverie and pointed out as a poor and eccentric
+mortal, but on the contrary as a thorough man of the world greatly
+liked by everybody.
+
+In 1860, when Smith was 34 years old, the Savilian professor of
+geometry at Oxford died. At that time the English universities
+were so constituted that the teaching was done by the college
+tutors. The professors were officers of the University; and before
+reform set in, they not only did not teach, they did not even
+reside in Oxford. At the present day the lectures of the
+University professors are in general attended by only a few
+advanced students. Henry Smith was the only Oxford candidate;
+there were other candidates from the outside, among them George
+Boole, then professor of mathematics at Queens College, Cork.
+Smith's claims and talents were considered so conspicuous by the
+electors, that they did not consider any other candidates. He did
+not resign as tutor at Balliol, but continued to discharge the
+arduous duties, in order that the income of his Fellowship might
+be continued. With proper financial sense he might have been
+spared from labors which militated against the discharge of the
+higher duties of professor.
+
+His freedom during vacation gave him the opportunity of attending
+the meetings of the British Association, where he was not only a
+distinguished savant, but an accomplished member of the social
+organization known as the Red Lions. In 1858 he was selected by
+that body to prepare a report upon the Theory of Numbers. It was
+prepared in five parts, extending over the years 1859-1865. It is
+neither a history nor a treatise, but something intermediate. The
+author analyzes with remarkable clearness and order the works of
+mathematicians for the preceding century upon the theory of
+congruences, and upon that of binary quadratic forms. He returns
+to the original sources, indicates the principle and sketches the
+course of the demonstrations, and states the result, often adding
+something of his own. The work has been pronounced to be the most
+complete and elegant monument ever erected to the theory of
+numbers, and the model of what a scientific report ought to be.
+
+During the preparation of the Report, and as a logical consequence
+of the researches connected therewith, Smith published several
+original contributions to the higher arithmetic. Some were in
+complete form and appeared in the \emph{Philosophical
+Transactions} of the Royal Society of London; others were
+incomplete, giving only the results without the extended
+demonstrations, and appeared in the Proceedings of that Society.
+One of the latter, entitled ``On the orders and genera of
+quadratic forms containing more than three indeterminates,''
+enunciates certain general principles by means of which he solves
+a problem proposed by Eisenstein, namely, the decomposition of
+integer numbers into the sum of five squares; and further, the
+analogous problem for seven squares. It was also indicated that
+the four, six, and eight-square theorems of Jacobi, Eisenstein and
+Lionville were deducible from the principles set forth.
+
+In 1868 he returned to the geometrical researches which had first
+occupied his attention. For a memoir on ``Certain cubic and
+biquadratic problems'' the Royal Academy of Sciences of Berlin
+awarded him the Steiner prize. On account of his ability as a man
+of affairs, Smith was in great demand for University and
+scientific work of the day. He was made Keeper of the University
+Museum; he accepted the office of Mathematical Examiner to the
+University of London; he was a member of a Royal Commission
+appointed to report on Scientific Education; a member of the
+Commission appointed to reform the University of Oxford; chairman
+of the committee of scientists who were given charge of the
+Meteorological Office, etc. It was not till 1873, when offered a
+Fellowship by Corpus Christi College, that he gave up his tutorial
+duties at Balliol. The demands of these offices and of social
+functions upon his time and energy necessarily reduced the total
+output of mathematical work of the highest order; the results of
+long research lay buried in note-books, and the necessary time was
+not found for elaborating them into a form suitable for
+publication. Like his master, Gauss, he had a high ideal of what a
+scientific memoir ought to be in logical order, vigor of
+demonstration and literary execution; and it was to his
+mathematical friends matter of regret that he did not reserve more
+of his energy for the work for which he was exceptionally fitted.
+
+He was a brilliant talker and wit. Working in the purely
+speculative region of the theory of numbers, it was perhaps
+natural that he should take an anti-utilitarian view of
+mathematical science, and that he should express it in exaggerated
+terms as a defiance to the grossly utilitarian views then popular.
+It is reported that once in a lecture after explaining a new
+solution of an old problem he said, ``It is the peculiar beauty of
+this method, gentlemen, and one which endears it to the really
+scientific mind, that under no circumstances can it be of the
+smallest possible utility.'' I believe that it was at a banquet of
+the Red Lions that he proposed the toast ``Pure mathematics; may
+it never be of any use to any one.''
+
+I may mention some other specimens of his wit. ``You take tea in
+the morning,'' was the remark with which he once greeted a friend;
+``if I did that I should be awake all day.'' Some one mentioned to
+him the enigmatical motto of Marischal College, Aberdeen: ``They
+say; what say they; let them say.'' ``Ah,'' said he, ``it
+expresses the three stages of an undergraduate's career. `They
+say'---in his first year he accepts everything he is told as if it
+were inspired. `What say they'---in his second year he is
+skeptical and asks that question. `Let them say' expresses the
+attitude of contempt characteristic of his third year.'' Of a
+brilliant writer but illogical thinker he said ``He is never right
+and never wrong; he is never to the point.'' Of Lockyer, the
+astronomer, who has been for many years the editor of the
+scientific journal \emph{Nature}, he said, ``Lockyer sometimes
+forgets that he is only the editor, not the author, of Nature.''
+Speaking to a newly elected fellow of his college he advised him
+in a low whisper to write a little and to save a little, adding
+``I have done neither.''
+
+At the jubilee meeting of the British Association held at York in
+1881, Prof. Huxley and Sir John Lubbock (now Lord Avebury)
+strolled down one afternoon to the Minster, which is considered
+the finest cathedral in England. At the main door they met Prof.\
+Smith coming out, who made a mock movement of surprise. Huxley
+said, ``You seem surprised to see me here.'' ``Yes,'' said Smith,
+``going in, you know; I would not have been surprised to see you
+on one of the pinnacles.'' Once I was introduced to him at a
+garden party, given in the grounds of York Minster. He was a tall
+man, with sandy hair and beard, decidedly good-looking, with a
+certain intellectual distinction in his features and expression.
+He was everywhere and known to everyone, the life and soul of the
+gathering. He retained to the day of his death the simplicity and
+high spirits of a boy. Socially he was an embodiment of Irish
+blarney modified by Oxford dignity.
+
+In 1873 the British Association met at Bradford; at which meeting
+Maxwell delivered his famous ``Discourse on Molecules.'' At the
+same meeting Smith was the president of the section of mathematics
+and physics. He did not take up any technical subject in his
+address; but confined himself to matters of interest in the exact
+sciences. He spoke of the connection between mathematics and
+physics, as evidenced by the dual province of the section. ``So
+intimate is the union between mathematics and physics that
+probably by far the larger part of the accessions to our
+mathematical knowledge have been obtained by the efforts of
+mathematicians to solve the problems set to them by experiment,
+and to create for each successive class of phenomena a new
+calculus or a new geometry, as the case might be, which might
+prove not wholly inadequate to the subtlety of nature. Sometimes
+indeed the mathematician has been before the physicist, and it has
+happened that when some great and new question has occurred to the
+experimenter or the observer, he has found in the armory of the
+mathematician the weapons which he has needed ready made to his
+hand. But much oftener the questions proposed by the physicist
+have transcended the utmost powers of the mathematics of the time,
+and a fresh mathematical creation has been needed to supply the
+logical instrument required to interpret the new enigma.'' As an
+example of the rule he points out that the experiments of Faraday
+called forth the mathematical theory of Maxwell; as an example of
+the exception that the work of Apollonius on the conic sections
+was ready for Kepler in investigating the orbits of the planets.
+
+At the time of the Bradford meeting, education in the public
+schools and universities of England was practically confined to
+the classics and pure mathematics. In his address Smith took up
+the importance of science as an educational discipline in schools;
+and the following sentences, falling as they did from a profound
+scholar, produced a powerful effect: ``All knowledge of natural
+science that is imparted to a boy, is, or may be, useful to him in
+the business of his after-life; but the claim of natural science
+to a place in education cannot be rested upon its usefulness only.
+The great object of education is to expand and to train the mental
+faculties, and it is because we believe that the study of natural
+science is eminently fitted to further these two objects that we
+urge its introduction into school studies. Science expands the
+minds of the young, because it puts before them great and
+ennobling objects of contemplation; many of its truths are such as
+a child can understand, and yet such that while in a measure he
+understands them, he is made to feel something of the greatness,
+something of the sublime regularity and something of the
+impenetrable mystery, of the world in which he is placed. But
+science also trains the growing faculties, for science proposes to
+itself truth as its only object, and it presents the most varied,
+and at the same time the most splendid examples of the different
+mental processes which lead to the attainment of truth, and which
+make up what we call reasoning. In science error is always
+possible, often close at hand; and the constant necessity for
+being on our guard against it is one important part of the
+education which science supplies. But in science sophistry is
+impossible; science knows no love of paradox; science has no skill
+to make the worse appear the better reason; science visits with a
+not long deferred exposure all our fondness for preconceived
+opinions, all our partiality for views which we have ourselves
+maintained; and thus teaches the two best lessons that can well be
+taught---on the one hand, the love of truth; and on the other,
+sobriety and watchfulness in the use of the understanding.''
+
+The London Mathematical Society was founded in 1865. By going to
+the meetings Prof.\ Smith was induced to prepare for publication a
+number of papers from the materials of his notebooks. He was for
+two years president, and at the end of his term delivered an
+address ``On the present state and prospects of some branches of
+pure mathematics.'' He began by referring to a charge which had
+been brought against the Society, that its Proceedings showed a
+partiality in favor of one or two great branches of mathematical
+science to the comparative neglect and possible disparagement of
+others. He replies in the language of a miner. ``It may be
+rejoined with great plausibility that ours is not a blamable
+partiality, but a well-grounded preference. So great (we might
+contend) have been the triumphs achieved in recent times by that
+combination of the newer algebra with the direct contemplation of
+space which constitutes the modern geometry---so large has been
+the portion of these triumphs, which is due to the genius of a few
+great English mathematicians; so vast and so inviting has been the
+field thus thrown open to research, that we do well to press along
+towards a country which has, we might say, been `prospected' for
+us, and in which we know beforehand we cannot fail to find
+something that will repay our trouble, rather than adventure
+ourselves into regions where, soon after the first step, we should
+have no beaten tracks to guide us to the lucky spots, and in which
+(at the best) the daily earnings of the treasure-seeker are small,
+and do not always make a great show, even after long years of
+work. Such regions, however, there are in the realm of pure
+mathematics, and it cannot be for the interest of science that
+they should be altogether neglected by the rising generation of
+English mathematicians. I propose, therefore, in the first
+instance to direct your attention to some few of these
+comparatively neglected spots.'' Since then quite a number of the
+neglected spots pointed out have been worked.
+
+In 1878 Oxford friends urged him to come forward as a candidate
+for the representation in Parliament of the University of Oxford,
+on the principle that a University constituency ought to have for
+its representative not a mere party politician, but an academic
+man well acquainted with the special needs of the University. The
+main question before the electors was the approval or disapproval
+of the Jingo war policy of the Conservative Government. Henry
+Smith had always been a Liberal in politics, university
+administration, and religion. The voting was influenced mainly by
+party considerations---Beaconsfield or Gladstone---with the result
+that Smith was defeated by more than 2 to 1; but he had the
+satisfaction of knowing that his support came mainly from the
+resident and working members of the University. He did not expect
+success and he hardly desired it, but he did not shrink when asked
+to stand forward as the representative of a principle in which he
+believed. The election over, he devoted himself with renewed
+energy to the publication of his mathematical researches. His
+report on the theory of numbers had ended in elliptic functions;
+and it was this subject which now engaged his attention.
