In researching this form I found two very different definitions.
First we have a verse form named for the algebraic equations that were introduced by Irish mathematician Sir William Rowan Hamilton in 1843 which had to do with the multiplication of three dimensional imaginary units by four dimensional objects. The connection between the algebraic concept and the form is they both have a 3 by 4 concept.*
Quaternion is a poetry style where the theme is divided into four. Each part explores the complementary natures of the theme or subject. The word quaternion is derived from the Latin word quaterni, meaning four by four. The poem may be in any poetic form.
By Anne Bradstreet
Another four I've left yet to bring on,
Of four times four the last Quarternion,
The Winter, Summer, Autumn & the Spring,
In season all these Seasons I shall bring;
Sweet Spring like man in his Minority,
At present claim'd, and had priority.
With Smiling face and garments somewhat green,
She trim'd her locks, which late had frosted been,
Nor hot nor cold, she spake, but with a breath,
Fit to revive, the nummed earth from death.
Three months (quoth she) are 'lotted to my share
March, April, May of all the rest most fair.
When Spring had done, the Summer did begin,
With melted tauny face, and garments thin,
Resembling Fire, Choler, and Middle age,
As Spring did Air, Blood, Youth in 's equipage.
Wiping the sweat from of her face that ran,
With hair all wet she pussing thus began;
Bright June, July and August hot are mine,
Of Autumn moneths September is the prime,
Now day and night are equal in each Clime,
The twelfth of this Sol riseth in the Line,
And doth in poizing Libra this month shine.
The vintage now is ripe, the grapes are prest,
Whose lively liquor oft is curs'd and blest:
For nought so good, but it may be abused,
But its a precious juice when well its used.
Cold, moist, young flegmy winter now doth lye
In swaddling Clouts, like new born Infancy
Bound up with frosts, and furr'd with hail & snows,
And like an Infant, still it taller grows;
December is my first, and now the Sun
To th' Southward Tropick, his swift race doth run:
The second cites this form as English, a twelve-line, three-quatrain poem with a fixed rhyme scheme.**
The Quaternion is metered at the discretion of the poet and is rhymed aabb ccdd abcd
By Judi Van Gorder
Forgotten in a tight airless attic,
a thing that was to prove problematic,
a Halloween reflective diorama
in reenactment of a psychodrama.
The shoebox, webbed and painted black,
a tiny coffin's cedar lid drawn back,
inside a body stiff in death, an axe
imbedded in the head, a final tax.
A challenge to create this poematic
with ghoulish display in frightful panorama.
The sounds with screeching tone, elegiac
and words don't always tell the lonely facts.
*Poetry Magnum Opus