Muses on the Gregorian

Ed Staples Erindale College, ACT

Pope Gregory XIII and the 7 1502 – 10 15851

he Gregorian calendar in use today is not so old. The United Kingdom, Tits colonies, including the American colonies, adopted the reforms in of 1752 during the reign of King George II. Before this, the realm was using the , and this had accounted fairly well for the Earth’s last quarter turn in its annual orbit. However, after one and one half millennia, the more accurate fraction of 0.24219 was beginning to reveal itself in the form of a concerning and a half retreat of the vernal . The simplicity of the Julian calendar is perhaps why it lasted for so long— one leap every four —but the effect of the 0.00781 difference was first noticed by the venerable as early as AD 725, who noted that the was ahead of its tabulated . Some 777 years later, in 1582, Gregory XIII signed a papal bull heralding a new and improved calendar for the world. While most Catholic countries adopted the changes immediately, the protestant countries, included the United Kingdom opted for deferral. It took another 170 years before the UK, on 22 1751, enacted the Gregorian calendar into law for itself and her colonies. The changes made in the United Kingdom were predominantly threefold. Strange as it may seem, New Years , under the old arrangements, was not 1 January of each year. Rather, it was celebrated on the 25th day of each year, the day known to Catholics as the Feast of the and colloquially as Lady’s Day. The new arrangements made 31 1751 the last day of that year, shortening it to a mere 282 days. The , as for all new years would then begin on 1 January as is still the custom today. However, that change did not fix the slip in the vernal equinox. The new law also determined that the day following Wednesday 2 September 1752 would be Thursday 14 September 1752, thereby shortening that year by 11 days. The new arrangements also put in place a mechanism that ensured that

Australian Senior Mathematics Journal vol. 27 no. 2 1 The day of the week that Pope Gregory XIII was born was determined using the Julian calendar.

36 Muses on the Gregorian calendar Australian Senior Mathematics Journal vol. 27 no. 2 37 (1) where the where n = 6, the left- x = 2 for ).

⎥ ⎥ ⎦ 2 + ] ) 8 2 = 1 , ⎥ ⎥ ⎦ 9 2 is given by: + 5 + ] n 5 3 ( = 1 for January, n = 1 for January, which, for non-negative numbers, for non-negative which, ⎦ [ d 3 o ⎣x 1 only. Readers might also be unfamiliar only. ⎢ ⎢ ⎣ m + 9, 12)] in the numerator. Think of this + 9, 12)] in the numerator. = [ 3 6 1 x ⎢ ⎢ ⎣ = ∑ = 29. The reader can check that this is in fact 2 n = x 6 Z = 6 (the number for , the sixth ) into ∑ = + 9 is divided by 12. So for example, if n n Z = 8 showing that there are 8 excess days in the completed the in days excess 8 are there that showing 8 = 6 Z is the month number (e.g., n = 2 into (1) to find Z Continuing, if we let n Zeller showed that the total excess of days that accrue from the beginning of Zeller showed that the total excess We can put Zeller’s function to the task of determining the number days can put Zeller’s We To understand how his function works, we need to think of each month’s to think of each month’s works, we need how his function understand To Again suppose we wish to know the total excess of days from the beginning Again suppose we wish to know the total excess of days from the beginning subscript n stumbled on a function that could be used to determine the number of the stumbled on a function that could as the left-overs when n announced at a meeting of the Mathematical Society of that he had announced at a meeting of the Christian Zeller equation (1), then (1), equation overs when 15 is divided by 12 is 3. correct. of March to just prior to the beginning of Februaryof beginning the to prior just to March of January). 31 to is, (that elapsed days of a year up to and including a specified date. elapsed days of a year up to and including are assigned a certain number of days is essentially a remnant of a certain number of days months are assigned months March, April and May (March and May have three of these and May months March, April and May (March and May have three of these to establish a simple mathematical formula for the pattern. However, on 16 on However, pattern. the for formula mathematical a simple establish to that have elapsed in any given (a non-) in the following that have elapsed in any given common year (a non-leap year) in the 21 September. up to and including, say, there would be virtually no more slippage of the in future . in future centuries. of the seasons no more slippage be virtually there would divisible by 400. they were takes the integer part of the number x takes the integer part of the number total days as made up of 28 days plus an excess that we can call can we that excess an plus days 28 of up made as days total way. Suppose we wish to know the number of days that have elapsed in the year way. n with the modular expression [mod( . The variations in the number of days in each month made it difficult of days in each month made The variations in the number history. has two). March 1883, a school inspector in by the name of Christian Zeller March 1883, a school inspector

