CONVERTING DATES INTO THE METONIC AND KALLIPPIC CALENDARS

CONVERTING DATES INTO THE METONIC AND KALLIPPIC CALENDARS G. R. F. Assar Oxford, UK (2 October 2008) In a seminal article on the link between the Greek astronomical and the Athenian civil calendars, B. L. Van der Waerden converted the dates of several ancient astronomical observations into the calendars of Euktemon-Meton and Kallippus.1 He took 13 Skirophorion = 21 Phamenoth in the archonship of Apseudes (27 June 432 BC)2 and 1 Hekatombaion = 28 June 330 BC as the epochs of these two astronomical calendars respectively and computed the current month and day numbers for each date. I have formulated his prescribed procedure as follows: The equivalent number of calendar days D in M months is:3 D = 30M – | 30M | 64 ………. Equ. 1 Conversely, the equivalent number of elapsed Metonic or Kallippic months M and remaining days R in a given number of elapsed days D from the above quoted two epochs are: 1 D (D + | | + d)| 30 63 D | + d) – 30M R = (D + | 63 M=| ………. Equ. 2 ………. Equ. 3 where d = 12 and d = 0 for the Metonic and Kallippic calendars respectively. Taking Hekatombaion as the first calendar month, the quotient Q from Equ. 2 gives the current month-number Mc in the Metonic calendar while Mc = Q + 1 in the Kallippic calendar. The current day-number Dc = R + 1 is the same for both calendars. However, Van der Waerden gave no reason for the 1-day discrepancy in some day-numbers in the month that immediately follows a set of 30-30 months. In describing the organisation of the Metonic cycle, Geminus (Eisagoge, 8.52) reveals the occurrence of such sets: ! "# $ 4 Within the 235 months, they made 110 hollow and 125 full, so that hollow and full months did not always follow one another alternately, but sometimes there would be two full months in 5 succession. This is now confirmed by the recurrent sets of 30-30 months in both the Metonic and Kallippic calendars discussed below.6 To demonstrate this and explain the reason for the 1-day discrepancy we must first identify the Metonic and Kallippic full and hollow months and their correct sequence. This could be done by counting from the epochs of the two cycles and, as prescribed by Geminus, dropping a day after every 63rd day.7 1 Van der Waerden (1960), 170-174. Fotheringham (1924), 383; Samuel (1972), 44-45. 3 The brackets | ….| signify the integer part of the quotients, i.e., rounded down to the nearest integer. 4 Manitius (1898), 120-121; Samuel (1972), 43; Aujac (1975), 56-57. 5 Heath (1913), 294. 6 With one 29-29 case in the Kallippic calendar, probably the last two months of the cycle. 7 Manitius (1898), 122-123; Heath (1913), 294; Samuel (1972), 43; Aujac (1975), 57. 2 1 Table 1 below sets out the Metonic months, using Fotheringham’s proposed intercalation in years 2, 5, 8, 10, 13, 16, and 18 of the cycle.8 This reveals 15 cases of 30-30 months whose positions in the calendar may be determined from the following equations: N1 = 9 + 15| n −1 n | + 17| | 2 2 N2 = N1 + 1 ………. Equ. 4 ………. Equ. 5 Where N1 and N2 are month numbers and 1 ≤ n ≤ 15 is the set number of each 30-30 month pair. The following examples demonstrate the discrepancy in some current day-numbers that fall in the month that comes immediately after a set of 30-30 months: Example 1: Counting from the epoch of the cycle, convert 7261 elapsed days into the calendar of