+
+In February, 1882, he was surprised to see in the \emph{Comptes
+rendus} that the subject proposed by the Paris Academy of Science
+for the \emph{Grand prix des sciences math\'ematiques} was the
+theory of the decomposition of integer numbers into a sum of five
+squares; and that the attention of competitors was directed to the
+results announced without demonstration by Eisenstein, whereas
+nothing was said about his papers dealing with the same subject in
+the Proceedings of the Royal Society. He wrote to M.\ Hermite
+calling his attention to what he had published; in reply he was
+assured that the members of the commission did not know of the
+existence of his papers, and he was advised to complete his
+demonstrations and submit the memoir according to the rules of the
+competition. According to the rules each manuscript bears a motto,
+and the corresponding envelope containing the name of the
+successful author is opened. There were still three months before
+the closing of the \emph{concours} (1 June, 1882) and Smith set to
+work, prepared the memoir and despatched it in time.
+
+Meanwhile a political agitation had grown up in favor of extending
+the franchise in the county constituencies. In the towns the
+mechanic had received a vote; but in the counties that power
+remained with the squire and the farmer; poor Hodge, as he is
+called, was left out. Henry Smith was not merely a Liberal; he
+felt a genuine sympathy for the poor of his own land. At a meeting
+in the Oxford Town Hall he made a speech in favor of the movement,
+urging justice to all classes. From that platform he went home to
+die. When he spoke he was suffering from a cold. The exposure and
+excitement were followed by congestion of the liver, to which he
+succumbed on February 9, 1883, in the 57th year of his age.
+
+Two months after his death the Paris Academy made their award. Two
+of the three memoirs sent in were judged worthy of the prize. When
+the envelopes were opened, the authors were found to be Prof.\
+Smith and M.\ Minkowski, a young mathematician of Koenigsberg,
+Prussia. No notice was taken of Smith's previous publication on
+the subject, and M.\ Hermite on being written to, said that he
+forgot to bring the matter to the notice of the commission. It was
+admitted that there was considerable similarity in the course of
+the investigation in the two memoirs. The truth seems to be that
+M.\ Minkowski availed himself of whatever had been published on
+the subject, including Smith's paper, but to work up the memoir
+from that basis cost Smith himself much intellectual labor, and
+must have cost Minkowski much more. Minkowski is now the chief
+living authority in that high region of the theory of numbers.
+Smith's work remains the monument of one of the greatest British
+mathematicians of the nineteenth century.
+
+\chapter [James Joseph Sylvester (1814-1897)]{JAMES JOSEPH
+SYLVESTER\footnote{This Lecture was delivered March 21,
+1902.---\textsc{Editors.}}}
+
+\large\begin{center}{(1814-1897)}\end{center}\normalsize
+
+James Joseph Sylvester was born in London, on the 3d of September,
+1814. He was by descent a Jew. His father was Abraham Joseph
+Sylvester, and the future mathematician was the youngest but one
+of seven children. He received his elementary education at two
+private schools in London, and his secondary education at the
+Royal Institution in Liverpool. At the age of twenty he entered
+St.\ John's College, Cambridge; and in the tripos examination he
+came out second wrangler. The senior wrangler of the year did not
+rise to any eminence; the fourth wrangler was George Green,
+celebrated for his contributions to mathematical physics; the
+fifth wrangler was Duncan F.\ Gregory, who subsequently wrote on
+the foundations of algebra. On account of his religion Sylvester
+could not sign the thirty-nine articles of the Church of England;
+and as a consequence he could neither receive the degree of
+Bachelor of Arts nor compete for the Smith's prizes, and as a
+further consequence he was not eligible for a fellowship. To
+obtain a degree he turned to the University of Dublin. After the
+theological tests for degrees had been abolished at the
+Universities of Oxford and Cambridge in 1872, the University of
+Cambridge granted him his well-earned degree of Bachelor of Arts
+and also that of Master of Arts.
+
+On leaving Cambridge he at once commenced to write papers, and
+these were at first on applied mathematics. His first paper was
+entitled ``An analytical development of Fresnel's optical theory
+of crystals,'' which was published in the \emph{Philosophical
+Magazine}. Ere long he was appointed Professor of Physics in
+University College, London, thus becoming a colleague of
+De~Morgan. At that time University College was almost the only
+institution of higher education in England in which theological
+distinctions were ignored. There was then no physical laboratory
+at University College, or indeed at the University of Cambridge;
+which was fortunate in the case of Sylvester, for he would have
+made a sorry experimenter. His was a sanguine and fiery
+temperament, lacking the patience necessary in physical
+manipulation. As it was, even in these pre-laboratory days he felt
+out of place, and was not long in accepting a chair of pure
+mathematics.
+
+In 1841 he became professor of mathematics at the University of
+Virginia. In almost all notices of his life nothing is said about
+his career there; the truth is that after the short space of four
+years it came to a sudden and rather tragic termination. Among his
+students were two brothers, fully imbued with the Southern ideas
+about honor. One day Sylvester criticised the recitation of the
+younger brother in a wealth of diction which offended the young
+man's sense of honor; he sent word to the professor that he must
+apologize or be chastised. Sylvester did not apologize, but
+provided himself with a sword-cane; the young man provided himself
+with a heavy walking-stick. The brothers lay in wait for the
+professor; and when he came along the younger brother demanded an
+apology, almost immediately knocked off Sylvester's hat, and
+struck him a blow on the bare head with his heavy stick. Sylvester
+drew his sword-cane, and pierced the young man just over the
+heart; who fell back into his brother's arms, calling out ``I am
+killed.'' A spectator, coming up, urged Sylvester away from the
+spot. Without waiting to pack his books the professor left for New
+York, and took the earliest possible passage for England. The
+student was not seriously hurt; fortunately the point of the sword
+had struck fair against a rib.
+
+Sylvester, on his return to London, connected himself with a firm
+of actuaries, his ultimate aim being to qualify himself to
+practice conveyancing. He became a student of the Inner Temple in
+1846, and was called to the bar in 1850. He chose the same
+profession as did Cayley; and in fact Cayley and Sylvester, while
+walking the law-courts, discoursed more on mathematics than on
+conveyancing. Cayley was full of the theory of invariants; and it
+was by his discourse that Sylvester was induced to take up the
+subject. These two men were life-long friends; but it is safe to
+say that the permanence of the friendship was due to Cayley's kind
+and patient disposition. Recognized as the leading mathematicians
+of their day in England, they were yet very different both in
+nature and talents.
+
+Cayley was patient and equable; Sylvester, fiery and passionate.
+Cayley finished off a mathematical memoir with the same care as a
+legal instrument; Sylvester never wrote a paper without
+foot-notes, appendices, supplements; and the alterations and
+corrections in his proofs were such that the printers found their
+task well-nigh impossible. Cayley was well-read in contemporary
+mathematics, and did much useful work as referee for scientific
+societies; Sylvester read only what had an immediate bearing on
+his own researches, and did little, if any, work as a referee.
+Cayley was a man of sound sense, and of great service in
+University administration; Sylvester satisfied the popular idea of
+a mathematician as one lost in reflection, and high above mundane
+affairs. Cayley was modest and retiring; Sylvester, courageous and
+full of his own importance. But while Cayley's papers, almost all,
+have the stamp of pure logical mathematics, Sylvester's are full
+of human interest. Cayley was no orator and no poet; Sylvester was
+an orator, and if not a poet, he at least prided himself on his
+poetry. It was not long before Cayley was provided with a chair at
+Cambridge, where he immediately married, and settled down to work
+as a mathematician in the midst of the most favorable environment.
+Sylvester was obliged to continue what he called ``fighting the
+world'' alone and unmarried.
+
+There is an ancient foundation in London, named after its founder,
+Gresham College. In 1854 the lectureship of geometry fell vacant
+and Sylvester applied. The trustees requested him and I suppose
+also the other candidates, to deliver a probationary lecture; with
+the result that he was not appointed. The professorship of
+mathematics in the Royal Military Academy at Woolwich fell vacant;
+Sylvester was again unsuccessful; but the appointee died in the
+course of a year, and then Sylvester succeeded on a second
+application. This was in 1855, when he was 41 years old.
+
+He was a professor at the Military Academy for fifteen years; and
+these years constitute the period of his greatest scientific
+activity. In addition to continuing his work on the theory of
+invariants, he was guided by it to take up one of the most
+difficult questions in the theory of numbers. Cayley had reduced
+the problem of the enumeration of invariants to that of the
+partition of numbers; Sylvester may be said to have revolutionized
+this part of mathematics by giving a complete analytical solution
+of the problem, which was in effect to enumerate the solutions in
+positive integers of the indeterminate equation:
+\begin{equation*}
+ax + by + cz + \ldots + ld = m.
+\end{equation*}
+\noindent Thereafter he attacked the similar problem connected
+with two such simultaneous equations (known to Euler as the
+problem of the Virgins) and was partially and considerably
+successful. In June, 1859, he delivered a series of seven lectures
+on compound partition in general at King's College, London. The
+outlines of these lectures have been published by the Mathematical
+Society of London.
+
+Five years later (1864) he contributed to the Royal Society of
+London what is considered his greatest mathematical achievement.
+Newton, in his lectures on algebra, which he called ``Universal
+Arithmetic'' gave a rule for calculating an inferior limit to the
+number of imaginary roots in an equation of any degree, but he did
+not give any demonstration or indication of the process by which
+he reached it. Many succeeding mathematicians such as Euler,
+Waring, Maclaurin, took up the problem of investigating the rule,
+but they were unable to establish either its truth or inadequacy.
+Sylvester in the paper quoted established the validity of the rule
+for algebraic equations as far as the fifth degree inclusive. Next
+year in a communication to the Mathematical Society of London, he
+fully established and generalized the rule. ``I owed my success,''
+he said, ``chiefly to merging the theorem to be proved in one of
+greater scope and generality. In mathematical research, reversing
+the axiom of Euclid and controverting the proposition of Hesiod,
+it is a continual matter of experience, as I have found myself
+over and over again, that the whole is less than its part.''
+
+Two years later he succeeded De~Morgan as president of the London
+Mathematical Society. He was the first mathematician to whom that
+Society awarded the Gold medal founded in honor of De~Morgan. In
+1869, when the British Association met in Exeter, Prof.\ Sylvester
+was president of the section of mathematics and physics. Most of
+the mathematicians who have occupied that position have
+experienced difficulty in finding a subject which should satisfy
+the two conditions of being first, cognate to their branch of
+science; secondly, interesting to an audience of general culture.
+Not so Sylvester. He took up certain views of the nature of
+mathematical science which Huxley the great biologist had just
+published in \emph{Macmillan's Magazine} and the \emph{Fortnightly
+Review}. He introduced his subject by saying that he was himself
+like a great party leader and orator in the House of Lords, who,
+when requested to make a speech at some religious or charitable,
+at-all-events non-political meeting declined the honor on the
+ground that he could not speak unless he saw an adversary before
+him. I shall now quote from the address, so that you may hear
+Sylvester's own words.
