# Talk:Astronomical year numbering

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## Year zero

+I have consulted with a local astronomers group and I have received the reply which unequivocally states that there was no year zero. They also referred me to the topic "Julian Day Numbers" for confirmation. Quizzically that article includes a year zero with further reference to Scalinger's explanation. In order to correct for an obvious problem Scalinger changed his original premise to account for the discrepancy. This provides a second misapplication of the contentious practice of placing Year Zero AD equal to the 1st year BC24.242.42.17 (talk) 17:56, 13 September 2008 (UTC)

+I believe I have been requested to clarify my comments. Scalinger set up a starting point for counting years at 4713. But since he also set Year zero AD equal to the first year BC he found that he had too many years. To correct for this complication he changed his magic number to 4712. Cassini and La Hire also invoked the equality between Year 0 AD and the first year BC. The sequence Year 1 BC, Year 0 AD, Year 1 AD gives an incorrect interval between years. J.C. Lerman's fence line analogy with index zero gives us the proper interval -1,0,+1.24.242.42.17 (talk) 00:18, 14 September 2008 (UTC)Sam Hastings

Scaliger (no 'n') selected only one year as the first year of his Julian period and never changed his mind. The modern designations for that year are either 4713 BC or −4712, depending on whether the modern writer excludes or includes a year zero within his own timeline. Scaliger himself never used either designation nor did he ever use a year zero. Scaliger referred to that first year by stating that the year before the first year of our era was the year 4713 of the Julian period.
The article should make it clear that astronomers use the proleptic Gregorian calendar, if true. If not, this articles has little content. — Arthur Rubin (talk) 15:48, 2 March 2009 (UTC)
Astronomers use the Julian calendar before 1582 and the Gregorian calendar thereafter as exemplified by Cassini (1740).[1]
I believe if one examines the quote from Cassini's "Astronomical Tables" closely it will become abundantly clear that there has been a misinterpretation of the quote. Both of the subsequent "which" statements refer to "The year 0 is that in which one supposes that Jesus Christ was born". Thus, "several chronologists mark 1 before the birth of Jesus" as that year whereas Cassini marked 0 as year 0. He did not mark 1 BC as 0. Now we have unequivocally the sequence -1,0,+1 (first negative year, point 0, first positive year) with no need to laboriously try to justify an imaginary "missing year zero".24.242.44.239 (talk) 20:43, 14 July 2009 (UTC)
Regardless of any interpretation of the quotation from Cassini (quote in French), his tables of the mean motion of the Sun, Moon, and planets indicate that year 0 as a bissextile year (B), that is, the year has a day labeled 29 February making it 366 days long. Cassini stated that the mean longitude given is at noon 1 January (1er Janvier à midi) of bissextile years, hence two days within the year are identified by name so it must be a full year, not a point in time. With a full year 0, all centuries, both BC and AD, contain 100 years beginning with a centennial year (nn00) and ending with the year immediately before a centennial year (nn99 AD or nn01 BC). (Note that Cassini used "avant Jesus-Christ" or BC for all years before his year 0 and "après Jesus-Christ" or AD for all years thereafter, in contrast to modern astronomers who use − for years before year 0 and + for years after year 0. This means that his BC years are shifted by one year from modern BC years, that is, his year 1 BC, the year immediately before year 0, is the modern year 2 BC.) If 0 were a point in time then its indicated mean longitude of the Sun, 9Sig8D1M3S (the superscripts indicate zodiacal signs of 30 degrees each, degrees, minutes and seconds) would be at the beginning of year 1 causing the first century AD to have only 99 years, because Cassini stated that the mean longitude of the Sun for all centennial years, including year 100, is given at the beginning of that year, so he did not include the rest of year 100 in his first century AD.
Furthermore, the given mean longitude of the Sun at the beginning of all centennial years before 1600 decreases by either 45M41S or 45M42S every century. Because all centuries have 100 years, all differences include 100 revolutions of 12 zodiacal signs or 360° each, that is, the Sun's mean longitude increases by 100Rev0Sig0D45M41.5S where Rev indicates revolutions. If the first century AD had only 99 years, then the Sun's mean longitude would increase by only 99 revolutions, specifically 99/100 of 100Rev0Sig0D45M41.5S or 99Rev0Sig0D45M14.1S. The table does not show this difference, so the first century AD cannot have only 99 years and 0 must be a full year to give the first century 100 years (0–99). — Joe Kress (talk) 05:27, 16 July 2009 (UTC)
Joe - Perhaps my suggestion that the Cassini quote has been misinterpreted needs clarification. Let me first eliminate the reference to BC years and present the following: "The year 0 is that year in which one supposes that Jesus Christ was born which we marked 0, so that the sum of the years before and after Jesus Christ gives the interval between these years, and where numbers divisible by 4 mark the leap years as so many before or after Jesus Christ". This incomplete quote can stand alone, giving the required interval of -1,0,+1. Thus the first century runs from January 1, 100 to December 31st 099. With regard to the eliminated quote I refer you to my previous comments. Year 1 BC is not a viable year in its usage next to Year 0 and Year AD 1. The other possibility is that "several chronologists" referred to the proposition that Jesus Christ was conceived and born in the FIRST year prior to Anno Domini 1.24.242.44.239 (talk) 18:51, 25 July 2009 (UTC)
Sorry about what I will call a typo (bad proofing?). The first century runs from Jan. 1, 000 to Dec. 31, 099. 24.242.44.239 (talk) 02:57, 26 July 2009 (UTC)
Your "incomplete quote" is unacceptable because the phrase "which several chronologists mark 1 before the birth of Jesus Christ" is critical to understanding Cassini. Its removal distorts his words to fit your view that the astronomer's year −1 is the historian's year 1 BC. I have already stated that Cassini's 1 BC (avant Jesus-Christ) is identical to modern astronomers' year −1. Furthermore, Cassini states that he inserted a year 0 so that years divisible by 4 are leap years. Your view is incompatible with this because the year that Cassini placed before 1 AD, year 0, is four years before year +4 and so must be divisible by 4 (your year −1 is not divisible by 4). His "1" cannot be translated as "first" because Cassini always places the superscript "er" after 1 (1er) if he means the ordinal "first (premier)".