Euktemon-Meton: Since 7261 > 6940, the quotient Q1 = 1 and remainder R1 = 321 from 7261 ÷ 6940 give the numbers of elapsed cycles and days in the current cycle respectively. From R1 ÷ 63 we get Q2 = 5 and T = 321 + 5 + 12 = 338. Dividing T by 30 gives Q3 = 11 and R2 = 8. Using equations 2 and 3, we get Mc = Q3 = 11 as the current month-number, following immediately the first 30-30 month pair 9-10 (n = 1 in Equ. 4), and Dc = R2 + 1 = 9 as the current day-number. However, counting from 13 Skirophorion in the current cycle, we have 17 + 6 × 30 + 4 × 29 = 313 days up to the end of month 10. This places the 321st day in day 8 of the 11th month, i.e., one day earlier than the day-number from Equ. 3 above. To explain the discrepancy, we must determine the position of the eliminated day (Pe) in the 11th Metonic (hollow) month. This may be obtained from the following Equations: Ne = | 30M + 18 | 64 Pe = 64Ne – [18 + 30(M – 1)] ………. Equ. 6 ………. Equ. 7 Where M = month number. From these we get Ne = 5 for M = Mc = 11 and Pe = 64 × 5 – (18 + 30 × 10) = 2, the position of the eliminated day in the hollow month 11. Since day 2 of the 11th month is eliminated, the count goes from 1 to 3 and so pushes the 8th day forward by 1 to day 9 of month. We can check this by the following two examples: Example 2: Counting from the epoch of the cycle, convert 7255 elapsed days into the calendar of Euktemon-Meton: Since 7255 > 6940, the quotient Q1 = 1 and remainder R1 = 315 from 7255 ÷ 6940 give the numbers of elapsed cycles and days in the current cycle respectively. From R1 ÷ 63 we get Q2 = 5 and T = 315 + 5 + 12 = 332. Dividing T by 30 gives Q3 = 11 and R2 = 2. Using equations 2 and 3, we get Mc = Q3 = 11 as the current month-number, following immediately the 30-30 month pair 9-10, and Dc = R2 + 1 = 3 as the current day-number. However, counting from 13 Skirophorion in the current cycle, we have 17 + 6 × 30 + 4 × 29 = 313 days up to the end of month 10. This 8 Fotheringham (1924), 387. Van der Waerden (1960), 175-177, argues that years 2 or 3, 10 or 11, and 18 or 19 could have been intercalary. In any case, the disposition of the embolismic years does not affect the sequence of full and hollow months. In a future note on Seleucid time-reckoning I will demonstrate that the Macedonian equivalent of the Metonic calendar intercalated a month in year 19 of the cycle. 2 places the 315th day in the 2nd day of the 11th month, i.e., one day before day-number 3 from Equ. 3 above. As shown earlier, because day 2 of the 11th month is omitted, the count goes from 1 to 3. The following example further clarifies the situation: Example 3: Counting from the epoch of the cycle, convert 7254 elapsed days into the calendar of Euktemon-Meton: Since 7254 > 6940, the quotient Q1 = 1 and remainder R1 = 314 from 7254 ÷ 6940 give the numbers of elapsed cycles and days in the current cycle respectively. Since this is 1 day less than the previous figure 7255 days, one expects to get the current day number as 3 – 1 = 2. However, from R1 ÷ 63 we get Q2 = 4 and T = 314 + 4 + 12 = 330. Dividing T by 30 gives Q3 = 11 and R2 = 0. Using equations 2 and 3, we get Mc = Q3 = 11 as the current month-number, following immediately the 30-30 month pair 9-10, and Dc = R2 + 1 = 1 as the current day-number. This time, counting from 13 Skirophorion in the current cycle, we get 17 + 6 × 30 + 4 × 29 = 313 