# Glimpses of History

Part 2. Renaissance to nineteenth century

Raise two to a prime p and then
Take off one — is it prime, asks MERSENNE?
What MERSENNE never knew,
It looks nice in base two,
And an analogue’s stated base ten.

Nor FERMAT nor MERSENNE really shone
With a listing of primes that’s spot on,
And students now snigger
At an incorrect figure,
Dividing by six forty-one.

A reduction step acting on pairs
Gets a prime to be sum of two squares;
FERMAT’s own device,
That he used once or twice,
As he listened to legal affairs.

“No sum of two cubes makes a third,
And higher pow’rs, also absurd,
But the space in this marge
Being not very large…”;
No more from FERMAT has been heard.

“On the shoulders of giants,” said haughty
Sir ISAAC. Few heard this as naughty.
But he based calculation
On whose observations?
Astronomer HOOKE was a shorty.    [NEWTON]

But at school we learned two of his fruits:
And (opposite notion)
The methods for putting down roots.    [NEWTON]

A fact that has long been suppressed
Is that NEWTON’s old polygon’s best.
Its factor detection
Proceeds by inspection,
Anticipates EISENSTEIN’s test.

Each night an hour longer in bed
Was the plight of DE MOIVRE, it’s said.
“In two weeks, it’s plain,
I shall not wake again
By induction.” And then he was dead.

His astronomy book must be shewn
To the King, and LAPLACE stands alone.
“We regard it as odd,
“there’s no mention of GOD!”
“I neglect higher Pow’rs of Unknown!”

Buying horses and cows at the fair,
One sev’n seventy, nothing to spare.
“How many bought now?”
“Twenty-one for a cow,
Thirty-one roubles each for a mare!”    [EULER]

Is insight notationed or notioned?
The parts out of n, what’s the quotient?
In EULER’s construction,
No further reduction,
Phi (n) – but why call it the totient?

There are several proofs that GAUSS saw
For the Great Reciprocity Law,
Of the Pearly-Bound Book,
So he sat down to find even more.

A progression from zero, steps q,
Reduce mod p, stop halfway through.
How many are minus
Determines the sign as
Pow’r p minus one over two.    [GAUSS]

The last proof of GAUSS reciprocity
Depends on the sort of atrocity
Each Little-Go student,
However imprudent,
Was accustomed to solve with velocity.

You innocents, listen in time:
Not ev’rything’s true that’s sublime.
If it’s certain that GAUSS
Did not keep a pet mouse,
We would still make it up for the rhyme.

With CARL GUSTAF JACOB JACOBI,
No integral’s safe from his probe. He
Filled leather-bound Works
With notations and quirks,

When CARL GUSTAF JACOB JACOBI
Left Potsdam to travel the globe, he
Crossed over the Oder
To train under YODA
As Jedi, or be one can no be.

The fractions we now know as FAREY,
He looked at a table, and there he
At properties hinted
Nomenclature’s quite arbitrary.

Analysis, rigorous, grounded,
His papers were tending
Towards the unending;

In ABEL’s `Quintessence’ one stage
Was quoted from CAUCHY the sage,
“Comptes, last year”. You look:
It’s a very thick book,
And CAUCHY wrote every page.

Of NAPOLEON’s soldiers, most fell;
Field of honour or snowfield, both Hell.
That campaign, it gave birth
To Operations Research,
And to MERTENS’s mother as well.

LEGENDRE wrote `easily seen’;
We know what such weasel words mean.
To set matters straight
L(s, chi) and the modern machine.

Just the primes in A. P. are officially
Ascribed to the younger DIRICHLET.
Both residue symbol
And HOOLEY-fied nimble
Interchange were DIRICHLET initially.

DIRICHLET assumed the existence
If it comes from some physical systems,
Of pulleys and pistons,
Or water in cisterns
Or forces that act at a distance.

In the integral studied by AIRY,
The sine of x cubed appears scary.
There’s an interval small
Which approximates all,
Where the angle, or phase, doesn’t vary.

Conspiracy theorists thrill
How GALOIS wrote out, not a will,
But his ultimate draft
(Two rejected as daft)
On Groups, and then went to the kill.    (see BORGES, `The Legend of the Hero and the Assassin’)

Or shot first and gone back to College, he
Would have done so much more,
And, surer than SCHUR,
Beaten STEINROD to found cohomology.

Is RIEMANN’s last joke still a laugh?
He carefully plots on a graph
The sum of the series,
Deducing nine zeros,
“And probably x is one half.”

Under gravity, gas in a ball,
Do the free oscillations stay small?
The stable condition:
First order HERMIT’ian,
And RIEMANN discovered it all.

If RIEMANN had spoken his heart,
We would have POLYA’s clue from the start:
His manuscript revels
In energy levels,
Then zeta is taken apart.    (thanks to KEATING for finding this)

“What are numbers and what should they be?”
Was DEDEKIND’s philosophy.
He cut to the real,
Imposed the ideal,
And his sum is a rum cup of tea.

To accompany JACOBI’s theta,
And its transform that RIEMANN called zeta,
This DEDEKIND chap
Filled a much-needed gap
In the alphabet: DEDEKIND’s eta.

When LINDEL”OF lightly let slip
A conjecture on zeta, his tip
Was carefully sounded:
“I don’t think it’s bounded
All over the critical strip.”

Success seemed to bless ARTHUR CAYLEY,
Who climbed ev’ry mountain-top gaily,
He ate strawberry jam
In his house on the Cam,
And published on Algebra daily.

He gave us three square roots of neg,
Was rude to his wife
For most of his life,
In his papers they found a fried egg.    [HAMILTON]

HERMITE was picked up as a spy,
“No, it’s Maths. I’ll explain. Let me try…”
“What I understand
Can’t be Maths” (gun in hand).
Good thing some top brass wandered by.

When Logic abandoned its lodging
At the hand of the artful CHARLES DODGSON,
Was `Alice’ a cover
For trysts with her mother,
An urgent erogenous sojourn?

LEWIS CARROLL found permanent fame
When he played a determinant game,
When studying minors,
Attaching a sign, as
The Matrix herself was his aim.    [DODGSON]

To render the ancients their due,
It has long been regarded as true,
(Implicit in GAUSS
And some work of LANDAU’s)
That unity doubled makes two.

Is the RIEMANN Hypothesis true?
“The numbers called `primes’
You can’t build up from `times’
Can be found from factorials too”.

Was to fathom where no zeros are,
A domain which essentially
Goes back for a century,
But it doesn’t go back very far.

The inversions that M”OBIUS knew:
The surfaces, one side or two,
With az+b
Over cz+d,
And mu (n), which does not require glue.

By a process he cannot discuss,
As POINCARE sat on the bus,
He saw the deduction
Of laws theta-FUCHS-ian.
What if he used the Metro like us?

Great ideas without names we can see
In the busy nineteenth century.
GRASSMAN used tensors daily,
There are quotients in CAYLEY,
And SCHUR found Co-Homology.