He titles Table III as "des e'poques des moyens mouvements du Soleil = the epochs of the mean motion of the Sun" (epoch in astronomy means a precise point in time) and gives precise values for the "longitude moyenne = mean longitude" and "lieu de l'Apogée = place of the apogee". He states on the page immediately before the quote you want to limit that these values are for 1er Janvier à midi = noon 1st January of the stated bissextile years. Hence his "Années = years" refer to these epochs (points in time), not to full years. So his interval is also between this points, and the interval between noon 1 January year 0 and noon 1 January year 100 includes a 366-day year 0, but does not include any part of year 100 (Cassini's years began at noon, not at midnight). Thus the interval between years 0 and 100 is 0 + 100 = 100 years exactly (Cassini adds his years). Similarly, the values given for 100 BC (avant Jesus-Christ) are applicable only to noon 1 January year 100 BC (−100), so the interval between noon 1 January year 100 BC and noon 1 January year 0 is also 100 + 0 = 100 years exactly, because it includes a 366-day year 100 BC but no part of year 0. Modern astronomers would label Cassini's 100 avant Jesus-Christ as −100 and 100 après Jesus-Christ as +100, and state that the intervals are the differences between similar points in time, not years, are 0 − (−100) = 100 years exactly and +100 − 0 = 100 years exactly (placing more positive years in the first position (minuend) and more negative years in the second position (subtrahend)).
To emphasize that modern astronomers treat 0 as a full 366-day year see the last year given in Fred Espenak's Phases of the Moon: −99 to 0 (100 to 1 BCE) which assigns 50 lunar phases to year 0, so it cannot be a single point in time. Espenak also states on the same page that "the year '0' here corresponds to '1 BCE'", but does not discuss intervals between years. — Joe Kress (talk) 08:45, 30 July 2009 +
Joe - We can go round and round in this discussion repeating ad infinitum our particular positions. Let us clear up, if possible, the question of what year in the passage of time the cardinal year 1 BC actually refers to. It is my contention that it appears after year 0 BC at the beginning of all calendars existing at the time Dionysius established his calendar and has no place in supporting the Cassini quote.24.242.44.239 (talk) 19:18, 30 July 2009 (UTC)
When Dionysius established his method of counting years, the standard method of identifying years in the Roman Empire was to name the consuls who entered office on 1 January, that is, years were named, not numbered. That had been the standard method for almost 1000 years in those areas under Roman domination. For example, the Roman historian Suetonius stated that when Julius Caesar began to conduct all affairs of state alone, a year which should have been identified as when Bibulus and Caesar were consuls was sometimes sarcastically identified as the year of Julius and Caesar.[2] Renaissance scholars deduced that that year was 59 BC.
I've already given the subsequent history of BC/AD years at Year zero#Historians. Your argument does not rest with Dionysius since he never used BC years nor with calendars in existence at that time because none of them used BC years. I don't think you have any disagreement with the historical BC/AD scale of years constructed by Renaissance scholars and used ever since, which places 1 BC immediately before year AD 1 without any year 0. Your argument rests solely with modern astronomers from Cassini to the present and where they place their year 0 relative to historical BC/AD years. Cassini confused the issue by forcing a year 0 between 1 BC and AD 1. Astronomers since 1850 avoided this confusion by not using the terms "BC" and "AD", replacing them with negative and positive years. So no year "0 BC" is used by either historians or astronomers. Astronomers since 1850 have used a year 0 without any era designation. Modern astronomers have never used either 0 BC or 0 AD, and historians certainly don't use a year 0 in any form.
Your question was explicitly answered by Fred Espenak when he stated "the year '0' here corresponds to '1 BCE'".[3] This places year 0 between 2 BCE and 1 CE, renaming them as −1 and +1, creating the scale −1, 0, +1 corresponding to 2 BCE, 1 BCE, 1 CE, or if you prefer, to 2 BC, 1 BC, AD 1. The article should explain that Cassini's intervals are not between years at the ends of any interval, but between arbitrary points in time within those end years. — Joe Kress (talk) 01:19, 31 July 2009 (UTC)
The explanation of Jean Meeus in Astronomical Algorithms (p.60) follows:
There is a disagreement between astronomers and historians about how to count the years preceding year 1. In this book, the 'B.C.' years are counted astronomically. Thus, the year before the year +1 is the year zero, and the year preceding the latter is the year −1. The year which historians call 585 B.C. is actually the year −584.
The astronomical counting of the negative years is the only one suitable for arithmetical purpose. For example, in the historical practice of counting, the rule of divisibility bt 4 revealing Julian leap-years no longer exists; these years are, indeed, 1, 5, 9, 13, ... B.C. In the astronomical sequnce, however, these leap-years are called 0, −4, −8, −12, ..., and the rule of divisibility by 4 subsists.
Meeus does not rely on year numbers to calculate intervals. Instead he relies on Julian centuries (each having 36,525 days) and Julian days (pp.125–126). First the two dates (including the time of day) between which the interval is desired are converted to Julian days, a count of days and fractions thereof since noon 1 January 4713 BCE (Julian). In particular, noon (12:00) 1 January 2000 (Gregorian) is JD 2451545.0 (J2000.0). If JD0 is the initial epoch and JD is the final epoch, then ${\displaystyle t={\tfrac {JD-JD_{0}}{36525}}}$ is the interval in Julian centuries. A specific kind of interval which is widely used is the number of Julian centuries since J2000.0: ${\displaystyle T={\tfrac {JD-2451545.0}{36525}}}$Joe Kress (talk) 05:27, 31 July 2009 (UTC)