days up to the end of month 10. Our 314th day thus becomes the 1st day of the 11th month, i.e., no earlier than the day-number from Equ. 3 above. This is because day 1 of the 11th month falls before the omitted day and so remains unaffected by the elimination of the 2nd day. I should point out that this 1-day discrepancy occurs irrespective of both the epoch-date of the cycle and the length of Skirophorion in 432 BC. For example, if the latter is considered hollow, the month-numbers in the 30-30 sets may be obtained from the following equations without affecting the 1-day difference: n n −1 | ………. Equ. 8 N1 = 9 + 15| | + 17| 2 2 N2 = N1 + 1 ………. Equ. 9 If, on the other hand, the cycle begins with 1 Hekatombaion 432 BC, the below equations determine the positions of its 30-30 months, again with no effect on the 1-day discrepancy: N1 = 1 + 15| n n −1 | + 17| | 2 2 N2 = N1 + 1 ………. Equ. 10 ………. Equ. 11 Equations 10 and 11 may be used to find the 30-30 sets of months in the Kallippic calendar also, beginning on 1 Hekatombaion 330 BC (28 June) with 1 ≤ n ≤ 59 . The position of the eliminated day (Pe) in month M of the Kallippic calendar may be obtained for the following equations: Ne = | 30M | 64 Pe = 64Ne – 30(M – 1) ………. Equ. 12 ………. Equ. 13 In the absence of lists of calendar months similar to those in Tables 1 and 2 below, equations 113 allow a speedy reconstruction of the sequence of Metonic and Kallippic months. Apart from the 30-30 sets, the remainder are alternating 30-29 months, save the last month of the Kallippic cycle which is taken to be hollow, not full.9 Unfortunately, we have no ancient astronomical records dated specifically on the Metonic calendar with day, month and year numbers. Fotheringham considered three lunar eclipses cited 9 This is assumed by Fotheringham (1924), 389. On the elimination of the 441st day in the Kallippic cycle cf. Jones (2000), 154, who proposes removal of the 63rd day at 24 points of the cycle. However, for the sake of simplicity, it is possible that elimination followed the pattern adduced by Fotheringham. 3 by Ptolemy in the Almagest and concluded that Hipparchus (ca. 190-120 BC) had mapped their original Babylonian dates on to the Metonic calendar with Attic archons and months.10 However, since these converted dates lack day-numbers, it is impossible to confirm Fotheringham and plead that Hipparchus indeed translated the Babylonian dates into the Metonic calendar. As three later examples from the Seleucid period indicate, Hipparchus may have employed the Kallippic calendar in converting the Babylonian lunisolar dates.11 The Seleucid examples are, according to Ptolemy, , that is, on the Chaldean/Babylonian calendar. They appear in the Almagest and are dated 5 Apellaios 67 SEB, 14 Dios 75 SEB, and 5 Xandikos 82 SEB.12 The first two relate to the early morning observations of Mercury on 19 Nov. 245 BC and 30 Oct. 237 BC respectively. The third concerns an evening observation of Saturn on 1 Mar. 229 BC. Given that the original lunisolar dates of these observations would have had little practical application for non-Babylonian astronomers, they must have been subsequently mapped on to an astronomical calendar. Simple computations show that these particular Chaldean dates conform to a Kallippic calendar that started off with midnight 1 Hyperberetaios on 27 August 311 BC (JDN = 1608069), i.e. a Kallippic version of the Seleucid era on the Babylonian count.13 As for the Kallippic calendar, Fotheringham used four stellar occultations by the moon that Timocharis had observed and found that they had a common epoch. 