François-Joseph Servois (1767-1847) was a priest, artillery officer, and professor of mathematics. His research spanned many areas, including geometry and the foundations of calculus. In this article, we introduce readers to a perpetual calendar created by Servois (see Figure 1), provide an annotated translation of Servois' 1813 “Calendrier perpétuel" [1813], and suggest some possible uses for it in the classroom.

**Figure 1.** Servois' perpetual calendar appeared on page 92 of his 1813 article, “Calendrier perpétuel." The signature in the lower right corner is that of Joseph Diaz Gergonne, the editor of the journal, *Annales des mathématiques pures et appliquées* (also known as "Gergonne's journal"), in which the article appeared. (Image used courtesy of the Science, Industry and Business Library of the New York Public Library.)

Before we explore Servois' perpetual calendar further, we provide a brief review of the history of calendars.

François-Joseph Servois (1767-1847) was a priest, artillery officer, and professor of mathematics. His research spanned many areas, including geometry and the foundations of calculus. In this article, we introduce readers to a perpetual calendar created by Servois (see Figure 1), provide an annotated translation of Servois' 1813 “Calendrier perpétuel" [1813], and suggest some possible uses for it in the classroom.

**Figure 1.** Servois' perpetual calendar appeared on page 92 of his 1813 article, “Calendrier perpétuel." The signature in the lower right corner is that of Joseph Diaz Gergonne, the editor of the journal, *Annales des mathématiques pures et appliquées* (also known as "Gergonne's journal"), in which the article appeared. (Image used courtesy of the Science, Industry and Business Library of the New York Public Library.)

Before we explore Servois' perpetual calendar further, we provide a brief review of the history of calendars.

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Throughout history, there have been many attempts at creating calendars. Some early systematic endeavors were made by the ancient Babylonians, Chinese, and Egyptians. The Babylonians (ca. 1900–1600 BCE) had a lunar calendar with 12 months, which alternated between lengths of 29 and 30 days [Richards 1998], and they approximated the length of a year to be 365 days, 6 hours, and 11 minutes [Smith 1953]. The Babylonian calendar went through several reforms, but survived until the rise of Islam in the seventh century [Richards 1998]. Astronomy and the creation of calendars also were of great importance in China and East Asia. According to Smith [1953], Chinese calendars changed significantly depending on the Emperor at the time. However, as early as the Shang Dynasty (1600 BCE), most Chinese believed that the mean length of a year was 365.25 days [Richards 1998]. The ancient Egyptians had a calendar that consisted of 12 months of 30 days each, and five extra days added at the end of the year [Dershowitz and Reingold 1997]. Like those of the Babylonians and of the Chinese, the Egyptian calendar went through several reforms.

David Eugene Smith, in his classic *History of Mathematics* [1953], stated that the oldest Roman calendar was probably created in about 753 BCE, supposedly by Romulus, the legendary founder of Rome. This calendar had a year of about 304 days divided into ten months of varying lengths, with each year beginning with the month of March. The legend continued with Numa Pompilius, Romulus' successor as King of Rome, who added the months of January and February to this calendar in about 713 BCE, and declared that each year would begin on January 1.

**Figure 2.** The *Tusculum Portrait* of Julius Caesar, photograph by Tataryn 77. (Picture and data courtesy of the Yorck Project, under Creative Commons Attribution-Share-Alike License.)

According to Richards [1998], the *Decemvirs,* a committee in charge of writing laws for the Roman Republic, made reforms to the intercalation process (insertion of leap days) for the Roman calendar in 450 BCE, the administration of which was given to the priests, who kept the process secret [Smith 1953]. However, this matter soon became a political competition which caused the process to become disorganized [Richards 1998]. By the time of Julius Caesar, this mismanagement had caused the Roman calendar to be eighty days out of its astronomical place [Smith 1953]. Due to this error, astronomers under Caesar's administration created the so-called Julian calendar, which was implemented in 46 BCE. It was declared that the year 46 BCE was to have 445 days. However, every year thereafter would have months of fixed lengths, where one year had 365 days divided into 12 months, and every fourth year was a leap year with 366 days. Caesar's original calendar began in March, but he later decided to move the beginning of the year to January 1, a return to the practice in Pompilius' calendar. Additionally, he changed the names of several months, such as Quintilis to Julius (July was Caesar's birth month) [Smith 1953]. After Caesar's death, confusion arose among the priests regarding Caesar's orders as to when the next leap year would occur [Smith 1953]. Gaius Octavius Thurinus, more commonly known as Augustus Caesar, corrected this error in 9 BCE by omitting the next three leap years, whence there would be no leap year observed until 8 CE [Richards 1998]. In Augustus' honor, the summer month, which the Romans called Sextilis, was renamed August after him [Smith 1953]. The Julian calendar was a solar calendar [Norby 2000] and had a mean of 365.25 days.