Joe: In my remarks of July 30 I suggested that we give our positions with regard to our interpretation of the Year 1 BC (or 1 BCE). You say Fred Espenak answered my question explicitly. He did not; he simply states the same sequence to which I have shown to be unsupportable. You then set up two sequences - 1) 2BCE, Year 0, 1CE, and 2) 2BCE, 1BCE, 1CE. In both sequences your years BCE are cardinal years and are obviously running backwards. You have to flip the sequences with regard to years before the Christian era. Time marches FORWARD relentlessly. Sequence 1 becomes incomplete because 2BCE is retired to its rightful place in antiquity. Sequence 2 should be ordinal: 2nd negative year, 1st negative year, (point 0), 1st positive year. With reference to your comment about Cassini's use of "iere" for ordinal numbers I note that he simply "marked" the years with numerals (0 aqnd 1). I know you have said Cassini used 1BC as ordinal as well as cardinal. You also mentioned that he "forced" a year 0 into the sequence 1BC, 0, AD1 used in this article. Did they have bissextile years in the eras before the Common Era? Of course not. With AD1 being the first year AD we now have the desired leap year AD0 and the required years divisible by 4 following.24.242.44.239 (talk) 16:59, 4 August 2009 (UTC)
You have made an astonishing statement concerning the numbering of years used by all historians which have nothing to do with astronomical year numbering. If this statement represents your views, then we have no common ground for discussing any changes made by astronomers.
Backwards numbering of years before the Christian era is incorrect.
For example, the article Julius Caesar states that he was born in 100 BC and assassinated in 44 BC. Obviously, the years in between must be numbered backwards. — Joe Kress (talk) 20:07, 4 August 2009 (UTC)
Joe - I have been of the opinion that historians use the sequence: -1,0,+1. Thus backwards numbering of years before the Christian era is justified and is not INCORRECT as you state. It depends upon the fact that these numerals refer to instantaneous points in time (ordinal numbers). You, yourself, have said that astronomers often take this approach. The sequence you use to justify a Year 1 BC equal to the Year 0 AD (cardinal) demands that the years run backwards and is thus MORE ASTONISHING. I suggest that you give serious consideration to the fence line hypothesis with index 0. Each post represents a point in time (-1,0,+1) (ordinal-counting years). The first year AD is Year 0 AD a leap year with every fourth year thereafter becoming a leap year. And I repeat; the first negative year can only be the last year of any existing calendar at point 0. I do not mind calling that year 1 BCE as long as we recognize it as an ordinal year.24.242.44.239 (talk) 17:21, 5 August 2009 (UTC)
Historians DO NOT use the sequence -1,0,+1. Just constult any discussion concerning some person or event that happened before the Christian era on Wikipedia or the Internet in general. Historians always assign a named era to the numbers, such as BC or BCE. Historians NEVER use bare negative numbers. Hence we have no common basis for any further discussion. Bye. — Joe Kress (talk) 20:02, 5 August 2009 (UTC)
Hello again Joe -I suggest you check www.spiritus-temporis.com/year-zero. You can't stop me with a mis-statement.24.242.44.239 (talk) 20:42, 5 August 2009 (UTC)
I have pointed out that the sequence used in this article (Year 1 BC equals Year 0 AD) runs backwards and is therefore invalid. Since you put Year 2 BC after Year 1 BC (in a backwards direction) you have established the necessary conclusion that Year 1BC starts at January 1st, Year 1 BC and runs backward to December 31st, 1 BC. But Year 0 AD starts at January 1st, Year 0 AD and runs forward to December 31st, Year 0 AD. It is obvious that the two cardinal years cannot be equivalent. Reversing the direction of Year 1 BC (and of Year 2 BC) by invoking the concept of negative years results in the placing of Year 2 BC after Year 1 BC (in the forward direction) and has the same effect as "flipping" the sequence as I have advocated.24.242.44.239 (talk) 21:44, 6 August 2009 (UTC)
Joe: The article "Astronomical year numbering" is still incorrect. I believe that I have successfully demonstrated that the sequence 2BC, 1BC,0,1AD, 2AD, etc. as employed in the article must be construed as a series of cardinal numbers. As such the series has time going bqckwards. The sequence when considered as using ordinal numbers is quite correct: 2nd year BC, 1st year BC, point 0, 1st year AD, 2nd year AD, etc. You, yourself, support the net effect of this interpretation when you state that "For normal calculation a number zero is often needed, here most notably when calculating the number of years in a period that spans the epoch, the end years need only to be subtracted from each other". Lo and behold, with the AD era starting with point 0 the first year AD is cardinal Year 0 which is bisextile and constitutes a leap year. An added benefit is that without the cardinal Year 0 the problem of losing leap years disappears. I feel sure you know what I mean by that statement.24.242.42.19 (talk) 18:02, 29 August 2009 (UTC)
Another boo-boo. The next-to-last sentence should have read "with the cardinal Year 0, etc". 24.242.42.19 (talk) 18:11, 29 August 2009 (UTC)
Joe: When I suggested the benefits of looking at your sequence as a series of ordinal numbers it didn't immediately occur to me that such a series ought to be considered very seriously. It seems to me that we are desperately trying to find a SINGLE numbering system when TWO are required to satisfy differing situations. This conclusion is reinforced by your suggestion that Cassini tried to "force" a cardinal Year 0AD to be equal to the first year BC. Attempts to prove this are difficult to accept. I believe that it is now appropriate to take a long, hard look at the fence line analogy. With index 0 this labels the posts as -1,0,+1 (ordinal numbering system) satisfying "some" astronomers. This is exactly what one arrives at by considering your sequence as ordinal numbers. The rails represent cardinal Years with the first rail following post 0 being Year 0AD. This is followed by Year 1AD (or CE). The first negative Year would be Year 1BCE. It is important to note that the negative Years BCE have no dependence upon Years BC except that the starting point is post 0. OK, now we have two complementary and independent sequences satisfying two important but different situations. The only point of contention that I foresee is that Year 0AD is not set equal to the first year BC (or BCE). From my contact with an astronomers group this is probably not an issue. They refer to the Julian day numbers "to determine the number of days between two arbitrary dates". I have just located my copy of those emails. I would be happy to email or snail mail a copy of my question and their reply. But I do not want to draw them into our discussion. 24.242.42.19 (talk) 03:36, 4 September 2009 (UTC)