14 These are dated 25 Poseideon, year 36 of the first Kallippic cycle = 16/17 Phaophi, year 454 of Nabunassar, (20/21 Dec. 295 BC), dawn observation (JDN = 1614029); 15 Elaphebolion of the same Kallippic year = 5/6 Tybi (9/10 Mar. 294 BC), evening observation (JDN = 1614107); 8 Anthesterion, year 47 of the first Kallippic cycle = 29/30 Hathyr, year 465 of Nabunassar (29/30 Jan. 283 BC), evening observation (JDN = 1618087); and 25 Pyanepsion, year 48 of the first Kallippic cycle = 7/8 Thoth, year 466 of Nabunassar (8/9 Nov. 283 BC), dawn observation (JDN = 1618370). Fotheringham then concluded that the first Kallippic month, Hekatombaion, of the first cycle began at the sunset following both the summer solstice and the mean new moon in the early morning on 28 June 330 BC (JDN = 1601069).15 The first Kallippic year, therefore, fell in year 8 of the 6th Metonic cycle. Table 2 below gives the 940 months of the Kallippic calendar, including 59 pairs of 30-30 months whose numbers may be computed from equations 10 and 11 with 1 ≤ n ≤ 59 . The cause of the 1-day discrepancy between some of the computed and manually determined day-numbers in the month immediately following the Kallippic sets of 30-30 months is similar to the one under the Metonic calendar. I should, nevertheless, point out that the four Timocharian observations do not have a unique Kallippic epoch. Table 3 below gives other dates in the first Kallippic cycle, beginning with 16 July 329 BC and ending with 3 May 295 BC. These too satisfy equations 2 and 3 above and thus yield the same month and day-numbers for Timocharis’ observations as those reckoned from 28 June 330 BC. However, with the exception of the latter, the remaining dates have little or no astronomical and/or calendrical significance and are, therefore, superfluous.16 A final point concerns the 1-day discrepancy between those computed and manually counted current day-numbers that fall after the eliminated day in every hollow month of the Metonic and 10 Fotheringham (1924), 386, argues that the dates of these eclipses fall between the inceptions of the Metonic and Kallippic cycles and so their Attic months must be Metonic. Yet these records could have reached the Greek astronomers years after their compilation and not necessarily within the period 432-330 BC. Cf. also Samuel (1972), 46, and Toomer (1984), 211-213, who concur with Fotheringham. 11 Jones (2000), 151-152, suggests that these may be on the Kallippic reckoning. 12 Toomer (1984), 452-453 and 541. 13 The calendrical significance of these dates will be discussed in some detail in a forthcoming paper. 14 Fotheringham (1924), 387-392. For Ptolemy’s treatment of these observations in the Almagest cf. Toomer (1984), 334-338. 15 Fotheringham (1924), 390-391; NASA’s JPL HORIZONS Web-Interface at: http://ssd.jpl.nasa.gov/horizons.cgi#top gives 01:22-01:23 UT and 01:39 UT as the moments of the summer solstice and new moon, respectively, on 28 June 330 BC. 16 A series of dates in the Almagest place the first year of the first Kallippic cycle in 330/29 BC. Cf. Fotheringham (1924), 388; Samuel (1972), 47. 