+
+``In obedience,'' he said, ``to a somewhat similar combative
+instinct, I set to myself the task of considering certain
+utterances of a most distinguished member of the Association, one
+whom I no less respect for his honesty and public spirit, than I
+admire for his genius and eloquence, but from whose opinions on a
+subject he has not studied I feel constrained to differ. I have no
+doubt that had my distinguished friend, the probable
+president-elect of the next meeting of the Association, applied
+his uncommon powers of reasoning, induction, comparison,
+observation and invention to the study of mathematical science, he
+would have become as great a mathematician as he is now a
+biologist; indeed he has given public evidence of his ability to
+grapple with the practical side of certain mathematical questions;
+but he has not made a study of mathematical science as such, and
+the eminence of his position, and the weight justly attaching to
+his name, render it only the more imperative that any assertion
+proceeding from such a quarter, which may appear to be erroneous,
+or so expressed as to be conducive to error should not remain
+unchallenged or be passed over in silence.
+
+``Huxley says `mathematical training is almost purely deductive.
+The mathematician starts with a few simple propositions, the proof
+of which is so obvious that they are called self-evident, and the
+rest of his work consists of subtle deductions from them. The
+teaching of languages at any rate as ordinarily practised, is of
+the same general nature---authority and tradition furnish the
+data, and the mental operations are deductive.' It would seem from
+the above somewhat singularly juxtaposed paragraphs, that
+according to Prof.\ Huxley, the business of the mathematical
+student is, from a limited number of propositions (bottled up and
+labelled ready for use) to deduce any required result by a process
+of the same general nature as a student of languages employs in
+declining and conjugating his nouns and verbs---that to make out a
+mathematical proposition and to construe or parse a sentence are
+equivalent or identical mental operations. Such an opinion
+scarcely seems to need serious refutation. The passage is taken
+from an article in \emph{Macmillan's Magazine} for June last,
+entitled, `Scientific Education---Notes of an after-dinner
+speech'; and I cannot but think would have been couched in more
+guarded terms by my distinguished friend, had his speech been made
+\emph{before} dinner instead of \emph{after}.
+
+``The notion that mathematical truth rests on the narrow basis of
+a limited number of elementary propositions from which all others
+are to be derived by a process of logical inference and verbal
+deduction has been stated still more strongly and explicitly by
+the same eminent writer in an article of even date with the
+preceeding in the \emph{Fortnightly Review}; where we are told
+that `Mathematics is that study which knows nothing of
+observation, nothing of experiment, nothing of induction, nothing
+of causation.' I think no statement could have been made more
+opposite to the undoubted facts of the case, which are that
+mathematical analysis is constantly invoking the aid of new
+principles, new ideas and new methods not capable of being defined
+by any form of words, but springing direct from the inherent
+powers and activity of the human mind, and from continually
+renewed introspection of that inner world of thought of which the
+phenomena are as varied and require as close attention to discern
+as those of the outer physical world; that it is unceasingly
+calling forth the faculties of observation and comparison; that
+one of its principal weapons is induction; that is has frequent
+recourse to experimental trial and verification; and that it
+affords a boundless scope for the exercise of the highest efforts
+of imagination and invention.''
+
+Huxley never replied; convinced or not, he had sufficient sagacity
+to see that he had ventured far beyond his depth. In the portion
+of the address quoted, Sylvester adds parenthetically a clause
+which expresses his theory of mathematical knowledge. He says that
+the inner world of thought in each individual man (which is the
+world of observation to the mathematician) may be conceived to
+stand in somewhat the same general relation of correspondence to
+the outer physical world as an object to the shadow projected from
+it. To him the mental order was more real than the world of sense,
+and the foundation of mathematical science was ideal, not
+experimental.
+
+By this time Sylvester had received most of the high distinctions,
+both domestic and foreign, which are usually awarded to a
+mathematician of the first rank in his day. But a discontinuity
+was at hand. The War Office issued a regulation whereby officers
+of the army were obliged to retire on half pay on reaching the age
+of 55 years. Sylvester was a professor in a Military College; in a
+few months, on his reaching the prescribed age, he was retired on
+half pay. He felt that though no longer fit for the field he was
+still fit for the classroom. And he felt keenly the diminution in
+his income. It was about this time that he issued a small
+volume---the only book he ever published; not on mathematics, as
+you may suppose, but entitled \emph{The Laws of Verse}. He must
+have prided himself a good deal on this composition, for one of
+his last letters in \emph{Nature} is signed "J.~J.\ Sylvester,
+author of The Laws of Verse." He made some excellent translations
+from Horace and from German poets; and like Sir W.~R.\ Hamilton he
+was accustomed to express his feelings in sonnets.
+
+The break in his life appears to have discouraged Sylvester for
+the time being from engaging in any original research. But after
+three years a Russian mathematician named Tschebicheff, a
+professor in the University of Saint Petersburg, visiting
+Sylvester in London, drew his attention to the discovery by a
+Russian student named Lipkin, of a mechanism for drawing a perfect
+straight line. Mr.\ Lipkin received from the Russian Government a
+substantial award. It was found that the same discovery had been
+made several years before by M.\ Peaucellier, an officer in the
+French army, but failing to be recognized at its true value had
+dropped into oblivion. Sylvester introduced the subject into
+England in the form of an evening lecture before the Royal
+Institution, entitled ``On recent discoveries in mechanical
+conversion of motion.'' The Royal Institution of London was
+founded to promote scientific research; its professors have been
+such men as Davy, Faraday, Tyndall, Dewar. It is not a teaching
+institution, but it provides for special courses of lectures in
+the afternoons and for Friday evening lectures by investigators of
+something new in science. The evening lectures are attended by
+fashionable audiences of ladies and gentlemen in full dress.
+
+\begin{center}
+\includegraphics[width=50mm]{images/JJSfig1.png}
+\end{center}
+
+Euclid bases his \emph{Elements} on two postulates; first, that a
+straight line can be drawn, second, that a circle can be
+described. It is sometimes expressed in this way; he postulates a
+ruler and compass. The latter contrivance is not difficult to
+construct, because it does not involve the use of a ruler or a
+compass in its own construction. But how is a ruler to be made
+straight, unless you already have a ruler by which to test it? The
+problem is to devise a mechanism which shall assume the second
+postulate only, and be able to satisfy the first. It is the
+mechanical problem of converting motion in a circle into motion in
+a straight line, without the use of any guide. James Watt, the
+inventor of the steam-engine, tackled the problem with all his
+might, but gave it up as impossible. However, he succeeded in
+finding a contrivance which solves the problem very approximately.
+Watt's parallelogram, employed in nearly every beam-engine,
+consists of three links; of which \emph{AC} and \emph{BD} are
+equal, and have fixed pivots at \emph{A} and \emph{B}
+respectively. The link \emph{CD} is of such a length that
+\emph{AC} and \emph{BD} are parallel when horizontal. The tracing
+point is attached to the middle point of \emph{CD}. When \emph{C}
+and \emph{D} move round their pivots, the tracing point describes
+a straight line very approximately, so long as the arc of
+displacement is small. The complete figure which would be
+described is the figure of 8, and the part utilized is near the
+point of contrary flexure.
+
+\begin{center}
+\includegraphics[width=50mm]{images/JJSfig2.png}
+\end{center}
+
+A linkage giving a closer approximation to a straight line was
+also invented by the Russian mathematician before
+mentioned---Tschebicheff; it likewise made use of three links. But
+the linkage invented by Peaucellier and later by Lipkin had seven
+pieces. The arms \emph{AB} and \emph{AC} are of equal length, and
+have a fixed pivot at \emph{A}. The links \emph{DB}, \emph{BE},
+\emph{EC}, \emph{CD} are of equal length. \emph{EF} is an arm
+connecting \emph{E} with the fixed pivot \emph{F} and is equal in
+length to the distance between \emph{A} and \emph{F}. It is
+readily shown by geometry that, as the point \emph{E} describes a
+circle around the center \emph{F}, the point \emph{D} describes an
+exact straight line perpendicular to the line joining it and
+\emph{F}. The exhibition of this contrivance at work was the
+climax of Sylvester's lecture.
+
+In Sylvester's audience were two mathematicians, Hart and Kempe,
+who took up the subject for further investigation. Hart perceived
+that the contrivances of Watt and of Tschebicheff consisted of
+three links, whereas Peaucellier's consisted of seven. Accordingly
+he searched for a contrivance of five links which would enable a
+tracing point to describe a perfect straight line; and he
+succeeded in inventing it. Kempe was a London barrister whose
+specialty was ecclesiastical law. He and Sylvester worked up the
+theory of linkages together, and discovered among other things the
+skew pantograph. Kempe became so imbued with linkage that he
+contributed to the Royal Society of London a paper on the ``Theory
+of Mathematical Form,'' in which he explains all reasoning by
+means of linkages.
+
+About this time (1877) the Johns Hopkins University was organized
+at Baltimore, and Sylvester, at the age of 63, was appointed the
+first professor of mathematics. Of his work there as a teacher,
+one of his pupils, Dr.\ Fabian Franklin, thus spoke in an address
+delivered at a memorial meeting in that University: ``The one
+thing which constantly marked Sylvester's lectures was
+enthusiastic love of the thing he was doing. He had in the fullest
+possible degree, to use the French phrase, the defect of this
+quality; for as he almost always spoke with enthusiastic ardor, so
+it was almost never possible for him to speak on matters incapable
+of evoking this ardor. In other words, the substance of his
+lectures had to consist largely of his own work, and, as a rule,
+of work hot from the forge. The consequence was that a continuous
+and systematic presentation of any extensive body of doctrine
+already completed was not to be expected from him. Any unsolved
+difficulty, any suggested extension, such as would have been
+passed by with a mention by other lecturers, became inevitably
+with him the occasion of a digression which was sure to consume
+many weeks, if indeed it did not take him away from the original
+object permanently. Nearly all of the important memoirs which he
+published, while in Baltimore, arose in this way. We who attended
+his lectures may be said to have seen these memoirs in the making.
+He would give us on the Friday the outcome of his grapplings with
+the enemy since the Tuesday lecture. Rarely can it have fallen to
+the lot of any class to follow so completely the workings of the
+mind of the master. Not only were all thus privileged to see `the
+very pulse of the machine,' to learn the spring and motive of the
+successive steps that led to his results, but we were set aglow by
+the delight and admiration which, with perfect na\"{\i}vet\'e and
+with that luxuriance of language peculiar to him, Sylvester
+lavished upon these results. That in this enthusiastic admiration
+he sometimes lacked the sense of proportion cannot be denied. A
+result announced at one lecture and hailed with loud acclaim as a
+marvel of beauty was by no means sure of not being found before
+the next lecture to have been erroneous; but the Esther that
+supplanted this Vashti was quite certain to be found still more
+supremely beautiful. The fundamental thing, however, was not this
+occasional extravagance, but the deep and abiding feeling for
+truth and beauty which underlay it. No young man of generous mind
+could stand before that superb grey head and hear those
+expositions of high and dear-bought truths, testifying to a
+passionate devotion undimmed by years or by arduous labors,
+without carrying away that which ever after must give to the
+pursuit of truth a new and deeper significance in his mind.''
+
+One of Sylvester's principal achievements at Baltimore was the
+founding of the \emph{American Journal of Mathematics}, which, at
+his suggestion, took the quarto form. He aimed at establishing a
+mathematical journal in the English language, which should equal
+Liouville's \emph{Journal} in France, or Crelle's \emph{Journal}
+in Germany. Probably his best contribution to the \emph{American
+Journal} consisted in his ``Lectures on Universal Algebra'';
+which, however, were left unfinished, like a great many other
+projects of his.
