Editing Eclipse cycle (section)

== Eclipse cycles ==
This table summarizes the characteristics of various eclipse cycles, and can be computed from the numerical results of the preceding paragraphs; ''cf.'' Meeus (1997) Ch.9. More details are given in the comments below, and several notable cycles have their own pages. Many other cycles have been noted, some of which have been named.<ref name=vanGent>{{cite web |last1=Rob van Gent |title=A Catalogue of Eclipse Cycles |url=https://webspace.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm |publisher=Utrecht University}}</ref>

The number of days given is the average. The actual number of days and fractions of days between two eclipses varies because of the variation in the speed of the Moon and of the Sun in the sky. The variation is less if the number of anomalistic months is near a whole number, and if the number of anomalistic years is near a whole number. (See graphs lower down of semester and Hipparchic cycle.)

Any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros (''s'') and inex (''i'') intervals. These are listed in the column "formula".

{| class="wikitable sortable"
! Cycle!! Formula!!Days!!Synodic<br>months!!Draconic<br>months!!Anomalistic<br>months!!Eclipse<br>years!!Julian<br>years!!Anomalistic<br>years!!Eclipse<br>seasons!!Node
|-
| fortnight ||19''i'' − {{frac|30|1|2}}''s''||14.77|| 0.5 || 0.543 || 0.536 || 0.043 || 0.040 || 0.040 || 0.086 || alternate
|-
|[[synodic month]]||38''i'' − 61''s''|| 29.53  || 1 || 1.085 || 1.072 || 0.085 || 0.081 || 0.081 || 0.17 || same
|-
| pentalunex ||    53''s'' − 33''i'' || 147.65 || 5 || 5.426 || 5.359 || 0.426 || 0.404 || 0.404 || 0.852 || alternate
|-
| semester   ||      5''i'' − 8''s'' || 177.18 || 6 || 6.511 || 6.430 || 0.511 || 0.485 || 0.485 || 1     || alternate
|-
| lunar year ||    10''i'' − 16''s'' || 354.37 || 12 || 13.022 || 12.861 || 1.022 || 0.970 || 0.970 || 2  || same
|-
| hexon      ||     13''s'' - 8''i'' || 1,033.57 || 35 || 37.982 || 37.510 || 2.982 || 2.830 || 2.830 || 6 || same
|-
| hepton     ||      5''s'' − 3''i'' || 1,210.75 || 41 || 44.493 || 43.940 || 3.493 || 3.315 || 3.315 || 7 || alternate
|-
|[[Octon (astronomy)|octon]]||2''i'' − 3''s''||1,387.94||47||51.004||50.371|| 4.004 || 3.800 || 3.800 || 8 || same
|-
|[[tzolkinex]]||       2''s'' − ''i''|| 2,598.69 || 88 || 95.497 || 94.311 || 7.497 || 7.115 || 7.115 || 15 || alternate
|-
| Hibbardina ||     31''s'' − 19''i''|| 3,277.90 ||111 ||120.457 ||118.960 || 9.457 || 8.974 || 8.974 || 19 || alternate
|-
|[[sar (astronomy)|sar (half saros)]]||{{frac|2}}''s''||3,292.66||111.5||120.999||119.496||9.499||9.015||9.015||19|| same
|-
| [[tritos]] ||        ''i'' − ''s'' || 3,986.63 || 135 || 146.501 || 144.681 || 11.501 || 10.915 || 10.915 || 23 || alternate
|-
|[[saros (astronomy)|saros]] (''s'') ||''s''||6,585.32||223||241.999||238.992 || 18.999 || 18.030 || 18.029 || 38 || same
|-
|[[Metonic cycle]]||10''i'' − 15''s''|| 6,939.69  ||235|| 255.021 || 251.853 || 20.021 || 19.000 || 18.999 || 40 || same
|-
| semanex    ||       3''s'' - ''i'' || 9,184.01 || 311 || 337.496 || 333.303 || 26.496 || 25.145 || 25.144 || 53 || alternate
|-
| thix    ||       4''i'' - 5''s'' || 9,361.20 || 317 || 344.007 || 339.733 || 27.007 || 25.630 || 25.629 || 54 || same
|-
|[[inex]] (''i'')||             ''i''|| 10,571.95 || 358 || 388.500 || 383.674 || 30.500 || 28.944 || 28.944 || 61 || alternate
|-
|[[exeligmos]]||              3''s'' || 19,755.96 || 669 || 725.996 || 716.976 || 56.996 || 54.089 || 54.087 || 114 || same
|-
|Aubrey cycle||''i'' + {{frac|2}}''s''|| 20,449.93||692.5|| 751.498 || 742.162 || 58.998 || 55.989 || 55.987 || 118 || alternate
|-
| unidos     ||        ''i'' + 2''s''|| 23,742.59 || 804 || 872.497 || 861.658 || 68.497 || 65.004 || 65.002 || 137 || alternate
|-
|[[Callippic cycle]] ||40''i'' − 60''s''|| 27,758.75||940|| 1020.084 || 1007.411 || 80.084 || 75.999 || 75.997 || 160 || same
|-
| triad      ||               3''i'' || 31,715.85 || 1074 || 1165.500 || 1151.021 || 91.500 || 86.833 || 86.831 || 183 || alternate
|-
|quarter Palmen cycle||4''i'' - 1''s''||35,702.48 || 1209 || 1312.002 || 1295.702 || 103.002 || 97.748 || 97.745 || 206 || same
|-