4 Callippic cycles. According to Geminus (Eisagoge, 8.56), it was not always the 30th day of the month that was dropped.17 Each omission therefore led to a difference of one in two successive day-numbers when they fell either side of an eliminated day in the same calendar month. The following example clarifies the discrepancy: Example 4: Convert 6299 and 6301 elapsed days into the Metonic or Kallippic calendar: Using the above equations 2 and 3, we get Mc = 214 and Dc = 9 for D = 6299. However, although the 2-day difference between 6299 and 6301 means we should expect Dc = 11 for the latter, we get Mc = 214 but Dc = 12 for D = 6301. In other words, for a difference of 2 days in the number of elapsed days, there is a 3-day difference between the corresponding current day-numbers. This is caused by the elimination of the 6300th day of the cycle which would have been the 10th day of the 214th month. It should, therefore, be kept in mind that a manual count of days from the beginning of the cycle does not always lead to the correct day-number in the calendar if the effect of the eliminated day in the current month is overlooked. In the above example, because both the Metonic and Kallippic months 214 lack day 10, the numbers associated with the remaining days of that month after its 10th day are one ahead of those obtained through a manual day count from the beginning of the cycle. Regrettably, with one exception,18 we have no Metonic or Kallippic date that shares its month with an omitted day to ascertain whether the ancient astronomers employed the methods briefly described here to convert a given number of elapsed days into their calendars. In any case, it is possible that a competent astronomer-mathematician like Hipparchus would have had little trouble locating the 110 and 441 eliminated days in the Metonic and Kallippic calendars respectively, using some simple-to-operate equations as those given above. It is, therefore, unlikely that Hipparchus would have neglected the impact of the eliminated days even if we concede that he did not exactly follow our prescribed equations. 17 Manitius (1898), 122-123; Heath (1913), 294; Aujac (1975), 57. The Kallippic date 14 Dios of the second Chaldean record in the Almagest, concerning the observation of Mercury at dawn on 30 Oct. 237 BC (JDN = 1635162), falls after the 430th eliminated day of the cycle (Ne = 430 from Equ. 12)) which removed the 10th day (Pe = 10 from Equ. 13) of the 918th calendar month. Its day-number 14 is, therefore, one more than that obtained from a direct count of days, starting with the inception of the Kallippic version of the Seleuco-Babylonian calendar on 1 Hyperberetaios = 27 August 311 BC. As explained earlier, because the day-count in the 918th Kallippic month goes from 9 to 11, missing the 10th day, the correct Kallippic day of observation would be the 14th day of the 918th month (= Dios). Cf. also Jones (2000), 147-148, giving a series of Kallippic dates in the period 28 Dec. 85 BC – 3(?) Jun. 74 BC. These carry Egyptian months and days, not Kallippic, and so cannot identify the eliminated days of the corresponding Kallippic months. 