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\ctrline{\it What the Tortoise Said to Achilles}
\ctrline{by Lewis Carroll}
\vskip 12pt
\noindent Achilles had overtaken the Tortoise, and had seated
himself comfortably on its back.

``So you've got to the end of our race-course?'' said the Tortoise.
``Even though it {\sl DOES} consist of an infinite series of distances?
I thought some wiseacre or other had proved that the thing
couldn't be done?''

``It {\sl CAN} be done,'' said Achilles.  ``It {\sl HAS} been done!  
{\it Solvitur ambulando.}
You see the distances were constantly {\sl DIMINISHING}; and so---''


``But if they had been constantly {\sl INCREASING}?'' 
the Tortoise interrupted.  ``How then?''

``Then I shouldn't be here,'' Achilles modestly replied; ``and {\sl YOU}
would have got several times round the world, by this time!''

``You flatter me---{\sl FLATTEN}, I mean,'' said the Tortoise;
``for you {\sl ARE} a heavy weight, and {\sl NO} mistake!  
Well now, would you like to hear of a race-course, that most people
fancy they can get to the end of in two or three steps, while it
{\sl REALLY} consists of an infinite number of distances, each
one longer than the previous one?''

``Very much indeed!'' said the Grecian warrior, as he drew from
his helmet (few Grecian warriors possessed {\sl POCKETS} in those days)
an enormous note-book and pencil.  ``Proceed!  And speak {\sl SLOWLY},
please! {\sl SHORTHAND} isn't invented yet!''

``That beautiful First Proposition by Euclid!'' the Tortoise
murmured dreamily.  ``You admire Euclid?''

``Passionately!  So far, at least, as one {\sl CAN} admire a treatise
that won't be published for some centuries to come!''

``Well, now, let's take a little bit of the argument in that First
Proposition---just {\sl TWO} steps, and the conclusion drawn from them.
Kindly enter them in your note-book.  And in order to refer to them
conveniently, let's call them A, B, and Z:---

(A)  Things that are equal to the same are equal to each other.

(B)  The two sides of this Triangle are thins that are equal to the same.

(Z)  The two sides of this Triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically from 
A and B, so that any one who accepts A and B as true, {\sl MUST} accept
Z as true?''

``Undoubtedly!  The youngest child in a High School---as soon as High
Schools are invented, which will not be till some two thousand years
later---will grant {\sl THAT}.''

``And if some reader had {\sl NOT} yet accepted A and B as true, he 
might still accept the {\sl SEQUENCE} as a {\sl VALID} one, I suppose?''

``No doubt such a reader might exist.  He might say, `I accept as true the
Hypothetical Proposition that, {\sl IF} A and B be true, Z must be true;
but I {\sl DON'T} accept A and B as true.'  Such a reader would do 
wisely in abandoning Euclid, and taking to football.''

``And might there not {\sl ALSO} be some reader who would say
`I accept A and B as true, but I {\sl DON'T} accept the Hypothetical'?''

``Certainly there might.  {\sl HE}, also, had better take to football.''

``And {\sl NEITHER} of these readers,'' the Tortoise continued,
``is under any logical necessity to accept Z as true?''

``Quite so,'' Achilles assented.

``Well, now, I want you to consider {\sl ME} as a reader of the
{\sl SECOND} kind, and to force me, logically, to accept Z as true.''

``A tortoise playing football would be---'' Achilles was beginning.

``---an anomaly, of course,'' the Tortoise hastily interrupted.
``Don't wander from the point.  Let's have Z first, and football 
afterwards!''

``I'm to force you to accept Z, am I?''  Achilles said musingly.
``And your present position is that you accept A and B, but you
{\sl DON'T} accept the Hypothetical---''

``Let's call it C,'' said the Tortoise.

``---but you {\sl DON'T} accept

(C)  If A and B are true, Z must be true.

``That is my present position,'' said the Tortoise.

``Then I must ask you to accept C.''