| Mercury cycle||    2''i'' + 3''s'' || 40,899.87 || 1385 || 1502.996 || 1484.323 || 117.996 || 111.978 || 111.975 || 236 || same
|-
| tritrix    ||      3''i'' + 3''s'' || 51,471.82 || 1743 || 1891.496 || 1867.997 || 148.496 || 140.922 || 140.918 || 297 || alternate
|-
|de la Hire cycle||           6''i'' || 63,431.70 || 2148 || 2331.001 || 2302.041 || 183.001 || 173.667 || 173.662 || 366 || same
|-
| trihex     ||      3''i'' + 6''s'' || 71,227.78 || 2412 || 2617.492 || 2584.973 || 205.492 || 195.011 || 195.006 || 411 || alternate
|-
|Lambert II cycle||   9''i'' + ''s'' || 101,732.88 || 3445 || 3738.500 || 3692.054 || 293.500 || 278.529 || 278.522 || 587 || alternate
|-
|Macdonald cycle||   6''i'' + 7''s'' || 109,528.95 || 3709 || 4024.991 || 3974.986 || 315.991 || 299.874 || 299.866 || 632 || same
|-
|Utting cycle||      10''i'' + ''s'' || 112,304.83 || 3803 || 4127.000 || 4075.727 || 324.000 || 307.474 || 307.466 || 648 || same
|-
| selebit    ||      11''i'' + ''s'' || 122,876.78  || 4161 || 4515.500 || 4459.401 || 354.500 || 336.418 || 336.409 || 709 || alternate
|-
|[[Hipparchic cycle|Cycle of Hipparchus]]|| 25''i'' − 21''s''||126,007.02||4267||4630.531||4573.002||363.531||344.988||344.979||727|| alternate
|-
| Square year||      12''i'' + ''s'' || 133,448.73  || 4519 || 4904.000 || 4843.074 || 385.000 || 365.363 || 365.353 || 770 || same
|-
| Gregoriana ||     6''i'' + 11''s'' || 135,870.24 || 4601 || 4992.986 || 4930.955 || 391.986 || 371.992 || 371.983 || 784 || same
|-
| hexdodeka  ||     6''i'' + 12''s'' || 142,455.56 || 4824 || 5234.985 || 5169.947 || 410.985 || 390.022 || 390.012 || 822 || same
|-
|Grattan Guinness cycle||12''i'' - 4''s''||142,809.92||4836|| 5248.007 || 5182.807 || 412.007 || 390.992 || 390.982 || 824 || same
|-
| Hipparchian||     14''i'' + 2''s'' || 161,177.95  || 5458 || 5922.999 || 5849.413 || 464.999 || 441.281 || 441.270 || 930 || same
|-
|Basic period||              18''i'' || 190,295.11 || 6444 || 6993.002 || 6906.123 || 549.002 || 521.000 || 520.986 || 1098 || same
|-
| Chalepe    ||     18''i'' + 2''s'' || 203,465.76 || 6890 || 7476.999 || 7,384.107 || 586.999 || 557.059 || 557.044 || 1174 || same
|-
|tetradia (Meeus III)||22''i'' − 4''s''||206,241.63 || 6984 || 7579.008 || 7484.849 || 595.008 || 564.659 || 564.644 || 1190 || same
|-
|tetradia (Meeus I) ||19''i'' + 2''s''|| 214,037.71 || 7248 || 7865.499 || 7767.781 || 617.499 || 586.003 || 585.988 || 1235 || alternate
|-
|hyper exeligmos|| 24''i'' + 12''s'' || 332,750.68 || 11268 || 12227.987 || 12076.070 || 959.987 || 911.022 || 910.998 || 1920 || same
|-
| cartouche  ||              52''i'' || 549,741.44 || 18616 || 20202.006 || 19951.022 || 1586.006 || 1505.110 || 1505.070 || 3172 || same
|-
|Palaea-Horologia || 55''i'' + 3''s''|| 601,213.26 || 20359 || 22093.502 || 21819.019|| 1734.502 || 1646.032 || 1645.989 || 3469 || alternate
|-
| hybridia   ||     55''i'' + 4''s'' || 607,798.58 || 20582 || 22335.501 || 22058.012|| 1753.501 || 1664.062 || 1664.018 || 3507 || alternate
|-
| Selenid 1  ||     55''i'' + 5''s'' || 614,383.90 || 20805 || 22577.499 || 22297.004 || 1772.499 || 1682.091 || 1682.047 || 3545 || alternate
|-
| Proxima    ||     58''i'' + 5''s'' || 646,099.75 || 21879 || 23743.000 || 23448.024 || 1864.000 || 1768.925 || 1768.878 || 3728 || same
|-
| heliotrope ||     58''i'' + 6''s'' || 652,685.07 || 22102 || 23984.998 || 23687.016|| 1882.998 || 1786.954 || 1786.907 || 3766 || same
|-
| Megalosaros||     58''i'' + 7''s'' || 659,270.40 || 22325 || 24226.997 || 23926.009 || 1901.997 || 1804.984 || 1804.936 || 3804 || same
|-
| immobilis  ||     58''i'' + 8''s'' || 665,855.72 || 22548 || 24468.996 || 24165.001 || 1920.996 || 1823.014 || 1822.966 || 3842 || same
|-
|accuratissima||    58''i'' + 9''s'' || 672,441.04 || 22771 || 24710.994 || 24403.993 || 1939.994 || 1841.043 || 1840.995 || 3880 || alternate
|-
|Mackay cycle||     76''i'' + 9''s'' || 862736.15 || 29215 ||  31703.996 || 31310.116 || 2488.996 || 2362.043 || 2361.981 || 4978 || alternate
|-
| Selenid 2  ||    95''i'' + 11''s'' || 1,076,773.86 || 36463 || 39569.496 || 39077.897 || 3106.496 || 2948.046 || 2947.968 || 6213 || alternate
|-
| Horologia  ||    110''i'' + 7''s'' || 1,209,011.84 || 40941 || 44429.003 || 43877.031 || 3488.003 || 3310.094 || 3310.007 || 6976 || same
|}
===Notes===