18 5 Table 1 – The Full and Hollow Metonic Months (19-year Cycle begins with 13 Skirophorion. First complete month = Hecatombaion) A Month-number B 1 2* 3 4 5* 6 7 8* 9 10* 11 12 13* 14 15 16* 17 18* 19 1 13h 26 38h 50 63 75h 87h 100h 112 125 137 149h 162h 174 186 199 211h 224h 2h 14 27 39 51h 64h 76 88 101 113h 126h 138 150 163 175h 187 200h 212 225 3 4h h 15 16 29 28h 40h 41 52 53h 65 66h h 77 78 89h 90 102h 103 114 115h 127 128h 139h 140 151h 152 164h 165 176 177h 188h 189 201 202 213h 214 226h 227 5 6h h 17 18 30h 31 42 43h 54 55h 67 68h h 79 80 91 92h 104h 105 116 117h 129 130h 141h 142 153h 154 166h 167 178 179h 190h 191 203h 204 215h 216 228h 229 8h 20 33 45h 57h 70h 82 94h 107h 119h 132h 144 156h 169 181h 193 206 218 231 7 19h 32h 44 56 69 81h 93 106 118 131 143h 155 168h 180 192h 205h 217h 230h Key to Tables 1: A = year of the cycle B = cumulative number of omitted days C = days in year D = cumulative number of days * indicates embolismic year h indicates hollow month 6 9 21h 34h 46 58 71 83h 95 108 120 133 145h 157 170 182 194h 207h 219 232h 10 11h 22 23h 35 36h h 47 48 60h 59 72h 73 84 85h 96h 97 109h 110 121h 122 134h 135 146 147h 158h 159 171h 172 183h 184 195 196h 208 209h 220h 221 233 234 12 24 25h 37 49h 61 62h 74 86 98h 99 111h 123 124h 136h 148 160h 161 173h 185h 197 198h 210 222h 223 235h 5 11 17 23 29 34 40 46 52 58 64 69 75 81 86 93 98 104 110 C D 355 383 355 354 384 355 354 384 354 384 354 355 384 354 354 384 355 384 354 355 738 1093 1447 1831 2186 2540 2924 3278 3662 4016 4371 4755 5109 5463 5847 6202 6586 6940 Table 2 – The Full and Hollow Kallippic Months (76-year Cycle begins with 1 Hekatombaion) A Month-number B C D 1* 2 3* 4 5 6* 7 8 9* 10 11* 12 13 14* 15 16 17* 18 19 20* 21 22* 23 24 25* 26 27 28* 29 30* 31 32 33* 34 35 36* 37 38 39* 40 41* 42 43 44* 45 46 47* 48 49* 50 51 52* 53 54 55* 56 57 1 14 26h 39h 51 63 76 88h 100 113 125 138 150h 162 175h 187 199h 212h 224h 236 249 261h 274h 286h 298 311 323h 335h 348h 360 373 385 397h 410h 422 434h 447 459h 471 484 496 509 521h 533 546 558 570h 583h 595 608h 620 632h 645h 657 669 682 694h 2 15h 27 40 52h 64h 77h 89 101h 114h 126h 139h 151 163h 176 188h 200 213 225 237h 250h 262 275 287 299h 312h 324 336 349 361h 374h 386 398 411 423h 435 448h 460 472h 485h 497 510h 522 534h 547h 559h 571 584 596h 609 621h 633 646 658h 670h 683h 695 3h 16 28h 41h 53 65 78 90h 102 115 127 140 152h 164 177 189 201h 214h 226 238 251 263h 276h 288h 300 313 325h 337 350h 362 375 387h 399h 412h 424 436h 449 461h 473 486 498h 511 523h 535 548 560 572h 585h 597 610 622 634h 647h 659 671 684 696h 4 17 29 42 54h 66 79h 91 103h 116h 128h 141h 153 165h 178h 190h 202 215 227h 239h 252h 264 277 289 301h 314h 326 338h 351 363h 376h 388 400 413 425h 437 450 462 474h 487h 499 512h 524 536h 549h 561 573 586 598h 611h 623h 635 648 660h 672h 685h 697 5h 18h 30h 43h 55 67h 80 92h 104 117 129 142 154h 166 179 191 203h 216h 228 240 253 265h 278h 290 302 315 327h 339 352h 364 377 389h 401 414h 426 438h 451h 463h 475 488 500h 513 525h 537 550 562h 574h 587h 599 612 624 636h 649h 661 673 686 698h 6 19 31 44 56h 68 81 93 105h 118h 130 143h 155 167h 180h 192h 204 217 229h 241 254h 266 279 291h 303h 316h 328 340h 353 365h 378h 390 402h 415 427h 439 452 464 476h 489h 501 514 526 538h 551h 563 575 588 600h 613h 625 637 650 662h 674 687h 699 7h 20h 32h 45h 57 69h 82h 94h 106 119 131h 144 156h 168 181 193 205h 218h 230 242h 255 267h 280h 292 304 317 329h 341 354 366 379 391h 403 416h 428 440h 453h 465 477 490 502h 515h 527h 539 552 564h 576h 589h 601 614 626h 638h 651h 663 675h 688 700h 8 21 33 46 58h 70 83 95 107h 120h 132 145 157 169h 182h 194 206 219 231h 243 256h 268 281 293h 305 