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\begin{document}

\title{Discrete Logarithms}
\subtitle{Mathematics and Art}
\author{David Pierce}
\date{December 17, 2017--April 4, 2018}
\publishers{Matematics Department\\
Mimar Sinan Fine Arts University, Istanbul\\
\url{mat.msgsu.edu.tr/~dpierce/}\\
\url{polytropy.com}}

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\addchap{Introduction}

The core of this document is the two tables,
of logarithms and antilogarithms respectively,
constituting Chapter \ref{ch:lists}.
The numbers in the tables may appear to be random.
However, you can check in specific cases
that each table undoes the other:
for example, since the first table gives 564 as the logarithm of 319,
the second table inevitably gives 319 as the antilogarithm of 564.
The antilogarithm of 1 is 7,
and each successive antilogarithm
is either 7 times the previous,
or else it is the remainder of that multiple
after division by 997.
Therefore the logarithms can be used
as Briggsian or common logarithms once were,
for computing products by taking sums.
The logarithm of the product is the sum of the logarithms,
though sums now are taken \emph{modulo} 996,
and products \emph{modulo} 997.

In the terminology of Euler,
7 is a \emph{primitive root} of 997.
Chapter \ref{ch:math} 
reviews the mathematics,
from Euclid to Gauss and beyond.
If sufficiently interested, 
the layperson may follow the review,
while the professional may still find something new.

Anyone may contemplate the tables as conceptual art.
I consider art as such
in Chapter \ref{ch:art},
mainly through the work of R. G. Collingwood,
but also Mary Midgley, Arthur Danto, and others.
I review other examples of conceptual art.

I quote theory and scholarship,
poetry and fiction,
mostly from books in my personal collection.
The quotations may be considered
as if they were readymades of Marcel Duchamp,
or pictures in the exhibition that I am curating.
\chapter{Art}\label{ch:art}

\section{Creation}

What counts as art today is broader than Collingwood contemplated
in 1938 in \emph{The Principles of Art} \cite{Collingwood-PA}. 
Nonetheless, the book remains invaluable.

In writing poems, 
or painting pictures, 
or composing quartets, 
or even---I would add---proving  mathematical theorems,
before you can employ a \emph{technique,}
according to a \emph{plan,}
you have to discover how to do everything in the first place.
This need seems easily overlooked.
Collingwood points it out.
In creating your work of art,
you cannot say---%
you cannot \emph{express}---%
in any precise way, 
what you are trying to do,
before figuring out how to it.
The figuring out is precisely the expressing of it.

Expression is the key word.
As Collingwood says on his page 151,
\begin{quote}
  By creating for ourselves an imaginary experience or activity, 
we express our emotions; and this is what we call art. 
\end{quote}
This is not a conclusion, but a halfway point;
the text will end on page 336.
It is important to read further, here into page 152:
\begin{quote}
  What this formula means, we do not yet know. We can annotate it word
  by word; but only to forestall misunderstandings, thus. 
`Creating' refers to a productive activity
which is not technical in character. 
`For ourselves' does not exclude `for others'; 
on the contrary, it seems to include that; 
at any rate in principle. 
`Imaginary' does not mean anything in the least like `make-believe', 
nor does it imply that what goes by that name 
is private to the person who imagines. 
The `experience or activity' seems not to be sensuous, 
and not to be in any way specialized: 
it is some kind of general activity in which the whole self is involved. 
`Expressing' emotions is certainly not the same thing as arousing them. 
There is emotion there before we express it\lips 
\end{quote}
We are faced now with three problems: to understand
(1) imagination, 
(2) emotion, and 
(3) their connection.
\begin{quote}
  These problems must be dealt with\lips not by continuing to concentrate
  our attention on the special characteristics of aesthetic
  experience, but by broadening our view, so far as we can, until it
  covers the general characteristics of experience as a whole.
\end{quote}
I propose to consider this broadened view as encompassing mathematics.

I say that art and mathematics are creations.
You may disagree.
In \emph{Heart and Mind} from 1981,
in the chapter called Creation and Originality, 
Mary Midgley takes issue with the treatment of creation 
by Collingwood and others, especially Nietzsche and Sartre 
\cite[pp.\ 49--67]{Midgley-HM}.
She begins with the importance of her subject,
which is morality rather than art as such.
\begin{quote}
  The creation of moral values is a pressing topic because, 
whether we use words like \emph{creation} or not, 
we all need to find new moral ideas 
to help us deal with a confused and changing world. 
The notion that these ideas must be totally new, 
that they should not rest at all on traditional supports, 
exists and concerns us all. 
\end{quote}

The God of Genesis calls light into existence
and \emph{then} sees that it is good
\cite{KJV-Oxford}.
God causes dry land to appear and \emph{then} sees it as good.
Likewise with grass, herb, and tree, and with the lights of heaven, 
and so forth:
first they are created, and then they are evaluated.
Not even God just \emph{declares} what is good:
its existence is by fiat, but not its goodness.

As for ourselves,
if we are no longer going to take our values from heaven,
there is no sense in trying to do what not even its mythical ruler can do.
This is what I understand Midgley to argue.
``If God is really dead,'' she says, 
``why should we dress up in his clothes?''
We cannot just will things into existence,
especially not goodness:
\begin{quote}\sloppy
  The human will is not a mechanism 
for generating new thoughts out of nothing. 
It is a humble device for holding onto the thoughts which we have got 
and using them. 
\end{quote}

The will then is not creative, but preservative.
It may thus be humble, but it is still essential.
Students need it,
especially when they carry around the little electronic devices 
that are designed to draw their attention---to draw and quarter it,
one might say.
The student needs attention, application, \emph{persistence,}
as I observed elsewhere \cite[p.\ 245]{MR3312989}.
As expressing the thought,
I quoted one of William Blake's ``Proverbs of Hell''
from \emph{The Marriage of Heaven and Hell} \cite[plate 7]{Blake}:
\begin{quote}
\centering
  If the fool would persist in his folly he would become wise.
\end{quote}

That the creativity of civilization depends on persistence
is an argument for the \emph{rise}
of what Julian Jaynes calls the bicameral mind,
although the title of his 1976 book is
\emph{The Origin of Consciousness in the Breakdown of the Bicameral Mind.}
Other animals go about their business naturally,
but civilization is an unnatural business.
It requires us, in youth and later,
to do things that we do not see the point of.
The \emph{will} to do these things has needed to evolve.
For Jaynes, one stage in this evolution
was the hearing of voices that kept us at work.
``Let us consider a man,'' he says \cite[pp.\ 134 f.]{Jaynes},
\begin{quote}
  commanded by himself or his chief to set up a fish weir far upstream
  from a campsite.  If he is not conscious, and cannot therefore
  narratize the situation and so hold his analog `I' in a spatialized
  time with its consequences fully imagined, how does he do it?  It is
  only language, I think, that can keep him at this time-consuming
  all-afternoon work.  A Middle Pleistocene man would forget what he
  was doing.  But lingual man would have language to remind him,
  either repeated by himself, which would require a type of volition
  which I do not think he was then capable of, or, as seems more
  likely, by a repeated `internal' verbal hallucination telling him
  what to do\lips learned activities with no consummatory closure do
  need to be maintained by something outside of themselves.  This is
  what verbal hallucinations would supply.
\end{quote}

How we have come to be where we are is indeed a puzzle,
though I shall not dwell on Jaynes's attempt at a solution.
We can think of the puzzle both on a ``special'' scale---%
the scale of our species---%
and on a personal scale,
as Collingwood does in his last book, from 1942,
\emph{The New Leviathan: 
Or Man, Society, Civilization, and Barbarism.}
Here Collingwood takes issue with the notion of Rousseau that
``Man is born free, and everywhere he is in chains.''

``I do not doubt,'' says Collingwood
\cite[p.\ 176]{Collingwood-NL},
``that truths, and important truths, can be told in Rousseau's language.''
However,
\begin{quotation}
  \textbf{23.\ 93.} In human infancy the fact, 
as known to me at least, 
is that a man is born neither free nor in chains. 

\textbf{23.\ 94.} To be free is to have a will unhampered by external force, 
and a baby has none. 

\textbf{23.\ 95.} To be in chains is to have a will hampered by
  something which prevents it from expressing itself in action; and a
  baby has none. 

\textbf{23.\ 96.} A man is born a red and wrinkled lump of flesh
  having no will of its own at all, absolutely at the mercy of the
  parents by whose conspiracy he has been brought into
  existence. 

\textbf{23.\ 97.} That is what no science of human community, social
  or non-social, must ever forget.
\end{quotation}

I wonder whether Midgley forgets these facts
in \emph{Heart and Mind.}
She does recognize that creation can be perceived 
on a smaller scale than Genesis.
Indeed, she quotes Collingwood from
\emph{The Principles of Art} as showing this.
Here he is,
in an expansion of Midgley's quotation \cite[pp.\ 128 f.]{Collingwood-PA}.
\begin{quotation}\noindent
Readers suffering from theophobia will certainly by now have taken offence\lips
Perhaps some day, with an eye on the Athanasian Creed, 
they will pluck up courage to excommunicate an arithmetician 
who uses the word three. 
Meanwhile, readers willing to understand words instead of shying at them 
will recollect that the word `create' is daily used in contexts 
that offer no valid ground for a fit of \emph{odium theologicum}\lips

To create something means to make it non-technically, 
but yet consciously and voluntarily. 
Originally, \emph{creare} means to generate, or make offspring, 
for which we still use its compound `procreate'\lips 
The act of procreation is a voluntary act, 
and those who do it are responsible for what they are doing; 
but it is not done by any specialized form of skill\lips
It is in this sense that we speak of creating 
a disturbance or a demand or a political system. 
The person who makes these things is acting voluntarily; 
he is acting responsibly; 
but he need not be acting in order to achieve any ulterior end; 
he need not be following a preconceived plan; 
and he is certainly not transforming anything 
that can properly be called a raw material. 
It is in the same sense that Christians asserted, 
and neo-Platonists denied, that God created the world. 
\end{quotation}
Midgley objects,
in a way that suggests to me 
that she has not really thought about what it means to grow up,
or even what it means to compose an essay such as her own.

\section{Gender}

Midgley's experience of writing and life is no doubt different from mine.
An important difference
is connected to the English gendered pronouns.
Our \emph{first}-person pronouns are epicene;
but in the third person, I become he, while Midgley is she.

The distinction is not imposed on us by nature.
Each of us,
including objects thought to be inanimate, 
is simply \emph o in Turkish,
which is a language ``born free'' of 
``the curse of grammatical gender''
\cite[II.26, p.\ 48]{Lewis2}.
In English, we have a vestige of the curse,
a vestige that can either reflect
differences in experience,
or \emph{effect} them.

In 2013,
Midgley wrote to \emph{The Guardian} as follows
\cite{Midgley-Guardian}.
She was responding to the question of
``why, 
though five quite well-known female philosophers emerged from Oxford 
soon after the war, few new ones are doing so today.''
\begin{quotation}
As a survivor from the wartime group, I can only say: 
sorry, but the reason was indeed that there were fewer men about then. 
The trouble is not, of course, men as such---%
men have done good enough philosophy in the past. 
What is wrong is a particular style of philosophising 
that results from encouraging a lot of clever young men 
to compete in winning arguments. 
These people then quickly build up a set of games 
out of simple oppositions and elaborate them until, in the end, 
nobody else can see what they are talking about. 
All this can go on 
until somebody from outside the circle finally explodes it\lips
By contrast, in those wartime classes---which were small---men 
(conscientious objectors etc) were present as well as women, 
but they weren't keen on arguing.

It was clear that we were all more interested 
in understanding this deeply puzzling world than in putting each other down. 
That was how Elizabeth Anscombe, Philippa Foot, Iris Murdoch, 
Mary Warnock and I, in our various ways, 
all came to think out alternatives to the brash, 
unreal style of philosophising---based essentially on logical positivism---%
that was current at the time. 
\end{quotation}

It is unfortunate that war had to create an opportunity,
both for women to pursue and develop their thoughts,
and for men to learn from them, as I have learned from Midgley.
In \emph{Evolution As a Religion,}
she rightfully critiques the presumption of some scientists
(generally male) in making grand pronouncements on the meaning of life
from physical theories.
She quotes Steven Weinberg as saying,
in an ``excellent and informative little book,''
\begin{quotation}\noindent
The more the universe seems comprehensible, the more it also seems pointless. 

  But\lips The effort to understand the universe is one of the very few things
  that lifts [\emph{sic}] human life a little above the level of
  farce, and gives it some of the grace of tragedy.
\end{quotation}
Midgley observes
\cite[p.\ 87]{Midgley-ev},
\begin{quote}
Since virtually the whole book has been devoted to expounding astrophysics, 
not to discussing it as an occupation, 
and certainly not to discussing other occupations with which it might compete, 
Weinberg's readers might find this an unexpected blow. 
They might feel rather shaken and degraded by the sudden revelation 
that their lives are probably valueless, 
and they might also ask the reasonable question: how does Weinberg know?
\end{quote}
Obviously Weinberg is only giving his opinion.
The problem is not the rudeness of stating such an opinion,
but the unscientific practice of deriving the opinion \emph{from} science,
rather than recognizing it as connected with 
why one has done science in the first place.

Here I may have passed to my own thought, only prompted by Midgley.
Our subject was art and creation,
and I still wonder whether Midgley has understood Collingwood
when she says
\cite[pp.\ 64 f.]{Midgley-HM},
\begin{quote}
  It may seem that at this point the word `create' has been diluted
  into complete triviality, that it simply means `make'. But it still
  keeps an awkward core of special meaning, and one that is important
  for Collingwood's theory of art. On his view, creators need not,
  indeed characteristically do not, know in advance what they are
  going to make. He sees the absence of a `preconceived end' as a mark
  of real art, a mark which distinguishes it from mere craft. But if
  you really do not know what you are trying to bring about, it is
  hard to see how you can do it, and harder still to see how you can
  be called responsible. Artists don't in fact often talk in this
  way. They are often quite willing to discuss their aims and
  problems. But whether or not sense can be made of this for art, in
  morals it is surely a non-starter.
\end{quote}
This shows the difficulty of understanding Collingwood.
He does indeed distinguish art from craft;
but there is no X-ray machine
that you can feed artefacts into,
and a light flashes green for art, red for craft.
The same object has aspects of both.
It is not even the \emph{physical}
object that can be a work of art at all.
We shall come back to this later, on page \pageref{physical}.

After taking an examination,
students want to know how they did.
If they do not already know this,
just from what they themselves have written on their papers,
then they must not have had a preconceived end in any precise sense.
They want a good grade, but they do not know what this really means.
If they have done well, according to their teacher,
they may still be proud, and they have some right to be,
since they are responsible for what they did.

I had an aim when I set out to write this essay.
I could have talked about the aim in general terms.
But the aim has grown, and grown precise,
just as the essay has taken shape.
In particular, at the beginning,
I had no idea of the current sectional divisions of this essay.

\section{Individualism}

This essay is an \emph{expression.}
The term was key for art of Collingwood's time,
notably that of the \emph{Blaue Rieter} group, 
formed in Munich by Franz Marc in 1911.
According to Herbert Read in \emph{A Concise History of Modern Painting}
\cite[p.\ 228]{Read},
\begin{quote}
  \emph{Blaue Reiter} was the first coherent attempt 
to show that what matters in art---%
what gives art its vitality and effect---%
is not some principle of composition or some ideal of perfection, 
but a direct expression of feeling, 
the form corresponding to the feeling, 
as spontaneous as a gesture, but as enduring as a rock.
\end{quote}

Read begins his book with a long quotation from Collingwood's 1924 book,
\emph{Speculum Mentis or The Map of Knowledge.}
The idea is that,
``in art, a school once established normally deteriorates as it goes on''
 \cite[p.\ 82]{Collingwood-SM}.
Collingwood's ideas themselves continued to develop.
He published \emph{Outlines of a Philosophy of Art} in 1925,
but updated his views a dozen years later in \emph{The Principles of Art.}
Concerning the quotation that Read makes, but does not really analyze,
from \emph{Speculum Mentis,}
I suggest that a school of art, once founded, declines,
precisely because its very foundation 
constitutes the identification of a technique,
and technique is not art.

In its article on Aesthetics, 
the \emph{Internet Encyclopedia of Philosophy}
\cite{iep-collingwood} is misleading to suggest that Collingwood
``took art to be a matter of self-expression.'' 
There was no need to add the restriction to the self.
This assertion in the \emph{Encyclopedia} 
is indeed followed by the formula from 
\emph{The Principles of Art}
quoted above,
whereby art is a creating for our \emph{selves.}
However, if one reads beyond the formula, also as above,
then one sees how Collingwood was at pains to keep references to the self
from being misunderstood.
Creating art for ourselves
includes doing it for others.
One's imagination need not be private to oneself.

The central lesson of \emph{mathematics} 
is that each of us has the right to decide, 
for her- or himself, what is true.
Mathematical truth does not come down from heaven,
but comes up from within each of us.
It is like art in this way.

Mathematical truth is nonetheless common.
In mathematics,
we have the responsibility of resolving disputes amicably,
because anything on which there is fundamental disagreement
is not mathematics.

It may not be art either.

What is \emph{liked} may differ from person to person,
whether we are talking about art or mathematics.
Some mathematicians do not like the method of proof by contradiction.
They should still agree on 
whether a given proof by contradiction is \emph{correct}
as a proof by contradiction.
Likewise should we all be able to agree on whether something is art;
but the truth of this assertion 
is not so clear as the corresponding one for mathematics.
This is a \emph{practical} reason
why everybody should learn some mathematics:
it teaches the possibility, if not the obligation,
of peaceful resolution of differences.

