Editing Julian day (section)

===Julian Period <span class="anchor" id="Julian period"></span>===
The ''Julian day number'' is based on the ''Julian Period'' proposed by [[Joseph Justus Scaliger|Joseph Scaliger]], a classical scholar, in 1583 (one year after the Gregorian calendar reform) as it is the product of three calendar cycles used with the Julian calendar:

{{block indent|1=28 ([[Solar cycle (calendar)|solar cycle]]) × 19 ([[Metonic cycle|lunar cycle]]) × 15 ([[indiction|indiction cycle]]) = 7980 years}}

Its epoch occurs when all three cycles (if they are continued backward far enough) were in their first year together. Years of the Julian Period are counted from this year, {{nowrap|4713 BC}}, as {{nowrap|year 1}}, which was chosen to be before any historical record.<ref>Richards 2013, pp. 591–592.</ref>

Scaliger corrected chronology by assigning each year a tricyclic "character", three numbers indicating that year's position in the 28-year solar cycle, the 19-year lunar cycle, and the 15-year indiction cycle. One or more of these numbers often appeared in the historical record alongside other pertinent facts without any mention of the Julian calendar year. The character of every year in the historical record was unique – it could only belong to one year in the 7980-year Julian Period. Scaliger determined that 1&nbsp;BC or year&nbsp;0 was Julian Period {{nowrap|(JP) 4713}}. He knew that 1&nbsp;BC or year&nbsp;0 had the character 9 of the solar cycle, 1 of the lunar cycle, and 3 of the indiction cycle. By inspecting a 532-year [[Paschal cycle]] with 19 solar cycles (each of 28&nbsp;years, each year numbered 1–28) and 28 lunar cycles (each of 19&nbsp;years, each year numbered 1–19), he determined that the first two numbers, 9 and 1, occurred at its year&nbsp;457. He then calculated via [[modulo operation|remainder division]] that he needed to add eight 532-year Paschal cycles totaling 4256&nbsp;years before the cycle containing 1&nbsp;BC or year&nbsp;0 in order for its year&nbsp;457 to be indiction 3. The sum {{nowrap|4256 + 457}} was thus JP&nbsp;4713.<ref>Grafton 1975, p. 184</ref>

A formula for determining the year of the Julian Period given its character involving three four-digit numbers was published by [[Jacques de Billy]] in 1665 in the ''[[Philosophical Transactions of the Royal Society]]'' (its first year).<ref>de Billy 1665</ref> [[John Herschel|John F. W. Herschel]] gave the same formula using slightly different wording in his 1849 ''Outlines of Astronomy''.<ref>Herschel 1849</ref>

{{blockquote|text=Multiply the ''Solar'' Cycle by 4845, and the ''Lunar'', by 4200, and that of the ''Indiction'', by 6916. Then divide the Sum of the products by 7980, which is the ''Julian Period'': The ''Remainder'' of the Division, without regard to the ''Quotient'', shall be the year enquired after.|author=Jacques de Billy}}

[[Carl Friedrich Gauss]] introduced the [[modulo operation]] in 1801, restating de Billy's formula as: 
{{block indent|1=Julian Period year = (6916''a'' + 4200''b'' + 4845''c'') MOD 15×19×28}}
where ''a'' is the year of the indiction cycle, ''b'' of the lunar cycle, and ''c'' of the solar cycle.<ref>Gauss 1966</ref><ref>Gauss 1801</ref>

