09 28


Churches (and castles) on equal distances.

Group 1: ± 18,3 centimeter (± 4575 meter)
  Montazels - Campagne sur Aude  (18,2); Rouvenac - La Serpent  (18,2); Château Montferrand* - Arques   (18,25); Antugnac - Campagne sur Aude  (18,3); Peyrolles - Rennes-les-Bains  (18,35); St.Ferriol - Cavirac  (18,35); Sougraigne - Château le Bézu*  (18,4); Coustaussa - Château Montferrand*  (18,4); St.Just - Laval  (18,4); Terrolles - Serres  (18,45)

Group 2: ± 18,6 centimeter (± 4650 meter)
  Antugnac - Roquetaillade  (18,5); Arques - Terrole   (18,55); Serres - Luc sur Aude  (18,55); Espéraza - Coustaussa  (18,6); Espéraza - Granes  (18,65); Espéraza - St.Ferriol  (18,65); Espéraza - Les Sauzils  (18,7); St.Salvaires - Terroles  (18,65); Bouriège - Croux  (18,7); Rennes-les-Bains – Château Rennes-le-Château  (18,7); Château Castillou* - Tombe Poussin  (18,7); La Serpent - Bourigeole  (18,75); La Serpent - St.André  (18,8); Véraza - Tombe Poussin  (18,9)

Group 3: ± 19 centimeter (± 4750 meter)
  Rennes-le-Château - Rennes-les-Bains   (19); Rennes-le-Château - Campagne sur Aude   (19); Rennes-le-Château - Le Bézu  (19); Château Blanchefort* - Couiza   (19); Terrolles - Château Castillou*   (19); Les Sauzils - St.Ferriol  (19,15); Arques - Peyrolles  (19,15); Espéraza - Luc sur Aude  (19,2)


Position Tombe Poussin.

    Alongside you see two red lines on the map:

1: The line from the donjon of Arques to the church of Cassaignes

2: The perpendicular line on Peyrolles-Arques from the middle of Blanchefort-Arques

On the intersection is the Poussin Tomb

 



Distances of the Blanchefort-geometry.

Peyrolles - Arques: 16,85 cm on the map (4,21 km); Peyrolles - Blanchefort: 9,73 (2,43); Blanchefort - Arques: 19,46 (4,86); Blanchefort - Serres: 5,62 (1,40); Cassaignes - Blanchefort: 5,62 (1,40); Cassaignes - Serres: 7,94 (1,98); Peyrolles - Point X: 11,92 (2,98); Arques - Point X: 17,94 (4,48); Blanchefort - Point X: 3,56 (0,89); Cassaignes - Point X: 9,18 (2,29); Serres - Point X: 6,65 (1,66); Point X - Meridian: 5,50 (1,38); Middle PS - Meridian: 5,60 (1,40); Rennes-les-B. - Meridiaan: 5,60 (1,40).

  P = Peyrolles, B = Blanchefort, A = Arques
S = Serres, C= Cassaignes, X = Point X
D = middle BA, E = middle PB, F = middle BD

 

Calculations:

  Peyrolles - Point X: Triangle PBX. BB' is perpendicular on PX. PB = 9,73 cm ; Angle BPX = 15° ; Angle BXP = 45°. Calculation:  BB' : PB = sin 45°  »  BB' = 0,258819 x 9,73 = 2,5183 ; PB' : PB = cos 15°  »  PB' = 0,965925 x 9,73 = 9,3984 ; BB' : XB' = tan 45° = 1  »  XB' = 2,5183 ; PX = PB' + XB' = 9,3984 + 2,5183 = 11,9167 cm.




  Arques - Point X: Triangle PAX. XX' is perpendicular on PA. Angle APX = 75° ; PA = 16,85 cm ; PX = 11,92 cm. Calculation: XX' : PX = sin 75°  »  XX' = 0,965925 x 11,92 = 11,5138 ; PX' : PX = cos 75°  »  PX' = 0,258819 x 11,92 = 3,0851 ; AX' = PA - PX' = 16,85 - 3,0851 = 13,7649 ; XX' : AX' - tan angle PAX = 11,5138 : 13,7649 = 0,836460   »  Angle PAX = 39,91° » The angle of Arques - Point X on the meridian: 180° - 80° - 39,91° = 60,09°. Arques - Point X:  XX' : AX = sin 39,91° ; AX = 11,5138 : 0,641583  »  AX = 17,9459 cm.




  Blanchefort - Point X: Triangle PBX. BB' is perpendicular on PX. PB = 9,73 cm ; Angle BPX = 15° ; Angle BXP = 45°. Calculation:  BB' : PB = sin 45°  »  BB' = 0,258819 x 9,73 = 2,5183 ; BB' : BX = sin 45°  »  BX = 2,5183 : 0,0707106 = 3,5614 cm.



  Serres - Point X: Triangle BSX. Angle SBX = 90° ; BS = 5,62 cm ; BX = 3,56 cm. Calculation: SX² = BS² + BX² ; SX² = 31,5844 + 12,6736 = 44,258 ; SX = 6,6526 cm.




The angles of the Blanchefort-geometry.

  The angles on the Meridian of Paris are (considering an angle of 80° of the line Peyrolles-Arques): Peyrolles-Arques (PA): 80°; Cassaignes-Blanchefort (CX): 50°; Blanchefort-Point X (BX): 50°.

The angles on the parallel are: Peyrolles-Blanchefort (PB): 80°; Peyrolles-Serres (PS): 70°; Peyrolles-Point X (PX): 85°; Peyrolles-middle BA (PD): 40°; Cassaignes-Blanchefort (CX): 40°; Blanchefort-Point X (BX): 40°; Middle PS - Point X (MX): 89,5°.