+
+Sylvester had that quality of absent-mindedness which is popularly
+supposed to be, if not the essence, at least an invariable
+accompaniment, of a distinguished mathematician. Many stories are
+related on this point, which, if not all true, are at least
+characteristic. Dr.\ Franklin describes an instance which actually
+happened in Baltimore. To illustrate a theory of versification
+contained in his book \emph{The Laws of Verse}, Sylvester prepared
+a poem of 400 lines, all rhyming with the name Rosal\u\i{}nd or
+Rosal\=\i{}nd; and it was announced that the professor would read
+the poem on a specified evening at a specified hour at the Peabody
+Institute. At the time appointed there was a large turn-out of
+ladies and gentlemen. Prof.\ Sylvester, as usual, had a number of
+footnotes appended to his production; and he announced that in
+order to save interruption in reading the poem itself, he would
+first read the footnotes. The reading of the footnotes suggested
+various digressions to his imagination; an hour had passed, still
+no poem; an hour and a half passed and the striking of the clock
+or the unrest of his audience reminded him of the promised poem.
+He was astonished to find how time had passed, excused all who had
+engagements, and proceeded to read the Rosalind poem.
+
+In the summer of 1881 I visited London to see the Electrical
+Exhibition in the Crystal Palace---one of the earliest exhibitions
+devoted to electricity exclusively. I had made some investigations
+on the electric discharge, using a Holtz machine where De LaRue
+used a large battery of cells. Mr.\ De LaRue was Secretary of the
+Royal Institution; he gave me a ticket to a Friday evening
+discourse to be delivered by Mr.\ Spottiswoode, then president of
+the Royal Society, on the phenomena of the intensive discharge of
+electricity through gases; also an invitation to a dinner at his
+own house to be given prior to the lecture. Mr.\ Spottiswoode, the
+lecturer for the evening, was there; also Prof.\ Sylvester. He was
+a man rather under the average height, with long gray beard and a
+profusion of gray locks round his head surmounted by a great dome
+of forehead. He struck me as having the appearance of an artist or
+a poet rather than of an exact scientist. After dinner he
+conversed very eloquently with an elderly lady of title, while I
+conversed with her daughter. Then cabs were announced to take us
+to the Institution. Prof.\ Sylvester and I, being both bachelors,
+were put in a cab together. The professor, who had been so
+eloquent with the lady, said nothing; so I asked him how he liked
+his work at the Johns Hopkins University. ``It is very pleasant
+work indeed,'' said he, ``and the young men who study there are
+all so enthusiastic.'' We had not exhausted that subject before we
+reached our destination. We went up the stairway together, then
+Sylvester dived into the library to see the last number of
+\emph{Comptes Rendus} (in which he published many of his results
+at that time) and I saw him no more. I have always thought it very
+doubtful whether he came out to hear Spottiswoode's lecture.
+
+We have seen that H.~J.~S.\ Smith, the Savilian professor of
+Geometry at Oxford, died in 1883. Sylvester's friends urged his
+appointment, with the result that he was elected. After two years
+he delivered his inaugural lecture; of which the subject was
+differential invariants, termed by him reciprocants. An elementary
+reciprocant is $\frac{d^{2}y}{dx^{2}}$, for if
+$\frac{d^{2}y}{dx^{2}}=0$ then $\frac{d^{2}x}{dy^{2}}=0$. He
+looked upon this as the ``grub'' form, and developed from it the
+``chrysalis''
+\begin{equation*}
+\left\vert
+\begin{array}{ccc}
+\frac{d^{2}\phi}{dx^{2}}&\frac{d^{2}\phi}{dxdy}&\frac{d\phi}{dx},\\
+\frac{d^{2}\phi}{dxdy}&\frac{d^{2}\phi}{dy^{2}}&\frac{d\phi}{dy},\\
+\frac{d\phi}{dx}&\frac{d\phi}{dy}&\cdot
+\end{array}
+\right\vert
+\end{equation*}
+\noindent and the ``imago''
+\begin{equation*}
+\left\vert
+\begin{array}{ccc}
+\frac{d^{2}\Phi}{dx^{2}}&\frac{d^{2}\Phi}{dxdy}&\frac{d^{2}\Phi}{dxdr},\\
+\frac{d^{2}\Phi}{dxdy}&\frac{d^{2}\Phi}{dy^{2}}&\frac{d^{2}\Phi}{dydr},\\
+\frac{d^{2}\Phi}{dxdr}&\frac{d^{2}\Phi}{dydr}&\frac{d^{2}\Phi}{dr^{2}}.
+\end{array}
+\right\vert
+\end{equation*}
+\noindent You will observe that the chrysalis expression is
+unsymmetrical; the place of a ninth term is vacant. It moved
+Sylvester's poetic imagination, and into his inaugural lecture he
+interjected the following sonnet:
+
+\begin{center}
+\textsc{To a Missing Member of a Family Group of Terms in an
+Algebraical Formula:}
+\end{center}
+
+\begin{verse}
+Lone and discarded one! divorced by fate, \\
+Far from thy wished-for fellows---whither art flown? \\
+Where lingerest thou in thy bereaved estate, \\
+Like some lost star, or buried meteor stone? \\
+Thou minds't me much of that presumptuous one, \\
+Who loth, aught less than greatest, to be great, \\
+From Heaven's immensity fell headlong down \\
+To live forlorn, self-centred, desolate: \\
+Or who, new Heraklid, hard exile bore, \\
+Now buoyed by hope, now stretched on rack of fear, \\
+Till throned Astr\ae{}a, wafting to his ear \\
+Words of dim portent through the Atlantic roar, \\
+Bade him ``the sanctuary of the Muse revere \\
+And strew with flame the dust of Isis' shore.''
+\end{verse}
+
+This inaugural lecture was the beginning of his last great
+contribution to mathematics, and the subsequent lectures of that
+year were devoted to his researches in that line. Smith and
+Sylvester were akin in devoting attention to the theory of
+numbers, and also in being eloquent speakers. But in other
+respects the Oxonians found a great difference. Smith had been a
+painstaking tutor; Sylvester could lecture only on his own
+researches, which were not popular in a place so wholly given over
+to examinations. Smith was an incessantly active man of affairs;
+Sylvester became the subject of melancholy and complained that he
+had no friends.
+
+In 1872 a deputy professor was appointed. Sylvester removed to
+London, and lived mostly at the Athen\ae{}um Club. He was now 78
+years of age, and suffered from partial loss of sight and memory.
+He was subject to melancholy, and his condition was indeed
+``forlorn and desolate.'' His nearest relatives were nieces, but
+he did not wish to ask their assistance. One day, meeting a
+mathematical friend who had a home in London, he complained of the
+fare at the Club, and asked his friend to help him find suitable
+private apartments where he could have better cooking. They drove
+about from place to place for a whole afternoon, but none suited
+Sylvester. It grew late: Sylvester said, ``You have a pleasant
+home: take me there,'' and this was done. Arrived, he appointed
+one daughter his reader and another daughter his amanuensis.
+``Now,'' said he, ``I feel comfortably installed; don't let my
+relatives know where I am.'' The fire of his temper had not dimmed
+with age, and it required all the Christian fortitude of the
+ladies to stand his exactions. Eventually, notice had to be sent
+to his nieces to come and take charge of him. He died on the 15th
+of March, 1897, in the 83d year of his age, and was buried in the
+Jewish cemetery at Dalston.
+
+As a theist, Sylvester did not approve of the destructive attitude
+of such men as Clifford, in matters of religion. In the early days
+of his career he suffered much from the disabilities attached to
+his faith, and they were the prime cause of so much ``fighting the
+world.'' He was, in all probability, a greater mathematical genius
+than Cayley; but the environment in which he lived for some years
+was so much less favorable that he was not able to accomplish an
+equal amount of solid work. Sylvester's portrait adorns St.\
+John's College, Cambridge. A memorial fund of \pounds1500 has been
+placed in the charge of the Royal Society of London, from the
+proceeds of which a medal and about \pounds100 in money is awarded
+triennially for work done in pure mathematics. The first award has
+been made to M.\ Henri Poincar\'e of Paris, a mathematician for
+whom Sylvester had a high professional and personal regard.
+
+\chapter [Thomas Penyngton Kirkman (1806-1895)]{THOMAS \\
+PENYNGTON KIRKMAN\footnote{This Lecture was delivered April 20,
+1903.---\textsc{Editors.}}}
+
+\large\begin{center}{(1806-1895)}\end{center}\normalsize
+
+Thomas Penyngton Kirkman was born on March 31, 1806, at Bolton in
+Lancashire. He was the son of John Kirkman, a dealer in cotton and
+cotton waste; he had several sisters but no brother. He was
+educated at the Grammar School of Bolton, where the tuition was
+free. There he received good instruction in Latin and Greek, but
+no instruction in geometry or algebra; even Arithmetic was not
+then taught in the headmaster's upper room. He showed a decided
+taste for study and was by far the best scholar in the school. His
+father, who had no taste for learning and was succeeding in trade,
+was determined that his only son should follow his own business,
+and that without any loss of time. The schoolmaster tried to
+persuade the father to let his son remain at school; and the vicar
+also urged the father, saying that if he would send his son to
+Cambridge University, he would guarantee for sixpence that the boy
+would win a fellowship. But the father was obdurate; young Kirkman
+was removed from school, when he was fourteen years of age, and
+placed at a desk in his father's office. While so engaged, he
+continued of his own accord his study of Latin and Greek, and
+added French and German.
+
+After ten years spent in the counting room, he tore away from his
+father, secured the tuition of a young Irish baronet, Sir John
+Blunden, and entered the University of Dublin with the view of
+passing the examinations for the degree of B.A. There he never had
+instruction from any tutor. It was not until he entered Trinity
+College, Dublin, that he opened any mathematical book. He was not
+of course abreast with men who had good preparation. What he knew
+of mathematics, he owed to his own study, having never had a
+single hour's instruction from any person. To this self-education
+is due, it appears to me, both the strength and the weakness to be
+found in his career as a scientist. However, in his college course
+he obtained honors, or premiums as they are called, and graduated
+as a moderator, something like a wrangler.
+
+Returning to England in 1835, when he was 29 years old, he was
+ordained as a minister in the Church of England. He was a curate
+for five years, first at Bury, afterwards at Lymm; then he became
+the vicar of a newly-formed parish---Croft with Southworth in
+Lancashire. This parish was the scene of his life's labors. The
+income of the benefice was not large, about \pounds200 per annum;
+for several years he supplemented this by taking pupils. He
+married, and property which came to his wife enabled them to
+dispense with the taking of pupils. His father became poorer, but
+was able to leave some property to his son and daughters. His
+parochial work, though small, was discharged with enthusiasm; out
+of the roughest material he formed a parish choir of boys and
+girls who could sing at sight any four-part song put before them.
+After the private teaching was over he had the leisure requisite
+for the great mathematical researches in which he now engaged.
+
+Soon after Kirkman was settled at Croft, Sir William Rowan
+Hamilton began to publish his quaternion papers and, being a
+graduate of Dublin University, Kirkman was naturally one of the
+first to study the new analysis. As the fruit of his meditations
+he contributed a paper to the \emph{Philosophical Magazine} ``On
+pluquaternions and homoid products of sums of \emph{n} squares.''