We let n We Turning our discussion to the calendar itself, we find that the way in which in way the that find we itself, calendar the to discussion our Turning The formula uses the floor function floor the uses formula The All end of centuries (1800, 1900, 2000, etc.) would not be leap years unless be leap years unless etc.) would not 1900, 2000, of centuries (1800, All end beginning of month n to just prior to the March Monday 24 June 1822 – Wednesday 31 May 1899 – Wednesday Monday 24 June 1822 38 Australian Senior Mathematics Journal vol. 27 no. 2 Staples E E d This accounts for the 59 days of January and February plusthe(n andFebruary This accountsforthe59daysofJanuary To seewhy, thinkaboutthenumberofelapseddaysto,say, 16February. Here March totheendofAugust.Thismeansthattotalnumberdaysduring January falls.Forexample,supposewewishedto knowwhatdayofthe January February andthe21daysofSeptembertothisyieldsatotalelapseddaycount February week 21September2011fallson.UsingE number of days from 1 January to16February.number ofdaysfrom1January Againtheformulasimplifies to and including the day to E that period is 6 × 28 + 16 = 184 days. Adding the 59 days of January and that timeperiodis6×28+16=184days.Addingthe59daysofJanuary total numberofdaysthathaveelapsedinanycommonyearuptoandincluding between 1 March and 16 February inclusive. If we add the 59 days of January inclusive.Ifweaddthe59daysofJanuary between 1Marchand16February 9 +2 or11months.Thismeansthatthereare308Z consider that for months after February wehaveE consider thatformonthsafterFebruary of 264. packages of28plusZ day d for anyparticulardateoftheyear, providedweknowonwhich day 1 four daysafterSaturday, i.e., Wednesday. and February, andthensubtractafull365days,weareleftwith47,theprecise and knowingthat264≡5(mod7),wecanconclude21September was NB: Forleapyears,thisformulawillneedtobeadjustedupby1forn then thetotalnumberofdaysuptoandincludingd month ofacommonyearisgivenby: If =16andn 21,9 d , n We could use the equations in Box 1 to determine the day of the week Can wenowconstructaformulaforE Putting italltogetherwecanstatethefollowing.To determineE First notethatZ For January orFebruary,For January theformulaadjuststoE Z =Z =Z d n , ofmonthn n = =Z n ⎣ ⎢ ⎢ 9 +28n 264 as before. Because 1 January wasaSaturday, +28×921−25=264 asbefore.Because1January 1 3 n [ +28n d–54. m o = 2. From the beginning of March to the end of January is =2.FromthebeginningofMarchtoendJanuary −25+d d ( n wehavethefollowing: 5 + 9 =16sothatthereareexcessdaysfromthebeginningof 9 , E 1 . 2 d n , andd ) n d ] of the month = + ⎩ ⎨ ⎧ 2 Z Z ⎦ ⎥ ⎥ n n