You refuse to accept historical year numbering as used by historians and astronomical year numbering as used by astronomers, insisting that both are wrong. Both historians and astronomers use years containing 365 or 366 days—none of their years are points containing zero days. Historical years of 365/366 days are assigned cardinal numbers labeled either BC/AD or BCE/CE. The labels BC/BCE are equivalent to minus signs, so the numbers associated with them are not backwards. Astronomical years of 365/366 days are assigned cardinal numbers labeled −/+ (Cassini used BC/AD in French). Their equivalence is:

 Historical: (Jan 1 → Dec 31) 2 BC || (Jan 1 → Feb 29 → Dec 31) 1 BC || (Jan 1 → Dec 31) 1 AD Historical: (Jan 1 → Dec 31) 2 BCE || (Jan 1 → Feb 29 → Dec 31) 1 BCE || (Jan 1 → Dec 31) 1 CE Cassini: (Jan 1 → Dec 31) 1 BC || (Jan 1 → Feb 29 → Dec 31) 0 || (Jan 1 → Dec 31) 1 AD Astronomical: (Jan 1 → Dec 31) −1 || (Jan 1 → Feb 29 → Dec 31) 0 || (Jan 1 → Dec 31) +1

Joe Kress (talk) 05:48, 5 September 2009 (UTC)

Joe - It has been nearly a month since I have heard from you and I now have what I sincerely believe is the correct interpretation of the Cassini quote. He states that he marked the year in which one supposes that Christ was born is 0. That gives us year 0 AD (a leap year) and it follows that point 0 is its beginning. Obviously its ending is point +1. He also states that several chronologists mark the year in which one supposes Christ was born as year 1 BC (or -1). Thus we have the sequence -1,0,+1. Keeping in mind that these are ordinal numbers useful in counting they can easily be visualized as posts in an unending fence line as post-1, post 0 and post +1. Now it is obvious that Cassini's "so thats" are perfectly accomplished: the sum of the years before and after Jesus Christ gives the interval which is between these years and where numbers divisible by four mark the leap years BC and AD.

