In 1781 Longman & Broderip published a dance with this name but a different figure. Book 1 attributes this to Nathaniel Gow, but I have been unable to find any choreographies published by him (though lots of music). The earliest source I can find is from the Lowes who write:

Down the middle, and up again; reel three and three across the dance, the Lady with the first couple, and the Gentleman with the second; set and turn corners; then set, and turn partners.

The first dance in the first book of the (R)SCDS is Petronella.

According to the RSCDS (Book 1) this dance was "introduced" by Nathaniel Gow, a composer and musician of Edinburgh, in 1820. Although Gow published many tunes (including this one), I have been unable to find any choreographies by him.

Book 1 also says the dance was published in a book called The Ball-Room, 1827 (no editor, or publisher, or location given), actually the book is titled The Ball-Room, or the Juvenile Pupil's Assistant; Containing the Most Fashionable Quadrilles, with Les Lanciers of Sixteen, As Danced in the Public & Private Assemblies in Paris and the author is J. P. Boulogne, but, again, I can't find this work.

The earliest description of the figure I've found was published by the Lowes in Lowes' Ball-Conductor and Assembly Guide. of 1831. The Lowes published in Edinburgh, and their book has three sub-categories of country dance: English, Scotch and Irish. Petronella is categorized as an "English Country Dance". They describe it thusly:

First couple chassé round to the right, and set in the middle; round to the right again, and set on the sides; to the right again, and set in places; down the middle, up again, and pousette.

By 1853, the Lowes had become just Joseph Lowe, and he published a somewhat different version in his Lowe's Selection of Popular Country Dances (courtesy of the National Library of Scotland):

First couple turn round to the right, and set opposite to each other in the middle of the Dance (4 Bars). Again round to the right, and set on the sides of the Dance, (4 Bars) Again round to the right, and set in the middle, (4 Bars) Again to the right and set in places. (4 Bars) First couple down the middle and up again. (8 Bars) Poussette with top couple. (8 Bars)

Courtesy of National Library of Scotland, licensed under:

In ~1868 H. D. Wilcock in Ball-Room Guide; A Manual of Dancing page 70, writes:

1. First couple turn round to the right, and set in centre of dance (4)
2. Turn to the opposite side and set (4)
3. Turn to centre and set (4)
4. Turn and set in places (4)
5. Lead down the middle and up again (8)
6. Pousette (8)

In 1868 Elias Howe in The Pianist's Matinee: A Collection of Music for the Piano-Forte, writes:

First couple turn round to the right, and balance opposite to each other in the middle of the dance. (4 bars) Again round to the right, and balance on the sides of the dance. (4 bars) Again to the right, and balance in the middle. (4 bars) Again to the right, and balance in places. (4 bars) First couple down the middle and up again (8 bars) Right and left with the next couple. (8 bars)

In ~1872 J. F. Wallace in The Excelsior Manual of Dancing, (Glasgow), writes:

1. First couple turn to their right and set in centre of dance (4 bars)
2. Turn into each other's place and set (4)
3. Again turn to centre by right and set (4)
4. Turn to places and set, (4)
5. Down the centre and back, (8)
6. Poussette, (8)

In 1897 in D. Anderson's Ball-Room and Solo Dance Guide, (Dundee)

Time 2-4

Turn to right in centre with two steps facing each other, then set to each other with two steps.

Turn into each other's places with next two steps then set with other two steps.

Turn to right in centre facing each other with two steps then set to each other with two steps.

Turn into places next two steps then set with other two steps (the four quarter turns describe the figure of a diamond).

First gent then leads his partner down the centre and back to places, the first and second couples pousette.

Repeat from the beginning other three times

Courtesy of National Library of Scotland, licensed under:

Note: Anderson very explicitly says the lead down and back returns to places, while fifty years earlier it had an implied cast off. That means his "pousette" must be progressive, though he does not say that explicitly.

Many pages later in the book Anderson defines a "Petronella Step". He does not explicitly say it is for petronella turns, rather he says This Step may also be used for Contra Dances, Circassian Circle, etc., in setting to partners, but must never be used for Quadrilles.

PETRONELLA STEP.

Stand in 1st position.

Hop out on right foot in 2nd position.

Bring left foot in front in 5th position.

Beat behind with right.

Hop out on left foot in 2nd position.

Bring right foot down in front in 5th position.

Beat behind with left.

In ~1898 J. Scott Skinner in The People's Ball Room Guide, (Dundee), writes:

This is one of the very prettiest Country Dances we have.

The dancers are arranged in the usual way.

First lady and opposite gentleman advance to the centre, one, two, three, and turn with toes to right. Then they execute the first quickstop twice.

They advance to the opposite side and set.

They advance to the centre and set.

They advance and set in places, having thus, with their successive movements described the figure of a diamond.

Then down the centre, back, and pousette.

There is a tune "Petronella" for this dance. "Meg Merrilees" suits it admirably.

While in the book Prompting: How To Do It by John M. Schell, published Boston, 1890.

First couple balance to side (4); balance to center (4); balance to side (4); balance to center (4); first couple down centre and back, cast off (8); right and left (8).

Back in Scotland, in Allan's Reference Guide to the Ball-Room, says:

1. First couple advance to the centre of the dance, turning round to the right and set.
2. Advance to the opposite side and set.
3. Advance to the centre side and set.
4. Advance and set in places.
5. Down the center and up again.
6. Pousette.

Courtesy of National Library of Scotland, licensed under:

Note that in 1831 the 1s simply chassé into position with no turn, but by 1868 they are turning into the middle, then in 1924 the (R)SCDS describes the figure:

1st woman moves from A to B with two pas de basque, leading off with the right foot and making a three-quarter turn by the right, in the middle of which she has her back to her partner.
1st man moves similarly from C to D.

The modern contra figure seems similar to that described by Wilcock in 1868.

As this dance is rather boring for the 2s a number of variations have arisen.

The Scottish Country Dancing Dictionary attributes this to Gow, 1822, while the SCDDB says "Unknown" (1822). As far as I can tell, Gow did not publish any choreographies, though he did publish music. I can't find any publications from 1822 which appear appropriate.

However, in Lowes' Ball-Conductor and Assembly Guide, Edinburgh, ~1831 the the Lowes write:

First couple join hands, and set to second Lady; same to second Gentleman; down the middle, and up again; four hands across, and back again, and pousette at top.

RSCDS Book 12 says "collected in the Border Country."