Dance The Mathematicians 7457

{{explanation}}
A new couple starts every 40 bars. For [1–40] only 1c dance, for [161–200] only 4c dance

1–
: 1c lead down {3}, up {3} and cast (2c up) (2,1,3,4)
9–
: 1c ½Fig8 up ; set advancing | turn BH ½ to own side into DTr formation (2,1,3,4)
17–
: 2c+1c+3c Set as in DTr | 1c+2c change place EH ; all set as in DTr | all set, 1c+2c advancing for
25–
: 1c+2c Allemande
33–
: All Circle8 and back (2,1,3,4)

{{explanation}}
New top cpl (2c) repeats [1–40] while original 1c continues as follows

41–
: 1c cast to bottom (3c up) , turn RH and cast up into 3pl (3,2,1,4)
49–
: 1c ½Fig8 down round 4c ; set advancing | turn BH into extended DTr formation (3,2,1,4), (2c join NH with 1c in the middle and join hands with 3c+4c on own sides
57–
: Set in DTr | 1c change places with 4c EH (2,3,4,1) ; all set | bottom couples change places EH again into allemande position
65–
: 2c+3c & 1c+4c Allemande
73–
: All Circle8 and back (3,2,4,1)

{{explanation}}
Each couple dances 80 bars. A new couple starts after 40 bars (ie on bars 41, 81 & 121)

1-8
: 1s lead down for 3 bars, back to top & cast to 2nd place
9-16
: 1s dance 1/2 Fig of 8 round 2s, set advancing & 1/2 turn 2H into double
  triangle position
17-24
: 2s+1s+3s set & 1s+2s change places, set again with 1s+2s advancing into
  middle facing up
25-32
: 1s+2s dance Allemande
33-40
: All dance 8H round & back
41-48
: (New Couple starts as at bar 1) **while** 1s (original) continue casting
  to bottom, turn RH & cast up to 3rd place
49-56
: 1s (2s) dance 1/2 Fig of 8 round 4s (3s), set advancing & 1/2 turn 2H
  into a double triangle position (nearer hand joined with new dancing
  couple - 2s)
57-64
: All set & 1s+4s change places, All set again & all advance into middle
  facing up
65-72
: 1s+4s (also 2s+3s) dance Allemande
73-80
: All dance 8H round & back.

{{explanation}}
Note: First time through only 1st couple dance bars 1-40.  Last time through only 4s dance bars 41-80