The theme that what is \emph{mental}
need not be merely \emph{personal}
goes back to Collingwood's first book,
\emph{Religion and Philosophy} of 1916 \cite[p.\ 93]{Collingwood-RP}.
In the chapter called ``Matter,''
concerning this as distinguished from mind,
Collingwood wrote,
\begin{quote}
  A boot is more adequately described
in terms of mind---by saying who made it and what he
made it for---than in terms of matter.
 And in the case
of all realities alike, it seems that the materialistic
insistence on their objectivity is too strong; for it is
not true that we are unable to alter or create facts,
or even that we cannot affect the course of purely
``inanimate'' nature. Materialism, in short, is right
as against those theories which make the world an
illusion or a dream of my own individual mind; but
while it is right to insist on objectivity, it goes too far
in describing the objective world not only as something
different from, and incapable of being created or
destroyed by, my own mind, but as something different
and aloof from mind in general.
\end{quote}
Again, though art be expression, 
it is not \emph{self}-expression as such.

In \emph{The Principles of Art,}
even before formulating the tentative definition of art that we have seen,
Collingwood argues that art is not merely a private concern.
Art is for the world, for civilization,
even though civilization may not respect this 
\cite[pp.\ 33 f.]{Collingwood-PA}:
\begin{quote}
Here lies the peculiar tragedy of the artist's position in the modern world. 
He is heir to a tradition from which he has learnt what art should be; 
or at least, what it cannot be. 
He has heard its call and devoted himself to its service. 
And then, when the time comes for him to demand of society 
that it should support him in return for his devotion to a purpose which, 
after all, is not his private purpose 
but one among the purposes of modern civilization, 
he finds that his living is guaranteed 
only on condition that he renounces [\emph{sic}] his calling 
and uses [\emph{sic}] the art which he has acquired 
in a way which negates its fundamental nature, 
by turning journalist or advertisement artist or the like; 
a degradation far more frightful 
than the prostitution or enslavement of the mere body. 
\end{quote}
It \emph{is} disappointing that,
in closing this passage,
Collingwood takes up the mind-body dualism 
that he refuted in \emph{Religion and Philosophy.}
One might say, echoing him there,
``Prostitution is more adequately described in terms of mind---%
by saying it compromises one's capacity to love and be loved---%
than in terms of matter.''

Collingwood reiterates the universality of art
at the end of \emph{The Principles of Art} \cite[p.\ 333]{Collingwood-PA},
where he observes first (writing before 1938)
that English painting and literature aim
no longer just to amuse the wealthy, but to be competent as art.
\begin{quotation}\noindent
  But the question is whether this ideal of artistic competence
is directed backwards into the blind alley of nineteenth-%
century individualism, where the artist's only purpose was to
express himself, or forwards into a new path where the
artist, laying aside his individualistic pretensions, walks as
the spokesman of his audience.

In literature, those who chiefly matter have made the
choice, and made it rightly. The credit for this belongs in
the main to one great poet, who has set the example by
taking as his theme in a long series of poems a subject that
interests every one, the decay of our civilization.
\end{quotation}
The poet is T. S. Eliot.
Collingwood's conclusion is preceded by theory.
After the formula for art from his page 151 quoted earlier,
in starting to develop a theory of the imagination, 
Collingwood distinguishes thought from feeling.
One distinction is that while feelings are private,
thoughts are potentially public, or held in common
\cite[p.\ 157]{Collingwood-PA}.
One's own feeling of cold has no relation to anybody else's;
but the thought that a house is ten degrees Celsius
is the same for everybody in the house who has the thought.

\section{Eros}

By bringing feelings into consciousness, 
art allows them to be shared.
Art is ultimately identified with \emph{language.}
This is not language as a system for communication:
developing such a system requires language in the first place.

We are talking about language such as Archimedes uttered,
when he exclaimed ``Eureka!''\
in the story told by Vitruvius \cite[pp.\ 36 f.]{MR13:419b}.
The expression of the mathematician 
was not just the first-person singular perfect form \gr{e<'urhka}
of the verb \gr{e<ur'iskw} ``find'';
it was the cry of a thinker who had just understood
how to test the golden crown of King Hiero
for adulteration with silver.
``And if there had been among the passers-by,''
suggests Collingwood \cite[p.\ 267]{Collingwood-PA},
\begin{quote}
  a physicist as great as Archimedes himself, who had come to Syracuse
  in order to tell Archimedes that he had discovered specific gravity,
  it is not impossible that he might have understood the whole thing,
  and burst from the crowd, shouting, `So have I!'
\end{quote}

Collingwood admits that the imaginary example involving Archimedes is
``extreme and fantastic.''
So is John Donne's argument about language and perception 
in his poem 
``The Extasie''
(of the early seventeenth century),
comprising 76 lines \cite[pp.\ 39--41]{Donne}.
Donne and his beloved sit all day,
holding hands,
staring into one another's eyes,
``Our eye-beams twisted'':
\begin{quote}
  \begin{verse}
\poemlines2\setverselinenums{21}{22}
    \begin{altverse}
      If any, so by love refin'd,\\
That he soules language understood,\\
And by good love were growen all minde,\\
Within convenient distance stood,\\
He (though he knew not which soul spake,\\
Because both meant, both spake the same)\\
Might thence a new concoction take,\\
And part farre purer than he came.\\!
    \end{altverse}
  \end{verse}
\end{quote}
The refined soul speaks the language 
in which the love of the chaste couple is expressed;
but many souls are not so refined,
and so, for \emph{their} sake,
the couple ought to be more physically entwined.
\begin{quote}
  \begin{verse}
\poemlines2\setverselinenums{69}{70}
    \begin{altverse}
To'our bodies turne we then, that so\\
Weake men on love reveal'd may looke;\\
Loves mysteries in soules doe grow,\\
But yet the body is his booke.\\
And if some lover, such as wee,\\
Have heard this dialogue of one,\\
Let him still marke us, he shall see\\
Small change, when we'are to bodies gone.\\!
    \end{altverse}
  \end{verse}
\end{quote}

Let me suggest in passing that,
if a man today really does fear to approach a woman,
lest he be accused of harrassment,
then let him try writing a poem like Donne's.
It may not get him what he wants,
but he may learn something else.

Language may be used for \emph{self}-expression,
but this was not any more commendable for Collingwood
than it was for E. B. White,
who wrote in his contribution to \emph{The Elements of Style}
in the 1950s \cite[p.\ 59]{Strunk-White},
\begin{quote}
  The volume of writing is enormous,
these days,
and much of it has a sort of windiness about it,
almost as though the author were in a state of euphoria.
``Spontaneous me,'' sang Whitman,
and in his innocence let loose the hordes of uninspired scribblers
who would one day confuse spontaneity with genius.
\end{quote}

I do not know whether White meant to allude to the erotic \emph{content}
of Whitman's actual poem.
Any poem is a list of lines;
most of the 45 lines of ``Spontaneous Me'' \cite[pp.\ 89--91]{Whitman}
are longer than an ordinary printed page is wide,
and most of them are noun phrases, or series of noun phrases,
serving as the subject, or rather as an appositive to the subject,
of one long sentence,
whose verb does not come till the last line:
\begin{quote}
%\setlength{\leftmarginii}{0em}
  \begin{verse}
%\relscale{0.9}
\poemlines2\setverselinenums11
%\verselinenumbersleft
    Spontaneous me, Nature,\\
The loving day, the mounting sun, the friend I am happy with,\\
The arm of my friend hanging idly over my shoulder,\\
The hillside whiten'd with blossoms of the mountain ash,\\
The same late in autumn, the hues of red, yellow, drab, purple, and light and dark green,\\
\poemlines0\makebox[\linewidth]{\dotfill}\\
\poemlines2\setverselinenums{39}{39}
The consequent meanness of me should I skulk or find myself indecent, while birds and animals never once skulk or find themselves indecent,\\
The great chastity of paternity to match the great chastity of maternity,\\
The oath of procreation I have sworn, my Adamic and fresh daughters,\\
The greed that eats me day and night with hungry gnaw, till I saturate what shall produce boys to fill my place when I am through,\\
The wholesome relief, repose, content,\\
And this bunch pluck'd at random from myself,\\
It has done its work---I toss it carelessly to fall where it may.\\!
  \end{verse}
\end{quote}
The ellipsis stands for lines 
that are likewise interesting and graphic in themselves,
but that go on and on,
with a logic that may be as obscure as the logic 
of the list of logarithms
excerpted below in Table \ref{tab:log}
(page \pageref{tab:log}) 
and given subsequently in full in Chapter \ref{ch:lists}.

\section{Analysis}

I used \emph{Religion and Philosophy}
to illustrate \emph{The Principles of Art.}
I think one can do this,
even though Collingwood disavowed the earlier book,
soon after publication.  Around 1918, 
he added the following remarks to the proofs, 
which he had saved and bound
\cite[pp.\ xxii f.]{Collingwood-EPM}:
\begin{quote}
  This book 
was written in (and before) 1914 (begun 1912) and 
represents the high-water mark of my earliest line of 
thought---dogmatic belief in New Realism in spite of an 
insight into its difficulties which I think none of my teachers
shared\lips The whole thing represents a point of view
I should entirely repudiate, and its complete failure with the 
public gives me great satisfaction.
\end{quote}
The ``new realists'' 
were apparently the early exponents of so-called analytic philosophy.
I wonder if Collingwood isn't little known today,
precisely because of his distancing of himself
from what became analytic philosophy.

Stephen Trombley describes the general situation in 
\emph{Fifty Thinkers Who Shaped the Modern World.}
Unfortunately the book has but a single bibliography, 
and no notes, 
and so Trombley's sources are not clear; 
neither is there an index,
but Trombley seems not to name Collingwood.
Nonetheless,
some of what Collingwood has to say in his 1939 autography
is reflected in Trombley's chapter on F. H. Bradley
\cite[p.\ 115]{Trombley}:
\begin{quotation}\noindent
In the period between 1850 and 1903
there wasn't a school of British \emph{idealism,}
there was simply \emph{British} philosophy,
the general tendency of which was idealist.
`British idealism' is better regarded as a pejorative term
created by early analytic philosophers
to identify the status quo they wished to supplant with their own brand of thinking.
The strange death of idealism in British philosophy
goes hand in hand with philosophy's transformation 
from a gentleman's pastime into a profession\lips
[T. H.] Green's career is a milestone in the history of philosophy
because, according to the utilitarian Henry Sidgwick (1838--1900),
he was the first \emph{professional} philosopher in the English-speaking world.

The early analytic philosophers' war on British idealism
can be seen to involve much more than 
the desire to supplant neo-Hegelian idealism and metaphysics in its entirety
with logicism:
they also wanted the idealists' jobs.
The analytic side won both battles.
The professionalization of philosophy in Britain and the United States
resulted in the death of idealism and the erection of analytic philosophy
as the official way of thinking;
in this way a generation of teachers led by Russell,
Moore and Wittgenstein spawned a new generation of followers,
who in turn kept the analytic torch burning brightly
in the English-speaking world throughout the twentieth century
as their students and their students' students took up university teaching jobs.
(There are notable exceptions\lips)
\end{quotation}
In \emph{An Autobiography} \cite{Collingwood-Auto},
Collingwood admires what he calls the school of Green.
Those who charted a different course from Green's,
by devaluing thought,
by teaching such doctrines
as Cook Wilson's ``knowing makes no difference to what is known'':
they laid the ground for British support 
of Spanish fascism and German Nazism,
at least as of November 2, 1938, the date of the Preface of
\emph{An Autobiography.}
(The Munich agreement was signed on September 29 of that year
\cite[p.\ 250]{WDBH}.)

In \emph{What Art Is} of 2013,
Arthur Danto
considers art that Collingwood did not live to see.
However, Danto works in the analytic tradition quite literally,
dividing up philosophy into components of ontology and epistemology
 \cite[p.\ 5]{Danto}.
\begin{quote}
  When they see work that puzzles them, people ask, ``But is it art?''
  At this point I have to say that there is a difference between
  \emph{being} art and knowing whether something \emph{is} art.
  Ontology is the study of what it means to be something.  But knowing
  whether something is art belongs to epistemology---%
  the theory of knowledge---%
  though in the study of art it is called connoisseurship.  This book
  is intended mostly to contribute to the ontology of Art,
  capitalizing the term that it applies to widely---%
  really to everything that members of the art world deem worthy of
  being shown and studied in the great encyclopedic museums.
\end{quote}
The encyclopedic museums are such as the Metropolitan in New York
or the National Gallery in Washington,
as Danto has said on the previous page.

\section{Concepts}

What then is art?
Danto wants a \emph{definition.}
He is not satisfied with the idea from Wittgenstein
that works of art need share only a family resemblance
\cite[pp.\ 29--34]{Danto}.
Neither does Danto seem to like the idea of the ``open concept,''
attributed to Morris Weitz in 1956.
The Institutional Theory of art developed by George Dickie in the 1960s
is inadequate since, in Danto's example,
the head of the National Museums of Canada,
despite his leading position in the Art World, 
was able to be wrong in denying artistic status
to those peculiar works, discussed below,
called \emph{readymades.}

We might show further the inadequacy of the Institutional Theory
by observing that poems and music can be art,
but are not the kind of thing that is displayed in a museum.
Of course they may be given official status in other ways.
However, despite or because of this official status, 
a national anthem, or the output of a poet laureate,
is not art;
it is the kind of craft called \emph{magic}
in \emph{The Principles of Art.}
We shall return to this on page \pageref{magic}.
Meanwhile, even though Danto uses the term \emph{art}
to mean \emph{visual} art,
implicitly excluding poetry and music,
his theme is that what makes something art is invisible.

\begin{sloppypar}
Key works for Danto's considerations
are 
(1) Marcel Duchamp's 1915 readymade called 
\emph{In Advance of the Broken Arm,}
which was a snow shovel from a hardware store on Columbus Avenue in New York,
and 
(2) Andy Warhol's \emph{Brillo Box,} or boxes, of the 1960s.
How can these be art,
when they look just like things that are not art?
For Danto \cite[p.\ 37]{Danto},
\begin{quote}
  My sense is that, if there were no visible differences, there had to
  have been \emph{invisible differences}---%
  not invisible like the Brillo pads packed in the Brillo boxes [but
  not in Warhol's boxes], but properties that were \emph{always}
  invisible.  I've proposed two such properties that are invisible in
  their nature.  In my first book on the philosophy of art I thought
  that works of art are \emph{about} something, and I decided that
  works of art accordingly have meaning.  We infer meanings, or grasp
  meanings, but meanings are not at all material.  I then thought
  that, unlike sentences with subjects and predicates, the meanings
  are \emph{embodied} in the object that had them.  I then declared
  that works of art are \emph{embodied meanings.}
\end{quote}
As far as I can tell,
meaning is \emph{one} of the two invisible properties
that Danto has proposed for the work of art.
The other property is being a waking dream \cite[p.\ 48]{Danto}:
\begin{quote}
  I have decided to enrich my earlier definition of art---%
embodied meaning---%
with another condition that captures the skill of the artist.
Thanks to Descartes and Plato,
I will define art as ``wakeful dreams.''
\end{quote}
Danto has turned to Plato and Descartes---%
to the \emph{Meditations} of the latter
and the Divided Line in the \emph{Republic} of the former---%
because they deal with the distinction between dreaming and perceiving,
and this is like the distinction between
Warhol's Brillo boxes and the real thing.
\end{sloppypar}

We all have to make our own way in the world.
In his 1991 philosophical novel 
\emph{Lila} \cite[ch.\ 26, pp.\ 370--2]{Pirsig-L},
Robert Pirsig coins a useful word,
defined by an analogy:
\begin{quote}
  Philosophology is to philosophy
as musicology is to music,
or as art history and art appreciation are to art,
or as literary criticism is to creative writing.
\end{quote}
One might add two more terms to the analogy:
history and philosophy of mathematics, and mathematics itself.
According to Pirsig,
``philosophologists'' put
\begin{quote}\sloppy
a philosophological cart before the philosophical horse.
Philosoph\-ologists not only start by putting the cart first;
they usually forget the horse entirely.
They say first you should read 
what all the great philosophers of history have said
and \emph{then} you should decide what \emph{you} want to say.
The catch here is that by the time you've read 
what all the great philosophers of history have said 
you'll be at least two hundred years old.
\end{quote}
You have to do your own work.
It still seems to me that Arthur Danto
might have saved himself some trouble
by reading a philosopher of art from the previous generation.
If a work of art is an \emph{expression,}\label{physical}
as Collingwood observes,
then it is simply not a physical object.
In particular, it should not be expected 
to have \emph{properties} of physical objects.
Perhaps Danto need not have spent years figuring this out again.

Collingwood's ultimate expression of the idea
is in the first two chapters of his last book, 
quoted earlier, namely \emph{The New Leviathan.}
We are not made up of two parts, called body and mind.
We rather have two ways of thinking.
In their most refined forms, these ways can be called, respectively,
(1) sciences of nature, physical sciences, or sciences of body, and
(2) sciences of mind.
Here is Collingwood \cite[pp.\ 7--11]{Collingwood-NL}.
\begin{quotation}
  \textbf{1.\ 83.} 
Man as body is \emph{whatever the sciences of body say that he is.}
Without their help nothing can be known on that subject: 
their authority, therefore, is absolute. 

\textbf{1.\ 84.} Man as mind is \emph{whatever he is conscious of being.} 

\dotfill

\textbf{2.\ 43.}
 For man's body and man's mind are not two different things. 
They are one and the same thing, man himself, as known in two different ways. 

\textbf{2.\ 44.} 
Not a part of man, but the whole of man, 
is body in so far as he approaches the problem of self-knowledge 
by the methods of natural science. 