;Fortnight: Half a synodic month (29.53 days). When there is an eclipse, there is a fair chance that at the next [[Syzygy (astronomy)|syzygy]] there will be another eclipse: the Sun and Moon will have moved about 15° with respect to the nodes (the Moon being opposite to where it was the previous time), but the luminaries may still be within bounds to make an eclipse. For example, the penumbral [[May 2002 lunar eclipse|lunar eclipse of May 26, 2002]] is followed by the annular [[solar eclipse of June 10, 2002]] and penumbral [[June 2002 lunar eclipse|lunar eclipse of June 24, 2002]]. The shortest lunar fortnight between a new moon and a full moon lasts only about 13 days and 21.5 hours, while the longest such lunar fortnight lasts about 15 days and 14.5 hours. (Due to [[evection]], these values are different going from quarter moon to quarter moon. The shortest lunar fortnight between first and last quarter moons lasts only about 13 days and 12 hours, while the longest lasts about 16 days and 2 hours.)
:For more information see [[eclipse season]].

;[[Synodic month]]: Similarly, two events one synodic month apart have the Sun and Moon at two positions on either side of the node, 29° apart: both may cause a partial solar eclipse. For a lunar eclipse, it is a penumbral lunar eclipse.

;Pentalunex: 5 synodic months. Successive solar or lunar eclipses may occur 1, 5 or 6 synodic months apart.<ref name=vanGent/> When two solar eclipses are one month apart, one will be seen near the [[Arctic Circle]] and the other near the [[Antarctic Circle]]. When they are five months apart, they are both seen near the Arctic Circle or both near the Antarctic Circle.