Remember that 1 BC is ordinal and means only the first year BC and refers only to years in calendars extant at the time Dionysius established his calendar. 24.242.45.134 (talk) 19:45, 6 October 2009 (UTC)

Joe - It should be obvious but I'm going to point out anyway that the fence line analogy applies with equal force to the rails

(cardinal years) as well. Rail +1 is the year 0 AD and rail -1 is the last year of each and every calendar in existence when Dionysius established his calendar. It is impossible and unnecessary to attempt to integrate another rail into the sequence.24.242.42.230 (talk) 17:54, 12 October 2009 (UTC)

Joe - Let's look at the fundamental issue that prevents us from reaching any kind of rapprochement. I emphatically refute your argument that the Cassini quote supports the setting of Year 0 AD as equal to the first year BC and have given my reasons for taking that position. I stand by my own argument that it is impossible and unnecessary to attempt to force another rail ( Year 0 AD ) into the sequence of cardinal years. If we remain at odds I respectfully request a third opinion.24.242.42.230 (talk) 15:18, 21 October 2009 (UTC)
Joe - Enough of this monologue - it is obvious that we have reached an impasse. I request a third opinion and since we have been discussing the topic of "Astronomical year numbering" I further request that it be given by an astronomer.24.242.42.230 (talk) 20:17, 30 October 2009 (UTC)+
1. ||Talk:Astronomical year numbering||.Disagreement about the interpretation of the quote from Cassini.18:37, 17 November 2009 (UTC)
1. ||Talk:Astronomical year numbering||.Difference of opinion regarding the interpretation of the quote from Cassini. Editor 1 interprets the quote to mean that the Year 0 AD is equal to the first year BC. Editor 2 maintains that Cassini marked the year that Christ was born as Year 0 AD (the first year AD).00:08, 18 December 2009 (UTC)
Joe I assume that my request for a second Third opinion has been denied. Therefore I assume also that I am free to continue this discussion.

Let's look at where applying the decimal numbering system to the Cassini quote leads us.

This numbering system is employed at some time by both historians and astronomers. It is ....-2,-1,0,+1,+2.......... Even though historians apparently prefer the sequence .....-1,+1 ..... they cannot escape the point zero between these years. Now Cassini marked the year that Jesus was born as 0 and "some chronologists" (undocumented) marked the year as -1. The number 0 is like Janus who looks forward and backward. The number +1 marks the end of the first year AD. which is Year 0 AD. The number -1 marks the beginning of the first year BC, which is also a leap year. Now all of the results of this analysis fit Cassini----"so that the sum of the years before and after Jesus Christ gives the interval which is between these years, and where numbers divisible by 4 mark the leap years as so many before and after Jesus Christ". Notice that Cassini used the terms "numbers" and "mark", terms suitable for the decimal numbering system. The following table illustrates these conclusions:

I hope this table helps clarify my position regarding what I consider to be the most reasonable interpretation of the quote from Cassini.I

Joe - It has been increasingly evident that my failure to produce a table supporting my interpretation of the Cassini quote just leaves me hanging. By dint of much trial and error I was able to produce the desired table in my personal sandbox. I am hopeful that it now comes through the final test unscathed.

As I feared, what worked in my personal sandbox did not work here. If anyone is interested they can simply go to "edit this page" above. There they can see an approximation of the table I used to create the table I wanted. Then go to

User:Samhastings/Sandbox and see what that approximation produced there. Why does something presented to the personal sandbox give gibberish to the Talk page? 24.242.42.230(talk) 02:57, 7 February 2010 (UTC)

In my first comments at the start of the present Talk:Year Zero (Chronologists) I was close to the answer but was so obsessed with cardinal years that I didn't stop to think about what the answer would have really been. Ordinal years provide the necessary results. That is to say consider points in time not years. We both agree that a point in time is the instantaneous moment between the end of one year and the beginning of the next. Cassini, himself, gave us the answer in a nutshell. He said he marked the year that Jesus Christ was born 0 and that some chronologists marked that year as -1. This gives us the lineup of -1,0+1. We both agree that this is correct when astronomers are counting years by adding the endpoints without regard to sign. I have already stated that point 0 is like Janus and looks both ways (+ and -). Point -1 marks the beginning of Year 0. Likewise, point 0 marks the beginning of Year 0 and point +1 marks the end of Year 0 and the beginning of Year +1.