+He proposed the appellation "pluquaternions" for a linear
+expression involving more than three imaginaries (the $i$, $j$,
+$k$ of Hamilton), ``not dreading'' he says, ``the pluperfect
+criticism of grammarians, since the convenient barbarism is their
+own.'' Hamilton, writing to De~Morgan, remarked ``Kirkman is a
+very clever fellow,'' where the adjective has not the American
+colloquial meaning but the English meaning.
+
+For his own education and that of his pupils he devoted much
+attention to mathematical mnemonics, studying the \emph{Memoria
+Technica} of Grey. In 1851 he contributed a paper on the subject
+to the Literary and Philosophical Society of Manchester, and in
+1852 he published a book, \emph{First Mnemonical Lessons in
+Geometry, Algebra, and Trigonometry}, which is dedicated to his
+former pupil, Sir John Blunden. De~Morgan pronounced it ``the most
+curious crochet I ever saw,'' which was saying a great deal, for
+De~Morgan was familiar with many quaint books in mathematics. In
+the preface he says that much of the distaste for mathematical
+study springs largely from the difficulty of retaining in the
+memory the previous results and reasoning. ``This difficulty is
+closely connected with the unpronounceableness of the formul\ae{};
+the memory of the tongue and the ear are not easily turned to
+account; nearly everything depends on the thinking faculty or on
+the practice of the eye alone. Hence many, who see hardly anything
+formidable in the study of a language, look upon mathematical
+acquirements as beyond their power, when in truth they are very
+far from being so. My object is to enable the learner to `talk to
+himself,' in rapid, vigorous and suggestive syllables, about the
+matters which he must digest and remember. I have sought to bring
+the memory of the vocal organs and the ear to the assistance of
+the reasoning faculty and have never scrupled to sacrifice either
+good grammar or good English in order to secure the requisites for
+a useful \emph{mnemonic}, which are smoothness, condensation, and
+jingle.''
+
+As a specimen of his mnemonics we may take the cotangent formula
+in spherical trigonometry:
+
+\begin{equation*}
+\cot A \sin C + \cos b \cos C = \cot a \sin b
+\end{equation*}
+
+To remember this formula most masters then required some aid to
+the memory; for instance the following: If in any spherical
+triangle four parts be taken in succession, such as \emph{AbCa},
+consisting of two means \emph{bC} and two extremes \emph{Aa}, then
+the product of the cosines of the two means is equal to the sine
+of the mean side $\times$ cotangent of the extreme side minus sine
+of the mean angle $\times$ cotangent of the extreme angle, that is
+
+\begin{equation*}
+\cos b \cos C = \sin b \cot a - \sin C \cot A.
+\end{equation*}
+
+This is an appeal to the reason. Kirkman, however, proceeds on the
+principle of appealing to the memory of the ear, of the tongue,
+and of the lips altogether; a true \emph{memoria technica}. He
+distinguishes the large letter from the small by calling them
+\emph{Ang, Bang, Cang} (\emph{ang} from angle in contrast to
+side). To make the formula more euphoneous he drops the s from cos
+and the n from sin. Hence the formula is
+\medskip
+\begin{center}
+cot \emph{Ang} si \emph{Cang} and co \emph{b} co \emph{Cang} are
+cot \emph{a} si \emph{b}
+\end{center}
+\medskip
+\noindent which is to be chanted till it becomes perfectly
+familiar to the ear and the lips. The former rule is a hint
+offered to the judgment; Kirkman's method is something to be
+taught by rote. In his book Kirkman makes much use of verse, in
+the turning of which he was very skillful.
+
+In the early part of the nineteenth century a publication named
+the \emph{Lady's and Gentlemen's Diary} devoted several columns to
+mathematical problems. In 1844 the editor offered a prize for the
+solution of the following question: ``Determine the number of
+combinations that can be made out of $n$ symbols, each combination
+having $p$ symbols, with this limitation, that no combination of
+$q$ symbols which may appear in any one of them, may be repeated
+in any other.'' This is a problem of great difficulty; Kirkman
+solved it completely for the special case of $p=3$ and $q=2$ and
+printed his results in the second volume of the \emph{Cambridge
+and Dublin Mathematical Journal}. As a chip off this work he
+published in the \emph{Diary} for 1850 the famous problem of the
+fifteen schoolgirls as follows: ``Fifteen young ladies of a school
+walk out three abreast for seven days in succession; it is
+required to arrange them daily so that no two shall walk abreast
+more than once.'' To form the schedules for seven days is not
+difficult; but to find all the possible schedules is a different
+matter. Kirkman found all the possible combinations of the fifteen
+young ladies in groups of three to be 35, and the problem was also
+considered and solved by Cayley, and has been discussed by many
+later writers; Sylvester gave 91 as the greatest number of days;
+and he also intimated that the principle of the puzzle was known
+to him when an undergraduate at Cambridge, and that he had given
+it to fellow undergraduates. Kirkman replied that up to the time
+he proposed the problem he had neither seen Cambridge nor met
+Sylvester, and narrated how he had hit on the question.
+
+The Institute of France offered several times in succession a
+prize for a memoir on the theory of the polyedra; this fact
+together with his work in combinations led Kirkman to take up the
+subject. He always writes \emph{polyedron} not \emph{polyhedron};
+for he says we write \emph{periodic} not \emph{perihodic}. When
+Kirkman began work nothing had been done beyond the very ancient
+enumeration of the five regular solids and the simple combinations
+of crystallography. His first paper, ``On the representation and
+enumeration of the polyedra,'' was communicated in 1850 to the
+Literary and Philosophical Society of Manchester. He starts with
+the well-known theorem $P+S = L+2$, where $P$ is the number of
+points or summits, $S$ the number of plane bounding surfaces and
+$L$ the number of linear edges in a geometrical solid. "The
+question---how many $n$-edrons are there?---has been asked, but it
+is not likely soon to receive a definite answer. It is far from
+being a simple question, even when reduced to the narrower
+compass---how many $n$-edrons are there whose summits are all
+trihedral"? He enumerated and constructed the fourteen 8-edra
+whose faces are all triangles.
+
+In 1858 the French Institute modified its prize question. As the
+subject for the \emph{concours} of 1861 was announced:
+``Perfectionner en quelque point important la th\'eorie
+g\'eom\'etrique des poly\`edres,'' where the indefiniteness of the
+question indicates the very imperfect state of knowledge on the
+subject. The prize offered was 3000 francs. Kirkman appears to
+have worked at it with a view of competing, but he did not send in
+his memoir. Cayley appears to have intended to compete. The time
+was prolonged for a year, but there was no award and the prize was
+taken down. Kirkman communicated his results to the Royal Society
+through his friend Cayley, and was soon elected a Fellow. Then he
+contributed directly an elaborate paper entitled ``Complete theory
+of the Polyedra.'' In the preface he says, ``The following memoir
+contains a complete solution of the classification and enumeration
+of the $P$-edra $Q$-acra. The actual construction of the solids is
+a task impracticable from its magnitude, but it is here shown that
+we can enumerate them with an accurate account of their symmetry
+to any values of \emph{P} and \emph{Q}.'' The memoir consisted of
+21 sections; only the two introductory sections, occupying 45
+quarto pages, were printed by the Society, while the others still
+remain in manuscript. During following years he added many
+contributions to this subject.
+
+In 1858 the French Academy also proposed a problem in the Theory
+of Groups as the subject for competition for the grand
+mathematical prize in 1860: ``Quels peuvent \^etre les nombres de
+valeurs des fonctions bien d\'efinies qui contiennent un nombre
+donn\'e de lettres, et comment peut on former les fonctions pour
+lesquelles il existe un nombre donn\'e de valeurs?'' Three memoirs
+were presented, of which Kirkman's was one, but no prize was
+awarded. Not the slightest summary was vouchsafed of what the
+competitors had added to science, although it was confessed that
+all had contributed results both new and important; and the
+question, though proposed for the first time for the year 1860,
+was withdrawn from competition contrary to the usual custom of the
+Academy. Kirkman contributed the results of his investigation to
+the Manchester Society under the title ``The complete theory of
+groups, being the solution of the mathematical prize question of
+the French Academy for 1860.'' In more recent years the theory of
+groups has engaged the attention of many mathematicians in Germany
+and America; so far as British contributors are concerned Kirkman
+was the first and still remains the greatest.
+
+In 1861 the British Association met at Manchester; it was the last
+of its meetings which Sir William Rowan Hamilton attended. After
+the meeting Hamilton visited Kirkman at his home in the Croft
+rectory, and that meeting was no doubt a stimulus to both. As
+regards pure mathematics they were probably the two greatest in
+Britain; both felt the loneliness of scientific work, both were
+metaphysicians of penetrating power, both were good versifiers if
+not great poets. Of nearly the same age, they were both endowed
+with splendid physique; but the care which was taken of their
+health was very different; in four years Hamilton died but Kirkman
+lived more than 30 years longer.
+
+About 1862 the \emph{Educational Times}, a monthly periodical
+published in London, began to devote several columns to the
+proposing and solving of mathematical problems, taking up the work
+after the demise of the \emph{Diary}. This matter was afterwards
+reprinted in separate volumes, two for each year. In these
+reprints are to be found many questions proposed by Kirkman; they
+are generally propounded in quaint verse, and many of them were
+suggested by his study of combinations. A good specimen is ``The
+Revenge of Old King Cole''
+
+\begin{verse}
+``Full oft ye have had your fiddler's fling, \\
+For your own fun over the wine; \\
+And now'' quoth Cole, the merry old king, \\
+``Ye shall have it again for mine. \\
+My realm prepares for a week of joy \\
+At the coming of age of a princely boy--- \\
+Of the grand six days procession in square, \\
+In all your splendour dressed, \\
+Filling the city with music rare \\
+From fiddlers five abreast,'' etc.
+\end{verse}
+
+The problem set forth by this and other verses is that of 25 men
+arranged in five rows on Monday. Shifting the second column one
+step upward, the third two steps, the fourth three steps, and the
+fifth four steps gives the arrangement for Tuesday. Applying the
+same rule to Tuesday gives Wednesday's array, and similarly are
+found those for Thursday and Friday. In none of these can the same
+two men be found in one row. But the rule fails to work for
+Saturday, so that a special arrangement must be brought in which I
+leave to my hearers to work out. This problem resembles that of
+the fifteen schoolgirls.