;Semester: Half a lunar year. Eclipses will repeat exactly one semester apart at alternating nodes in a cycle that lasts for 8 eclipses. Because it is close to a half integer of anomalistic, draconic months, and tropical years, each solar eclipse will (usually) alternate between hemispheres each semester, as well as alternate between total and annular. Hence there is usually a maximum of one total or annular eclipse each in a given lunar year. (However, in the middle of an eight-semester series the hemispheres switch, and there is a switch during the series between whether the odd ones or the even ones are total.) It is possible to have two eclipses separated by a semester and a third eclipse one month before or after, so that two of the three are separated by seven months, but this only happens during certain centuries (see graph of inex versus date below). Because it is close to a half integer of anomalistic, draconic months, and tropical years, each lunar eclipse will usually alternate between edges of Earth's shadow each semester, as well as alternate between Lunar eclipses with the moon’s penumbral and umbral shadow difference less or greater than 1. Hence there is usually a maximum of one Lunar eclipse with Moon’s penumbral and umbral shadow difference less or greater than 1 each in a given lunar year.

;Lunar year: Twelve (synodic) months, a little longer than an eclipse year: the Sun has returned to the node, so eclipses may again occur.

;Hexon: 6 eclipse seasons, and a fairly short eclipse cycle. Each eclipse in a hexon series (except the last) is followed by an eclipse whose saros series number is 8 lower, always occurring at the same node. It is equal to 35 synodic months, 1 less than 3 lunar years (36 synodic months). At any given time there are six hexon series active.

;Hepton: 7 eclipse seasons, and one of the less noteworthy eclipse cycles. Each eclipse in a hepton is followed by an eclipse 3 saros series before, always occurring at alternating nodes. It is equal to 41 synodic months. The interval is nearly a whole number of weeks (172.96), so each eclipse is followed by another that is usually on the same day of the week (moving backwards irregularly by an average of a quarter day). At any given time there are seven hepton series active.

;[[Octon (astronomy)|Octon]]: 8 eclipse seasons, {{frac|5}} of the Metonic cycle, and a fairly decent short eclipse cycle, but poor in anomalistic returns. Each octon in a series is 2 saros apart, always occurring at the same node. It is equal to 47 synodic months. At any given time there are eight octon series active.

;[[Tzolkinex]]: Includes a half draconic month, so occurs at alternating nodes and alternates between hemispheres. Each consecutive eclipse is a member of preceding saros series from the one before. Equal to nearly ten [[tzolk'in]]s. Every third tzolkinex in a series is near an integer number of anomalistic months and so will have similar properties.

;Hibbardina: An eclipse "cycle" of at most 3 eclipses, but in fact meant as a period separating a pair of similar eclipses with opposite gamma values. Adding 1 lunation (for 112 synodic months) gives another period with the same property, the other half of a saros. The two surround a sar (half-saros). Named for William B. Hibbard who identified it in 1956.<ref name=vanGent/> One lunar year less than a Hibbardina, 99 lunations, is only about a day and a half more than eight years.

;Sar (half saros): Includes an odd number of fortnights (223). As a result, eclipses alternate between lunar and solar with each cycle, occurring at the same node and with similar characteristics. A solar eclipse with small [[Gamma (eclipse)|gamma]] will be followed by a very central total lunar eclipse. A solar eclipse where the Moon's penumbra just barely grazes the southern limb of Earth will be followed half a saros later by a lunar eclipse where the Moon just grazes the southern limb of the Earth's penumbra.<ref name=vanGent/>

;[[Tritos]]: Equal to an inex minus a saros. A triple tritos is close to an integer number of anomalistic months and so will have similar properties.

;[[Saros (astronomy)|Saros]]: The best known eclipse cycle (described in the [[Almagest]] but not given this name), and one of the best for predicting eclipses, in which 223 synodic months equal 242 draconic months with an error of only 51 minutes. It is also very close to 239 anomalistic months, which makes the circumstances between two eclipses one saros apart very similar. Being a third of a day more than a whole number of days, each succeeding eclipse is centered about 120° further west over the Earth. If the Earth's orbit around the sun were circular, the saros cycle would be very close to a periodic orbit that would repeat exactly every 223 months.ὤ<ref name=Valsecchi>{{cite journal |last1=Giovanni Valsecchi, Ettore Perozzi, Archie Roy, Bonnie Steves |title=Periodic orbits close to that of the Moon |journal=Astronomy and Astrophysics |date=Mar 1993 |page=311 |url=https://www.researchgate.net/publication/234420602}}</ref>