\textbf{2.\ 45.} Not a part of man, but the whole of man, 
is mind in so far as he approaches the problem of self-knowledge 
by expanding and clarifying the data of reflection.

\dotfill

\textbf{2.\ 48\lips} In the natural sciences, 
mind is not that which is left over when explaining has broken down; 
it is what does the explaining\lips
\end{quotation}

Sciences of mind are \emph{criteriological} sciences,
like logic, ethics, history, economics.
They study whether something---some instance of \emph{thinking}---%
is going well or ill.
How this thinking is proceeding is judged
not only by an external standard
(in which case, for its study, 
the term \emph{normative science} might be sufficient);
it is judged by the standards or \emph{criteria} of the thinking itself.

Collingwood introduces the term \emph{criteriological}
in a note in \emph{The Principles of Art} \cite[p.\ 171]{Collingwood-PA},
though the concept itself is found in
\emph{An Essay on Philosophical Method} of 1933.
In this \emph{Essay} 
is also found the reason why it is hard to stick with one subject
when thinking about Collingwood;
for here is where the doctrine of the \emph{overlap of classes}
is introduced \cite[p.\ 35]{Collingwood-EPM}:
\begin{quote}
   Thus art, for the critic, is a highly specialized thing, 
limited to a small and select body of works 
outside which lie all the pot-boilers and failures of artists, 
and the inartistic expressions of everyday life; 
for the aesthetic philosopher, these too are art, 
which becomes a thread running all through 
the fabric of the mind's activity\lips
when a concept has a dual significance, 
philosophical and non-philosophical, 
in its non-philosophical phase it qualifies a limited part of reality, 
whereas in its philosophical it leaks or escapes out of these limits 
and invades the neighbouring regions, 
tending at last to colour our thought of reality as a whole. 
\end{quote}

\section{Practice}

The leakage of concepts is not very satisfactory 
for one who likes things tidy.
Nonetheless, it happens.
In particular, the ``inartistic expressions of everyday life''
have come to be considered as art by practicing artists.

Danto already knew that art could be considered as immaterial.
At least he was aware of the idea,
attributed to Harold Rosenberg,
``that what abstract painters did was perform an action on a canvas,
the way a bullfighter performs an action in the ring''
\cite[p.\ 11]{Danto}.
One could let this idea leak out,
so that all art would become an act of expression,
as it is for Collingwood;
but Danto does not seem to have been quite ready for this.

Collingwood spends half of \emph{The Principles of Art}
in formulating a sort of definition of art,
because the concept needs to be distinguished from overlapping concepts 
such as craft, amusement, and magic.\label{magic}
Craft is doing things with a technique, for a purpose.
Craft may \emph{arouse} emotion,
either for its own sake, as in amusement,
or else, as in \emph{magic,}
for something useful beyond itself, such as social control.

Danto mentions some forgeries of Warhol Brillo boxes.
Apparently they were intended to deceive,
for pecuniary gain,
since bidding on authentic Warhol boxes at auction,
when possible at all, started at two million dollars
\cite[p.\ 50]{Danto}.
Here we are in the realm of magic,
where an industry has been created to manipulate feelings about art,
and people care about the \emph{provenance} of a box,
regardless of whether the box itself
helps them to express some artistic feeling.

As I suggested at the beginning,
Collingwood did not live to see the term \emph{art} broadened
to cover examples like \emph{Brillo Box} that Danto considers.
Walking into a building of the University of California at Berkeley,
in order hold an informal seminar
\cite[p.\ 19]{Danto},
\begin{quote}
  I walked past a large classroom which was being painted.
The room contained ladders, drop clothes,
cans of wall paint and turpentine,
and brushes and rollers.
I suddenly thought:
what if this is an installation titled \emph{Paint Job}?
\end{quote}
Danto mentions just such an installation by
``the Swiss artistic duo Fischli and Weiss.''
It seems to me that Danto has the right spirit here.
Such installations should be seen as a way to find art
in our own ordinary lives.

When I was a sophomore in Santa Fe in 1984--5,
at the college called St John's that I have described elsewhere
\cite{Pierce-SJC},
a guest lecturer mentioned an artist 
who had asked maintenance workers
to consider one hour of their daily work as art.
Their work could thus have been the kind of thing
that Danto imagined in Berkeley as \emph{Paint Job.}

I did not remember the name of the artist,
but rediscovered her work in 2013,
in the 13th Istanbul Biennial
\cite[pp.\ 184--7]{IB-2013}.
After the labor of giving birth,
Mierle Laderman Ukeles came to think of maintenance work as art.
She issued \emph{Manifesto for Maintenance Art 1969!}
Her work called \emph{I Make Maintenance Art One Hour Every Day}
was carried out over seven weeks in 1976 with 
``300 sky-rise service personnel.''

I do not know what those service personnel made of their service as artists.
Possibly they acted as if serving a deity,
as enjoined by Jesus of Nazareth
when describing Judgment Day in Matthew 25:
\begin{quotation}
40  And the King shall answer and say unto them,
Verily I say unto you,
Inasmuch as he have done \emph{it}
unto one of the least of my brethren,
ye have done \emph{it} unto me.
\end{quotation}
This is why, as Zooey recalls to Franny, 
Seymour told him to shine his shoes,
even when appearing on a \emph{radio} program,
in the story of J. D. Salinger.
Zooey should shine his shoes for the Fat Lady
\cite[pp.\ 198--200]{Salinger-FZ}.
\begin{quote}
  But I'll tell you a terrible secret---%
\emph{There isn't anyone out there who isn't Seymour's Fat Lady}
[\lips] And don't you know---\emph{listen} to me, now---%
\emph{don't you know who that Fat Lady really is?}
\lips Ah, buddy.  Ah buddy.
It's Christ Himself.  Christ Himself, buddy.
\end{quote}
Service to a deity is presumably why,
by the account of the artist David Macauley
that I have remembered from childhood \cite[p.\ 63]{Macauley},
in the construction of the cathedral of the make-believe
or imaginary town of Chutreaux,
\begin{quote}
  While the windows were being installed,
plasterers covered the underside of the vault
and painted red lines on it
to give the impression that all the stones of the web
were exactly the same size.
They were eager for the web
to appear perfect
even if no one could see the lines from the ground.
\end{quote}
God would see the lines.

Workers as artists could add decorative flourishes,
as in latte art,
or the shamrock in a head of Guinness stout,
or a towel rolled into a swan on a hotel bed.
Workers might only scrub the floors extra hard,
if that is their job.
Is this what Mierle Laderman Ukeles had in mind?

At the 1985 show at the Hirshhorn Museum
called \emph{Representation Abroad} \cite{Rep-Abroad},
I was inspired by the Spanish realists
Antonio L\'opez-Garcia and Isabel Quintanilla
to find artistic visions in everyday life,
even in a bathroom sink or the corner of a basement.
However, Ukeles
enjoined maintenance workers not to \emph{see} art,
but to \emph{be} artists.

Perhaps one cannot just decide to be an artist.
In introducing \emph{Selected Poems of Robert Frost,}
Robert Graves writes \cite[p.\ x]{Frost},
\begin{quote}
  I agree with Frost that a poem planned beforehand never comes off. 
Real ones appear unexpectedly, 
and always at a time when the poet is in a so-called state of grace: 
which means a clear mind, tense heart, 
and no worries about fame, money, or other people, 
but only the excitement of a unique revelation about to be given.
\end{quote}
Can one watch for that state of grace,
to be ready for it, if it should come?

As he describes in \emph{Surely You're Joking} 
\cite[p.\ 166]{Feynman-Joking},
Richard Feynman would seem to have approached the job of teaching
as a chance to receive a state of grace.
\begin{quote}
If you're teaching a class, 
you can think about the elementary things that you know very well.  
These things are kind of fun and delightful.  
It doesn't do any harm to think them over again.  
Is there a better way to present them?  
Are there any new problems associated with them?  
Are there any new thoughts you can make about them?  
The elementary things are \emph{easy} to think about; 
if you can't think of a new thought, no harm done; 
what you thought about it before is good enough for the class.  
If you \emph{do} think of something new, 
you're rather pleased that you have a new way of looking at it.
\end{quote}
The present work itself comes out of teaching.

\section{Numbers}

I have taught number theory a few times 
as an upper-level undergraduate elective,
covering arithmetical functions and their convolution,
primitive roots of all numbers that have them, 
and quadratic reciprocity.
In the first-year course that I recently taught,
I could not go so far.
The main aim
was for the students to learn about proofs, 
perhaps for the first time, 
in the context of real mathematics.
The students were doing the same thing concurrently
in another course,
by reading and presenting to one another
the proofs in Book \textsc i of Euclid's \emph{Elements,}
in the manner of my own aforementioned \emph{alma mater,}
St John's College.

In the number-theory course,
induction yields the basic form of what we call Fermat's Theorem:
for every prime number $p$,
for every number $a$ that it is not itself a multiple of $p$,
the product of $p-1$ instances of $a$,
namely the power $a^{p-1}$,
exceeds by $1$ a multiple of $p$.
Playing around with special cases suggests more:
that for each prime $p$,
for each of some numbers $a$ called \emph{primitive roots} of $p$,
the power $a^{p-1}$ is the \emph{least} of the powers of $a$ 
with the indicated property.
One can prove this with the help of Euler's $\euphi$-function,
which counts the numbers less than its argument 
that are prime to that argument.

I am old enough that pocket calculators started coming out 
only after I was in school.
We still had to learn to use the trig and log tables at the end of our
algebra and geometry books \cite{Weeks-Adkins-alg-2,Weeks-Adkins}.
To satisfy my own curiosity,
I asked for, and received as a gift, 
a slide rule from a relative in engineering.
To me it is a source of fascination and delight that,
using primitive roots, 
one can compose log tables for \emph{exact} computations.

One can also construct ``discrete'' slide-rules,
corresponding to those tables.
I did this for my class, crudely, 
with the stiff cardboard of an old notebook cover,
for the small primes 7 and 11;
for 13, I cut a circle out of the side of a cardboard box
and arranged the numbers like hours on a clockface,
as in Figure \ref{fig:clock},
\begin{figure}
\begin{center}
\psset{unit=2cm}
  \begin{pspicture}(-1.2,-1.2)(1.2,1.2)
%\psgrid
    \pscircle(0,0)1
\uput[u](0,1){12}
\uput[60](! 0.5 3 sqrt 2 div){11}
\uput[30](! 3 sqrt 2 div 0.5){9}
\uput[r](1,0){5}
\uput[-30](! 3 sqrt 2 div -0.5){10}
\uput[-60](! 0.5 3 sqrt -2 div){7}
\uput[d](0,-1){1}
\uput[-120](! -0.5 3 sqrt -2 div){2}
\uput[-150](! 3 sqrt -2 div -0.5){4}
\uput[l](-1,0){8}
\uput[150](! 3 sqrt -2 div 0.5){3}
\uput[120](! -0.5 3 sqrt 2 div){6}
%%%%%%%%%%%%%%%%%%%%%%%%
\uput[d](0,1){8}
\uput[-120](! 0.5 3 sqrt 2 div){3}
\uput[-150](! 3 sqrt 2 div 0.5){6}
\uput[l](1,0){12}
\uput[150](! 3 sqrt 2 div -0.5){11}
\uput[120](! 0.5 3 sqrt -2 div){9}
\uput[u](0,-1){5}
\uput[60](! -0.5 3 sqrt -2 div){10}
\uput[30](! 3 sqrt -2 div -0.5){7}
\uput[r](-1,0){1}
\uput[-30](! 3 sqrt -2 div 0.5){2}
\uput[-60](! -0.5 3 sqrt 2 div){4}
  \end{pspicture}
\end{center}
\caption{Powers of 2 \emph{modulo} 13}\label{fig:clock}
\end{figure}
where the dial is set to show multiplication by 5,
\emph{modulo} 13.
Since 5 and 3 on the inner circle
line up with 1 and 11 on the outer circle,
5 times 11 should exceed by 3 a multiple of 13;
and this is true, since
\begin{equation*}
  5\times 11=55
  %=52+3
  =4\times13+3.
\end{equation*}
One could construct similarly 
a finely machined rotating device,
perhaps based on the prime 181,
so that the 180 noncongruent non-multiples of this number
would be positioned every two degrees.

Such a construction would partake of some of the spirit of Duchamp's
\emph{3 Stoppages Etalon (3 Standard Stoppages)} of 1913
\cite[pp.\ 78 f.]{ACH-Duchamp}:
\begin{quote}
  Duchamp took three one-metre lengths of string
and dropped them from a height of one metre onto a canvas.
He then stuck the threads down and thereby fixed the new lengths that chance,
gravity and the `whims' of the threads had created\lips
%The term \emph{stoppages}---a French word
%meaning the invisible mending of a garment---%
%which he had glimpsed on a shop sign,
%seemed appropriate for his impersonal threading;
%when he rendered the title into English, however,
%he used the same word, which thus takes on new connotations.
Duchamp then proceeded to make three `rulers' 
that followed the exact contours of the threads 
and went on to box them like technical instruments 
(but in a wooden box resembling a case for croquet sets).
\end{quote}
Duchamp's practice may recall
what Julian Jaynes describes as
``sortilege or the casting of lots\lips
designed to provoke the gods' answers 
to specific questions in novel situations''
\cite[p.\ 239]{Jaynes}.
According to Jaynes's proposal,
this is what we did when we could no longer directly hear 
the voices of the gods \cite[p.\ 236]{Jaynes}:
\begin{quote}
  Subjective consciousness,
that is, the development on the basis of linguistic metaphors
of an operation space in which an `I'
could narratize out alternative actions to their consequences,
was of course the great world result of this dilemma.
But a more primitive solution,
and one that antedates consciousness 
as well as paralleling it throughout history,
is that complex of behaviors known as divination.
\end{quote}
To multiply numbers by means of their discrete logarithms
might seem as mysterious as divination.

I may myself be suggesting things that are beyond my comprehension,
as artist Bob Deweese thought Robert Pirsig's 
\emph{alter ego} Phaedrus was doing,
in Pirsig's fictionalized recollections
in \emph{Zen and the Art of Motorcycle Maintenance}
\cite[ch.\ 12, p.\ 140]{ZAMM}.
\begin{quotation}\noindent
  Phaedrus would say something he thought was pretty funny 
and DeWeese would look at him in a puzzled way or else take him seriously\lips

For example, there is the fragment of memory about a dining-room table 
whose edge veneer had come loose and which Phaedrus had reglued. 
He held the veneer in place while the glue set 
by wrapping a whole ball of string around the table, round and round and round. 

DeWeese saw the string and wondered what that was all about. 

``That's my latest sculpture,'' Phaedrus had said. 
``Don't you think it kind of builds?'' 

Instead of laughing, DeWeese looked at him with amazement, 
studied it for a long time and finally said, 
``Where did you learn all this?'' 
For a second Phaedrus thought he was continuing the joke, 
but he was serious.
\end{quotation}
Phaedrus treated modern art flippantly,
but practitioners like DeWeese would not do so.

Or perhaps they might.
The descriptively titled work called
``The first thousand numbers classified in alphabetical order,''
dated 1989,
by Claude Closky
\cite{Closky}---is it a prose poem, or just a joke?
One can reconstruct for oneself as much of the work as desired:
\begin{quote}\sloppy
Eight, 
eight hundred and eight, 
eight hundred and eighteen,
eight hundred and eighty,
eight hundred and eighty-eight,
eight hundred and eighty-five,
eight hundred and eighty-four,
eight hundred and eighty-nine,
eight hundred and eighty-one,
eight hundred and eighty-seven,
eight hundred and eighty-six,
eight hundred and eighty-three,
eight hundred and eighty-two,
eight hundred and eleven,
\lips
two hundred and twelve,
two hundred and twenty,
two hundred and twenty-eight,
two hundred and twenty-five,
two hundred and twenty-four,
two hundred and twenty-nine,
two hundred and twenty-one,
two hundred and twenty-seven,
two hundred and twenty-six,
two hundred and twenty-three,
two hundred and twenty-two,
two hundred and two.
\end{quote}
I seem to recall being taught in the third grade
that there was no need to say ``and'' after the number of hundreds.
Thus the 891 instances of this word might be removed from Closky's work,
in an act of what might be called cleaning.
Arthur Danto reports that the cleaning of the Sistine Ceiling in the 1990s
was thought by some to remove a dimness 
that had been intended by Michelangelo 
to suggest the Allegory of the Cave in the \emph{Republic}
\cite[pp.\ 55 f.]{Danto}. Danto himself concludes not.

Meanwhile,
back when New Math was the prevalent educational philosophy 
in the United States,
my third-grade classmates and I were also taught to distinguish 
a \emph{number} from the \emph{numeral} whereby it was expressed.
``The first thousand numbers classified in alphabetical order''
might be understood to teach the lesson that there \emph{is} a distinction.
The lesson would be more explicit
if each number, as written out,
were followed by its expression in Arabic numerals.
This would make Closky's work notionally useful,
like a dictionary.