The angles from Arques.

  P = Peyrolles, B = Blanchefort, A = Arques
S = Serres, M = middle PS, X = Point X



  On the meridian: Arques - Peyrolles (AP): 80°; Arques - middle PS (AM): 88,95°; Arques - Serres (AS): 80,88°; Arques - Blanchefort (AB): 70°; Arques - Point X (AX): 60,09°.

On the parallel this angle is 29,91°. So the angle from Arques to the Secret Place, which can be seen on the grave of Marie de Negri, is not 60° but 60,09°. This is a very tiny aberration: the Secret Place should be no more than 6 meter to the north.




Calculations:


  Angle Arques - "middle PS" on the Meridian: Triangle PAM. MM' is perpendicular on PA. PA = 16,58 ; PS = 1/3 PA = 5,62 ; PM = 1/2 PS = 2,81 ; PM = 1/6 PA = 2,81 ; PM' = 1/2 PM = 1,40 ; AM' = PA - PM' = 15,45. MM' : PM = sin 60°  »  MM' = 0,866025 x 2,81 = 2,4335 ; MM' : AM' = tan hoek PAM = 2,4335 : 15,45 = 0,1575  »   Angle PAM = 8,95°. Angle (A-M): 180° - 80° - 8,95° = 91,05°  (of 88,95°).



  Angle Arques - Serres on the Meridian: Triangle PAS. SS' is perpendicular on PA. PA = 16,85 ; PS = 5,62 ; PS' = 2,81 ; AS' = 14,04. SS' : PS = sin 60°  »  SS' = 0,866025 x 5,62 = 4,8670 ; SS' : AS' = tan anglePAS = 4,8670 : 14,04 = 0,3466  »  angle PAS = 19,12°. Angle (A-S): 180° - 80° - 19,12° = 80,88°.



  Angle "Rose Line" on the parallel: Triangle PMX. MM’ is perpendicular on PX. PM = 2,81 cm ; PX = 11,92 cm ; Angle MPX = 15°. MM’: PM = sin 15° » MM’= 0,258819 x 2,81 = 0,7272 ; PM’ : PM = cos 15° » PM’ = 0,965925 x 2,81 = 2,7142 ; XM’ = PX – PM’ = 11,92 – 2,7142 = 9,2058 ; MM’ : XM’ = tan Angle PXM = 0,7272 : 9,2058 = 0,078993 » Angle PXM = 4,5165°. In driehoek BXM geldt: Hoek BXM = Angle PXM + Angle PXB = 4,5° + 45° = 49,5°. In the triangle, formed by the extension of PX, the extension of BX and the parallel: The Angle of the extension of BX on the parallel = 40° ; The angle of the "Rose Line" = 180° - 49,5° - 40° = 90,5° (of 89,5°).





The paintings.


"Les Bergers d'Arcadie" of Nicolas Poussin.
 

  Poussin used a double "Blanchefort-geometry" for his painting.

AA' is the common line through Point X from Arques. It's the line on the grave of Marie de Negri. P = Peyrolles (below) / P' = Peyrolles (on top) A= Arques (left below) / A' = Arques (right on top) / A'' = Arques right below. H = the angle left below / H' = the angle right on top.

According to the Louvre the painting is 85 x 121 centimeter. The "taller" version is: Angle HPA = 90° - 80° = 10°; AH : PH = tan 10°  »  AH = 0,716326 x 60,5 = 10,6677 cm; A'H' = AH = 10,6677 cm; Angle A''AA' = 39.91° - 10° = 29,91°; AA'' = 121 cm; A'A'' : AA'' = tan 29,91°  »  A'A'' = 0,575257 x 121 = 69,6060 cm; The total height = AH + A'A'' + A'H' = 90,9414 centimeter.

In my opinion the painting is 3 : 4 (height : width). So the height is: 3/4 x 121 = 90,75 centimeter (a difference of ± 2 milimeter). Suppose the height is 90,75 cm and AH and A'H' are both 10,6677 cm: A'A'' = 90,75 - 21,3354 = 69,4146; A'A'' : AA'' = tan angle A''AA'  »  69,4146 : 121 = 0,573674  »  angle A''AA' = 29,8417°. An other calculation says this angle must be 29,91°. This is an abbaeration of 0,07°. Both aberations (2 mm of 0,07°) are very small. So I think my opinion is right that the painting is 3 : 4.


"The Adoration of the Lamb" of Jan van Eijck.

  "The Just Judges": 51 x 145 centimeter; Tan angle = 145 : 51 = 2,843137 » The angle is 70,62°.
"The Knights of Christ": 54 x 149,2 centimeter; angle = 149,2 : 54 = 2,762929 » The angle is 70,10°.
"The Hermits": 53,9 x 148,6 centimeter; Tan angle = 148,6 : 53,9 = 2,756957 » The angle is 70,06°.
"The Pelgrims": 54,2 x 148,7 centimeter; Tan angle = 148,7 : 54,2 = 2,743542 » The angle is 69,97°.

Conclusion: The aberations are very small, so the angle (used by Van Eijck) must be 70°. The panel with the "greatest" aberation (0,62°), is the copy of the Just Judges.


The central panel.

The diagonals of the central panel are on an angle of 30°. So the painting consist of two triangles of 30°-60°-90°. The central panel is 242,3 x 137,7 centimeter; Tan angle = 137,7 : 242,3 = 0,568303 » The angle is 29,60°. If the angle is 30°, the height of the painting is 139,9 centimeter, an aberation of 2,2 centimeter. This is the size of the inner edge of the panel !!!