[[File:Dates of solar eclipses in XXI century.png|thumb|600px| [[Histogram]] of dates of solar eclipses in 21st century. The dates form 35 clusters. Each cluster contains eclipses separated by Metonic cycles of 19 years. Each series contains four or five eclipses, and 46 or 65 or 84 years after the first one another series starts about a day and a half later in the (Julian) year. This means that the clusters slowly move forward to later dates. In a saros series, every 18 years the eclipse moves to the next later cluster. After 631 years (35 saros) it comes back to the original cluster, which by then has moved, in the Julian calendar, to a date about 13 or 14 days later, or about 18 days later in the Gregorian calendar.]]
;[[Metonic cycle]] or enneadecaeteris: Nearly 6940 days, but as an eclipse cycle can be taken as 235 synodic months. This is just an hour and a half less than 19 years of {{frac|365|1|4}} days. It is also 5 "octon" periods and close to 20 eclipse years, so it yields a short series of four or five eclipses on the same calendar date or on two calendar dates. It is equivalent to 110 "hollow months" of 29 days and 125 "full months" of 30 days.

;Semanex: Equal to a whole number of weeks plus a hundredth of a day, so consecutive eclipses of the cycle are usually on the same day of the week. Each eclipse in this period is a member of a preceding saros series, always occurring on alternating nodes.<ref name=vanGent/>

;Thix: This eclipse cycle is just over 36 tzolk'ins, lasting 317 lunations. Each eclipse in this period is followed by an eclipse 4 saros series' later, always occurring on the same node.<ref name=vanGent/>

;[[Inex]]: Very convenient in the classification of eclipse cycles. One inex after an eclipse, another eclipse takes place at the opposite latitude. Inex series, after a sputtering beginning, go on for many thousands of years giving eclipses every 29 years minus 20 days, or 21 days if the last year has 366 days. Eighteen inex cycles (see "Basic period") are equal to 520.9996 [[Julian year (astronomy)|Julian years]] so an inex is {{sfrac|28|17|18}} Julian years. The inex cycle is the cycle that produces the highest number of eclipses while it lasts. Inex series 30 first produced a solar eclipse in saros series -245 (in 9435 BC), has been producing eclipses every 29 years since saros series -197 (in 8045 BC), and will continue long past AD 15,000,<ref name=Panorama/> by which time it will have produced 707 consecutive eclipses. The name was introduced by [[George van den Bergh]] in 1951.<ref name=vanGent/>

;Exeligmos: A triple saros, with the advantage that it has nearly an integer number of days, so the next eclipse will be visible at locations near the eclipse that occurred one exeligmos earlier, in contrast to the saros, in which the eclipse occurs about 8 hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier. Ptolemy in the [[Almagest]] mentions it after discussing what we now call the saros, and says that it is called the exeligmos (ἐξελιγμός, meaning "unrolling").

;Aubrey cycle: Named for the calculation of eclipses measured with [[Aubrey holes]], located at [[Stonehenge]]. With 1385 fortnights, eclipses alternate between lunar and solar in 56 years minus 3.5 days.<ref name=vanGent/>

;Unidos: Very close to 65 years. Equals 67 lunar years and exceeds 65 [[Julian year (astronomy)|Julian years]] by only 1.3 days (1.8 days over 65 average Gregorian years). Name suggested by Karl Palmen in that 2 saros are added over an inex.<ref name=vanGent/> A period of three Unidos (195 years, a "Trihex") is quite close to both a whole number of anomalistic years and a whole number of anomalistic months, which means the interval between two eclipses is quite constant.

;[[Callippic cycle]]: Originally defined as 4 Metonic cycles minus one day or precisely 76 years of {{frac|365|1|4}} days. In the table, taken as 940 synodic months, equivalent to 441 hollow months and 499 full months. This cycle, though useful for example in the calculation of the [[date of Easter]], can produce at most two solar eclipses (both partial) and at most two lunar eclipses (both penumbral). The Callipic cycle is 20 octons, and series of octons often produce only 21 eclipses, so only the first and the last of such a series are separated by a Callipic cycle. Most eclipses are not followed by another eclipse 940 lunations later, but rather 939 lunations later (two inex and a saros), which comes near an integer number of draconic months, producing similar eclipses. This is called a Short Callippic Period.<ref name=vanGent/>