. Withaslightsimplification,thisbecomes + + 2 2 8 8 n n Box 1 + + d d d , n − − n , thetotalnumberofelapseddaysup in a common (non-leap) year? First 2 5 d , 5 4 n =Z n d +28n , d n , =Z n n n =Z > ≤ n +28(n +d 2 2 n +28(n 2 +16or353days th dayofthen –25, wefindthat +9)59−365 d −3)+59d d >2. , n −3) , the th . . Muses on the Gregorian calendar Australian Senior Mathematics Journal vol. 27 no. 2 39 corresponding y c d m and d m , y , Carroll Number 21 20 17 Example September Table 1. Illustration of parts 1. of the date. Table , determine the remainder when the centurythe when remainder the determine c, is part Day Year Parts Month divided by 4 (remainder 0), then take this remainder away from divided by 4 (remainder 0), then take this remainder away from of twelves in 17 is 1, the left-overs are 5 and there is 1 four in the of twelves in 17 is 1, the left-overs are 5 and there is 1 four in the the year part, how many are left over after these sets of twelve are the century number 6 given as 2 × (3 − 0). part 20 has the Carroll removed, and how many complete sets of four are in those left-overs. left-overs—that is a total of 7. To find y, first determine how many complete sets of twelve are in To To find To 3, and finally double the result. For our example, you should find 3, and finally double the result. For our example, you should find Then add those three numbers up. For our example, the number Then add those three numbers up. For our example, the number the century part (that is the first two digits of the year), the year) the year part (the last two digits of etc.) February, the month part (January, the 17th, etc.) the day of the month part (the 5th, So for example, in 21 September 2017, the centurySo for example, in 21 September part part is 20, the year From here Dodgson determines the numbers c To use the method, Dodgson first considered that any date could be broken use the method, Dodgson first considered To arithmetic skills, involving a little mental computation with multiplication arithmetic skills, involving a little and division. Charles Dodgson (aka Lewis Carroll) (aka Lewis Dodgson Charles famous dates in history but and other events. The method is simple enough, for determining the day of the week given any date in the Gregorian calendar. in the Gregorian calendar. the day of the week given any date for determining is 17, the month part is September and the day part is 21. We could imagine a and the day part is 21. We is 17, the month part is September day of the week for things like dates of births, future dates for the current year, dates for the current year, things like dates of births, future day of the week for associated with the calendar months. does require memorising a few numbers it was one of Dodgson’s favourite party tricks, usually announcing that he favourite party tricks, it was one of Dodgson’s could determine the correct day of the week within 20 . The author the correct day of the week within could determine can provide a little entertainment value to others. Indeed, legend has it that entertainment value to others. can provide a little up into four parts. These are: 1. table of parts as shown in Table to each of the four parts as follows: has learned the technique himself and enjoyed pronouncing to others the others to pronouncing enjoyed and himself technique the learned has Year: 2. Century: Charles Dodgson (also known as Lewis Carroll) devised an efficient method (also known as Lewis Carroll) Charles Dodgson 3. With sufficient practice, the technique becomes an extremely useful one, and practice, the technique becomes With sufficient Friday 27 JanuaryFriday 27 14 January 1832 – Friday 1898 As a trick for students at most year levels, the technique strengthens basic As a trick for students at most 4. 1. 