+
+\begin{center}
+\begin{tabular}{ccccc}
+\multicolumn{5}{c}{Monday} \\
+A&B&C&D&E \\
+F&G&H&I&J \\
+K&L&M&N&O \\
+P&Q&R&S&T \\
+U&V&W&X&Y
+\end{tabular} \hspace{10 mm}
+\begin{tabular}{ccccc}
+\multicolumn{5}{c}{Tuesday} \\
+A&G&M&S&Y \\
+F&L&R&X&E \\
+K&Q&W&D&J \\
+P&V&C&I&O \\
+U&B&H&N&T
+\end{tabular}
+
+\medskip
+\begin{tabular}{ccccc}
+\multicolumn{5}{c}{Wednesday} \\
+A&L&W&I&T \\
+F&Q&C&N&Y \\
+K&V&H&S&E \\
+P&B&M&X&J \\
+N&G&R&D&O
+\end{tabular} \hspace{10 mm}
+\begin{tabular}{ccccc}
+\multicolumn{5}{c}{Thursday} \\
+A&Q&H&X&O \\
+F&V&M&D&T \\
+K&B&R&I&Y \\
+P&G&W&N&E \\
+U&L&C&S&J
+\end{tabular}
+\end{center}
+
+The Rev.\ Kirkman became at an early period of his life a broad
+churchman. About 1863 he came forward in defense of the Bishop of
+Colenso, a mathematician, and later he contributed to a series of
+pamphlets published in aid of the cause of ``Free Enquiry and Free
+Expression.'' In one of his letters to me Kirkman writes as
+follows: ``\emph{The Life of Colenso} by my friend Rev.\ Sir
+George Cox, Bart., is a most charming book; and the battle of the
+Bishops against the lawyers in the matter of the vacant see of
+Natal, to which Cox is the bishop-elect, is exciting. Canterbury
+refuses to ask, as required, the Queen's mandate to consecrate
+him. The Natal churchmen have just petitioned the Queen to make
+the Primate do his duty according to law. Natal was made a See
+with perpetual succession, and is endowed. The endowment has been
+lying idle since Colenso's death in 1883; and the bishops who have
+the law courts dead against them here are determined that no
+successor to Colenso shall be consecrated. There is a Bishop of
+South African Church there, whom they thrust in while Colenso
+lived, on pretense that Colenso was excommunicate. We shall soon
+see whether the lawyers or the bishops are to win.'' It was
+Kirkman's own belief that his course in this matter injured his
+chance of preferment in the church; he never rose above being
+rector of Croft.
+
+While a broad churchman the Rev.\ Mr.\ Kirkman was very vehement
+against the leaders of the materialistic philosophy. Two years
+after Tyndall's Belfast address, in which he announced that he
+could discern in matter the promise and potency of every form of
+life, Kirkman published a volume entitled \emph{Philosophy without
+Assumptions}, in which he criticises in very vigorous style the
+materialistic and evolutional philosophy advocated by Mill,
+Spencer, Tyndall, and Huxley. In ascribing everything to matter
+and its powers or potencies he considers that they turn philosophy
+upside down. He has, he writes, first-hand knowledge of himself as
+a continuous person, endowed with will; and he infers that there
+are will forces around; but he sees no evidence of the existence
+of matter. Matter is an assumption and forms no part of his
+philosophy. He relies on Boscovich's theory of an atom as simply
+the center of forces. Force he understands from his knowledge of
+will, but any other substance he does not understand. The obvious
+difficulty in this philosophy is to explain the belief in the
+existence of other conscious beings---other will forces. Is it not
+the \emph{great} assumption which everyone is obliged to make;
+verified by experience, but still in its nature an assumption?
+Kirkman tries to get over this difficulty by means of a syllogism,
+the major premise of which he has to manufacture, and which he
+presents to his reason for adoption or rejection. How can a
+universal proposition be easier to grasp than the particular case
+included in it?  If the mind doubts about an individual case, how
+can it be sure about an infinite number of such cases? It is a
+\emph{petitio principii}.
+
+As a critic of the materialistic philosophy Kirkman is more
+successful. He criticises Herbert Spencer on free will as follows:
+``The short chapter of eight pages on Will cost more philosophical
+toil than all the two volumes on Psychology. The author gets
+himself in a heat, he runs himself into a corner, and brings
+himself dangerously to bay. Hear him: `To reduce the general
+question to its simplest form; psychical changes either conform to
+law, or they do not. If they do not conform to law, this work, in
+common with all other works on the subject, is sheer nonsense;  no
+science of Psychology is possible. If they do conform to law,
+there cannot be any such thing as free will.' Here we see the
+horrible alternative. If the assertors of free will refuse to
+commit suicide, they must endure the infinitely greater pang of
+seeing Mr.\ Spencer hurl himself and his books into that yawning
+gulf, a sacrifice long devoted, and now by pitiless Fate
+consigned, to the abysmal gods of nonsense. Then pitch him down
+say I. Shall I spare him who tells me that my movements in this
+orbit of conscious thought and responsibility are made under
+`parallel conditions' with those of yon driven moon? Shall I spare
+him who has juggled me out of my Will, my noblest attribute; who
+has hocuspocused me out of my subsisting personality; and then, as
+a refinement of cruelty, has frightened me out of the rest of my
+wits by forcing me to this terrific alternative that either the
+testimony of this Being, this Reason and this Conscience is one
+ever-thundering lie, or else he, even he, has talked nonsense? He
+has talked nonsense, I say it because I have proved it. And every
+man must of course talk nonsense who begins his philosophy with
+abstracts in the clouds instead of building on the witness of his
+own self-consciousness. `If they do conform to law,' says Spencer,
+`there cannot be any such thing as free will.' The force of this
+seems to depend on his knowledge of `law.' When I ask, What does
+this writer know of law---definite working law in the
+Cosmos?---the only answer I can get is---Nothing, except a very
+little which he has picked up, often malappropriately, as we have
+seen, among the mathematicians. When I ask---What does he know
+\emph{about} law?---there is neither beginning nor end to the
+reply. I am advised to read his books \emph{about} law, and to
+master the differentiations and integrations of the coherences,
+the correlations, the uniformities, and universalities which he
+has established in the abstract over all space and all time by his
+vast experience and miraculous penetration. I have tried to do
+this, and have found all pretty satisfactory, except the lack of
+one thing---something like proof of his competence to decide all
+that scientifically. When I persist in my demand for such proof,
+it turns out at last---that he knows by heart the whole Hymn Book,
+the Litanies, the Missal, and the Decretals of the Must-be-ite
+religion! `Conform to law.' Shall I tell you what he means by
+that? Exactly ninety-nine hundredths of his meaning under the word
+\emph{law} is \emph{must be}.''
+
+Kirkman points out that the kind of proof offered by these
+philosophers is a bold assertion of \emph{must-be-so}. For
+instance he mentions Spencer's evolution of consciousness out of
+the unconscious: ``That an effectual adjustment may be made they
+(the separate impressions or constituent changes of a complex
+correspondence to be coordinated) \emph{must be} brought into
+relation with each other. But this implies some center of
+communication common to them all, through which they severally
+pass; and as they \emph{cannot} pass through it simultaneously,
+they \emph{must} pass through it in succession. So that as the
+external phenomena responded to become greater in number and more
+complicated in kind, the variety and rapidity of the changes to
+which this common center of communication is subject \emph{must}
+increase, there \emph{must} result an unbroken series of those
+changes, there \emph{must} arise a consciousness.''
+
+The paraphrase which Kirkman gave of Spencer's definition of
+Evolution commended itself to such great minds as Tait and
+Clerk-Maxwell. Spencer's definition is: ``Evolution is a change
+from an indefinite incoherent homogeneity to a definite coherent
+heterogeneity, through continuous differentiations and
+integrations.'' Kirkman's paraphrase is ``Evolution is a change
+from a nohowish untalkaboutable all-likeness, to a somehowish and
+in-general-talkaboutable not-all-likeness, by continuous
+somethingelseifications and sticktogetherations.'' The tone of
+Kirkman's book is distinctly polemical and full of sarcasm. He
+unfortunately wrote as a theologian rather than as a
+mathematician. The writers criticised did not reply, although they
+felt the edge of his sarcasm; and they acted wisely, for they
+could not successfully debate any subject involving exact science
+against one of the most penetrating mathematicians of the
+nineteenth century.
+
+We have seen that Hamilton appreciated Kirkman's genius; so did
+Cayley, De~Morgan, Clerk-Maxwell, Tait. One of Tait's most
+elaborate researches was the enumeration and construction of the
+knots which can be formed in an endless cord---a subject which he
+was induced to take up on account of its bearing on the vortex
+theory of atoms. If the atoms are vortex filaments their
+differences in kind, giving rise to differences in the spectra of
+the elements, must depend on a greater or less complexity in the
+form of the closed filament, and this difference would depend on
+the knottiness of the filament. Hence the main question was ``How
+many different forms of knots are there with any given small
+number of crossings?'' Tait made the investigation for three,
+four, five, six, seven, eight crossings. Kirkman's investigations
+on the polyedra were much allied. He took up the problem and, with
+some assistance from Tait, solved it not only for nine but for ten
+crossings. An investigation by C.~N.\ Little, a graduate of Yale
+University, has confirmed Kirkman's results.
+
+Through Professor Tait I was introduced to Rev.\ Mr.\ Kirkman; and
+we discussed the mathematical analysis of relationships, formal
+logic, and other subjects. After I had gone to the University of
+Texas, Kirkman sent me through Tait the following question which
+he said was current in society: ``Two boys, Smith and Jones, of
+the same age, are each the nephew of the other; how many legal
+solutions?'' I set the analysis to work, wrote out the solutions,
+and the paper is printed in the \emph{Proceedings} of the Royal
+Society of Edinburgh. There are four solutions, provided Smith and
+Jones are taken to be mere arbitrary, names; if the convention
+about surnames holds there are only two legal solutions. On seeing
+my paper Kirkman sent the question to the \emph{Educational Times}
+in the following improved form:
+
+\begin{verse}
+Baby Tom of baby Hugh \\
+The nephew is and uncle too; \\
+In how many ways can this be true?
+\end{verse}
+
+Thomas Penyngton Kirkman died on February 3, 1895, having very
+nearly reached the age of 89 years. I have found only one printed
+notice of his career, but all his writings are mentioned in the
+new German Encyclop\ae{}dia of Mathematics. He was an honorary
+member of the Literary and Philosophical Societies of Manchester
+and of Liverpool, a Fellow of the Royal Society, and a foreign
+member of the Dutch Society of Sciences at Haarlem. I may close by
+a quotation from one of his letters: ``What I have done in helping
+busy Tait in knots is, like the much more difficult and extensive
+things I have done in polyedra or groups, not at likely to be
+talked about intelligently by people so long as I live. But it is
+a faint pleasure to think it will one day win a little praise.''
+
+
+\chapter [Isaac Todhunter (1820-1884)]{ISAAC
+TODHUNTER\footnote{This Lecture was delivered April 13,
+1904.---\textsc{Editors.}}}
+
+\large\begin{center}{(1820-1884)}\end{center}\normalsize
+
+Isaac Todhunter was born at Rye, Sussex, 23 Nov., 1820. He was the
+second son of George Todhunter, Congregationalist minister of the
+place, and of Mary his wife, whose maiden name was Hume, a
+Scottish surname. The minister died of consumption when Isaac was
+six years old, and left his family, consisting of wife and four
+boys, in narrow circumstances. The widow, who was a woman of
+strength, physically and mentally, moved to the larger town of
+Hastings in the same county, and opened a school for girls. After
+some years Isaac was sent to a boys' school in the same town kept
+by Robert Carr, and subsequently to one newly opened by a Mr.\
+Austin from London; for some years he had been unusually backward
+in his studies, but under this new teacher he made rapid progress,
+and his career was then largely determined.
+
+After his school days were over, he became an usher or assistant
+master with Mr.\ Austin in a school at Peckham; and contrived to
+attend at the same time the evening classes at University College,
+London. There he came under the great educating influence of
+De~Morgan, for whom in after years he always expressed an
+unbounded admiration; to De~Morgan ``he owed that interest in the
+history and bibliography of science, in moral philosophy and logic
+which determined the course of his riper studies.'' In 1839 he
+passed the matriculation examination of the University of London,
+then a merely examining body, winning the exhibition for
+mathematics (\pounds30 for two years); in 1842 he passed the B.A.\
+examination carrying off a mathematical scholarship (of \pounds50
+for three years); and in 1844 obtained the degree of Master of
+Arts with the gold medal awarded to the candidate who gained the
+greatest distinction in that examination.