;Triad: A triple inex, with the advantage that it has nearly an integer number of anomalistic months, which makes the circumstances between two eclipses one Triad apart very similar, but at the opposite latitude. Almost exactly 87 calendar years minus 2 months. The triad means that every third saros series will be similar (central eclipses mostly total or mostly annular for example). Solar saros [[Solar Saros 130|130]], [[Solar Saros 133|133]], [[Solar Saros 136|136]], [[Solar Saros 139|139]], [[Solar Saros 142|142]] and [[Solar Saros 145|145]], for example, all produce mainly total eclipses when they are central, because the moon is close to perigee. In fact, at the solar eclipse of October 17, 1781, which was in saros series 130 and inex 50, was both very central and at perigee.<ref name=Valsecchi/> But this repetition is not perfect. In about 2460 years, the above-mentioned series, 130, 133... (equivalent to 1 modulo 3) will give central solar eclipses that are annular, near apogee. In about 820 years central lunar eclipses, but not solar ones, will be near perigee every three saros series, and in around 1640 years the solar saros series with index equivalent to 2 modulo 3 will give central eclipses near perigee.<ref name=Duke/>

;Quarter Palmen cycle: Named after Karl Palmen in that a saros is subtracted from 4 inex. Each eclipse is followed by an eclipse 4 saros series later, always occurring at the same node. It equals 97 years 9 months or 1209 lunations.<ref name=vanGent/>

;Mercury cycle: Equals approximately 353 synodic periods of [[Mercury (planet)|Mercury]],<ref>[http://mreclipse.com/SENL/SENL9902/SENL902bd.htm SE Newsletter February 1999],</ref> so that eclipses synchronize with the timing of Mercury's position in its orbit during each period, equaling 112 years minus one week or 1385 lunations.<ref name=vanGent/>

;Tritrix: Equals 3 inex plus 3 saros, which is 140 years 11 months or 1743 lunations, always occurring on alternating nodes.<ref name=vanGent/> The tritrix is very close to a whole number of anomalistic months ((1867.9970) and close to a whole number of anomalistic years, which means the interval between two eclipses is quite constant. Two tritrix minus a saros (3263 lunations) is even closer to a whole number of anomalistic months (3497.0018), being exactly thirteen seventeenths of the [[Hipparchic cycle|Cycle of Hipparchus]] (see below).

;de la Hire cycle: A sextuple inex, adopted by [[Philippe de La Hire|Phillippe de la Hire]] in his Tabularum Astronomicarum in 1687. It equals 6 inex periods, which is 173 years and around 8 months, or 2148 lunations, equaling 179 lunar years, always occurring on the same node at nearly an integer number of anomalistic months, as it equals 2 triads.<ref name=vanGent/>

;Trihex: Equals 3 inex plus 6 saros, lasting 195 Julian years and 4 days or 2412 lunations, equaling 201 lunar years, always occurring at alternating nodes. Just two days over a whole number of anomalistic years and near a whole number of anomalistic months, which means the interval between two eclipses is quite constant.

;Lambert II cycle: An eclipse cycle in which eclipses occur in similar circumstances, according to [[Johann Heinrich Lambert]] in 1765. (The "Lambert I cycle is what we also call the inex.) Very close to a half-integer number of draconic months. It equals about 278 and a half years.<ref name=vanGent/>

;Macdonald cycle: An eclipse cycle equal to 299 years and about ten and a half months, always occurring on the same node. Peter Macdonald found that a series of eclipses of especially long duration visible from Britain occurs with this interval in the period AD 1 to 3000.<ref name=vanGent/> A Macdonald series has around ten eclipses and lasts about 3000 years. All or most are on the same day of the week, since the interval is only about an hour less than a whole number of weeks and the length is fairly constant becauses the anomaly of the moon is almost constant.

;Utting cycle: The seventh convergent in the continued fractions development between the ratio of the eclipse year and synodic month, if this ratio is approximated as between 2.17039173 and 2.17039179. Discussed by James Utting in 1827.<ref name=vanGent/>

;Selebit: An eclipse cycle where the number of eclipse years (354.5) closely matches (by chance) the number of days in a lunar year (354.371). It equals approximately 336 years 5 months 6 days or 4161 lunations. It is a convergent in the continued fractions development of the ratio between the eclipse year and the synodic month, giving a series of eclipses one selebit apart a life expectancy of thousands of years.