40 Australian Senior Mathematics Journal vol. 27 no. 2 Staples Month: Day: when thatsumisdividedby7.Usingmodulararithmetic,wedetermine work out. 6 +7521=4(mod 7). nothing totheremainderandthus6 +5=114(mod7)isslightlyeasier four numbersmodulo7beforeweaddthem.Soand21inthisexample So 21September2017will beaThursday. as showninTable 4: The greatnewsaboutmodulararithmeticisthatwecanreduceeachofthe The lasthurdleinvolvesconvertingthefinalremaindertoadayofweek From here,wesimplyaddtheCarrollnumbersupandfindleft-overs Table 1iscompletedasshowninTable 3. Commit to memory themonthTableCommit tomemory 2form The daypartd Table 4.Relationshipbetween final remainder anddayofweek. Century Month Parts Year Day 1 4 2 3 6 m 5 0 Table numbers. toCarroll 3.From dateparts Final remainder Table 2.Value ofmforeachmonth. isthedaynumberitself.Sofor21st,d April, 4 Month May June January, February, March, 3 2 1 6 0 September, September Example 17 20 21 Wednesday Thursday Saturday Monday Sunday Friday Day Carroll number d m c y =21 =6 =7 =5 , asshown =21.So Muses on the Gregorian calendar Australian Senior Mathematics Journal vol. 27 no. 2 41 For example, the first two Carroll numbers associated with 2014 are 6 and For example, the first two Carroll For example, the date 21 February the date 21 For example, 6, 11, 3, and Carroll numbers 2020 has Now while the explanation looks lengthy, in practice, days of the week can Now while the explanation looks lengthy, It is a nice idea to determine, for each month, a Carroll Value which It is a nice idea to determine, for each month, a Carroll Value One slight occurs when dealing with leap years. When dealing When dealing with leap years. occurs when dealing complication One slight We can note other things as well. For example, the calendar for the two can note other things as well. We The author has thought of a few tricks to help with the method. First of all, of a few tricks to help with The author has thought A couple of other points can be made. When dealing with remainders the remainders with When dealing made. can be points other couple of A so adds nothing to the final remainder. Also, most of the useful determinations Also, most of the useful determinations so adds nothing to the final remainder. a Sunday and the 1 December 2014 is a Monday. a Sunday and the 1 December 2014 is a Monday. any common year have the same Carroll value of 1 (0 + 1 = 5 + 31 (mod 7)). any common year have the same Carroll value of 1 (0 + 1 = 5 + 31 add 7 + 2 = 9 and then add the 2 (for 2014) to get 11 or in other words add 7 + 2 = 9 and then add the Similarly day is 4 and 5 and 2, or 4 (mod 7), or Thursday. In fact In Thursday. 7), or 4 (mod 2, or 5 and 4 and day is Christmas Similarly the date September and December as 5, together with 2 for 2014, means that fall on the same day of the week. four Carroll numbers can be reduced modulo 7 at any time. For example, the 21 four Carroll numbers can be reduced incorporates both the month and year. For example, suppose we are just about just are we suppose example, For year. and month the both incorporates in those months indicates the day. That is, for example, 14 September 2014 is in those months indicates the day. in our heads. Suppose the day of the week for 7 August is required. I simply in our heads. Suppose the day of involve the current year, and so we can prepare a corresponding number that and so we can prepare a corresponding number involve the current year, in our example above could have immediately been reduced to 0 (mod 7) and 7) (mod to 0 been reduced immediately could have above example our in months September and December and for the two months April and July months September and December days aligned—the same applies for February, must always have their dates and combines the first two Carroll numbers. correct day is Friday. (February, March and and 3 is a FeMiNine number (February, months, May Day was 1 May, be determined very quickly, particularly so for a given year. Let me give you a particularly so for a given year. be determined very quickly, better still, the years of the previous centuryprevious the of years the still, better Carroll the have all ) (the to commence the month of March in 2014. The Carroll Value is 3 + 2 = 5 in 2014. The Carroll Value to commence the month of March taste of how to think using the method. As stated above, 2014 has a combined the letter l, June has four letters, September and December are the ‘three e’ and December are the ‘three e’ has four letters, September the letter l, June the century in