**Figure 3.** Christopher Clavius. Painting by Francisco Villamena (1566–1624) (public domain).

The Julian Reform did not solve all of the problems with the calendar, because the mean number of days per year is not 365.25 days. Actually, it is about 11 minutes and 14 seconds shorter [Smith 1953]. This over-estimate led to the vernal equinox slipping by about a day every 128 years during the Julian period [Norby 2000] and this error became very apparent by the end of the fourteenth century. Finally, in order to rectify this issue, Pope Gregory XIII, along with the mathematicians Christopher Clavius (1538-1612) and Aloysius Lilius (ca. 1510-1576), created the Gregorian Calendar. This calendar was established in October 1582, when Gregory decreed that October 4 of 1582 would be immediately followed by October 15 [Norby 2000]. To compensate for the error in approximating the vernal equinox, Gregory instituted the 100 years and 400 years rule. The rule – namely, “Each year which is divisible by 4 but not by 100 and each year which is divisible by 400 is a leap year with 366 days. All other years contain 365 days” [Morris 1921, p. 127] – gives a mean year of approximately 365.24 days and would require no further adjustment for about 3000 years [Smith 1953].

**Figure 4.** Pope Gregory XIII. Painting by Lavinia Fontana (1552–1614) (public domain).

The purpose of the present article is to focus on a *perpetual calendar* created in 1813. Therefore, our brief journey through the history of calendars ends here. Readers interested in a more detailed account of the history of the calendar can refer to Smith [1953], Richards [1998], and Tondering [2012]. Now, we turn to the concept of a perpetual calendar.

A perpetual calendar, also known as a "forever calendar," is a compact and efficient calendar which is intended to be accurate and require no adjustments for many years. This seems to be a vague definition; however, a perpetual calendar's lifespan is determined by its creator. Perpetual calendars date as far back as the Julian era.

Perpetual calendars can come in many different flavors. For instance, there are perpetual calendars that require the user to know on which day of the week January 1 falls. There are also calendars, such as the one we will discuss, that require no special auxiliary information. Perpetual calendars that require little or no information seem great at face value; however, their downfall is that they are not efficient for everyday use. That is to say, if a specific perpetual calendar requires you to begin with very little information, then that calendar will involve more calculations for its use.

According to Richards [1998], the Gregorian calendar had one element that almost everyone disliked: the fact that there are no fewer than 14 different calendars required to deal with the distribution of days in the week and leap years. Thus, there seemed to be no way to create an “efficient" perpetual calendar under the Gregorian Reform. Despite this fact, many individuals still create perpetual calendars to this day.

**Figure 5.** Perpetual calendar owned by Rob Bradley and Susan Petry.

François-Joseph Servois was born on July 19, 1767, in the village of Mont-de-Laval, located in the Department of Doubs close to the Swiss border. Servois attended several religious schools in Mont-de-Laval and Besançon, and studied to become a priest. He was ordained at Besançon near the beginning of the French Revolution. His religious career, however, was destined to be a short one. With the outbreak of the revolution, Servois left the priesthood in 1793 and became an officer in the Heavy Artillery. In his leisure time he studied mathematics.

During his first few years in the military, Servois began to suffer from poor health and he requested a transfer into the field of academia. By virtue of a recommendation from the great mathematician Adrien-Marie Legendre (1752-1833), he was assigned his first teaching position at the artillery school in Besançon on July 7, 1801. He served on the faculty at numerous artillery schools during his academic career, including at Besançon (1801), Châlons (March 1802 - December 1802), Metz (December 1802 - February 1808, 1815-1816), and La Fére (February 1808-1814, 1814-1815). Additionally, Servois fought in several well-known battles, such as the crossing of the Rhine and the battle at Neuwied (1796-97), and the final battle in March of 1814 to defend Paris from the Austrian and Prussian armies. Interestingly, the official French records indicate that he participated in the Siege of Maastricht; however, the history of the First Foot Artillery Regiment, the attachment with which he served, does not mention it having taken part in the battle. Either there is an error in the records or Servois was sent alone to Maastricht, rather than with the entire Regiment.