``I'll do so,'' said the Tortoise, ``as soon as you've entered it in that
notebook of yours.  What else have you got in it?''

``Only a few memoranda,'' said Achilles, nervously fluttering the
leaves:  ``a few memoranda of---of the battles in which I have 
distinguished myself!''

``Plenty of blank leaves, I see!'' the Tortoise cheerily remarked.  
``We shall need them {\sl ALL}!''  (Achilles shuddered.)  
``Now write as I dictate:---

(A)  Things that are equal to the same are equal to each other.

(B)  The two sides of this Triangle are things that are equal to the same.

(C)  If A and B are true, Z must be true.

(Z)  The two sides of this Triangle are equal to each other.

``You should call it D, not Z,'' said Achilles.  ``It comes {\sl NEXT}
to the other three.  If you accept A and B and C, you {\sl MUST}
accept Z.''

``And why must I?''

``Because it follows {\sl LOGICALLY} from them.  If A and B and C
are true, Z {\sl MUST} be true.  You can't dispute {\sl THAT}, I imagine?''

``If A and B and C are true, Z {\sl MUST} be true,'' the Tortoise
thoughtfully repeated.  ``That's {\sl ANOTHER} Hypothetical, isn't it?
And, if I failed to seet its truth, I might accept A and B and C, and
{\sl STILL} not accept Z, mightn't I?''

``You might,'' the candid hero admitted; ``though such obtuseness
would certainly be phenomenal.  Still, the event is {\sl POSSIBLE}.
So I must ask you to grant {\sl ONE} more Hypothetical.''

``Very good, I'm quite willing to grant it, as soon as you've written
it down.  We will call it

(D)  If A and B and C are true, Z must be true.

\noindent Have you entered that in your note-book?''

``I {\sl HAVE}!'' Achilles joyfully exclaimed, as he ran the pencil
into its sheath.  ``And at last we've got to the end of this ideal
race-course!  Now that you accept A and B and C and D, {\sl OF COURSE}
you accept Z.''

``Do I?'' said the Tortoise innocently.  ``Let's make that quite clear.
I accept A and B and C and D.  Suppose I {\sl STILL} refused to
accept Z?''

``Then Logic would take you by the throat and {\sl FORCE} you to do it!''
Achilles triumphantly replied.  ``Logic would tell you, `You can't help
yourself.  Now that you've accepted A and B and C and D, you {\sl MUST}
accept Z!'  So you've no choice, you see.''

``Whatever {\sl LOGIC} is good enough to tell me is worth {\sl WRITING DOWN},''
said the Tortoise.  ``So enter it in your book, please.  We will call it

(E)  If A and B and C and D are true, Z must be true.

\noindent Until I've granted {\sl THAT}, of course I needn't grant Z.
So it's quite a {\sl NECESSARY} step, you see?''

``I see,'' said Achilles; and there was a touch of sadness in his tone.

Here the narrator, having pressing business at the Bank, was obliged
to leave the happy pair, and did not again pass the spot until some months
afterwards.  When he did so, Achilles was still seated on the back of the
much-enduring Tortoise, and was writing in his notebook, which appeared
to be nearly full.  The Tortoise was saying, ``Have you got that last
step written down?  Unless I've lost count, that makes a thousand and one.
There are several millions more to come.  And {\sl WOULD} you mind, as a
personal favour, considering what a lot of instruction this colloquy
of ours will provide for the Logicians of the Nineteenth Century---\!
{\sl WOULD} you mind adopting a pun that my cousin the Mock-Turtle will
then make, and allowing yourself to be renamed {\sl TAUGHT-US}?''

``As you please,'' replied the weary warrior, in the hollow tones of despair,
as he buried his face in his hands.  ``Provided that {\sl YOU}, for 
{\sl YOUR} part, will adopt a pun the Mock-Turtle never made, and allow
yourself to be re-named {\sl A KILL-EASE}!''

\vfill
\end