the ordered sequence can only be one of four numbers number number 0. To help with Table 2, April and July are the only months with and July are the only months 2, April help with Table number 0. To 6, 4, 2, 0. The years of the current century6, 4, 2, 0. The years the Carroll number 6, and all have with January or February of a leap year, subtract one from the final remainder. remainder. from the final subtract one with January or February a leap year, of November). March and November in common years. Also, 1 January and 31 December in 25 (which is really 4) and 3 and 2, or a total of 9, or 2 (mod 7), or Tuesday. or Tuesday. 25 (which is really 4) and 3 and 2, or a total of 9, or 2 (mod 7), 21 and these sum to 6(mod 7). However, because 2020 is a leap year, the is a leap year, because 2020 to 6(mod 7). However, 21 and these sum Carroll Value of 2 (mod 7) so for example the day for 25th March 2014 is of 2 (mod 7) so for example the Carroll Value 3, so for that year we hold what we might call the Carroll3, so for that year we hold what we value of 9(mod7) = 2 This implies that in any common year, the first and last days of the year must the This implies that in any common year, 7) or Thursday. 4(mod 7) or Thursday. (mod 7). If you commit 5 to memory a month, then the day for any date in for 42 Australian Senior Mathematics Journal vol. 27 no. 2 Staples August, 10October, 12December, 9May, 5September, 7Novemberand11 John Conwayonthedayofweek Carroll numbers work, and could a similar system be devised for the Julian Imagine marking the Calendar with the yearly Carroll value, and getting In fact,afterawhile,withpractice,knowingtheCarrollvalueforparticular 2003, so6forthecentury, andno12sin3,so3isleftover, andno4sin3,so Perhaps foraslightlyoldergroupwehear, “IwasbornonAprilFool’s Day John Conway, ProfessorofMathematicsatPrincetonUniversity, notesthe July areallFridaysin2014. Dodgson hasalsoprovideduswithaninterestingdevicethatcanbeappliedin year, theCarrollmonthnumbers(whichnever andcommittingtomemory versa (easilyrememberedbythephrase“Working 9to5ina7/11store”)—all levels, wecouldexploremodulararithmeticelements,andinvestigatewhythe value of3(4+6),alongwith 6June,8August,etc. the dayisconfirmedwithsomethinglike,“It’s 2016,so23 the23rdFebruary that monthbecomesobvious.Forexample,16MarchisaSunday(amultiple the classroom.Thepossibilities,forexample,withyoungstudentsareendless. that year. For example, the Doomsday for 2014 isFriday. So4April,6 June, 8 useful keyfordeterminingthedayofweek. Forexample,in2011,the many calendarsarepossible?Thelistofquestionsisendless.Athigheryear month andviceversatheseventhdayofeleventh calendar. At all levels though, learning these things provides a simple and example—interesting becausetheCarrollvalueis0plus0.Howoften could introduce some analysis ofCarroll numbers. Take 25thMarch1956for out is3,soitwouldbeWednesday, butthisyearisaleapyear, soitsTuesday!” of 7),themonthstartsonSaturday, therearefiveSaturdaysforMarch, etc. change), gives you a really practical tool to use on a day-to-day basis. Charles corresponds totheCarroll numbersaswell.ThefourthofAprilhasaCarroll powerful toolforlifeand strengthensbasicmentalcomputationinafunway.powerful does thatoccur?Howmanytimes will yourbirthday fall onaSaturday? How is 7, and with the 2 is 9, which is really 2. I was born on Tuesday!” Teachers it’s 9for2003,whichisthesameas2,then1stofApril1plus6 plus 35fortheyearequals31andleft-overaftersevensarecast days aheadofMonday5 September—Wednesday. Ofcourse,Conway’s rule doomsday wasMonday. Adatesuchas21September2011istherefore16 fall onthesamedayirrespectiveofyear!Soalsofifthninth falling onthesamedayincalendar. Infact,henamesittheDoomsday for amazing factthatthek students to commit to memory theCarrollmonthnumbers.Eachmorning, students tocommitmemory Conway suggeststhatknowingtheDoomsdayfor anyparticularyearisa tℎ dayofthek tℎ month,fork =4,6,8,10,12willalways Muses on the Gregorian calendar Australian Senior Mathematics Journal vol. 27 no. 2 43 1 21 22 24 23 20 19 17 15 12 14 18 13 16 Jeudi Mardi Lundi Samedi 5 8 6 7 9 Vendridi Mercredi 11 10 Dimanche French names 1 2 3 4 22 24 23 Assigned 21 20 17 15 18 