Thus we have -1,0,+1 Remember that these are ordinal numbers. The number -1 represents the first negative year.This year starts at 0 and proceeds to -1. The first positive year starts at 0 and proceeds to +1. This numbering system is well illustrated by the rectilinear graph - the x-axis (-2,-1,0,+1,+2). So now we have the first negative year as Year 0(-) and the first positive year as Year 0(+). This gives us 2 leap years zero spanning the epoch and the attendant leap years every four years.24.242.42.230(talk) 02:49, 16 March 2010 (UTC)+

Sorry! Instead of "Point -1 marks the beginning of Year 0". I meant to say that "Point 0 marks the beginning of Year -1".24.242.42.230 (talk) 02:41, 20 March 2010 (UTC)
Please disregard my "Sorry!" comment above. 24.242.42.230 (talk) 22:11, 26 March 2010 (UTC)
Joe: Let me restate my current position re the Cassini quote. He marks the year that Jesus Christ was born as 0 and some chronologists mark the birth year as -1. I believe when he used the term "mark" he referred to the instantaneous point in time between two adjacent years. Thus we have the following (where the upright slashes represent these points).

........|-1........|0........|+1........|+2........ This sequence is readily identifiable as the x axis in a rectilinear graph with point 0 being where the y axis intersects the x axis. It is also the representation of how astronomers often count the intervals between years + and - (regardless of sign}. Thus we have an interval of two years between years -1 and +1. But point 0 is like Janus and is looking both ways. Thus we have a Year 0+ running from point 0 to point +1 and a Year 0- running from point 0 to point -1. ........|-2 (Year -1) |-1 (Year 0-) |0 ( Year 0+ |+1 (Year +1) |+2........ and we have a satisfactory interval of four Years. All of this is in agreement with Cassini's conclusion that we have the necessary leap year 0+ and subsequent leap years every fourth year. And it satisfies your explicit requirement for a leap Year 0-. Can we finally agree that the third millennium started on January 1, 2000?Sammy 22:29, 6 April 2010 (UTC) —Preceding unsigned comment added by Samhastings (talkcontribs)

Joe: As for support of my position that there is no year zero but there is a point zero see the website "Calendars through the Ages",

"History of the Calendar", "Was Jesus born in the year zero?" Sam86.148.196.154(talk) 20:00, 19 March 2011 (UTC) —Preceding unsigned comment added by 24.242.42.230 (talk) 23:03, 31 January 2010 (UTC)

 Year 3758 | Year 0 AD | Year 1 AD | Year 0 AD | Year 1 AD | Year 2 AD |

## Third opinion

Bradv (talk · contribs) wants to offer a third opinion. To assist with the process, editors are requested to summarize the dispute in a short sentence below.

Viewpoint by Sam Hastings

The editor of Astronomical year numbering insists that Cassini's quote meant that he set Year 0 AD equal to the first year BC. I contend that he marked the year Christ was born as 0 and that some chronologists marked the year that Christ was born as -1. These interpretations lead to different conclusions.Sammy 15:28, 18 April 2010 (UTC)

Viewpoint by (Jc3s5h)

As explained above by Joe Kress, the values in tables force the conclusion that Cassini considered the year 0 to be equivalent to the year 0 used by modern astronomers and equivalent to the year 1 BC, with the alternate name 1 BCE, as used by modern historians. My French is not good enough to tell if Cassini expresses any opinion about when Jesus was born, or if he expresses any opinion about when Dionysius Exiguus thought Jesus was born. Perhaps Cassini was merely adopting conventional phrases without any deep thought; I can't tell. Jc3s5h (talk) 16:17, 18 April 2010 (UTC)

 Response to third opinion request: It is not the job of Wikipedia editors to interpret primary sources, or to draw conclusions from these sources. If there is ambiguity in the quote, or interpretation required to understand the quote, secondary sources should be employed to establish this interpretation. As it stands right now, the entire second paragraph under "Year zero usage" is unsourced, and could be considered original research. I recommend finding sources that draw the conclusions that you each have drawn, and employing these sources to either detail the ambiguity in Cassini's quote, or to resolve the ambiguity altogether. I'll continue watching this page, and can assist in this process if you like. I hope this helps.—Brad 16:51, 18 April 2010 (UTC)
Viewpoint by Joe Kress

This is the second third opinion on this subject. The first is at Talk:0 (year)#Third opinion. Samhastings (usually editing anonymously as 24.242...) is a tendentious editor who has been pushing his personal point of view (on talk pages, not articles) since 2007 in Talk:Year zero/Archive 2 where he refused to accept explanations by several editors that astronomers equate their year zero with a historian's year 1 BC. Since then he has continued to argue his position that year 0 is only a single point in time to the extent that it dominates talk pages (at Talk:0 (year) and this talk page). This is not allowed according to an Arbitration Committee ruling in another case.

Cassini's term "year 0" means just that, a year (not a point) numbered 0. Cassini marks it with a "B" [4] in his tables meaning it is a bissextile year (a leap year) containing 366 days, including February 29 [5]. Cassini's phrase "which several chronologists mark 1 before the birth of Jesus Christ and which we marked 0" means (in modern terms) that the year that historians mark 1 BC, Cassini marked year 0. In the table below I equate historical (BC/BCE and AD/CE), Cassini, and modern astronomical years. I've added a line of dates to show that all years have 365 or 366 days, including year 0; and that time advances from Jan 1 to Dec 31 even in BC years (Samhastings apparently cannot accept either). I've also added the Julian day number at Jan 1 noon of each year which also shows that each has 365 or 366 days.