[[John Collins (mathematician)|John Collins]] described the details of how these three numbers were calculated in 1666, using many trials.<ref>Collins 1666</ref> A summary of Collin's description is in a footnote.<ref>
{| class=wikitable
|+ Calculation of 4845, 4200, 6916 <br> by Collins
|-
| || align=center | ''Try'' 2+ until ||
|-
| align=center | {{sfrac|7980|28}} = 19×15 = 285 
| align=center | {{sfrac|285×''Try''|28}} = <br> {{nowrap|remainder 1}} 
| align=center | 285×17 = 19×15×17 = 4845
|-
| align=center | {{sfrac|7980|19}} = 28×15 =  420 
| align=center | {{sfrac|420×''Try''|19}} = <br> {{nowrap|remainder 1}} 
| align=center | 420×10 = 28×15×10 = 4200
|-
| align=center | {{sfrac|7980|15}} = 28×19 = 532 
| align=center | {{sfrac|532×''Try''|15}} = <br> {{nowrap|remainder 1}} 
| align=center | 532×13 = 28×19×13 = 6916
|}</ref>
Reese, Everett and Craun reduced the dividends in the ''Try'' column from 285, 420, 532 to 5, 2, 7 and changed remainder to modulo, but apparently still required many trials.<ref name="Reese, Everett and Craun 1981">Reese, Everett and Craun 1981</ref>

The specific cycles used by Scaliger to form his tricyclic Julian Period were, first, the indiction cycle with a first year of 313.{{efn|All years in this paragraph are those of the Anno Domini Era at the time of Easter}}<ref>Depuydt 1987</ref> Then he chose the dominant 19-year Alexandrian lunar cycle with a first year of 285, the [[Era of Martyrs]] and the Diocletian Era epoch,<ref>Neugebauer 2016, pp. 72–77, 109–114</ref> or a first year of 532 according to [[Dionysius Exiguus]].<ref name="Dionysius Exiguus 2003/525">Dionysius Exiguus 2003/525</ref> Finally, Scaliger chose the post-Bedan solar cycle with a first year of 776, when its first quadrennium of [[concurrent (Easter)|concurrent]]s, {{nowrap|1 2 3 4}}, began in sequence.{{efn|1=The concurrent of any Julian year is the weekday of its March{{nbsp}}24, numbered from Sunday=1.}}<ref>''De argumentis lunæ libellus'', col. 705</ref><ref>Blackburn and Holford-Strevens, p. 821</ref><ref>Mosshammer 2008, pp. 80–85</ref> Although not their intended use, the equations of de Billy or Gauss can be used to determined the first year of any 15-, 19-, and 28-year tricyclic period given any first years of their cycles. For those of the Julian Period, the result is AD&nbsp;3268, because both remainder and modulo usually return the lowest positive result. Thus 7980&nbsp;years must be subtracted from it to yield the first year of the present Julian Period, −4712 or 4713&nbsp;BC, when all three of its sub-cycles are in their first years.

Scaliger got the idea of using a tricyclic period from "the Greeks of Constantinople" as Herschel stated in his quotation below in [[#Julian day numbers|Julian day numbers]].<ref name="Herschel 1849, p. 634">Herschel 1849, p. 634</ref> Specifically, the monk and priest Georgios wrote in 638/39 that the Byzantine year 6149&nbsp;AM (640/41) had indiction 14, lunar cycle 12, and solar cycle 17, which places the first year of the [[Byzantine Era]] in 5509/08&nbsp;BC, the Byzantine Creation.<ref>Diekamp 44, 45, 50</ref> Dionysius Exiguus called the Byzantine lunar cycle his "lunar cycle" in argumentum 6, in contrast with the Alexandrian lunar cycle which he called his "nineteen-year cycle" in argumentum 5.<ref name="Dionysius Exiguus 2003/525"/>

Although many references say that the ''Julian'' in "Julian Period" refers to Scaliger's father, [[Julius Caesar Scaliger|Julius Scaliger]], at the beginning of Book V of his ''{{lang|la|Opus de Emendatione Temporum}}'' ("Work on the Emendation of Time") he states, "{{lang|la|Iulianam vocauimus: quia ad annum Iulianum accomodata}}",<ref>Scaliger 1629, p. 361</ref>{{refn|Scaliger used these words in his 1629 edition on p. 361 and in his 1598 edition on p. 339. In 1583 he used "{{lang|la|Iulianam vocauimus: quia ad annum Iulianum duntaxat accomodata est}}" on p. 198.}} which Reese, Everett and Craun translate as "We have termed it Julian because it fits the Julian year".<ref name="Reese, Everett and Craun 1981"/> Thus ''Julian'' refers to the [[Julian calendar]].