;[[Hipparchic cycle|Cycle of Hipparchus]]: Not a long-lasting eclipse cycle, but [[Hipparchus]] constructed it to closely match an integer number of synodic and anomalistic months, years (345), and days. Because of it being close to a whole number of both anomalistic months and anomalistic years, its length is always within about an hour of 126007 days and half an hour. (See graphs lower down of semester and Hipparchic cycle.) This means that at the time of the second eclipse same side of the earth will be facing the sun as at the first eclipse (but the value of gamma will be different). By comparing his own eclipse observations with Babylonian records from 345 years earlier, Hipparchus could verify the accuracy of the various periods that the Chaldeans used. Ptolemy points out that dividing it by 17 still gives a whole number of synodic months (251) and anomalistic months (269), but this is not an eclipse interval because it is not near a whole or half integer number of draconic months.

;Square year: An eclipse cycle where the number of solar years (365.371) closely matches (by chance) the number of days in 1 solar year (365.242). Lasting 365 years 4.5 months or 4519 lunations. It is the eighth convergent in the continued fractions development of the ratio between the eclipse year and the synodic month, giving a series of eclipses one square year apart a life expectancy of thousands of years. Many eclipses of our day belong to "square year" series or selebit series that have been going for over 13,000 years, and many will continue for over 13,000 years.<ref name=Panorama>See Panorama of Quaglia and Tilley.</ref><ref name=vanGent/>

;Gregoriana: Known for returning toward the same day of the week and Gregorian calendar date, as approximately an integer number of years, months, and weeks, are achieved, usually moving only a quarter day later in the Gregorian calendar.<ref name=vanGent/><ref>[https://earthsky.org/space/how-often-do-we-have-a-march-equinox-solar-eclipse/ How often does a solar eclipse happen on the March equinox?],</ref>

;Hexdodeka: Equal to six Unidos or two Trihex. Useful for giving accurate calculations of the timing of lunisolar syzygies.<ref name=vanGent/>

;Grattan Guinness cycle: The shortest cycle that gives eclipses on the same date (more or less) in both the Gregorian and in a 12-month lunar calendar, because it is almost exactly a whole number of Gregorian years (391.00029) as well as being exactly 403 12-month lunar years. Discovered by [[Henry Grattan Guinness]] in a speculative interpretation of {{Bibleverse|Revelation|9:15}}.<ref>[https://books.google.com/books?id=X5sVwabpBt4C&pg=PA197 Eminent Lives in Twentieth-century Science & Religion],</ref><ref name=vanGent/>

;Hipparchian: Fourteen inex plus two saros. The [[Almagest]] attributes this cycle to [[Hipparchus]]. George van den Bergh called it the "Long Babylonian Period" or the "Old Babylonian Period", but there is no evidence that the Babylonians were aware of it.<ref name=vanGent/>

;Basic period: Achieves nearly an integer number (521) of [[Julian year (astronomy)|Julian years]], [[anomalistic year]]s (521 anomalistic years minus 5 days), and weeks (27185 weeks plus 0.1 day), leading to eclipses on the same day of the Julian calendar and week.<ref name=vanGent/>

;Chalepe: Equals 18 inex plus 2 saros, therefore 557 years plus about 1 month.<ref name=vanGent/>

[[File:Inex and saros for tetrads between AD 1000 and 2500.png|thumb|400px|Inex and saros for tetrads between AD 1000 and 2500, showing the tetradia]]
;Tetradia: Sometimes 4 total lunar eclipses occur in a row with intervals of 6 lunations (one semester) between them, and this is called a [[Tetrad (astronomy)|tetrad]]. [[Giovanni Schiaparelli]] noticed that there are eras when such tetrads occur comparatively frequently, interrupted by eras when they are rare. This variation takes about 6 centuries. [[Antonie Pannekoek]] (1951) offered an explanation for this phenomenon and found a period of 591 years. Van den Bergh (1954) from [[Theodor von Oppolzer]]'s ''Canon der Finsternisse'' found a period of 586 years. This happens to be an eclipse cycle; see Meeus [I] (1997). The phenomenon is related to the elliptical orbit of the Earth, as explained below. Recently Tudor Hughes explained that secular changes in the [[orbital eccentricity|eccentricity]] of the Earth's [[orbit (celestial mechanics)|orbit]] cause the period for occurrence of tetrads to be variable, and it is currently about 565 years; see Meeus III (2004) for a detailed discussion. The Tetradia period also shows up in the distance between eras in which there are pairs of (non-consecutive) eclipses seven months apart, or eras in which there are more pairs of eclipses one month apart, or eras in which there are saros series in which gamma is fairly constant for many decades, or eras with more low-gamma eclipses.<ref name=Duke/>

;Hyper exeligmos: Equals 12 "Short Callippic Periods" (each a month shorter than a Callipic cycle), or 12 Callippic cycles minus 1 lunar year, so therefore a bit over 911 years or 11268 lunations, which is 939 lunar years. First mentioned by Alexander Pogo in 1935.<ref name=vanGent/>

'''The next nine cycles, Cartouche through Accuratissima,''' are all similar, being equal to 52 inex periods plus up to two triads and various numbers of saros periods. This means they all have a near-whole number of anomalistic months. They range from 1505 to 1841 years, and each series lasts for many thousands of years.