Servois' research spanned several areas, such as mechanics, geometry, and calculus. However, his best known work is his “Essay on a New Method of Exposition of the Principles of Differential Calculus” [Servois 1814b], which treated the foundations of the differential calculus. In this essay, he gave the terms “commutative” and “distributive” their first mathematical meanings. On May 2, 1817, Servois was assigned his final position as Curator of the Artillery Museum, which is currently part of the Museum of the Army in Paris. He applied for retirement in 1827, and was finally granted his retirement in 1828 [Aebischer and Languereau]. He retired to his hometown of Mont-de-Laval and lived with his sister and his two nieces until his death on April 17, 1847. Interested readers can refer to Petrilli [2010] for further biographic information and a review of Servois' other mathematical works.

Servois' “Calendrier perpétuel” (“Perpetual Calendar”) [1813] strays from his other research areas. The majority of his publications fall under the category of “algebraic formalism.” (See Bradley and Petrilli [2010]). Additionally, Servois downplayed the importance of this work on the perpetual calendar to Joseph Diaz Gergonne (1771-1859), editor of the *Annales des mathématiques pures et appliquées*. However, Gergonne thought otherwise. To the publication of Servois' 1813 article on the perpetual calendar, Gergonne attached the following footnote:

It was only through the plea of the editor of the

Annalesthat Mr. Servois who had sent him this ingenious calendar, without attaching the least importance to it, was willing to permit it to appear in this journal, in which we think that it would not at all be out of place [p. 84].

Servois stated that his calendar was designed to answer the following four general questions:

- To determine which day of the week corresponds to a certain day of a designated month of a given year.
- To determine which days of a designated month, in a given year, correspond to a certain day of the week.
- To determine which are the months of a designated year, in which a certain day of the week corresponds to a certain date.
- To determine which are the years in which a certain day of the week coincides with a given date of a designated month.

**Figure 6.** Servois' Perpetual Calendar.

We now present a discussion of the layout of his calendar. Figure 6 presents a translated version of Servois' perpetual calendar. Within each of the circles Servois placed the names of the days of the week and a systematic grouping of the months. The ordering of the days of the week is obvious; however, a little explanation is required for the groupings of the months. The groupings are called “corresponding months.” Two months correspond in a grouping if the number of days between their first days is \(0\mod 7.\) For example: during a non-leap year, January and October correspond to each other, and June corresponds with no other months. However, during a leap year, January, April and July correspond, and October corresponds with no other months. As we shall see below, Servois' calendar was designed to provide a calendar for a common (non-leap) year; however, he described adjustments that would allow reckoning for a leap year as well. Interestingly, during any year, September and December always correspond; however, no months ever correspond with May or June. Let us illustrate by means of one example: between September 1 and November 31 there are 91 days and \(91\mod 7 = 0,\) which shows that September and December correspond.

The numbers which surround each of the circles in the table, such as 00., 01., 02., etc., represent the years in question. For example, 00., 01., and 02. represent 1800, 1801, and 1802, respectively. The numbers located at the bottom of the table, in what Servois called “medallions,” represent the calendar dates.

Let's try to answer Servois' first general question using his calendar.

**Example A.** We wish to know which day of the week corresponded to May 2, 1817 (the date of Servois' appointment as Curator to the Artillery Museum in Paris). Begin by locating the number “17.,” which represents the year 1817. You will find this in the third column of the table. Now, within the third column, find the circle which contains the month of May. Finally, move horizontally to the left, until you are above the “medallion” that contains the number “2.” Within the target circle is printed the day “Friday.” You will find that May 2, 1817 was a Friday.

But what if the year in question is a leap year, namely, a year number divisible by 4? If you wish to determine the day of the week during a leap year, then you will need to do a conversion only for dates in January and February, because all dates in those months are shifted backwards by one day. Therefore, if the year in question is a leap year, and you are examining dates in January and February, then you need to use the column that immediately precedes the one containing this date. Servois [1813, p. 85] stated that “this is a general remark,” which means you always follow this rule for leap years.

**Example B.** We wish to know which day of the week corresponded to January 7, 1828. Locate “28.” in the table. You will find it in the third column, so we use the second column instead. Now, find the circle which contains January and move horizontally until you are over the “medallion” which contains “7.” You will find that January 7, 1828 was a Monday.

