"Winthrope" is a cotillion (dance party game) figure found in Twentieth Century Cotillion Figures, by H. Layton Walker (Two Step Publishing Company, Buffalo, New York, 1912). It's not much of a game, however, though it includes the basic cotillion elements: a group of couples dancing, separating, selecting new partners, and bestowing favors. The figure itself, however, is simply an interesting way to transition from a typical grand march into a circle of couples for couple dancing. And that is actually useful.
I have no idea why it's called Winthrope. [see note at the bottom of the post for new information]
Here are the four diagrams (click to enlarge), followed by a transcription of the instructions:
WINTHROPE
Sixteen couples up and dance. Favor, thirty-two couples now on the floor. March down the centre of hall, etc. Continue to march around hall then down the centre in fours. First four to right, second four to left; around hall again, down in eights; halt; second and fourth line of ladies about face and march, as in figure 1. Lead to the lower end of gentlemen and follow lines of figure No. 2. Which when you have completed, will leave you in position as figure 3. Now the second and fourth line of gentlemen about face; march as in figure 3, finishing as per figure 4, which will bring you opposite your original partners. Face your partner and all dance.
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Note that the first rank of couples is at the bottom of each diagram. The Xs represent the gentlemen and the Os the ladies. After the opening dance (probably a two-step, which is common in this source and conveniently often done to march music), the sixteen couples separate and choose new partners, bestowing favors, and form a column for a grand march. In conventional sequence, the couples peel off to either side, circle around to meet, and march down the hall in fours. The fours then peel off, circle around, and march down the hall in eights, halting when all the couples are in ranks of eight. That is the situation in Figure 1 above. The leading couple is at the bottom right of the diagram.
The first march is done by the ladies; the gentlemen stand still. The ladies form a serpentine, with the leading lady and those behind facing down the room, the next file of ladies facing up the room, the next down, and the next up. The leading lady turns to the left and marches down the outside of the formation; all the other ladies follow, weaving up and down the room as in Figure 1. The leading lady marches to the top of the room (the back of the formation), across the back to the corner diagonally opposite from where she started, where she begins to weave between the ranks of gentlemen, the other ladies still following, as shown in Figure 2. I suspect that it might be necessary for her to march once around the entire formation in order to get all the ladies out from among the gentlemen before she starts to weave through the ranks, lest she get tangled in the tail end of the ladies still weaving up and down.
As the line of ladies winds its way across and back between the ranks, the leading lady eventually arrives back at the top of the room, behind her partner, where she turns back down the outside again and marches counter-clockwise around the entire formation until the ladies have formed a rectangle around the gentlemen, as shown in Figure 2. The ladies then stop.
The second and fourth files of gentlemen then turn to face down the room and the gentlemen follow the leading gentleman, weaving up and down, as shown in Figure 3. There is no second weave back and forth across the room for the gentlemen, since there are no ranks of ladies to move between. The leader simply turns left and marches to the top of the room and continuing counter-clockwise inside the rectangle of ladies until he reaches his partner, all the other gentlemen following and ending up with their own partners in a large circle of couples.
Each couple then begins the two-step or whatever dance is appropriate to the music.
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The reconstruction of the figure is entirely straightforward.
There are two possible practical issues in using it: (1) the aforementioned potential tangle of the ladies' line if they move too quickly down the set and catch up with the end of their line when starting to weave across, which can be prevented by circling the set once before starting the crosswise weaving, and (2) making sure that the correct files of ladies and gentlemen turn to face up the room and that the resulting serpentine line is properly assembled at the top and bottom. This is easy enough of the dancers know the figure or if there are floor managers to assist. It might be a bit tricky if not; most dancers in rows of eight are probably not mentally labeling themselves by what file they're in. It would behoove the leader and/or floor managers to make sure everyone is facing the correct way before starting to weave.
While this was published as a cotillion figure, as noted above, I think it's real utility is that if the dancers can be guided through it, it's a clever and interesting way to end a grand march with the couples in a circle, much more interesting than simply having them circle the room. It should scale up easily for larger groups, though I don't think I'd try it with an unrehearsed group without floor managers on hand to make sure the files turn correctly.
[Edited 2/19/21 to add everything below]
Since I originally wrote this post I've come across an odd little book of original dances and cotillion figures with no title page or other front matter. From internal references, it can be no earlier than the summer of 1906, and I would guess that it is in fact from that year. The Winthrope, under the name "Winthrop", appears there with the very same diagrams, which Walker obviously copied. It is credited specifically to Birmingham, Alabama, dancing master Joseph C. Weissner, who appears on the list of dancing masters that Walker thanked at the end of his book. An advertisement for Weissner's dancing school from the October 9, 1906, issue of the Birmingham Age-Herald is at left (click to enlarge). The Hotel Hillman was a luxury hotel only a few years old at the time; detailed information about it and a photo of it in 1906 may be found at Birmingham history site Bhamwiki.
The little mystery book also provides a strong hint about the name of the figure. Multiple items in the book, including a dance by Weissner, are associated with the annual convention of the National Association of Dancing Masters, which opened in Winthrop, Massachusetts, on June 11, 1906, as reported in The Daily Morning Journal and Courier (New Haven, Connecticut) on June 12th. It seems likely that the figure was introduced at this convention and named Winthrop in honor of the hosting town.
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