;Cartouche: Equals 52 inex, therefore 1505 years and between 1 and 2 months. Eclipses in this period occur at a similar distance as nearly an integer number of anomalistic months are achieved.<ref name=vanGent/>

;Palaea-Horologia: Equals 55 inex plus 3 saros, which is over 1646 years. Useful for calculating the timing of eclipses. Close to a whole number of anomalistic months. A series lasts tens of thousands of years.<ref name=vanGent/>

;Hybridia: Equals 55 inex plus 4 saros, one saros more than a Palaea-Horologia, therefore over 1664 years, near an integer number of anomalistic months, therefore having similar properties, but at the opposite latitude.<ref name=vanGent/>

;Selenid: One saros more than a Hybridia. The name for eclipse cycles useful for calculating the magnitudes of eclipses in the [[3rd millennium]]. George van den Bergh first mentioned a period of 55 inex plus 5 saros (over 1682 years) before mentioning a period of 95 inex plus 11 saros (over 2948 years) in 1951.<ref name=vanGent/>

;Proxima: Equals 58 inex plus 5 saros, therefore a bit less than 1769 years, always occurring at the same node and toward an integer number of draconic and anomalistic months and weeks, making the circumstances of each eclipse a proxima apart similar in character.<ref name=vanGent/>

;Heliotrope: Equals 58 inex plus 6 saros, one saros more than a Proxima, therefore about 1787 years. Useful for calculating the longitudinal positions of the central lines of eclipses on Earth's surface near an integer number of years (1786.954 Julian years, 1786.991 Gregorian).<ref name=vanGent/>

;Megalosaros: Equals 58 inex plus 7 saros (one saros more than a Heliotrope), which is 95 Metonic cycles, or 95 saros plus 95 lunar years, or 100 saros plus 25 lunations, or a bit over 1805 years, always occurring on the same node, and revealing the Metonic cycle's mismatch from 19 years as 95 repeats accumulates the mismatch to about three years. The extra 25 lunations are needed because 100 saros cycles exceeds the life expectancy of a saros series.<ref name=vanGent/><ref>[https://articles.adsabs.harvard.edu//full/1901Obs....24..379C/0000382.000.html 29-Year-Eclipse-Cycle],</ref>

;Immobilis: Equals 58 inex plus 8 saros (one saros more than a Megalosaros), which is exactly 1879 lunar years. Always occurs on the same node. Very close to a whole number of anomalistic months, although 43 inex minus 5 saros (14279 months, 1154.5 years) is even closer.<ref name=vanGent/>

;Accuratissima: Equals 58 inex plus 9 saros (one saros more than an Immobilis), therefore 1841 years 1 month or 22771 lunations, which is presently about an hour more than a whole number of weeks, allowing eclipses to occur the same day of the week. Because of the slowing of the Earth's rotation, the length of the Accuratissima will become exactly equal to a whole number of days or weeks in around AD 2100, meaning that an eclipse around AD 1200 will be repeated at the same time of day on the same day of the week 1841 years later. The Accuratissima is also useful for calculating the magnitude and character of eclipses.<ref name=vanGent/> An Accuratissima plus a Tritrix plus a saros makes an eclipse cycle 1.8 days short of 2000 Julian years, or 13.2 days longer than 2000 Gregorain years. It is only half a day less than a whole number of anomalistic months, whereas the Accuratisssima is only 0.2 days short of a whole number of anomalistic months.

;Mackay cycle: Equals 76 inex plus 9 saros, therefore 2362 years and about a month, always occurring on the same node. Mentioned by A. Mackay in the 1800s.<ref name=vanGent/>

;Horologia: Equals 110 inex plus 7 saros, therefore 3310 years and about 2 months, always occurring on the same node. It is useful for calculating the timing and magnitudes of eclipses as they are approximately an integer number of draconic and anomalistic months and weeks apart (172,715.97 weeks), leading to similar eclipses in character and week timing.<ref name=vanGent/>