In 2013,
I translated this work, or the \emph{concept} of the work, into Turkish:
\begin{quote}
  Alt\i, 
alt\i\ y\"uz,
alt\i\ y\"uz alt\i, 
alt\i\ y\"uz altm\i\c s, 
alt\i\ y\"uz altm\i\c s alt\i, 
alt\i\ y\"uz altm\i\c s be\c s, 
alt\i\ y\"uz altm\i\c s bir, 
alt\i\ y\"uz altm\i\c s dokuz, 
alt\i\ y\"uz altm\i\c s d\"ort, 
alt\i\ y\"uz altm\i\c s iki, 
alt\i\ y\"uz altm\i\c s sekiz, 
alt\i\ y\"uz altm\i\c s \"u\c c, 
alt\i\ y\"uz altm\i\c s yedi, 
alt\i\ y\"uz be\c s, 
alt\i\ y\"uz bir, 
alt\i\ y\"uz doksan, 
alt\i\ y\"uz doksan alt\i, 
\lips
y\"uz yetmi\c s yedi, 
y\"uz yirmi,
y\"uz yirmi alt\i, 
y\"uz yirmi be\c s, 
y\"uz yirmi bir, 
y\"uz yirmi dokuz, 
y\"uz yirmi d\"ort, 
y\"uz yirmi iki, 
y\"uz yirmi sekiz, 
y\"uz yirmi \"u\c c, 
y\"uz yirmi yedi.
\end{quote}
Then I created a dictionary of Roman numerals,
summarized in Table \ref{tab:Roman}.
\begin{table}\centering
    \begin{longtable}{rl}
    1.&\rn{C}{100}\\
    2.&\rn{CC}{200}\\
    3.&\rn{CCC}{300}\\
    4.&\rn{CCCI}{301}\\
    5.&\rn{CCCII}{302}\\
    6.&\rn{CCCIII}{303}\\
    7.&\rn{CCCIV}{304}\\
    8.&\rn{CCCIX}{309}\\
    9.&\rn{CCCL}{350}\\
    10.&\rn{CCCLI}{351}\\
    11.&\rn{CCCLII}{352}\\
    12.&\rn{CCCLIII}{353}\\
    13.&\rn{CCCLIV}{354}\\
    14.&\rn{CCCLIX}{359}\\
%    15.&\rn{CCCLV}{355}\\
    \multicolumn2c{\makebox[54mm]{\dotfill}}\\
%    3985.&\rn{XXIX}{29}\\
    3986.&\rn{XXV}{25}\\
    3987.&\rn{XXVI}{26}\\
    3988.&\rn{XXVII}{27}\\
    3989.&\rn{XXVIII}{28}\\
    3990.&\rn{XXX}{30}\\
    3991.&\rn{XXXI}{31}\\
    3992.&\rn{XXXII}{32}\\
    3993.&\rn{XXXIII}{33}\\
    3994.&\rn{XXXIV}{34}\\
    3995.&\rn{XXXIX}{39}\\
    3996.&\rn{XXXV}{35}\\
    3997.&\rn{XXXVI}{36}\\
    3998.&\rn{XXXVII}{37}\\
    3999.&\rn{XXXVIII}{38}
    \end{longtable}
\caption{Roman numerals in alphabetical order}\label{tab:Roman}
\end{table}
One who knows Roman numerals may recognize
that the greatest Roman numeral in the dictionary is MMMCMXCIX;
but this is entry number 3241.
One who never got the hang of Roman numerals
might conceivably find the dictionary useful.
I even allowed the MakeIndex program that accompanies \LaTeX\
to produce an index of the page number 
where each Arabic numeral appeared.

There was a dictionary in the 12th Istanbul Biennial, in 2011.
Born in Istanbul,
living in Stockholm,
Meri\c c Alg\"un Ringborg created
\emph{\"O (The Mutual Letter),}
a Swedish-Turkish dictionary,
consisting only of the 1270 words 
that are spelled the same in Turkish as in Swedish \cite[p.\ 252]{IB-2011}.
Some of the words feature the letter \"O,
which is common to the two languages,
though it has different places in the alphabetical order;
in Turkish it lies between O and P.
Distributed as a saddle-bound booklet of 40 blue pages of size A6,
the dictionary is summarized in Table \ref{tab:O}.
\begin{table}\centering
%  \poemlines0
%  \settowidth{\versewidth}{\dict{abrakadabra}}
%  \begin{verse}[\versewidth]
  \begin{tabular}{l}
    \dict{abdomen}\\
    \dict{abdominal}\\
    \dict{abort}\\
    \dict{abrakadabra}\\
    \dict{absorbent}\\
    \dict{adenin}\\
    \dict{adenit}\\
    \dict{adenoid}\\
    \dict{adenom}\\
    \dict{adrenalin}\\
    \dict{aerosol}\\
    \dict{agoni}\\
    \dict{agorafobi}\\
    \dict{agronomi}\\
    \dotfill\\
    \dict{vokalist}\\
    \dict{volt}\\
    \dict{volta}\\
    \dict{yen}\\
    \dict{yoga}\\
    \dict{zebra}\\
    \dict{zenit}\\
    \dict{zeolit}\\
    \dict{zirkon}\\
    \dict{zon}\\
    \dict{zoolog}\\
    \dict{zootomi}\\
    \dict{\"odem}\\
    \dict{\"ostron}
    \end{tabular}
%  \end{verse}
\caption{Alg\"un Ringborg, \emph{\"O (The Mutual Letter)}}\label{tab:O}
\end{table}
The artist stresses that, despite appearances,
the paired words belong to different languages
and are pronounced accordingly;
she suggests that this could be heard in the aural component
of the display in the Biennial,
though I do not personally remember it.

Before passing to the logarithm ``dictionaries'' 
or tables of Chapter \ref{ch:lists},
let me quote them too elliptically, as in Table \ref{tab:log}.
\begin{table}\centering
  \begin{tabular}{rr||rr}
    \multicolumn2{c||}{logarithms}&\multicolumn2{c}{antilogs}\\\hline
     2.&201& 1.&  7\\
     3.&  6& 2.& 49\\
     4.&402& 3.&343\\
     5.&465& 4.&407\\
     6.&207& 5.&855\\
     7.&  1& 6.&  3\\
     8.&603& 7.& 21\\
     9.& 12& 8.&147\\
    10.&666& 9.& 32\\
    11.&817&10.&224\\
    12.&408&11.&571\\
    13.&580&12.&  9\\
    14.&202&13.& 63\\
    15.&471&14.&441\\
    \multicolumn2{c||}{\dotfill}&\multicolumn2{c}{\dotfill}\\
    983.&700&982.& 52\\
    984.& 82&983.&364\\
    985.&906&984.&554\\
    986.&319&985.&887\\
    987.&168&986.&227\\
    988.&510&987.&592\\
    989.&105&988.&156\\
    990.&499&989.& 95\\
    991.&705&990.&665\\
    992.&963&991.&667\\
    993.&900&992.&681\\
    994.&504&993.&779\\
    995.&699&994.&468\\
    996.&498&995.&285
  \end{tabular}
\caption{Discrete logarithms}\label{tab:log}
\end{table}
The need for a distinct table of antilogarithms
(at least if practical use is contemplated)
should be contrasted with the case of common logarithms,
which proceed in order as in Table \ref{tab:common}
\cite[pp.\ 606 f.]{Weeks-Adkins-alg-2}.
\begin{table}\centering
  \begin{tabular}{rc}
$N$&$\log N$\\\hline
1&0.0000\\
2&0.3010\\
3&0.4771\\
4&0.6021\\
5&0.6990\\
6&0.7782\\
7&0.8451\\
8&0.9031\\
9&0.9542\\
10&1.0000
  \end{tabular}
\caption{Common logarithms, coarsely}\label{tab:common}
\end{table}
Gaps can be filled in as finely as wished,
as for example in Table \ref{tab:fine}.
\begin{table}
  \centering
  \begin{tabular}{cc|cc|cc|cc}
$N$&$\log N$&$N$&$\log N$&$N$&$\log N$&$N$&$\log N$\\\hline
1.00&0.0000&1.10&0.0414&1.20&0.0792&9.90&0.9956\\
1.01&0.0043&1.11&0.0453&1.21&0.0828&9.91&0.9961\\
1.02&0.0086&1.12&0.0492&1.22&0.0864&9.92&0.9965\\
1.03&0.0128&1.13&0.0531&1.23&0.0899&9.93&0.9969\\
1.04&0.0170&1.14&0.0569&1.24&0.0934&9.94&0.9974\\
1.05&0.0212&1.15&0.0607&\multicolumn2{c|}{\dotfill}&9.95&0.9978\\
1.06&0.0253&1.16&0.0645&9.86&0.9939&9.96&0.9983\\
1.07&0.0294&1.17&0.0682&9.87&0.9943&9.97&0.9987\\
1.08&0.0334&1.18&0.0719&9.88&0.9948&9.98&0.9991\\
1.09&0.0374&1.19&0.0755&9.89&0.9952&9.99&0.9996
  \end{tabular}
  \caption{Common logarithms, finely}
  \label{tab:fine}
\end{table}
But the number of discrete logarithms 
is fixed by the modulus that they are based on---%
997, in the next chapter.

\chapter{Tables}\label{ch:lists}

\section{Logarithms}\label{sect:logs}

\begin{multicols}{5}
\begin{compactitem}
%  \dlog{1}{0}
  \dlog{2}{201}
  \dlog{3}{6}
  \dlog{4}{402}
  \dlog{5}{465}
  \dlog{6}{207}
  \dlog{7}{1}
  \dlog{8}{603}
  \dlog{9}{12}
  \dlog{10}{666}
  \dlog{11}{817}
  \dlog{12}{408}
  \dlog{13}{580}
  \dlog{14}{202}
  \dlog{15}{471}
  \dlog{16}{804}
  \dlog{17}{371}
  \dlog{18}{213}
  \dlog{19}{524}
  \dlog{20}{867}
  \dlog{21}{7}
  \dlog{22}{22}
  \dlog{23}{248}
  \dlog{24}{609}
  \dlog{25}{930}
  \dlog{26}{781}
  \dlog{27}{18}
  \dlog{28}{403}
  \dlog{29}{743}
  \dlog{30}{672}
  \dlog{31}{954}
  \dlog{32}{9}
  \dlog{33}{823}
  \dlog{34}{572}
  \dlog{35}{466}
  \dlog{36}{414}
  \dlog{37}{183}
  \dlog{38}{725}
  \dlog{39}{586}
  \dlog{40}{72}
  \dlog{41}{469}
  \dlog{42}{208}
  \dlog{43}{853}
  \dlog{44}{223}
  \dlog{45}{477}
  \dlog{46}{449}
  \dlog{47}{161}
  \dlog{48}{810}
  \dlog{49}{2}
  \dlog{50}{135}
  \dlog{51}{377}
  \dlog{52}{982}
  \dlog{53}{142}
  \dlog{54}{219}
  \dlog{55}{286}
  \dlog{56}{604}
  \dlog{57}{530}
  \dlog{58}{944}
  \dlog{59}{832}
  \dlog{60}{873}
  \dlog{61}{697}
  \dlog{62}{159}
  \dlog{63}{13}
  \dlog{64}{210}
  \dlog{65}{49}
  \dlog{66}{28}
  \dlog{67}{132}
  \dlog{68}{773}
  \dlog{69}{254}
  \dlog{70}{667}
  \dlog{71}{302}
  \dlog{72}{615}
  \dlog{73}{728}
  \dlog{74}{384}
  \dlog{75}{936}
  \dlog{76}{926}
  \dlog{77}{818}
  \dlog{78}{787}
  \dlog{79}{92}
  \dlog{80}{273}
  \dlog{81}{24}
  \dlog{82}{670}
  \dlog{83}{90}
  \dlog{84}{409}
  \dlog{85}{836}
  \dlog{86}{58}
  \dlog{87}{749}
  \dlog{88}{424}
  \dlog{89}{890}
  \dlog{90}{678}
  \dlog{91}{581}
  \dlog{92}{650}
  \dlog{93}{960}
  \dlog{94}{362}
  \dlog{95}{989}
  \dlog{96}{15}
  \dlog{97}{846}
  \dlog{98}{203}
  \dlog{99}{829}
  \dlog{100}{336}
  \dlog{101}{910}
  \dlog{102}{578}
  \dlog{103}{157}
  \dlog{104}{187}
  \dlog{105}{472}
  \dlog{106}{343}
  \dlog{107}{62}
  \dlog{108}{420}
  \dlog{109}{294}
  \dlog{110}{487}
  \dlog{111}{189}
  \dlog{112}{805}
  \dlog{113}{855}
  \dlog{114}{731}
  \dlog{115}{713}
  \dlog{116}{149}
  \dlog{117}{592}
  \dlog{118}{37}
  \dlog{119}{372}
  \dlog{120}{78}
  \dlog{121}{638}
  \dlog{122}{898}
  \dlog{123}{475}
  \dlog{124}{360}
  \dlog{125}{399}
  \dlog{126}{214}
  \dlog{127}{917}
  \dlog{128}{411}
  \dlog{129}{859}
  \dlog{130}{250}
  \dlog{131}{770}
  \dlog{132}{229}
  \dlog{133}{525}
  \dlog{134}{333}
  \dlog{135}{483}
  \dlog{136}{974}
  \dlog{137}{226}
  \dlog{138}{455}
  \dlog{139}{110}
  \dlog{140}{868}
  \dlog{141}{167}
  \dlog{142}{503}
  \dlog{143}{401}
  \dlog{144}{816}
  \dlog{145}{212}
  \dlog{146}{929}
  \dlog{147}{8}
  \dlog{148}{585}
  \dlog{149}{448}
  \dlog{150}{141}
  \dlog{151}{872}
  \dlog{152}{131}
  \dlog{153}{383}
  \dlog{154}{23}
  \dlog{155}{423}
  \dlog{156}{988}
  \dlog{157}{577}
  \dlog{158}{293}
  \dlog{159}{148}
  \dlog{160}{474}
  \dlog{161}{249}
  \dlog{162}{225}
  \dlog{163}{815}
  \dlog{164}{871}
  \dlog{165}{292}
  \dlog{166}{291}
  \dlog{167}{258}
  \dlog{168}{610}
  \dlog{169}{164}
  \dlog{170}{41}
  \dlog{171}{536}
  \dlog{172}{259}
  \dlog{173}{262}
  \dlog{174}{950}
  \dlog{175}{931}
  \dlog{176}{625}
  \dlog{177}{838}
  \dlog{178}{95}
  \dlog{179}{611}
  \dlog{180}{879}
  \dlog{181}{683}
  \dlog{182}{782}
  \dlog{183}{703}
  \dlog{184}{851}
  \dlog{185}{648}
  \dlog{186}{165}
  \dlog{187}{192}
  \dlog{188}{563}
  \dlog{189}{19}
  \dlog{190}{194}
  \dlog{191}{241}
  \dlog{192}{216}
  \dlog{193}{42}
  \dlog{194}{51}
  \dlog{195}{55}
  \dlog{196}{404}
  \dlog{197}{441}
  \dlog{198}{34}
  \dlog{199}{234}
  \dlog{200}{537}
  \dlog{201}{138}
  \dlog{202}{115}
  \dlog{203}{744}
  \dlog{204}{779}
  \dlog{205}{934}
  \dlog{206}{358}
  \dlog{207}{260}
  \dlog{208}{388}
  \dlog{209}{345}
  \dlog{210}{673}
  \dlog{211}{479}
  \dlog{212}{544}
  \dlog{213}{308}
  \dlog{214}{263}
  \dlog{215}{322}
  \dlog{216}{621}
  \dlog{217}{955}
  \dlog{218}{495}
  \dlog{219}{734}
  \dlog{220}{688}
  \dlog{221}{951}
  \dlog{222}{390}
  \dlog{223}{568}
  \dlog{224}{10}
  \dlog{225}{942}
  \dlog{226}{60}
  \dlog{227}{986}
  \dlog{228}{932}
  \dlog{229}{120}
  \dlog{230}{914}
  \dlog{231}{824}
  \dlog{232}{350}
  \dlog{233}{145}
  \dlog{234}{793}
  \dlog{235}{626}
  \dlog{236}{238}
  \dlog{237}{98}
  \dlog{238}{573}
  \dlog{239}{565}
  \dlog{240}{279}
  \dlog{241}{919}
  \dlog{242}{839}
  \dlog{243}{30}
  \dlog{244}{103}
  \dlog{245}{467}
  \dlog{246}{676}
  \dlog{247}{108}
  \dlog{248}{561}
  \dlog{249}{96}
  \dlog{250}{600}
  \dlog{251}{595}
  \dlog{252}{415}
  \dlog{253}{69}
  \dlog{254}{122}
  \dlog{255}{842}
  \dlog{256}{612}
  \dlog{257}{552}
  \dlog{258}{64}
  \dlog{259}{184}
  \dlog{260}{451}
  \dlog{261}{755}
  \dlog{262}{971}
  \dlog{263}{880}
  \dlog{264}{430}
  \dlog{265}{607}
  \dlog{266}{726}
  \dlog{267}{896}
  \dlog{268}{534}
  \dlog{269}{686}
  \dlog{270}{684}
  \dlog{271}{347}
  \dlog{272}{179}
  \dlog{273}{587}
  \dlog{274}{427}
  \dlog{275}{751}
  \dlog{276}{656}
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\end{compactitem}
\end{multicols}


