### Foundations of arithmetic

#### J. A. Lindon

One day when Mugg the Missing Link was prowling through the woods,
In search of wives and mammoth-meat and other usefull goods,
Whom should he see, on pushing out from deep arboreal shade,
But Ogg, the Paleolithic Man, cross-legged in a glade.

This Ogg had made a neat array of pebbles on the ground,
In number they were twenty-one, the most that could be found,
And Ogg, with one red-hairy hand pressed to his bony brow,
Was staring at hese pebbles like a ruminating cow.

```           o o o o o o o
o o o o o o o
o o o o o o o
```

Thought Mugg - for he was Primitive - I should be very dull
To lose this opportunity of busting in his skull;
My club weighs half a hundredweigh, he doesn't wear a hat
(And here he wondered) Yes, but what the devil is he at?

For Ogg was touching pebbles and then prodding at his digits,
Until the weirdness of it all afflicted Mugg with fidgets:
"Invented any goodish wheels just recently?" he hollered,
And doubled up in merriments, his face raw-beefy coloured.

Ogg looked at him in pity, then he drummed upon his chest:
"I've don a Think!" he bellowed "Monkey Mugg. I've done a think!
And I would write it down, but no one's yet invented ink."

Mugg moved a little closer, and his eyes and mouth were round,
And stared in trepidation at those pebbles on the ground.
Ogg pointed with a nailed red-hairy sausage at the rows
And said, "Three people's hand-plus-two is hand-plus-feet-plus-nose."

```oooo       oooo       oooo       oooo     oooo
o   +      o   +      o   =      o o       + ooooO Oooo + A
o          o          o
o          o          o
```

"And this is hand-plus-two of people's three-for-each-by-name,
So three times hand-plus-two and hand-plus-to time three's the same!"
Mugg scratched his matted hairy head, not knowing what to say.
Said Ogg, "It's all made clear by this rectangular array."

```                             o o o
o o o
o o o o o o o     o o o
o o o o o o o  =  o o o
o o o o o o o     o o o
o o o
o o o
```

"Three rows of hand-plus-two and hand-plus-two short rows of three
are just the same according to which way you look, you see?
In brief, a tripe heptad is the same as seven trebles,
And may quite possibly be true of other things than pebbles."

Mugg viewed it from all angles, then he gave a raucous belch
And trod on a Batrachian that perished with a sqelch.
He growled, "I do not understand these arithmetic quirks,
But maybe we should try to discover if it works."

So home they went to get their wives and drag them by the hair,
But what with all their screeching and their running every way,
At first they would not form a neat rectangular array.

So Ogg he then positioned each by holding of her down
While Mugg with mighty club in hand, just dinted in her crown;
And when they had them all in place, like pebbles, they could see
That three times hand-plus-two in wives was hand-plus-two times three!

```      o<= o<= =>o =>o o<= o<= =>o

=>o o<= o<= =>o o<= o<= =>o

o<= =>o =>o o<= =>o o<= =>o
```

Then Ogg he roared in high delight, cartwheeling to and fro
(Carts had not been invented, but he did it just to show!),
And Mugg he grinned a shaggy grin and slapped a hairy thigh
And said, "It's true, as sure as Pterodactyls learned to fly!"

And then they feasted on their wives in unuxorious zest,
Exept for one whose skull was rather thicker than the rest,
And she was sent to dig a pit and bury every bone,
While Mugg and Ogg went off to find a flat unsullied stone.

Then Ogg he sharpened up a flint and scratched upon the rock:
First Arithmetic Theorem - by Ogg the son of Mok.
He drew his little diagram, and proved, with QED,
That three times hand-plus-two of x is hand-plus-two times three.

But Mugg the Missing Link grew bored, and left him there alone,
Still scratching with his silly flint upon his silly stone;
And belching, plunged back in the woods on feet toe's simple fives,
In search of wives and mammoth-meat, particularly wives!