+
+Sylvester was then professor of natural philosophy in University
+College, and Todhunter studied under him. The writings of Sir John
+Herschel also had an influence; for Todhunter wrote as follows
+(\emph{Conflict of Studies}, p.\ 66): ``Let me at the outset
+record my opinion of mathematics; I cannot do this better than by
+adopting the words of Sir J.\ Herschel, to the influence of which
+I gratefully attribute the direction of my own early studies. He
+says of Astronomy, `Admission to its sanctuary can only be gained
+by one means,---sound and sufficient knowledge of mathematics, the
+great instrument of all exact inquiry, without which no man can
+ever make such advances in this or any other of the higher
+departments of science as can entitle him to form an independent
+opinion on any subject of discussion within their range.'\,''
+
+When Todhunter graduated as M.A.\ he was 24 years of age.
+Sylvester had gone to Virginia, but De~Morgan remained. The latter
+advised him to go through the regular course at Cambridge; his
+name was now entered at St.\ John's College. Being somewhat older,
+and much more brilliant than the honor men of his year, he was
+able to devote a great part of his attention to studies beyond
+those prescribed. Among other subjects he took up Mathematical
+Electricity. In 1848 he took his B.A.\ degree as senior wrangler,
+and also won the first Smith's prize.
+
+While an undergraduate Todhunter lived a very secluded life. He
+contributed along with his brothers to the support of their
+mother, and he had neither money nor time to spend on
+entertainments. The following legend was applied to him, if not
+recorded of him: ``Once on a time, a senior wrangler gave a wine
+party to celebrate his triumph. Six guests took their seats round
+the table. Turning the key in the door, he placed one bottle of
+wine on the table asseverating with unction, `None of you will
+leave this room while a single drop remains.'\,''
+
+At the University of Cambridge there is a foundation which
+provides for what is called the Burney prize. According to the
+regulations the prize is to be awarded to a graduate of the
+University who is not of more than three years' standing from
+admission to his degree and who shall produce the best English
+essay ``On some moral or metaphysical subject, or on the
+existence, nature and attributes of God, or on the truth and
+evidence of the Christian religion.'' Todhunter in the course of
+his first postgraduate year submitted an essay on the thesis that
+``The doctrine of a divine providence is inseparable from the
+belief in the existence of an absolutely perfect Creator.'' This
+essay received the prize, and was printed in 1849.
+
+Todhunter now proceeded to the degree of M.A., and unlike his
+mathematical instructors in University College, De~Morgan and
+Sylvester, he did not parade his non-conformist principles, but
+submitted to the regulations with as good grace as possible. He
+was elected a fellow of his college, but not immediately, probably
+on account of his being a non-conformist, and appointed lecturer
+on mathematics therein; he also engaged for some time in work as a
+private tutor, having for one of his pupils P.~G.\ Tait, and I
+believe E.~J.\ Routh also.
+
+For a space of 15 years he remained a fellow of St.\ John's
+College, residing in it, and taking part in the instruction. He
+was very successful as a lecturer, and it was not long before he
+began to publish textbooks on the subjects of his lectures. In
+1853 he published a textbook on \emph{Analytical Statics}; in 1855
+one on \emph{Plane Coordinate Geometry}; and in 1858
+\emph{Examples of Analytical Geometry of Three Dimensions}. His
+success in these subjects induced him to prepare manuals on
+elementary mathematics; his \emph{Algebra} appeared in 1858, his
+\emph{Trigonometry} in 1859, his \emph{Theory of Equations} in
+1861, and his \emph{Euclid} in 1862. Some of his textbooks passed
+through many editions and have been widely used in Great Britain
+and North America. Latterly he was appointed principal
+mathematical lecturer in his college, and he chose to drill the
+freshmen in Euclid and other elementary mathematics.
+
+Within these years he also labored at some works of a more
+strictly scientific character. Professor Woodhouse (who was the
+forerunner of the Analytical Society) had written a history of the
+calculus of variations, ending with the eighteenth century; this
+work was much admired for its usefulness by Todhunter, and as he
+felt a decided taste for the history of mathematics, he formed and
+carried out the project of continuing the history of that calculus
+during the nineteenth century. It was the first of the great
+historical works which has given Todhunter his high place among
+the mathematicians of the nineteenth century. This history was
+published in 1861; in 1862 he was elected a Fellow of the Royal
+Society of London. In 1863 he was a candidate for the Sadlerian
+professorship of Mathematics, to which Cayley was appointed.
+Todhunter was not a mere mathematical specialist. He was an
+excellent linguist; besides being a sound Latin and Greek scholar,
+he was familiar with French, German, Spanish, Italian and also
+Russian, Hebrew and Sanskrit. He was likewise well versed in
+philosophy, and for the two years 1863-5 acted as an Examiner for
+the Moral Science Tripos, of which the chief founders were himself
+and Whewell.
+
+By 1864 the financial success of his books was such that he was
+able to marry, a step which involved the resigning of his
+fellowship. His wife was a daughter of Captain George Davies of
+the Royal Navy, afterwards Admiral Davies.
+
+As a fellow and tutor of St.\ John's College he had lived a very
+secluded life. His relatives and friends thought he was a
+confirmed bachelor. He had sometimes hinted that the grapes were
+sour. For art he had little eye; for music no ear. ``He used to
+say he knew two tunes; one was `God save the Queen,' the other
+wasn't. The former he recognized by the people standing up.'' As
+owls shun the broad daylight he had shunned the glare of parlors.
+It was therefore a surprise to his friends and relatives when they
+were invited to his marriage in 1864. Prof.\ Mayor records that
+Todhunter wrote to his fianc\'ee, ``You will not forget, I am sure,
+that I have always been a student, and always shall be; but books
+shall not come into even distant rivalry with you,'' and Prof.\
+Mayor insinuated that thus forearmed, he calmly introduced to the
+inner circle of their honeymoon Hamilton on \emph{Quaternions}.
+
+It was now (1865) that the London Mathematical Society was
+organized under the guidance of De~Morgan, and Todhunter became a
+member in the first year of its existence. The same year he
+discharged the very onerous duties of examiner for the
+mathematical tripos---a task requiring so much labor and involving
+so much interference with his work as an author that he never
+accepted it again. Now (1865) appeared his \emph{History of the
+Mathematical Theory of Probability}, and the same year he was able
+to edit a new edition of Boole's \emph{Treatise on Differential
+Equations}, the author having succumbed to an untimely death.
+Todhunter certainly had a high appreciation of Boole, which he
+shared in common with De~Morgan. The work involved in editing the
+successive editions of his elementary books was great; he did not
+proceed to stereotype until many independent editions gave ample
+opportunity to correct all errors and misprints. He now added two
+more textbooks; \emph{Mechanics} in 1867 and \emph{Mensuration} in
+1869.
+
+About 1847 the members of St.\ John's College founded a prize in
+honor of their distinguished fellow, J.~C.\ Adams. It is awarded
+every two years, and is in value about \pounds225. In 1869 the
+subject proposed was ``A determination of the circumstances under
+which Discontinuity of any kind presents itself in the solution of
+a problem of maximum or minimum in the Calculus of Variations.''
+There had been a controversy a few years previous on this subject
+in the pages of \emph{Philosophical Magazine} and Todhunter had
+there advocated his view of the matter. This view is found in the
+opening sentences of his essay: ``We shall find that, generally
+speaking, discontinuity is introduced, by virtue of some
+restriction which we impose, either explicitly or implicitly in
+the statement of the problems which we propose to solve.'' This
+thesis he supported by considering in turn the usual applications
+of the calculus, and pointing out where he considers the
+discontinuities which occur have been introduced into the
+conditions of the problem. This he successfully proves in many
+instances. In some cases, the want of a distinct test of what
+discontinuity is somewhat obscures the argument. To his essay the
+prize was awarded; it is published under the title ``Researches in
+the Calculus of Variations''---an entirely different work from his
+\emph{History of the Calculus of Variations}.
+
+In 1873 he published his \emph{History of the Mathematical
+Theories of Attraction}. It consists of two volumes of nearly 1000
+pages altogether and is probably the most elaborate of his
+histories. In the same year (1873) he published in book form his
+views on some of the educational questions of the day, under the
+title of \emph{The Conflict of Studies, and other essays on
+subjects connected with education}. The collection contains six
+essays; they were originally written with the view of successive
+publication in some magazine, but in fact they were published only
+in book form. In the first essay, that on the Conflict of
+Studies---Todhunter gave his opinion of the educative value in
+high schools and colleges of the different kinds of study then
+commonly advocated in opposition to or in addition to the old
+subjects of classics and mathematics. He considered that the
+Experimental Sciences were little suitable, and that for a very
+English reason, because they could not be examined on adequately.
+He says:
+
+``Experimental Science viewed in connection with education,
+rejoices in a name which is unfairly expressive. A real experiment
+is a very valuable product of the mind, requiring great knowledge
+to invent it and great ingenuity to carry it out. When Perrier
+ascended the Puy de D\^ome with a barometer in order to test the
+influence of change of level on the height of the column of
+mercury, he performed an experiment, the suggestion of which was
+worthy of the genius of Pascal and Descartes. But when a modern
+traveller ascends Mont Blanc, and directs one of his guides to
+carry a barometer, he cannot be said to perform an experiment in
+any very exact or very meritorious sense of the word. It is a
+repetition of an observation made thousands of times before, and
+we can never recover any of the interest which belonged to the
+first trial, unless indeed, without having ever heard of it, we
+succeeded in reconstructing the process of ourselves. In fact,
+almost always he who first plucks an experimental flower thus
+appropriates and destroys its fragrance and its beauty.''
+
+At the time when Todhunter was writing the above, the Cavendish
+Laboratory for Experimental Physics was just being built at
+Cambridge, and Clerk-Maxwell had just been appointed the professor
+of the new study; from Todhunter's utterance we can see the state
+of affairs then prevailing. Consider the corresponding experiment
+of Torricelli, which can be performed inside a classroom; to every
+fresh student the experiment retains its fragrance; the sight of
+it, and more especially the performance of it imparts a kind of
+knowledge which cannot be got from description or testimony; it
+imparts accurate conceptions and is a necessary preparative for
+making a new and original experiment. To Todhunter it may be
+replied that the flowers of Euclid's \emph{Elements} were plucked
+at least 2000 years ago, yet, he must admit, they still possess,
+to the fresh student of mathematics, even although he becomes
+acquainted with them through a textbook, both fragrance and
+beauty.
+
+Todhunter went on to write another passage which roused the ire of
+Professor Tait. ``To take another example. We assert that if the
+resistance of the air be withdrawn a sovereign and a feather will
+fall through equal spaces in equal times. Very great credit is due
+to the person who first imagined the well-known experiment to
+illustrate this; but it is not obvious what is the special benefit
+now gained by seeing a lecturer repeat the process. It may be said
+that a boy takes more interest in the matter by seeing for
+himself, or by performing for himself, that is, by working the
+handle of the air-pump; this we admit, while we continue to doubt
+the educational value of the transaction. The boy would also
+probably take much more interest in football than in Latin
+grammar; but the measure of his interest is not identical with
+that of the importance of the subjects. It may be said that the
+fact makes a stronger impression on the boy through the medium of
+his sight, that he believes it the more confidently. I say that
+this ought not to be the case.  If he does not believe the
+statements of his tutor---probably a clergyman of mature
+knowledge, recognized ability and blameless character---his
+suspicion is irrational, and manifests a want of the power of
+appreciating evidence, a want fatal to his success in that branch
+of science which he is supposed to be cultivating.''