\section{Antilogarithms}\label{sect:powers}

\begin{multicols}{5}
\begin{compactitem}
\dpow{7}{1}
\dpow{49}{2}
\dpow{343}{3}
\dpow{407}{4}
\dpow{855}{5}
\dpow{3}{6}
\dpow{21}{7}
\dpow{147}{8}
\dpow{32}{9}
\dpow{224}{10}
\dpow{571}{11}
\dpow{9}{12}
\dpow{63}{13}
\dpow{441}{14}
\dpow{96}{15}
\dpow{672}{16}
\dpow{716}{17}
\dpow{27}{18}
\dpow{189}{19}
\dpow{326}{20}
\dpow{288}{21}
\dpow{22}{22}
\dpow{154}{23}
\dpow{81}{24}
\dpow{567}{25}
\dpow{978}{26}
\dpow{864}{27}
\dpow{66}{28}
\dpow{462}{29}
\dpow{243}{30}
\dpow{704}{31}
\dpow{940}{32}
\dpow{598}{33}
\dpow{198}{34}
\dpow{389}{35}
\dpow{729}{36}
\dpow{118}{37}
\dpow{826}{38}
\dpow{797}{39}
\dpow{594}{40}
\dpow{170}{41}
\dpow{193}{42}
\dpow{354}{43}
\dpow{484}{44}
\dpow{397}{45}
\dpow{785}{46}
\dpow{510}{47}
\dpow{579}{48}
\dpow{65}{49}
\dpow{455}{50}
\dpow{194}{51}
\dpow{361}{52}
\dpow{533}{53}
\dpow{740}{54}
\dpow{195}{55}
\dpow{368}{56}
\dpow{582}{57}
\dpow{86}{58}
\dpow{602}{59}
\dpow{226}{60}
\dpow{585}{61}
\dpow{107}{62}
\dpow{749}{63}
\dpow{258}{64}
\dpow{809}{65}
\dpow{678}{66}
\dpow{758}{67}
\dpow{321}{68}
\dpow{253}{69}
\dpow{774}{70}
\dpow{433}{71}
\dpow{40}{72}
\dpow{280}{73}
\dpow{963}{74}
\dpow{759}{75}
\dpow{328}{76}
\dpow{302}{77}
\dpow{120}{78}
\dpow{840}{79}
\dpow{895}{80}
\dpow{283}{81}
\dpow{984}{82}
\dpow{906}{83}
\dpow{360}{84}
\dpow{526}{85}
\dpow{691}{86}
\dpow{849}{87}
\dpow{958}{88}
\dpow{724}{89}
\dpow{83}{90}
\dpow{581}{91}
\dpow{79}{92}
\dpow{553}{93}
\dpow{880}{94}
\dpow{178}{95}
\dpow{249}{96}
\dpow{746}{97}
\dpow{237}{98}
\dpow{662}{99}
\dpow{646}{100}
\dpow{534}{101}
\dpow{747}{102}
\dpow{244}{103}
\dpow{711}{104}
\dpow{989}{105}
\dpow{941}{106}
\dpow{605}{107}
\dpow{247}{108}
\dpow{732}{109}
\dpow{139}{110}
\dpow{973}{111}
\dpow{829}{112}
\dpow{818}{113}
\dpow{741}{114}
\dpow{202}{115}
\dpow{417}{116}
\dpow{925}{117}
\dpow{493}{118}
\dpow{460}{119}
\dpow{229}{120}
\dpow{606}{121}
\dpow{254}{122}
\dpow{781}{123}
\dpow{482}{124}
\dpow{383}{125}
\dpow{687}{126}
\dpow{821}{127}
\dpow{762}{128}
\dpow{349}{129}
\dpow{449}{130}
\dpow{152}{131}
\dpow{67}{132}
\dpow{469}{133}
\dpow{292}{134}
\dpow{50}{135}
\dpow{350}{136}
\dpow{456}{137}
\dpow{201}{138}
\dpow{410}{139}
\dpow{876}{140}
\dpow{150}{141}
\dpow{53}{142}
\dpow{371}{143}
\dpow{603}{144}
\dpow{233}{145}
\dpow{634}{146}
\dpow{450}{147}
\dpow{159}{148}
\dpow{116}{149}
\dpow{812}{150}
\dpow{699}{151}
\dpow{905}{152}
\dpow{353}{153}
\dpow{477}{154}
\dpow{348}{155}
\dpow{442}{156}
\dpow{103}{157}
\dpow{721}{158}
\dpow{62}{159}
\dpow{434}{160}
\dpow{47}{161}
\dpow{329}{162}
\dpow{309}{163}
\dpow{169}{164}
\dpow{186}{165}
\dpow{305}{166}
\dpow{141}{167}
\dpow{987}{168}
\dpow{927}{169}
\dpow{507}{170}
\dpow{558}{171}
\dpow{915}{172}
\dpow{423}{173}
\dpow{967}{174}
\dpow{787}{175}
\dpow{524}{176}
\dpow{677}{177}
\dpow{751}{178}
\dpow{272}{179}
\dpow{907}{180}
\dpow{367}{181}
\dpow{575}{182}
\dpow{37}{183}
\dpow{259}{184}
\dpow{816}{185}
\dpow{727}{186}
\dpow{104}{187}
\dpow{728}{188}
\dpow{111}{189}
\dpow{777}{190}
\dpow{454}{191}
\dpow{187}{192}
\dpow{312}{193}
\dpow{190}{194}
\dpow{333}{195}
\dpow{337}{196}
\dpow{365}{197}
\dpow{561}{198}
\dpow{936}{199}
\dpow{570}{200}
\dpow{2}{201}
\dpow{14}{202}
\dpow{98}{203}
\dpow{686}{204}
\dpow{814}{205}
\dpow{713}{206}
\dpow{6}{207}
\dpow{42}{208}
\dpow{294}{209}
\dpow{64}{210}
\dpow{448}{211}
\dpow{145}{212}
\dpow{18}{213}
\dpow{126}{214}
\dpow{882}{215}
\dpow{192}{216}
\dpow{347}{217}
\dpow{435}{218}
\dpow{54}{219}
\dpow{378}{220}
\dpow{652}{221}
\dpow{576}{222}
\dpow{44}{223}
\dpow{308}{224}
\dpow{162}{225}
\dpow{137}{226}
\dpow{959}{227}
\dpow{731}{228}
\dpow{132}{229}
\dpow{924}{230}
\dpow{486}{231}
\dpow{411}{232}
\dpow{883}{233}
\dpow{199}{234}
\dpow{396}{235}
\dpow{778}{236}
\dpow{461}{237}
\dpow{236}{238}
\dpow{655}{239}
\dpow{597}{240}
\dpow{191}{241}
\dpow{340}{242}
\dpow{386}{243}
\dpow{708}{244}
\dpow{968}{245}
\dpow{794}{246}
\dpow{573}{247}
\dpow{23}{248}
\dpow{161}{249}
\dpow{130}{250}
\dpow{910}{251}
\dpow{388}{252}
\dpow{722}{253}
\dpow{69}{254}
\dpow{483}{255}
\dpow{390}{256}
\dpow{736}{257}
\dpow{167}{258}
\dpow{172}{259}
\dpow{207}{260}
\dpow{452}{261}
\dpow{173}{262}
\dpow{214}{263}
\dpow{501}{264}
\dpow{516}{265}
\dpow{621}{266}
\dpow{359}{267}
\dpow{519}{268}
\dpow{642}{269}
\dpow{506}{270}
\dpow{551}{271}
\dpow{866}{272}
\dpow{80}{273}
\dpow{560}{274}
\dpow{929}{275}
\dpow{521}{276}
\dpow{656}{277}
\dpow{604}{278}
\dpow{240}{279}
\dpow{683}{280}
\dpow{793}{281}
\dpow{566}{282}
\dpow{971}{283}
\dpow{815}{284}
\dpow{720}{285}
\dpow{55}{286}
\dpow{385}{287}
\dpow{701}{288}
\dpow{919}{289}
\dpow{451}{290}
\dpow{166}{291}
\dpow{165}{292}
\dpow{158}{293}
\dpow{109}{294}
\dpow{763}{295}
\dpow{356}{296}
\dpow{498}{297}
\dpow{495}{298}
\dpow{474}{299}
\dpow{327}{300}
\dpow{295}{301}
\dpow{71}{302}
\dpow{497}{303}
\dpow{488}{304}
\dpow{425}{305}
\dpow{981}{306}
\dpow{885}{307}
\dpow{213}{308}
\dpow{494}{309}
\dpow{467}{310}
\dpow{278}{311}
\dpow{949}{312}
\dpow{661}{313}
\dpow{639}{314}
\dpow{485}{315}
\dpow{404}{316}
\dpow{834}{317}
\dpow{853}{318}
\dpow{986}{319}
\dpow{920}{320}
\dpow{458}{321}
\dpow{215}{322}
\dpow{508}{323}
\dpow{565}{324}
\dpow{964}{325}
\dpow{766}{326}
\dpow{377}{327}
\dpow{645}{328}
\dpow{527}{329}
\dpow{698}{330}
\dpow{898}{331}
\dpow{304}{332}
\dpow{134}{333}
\dpow{938}{334}
\dpow{584}{335}
\dpow{100}{336}
\dpow{700}{337}
\dpow{912}{338}
\dpow{402}{339}
\dpow{820}{340}
\dpow{755}{341}
\dpow{300}{342}
\dpow{106}{343}
\dpow{742}{344}
\dpow{209}{345}
\dpow{466}{346}
\dpow{271}{347}
\dpow{900}{348}
\dpow{318}{349}
\dpow{232}{350}
\dpow{627}{351}
\dpow{401}{352}
\dpow{813}{353}
\dpow{706}{354}
\dpow{954}{355}
\dpow{696}{356}
\dpow{884}{357}
\dpow{206}{358}
\dpow{445}{359}
\dpow{124}{360}
\dpow{868}{361}
\dpow{94}{362}
\dpow{658}{363}
\dpow{618}{364}
\dpow{338}{365}
\dpow{372}{366}
\dpow{610}{367}
\dpow{282}{368}
\dpow{977}{369}
\dpow{857}{370}
\dpow{17}{371}
\dpow{119}{372}
\dpow{833}{373}
\dpow{846}{374}
\dpow{937}{375}
\dpow{577}{376}
\dpow{51}{377}
\dpow{357}{378}
\dpow{505}{379}
\dpow{544}{380}
\dpow{817}{381}
\dpow{734}{382}
\dpow{153}{383}
\dpow{74}{384}
\dpow{518}{385}
\dpow{635}{386}
\dpow{457}{387}
\dpow{208}{388}
\dpow{459}{389}
\dpow{222}{390}
\dpow{557}{391}
\dpow{908}{392}
\dpow{374}{393}
\dpow{624}{394}
\dpow{380}{395}
\dpow{666}{396}
\dpow{674}{397}
\dpow{730}{398}
\dpow{125}{399}
\dpow{875}{400}
\dpow{143}{401}
\dpow{4}{402}
\dpow{28}{403}
\dpow{196}{404}
\dpow{375}{405}
\dpow{631}{406}
\dpow{429}{407}
\dpow{12}{408}
\dpow{84}{409}
\dpow{588}{410}
\dpow{128}{411}
\dpow{896}{412}
\dpow{290}{413}
\dpow{36}{414}
\dpow{252}{415}
\dpow{767}{416}
\dpow{384}{417}
\dpow{694}{418}
\dpow{870}{419}
\dpow{108}{420}
\dpow{756}{421}
\dpow{307}{422}
\dpow{155}{423}
\dpow{88}{424}
\dpow{616}{425}
\dpow{324}{426}
\dpow{274}{427}
\dpow{921}{428}
\dpow{465}{429}
\dpow{264}{430}
\dpow{851}{431}
\dpow{972}{432}
\dpow{822}{433}
\dpow{769}{434}
\dpow{398}{435}
\dpow{792}{436}
\dpow{559}{437}
\dpow{922}{438}
\dpow{472}{439}
\dpow{313}{440}
\dpow{197}{441}
\dpow{382}{442}
\dpow{680}{443}
\dpow{772}{444}
\dpow{419}{445}
\dpow{939}{446}
\dpow{591}{447}
\dpow{149}{448}
\dpow{46}{449}
\dpow{322}{450}
\dpow{260}{451}
\dpow{823}{452}
\dpow{776}{453}
\dpow{447}{454}
\dpow{138}{455}
\dpow{966}{456}
\dpow{780}{457}
\dpow{475}{458}
\dpow{334}{459}
\dpow{344}{460}
\dpow{414}{461}
\dpow{904}{462}
\dpow{346}{463}
\dpow{428}{464}
\dpow{5}{465}
\dpow{35}{466}
\dpow{245}{467}
\dpow{718}{468}
\dpow{41}{469}
\dpow{287}{470}
\dpow{15}{471}
\dpow{105}{472}
\dpow{735}{473}
\dpow{160}{474}
\dpow{123}{475}
\dpow{861}{476}
\dpow{45}{477}
\dpow{315}{478}
\dpow{211}{479}
\dpow{480}{480}
\dpow{369}{481}
\dpow{589}{482}
\dpow{135}{483}
\dpow{945}{484}
\dpow{633}{485}
\dpow{443}{486}
\dpow{110}{487}
\dpow{770}{488}
\dpow{405}{489}
\dpow{841}{490}
\dpow{902}{491}
\dpow{332}{492}
\dpow{330}{493}
\dpow{316}{494}
\dpow{218}{495}
\dpow{529}{496}
\dpow{712}{497}
\dpow{996}{498}
\dpow{990}{499}
\dpow{948}{500}
\dpow{654}{501}
\dpow{590}{502}
\dpow{142}{503}
\dpow{994}{504}
\dpow{976}{505}
\dpow{850}{506}
\dpow{965}{507}
\dpow{773}{508}
\dpow{426}{509}
\dpow{988}{510}
\dpow{934}{511}
\dpow{556}{512}
\dpow{901}{513}
\dpow{325}{514}
\dpow{281}{515}
\dpow{970}{516}
\dpow{808}{517}
\dpow{671}{518}
\dpow{709}{519}
\dpow{975}{520}
\dpow{843}{521}
\dpow{916}{522}
\dpow{430}{523}
\dpow{19}{524}
\dpow{133}{525}
\dpow{931}{526}
\dpow{535}{527}
\dpow{754}{528}
\dpow{293}{529}
\dpow{57}{530}
\dpow{399}{531}
\dpow{799}{532}
\dpow{608}{533}
\dpow{268}{534}
\dpow{879}{535}
\dpow{171}{536}
\dpow{200}{537}
\dpow{403}{538}
\dpow{827}{539}
\dpow{804}{540}
\dpow{643}{541}
\dpow{513}{542}
\dpow{600}{543}
\dpow{212}{544}
\dpow{487}{545}
\dpow{418}{546}
\dpow{932}{547}
\dpow{542}{548}
\dpow{803}{549}
\dpow{636}{550}
\dpow{464}{551}
\dpow{257}{552}
\dpow{802}{553}
\dpow{629}{554}
\dpow{415}{555}
\dpow{911}{556}
\dpow{395}{557}
\dpow{771}{558}
\dpow{412}{559}
\dpow{890}{560}
\dpow{248}{561}
\dpow{739}{562}
\dpow{188}{563}
\dpow{319}{564}
\dpow{239}{565}
\dpow{676}{566}
\dpow{744}{567}
\dpow{223}{568}
\dpow{564}{569}
\dpow{957}{570}
\dpow{717}{571}
\dpow{34}{572}
\dpow{238}{573}
\dpow{669}{574}
\dpow{695}{575}
\dpow{877}{576}
\dpow{157}{577}
\dpow{102}{578}
\dpow{714}{579}
\dpow{13}{580}
\dpow{91}{581}
\dpow{637}{582}
\dpow{471}{583}
\dpow{306}{584}
\dpow{148}{585}
\dpow{39}{586}
\dpow{273}{587}
\dpow{914}{588}
\dpow{416}{589}
\dpow{918}{590}
\dpow{444}{591}
\dpow{117}{592}
\dpow{819}{593}
\dpow{748}{594}
\dpow{251}{595}
\dpow{760}{596}
\dpow{335}{597}
\dpow{351}{598}
\dpow{463}{599}
\dpow{250}{600}
\dpow{753}{601}
\dpow{286}{602}
\dpow{8}{603}
\dpow{56}{604}
\dpow{392}{605}
\dpow{750}{606}
\dpow{265}{607}
\dpow{858}{608}
\dpow{24}{609}
\dpow{168}{610}
\dpow{179}{611}
\dpow{256}{612}
\dpow{795}{613}
\dpow{580}{614}
\dpow{72}{615}
\dpow{504}{616}
\dpow{537}{617}
\dpow{768}{618}
\dpow{391}{619}
\dpow{743}{620}
\dpow{216}{621}
\dpow{515}{622}
\dpow{614}{623}
\dpow{310}{624}
\dpow{176}{625}
\dpow{235}{626}
\dpow{648}{627}
\dpow{548}{628}
\dpow{845}{629}
\dpow{930}{630}
\dpow{528}{631}
\dpow{705}{632}
\dpow{947}{633}
\dpow{647}{634}
\dpow{541}{635}
\dpow{796}{636}
\dpow{587}{637}
\dpow{121}{638}
\dpow{847}{639}
\dpow{944}{640}
\dpow{626}{641}
\dpow{394}{642}
\dpow{764}{643}
\dpow{363}{644}
\dpow{547}{645}
\dpow{838}{646}
\dpow{881}{647}
\dpow{185}{648}
\dpow{298}{649}
\dpow{92}{650}
\dpow{644}{651}
\dpow{520}{652}
\dpow{649}{653}
\dpow{555}{654}
\dpow{894}{655}
\dpow{276}{656}
\dpow{935}{657}
\dpow{563}{658}
\dpow{950}{659}
\dpow{668}{660}
\dpow{688}{661}
\dpow{828}{662}
\dpow{811}{663}
\dpow{692}{664}
\dpow{856}{665}
\dpow{10}{666}
\dpow{70}{667}
\dpow{490}{668}
\dpow{439}{669}
\dpow{82}{670}
\dpow{574}{671}
\dpow{30}{672}
\dpow{210}{673}
\dpow{473}{674}
\dpow{320}{675}
\dpow{246}{676}
\dpow{725}{677}
\dpow{90}{678}
\dpow{630}{679}
\dpow{422}{680}
\dpow{960}{681}
\dpow{738}{682}
\dpow{181}{683}
\dpow{270}{684}
\dpow{893}{685}
\dpow{269}{686}
\dpow{886}{687}
\dpow{220}{688}
\dpow{543}{689}
\dpow{810}{690}
\dpow{685}{691}
\dpow{807}{692}
\dpow{664}{693}
\dpow{660}{694}
\dpow{632}{695}
\dpow{436}{696}
\dpow{61}{697}
\dpow{427}{698}
\dpow{995}{699}
\dpow{983}{700}
\dpow{899}{701}
\dpow{311}{702}
\dpow{183}{703}
\dpow{284}{704}
\dpow{991}{705}
\dpow{955}{706}
\dpow{703}{707}
\dpow{933}{708}
\dpow{549}{709}
\dpow{852}{710}
\dpow{979}{711}
\dpow{871}{712}
\dpow{115}{713}
\dpow{805}{714}
\dpow{650}{715}
\dpow{562}{716}
\dpow{943}{717}
\dpow{619}{718}
\dpow{345}{719}
\dpow{421}{720}
\dpow{953}{721}
\dpow{689}{722}
\dpow{835}{723}
\dpow{860}{724}
\dpow{38}{725}
\dpow{266}{726}
\dpow{865}{727}
\dpow{73}{728}
\dpow{511}{729}
\dpow{586}{730}
\dpow{114}{731}
\dpow{798}{732}
\dpow{601}{733}
\dpow{219}{734}
\dpow{536}{735}
\dpow{761}{736}
\dpow{342}{737}
\dpow{400}{738}
\dpow{806}{739}
\dpow{657}{740}
\dpow{611}{741}
\dpow{289}{742}
\dpow{29}{743}
\dpow{203}{744}
\dpow{424}{745}
\dpow{974}{746}
\dpow{836}{747}
\dpow{867}{748}
\dpow{87}{749}
\dpow{609}{750}
\dpow{275}{751}
\dpow{928}{752}
\dpow{514}{753}
\dpow{607}{754}
\dpow{261}{755}
\dpow{830}{756}
\dpow{825}{757}
\dpow{790}{758}
\dpow{545}{759}
\dpow{824}{760}
\dpow{783}{761}
\dpow{496}{762}
\dpow{481}{763}
\dpow{376}{764}
\dpow{638}{765}
\dpow{478}{766}
\dpow{355}{767}
\dpow{491}{768}
\dpow{446}{769}
\dpow{131}{770}
\dpow{917}{771}
\dpow{437}{772}
\dpow{68}{773}
\dpow{476}{774}
\dpow{341}{775}
\dpow{393}{776}
\dpow{757}{777}
\dpow{314}{778}
\dpow{204}{779}
\dpow{431}{780}
\dpow{26}{781}
\dpow{182}{782}
\dpow{277}{783}
\dpow{942}{784}
\dpow{612}{785}
\dpow{296}{786}
\dpow{78}{787}
\dpow{546}{788}
\dpow{831}{789}
\dpow{832}{790}
\dpow{839}{791}
\dpow{888}{792}
\dpow{234}{793}
\dpow{641}{794}
\dpow{499}{795}
\dpow{502}{796}
\dpow{523}{797}
\dpow{670}{798}
\dpow{702}{799}
\dpow{926}{800}
\dpow{500}{801}
\dpow{509}{802}
\dpow{572}{803}
\dpow{16}{804}
\dpow{112}{805}
\dpow{784}{806}
\dpow{503}{807}
\dpow{530}{808}
\dpow{719}{809}
\dpow{48}{810}
\dpow{336}{811}
\dpow{358}{812}
\dpow{512}{813}
\dpow{593}{814}
\dpow{163}{815}
\dpow{144}{816}
\dpow{11}{817}
\dpow{77}{818}
\dpow{539}{819}
\dpow{782}{820}
\dpow{489}{821}
\dpow{432}{822}
\dpow{33}{823}
\dpow{231}{824}
\dpow{620}{825}
\dpow{352}{826}
\dpow{470}{827}
\dpow{299}{828}
\dpow{99}{829}
\dpow{693}{830}
\dpow{863}{831}
\dpow{59}{832}
\dpow{413}{833}
\dpow{897}{834}
\dpow{297}{835}
\dpow{85}{836}
\dpow{595}{837}
\dpow{177}{838}
\dpow{242}{839}
\dpow{697}{840}
\dpow{891}{841}
\dpow{255}{842}
\dpow{788}{843}
\dpow{531}{844}
\dpow{726}{845}
\dpow{97}{846}
\dpow{679}{847}
\dpow{765}{848}
\dpow{370}{849}
\dpow{596}{850}
\dpow{184}{851}
\dpow{291}{852}
\dpow{43}{853}
\dpow{301}{854}
\dpow{113}{855}
\dpow{791}{856}
\dpow{552}{857}
\dpow{873}{858}
\dpow{129}{859}
\dpow{903}{860}
\dpow{339}{861}
\dpow{379}{862}
\dpow{659}{863}
\dpow{625}{864}
\dpow{387}{865}
\dpow{715}{866}
\dpow{20}{867}
\dpow{140}{868}
\dpow{980}{869}
\dpow{878}{870}
\dpow{164}{871}
\dpow{151}{872}
\dpow{60}{873}
\dpow{420}{874}
\dpow{946}{875}
\dpow{640}{876}
\dpow{492}{877}
\dpow{453}{878}
\dpow{180}{879}
\dpow{263}{880}
\dpow{844}{881}
\dpow{923}{882}
\dpow{479}{883}
\dpow{362}{884}
\dpow{540}{885}
\dpow{789}{886}
\dpow{538}{887}
\dpow{775}{888}
\dpow{440}{889}
\dpow{89}{890}
\dpow{623}{891}
\dpow{373}{892}
\dpow{617}{893}
\dpow{331}{894}
\dpow{323}{895}
\dpow{267}{896}
\dpow{872}{897}
\dpow{122}{898}
\dpow{854}{899}
\dpow{993}{900}
\dpow{969}{901}
\dpow{801}{902}
\dpow{622}{903}
\dpow{366}{904}
\dpow{568}{905}
\dpow{985}{906}
\dpow{913}{907}
\dpow{409}{908}
\dpow{869}{909}
\dpow{101}{910}
\dpow{707}{911}
\dpow{961}{912}
\dpow{745}{913}
\dpow{230}{914}
\dpow{613}{915}
\dpow{303}{916}
\dpow{127}{917}
\dpow{889}{918}
\dpow{241}{919}
\dpow{690}{920}
\dpow{842}{921}
\dpow{909}{922}
\dpow{381}{923}
\dpow{673}{924}
\dpow{723}{925}
\dpow{76}{926}
\dpow{532}{927}
\dpow{733}{928}
\dpow{146}{929}
\dpow{25}{930}
\dpow{175}{931}
\dpow{228}{932}
\dpow{599}{933}
\dpow{205}{934}
\dpow{438}{935}
\dpow{75}{936}
\dpow{525}{937}
\dpow{684}{938}
\dpow{800}{939}
\dpow{615}{940}
\dpow{317}{941}
\dpow{225}{942}
\dpow{578}{943}
\dpow{58}{944}
\dpow{406}{945}
\dpow{848}{946}
\dpow{951}{947}
\dpow{675}{948}
\dpow{737}{949}
\dpow{174}{950}
\dpow{221}{951}
\dpow{550}{952}
\dpow{859}{953}
\dpow{31}{954}
\dpow{217}{955}
\dpow{522}{956}
\dpow{663}{957}
\dpow{653}{958}
\dpow{583}{959}
\dpow{93}{960}
\dpow{651}{961}
\dpow{569}{962}
\dpow{992}{963}
\dpow{962}{964}
\dpow{752}{965}
\dpow{279}{966}
\dpow{956}{967}
\dpow{710}{968}
\dpow{982}{969}
\dpow{892}{970}
\dpow{262}{971}
\dpow{837}{972}
\dpow{874}{973}
\dpow{136}{974}
\dpow{952}{975}
\dpow{682}{976}
\dpow{786}{977}
\dpow{517}{978}
\dpow{628}{979}
\dpow{408}{980}
\dpow{862}{981}
\dpow{52}{982}
\dpow{364}{983}
\dpow{554}{984}
\dpow{887}{985}
\dpow{227}{986}
\dpow{592}{987}
\dpow{156}{988}
\dpow{95}{989}
\dpow{665}{990}
\dpow{667}{991}
\dpow{681}{992}
\dpow{779}{993}
\dpow{468}{994}
\dpow{285}{995}
%\dpow{1}{996}
\end{compactitem}
\end{multicols}