A final note about leap years: You will notice that Servois placed asterisks in all of the columns. We will work through an example to see why they appear in his calendar.

**Example C.** Consider the following question: in which years during the nineteenth century did February 7 fall on a Saturday? (This is an example of Servois' second general question.) Begin by locating the number “7” in the “medallions” at the bottom of the calendar. You will find it in the last column. Find the circle in that column which contains the month of February. Move horizontally to the left until you find Saturday. Every non-leap year, and no one of the leap years, that appears in this column would have February 7 fall on a Saturday. Due to the leap year rule, for every asterisk that appears in this column, substitute the year that appears in the corresponding position one column to the right. Therefore, the years that had February 7 on a Saturday were: 1801, 1807, 1818, 1824*, 1829, 1835, 1846, 1852*, 1857, 1863, 1874, 1880*, 1885, and 1891. The leap years listed here with asterisks are precisely those that correspond to the asterisks in the table. By the same token, note that the leap years 1812, 1840, 1868, and 1896, which are listed in this column, are omitted from our list of desired years because they correspond to asterisks in the column to the left of this column.

The reader will notice that Servois used dates only from the nineteenth century. Does his calendar work in other centuries? Yes! Servois stated that his calendar was truly “perpetual,” because it could be easily converted for use in any other century. (What Servois really meant here is any century after the Gregorian Reform.) The conversion works on a mod 4 system, wherein whenever a year number references a particular column in Servois' table, one should use instead the column \(4 - 2k\) positions to the right, where \(k\) is the remainder mod 4 of the number formed from the first two digits of the year. For instance, since \(18 \equiv 2 \mod 4,\) then for years in the 1800s, we are instructed to correct Servois' table by using the column 0 columns to the right of the one we are instructed to use, which is, of course, exactly what is to be expected. However, for years in the 2000s, \(k = 0\) implies that we should correct the table by moving 4 columns to the right (or, equivalently, 3 to the left) when locating a target column. Here is a summary of the rules: If the number formed from the first two digits of the year is:

- \(0\mod 4,\) then the first year (00.) of your century will begin in the seventh column (thus, all years are shifted four columns to the right);
- \(1\mod 4,\), then begin with the fifth column (all years are shifted two columns to the right);
- \(2\mod 4,\) begin in the third column (so, no modifications are needed);
- \(3\mod 4,\) begin in the first column (all years are shifted two columns to the left).

We can see that his calendar is designed on the basis of whether the first year of the century is a leap year or not. Thus, every century follows in these cycles of four.

**Example D.** Suppose we wish to know which day of the week corresponded to January 13, 1983 (the author's birthday). To begin, \(19\mod 4 = 3,\), so we begin in the first column. Thus, all columns for the 1900s are moved two columns to the left. Find “83.” in the table. You will notice it is in the first column, so “1983” really corresponds to the sixth column. Now, following Servois' general rules, find the circle which contains January and move horizontally until you are over the “medallion” that contains 13 (you do not need to move at all). You will notice that January 13, 1983 was a Thursday.

Our last three examples have demonstrated that, while this calendar may be fun to play with, it is probably not very efficient for everyday use because of the tedious calculations and the number of rules and special cases that are required for its use.

We have not given examples of Servois' third and fourth general questions. We encourage you to explore these questions on your own before consulting the translation of Servois' instructions and examples that appears on page 9.

Gergonne, editor of the *Annales,* attached the following footnote to page 88 of Servois' [1813] article:

I do not know if it has yet been remarked that the Persian intercalation, I mean that of 8 days over 33 years, slightly more accurate than the Gregorian intercalation, could be brought back in a completely remarkable way because of its precision and uniformity. To do this would require adding one day every four years, to delete it every century, to restore it every four centuries, to delete it every ten thousand years, to restore it every forty thousand years, and so on. Indeed, this would give the length of the mean year as, \[365^j + \frac{1}{4} - \frac{1}{100} + \frac{1}{400} - \frac{1}{10000} + \frac{1}{40000} - \ldots\] or \[365^j + \left(\frac{1}{4} + \frac{1}{400} + \frac{1}{4000} + \ldots \right) - \left(\frac{1}{100} + \frac{1}{10000} + \ldots \right)\] or \[365^j + \frac{25}{99} - \frac{1}{99} = 365^j + \frac{24}{99} = 365^j + \frac{8}{33}.\]

Here, Gergonne used the superscript \(j\) to abbreviate the word *jours* (days).