Historical Cassini Astronomical Dates 2 BC 2 BCE 1 BC 1 BCE 1 AD 1 CE 2 AD 2 CE 1 BC 0 1 AD 2 AD −1 0 +1 +2 Jan 1   (365 days)   Dec 31 Jan 1   (366 days)   Dec 31 Jan 1   (365 days)   Dec 31 Jan 1   (365 days)   Dec 31 1720693 1721058 1721424 1721789

Joe Kress (talk) 08:51, 3 May 2010 (UTC)

Viewpoint by Sam Hastings
I refer Joe to his own words in the article "Astronomical year numbering" (second paragraph, last sentence beginning with "For normal calculations a number zero.......). In his most recent table he has shown an interval of three years between -1 and +1 and no amount of dodging is going to destroy that fact. Furthermore, there are no leap years available in his AD years. He has created an anachronism.
When Cassini marked the year that Christ was born as 0 it makes no difference whether it refers to Year 0 or point 0, one gets the same result: point 0 is the beginning of Year 0 and point +1 is the end of the first year AD. He also states that several chronologists marked the year that Christ was born as -1. Thus Cassini has created the -1,0,+1 sequence that Joe approves. But 0 is not Year 0 it is point 0. And, finally, one must realize that point 0 is like Janus and looks in both directions in the decimal numbering system. The rectilinear graph is a useful example. Where the Y-axis crosses the X-axis is point 0. One proceeds from 0 to +1 (Year 0 AD). And it also proceeds from 0 to -1 (Year 0 BC). Thus we have the TWO necessary bisextile years and a complete agreement with Cassini's conclusions.|}20:03, 3 May 2010 (UTC) —Preceding unsigned comment added by 24.242.42.230 (talk)
1. ||Talk:Astronomical year numbering||. I moved my comments to the Talk:Astronomical year numbering talk page after losing a third opinion in another section. I did this because how one interprets the Cassini quote has a direct bearing on the question. I believe Joe had some interest in continuing the discussion because of my assertion that a certain local astronomers's group denied a year zero. Although this quote supports the use of point 0 it is an unacceptable reference. I do not intend to get the group saddled with my problems but I appreciated their support. After a gap of about 10 months I again tried to prove my interpretation of the Cassini quote as "marking the year that Christ was born" as point 0, not year 0 (but refer to the first sentence in the paragraph above). Joe again joined the discussion of his own volition. He abandoned the discussion in Sept. of 2009 without explanation. I continued with a monologue until I finally realized -1,0,+1 demands two years 0 (0- and 0+).

Brad offered to provide a third opinion. However, he has placed unreasonable restrictions on both editors. If either of us had been able to document our particular interpretations of the Cassini quote there would probably not be a dispute. The key problem resides in my insistence on a point 0 and Joe's insistence on a year 0 in understanding what the sequence -1,0,+1 means.22:01, 27 May 2010 (UTC) —Preceding unsigned comment added by 24.242.42.230 (talk)

1. It is apparent that Bradv has no desire to see this third opinion process to continue. In the meantime I have found another incorrect statement in the section "Counting intervals without a zero" in the article "(0)Year". The third paragraph of the section should have read as follows: If the Gregorian calendar had begun with a year 0 as its first year, then the year 9 would have been the tenth year of the calendar (completing the first decade) and the year 10 would have been the first year of the second decade. Similarly, the year 1999 would have been the 2000th year of the calendar, therefore the actual first year of the 21st century and the third millennium.18:13, 12 June 2010 (UTC) —Preceding unsigned comment added by 24.242.42.230 (talk)
1. I believe that my insistence on a point zero with special implications at the epoch is well worth consideration so J I have initiated a request for arbitration. Joe, I will need your cooperation for this attempt to go forward. We are both to submit statements of 500 words or less to show the arbitrators that there is a dispute requiring their intervention. We are not to try to prove our cases at this time. That will come later. Sam —Preceding unsigned comment added by Samhastings (talkcontribs) 21:28, 24 June 2010 (UTC)
Sam Hastings, I would like to give the requested third opinion on this but can you clarify where you stand on one issue? In his tables, Cassini provides a calculation of the mean longitude of the moon. The mean longitude of a heavenly body by definition increases by an amount which is directly proportional to time. (I have a lot of trouble with people who don't understand this and say, for example, that the time between mean vernal equinoxes is different from the time between mean autumnal equinoxes).
According to Cassini, in the interval from 100BC (leap year) to AD100 (leap year) the moon's mean longitude increases from 14 degrees 54 minutes 1 second to 300 degrees 33 minutes 46 seconds, which is a westward motion of 104 degrees, 20 minutes and 15 seconds. In the interval from AD100 (leap year) to AD300 (leap year) the mean longitude decreases to 166 degrees 13 minutes 30 seconds, which is a westward motion of 104 degrees 20 minutes and 16 seconds. Now the moon circles the earth in roughly thirty days, so she moves roughly 12 degrees per day, or about 43,200 seconds of arc in 86,400 seconds of time. To all intents and purposes, therefore, Cassini reckons the number of days between a given day and hour in 100BC (leap year) and AD100 (leap year) to be equal to the number of days between the same day and hour in AD100 and the same day and hour in AD300. Do you accept this proposition? I will be looking

out for your response here, of course, but I will also be watching my talk page User talk:Meletian in case you find it more convenient to post there.