\chapter{Mathematics}\label{ch:math}

\section{Practice and Theory}

In Chapter \ref{ch:lists},
if the entry \fbox{\dlogq{$x$}{$y$}} appears in the table of logarithms
(\S\ref{sect:logs}),
or \fbox{\dpowq{$x$}{$y$}} in the table of antilogarithms
(\S\ref{sect:powers}),
let us write
\begin{equation*}
  \log x=y.
\end{equation*}
As suggested in the Introduction, this means
\begin{equation*}
  7^{y}\equiv x\pmod{997},
\end{equation*}
in Gauss's notation for congruence, defined in this chapter.
Suppose in particular that the product of two numbers $a$ and $b$, 
each less than 997,
is desired.  One can find the product as follows.
\begin{compactenum}
  \item
    Look up $\log a$ and $\log b$.
  \item
    Compute the sum $\log a +\log b$.
  \item
    If this sum exceeds 996, subtract the latter.
  \item
Look up the antilogarithm of the result.
\end{compactenum}
The number so obtained is either the product $ab$ 
of the original numbers
or else its remainder after division by 997.

For example:
\begin{compactenum}
\item
  The logarithms of 23 and 31 are 248 and 954.
\item
  The sum of 248 and 954 is 1202.
\item
  This, less 996, is 206.
\item
  The antilogarithm of this is 713, which is 23 times 31.
\end{compactenum}

The rest of this chapter shows
why this procedure is possible.
In principle,
the review should be mostly accessible to the interested layperson.
In practice, 
the material might take several weeks of study.
Any reader must tolerate some quotations
(accompanied by translations) in Greek, Latin, and French.
For the mathematics itself,
a contemporary textbook is Burton's
\emph{Elementary Number Theory} \cite{Burton},
but everything is found---in Latin, originally---in Gauss's
\emph{Disquisitiones Arithmeticae} \cite{Gauss}.

The treatment of discrete logarithms given here
is terser than the laborious exposition of common logarithms
in Isaac Asimov's 1965
\emph{Easy Introduction to the Slide Rule.}
On the other hand, 
Asimov tacitly requires the reader to accept,
for example, that the number 10 has a square root
\cite[p.\ 49]{Asimov}.
This number is \emph{approximated} by 3.162120,
and the reader is supposed to be able to verify, by hand, 
that the square of this number is 9.9990028944.
(I have actually done this.)

Agreeing with David Fowler \cite{MR1186456},
I think Dedekind was right to say in the 1880s
\cite[pp.\ 22, 40]{MR0159773}
that he had been the first to \emph{prove,}
as a consequence of the construction of the real numbers,
the existence of square roots,
along with the rule for their multiplication,
whereby, for example,
\begin{equation*}
  \surd2\cdot\surd3=\surd6.
\end{equation*}
One can give a geometrical argument
for this particular equation,
as in Figure \ref{fig:236},
\begin{figure}
  \centering
\psset{unit=25mm}
  \begin{pspicture}(0,-0.125)(1.74,1.865)
\psset{PointSymbol=none,RightAngleSize=0.1}
    \pstGeonode[PosAngle=-90](0,0)A(1,0)B
\pstRotation[RotAngle=-90,PosAngle=-22.5]BA[C]
\pstRightAngle ABC
\pstRotation[RotAngle=-90,PointName=none]CA
\pstRightAngle AC{A'}
\pstInterLC[PosAngle=90,PointNameA={}]C{A'}CB{D'}D
\pstInterLC[PosAngle=-90,PointNameA={}]ABAD{E'}E
\pstRotation[RotAngle=-90,PosAngle=90]EA[F]
\pstRightAngle AEF
\ncline AE\ncline EF\ncline AF\ncline AD\ncline CD\ncline BC
%\pstSegmentMark BA
%\pstSegmentMark BC
%\pstSegmentMark CD
\psset{linestyle=dotted}
\pstArcOAB CDB
\pstArcOAB AED
\pstArcOAB BCA
  \end{pspicture}
  \caption{$\surd2\cdot\surd3=\surd6$}
  \label{fig:236}
\end{figure}
where $ABC$ is an isosceles right triangle,
and $CD$ is drawn perpendicular to $AC$ and equal to $CB$, 
and $AE=AD$,
and the perpendicular to $AE$ at $E$
meets the extension of $AC$ at $F$.
If $AB$ and therefore $BC$ and $CD$ are each counted as a unit,
then $AC$ has length $\surd2$, and so $AD$ has length $\surd3$.
Since again $AE=AD$, and $AEF$ is isosceles,
we conclude that $AF$, as hypoteneuse of $AEF$, has length $\surd6$.
By similar triangles and the result concerning them called Thales's Theorem
(mentioned also later, on page \pageref{Thales}), 
$AF$ also has length $\surd2\cdot\surd3$.
However, this conclusion assumes the geometrical theory of multiplication
suggested by Descartes in his \emph{G\'eometrie} \cite{Descartes-Geometrie},
but not rigorously justified, as far as I know,
until the 1890s, 
in Hilbert's \emph{Foundations of Geometry} \cite{MR0116216}.

Such theoretical matters are beyond Asimov's scope.
They would not be beyond my scope,
if common logarithms rather than discrete logarithms were my subject.
I start with the question of what a number is in the first place.

\section{Numbers}

The second mathematical activity of our lives is to count.
The first is to recognize the existence
of such individuals or unities as \emph{can} be counted.

Let us understand a \textbf{number} as a collection
whose members can be counted.
This would seem to be the sense of number in Euclid's 
\emph{Elements} \cite{MR17:814b},
where, at the head of Book \textsc{vii}, 
a number is said to be a multitude of individuals, 
or unities, 
or (transliterating the Greek) monads.
John Dee invented the word \emph{unit,}
precisely to translate Euclid's \gr{<h mon'as -'ados}
\cite[\S2.5]{Pierce-on-comm-symm}.

Euclid's numbers might be understood as being what
in modern terms are \emph{finite sets.}
When two sets are in one-to-one correspondence,
today we may say that they are \emph{equipollent};
for Euclid they are simply \textbf{equal} as numbers,
just as, by definition, 
two distinct sides of an isosceles triangle
(like $AB$ and $BC$ in Figure \ref{fig:236}) 
are equal as bounded straight lines.
This is the meaning of the Greek adjective \gr{>isoskel'hs -'es},
which combines \gr{>'isos -h -on} \emph{equal} 
with \gr{t'o sk'elos -ous} \emph{leg}.
In Euclid's diagrams,
a number is such a bounded straight line
as is implicitly divisible into units,
all being equal to one another or, 
in modern terms, having the same length.

\section{Multiplication}

It is possible to \textbf{multiply} one number, 
the \textbf{multiplicand,}
by another number, the \textbf{multiplier.}
This means to lay out the multiplicand
as many times as there are units in the multiplier,
so that a new number is obtained.
The new number is the \textbf{multiple} of the multiplicand by the multiplier,
and it is the \textbf{product} of the two numbers.

To obtain a product,
what we lay out is perhaps not strictly the multiplicand itself, 
but \emph{copies} of it,
namely numbers that are equal to it.
The distinction is lost in our notation.
Five times six would appear to be, literally, 
six, laid out five times;
this gives
\begin{equation*}
6+6+6+6+6,
\end{equation*}
the sum we know as $30$.
I propose to denote the product here as $6\cdot 5$,
to be understood as six, multiplied by five.

The multiplicand \textbf{measures} the product
and is a \textbf{submultiple} of it;
the multiplier \textbf{divides} the product.
We can measure thirty apples by six apples:
the result is five piles, each holding six apples.
This means we can divide the thirty apples among five children:
each child gets six apples.
Without using this terminology,
Alexandre Borovik discusses the distinction 
between measuring and dividing apples
in \emph{Metamathematics of Elementary Mathematics} \cite{Borovik-Meta}.

Using the results just discussed,
how can we show that the thirty apples
can \emph{also} be divided among six children?
Why should the sum
 \begin{equation*}
5+5+5+5+5+5
\end{equation*}
of six fives
be equal to the sum of five sixes as above?
We shall review Euclid's general proof
of what we call the \emph{commutativity} of multiplication.
The proof will involve \emph{ratios} of numbers.

\section{The Euclidean Algorithm}

Given a pair of numbers,
we may transform it by subtracting the less from the greater.
We can continue
until the two numbers become equal.
We call this process the \textbf{Euclidean Algorithm.}
In the first two propositions of Book \textsc{vii} of the \emph{Elements}
\cite{Euclid-homepage},
Euclid describes the process with the passive form of the verb
\gr{>anjufair'ew}, \emph{to take away alternately.}
It is a deficiency of the big Liddell--Scott--Jones lexicon \cite{LSJ}
that Euclid is not cited under this word,
from which can be derived the noun \emph{anthyphaeresis}
(\gr{>anjufa'iresis}), meaning
\emph{alternate subtraction.}

At the end of the anthyphaeresis,
either of the two equal numbers 
measures all of the numbers that came before,
and so it is in particular a \textbf{common measure} 
of the original two numbers.
Moreover, every common measure of these numbers
measures every number found in the course of the anthyphaeresis;
in particular, the common measure measures the last number,
which is therefore the \textbf{greatest common measure} 
of the first two numbers.
In the case where this greatest common measure
is properly speaking not a number but a single unit,
the two original numbers must be \textbf{prime to one another.}

%\section{Proportion}

I once considered teaching number theory on the pattern of Euclid,
but then I found his approach too strange for the modern student.
I did learn two things: 
(1) the implicit use of the Euclidean algorithm
in the definition of proportion of numbers, and
(2) the use of this definition 
in a rigorous proof of commutativity of multiplication.

