Circassia is (or was) a region on the northeastern shore of the Black Sea.

In Baltimore in 1820 George Willig Jr. published a dance called "The Circassian" in a pamphlet called Admired Cotillions (in the US the distinction between Quadrilles and Cotillions was often lost, and I think this is actually one set of a quadrille. It isn't in a circle and it is not progressive.

Forward two cross over. Chasse. cross. to place. Balance and turn partner. Ladies Chain. promenade and half right and left to place.

The figure for this dance is said to come from Le Pantalon in Payne's 1st Set of Quadrilles. The first 24 bars match and the last 8 have been changed to make the dance progressive. The idea of dancing in a circle, and the name "Circassian" come from G.M.S. Chivers.

In 1822 in his book The Modern Dancing Master, G.M.S. Chivers introduces a style of dancing he calls "Circassian Circle". He then publishes four dances to be done in this style, none of which is called "Circassian Circle" itself.

Chivers's version is different from the modern way of dancing. At the time, in longways sets, only the top two couples would start dancing, in this circle dance only the two couples closest to the music would start, they would pass each other, and then each would dance with the next couple in their direction of motion. And so on.

Any number of persons can join in the Dance, observing that there is an even number of couples; the odd numbers stand on one side, and the even numbers on the other, forming a circle, thus:

The 1st and 2nd couples face each other, who perform the Figure and exchange places: then the 1st and 4th couples, and at the same time, the 2nd and 3rd couples do the same, and so continued all round. Each Dance terminates when all have been through it.

Each lady to stand on the right of her partner throughout the Dance. All the odd numbers face the same as No. 1, and the even numbers face the same as No. 2.

— The Bodleian Libraries, University of Oxford, Harding Mus F 346

Note this is simply a general description of how to dance in a circle. It is followed by four figures for this format. the one which most closely matches the modern version is called "La Daphne":

Right and left set and turn partners ladies chain half pousette into each others place, and turn partners under the arm

— The Bodleian Libraries, University of Oxford, Harding Mus F 346

The next version I can find is Lowes' Ball-Conductor and Assembly Guide (Edinbourgh, ~1831) where they write:

The Circassian Circle

Is a circular Country Dance, for which the party is arranged, as for Quadrilles, across the room; the four at the top commence first, and change sides at the end of the figures, they continue to repeat them with all the couples down the opposite sides of the room, crossing over at the bottom, and continuing till they arrive at their original places; all the other couples must also continue, after they commence, till they make the circuit of the room; and when all have regained their places, the dance is finished

First Circassian Circle

Ladies chain, set and turn partners, right and left, hands four half round, and turn partners, changing places.

I think the Lowes are describing a hall full of two couple quadrille sets (that is improper duple minor lines), lined up along each side wall of a building. As with Chivers, and in keeping with the staggered start in use for country dances at the time, only one set start dancing, the one at the top, they pass each other, and one moves down the right wall, and one down the left, as they move down the lines the dancers behind them will also start, then when they reach the bottom instead of becoming 2s and working their way up, the cross over to the other line, and work their way up. When they reach the top of the other line they stop and wait until everyone else has reached their original position.

In ~1842 Tilt & Bogue published The Ball-Room Preceptor (Third Edition) containing a description of the Circassian Circle very like Chivers's. Only two couples dance at the start, and no specific figure is provided. (In ~1844 Coulon (London), in 1848 George Appleton, Philadelphia, and in ~1850 Kent & Co., Nottingham copy their description word for word)

The next version I've found is Wilson, 1852.

Wilson writes:

Couples vis-a-vis; form a circle round the room; figure, right and left or hands across; set and turn partners; ladies chain; waltz pass on to next couple; repeat.

Wilson defines "Right and Left" on page 15 of The Art of Dancing and it involves no hand clasps.

Sometime between 1850 and 1853 in Lowe's Selection of Popular Country Dances:

Any number of Couples can join in this Dance, the Couples arranging themselves along each side of the room, (the Ladies standing on the right of their partners) and all facing the top of the room, except the first Couples who face each other across the dance.

FIGURE

Ladies chain by top couples. (8 Bars) Balancez and turn partners. (8 bars) Right and left. (8 bars) Hands four half round, then turn parners and face the next couples on the sides of the dance (8 Bars) This figure is repeated with every couple, all down the sides of the dance across the bottom, and up the other sides to places, every couple after they commence keep going on till all have regained their own places.

Courtesy of National Library of Scotland, licensed under:

Joseph Lowe is still recommending the old style progression where only the top couples start. But he now calls this dance "Circassian Circle" rather than one of several dances in that style.

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

The couples stand in a circle round the room, two and two, couples facing each other, and backs to other couples. Gentlement have partners always on right hand.

Rights and lefts to places.

Set and turn partners.

Ladies' chain.

Pousette into opposite places.

Repeat figure with next couple round the room.

Courtesy of National Library of Scotland, licensed under:

Given the way he sets up the dance, Anderson must have everyone starting at once.

Note that in Le Pantalon of Payne's 1st Set of Quadrilles the four bars of setting to partner (which come after the rights and lefts) are two different setting steps: Balancé and Sissone Balotté Jetté and Assemblé. The first is an advance and retire movement (toward and away from partner) and the second a side to side movement (to the right, and back to the left).

Wilson published several figures named Rob Roy in his L'Assemblee of 1819 but none has a figure which matches the RSCDS version.

On page 73 of his The Dancers' Guide, 1821, Chivers writes:

Rob Roy Macgregor... First couple lead down the middle, the second follow, half right and left back again, and half right and left swing corners right and left at top

On page 69 of his The Excelsior Manual of Dancing, ~1872, J. F. Wallace writes:

1. First couple down the centre, the second couple follow, (4 bars)
2. Half right and left by couples No. 1 and 2 at bottom, (4)
3. Couples No. 1 and 2 back again and half right and left, (8)
4. Set at corners and turn partners, (8)
5. Right and left by couples No. 1 and 2, (8)