The process of intercalation refers to the insertion of a leap day in a given year to keep the calendar synchronized with the astronomical year [Richards 1998]. For instance, the Julian calendar had an intercalation of one leap day added every four years, which gave a mean of 365.25 days per year. The Gregorian intercalation consists of a leap day added only in years that are divisible by 4, except that years that are divisible by 100 do not get a leap day unless they are divisible by 400. This intercalation rule gives the Gregorian calendar a mean of 365.24 days per year. However, the Gregorian intercalation is off by 26 seconds per year, so that after about 3000 years, the calendar would be about one day ahead of the astronomical year [Richards].

In his footnote, Gergonne referred to the Persian intercalation, which is eight intercalations in thirty-three years. In other words, every thirty-three years eight leap days would need to be added. This gives an error of 19.45 seconds as opposed to the Gregorian error of 26 seconds [Richards]. This intercalation is attributed to the mathematician Omar Khayyam (1048–1131), who was appointed by Sultan Malik Shah of Khorasan (in Persia) to reform the calendar in about 1079 CE [O’Connor and Robertson 1999]. According to O'Connor and Robertson [1999], Khayyam measured the length of the year to be 365.24219858156 days, which was very precise given that modern astronomical precision allows us to measure the solar year at 365.242190 days.

The final portion of Servois' [1813] paper included a table, created by Servois, which could be used to find the date of Easter. His table was created using the algorithm introduced by Johann Carl Friedrich Gauss (177–1855) in 1800. We present a brief discussion of Gauss' algorithm, and then present Servois' table.

**Figure 7.** Carl Friedrich Gauss. Painting by G. Biermann (1824–1908) (public domain).

Easter is a moveable feast, which means it is not fixed to a particular day of the calendar year. The First Council of Nicaea (325 CE) defined the date of Easter to be the first Sunday after the occurrence of the Paschal Full Moon (the first full moon that occurs on or after March 21) [Richards 1998]. This definition led to the creation of the Easter calculation as a scientific problem for mathematicians and astronomers in the Christian West. Readers interested in an in depth discussion of the date of Easter Sunday should refer to [Richards, pp. 346-383]. There have been many attempts at creating algorithms to calculate the date of Easter; however, here we discuss only Gauss' attempts. Readers interested in a detailed history of Gauss' work should refer to Bien [2004].

In his 1800 paper, “Berechnung des jüdischen Osterfestes,” Gauss presented his calculation for the date of the Paschal Full Moon and the date of Easter. His method has the benefit of being simple, so that a person who is not savvy with mathematics can use it. It does not appear that Gauss created this formula for everyday use. It might be a legend, but some historians, including Bien [2004], claim that Gauss' motive for creating this formula was to calculate his own birth date. His mother had told him that he was born eight days before the Feast of the Ascension on a Wednesday in 1777. Using a slightly different method from the one that we describe below, Gauss was able to find that he was born on April 30.

The formulas for Gauss' algorithm can be found in any encyclopedia; however, some of them are a little mysterious. Reints [2009] provides concise but complete explanations of the meanings of the formulas and of the values used in them. We now provide a brief description of these formulas following Reints [2009]; readers interested in further details should refer to this source.

The details of the calculation are carried out by associating the year \(y\) with a sequence of values \(m,\) \(n,\) \(a,\) \(b,\) \(c,\) \(d,\) and \(e,\) which will result in the determination of the date of the Paschal Full Moon and Easter Sunday in year \(y.\) Gauss stated that for the nineteenth century, one should take \(m = 23\) and \(n = 4.\) What are \(m\) and \(n\)? The value of \(m\) is dependent on the century in question. This value is calculated from the cycle of the dates on which the Paschal Full Moon occurred in that century [Director 1997]. There are about 235 lunations (mean duration of one lunar phase) in 19 tropical years. (A tropical year is the time it takes for the sun to return to the same point in the sky relative to the background stars, as seen by an Earthbound observer.) But these two periods of time are not exactly equal. There is an error of about one day every 310 years, so the value of \(m\) takes this shift into account. The value of \(n\) is based on the cycle of the days of the week on which the Paschal moon occurs in that century. In other words, \(n\) works according to a mod 7 system, with 0 representing Sunday, 1 representing Monday, etc. Now, using the values of \(y,\) \(m,\) and \(n,\) you calculate the following:

- First, calculate \(a = y \mod 19.\) This identifies the location of the year within the Metonic cycle, the 19-year cycle of the phases of the moon alluded to above.
- Next, find \(b = y \mod 4.\) This formula accounts for the 4-year cycle of the leap years in the Julian calendar.
- Then, calculate \(c = y \mod 7.\) This takes into account that a non-leap year is one day longer than 52 weeks.
- Then, calculate \(d = (19a + m) \mod 30.\) This gives the number of days that need to be added to March 21 in order to find out the date of the Paschal Full Moon.
- Finally, find \(e = (2b + 4c + 6d + n) \mod 7.\) This is the number of days from the Paschal Full Moon to the next Sunday.