Suppose we apply the Euclidean Algorithm to two numbers, 
lying on the left and right respectively.
At each step of the algorithm,
we record first whether the left-hand or right-hand number is greater.
Thus we may obtain a sequence of letters L and R.
If this is the same as the sequence obtained from another pair of numbers,
then, by Euclid's definition at the head of Book \textsc{vii}, 
the four numbers are \textbf{in proportion,}
and the first two numbers have the \textbf{same ratio} 
as the second two numbers.

Let us pass to modern symbolism in an example.
If the first two numbers are $14$ and $10$,
then the steps of the algorithm give us
\begin{align*}
  &(14,10),&
  &(4,10),&
  &(4,6),&
  &(4,2),&
  &(2,2),
\end{align*}
whence $2$ is the greatest common measure of $14$ and $10$.
From $21$ and $15$ we obtain
\begin{align*}
  &(21,15),&
  &(6,15),&
  &(6,9),&
  &(6,3),&
  &(3,3),
\end{align*}
so $3$ is the greatest common measure of $21$ and $15$.
In either case, the pattern of larger entries is LRRL,
and therefore, by definition,
\begin{equation}\label{eqn:prop}
  14:10::21:15.
\end{equation}
This is not strictly an equation,
but an \emph{identity.}
The ratio $14:10$ is not \emph{equal} to $21:15$,
but the two ratios are the \emph{same} as one another:
they are one.
Euclid's language makes the distinction between equality and sameness;
the former is not used for ratios.

If we repeat the last letter in LRRL,
obtaining LRRLL,
and if we replace subsequences of repeated letters with their numbers,
we obtain the sequence $(1,2,2)$,
whose entries appear in the continued fraction
\begin{equation*}
  1+\cfrac1{2+\cfrac12}.
\end{equation*}
This then is a way to represent the ratio $14:10$ or $21:15$.
We may also note
\begin{align*}
&\begin{gathered}
  14=2\cdot 7,\\
	10=2\cdot5,
	\end{gathered}&
	&\begin{gathered}
	21=3\cdot7,\\
	15=3\cdot5,
	\end{gathered}
\end{align*}
where the repetition of the multipliers $7$ and $5$
is another way to verify the proportion \eqref{eqn:prop}.
However,
it is important that $7$ and $5$ are prime to one another,
so that they are uniquely determined by either of the pairs 
$(14,10)$ and $(21,15)$.
It will be a consequence of commutativity that
\begin{equation}\label{eqn:cross}
  14\cdot15=21\cdot10,
\end{equation}
that is,
$2\cdot7\cdot3\cdot5=3\cdot7\cdot2\cdot5$.
Nevertheless, in Euclidean mathematics, 
an equation like \eqref{eqn:cross} 
cannot serve as a \emph{definition} the proportion \eqref{eqn:prop},
simply because the equation does not immediately establish 
that something about the pair $(14,10)$
is the \emph{same} as for $(21,15)$.

\section{Commutativity}

Multiplication is certainly commutative in case one of the factors is unity;
for the product then is simply the other factor.

From the definition of proportionality,
all ratios of the form $x:x\cdot a$ are the same.
In saying this so compactly, 
we follow the convention established by Descartes \cite{Descartes-Geometrie},
whereby letters from the beginning of the alphabet denote constants,
and from the end, variables.
Since the ratio $1:1\cdot a$ is just $1:a$,
we can conclude
\begin{equation}\label{eqn:1:a::b:ba}
  1:a::b:b\cdot a.
\end{equation}
Suppose now $a:b::c:d$,
so that the steps of the Euclidean algorithm are the same,
whether applied to $(a,b)$ or to $(c,d)$.
These steps are then the same as for $(a+c,b+d)$,
by what we call the commutativity of addition.
For, assuming $a>b$, we must also have $c>d$,
and so $a+c>b+d$, and consequently
\begin{equation*}
  (a+c)-(b+d)=(a-b)+(c-d).
\end{equation*}
We conclude
\begin{equation}\label{eqn:implies}
  a:b::c:d\logicrel{implies}a:b::a+c:b+d.
\end{equation}
As a special case,
since $a:b::a:b$, we have $a:b::a\cdot2:b\cdot2$.
Likewise, repeated application of the implication 
\eqref{eqn:implies} gives
\begin{equation*}
  a:b::a\cdot c:b\cdot c.
\end{equation*}
As a special case,
\begin{equation*}
  1:a::b:a\cdot b.
\end{equation*}
Combining this with \eqref{eqn:1:a::b:ba} yields
\begin{equation*}
  b:a\cdot b::b:b\cdot a.
\end{equation*}
From this we conclude
\begin{equation*}
  a\cdot b=b\cdot a.
\end{equation*}
In modern symbolism and typography,
such is Euclid's rigorous proof of Proposition 16 of Book \textsc{vii}
of the \emph{Elements.}

\section{Congruence}

Let us henceforth employ the terminology and notation of Gauss,
born 1777,
who writes at the beginning of the 
\emph{Disquisitiones Arithmeticae} of 1801
\cite{Gauss-Latin},
\begin{quotation}
Si numerus $a$ numerorum $b$, $c$ differantiam metitur. 
$b$ et $c$
\emph{secundum $a$ congrui} dicuntur,
sin minus, \emph{incongrui:}
ipsum $a$ \emph{modulum} appellamus.
Uterque numerorum $b$, $c$ priori in casu alterius \emph{residuum,}
in posteriori vero \emph{nonresiduum} vocatur\lips

Omnia numeri dati $a$ residua secundum modulum $m$
sub formula $a+km$ comprehenduntur,
designante $k$ numerum integrum indeterminatum\lips

Numerorum congruentiam hoc signo, $\equiv$, in posterum denotabimus,
modulum ubi opus erit in clausulis adiungentes,
$-16\equiv9$ (mod.\ $5$), $-7\equiv15$ (mod.\ $11$).
\end{quotation}
In the English version of Arthur A. Clarke \cite{Gauss},
Gauss's words are rendered as follows.
\begin{quotation}
If a number $a$ divides the difference of the numbers $b$ and $c$,
$b$ and $c$ are said to be \emph{congruent relative to} $a$;
if not, $b$ and $c$ are \emph{noncongruent.}
The number $a$ is called the \emph{modulus.}
If the numbers $b$ and $c$ are congruent,
each of them is called a \emph{residue} of the other.
If they are noncongruent they are called \emph{nonresidues}\lips

Given $a$,
all its residues modulo $m$ are contained in the formula
$a+km$ where $k$ is an arbitrary integer\lips

Henceforth we shall designate congruences by the symbol $\equiv$,
joining to it in parentheses the modulus when it is necessary to do so;
e.\ g.\ 
$-7\equiv15$ (mod.\ $11$),
$-16\equiv9$ (mod.\ $5$).
\end{quotation}
It would be more faithful to Gauss, 
and to his predecessors Euclid and Fermat
(whom we shall consider presently),
to say ``measures'' where Clarke says ``divides.''
However,
we have shown that there is no mathematical difference.

Where Gauss has \emph{secundum modulum,}
Clarke has ``modulo.''
This is the ablative or dative case of the Latin \emph{modulus -i,}
which is the diminutive of \emph{modus -i} ``measure.''
In \emph{An Adventurer's Guide to Number Theory}
\cite[p.\ 116]{MR1314197},
after discussing the congruences
$5\equiv 12\equiv1083\pmod7$,
Richard Friedberg writes,
\begin{quote}
If you have studied Latin,
you will understand that ``modulo $7$''
is an ablative absolute and means
``$7$ being the modulus.''
In the eighteenth century,
when congruences were first studied,
most mathematical articles were written in Latin.
The phrase,
``modulo $7$,'' was so catchy that it still sticks.
\end{quote}
Friedberg is probably correct that \emph{modulo}
is in the ablative case;
he appears to be wrong about the reason.

Again, where we say ``modulo $7$,''
Gauss says \emph{secundum modulum $7$,}
``with respect to the modulus $7$.''
\emph{Secundum} is a preposition
taking the accusative case,
here \emph{modulum.}
The preposition
derives from the adjective \emph{secundus -a -um} ``following,''
which, in the form ``second,''
serves in English as the ordinal form of the cardinal number ``two.''
When doing duty for Gauss's \emph{secundum modulum,}
\emph{modulo} should probably be understood as an
\emph{instrumental} ablative.
The uses of the earlier Indo-European instrumental case
were apparently taken up by the Latin ablative.
In the present context,
the modulus is the instrument---%
the measuring stick---%
whereby congruence is to be determined.

Congruence is a geometric notion.
In \emph{Number Theory and Its History} \cite[p.\ 211]{MR939614},
after defining things as Gauss does, Oystein Ore writes simply,
\begin{quote}
These terms, as one sees, are derived from Latin,
\emph{congruent} meaning \emph{agreeing} or \emph{corresponding}
while \emph{modulus} signifies \emph{little measure.}
\end{quote}
We can say more.
Where Heath \cite{MR17:814b} translates one of Euclid's common notions as
\begin{quote}
Things which coincide with one another are equal to one another,
\end{quote}
the verb ``coincide'' 
might just as well be ``are congruent.''
Commandinus \cite{Euclid-Commandinus}
and Heiberg \cite{Euclid-Heiberg}
use the Latin source of our adjective ``congruent'' 
to translate Euclid's Greek, thus:
\begin{quote}\centering
  qu\textogonekcentered e sibi ipsis congruunt,
  inter se sunt \textogonekcentered equalia.\\
quae inter se congruunt, aequalia sunt.\\
\gr{t`a >efarm'ozonta >ep'' >all'hla >'isa >all'hlois >est'in.}
\end{quote}
(Commandinus's printer uses the
\textogonekcentered e or \emph{e caudata} for \emph{ae}.
The printer uses also the old-fashioned long ess, 
when the ess is not terminal, 
but I have not managed to print this with \LaTeX.)

\section{Divisibility}

We are now allowed 
to use the notions of division and measurement interchangeably.
We may also consider our objects of study to be not simply counting numbers,
but ``signed'' counting numbers, or \emph{integers}---%
of which the counting numbers are just the positive instances.

Thus for example the Euclidean Algorithm allows us to find
what is now called the \emph{greatest common divisor} or ``gee cee dee'' 
($\gcd$) of two numbers.
Moreover, the Algorithm allows us to solve the equation
\begin{equation*}
ax+by=\gcd(a,b),
\end{equation*}
where now one of $x$ and $y$ will be negative
when $a$ and $b$ are positive.
This result is called \textbf{B\'ezout's Lemma,}
perhaps by way of impressing on students the importance of the result;
such possibilities are discussed in\label{Thales}
``The Theorem of {T}hales:
{A} Study of the Naming of Theorems in School
Geometry Textbooks''
\cite{Thales-Theorem},
a source I used in my own study of Thales's Theorem \cite{Pierce-Thales-9}.
The connection of B\'ezout to the lemma named for him
does seem even more tenuous than in the case of Thales.

To symbolize that an integer $a$ measures or divides an integer $b$,
we may write
\begin{equation*}
  a\divides b.
\end{equation*}
I do not know the origin of this notation,
but Landau used it in 1927 \cite[p.\ 11]{MR0092794},
and Hardy and Wright (who also use it) 
say in 1938 \cite[p.\ vii]{MR568909},
\begin{quote}
  To Landau in particular we,
in common with all serious students of the theory of numbers,
owe a debt which we could hardly overstate.
\end{quote}
For Landau and for Hardy and Wright, unlike Gauss,
the symbolism of divisibility comes before that of congruence.
Hardy and Wright \cite[p.\ 49]{MR568909} observe of congruence,
\begin{quote}
  The definition does not introduce any new idea,
since `$x\equiv a\pmod m$' and `$m\divides x-a$'
have the same meaning, but each notation has its advantages.
\end{quote}

Strictly speaking, 
Landau's sign of divisibility is oblique,
like the solidus we use for denoting fractions.
For us, $a/b$ is a rational number;
for Landau, it is the assertion that $aq=b$ for some integer $q$.
This assertion has the consequence that Landau expresses as
$\size a/\size b$; we have to write,
more confusingly, $\size a\divides\size b$.
However, there are no other absolute values discussed in the present work.

The fraction that for us is $a/b$ is for Landau $\frac ab$ or $a:b$.
It so happens that Landau finds greatest common divisors,
not with the Euclidean Algorithm,
but by first observing that the least common multiple of $a$ and $b$
divides every common multiple
(since otherwise the remainder 
would be a common multiple less than the least).

The quotient of $ab$ by the least common multiple of $a$ and $b$
is shown to be the greatest common divisor.
Landau and Hardy and Wright denote this by $(a,b)$,
which is convenient for international use;
but I shall stick with $\gcd(a,b)$.
Hardy and Wright also use $\{a,b\}$,
with braces,
to denote the least common multiple of $a$ and $b$;
but I shall use $\lcm(a,b)$.
Thus
\begin{equation*}
  \frac{ab}{\lcm(a,b)}\cdot\frac{\lcm(a,b)}b=a,
\end{equation*}
and similarly with $a$ and $b$ interchanged, so $ab/\lcm(a,b)$
is a common divisor of $a$ and $b$.
If $d$ is a common divisor, then $ab/d$ is a common multiple, so
\begin{align*}
  \lcm(a,b)&\;\left\vert\;\frac{ab}d,\right.&
d&\;\left\vert\;\frac{ab}{\lcm(a,b)}.\right.
\end{align*}
Thus
\begin{equation*}
  \gcd(a,b)=\frac{ab}{\lcm(a,b)}.
\end{equation*}
We shall use this and its notation once later.
We shall have used braces as is customary today,
to delineate sets.

If $a\divides bc$ and $\gcd(a,b)=1$,
then,
since $ab$ is the least common multiple of $a$ and $b$,
and $bc$ is \emph{some} common multiple,
we have $ab\divides bc$ by what we have shown.
It now follows that $a\divides c$.
This is Landau's proof of what we shall call \textbf{Euclid's Lemma.} 
Strictly, Proposition 30
of Book \textsc{vii} of the \emph{Elements}
is the case where $a$ is prime.

B\'ezout's Lemma gives the neat proof of Euclid's Lemma
that may be more common than Landau's.
From $ax+by=1$, we obtain $acx+bcy=c$,
so that, since $a\divides acx$,
if also $a\divides bc$, we can conclude $a\divides c$.

Useful for us at present is indeed Euclid's special case.
With respect to a prime modulus $p$,
if $ab\not\equiv0$, then neither of $a$ and $b$
can be congruent to $0$.
This gives us \textbf{cancellation:}
if $a\not\equiv0$, and $ab\equiv ac$, then $b\equiv c$.
Thus the first $p-1$ multiples of $a$,
starting from $a$ itself,
are incongruent to one another and to $0$.
Any list of numbers with this property can have length at most $p-1$.
Thus if we add $1$ to the list of multiple of $a$,
it must be congruent to one of these multiples.
This means $a$ is \textbf{invertible} with respect to $p$.
With the Euclidean Algorithm,
we can actually find the inverse, since $ax+py=1$
means $ax\equiv1\pmod p$.



        

\section{Fermat's Theorem}

On Thursday, October 18, 1640, 
in a letter to Bernard Fr\'enicle de Bessy (1605--1675),
Pierre de Fermat (1601--65)
described as follows
what we now know as \textbf{Fermat's Theorem}
\cite[p.\ 209]{Fermat-Oeuvres-2}.
\begin{quotation}
  Tout nombre premier mesure infailliblement
  une des puissances $-$ 1
  de quelque progression que se soit,
  et l'exposant de la dite puissance
  est sous-multiple du nombre premier donn\'e $-$ 1;
  et, apr\`es qu'on a trouv\'e
  la premi\`ere puissance qui satisfait \`a la question,
  toutes celles dont les exposants sont multiples
  de l'exposant de la premi\`ere satisfont
  tout de m\`eme \`a la question.

  Exemple: soit la progression donn\'ee
  \begin{center}
\begin{tabular}{*7{c}}
\smaller1&\smaller2&\smaller3&\smaller4&\smaller5&\smaller6&\smaller\\
3&9&27&81&243&729&etc.
\end{tabular}
  \end{center}
  avec ses exposants en dessus.

  Prenez, par exemple, le nombre premier 13.
  Il mesure la troisi\`eme puissance $-$ 1, de laquelle 3,
  exposant, est sous-multiple de 12,
  qui est moindre de l'unit\'e que le nombre 13,
  et parce que l'exposant de 729,
  qui est 6,
  est multiple du premier exposant,
  qui est 3,
  il s'ensuit que 13 mesure aussi lat dite puissance 729 $-$ 1.

  Et cette proposition est g\'en\'eralement vraie
  en toutes progressions et en tous nombres premiers;
  de quoi je vous envoierois la d\'emonstration,
  si je n'appr\'ehendois d'\'etre trop long.
\end{quotation}
In his
\emph{Source Book in Mathematics, 1200--1800}
\cite[p.\ 28]{Fermat-theorem},
Struik translates Fermat as below.
Instead of \emph{measures,} Struik says ``is a factor of'';
instead of \emph{submultiple,} ``divisor.''
He also misdates the letter as being of October 10, 1640.
\begin{quotation}
Every prime number is always a factor
[\emph{mesure infailliblement}]
of one of the powers of any progression minus $1$,
and the exponent of this power is a divisor
of the prime number minus $1$.
After one has found the first power that satisfies the proposition,
all those powers of which the exponents
are multiples of the exponent of the first power
also satisfy the proposition. 

