## XII.—Notes

Achilles had overtaken the Tortoise, and had seated himself comfortably on its back. (¶ 1)

So you've got to the end of our race-course?

said the
Tortoise. Even though it

(¶ 2)*does* consist of an infinite series of
distances? I thought some wiseacre or another had proved that the thing couldn't
be done?

It

said Achilles; *can* be done,It

(¶ 3)*has*
been done! Solvitur ambulando. You see, the distances
were constantly *diminishing*; and so—

But if they had been constantly

the
Tortoise interrupted. *increasing*?How then?

(¶ 4)

Then I shouldn't be

Achilles modestly replied;
*here*,and

(¶ 5)*you* would have got several times round the world, by this
time!

You flatter me—

said the
Tortoise; *flatten*, I mean,for you

(¶ 6)*are* a heavy weight, and *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 *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 prossessed *pockets* in those days) an
enormous note-book and a pencil. Proceed! And speak

(¶ 7)*slowly*, please.
*Short-hand* isn't invented yet!

That beautiful First Proposition of Euclid!

the Tortoise
murmured dreamily. You admire Euclid?

(¶ 8)

Passionately! So far, at least, as one

(¶ 9)*can* admire a treatise that
wo'n't be published for some centuries to come!

Well, now, let's take a little bit of the argument in that First
Proposition—just

(¶ 10)*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 things 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, *must* accept `Z` as true?

Undoubtedly! The youngest child in High School—as soon as
High Schools are invented, which wlil not be till some two thousand years
later—will grant

(¶ 11)*that*.

And if some reader had

(¶ 12)*not* yet accepted `A` and `B`
as true, he might still accept the *sequence* as a *valid* one,
I suppose?

No doubt such a reader might exist. He might say,

(¶ 13)I accept as true the
Hypothetical Proposition that,

Such a reader would do wisely in abandoning Euclid, and taking to
football.*if* `A` and `B` be true,
`Z` must be true; but, I *don't* accept `A` and `B`
as true.

And might there not

(¶14)*also* be some reader who would say, I
accept

?`A` and `B` as true, but I *don't* accept the
Hypothetical

Certainly there might.

(¶ 15)*He*, also, had better take to
football.

And

the Tortoise
continued, *neither* of these readers,is

(¶ 16)*as yet* under any logical necessity to accept
`Z` as true?

Quite so,

Achilles assented. (¶ 17)

Well, now, I want you to consider

(¶ 18)*me* as a reader of the
*second* kind, and to force me, logically, to accept `Z` as
true.

A tortoise playing football would be—

Achilles was beginning (¶ 19)

—an anomaly, of course,

the Tortoise hastily interrupted. Don't
wander from the point. Let's have

(¶ 20)`Z` first, and football afterwards!

I'm to force you to accept

Achilles said musingly.
`Z`, am I?And your present position is that you accept

(¶ 21)`A` and `B`,
but you *don't* accept the Hypothetical—

Let's call it

said the Tortoise. (¶22)`C`,

—but you

(¶ 23)*don't* accept

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

That is my present position,

said the Tortoise. (¶ 24)

Then I must ask you to accept

(¶ 25)`C`.

I'll do so,

said the Tortoise, as soon as you've entered it in that
note-book of yours. What else have you got in it?

(¶ 26)

Only a few memoranda,

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

(¶ 27)

Plenty of blank leaves, I see!

the Tortoise cheerily remarked. We
shall need them

(Achilles shuddered.) *all*!Now write as I dictate:—

(¶ 28)

(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

said Achilles.
`D`, not `Z`,It comes

(¶ 29)*next* to the other three. If you accept `A` and
`B` and `C`, you *must* accept `Z`.

And why

(¶ 30)*must* I?

Because it follows

(¶ 31)*logically* from them. If `A`
and `B` and `C` are true, `Z` *must* be true.
You don't dispute *that*, I imagine?

If

the Tortoise thoughtfully repeated.
`A` and `B` and `C` are true,
`Z` *must* be true,That's

(¶ 32)*another* Hypothetical, isn't it? And, if I failed to see its
truth, I might accept `A` and `B` and `C`, and
*still* not accept `Z`, mightn't I?

You might,

the candid hero admitted; though such
obtuseness would certainly be phenomenal. Still, the event is

(¶ 33)*possible*.
So I must ask you to grant *one* more Hypothetical.

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

(¶ 34)

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

Have you entered that in your notebook?

I

Achilles joyfully exclaimed, as he ran the
pencil into its sheath. *have*!And at last we've got to the end of this ideal
race-course! Now that you accept

(¶ 35)`A` and `B` and `C`
and `D`, *of course* you accept `Z`.

Do I?

said the Tortoise innocently. Let's make that quite
clear. I accept

(¶ 36)`A` and `B` and `C` and `D`.
Suppose I *still* refused to accept `Z`?

Then Logic would take you by the throat, and

Achilles triumphantly replied. *force* you
to do it!Logic would tell you,

(¶ 37)You
ca'n't help yourself. Now that you've accepted

So
you've no choice, you see. `A` and `B` and
`C` and `D`, you *must* accept `Z`!

Whatever

said the Tortoise. *Logic* is good enough to tell me is worth
*writing down*,So enter it in your note-book,
please. We will call it

(¶ 38)

(E) If `A` and `B` and `C` and `D`
are true, `Z` must be true. Until I've granted *that*, of course
I needn't grant `Z`. So it's quite a *necessary* step, you
see?

I see,

said Achilles; and there was a touch of sadness in his
tone. (¶ 39)

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 note-book, 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

(¶ 40)*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—*would* you mind adopting a pun
that my cousin the Mock-Turtle will then make, and allowing yourself to be
re-named *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

(¶ 41)*you*,
for *your* part, will adopt a pun the Mock-Turtle never made, and allow
yourself to be re-named *A Kill-Ease*!