+
+Clear physical conceptions cannot be got by tradition, even from a
+clergyman of blameless character; they are best got directly from
+Nature, and this is recognized by the modern laboratory
+instruction in physics. Todhunter would reduce science to a matter
+of authority; and indeed his mathematical manuals are not free
+from that fault. He deals with the characteristic difficulties of
+algebra by authority rather than by scientific explanation.
+Todhunter goes on to say: ``Some considerable drawback must be
+made from the educational value of experiments, so called, on
+account of their failure. Many persons must have been present at
+the exhibitions of skilled performers, and have witnessed an
+uninterrupted series of ignominious reverses,---they have probably
+longed to imitate the cautious student who watched an eminent
+astronomer baffled by Foucault's experiment for proving the
+rotation of the Earth; as the pendulum would move the wrong way
+the student retired, saying that he wished to retain his faith in
+the elements of astronomy.''
+
+It is not unlikely that the series of ignominious reverses
+Todhunter had in his view were what he had seen in the physics
+classroom of University College when the manipulation was in the
+hands of a pure mathematician---Prof.\ Sylvester. At the
+University of Texas there is a fine clear space about 60 feet high
+inside the building, very suitable for Foucault's experiment. I
+fixed up a pendulum, using a very heavy ball, and the turning of
+the Earth could be seen in two successive oscillations. The
+experiment, although only a repetition according to Todhunter, was
+a live and inspiring lesson to all who saw it, whether they came
+with previous knowledge about it or no. The repetition of any such
+great experiment has an educative value of which Todhunter had no
+conception.
+
+Another subject which Todhunter discussed in these essays is the
+suitability of Euclid's \emph{Elements} for use as the elementary
+textbook of Geometry. His experience as a college tutor for 25
+years; his numerous engagements as an examiner in mathematics; his
+correspondence with teachers in the large schools gave weight to
+the opinion which he expressed. The question was raised by the
+first report of the Association for the Improvement of Geometrical
+Teaching; and the points which Todhunter made were afterwards
+taken up and presented in his own unique style by Lewis Carroll in
+``Euclid and his modern rivals.'' Up to that time Euclid's manual
+was, and in a very large measure still is, the authorized
+introduction to geometry; it is not as in this country where there
+is perfect liberty as to the books and methods to be employed. The
+great difficulty in the way of liberty in geometrical teaching is
+the universal tyranny of competitive examinations. Great Britain
+is an examination-ridden country. Todhunter referred to one of the
+most distinguished professors of Mathematics in England; one whose
+pupils had likewise gained a high reputation as investigators and
+teachers; his ``venerated master and friend,'' Prof.\ De~Morgan;
+and pointed out that he recommended the study of Euclid with all
+the authority of his great attainments and experience.
+
+Another argument used by Todhunter was as follows: In America
+there are the conditions which the Association desires; there is,
+for example, a textbook which defines parallel lines as those
+which \emph{have the same direction}. Could the American
+mathematicians of that day compare with those of England? He
+answered no.
+
+While Todhunter could point to one master---De~Morgan---as in his
+favor, he was obliged to quote another master---Sylvester---as
+opposed. In his presidential address before section A of the
+British Association at Exeter in 1869, Sylvester had said: ``I
+should rejoice to see \ldots Euclid honorably shelved or buried
+`deeper than did ever plummet sound' out of the schoolboy's reach;
+morphology introduced into the elements of algebra; projection,
+correlation, and motion accepted as aids to geometry; the mind of
+the student quickened and elevated and his faith awakened by early
+initiation into the ruling ideas of polarity, continuity,
+infinity, and familiarization with the doctrine of the imaginary
+and inconceivable.'' Todhunter replied: ``Whatever may have
+produced the dislike to Euclid in the illustrious mathematician
+whose words I have quoted, there is no ground for supposing that
+he would have been better pleased with the substitutes which are
+now offered and recommended in its place.  But the remark which is
+naturally suggested by the passage is that nothing prevents an
+enthusiastic teacher from carrying his pupils to any height he
+pleases in geometry, even if he starts with the use of Euclid.''
+
+Todhunter also replied to the adverse opinion, delivered by some
+professor (doubtless Tait) in an address at Edinburgh which was as
+follows: ``From the majority of the papers in our few mathematical
+journals, one would almost be led to fancy that British
+mathematicians have too much pride to use a simple method, while
+an unnecessarily complex one can be had. No more telling example
+of this could be wished for than the insane delusion under which
+they permit `Euclid' to be employed in our elementary teaching.
+They seem voluntarily to weight alike themselves and their pupils
+for the race.'' To which Todhunter replied: ``The British
+mathematical journals with the titles of which I am acquainted are
+the Quarterly Journal of Mathematics, the Mathematical Messenger,
+and the Philosophical Magazine; to which may be added the
+Proceedings of the Royal Society and the Monthly Notices of the
+Astronomical Society. I should have thought it would have been an
+adequate employment, for a person engaged in teaching, to read and
+master these periodicals regularly; but that a single
+mathematician should be able to improve more than half the matter
+which is thus presented to him fills me with amazement. I take
+down some of these volumes, and turning over the pages I find
+article after article by Profs.\ Cayley, Salmon and Sylvester, not
+to mention many other highly distinguished names. The idea of
+amending the elaborate essays of these eminent mathematicians
+seems to me something like the audacity recorded in poetry with
+which a superhuman hero climbs to the summit of the Indian Olympus
+and overturns the thrones of Vishnu, Brahma and Siva. While we may
+regret that such ability should be exerted on the revolutionary
+side of the question, here is at least one mournful satisfaction:
+the weapon with which Euclid is assailed was forged by Euclid
+himself. The justly celebrated professor, from whose address the
+quotation is taken, was himself trained by those exercises which
+he now considers worthless; twenty years ago his solutions of
+mathematical problems were rich with the fragrance of the Greek
+geometry. I venture to predict that we shall have to wait some
+time before a pupil will issue from the reformed school, who
+singlehanded will be able to challenge more than half the
+mathematicians of England.'' Professor Tait, in what he said, had,
+doubtless, reference to the avoidance of the use of the Quaternion
+method by his contemporaries in mathematics.
+
+More than half of the Essays is taken up with questions connected
+with competitive examinations. Todhunter explains the influence of
+Cambridge in this matter: ``Ours is an age of examination; and the
+University of Cambridge may claim the merit of originating this
+characteristic of the period. When we hear, as we often do, that
+the Universities are effete bodies which have lost their influence
+on the national character, we may point with real or affected
+triumph to the spread of examinations as a decisive proof that the
+humiliating assertion is not absolutely true. Although there must
+have been in schools and elsewhere processes resembling
+examinations before those of Cambridge had become widely famous,
+yet there can be little chance of error in regarding our
+mathematical tripos as the model for rigor, justice and
+importance, of a long succession of institutions of a similar kind
+which have since been constructed.'' Todhunter makes the damaging
+admission that ``We cannot by our examinations, \emph{create}
+learning or genius; it is uncertain whether we can infallibly
+\emph{discover} them; what we detect is simply the
+examination-passing power.''
+
+In England education is for the most part directed to training
+pupils for examination. One direct consequence is that the memory
+is cultivated at the expense of the understanding; knowledge
+instead of being assimilated is crammed for the time being, and
+lost as soon as the examination is over. Instead of a rational
+study of the principles of mathematics, attention is directed to
+problem-making,---to solving ten-minute conundrums. Textbooks are
+written with the view not of teaching the subject in the most
+scientific manner, but of passing certain specified examinations.
+I have seen such a textbook on trigonometry where all the
+important theorems which required the genius of Gregory and others
+to discover, are put down as so many definitions. Nominal
+knowledge, not real, is the kind that suits examinations.
+
+Todhunter possessed a considerable sense of humour. We see this in
+his Essays; among other stories he tells the following: A youth
+who was quite unable to satisfy his examiners as to a problem,
+endeavored to mollify them, as he said, ``by writing out book work
+bordering on the problem.'' Another youth who was rejected said
+``if there had been fairer examiners and better papers I should
+have passed; I knew many things which were not set.'' Again: ``A
+visitor to Cambridge put himself under the care of one of the
+self-constituted guides who obtrude their services. Members of the
+various ranks of the academical state were pointed out to the
+stranger---heads of colleges, professors and ordinary fellows; and
+some attempt was made to describe the nature of the functions
+discharged by the heads and professors. But an inquiry as to the
+duties of fellows produced and reproduced only the answer, `Them's
+fellows I say.' The guide had not been able to attach the notion
+of even the pretense of duty to a fellowship.''
+
+In 1874 Todhunter was elected an honorary fellow of his college,
+an honor which he prized very highly. Later on he was chosen as an
+elector to three of the University professorships---Moral
+Philosophy, Astronomy, Mental Philosophy and Logic. When the
+University of Cambridge established its new degree of Doctor of
+Science, restricted to those who have made original contributions
+to the advancement of science or learning, Todhunter was one of
+those whose application was granted within the first few months.
+In 1875 he published his manual \emph{Functions of Laplace, Bessel
+and Legendre}. Next year he finished an arduous literary
+task---the preparation of two volumes, the one containing an
+account of the writings of Whewell, the other containing
+selections from his literary and scientific correspondence.
+Todhunter's task was marred to a considerable extent by an
+unfortunate division of the matter: the scientific and literary
+details were given to him, while the writing of the life itself
+was given to another.
+
+In the summer of 1880 Dr.\ Todhunter first began to suffer from
+his eyesight, and from that date he gradually and slowly became
+weaker. But it was not till September, 1883, when he was at
+Hunstanton, that the worst symptoms came on. He then partially
+lost by paralysis the use of the right arm; and, though he
+afterwards recovered from this, he was left much weaker. In
+January of the next year he had another attack, and he died on
+March 1, 1884, in the 64th year of his age.
+
+Todhunter left a \emph{History of Elasticity} nearly finished. The
+manuscript was submitted, to Cayley for report; it was in 1886
+published under the editorship of Karl Pearson. I believe that he
+had other histories in contemplation; I had the honor of meeting
+him once, and in the course of conversation on mathematical logic,
+he said that he had a project of taking up the history of that
+subject; his interest in it dated from his study under De~Morgan.
+Todhunter had the same ruling passion as Airy---love of
+order---and was thus able to achieve an immense amount of
+mathematical work. Prof.\ Mayor wrote, ``Todhunter had no enemies,
+for he neither coined nor circulated scandal; men of all sects and
+parties were at home with him, for he was many-sided enough to see
+good in every thing. His friendship extended even to the lower
+creatures. The canaries always hung in his room, for he never
+forgot to see to their wants.''
+
+
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+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of Ten British Mathematicians of the 19th
+% Century, by Alexander Macfarlane                                        %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK TEN BRITISH MATHEMATICIANS ***  %
+%                                                                         %
+% ***** This file should be named 9942-t.tex or 9942-t.zip *****          %
+% This and all associated files of various formats will be found in:      %
+%         http://www.gutenberg.org/9/9/4/9942/                            %
+%                                                                         %
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+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #9942 (https://www.gutenberg.org/ebooks/9942)
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