PAGE F 58 – 59  /D 37 – 38/


A/         4 Ahau                                1,426,360 days

            -     12.11                                                        -        251 days

B/        /    13 Muluc                                1,426,109 days



C/     4 Ahau                               1,386,580 days

            -        1.7.11                                                    -        511 days

D/        /  13 Muluc                               1,386,069 days



A/        J.D. 2,048,621                                             October 26, 896

B/        J.D. 2,048,370                                             February 18, 896

C/        J.D. 2,008,841                                             November 28, 787

D/        J.D. 2,008,330                                             July 5, 786


   The multiples of (1 to 9)x78, (1 to 18)x780 and (1 to 9)x14,820 days are added to dates B and D.  Any multiple of 780 could be reached by mere addition through their mutual combination.


   The interval of 39,780 days is contained between dates A and C. It contains with little differences of just a few days: 

  68 times Venus synodic circulation length rounded to 585 days.

177 times Venus siderial circulation length (224.700 640 days).

  51 times Mars synodic circulation length rounded to 780 days.

  58 times Mars siderial circulation length (686.979800 days)

  17 times Venus and Mars basic conjunction cycle rounded to 2,340 days.

153 times the holy cycle tzolkin of 260 days.

   Venus synodic circulation length is moving between 577 to 592 days. Its real average length is 583.921 394 days. Seemingly irregular sling movement of the planet 542 days forwards and 42 days backwards arises from adding the movements of the Earth to the movements of the Venus. Nearly the same movement is performed by Mars. Mars synodic movement length is moving between 764 and 810 days. The direct movement is about 706 days and the reverse one about 74 days. Mars average synodic circulation length is 779.936 160 days. The basic theoretical cycle of the Venus and Mars conjunction cycle is 2,340 days. And again, it moves between 2,315 and 2,360 days.


   The Mayan astronomers were obviously aware of Venus and Mars mutual complicated movements, so they have been working with the rounded values, which enabled them determining the planets conjunctions for a long period with little mistake. The position of Venus and Mars is expressed in degrees of geocentric ecliptic co-ordinates.


A/ October 26, 896              Venus and Mars conjunction.

                                               The Venus 176.267°

                                               The Mars 175.374°

                                               The distance of the planets was 0.893°


                                               The real conjunction held 2 days ago – October 24,896

                                               The Venus 173.916°

                                               The Mars 174.117°

                                               The distance of both the planets was 0.201°.


C/ November 28, 787          Venus and Mars conjunction.

Venus 207.547°

                                               The Mars 204.765°

                                               The distance of the planets was 2.782°

                                               The real conjunction held 1 day ago – November 27, 787

                                               The Venus 206.870°

                                               The Mars 204.115°

                                               The distance of both the planets was 2.755°.


   In those cases the Mayan astronomers have determined Venus and Mars conjunction a day or two later. The angle distance has changed only slightly during this period, so it was not observable. Their mutual shift of distance was only 0.692° and 0.027°.


   If the conjunction is held during Venus being as a morning star close to the eastern elongation with the Sun, then the next conjunction is held around the western elongation, when Venus shines as a morning star in the sky. The interval between two following conjunctions is during those constellations very short and it moves between 206 and 295 days. The average value of this scatter is 251 days and it is to be found between dates A and B.


   The A date introduces Venus and the Mars conjunction. At the same time Venus was after western elongation with the Sun. Since this date, the time interval of 241 days (average time from the last conjunction – date B) is counted, till Venus was shortly before eastern elongation with the Sun.


B/ February 18, 896            Venus and Mars conjunction.

                                               The Venus 10.896°

                                               The Mars 9.966°

                                               The distance of the planets was 0.93°


                                               The real conjunction held 2 days ago – February 16,896

                                               The Venus 8.504°

                                               The Mars 8.507°

                                               The distance of both the planets was 0.003°.


   In this case, the Mayan astronomers have made a mistake of 0.897°, of which the planets receded since the real conjunction.


   In all three determined conjunctions the Mayan astronomers have made a mistake  moving from 0.027°to 0.897°, of which the planets receded since the real conjunction. It is so slight difference, that the planets movement on ecliptics in such small angle distance was undiscoverable for the Mayan astronomers.


   Mars and Jupiter conjunctions are in average repeating after 2.21 years. They are together around their oppositions with the Sun once in 49.14 years. It is the time of the best conditions for their observation. Repetition of triple conjunctions or close mutual approaches occurs during a short time period (tens of days) while the planets in this position.


   The D date, July 5, 786 expresses Mars and Jupiter short-period maximal distance between two following conjunctions.  At the same time the planets were close to their oppositions with the Sun.


                                               The first maximal approach – May 4, 786.

                                               The Mars 256.336°

                                               The Jupiter 267.078°

                                               The distance of both the planets was 10.742°.


D/ July 5, 786                       The maximal angle distance after the first approach.

                                               The Mars 243.621°

                                               The Jupiter 260.162°

                                               The distance of both the planets was 16.541°


                                               The second conjunction – August 22, 786.

                                               The Mars 258.101°

                                               The Jupiter 258.314°

                                               The distance of both the planets was 0.213°.


   The Mayan astronomers have been observing the mutual conjunctions of the outer planets (Mars, Jupiter, Saturn) during their oppositions with the Sun. In these positions were the planets excellently observable. We can say the same of dating Jupiter and the Saturn conjunction on page F 45 (D 74).


   The table of 780 days multiples follows after the opening Mayan dates. It is slightly rounded length of Mars synodic circulation (779.936 16 days). Through the mutual combination of those multiples we can get any multiple of 780 days. The table serves to determining the next conjunctions, or close approaches, of Mars and Venus, or the Mars and Jupiter.


   Mars and Venus conjunction is repeating after three synodic circulations of Mars, i.e. 2.340 days. By mere adding this cycle to the B date (February 18, 896), we get the time of the next conjunction.


   The mistake has increased and the planets receded more and more after every cycle because of the Mayan astronomers counting with the rounded values.  In spite of that the method was usable for 185.795 years with 29 basic cycles of conjunctions. The angle distance of both the planets was after this time only 7.685°. But for calculation of the next conjunctions is the triple multiple of 780 days no longer usable.


   For calculating the next conjunctions of Mars and Venus, it is necessary to change for another cycle. 140 synodic circulations of Mars are again added to the starting Mayan date B. 140 synodic circulations of Mars are 109,200 days, i.e. nearly 299 years, with 187 synodic circulations of Venus. The planets get into the near approach after this time again. For a certain time, it is possible to add a shorter basic interval of 2,340 days (3 times 780 days) to this conjunction to get their conjunctions or close approaches, before the mistake grow shows again. Out of the table of 780 days multiplies we can easily find out the multiply of 140 and 780 days and the total of 103,740 and 5,460 days is 109,200 days.


   The parallel  positions of Mars and Jupiter nearby their oppositions with the Sun (when they also get into mutual conjunction) are repeating every 49.14 years with 23 synodic circulations of Mars, i.e. 17,940  days. Out of the table of 780 days multiplies we can find this value by adding 14,820 days to 3,120 days, which is exactly 17,940 days.  By gradual adding the multiplies of this value to the initial Mayan date D (July 5, 786), Mars and Jupiter are always reaching the opposition with the Sun, when they are good observable, but they are also situated approximately between two following conjunctions. Because of working with average values of astronomical data, the mistake grows approximately after 147.4 years.