318h 330 342h 355h 367h 380h 392 404h 417 429h 441 454 466h 478h 491h 503 516 528 540h 553h 565 577 590 602h 615h 627 639 652 664h 676 689 701 7 9h 22h 34 47h 59 71h 84h 96h 108 121 133h 146h 158h 170 183 195h 207h 220h 232 244h 257 269h 282h 294 306h 319 331h 343 356 368 381 393h 405 418 430 442h 455h 467 479 492 504h 517h 529 541 554 566h 578 591h 603 616 628h 640h 653h 665 677h 690h 702h 10 23 35h 48 60h 72 85 97 109h 122h 134 147 159 171h 184h 196 208 221 233h 245 258 270 283 295h 307 320h 332 344h 357h 369 382h 394 406h 419h 431h 443 456 468h 480h 493h 505 518 530h 542h 555h 567 579h 592 604h 617h 629 641 654 666h 678 691 703 11h 24h 36 49 61 73h 86h 98 110 123 135h 148h 160h 172 185 197h 209 222h 234 246h 259h 271h 284h 296 308h 321 333h 345 358 370h 383 395h 407 420 432 444h 457h 469 481 494 506h 519h 531 543 556 568h 580 593 605 618 630h 642 655h 667 679h 692h 704h 12 25 37h 50h 62h 74 87 99h 111h 124h 136 149 161 173h 186h 198 210h 223 235h 247 260 272 285 297h 309 322 334 346h 359h 371 384h 396 408h 421h 433 445 458 470h 482 495h 507 520 532h 544h 557h 569 581h 594h 606h 619h 631 643h 656 668h 680 693 705 13h 38 75h 112 137h 174 211 248h 273 310h 347 372h 409 446h 483h 508h 545 582 607 644 681h 6 11 17 23 29 35 40 46 52 58 64 69 75 81 87 92 98 104 110 116 121 127 133 139 145 150 156 162 168 174 180 185 191 197 202 209 214 220 226 232 238 243 249 255 261 266 272 278 284 290 295 301 307 313 319 324 330 384 355 384 354 354 384 355 354 384 354 384 355 354 384 354 355 384 354 354 384 355 384 354 354 384 355 354 384 354 384 354 355 384 354 355 383 355 354 384 354 384 355 354 384 354 355 384 354 384 354 355 384 354 354 384 355 354 384 739 1123 1477 1831 2215 2570 2924 3308 3662 4046 4401 4755 5139 5493 5848 6232 6586 6940 7324 7679 8063 8417 8771 9155 9510 9864 10248 10602 10986 11340 11695 12079 12433 12788 13171 13526 13880 14264 14618 15002 15357 15711 16095 16449 16804 17188 17542 17926 18280 18635 19019 19373 19727 20111 20466 20820 Table 2 – Continued A 58* 59 60* 61 62 63* 64 65 66* 67 68* 69 70 71* 72 73 74* 75 76 706 719h 731 744 756h 768h 781h 793 805h 818h 830h 843h 855 867h 880 892h 904 917 929 707h 720 732h 745h 757 769 782 794h 806 819 831 844 856h 868 881 893 905h 918h 930 708 721 733 746 758h 770 783h 795 807h 820h 832h 845h 857 869h 882h 894h 906 919 931h Month-number 709h 722h 734h 747h 759 771h 784 796h 808 821 833 846 858h 870 883 895 907h 920h 932 710 723 735 748 760h 772 785 797 809h 822h 834 847h 859 871h 884h 896h 908 921 933h 711h 724h 736h 749h 761 773h 786h 798h 810 823 835h 848 860h 872 885 897 909h 922h 934 713h 726h 738 751h 763 775h 788h 800h 812 825 837h 850h 862h 874 887 899h 911h 924h 936 712 725 737 750 762h 774 787 799 811h 824h 836 849 861 873h 886h 898 910 923 935h 714 727 739h 752 764h 776 789 801 813h 826h 838 851 863 875h 888h 900 912 925 937h 715h 728h 740 753 765 777h 790h 802 814 827 839h 852h 864h 876 889 901h 913 926h 938 716 729 741h 754h 766h 778 791 803h 815h 828h 840 853 865 877h 890h 902 914h 927 939h 717h 718 730h 742 743h 755 767 779h 780 792h 804 816 817 829 841h 842 854h 866 878 879h 891 903h 915 916h 928h 940h B C D 336 342 348 353 359 365 371 376 382 388 394 400 405 412 417 423 429 435 441 384 354 384 355 354 384 354 355 384 354 384 354 355 383 355 354 384 354 354 21204 21558 21942 22297 22651 23035 23389 23744 24128 24482 24866 25220 25575 25958 26313 26667 27051 27405 