\emph{Example:}  Let the given progression be
\begin{equation*}
\begin{array}{*7{r}}
1&2& 3& 4&  5&  6&\\
3&9&27&81&243&729&\text{etc.}
\end{array}
\end{equation*}
with its exponents written on top.

Now take, for instance, the prime number $13$.
It is a factor of the third power minus $1$,
of which $3$ is the exponent and a divisor of $12$,
which is one less than the number $13$,
and because the exponent of $729$, which is $6$,
is a multiple of the first exponent, which is $3$,
it follows that $13$
is also a factor of this power $729$ minus $1$. 

And this proposition is generally true for all progressions and for
all prime numbers, of which I would send you the proof if I were not
afraid to be too long. 
\end{quotation}
According to Fermat then,
for every number $a$,
for every prime number $p$,
there is a positive exponent $\ell$ such that,
with respect to the modulus $p$,
\begin{equation*}
  a^{\ell}\equiv1;
\end{equation*}
moreover, if $k$ is the least such $\ell$,
then $k\divides p-1$ and $a^{kx}\equiv1$
(for every multiplier $x$).
Let us not fault Fermat for omitting the condition
$a\not\equiv0$ and for not strictly observing
that, conversely, $k$ divides every $\ell$.

Usually what is called \textbf{Fermat's Theorem}
is the special case that $a^{p-1}\equiv1$ when $p\ndivides a$.
This is the usage of Gauss,
who derives the result after proving $k\divides p-1$ as above.
He then observes 
that can prove the basic form of Fermat's Theorem by induction.
Indeed, the claim is trivially true when $a=1$.
If it is true when $a=b$,
then it is true when $a=b+1$,
since, as a consequence of Euclid's Lemma,
\begin{equation*}
  (b+1)^p\equiv b^p+1\pmod p.
\end{equation*}
Gauss attributes this proof to Euler.


%\section{The Lagrange Theorem}

Gauss also attributes to Euler
a proof of the more general assertion of Fermat.
We can summarize the proof as follows,
using the terminology of Landau \cite[p.\ 42]{MR0092794},
whereby, with respect to a modulus $n$,
a \textbf{complete set of residues}
has any two, and therefore all three, 
of the following properties:
\begin{compactenum}[1)]
\item
there are exactly $n$ members of the set,
  \item
no two members are congruent,
\item
every number is congruent to one of them.
\end{compactenum}
If from a complete set of residues
we select precisely those members that are prime to $n$,
we have a \textbf{reduced set of residues.}
For any prime number $p$
and any $a$ that is prime to $p$,
there must be numbers $b_i$ such that the entries in the table below
are incongruent from one another and
compose a reduced set of residues with respect to $p$.
\begin{equation*}
  \begin{array}{cccc}
    a&a^2&\cdots&a^k\\
    ab_1&a^2b_1&\cdots&a^kb_1\\
    ab_2&a^2b_2&\cdots&a^kb_2\\
    \hdotsfor4\\
    ab_n&a^2b_n&\cdots&a^kb_n
  \end{array}
\end{equation*}
In particular, the table has $k$ columns and $p-1$ entries,
and therefore $k\divides p-1$.


\section{Algebra}

Since a reduced set of residues with respect to a given modulus
is closed under both multiplication and inversion,
those residues compose a finite \emph{group.}
If the modulus is $n$,
then the size of the group of reduced residues
is the number recognized by Euler and denoted by Gauss by
\begin{equation*}
\euphi(n).
\end{equation*}
(Actually Gauss just wrote $\phi n$.)

The general form of Fermat's Theorem
is then a special case of the \textbf{Lagrange Theorem,}
which is that the order of a finite group
is divisible by the order of every subgroup.
Relevant sections of Lagrange's paper \cite{Lagrange}
are selected and translated in Struik's 
\emph{Source Book} \cite{Lagrange-Struik};
but as far as I can tell,
one can infer from the paper only that the ``Lagrange Theorem''
holds when the group is the group of permutations of finitely many objects.

Abstract algebra both illuminates and complicates 
the number theory or \emph{arithmetic} 
that is its origin.
Number theory involves various structures,
such as the groups of residues just mentioned;
algebra looks at these as wholes and gives them names.

As being ordered and being capable of being added and multiplied
as learned in school,
the integers compose a so-called \emph{ordered commutative ring,}
denoted by $\Z$, supposedly for the German \textfrak{\larger Zahl.}
If we are going to use the symbol $\Z$,
we might as well also allow the symbol $\N$
for the positive part of $\Z$,
consisting of the counting numbers.

One sometimes wants a name for the \emph{non-negative} part of $\Z$,
namely the positive part with zero.
Some writers use $\N$ for this part,
but the name $\upomega$ (omega) is already used in set theory,
and so I would use that, 
if I had a need, which I do not in the present work.

Landau and Hardy and Wright do not need even a symbol like $\Z$.
The term ``ring'' for what is symbolized by $\Z$ may be unfortunate,
but it seems to arise from the observation that, for example,
the real numbers $a+b\surd 2$, where $a$ and $b$ are integers,
also compose a ring, since the product of two such numbers
``circles back'' to being such a number as well:
\begin{equation*}
  (a+b\surd 2)(c+d\surd 2)=ac+2bd+(ad+bc)\surd 2.
\end{equation*}

Given $n$ in $\N$,
for the moment we let
\begin{equation*}
  [a]_n=\{x\colon x\equiv a\}\pmod n,
\end{equation*}
the \textbf{congruence class} of $a$ with respect to $n$.
We use this only to define
\begin{equation*}
  \Z_n=\{[x]_n\colon x\in\Z\},
\end{equation*}
the set of congruence classes with respect to $n$.
Here we are only replacing each element of a complete set of residues
with its congruence class.
Like $\Z$ itself,
$\Z_n$ is a commutative ring, though it is not ordered.

The multiplicatively invertible elements---the \textbf{units}---of $\Z_n$ 
compose the group denoted by
\begin{equation*}
  \units{\Z_n}.
\end{equation*}
Again, the size or \textbf{order} of this group is $\euphi(n)$.
For every prime $p$, since $\euphi(p)=p-1$,
this means both that $\Z_p$ is a \textbf{field}---%
a commutative ring, like the ring of rational or real numbers,
in which every nonzero element is invertible---%
and that (with the help of the Lagrange Theorem)
Fermat's Theorem holds.

\section{Primitive roots}

If $d$ and $n$ are counting numbers, $d$ dividing $n$,
then the integers that have with $n$ the greatest common divisor $d$
are in one-to-one correspondence with the integers that are prime to $n/d$.
The correspondence is between $dx$ and $x$,
where $\gcd(x,n/d)=1$.
Moreover, for every element $a$ of $\Z_n$,
$\gcd(a,n)$ is well-defined, and it divides $n$.
In symbols then,
for all $a$ in $\Z_n$,
\begin{equation}\label{eqn:iff}
	\gcd(a,n)=d\logicrel{if and only if}
	d\divides a\And\gcd\left(\frac ad,\frac nd\right)=1.  
\end{equation}

The foregoing conclusion will serve as a lemma for a theorem
that Gauss sets out \cite[\P39]{Gauss-Latin},
as we shall, in Euclid's \emph{protasis} style.
I take the terminology here from David Fowler \cite[p.\ 386]{MR2000c:01004},
as naming the style's
``distinctive and useful opening feature, the enunciation or
\emph{protasis.}''
In a proposition of Euclid,
first comes an enunciation, and only then comes the demonstration or proof
of what has been enunciated.
Students readily adopt this style,
first writing what they want to prove,
then writing down more things, which they hope will be considered as a proof.
It is not always clear that the students understand 
the logical relations involved.

Euclid avoids confusing his readers by following a consistent pattern.
Each of his propositions has up to six parts, 
always in the same order.
In his commentary on Euclid,
Proclus names the parts of a proposition as enunciation (\gr{pr'otasis}), 
exposition, specification, construction, demonstration, and conclusion
\cite[p.\ 159 (203)]{MR1200456}.

Neither Proclus nor Euclid has a name 
for what we call the proposition as a whole.
Proclus says, in Morrow's translation
\cite[p.\ 63 (77)]{MR1200456},
\begin{quote}
  Again the propositions that follow from the first principles
he divides into problems and theorems;
\end{quote}
but as in the King James Bible,
the words ``propositions that follow''
could be italicized, as having no explicit counterpart in the Greek,
which, for the passage just quoted, is \cite[p.\ 77]{1873procli}
\begin{quote}
  \gr{P'alin d'' a>~u t`a >ap`o t~wn >arq~wn e>is probl'hmata
diaire~itai ka`i jewr'hmata.}
\end{quote}
Pappus makes the etymology clear \cite[pp.\ 566 f.]{MR13:419b}:
in a problem (\gr{pr'oblhma}) it is proposed (\gr{prob'alleta'i}) 
to do something;
in a theorem (\gr{je'wrhma}),
the implications of hypotheses are contemplated (\gr{jewre~itai}).
Euclid signals the distinction between a problem and a theorem
by how he ends it,
using respectively the words that we translate into Latin and abbreviate as
\textsc{q.e.f.}\ (``which was to be done'')
and \textsc{q.e.d.}\ (``which was to be proved'').

A web edition of Euclid's Greek text labels each proposition as
a \gr{pr'otasis} \cite{Euclid-homepage}.
This conforms to the modern practice 
of treating the enunciation as a \emph{metonym} 
for the proposition as a whole.
Here the terminology is from Reviel Netz,
who argues that for Euclid the \emph{diagram} 
is the metonym of the proposition \cite[p.\ 38]{MR1683176}.
The handy little book called \emph{The Bones} \cite{bones}
does not choose sides,
but supplies both the enunciation and the diagram 
(and nothing else)
for each of Euclid's propositions.

For the proposition below,
Gauss simply italicizes his protasis:
\begin{quote}\itshape
  Si $a$, $a'$, $a''$, etc.\ sunt omnes divisores ipsius $A$
(unitate et ipso $A$ non exclusis), erit
\begin{equation*}
  \phi a+\phi a'+\phi a''+\text{etc.}= A.
\end{equation*}
\end{quote}
We give the protasis a bold label.

\begin{theorem}[Gauss]
The sum of the values of $\euphi(d)$,
as $d$ ranges over the positive divisors of $n$, is just $n$ itself;
in symbols,
\begin{equation}\label{eqn:Gauss}
  \sum_{d\divides n}\euphi(d)=n.  
\end{equation}
\end{theorem}

\begin{proof}
Suppose $d\divides n$.
By \eqref{eqn:iff}, the two sets
\begin{align*}
&\{x\in\Z_n\colon\gcd(x,n)=d\},&&\{y\in\Z_{n/d}\colon\gcd(y,n/d)=1\}
\end{align*}
have the same size.
The size of the latter set being $\euphi(n/d)$,
we can conclude
\begin{equation*}
n=\sum_{d\divides n}\euphi\left(\frac nd\right).
\end{equation*}
This yields \eqref{eqn:Gauss}, by symmetry.
\end{proof}

The \textbf{order} of an element $a$ of $\units{\Z_n}$
is the least positive exponent $\ell$
such that $a^{\ell}=1$.
In symbols,
\begin{equation*}
  \ord na=\min\{x\in\N\colon a^x=1\}.
\end{equation*}
If we think of $a$ as an integer,
rather than a congruence class,
we should perhaps write something like
\begin{equation*}
\ord na=\min\{x\in\N\colon a^x\equiv1\}\pmod n.
\end{equation*}
If it exists, a \textbf{primitive root} of $n$
is an element of $\units{\Z_n}$ having order $\euphi(n)$.
Euler gave a proof of the following, 
but there was a gap, which,
according to Burton \cite[p.\ 162]{Burton},
Legendre filled.
Gauss mentions the gap, but not Legendre.

\begin{theorem}
Every prime number has a primitive root.
\end{theorem}

\begin{proof}
Let $p$ be a prime number.
We shall show that the number of its primitive roots is $\euphi(p-1)$,
which is positive.
Since $\euphi(p)=p-1$,
the order of every element of $\units{\Z_p}$ measures this.
If $d\divides p-1$, let us denote by
\begin{equation*}
\uppsi_p(d)
\end{equation*}
the number of elements of $\units{\Z_p}$ having order $d$.
Since every element of $\units{\Z_p}$ 
has some such order $d$, we have
\begin{equation*}
p-1=\sum_{d\divides p-1}\uppsi_p(d).
\end{equation*}
By the previous theorem,
we shall be done when we show $\uppsi_p(d)\leq\euphi(d)$.
Suppose $\uppsi_p(d)>0$,
so that some $a$ in $\units{\Z_p}$ has order $d$.
The $d$ elements of the set $\{a^t\colon t\in\Z_d\}$
are solutions of the congruence
\begin{equation*}
x^d\equiv1\pmod p.
\end{equation*}
They must be the only solutions,
since the congruence can have at most $d$ solutions
(since $\Z_p$ is a field).
Moreover, with respect to $p-1$,
\begin{align*}
\ord p{a^k}
&=\min\{x\in\N\colon a^{kx}\equiv1\}\\
&=\frac1k\min\{y\in\N\colon k\divides y\And a^y\equiv1\}\\
&=\frac1k\min\{y\in\N\colon k\divides y\And d\divides y\}\\
&=\frac{\lcm(k,d)}k,
\end{align*}
and this is $d/\gcd(k,d)$,
by what we showed earlier.
Thus, if it is positive,
$\uppsi_p(d)$ must be the size of the set
\begin{equation*}
\{x\in\Z_d\colon\gcd(x,d)=1\},
\end{equation*}
and this size is by definition $\euphi(d)$.
\end{proof}

The foregoing is the \emph{first} of Gauss's two proofs.
It is the one that Hardy and Wright give
\cite[pp.\ 85 f.]{MR568909};
but they give it, unlike Gauss, 
\emph{after} proving Gauss's Law of Quadratic Reciprocity.
Like other writers,
they follow Gauss in using the notation $\psi$
where I have $\uppsi_p$.
It seems to me desirable to use the subscript $p$,
so that the ultimate independence of $\uppsi_p(d)$ from $p$
(as long as $d\divides p-1$) may be all the more remarkable.
I also make the subtle distinction of using an upright letter $\uppsi$
for something defined once for all;
an italic letter like $\psi$ may have different meanings in different settings,
even though the meaning may be considered constant in a particular setting.

\section{Two more proofs}

%\begin{sloppypar}
Landau gives Gauss's second proof that primes have primitive roots,
but he gives it, like Hardy and Wright,
only after Quadratic Reciprocity.
It
uses the Fundamental Theorem of Arithmetic,
that every prime number has a unique prime factorization.
Gauss seems to have been the first person to state this explicitly
\cite[p.\ 10]{MR568909}.
%\end{sloppypar}

Briefly,
suppose $p-1$ has the prime factorization $\prod_qq^{d(q)}$.
For each prime $q$ in the product, since the congruence
\begin{equation*}
  x^{(p-1)/q}\equiv1\pmod p
\end{equation*}
has at most $(p-1)/q$ solutions,
it has a non-solution, $a_q$, from $\units{\Z_p}$.
Then
$q^{d(q)}$ is the order of the power
\begin{equation*}
  a_q{}^{(p-1)/q^{d(q)}},
\end{equation*}
and the product $\prod_qa_q{}^{(p-1)/q^{d(q)}}$
of all of these powers has order $p-1$.

For a third proof that every prime number has a primitive root,
noting as we have that $\Z_p$ is a field,
we can just prove generally 
that the group of units of every finite field $K$ is \emph{cyclic.}
Again briefly,
if $a$ and $b$ in $\units K$ have orders $k$ and $m$,
then $ab$ must have order $km$,
if $k$ and $m$ are prime to one another.
As a result, if $\gcd(k,m)=d$, then $a^{k/d}b$ has order $\lcm(k,m)$.
If $a$ already has maximal order, then $k\divides m$.
In this case,
every element of $\units K$ is a root of the polynomial $x^m-1$;
in a field, this can have no more than $m$ roots;
therefore $m$ is less by $1$ than the size of $K$.

\section{Practicalities}

The number 997 is prime,
because (1) it is less than 1024, which is the square of 32,
and (2) it is indivisible by the primes less than 32,
namely 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. 
Now we know that 997 has a primitive root,
and Gauss's second proof suggests a procedure for finding one.
Alternatively, if $a$ is a candidate,
since 996 has the prime factorization $2^2\cdot3\cdot83$,
it is enough to check that none of the powers
\begin{align*}
  &a^{2^2\cdot3},&&a^{2^2\cdot83},&&a^{3\cdot83}
\end{align*}
is congruent to unity.  One can compute these by hand
by taking successive squares and using for example
\begin{equation*}
  83=64+16+2+1=2^6+2^4+2^2+2^0.
\end{equation*}
In fact a table of primes and their primitive roots in Burton 
\cite[p.\ 393]{Burton}
gives 7 as the least primitive root of 997.
For the table of antilogarithms in \S\ref{sect:powers},
I computed a list of exponents and the corresponding powers of 7 
\emph{modulo} 997 
with an electronic spreadsheet
(LibreOffice Calc),
relying on the rule
\begin{equation*}
  k^2\equiv\ell\logicrel{implies}(k+1)^2\equiv\ell+2k+1.
\end{equation*}
The spreadsheet might then order the list according to the powers,
rather than the exponents;
but for the table of logarithms in \S\ref{sect:logs}, 
I used the MakeIndex program coming with \LaTeX\ to do the reordering.


%\printindex

\AfterBibliographyPreamble{\smaller}
%\bibliographystyle{plain}
%\bibliography{../references}
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\end{thebibliography}


\end{document}
