% LaTeX source for Macfarlane's lecture on De Morgan
\documentclass{article}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\usepackage{times}
\begin{document}
\begin{center}
\Large{AUGUSTUS DE MORGAN}\footnote{This Lecture was delivered April 13,
1901.---\textsc{Editors}} \\
\Large{(1806--1871)}
\end{center}
\textsc{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 meet the difficult by
establishing in London a University on the principle of religious
neutrality. De Morgan, then 22 years of ace, 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 \textit{The Differential and Integral Calculus}
which was published by the Society; and he wrote one-sixth of the
articles in the \textit{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 of 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{center}
H$\cdot$O$\cdot$M$\cdot$O\,$\cdot$\,
P$\cdot$A$\cdot$U$\cdot$C$\cdot$A$\cdot$R$\cdot$U$\cdot$M\,$\cdot$\,
L$\cdot$I$\cdot$T$\cdot$E$\cdot$R$\cdot$A$\cdot$R$\cdot$U$\cdot$M.
\end{center}
He disliked the country, and while his family enjoyed the sea-side, 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 mecting 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
\textit{Trigonometry and Double Algebra}, published in I849; and his
earlier view of formal logic is found in a volume published in 1847. His
most unique work is styled a \textit{Budget of Paradoxes}; it originally
appeared as letters in the columns of the \textit{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
\textit{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
\textit{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 \textit{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, \textit{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:
\renewcommand{\theenumi}{\Roman{enumi}}
\renewcommand{\labelenumi}{\theenumi.}
\begin{enumerate}
\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. $a^b\times a^c=a^{b+c}$,
$(a^b)^c=a^{bc}$, $(ab)^c=a^cb^c$
\item $a-a=0$, $a\div a=1$.
\end{enumerate}
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 pre- ceding 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 law of association. If the commutative law fails, the
associative may hold good; but not \textit{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 \textit{a priori} knowledge of the mind.
De Morgan's work entitled \textit{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 \textit{arithmetic}, where
numbers only appear and symbols of operations such as +, $\times$, etc.
The next stage is \textit{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,
\textit{provided the operation is possible}. The third stage is
\textit{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 \textit{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 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
\textit{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 uutil the problem
was taken up by 1-lamilton. We now see the reason clearly: the symbol of
double algebra denotes not a length and a direction; but a multiplier
and \textit{an angle}. In it the angles are confined to one plane; hence
the next stage will be a \textit{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 \textit{History of
the Inductive Sciences}, and \textit{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 \textit{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 \textit{Some $M$'s are $A$'s}, and \textit{Some
$M$'s 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 \textit{Most $M$'s are $A$'s} and \textit{Most $M$'s 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
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 \textit{The whole of $A$ is the
whole of $B$}, and \textit{The whole of $A$ 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 \textit{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 \textit{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 \textit{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 \textit{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 paradoxer, and his system a 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 brouoht forward would now be called \textit{crotchets}, which is
the nearest word we have to old 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 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.\ .\ .\ .\ 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 \textit{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 \textit{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 lames Smith, a
successful merchant of Liverpool. He found $\pi=3\frac{1}{8}$. His mode
of reasoning was a curious caricature of the \textit{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 $3\frac{1}{8}$ is the true value. The following is a
specimen of De Morgan's dragaing 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 \textit{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 \textit{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 \textit{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 Moroan 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
\textit{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 \textit{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.
\begin{flushleft}
From A~Macfarlane, \textit{Lectures on Ten British Mathematicians of the
Nineteenth Century}, New York: Wiley and London: Chapman and Hall 1916,
pp.\,19--33.
\end{flushleft}
\end{document}
%