[)*-GSP!N  7Drawn by Michael Thwaites 11/10/94, mjt@stubbs.ucop.edudu hxLPstdfHTGS2.8Š    (01!gst$%`a*i#bX"hFrom: "John Conway" 1) Quadrisect the perimeter of a circle center O by 4 points N,E,W,S in the appropriate positions. 2) Quadrisect the line ON by a point A near O (ie., 4OA = ON) 3) Quadrisect the angle OAE by a line AB near AO (ie., 4OAB = OAE) this line to meet OE in B, and BA produced to meet the circle in C. 4) Quadrisect the straight angle BAC by a line AD near AB, the point D to lie on WO (ie., BAD = 45 degrees). 5) Let the circle on diameter DE cut NS in F and G. 6) Let the circle center B through FG cut EW in H and I 7) Then perpendiculars to WE through H and I will hit the original circle in 4 points of the 17-gon one of whose vertices is E. I first saw this in Hardy and Wright's book on The Theory of Numbers, which is where I've just checked up on it. H & W confirm my impression of the history. They say that Gauss worked out the general theory in Paragraphs 335-366 of his Disquisitiones, but that the first explicit construction was given by Erchinger, for whom they refer to Gauss' Werke, vol II, pp186-187. This "Quadruple Quadrisection" construction (my name) is due to Richmond, who gave it in the Quarterly Journal of Math, 1893. Of the four or five constructions I have seen, it is definitely the nicest. If you intersect the other quadrisectors of that straight angle with WE and treat the resulting points similarly, you can get more vertices in the same way - but it's easier to use your compasses to step around the circle from the ones given, for which the constructing points are the most conveniently situated. John Conway HHi0i@]VP[UH/ RBB OC[CD-sBgBg 1C[CDC.sX?5%F?5%F N C[Ae@W  mC[Ae@C[CD?']  jC[Ae@C[CD?kpV  GC[Bٌ']  kC[CDB22CD?pu S C[C9uV  cC[CDC[C9? j  nC[BٌC[CD? +0] W B22CD ] E CĹCD *V0  wB22CDC[CD?  rC[CDCĹCD? AC[Cc* *C  c1C[Cc*CĹCD?'  p1C[Cc*CJCSSu?9X>] J1 C4C'  q1C[Cc*Cx1 Cm5?V  B1CjcCD2V  v1CjcCDC[Cc*?V C1 C/֮AV  z1C/֮ACjcCD?oX'U a1C[Cc*BGxCR?Z_V` S1BCD'  b1C[Cc*C=iCT?V, D1CE\CD  b1C[Cc*CE\CD?  d1CE\CDCĹCD?&+V  U1CCD b' 21CCDBú?5%F?5%F! G1 C[Cv " F1 C[CN"* 41CjcCDBz?5%F?5%F$',VV H1 Ca1CD %" I1 C+BdCD%&', f1Ca1CDCa1C~? &'U  e1C+BdCDC+BdC9?'joT C1 C+BdC@) B1 C+BdA)^'c,& A1 Ca1C(&'+, Z1 Ca1BZa(V0 51Ca1BZaBC?5%F?5%F-+%&c,V  e1Ca1BZaCa1C?,-U E1C+BdA.|U D1CC/.| D1CCh1E> 61CC/B8?5%F?5%F1LZQ_U  G1CBb3 F1CĹCD3{Ct  a1CChCĹCD?52{  k1CĹCDCC/?15ijVD 81Ca1BZaB8?5%F?5%F-4U 71CBbB8?5%F?5%F41U J1CkAYS8LZQ_V| H1CBa8%&Q_33  k1CBaCa1BZa?-;KY  j1CC/CBa?;18Z=_ H1CCh;k] 10C+BdAB8?5%F?5%F0:W,Ux 90CkAYSB8?5%F?5%F:-Y=  z1CChCCh?2>7&c_?  y1Ca1CCCh?>,7p<u L1BtBc$Q? K1CkAYT?+,C  m1Ca1BZaCkAYT?D-pu  K1CkCkD2z 13BtBc$QB8?5%F?5%FC0]u,Ut  x1CkCkCa1C?,Fiuheir  w1C+BdC@CkCk?F*fEkJ Q1B[BSG P1C+BdAGCq+  n1CkAYTC+BdA?KDE#Ji@ Q1B[C J( 14B[BSB8?5%F?5%FJCo  d1C+BdAC+BdC@?*K.3V S1B>?C#N7p<ui@ R1BtBc$PNe-J335  r1B[BSB>?C#?PJ6Dku  q1BtBc$PB[BS?JQo<V  p1C+BdABtBc$P?QKMpRui@ R1BtCvQ.3 S1B>?CdPdq] 15B>?C#B8?5%F?5%FPJLoo33  v1BtCvC+BdC@?*UDRu  u1B[C BtCv?UM-#J33  t1B>?CdB[C ?MV-3c  s1B>?C#B>?Cd?VP