Joe's argument is clear enough, but effort is required to follow it. He says
the year "0" here corresponds to "1BCE". This places year 0 between 2BCE and 1CE, renaming them as -1 and +1, creating the scale -1, 0, +1 corresponding to 2BCE, 1BCE, 1CE, or if you prefer, to 2BC, 1BC, AD1.
The + sign is unnecessary and is not used in practice - this year's Astronomical Ephemeris (or whatever the volume is now called) is for 2010, not +2010.
This creates the following timeline:

Historical

Astronomical

-1, 0, (+)1

This arrangement is analogous to the method of using logarithms. Like you, it's many years since I left school but I recall a logarithm is made up of two parts. To the left of the decimal point we have the "characteristic", to the right the "mantissa". The characteristic displays the features of the timeline: 1 is ten to the power 1, 0 is ten to the power 0, and "bar 1" (i.e. 1 with a minus sign on top of it) is ten to the power -1. The minus sign is on top of the power (rather than to the left of it) because the mantissa is always positive, e.g. "bar 1 point one two three four" is further away from zero than "bar one point six seven eight nine".

This precisely mirrors chronology before Anno Domini. Chronology within a year (the mantissa) is the same as always (January of a particular year is still earlier in time than December of that year). The count from the epoch (the characteristic) is reversed - year 20 is earlier in time than year 10. This is something else which people don't appreciate - I am troubled by people who claim that the second sixth day before the calends of March (counted backwards by Romans, like shopping days before Christmas) is later in time than the first sixth day before the calends.

Astronomers could also tidy up their act by not describing years as, for example -100 (which has the connotation of a point in time). Something like "100ACN" (Astronomical Negative Count) to describe what we call 101BC would be clearer. Cassini's explanation of his system is also in error. He says

The year 0 is that in which one supposes that Jesus Christ was born, which several chronologers mark 1 before the birth of Jesus Christ and which we marked 0, so that the sum of the years before and after Jesus Christ gives the interval which is between these years, and where numbers divisible by 4 mark the leap years as so many before or after Jesus Christ.
There is some sloppy terminology here. AD2010 ends on 31 December, AD2010 and AD 2012 begins on 1 January, AD2012 so the interval between them is one year. However, the interval between 12h 1 January, 2010 and 12h 1 January, 2012 is two years. With a year zero, the second relationship is unchanged throughout recorded time. Without a year zero, the interval between a year BC and a year AD is two less than the sum of the numbers.

What you want to avoid is the introduction of a "point zero" or fence post between 1BC and AD1 (adjacent fence panels) because someone, somewhere, is going to get their panels and posts mixed up.

As far as cardinality and ordinality is concerned, you pays your money and you takes your choice. Provided you don't mix the one with the other you'll be all right. Counting from a point in time is one thing - a child's first year culminates in its first birthday. In chronology this translates to the difference between current and elapsed years. Our system uses current years - the second year to begin since the epoch is year 2. Some civilisations use elapsed years - the same year is year 1 because only one complete year has elapsed since the epoch at the point when it begins.
Another analogy. In Portugal, to stop a lift at the ground floor you press "R/C" (res - do - chao). To stop at the first floor, you press "1" and so on. In Spain, as in most of continental Europe, "1" stops you at the ground floor and "2" at the next floor up. These countries have no "ground zero". So you deal with it by remembering which side of the border you're on, just as you remember to switch from driving on the right to driving on the left when you cross the border from Spain into Gibraltar.
Best wishes. 81.159.32.4 (talk) 18:40, 18 July 2010 (UTC) —Preceding unsigned comment added by 212.85.12.219 (talk)
Cassini refers to years as complete years (the year in which Christ was born). Your measuring time from inside a year is not consistent with his concluding requirement of divisibility by four. My interpretation of the Cassini quote is that the second year of the epoch is

year +1 not year +2.—Preceding unsigned comment added by Samhastings (talkcontribs) 01:36, 2 September 2010 UTC

The above statement by Sam is correct, but irrespective of the fact that the third millennium begins in AD2000 from the astronomers' point of view, we don't use that system. All the systems which use elapsed years don't count back beyond the epoch, so we can't learn anything from them. Cassini's system, like ours, counts back beyond the epoch, but so what? We are saddled with a system that (a) counts back beyond the epoch (b) uses current years and (c) does not have a year zero. 156.61.160.1 (talk) 09:15, 2 September 2010 (UTC)