27759 Key to Table 2: A = year of the cycle B = cumulative number of omitted days C = days in year D = cumulative number of days * indicates embolismic year h indicates hollow month Table 3 – Possible Kallippic Epochs of the Four Timocharian Observations No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Date (BC) 28 Jun. 330 16 Jul. 329 13 Sep. 329 2 Oct. 328 30 Nov. 328 28 Jan. 327 16 Feb. 326 16 Apr. 326 4 May 325 2 Jul. 325 30 Aug. 325 18 Sep. 324 16 Nov. 324 5 Dec. 323 2 Feb. 322 2 Apr. 322 20 Apr. 321 18 Jan. 321 7 Jul. 320 4 Sep. 320 2 Nov. 320 21 Nov. 319 19 Jan. 318 JDN 1601069 1601453 1601512 1601896 1601955 1602014 1602398 1602457 1602841 1602900 1602959 1603343 1603402 1603786 1603845 1603904 1604288 1604347 1604731 1604790 1604849 1605233 1605292 No. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Date (BC) 7 Feb. 317 6 Apr. 317 4 Jun. 317 23 Jan. 316 21 Aug. 316 9 Sep. 315 7 Nov. 315 5 Jan. 314 24 Jan. 313 23 Mar. 313 11 Apr. 312 9 Jan. 312 7 Aug. 312 26 Aug. 311 24 Oct. 311 12 Nov. 310 10 Jan. 309 9 Mar. 309 28 Mar. 308 26 May 308 14 Jun. 307 12 Aug. 307 10 Oct. 307 JDN 1605676 1605735 1605794 1606178 1606237 1606621 1606680 1606739 1607123 1607182 1607566 1607625 1607684 1608068 1608127 1608511 1608570 1608629 1609013 1609072 1609456 1609515 1609574 No. 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 Date (BC) 29 Oct. 306 27 Dec. 306 14 Jan. 304 14 Mar. 304 12 May 304 31 May 303 29 Jul. 303 17 Aug. 302 15 Oct. 302 13 Dec. 302 31 Dec. 301 28 Feb. 300 19 Mar. 299 17 May 299 15 Jul. 299 3 Aug. 298 1 Oct. 298 19 Oct. 297 17 Dec. 297 14 Feb. 296 5 Mar. 295 3 May 295 JDN 1609958 1610017 1610401 1610460 1610519 1610903 1610962 1611346 1611405 1611464 1611848 1611907 1612291 1612350 1612409 1612793 1612852 1613236 1613295 1613354 1613738 1613797 Abbreviations: SEB Seleucid Era of the Babylonian calendar (epoch = 2/3 April 311 BC = 1 N s nu) JDN Julian Day Number 8 Bibliography: Aujac G, Géminos. Introduction aux Phénomènes, Paris, 1975. Fotheringham J. K., “The Metonic and Callippic Cycles”, Monthly Notices of the Royal Astronomical Society, 84 (1924), 383-392. Heath T., Aristarchus of Samos. The Ancient Copernicus. A History of Greek Astronomy to Aristarchus together with Aristarchus’s Treatise on the Sizes and Distances of the Sun and Moon. A New Greek Text with Translation and Notes. Oxford, 1913. Jones A., “Calendrica I: New Callippic Dates”, Zeitschrift für Papyrologie und Epigraphik, 129 (2000), 141-158. Manitius C., Gemini. Elementa Astronomiae. Ad Codicum Fidem Recensuit Germanica Interpretatione et Commentariis Instruxit. Lipsiae, 1898 (reprinted, Stuttgart, 1974). Samuel A. E., Greek and Roman Chronology. Calendars and Years in Classical Antiquity. Munich, 1972. Toomer G. J., Ptolemy’s Almagest. London, 1984. Van der Waerden B. L., “Greek Astronomical Calendars and their Relation to the Athenian Civil Calendar”, The Journal of Hellenic Studies, 80 (1960), 168-180. Copyright © 2 October 2008 by G. R. F. Assar 1- Adjusted page 1 on 3 October 2008: I’ve formulated Van der Waerden’s procedure. 2- Adjusted page 6 on 3 October 2008: Abbreviations: JDN = Julian Day Number. 3- Added footnote 16 on 3 October 2008. 4- Modified on 4-5 May 2013 some paragraphs concerning the 1-day discrepancy between the calendrical and manual counts, caused by the position of the eliminated day after each set of 30-30 months. 9