16 19 8 9 11 12 14 13 10 Friday Sunday Monday Tuesday Saturday Thursday Wednesday English day names 5 6 7 1 2 3 4 Table 6. Allocation of day names. Table Table 5. Assignment of Planets to days. 5. Assignment of Planets Table Orbital period 88 days 687 days 365 days 29 years 29 days 224 days 12 years Moon Venus Saturn Jupiter Mercury 1st planet According to the Roman historian Dion Cassius (AD 150–235), the first According to the Roman historian Planet Saturn Sun Jupiter Moon Mercury Mars Venus and since 7 and 24 are relatively prime, no planet can be assigned to the same and since 7 and 24 are relatively prime, no planet can be assigned assigned to the very repeating next planet on the list: the Sun. The cycle keeps although not officially sanctioned until much later, was becoming popular in was sanctioned until much later, although not officially away from the Earth, and this happens to corresponds to each planet’s orbital to each planet’s and this happens to corresponds away from the Earth, day more than once across the 168 hours of a week. Hence we arrive at Table at Table day more than once across the 168 hours of a week. Hence we arrive period as shown in Table 5. in Table period as shown of the heavenly wanderers. Each subsequent hour of that first day was assigned Each subsequent of the heavenly wanderers. of the sequence of day names in the modern calendar. The week of 7 days, week of The calendar. the modern in of day names sequence of the through to the 1st hour of the 3rd day (assigned to the Moon) as shown in Moon) the to (assigned 3rd day the of hour the 1st to through new planet, a assigned is day new each of hour first the Progressively, 4. table the planets in order of presumed distance from the Earth. After the 7th hour 7th the After Earth. the from distance presumed of order in planets the Then the first hour of the day is the 24th hour (assigned to Mars). the Roman world around the time of the birth of Christ. Each of the seven days around the time of the birth of Christ. the Roman world 6: was assigned to a ‘planet’ according to the assumed ‘distance’ each planet was ‘planet’ according to the assumed was assigned to a hour of the first 24 hour day was assigned the planet Saturn being the slowest hour of the first 24 hour day was As a final note, Richards (1998) gives us a fascinating insight into the origins origins the into insight fascinating a us gives (1998) Richards note, final a As (assigned to the Moon), the cycle begins to repeat, over and over again until (assigned to the Moon), the cycle The assignment of the days of the week the days of The assignment I have included the French naming equivalents in Table 5 to illustrate that Staples we only need two countries modern naming conventions to capture all of the remnants of these planet allocations. The calendar is a fascinating study of mans’ attempt to measure time by the observation of the motions of the Moon around the Earth and the Earth around the Sun. So much of existence—our life cycles, the seasons, our religions, our communications, agriculture and manufacturing, even our military campaigns—depends on measuring time well. An Earth that takes turns per cycle seems designed, and perhaps our adoption of the Gregorian calendar says something about an unspoken acceptance of a dawning truth - the resignation to a world that is not so square. The algorithms presented in this article could easily be adapted for the classroom. Students could develop computer programs around the Zeller congruencies and junior students in particular could strengthen basic arithmetic concepts by learning the Lewis Carroll technique as well as getting the chance to see mathematics as a powerful and practical pursuit.

References

Richards, E. G. (1998). Mapping time: The calendar and its history. New York, NY: Oxford University Press. HREF URL: http://en.wikipedia.org/wiki/Doomsday_rule URL: http://en.wikipedia.org/wiki/Pope_Gregory_XIII URL: http://en.wikipedia.org/wiki/Christian_Zeller URL: http://www-history.mcs.st-and.ac.uk/history/Biographies/Dodgson.html URL: http://www.merlyn.demon.co.uk/zeller-c.htm Australian Senior Mathematics Journal vol. 27 no. 2

44