The Paschal Full Moon will occur \(d\) days after March 21 and Easter Sunday will be March \((22 + d + e).\) If \((22 + d + e)\) is greater than 31, then you roll over into the month of April.

Now, the length of a lunar month is not 30 days, but rather 29.53 days. Therefore, when the value of \(d\) is large this causes the Paschal Full Moon to be one day late. Additionally, if it happens on a Sunday, then we calculate a date of Easter which will be one week overdue. To compensate for this possibility, you must watch for this condition:

- If \(d = 29\) and \(e = 6,\) then replace April 26 with April 19.
- If \(d = 28,\) \(e = 6,\) and \(a > 10,\) then replace April 25 with April 18.

Again, the preceding explanation was taken from Reints [2009], except where noted.

Richards [1998] states that Gauss' 1800 method had a defect in it, namely that Gauss did not correctly account for the lunar equations of the Paschal Full Moon. In other words, there were errors with the equations to calculate the values of \(m\) and \(n.\) (Interestingly, in the second edition of his article, Gauss presented no expressions for \(m\) and \(n,\) but rather gave a table for the values up to the year 2499.) However, his error would not affect his calculations until the year 4200 CE. Bien [2004] reports that there actually were numerous errors in Gauss' original algorithm, and Gauss did not correct all the errors until 1816.

Now, what did Servois do with Gauss' algorithm? Servois used Gauss' 1800 algorithm to create an efficient table which would calculate the date of the Paschal Full Moon without any calculations. (See Figure 8.) Once an individual had the date of the Paschal Full Moon, then she could use Servois' perpetual calendar to find the date of the Sunday after the Paschal Full Moon and that would be Easter Sunday. Therefore, for the nineteenth century, one would need no algorithm to determine the date of Easter Sunday. However, if others tried to imitate Servois' table and create similar tables for different centuries, then, like Gauss' algorithm, these tables would fail to work after the year 4200 CE.

**Figure 8.** Servois' table for Gauss' algorithm for the date of Easter. (Image used courtesy of the Science, Industry and Business Library of the New York Public Library.)

How does Servois' table (see Figure 8) work? From item (1.) of Gauss' algorithm, the phases of the moon occur on different calendar days from year to year; however, they repeat in cycles every 19 years [Director 1997] and this is the principle upon which Servois' table works. The decades are on the left hand side of the table and the units are above, which Servois compared to a table of logarithms. As a general note, dates that are less than 20 in the table belong to the month of April.

Let's work though an example using Gauss' algorithm and we shall see how efficient Servois' table really is. We will calculate the date of the Paschal Full Moon and the date of Easter Sunday for the year 1827 (the year of Servois' retirement). We let \(y = 1827,\) \(m = 23,\) and \(n = 4.\)

- \(a = 1827 \mod 19 = 3.\)
- \(b = 1827 \mod 4 = 3.\)
- \(c = 1827 \mod 7 = 0.\)
- \(d = \left(19(3) + 23\right) \mod 30 = 20.\)
- \(e = \left(2(3) + 4(0) + 6(20) + 4\right) \mod 7 = 4.\)

Therefore, the Paschal Full Moon occurred on April 10, 1827, and Easter Sunday was April 15, 1827.

Now, let's evaluate the efficiency of Servois' table. In Figure 8, go to the row “182” and over to the unit “7” and you will find that the date of the Paschal Full Moon was April 10, 1827. Turning to Servois' perpetual calendar, we find on which day of the week April 10, 1827 occurred. According to Servois' calendar (see Figure 6), April 10, 1827, was a Tuesday. Therefore, Easter Sunday would have occurred the following Sunday, which was April 15, 1827.

The mathematician Jacques Français (1775-1833) had the following to say regarding Servois' perpetual calendar:

The examination of the ingenious table of triple entries given in this volume (page 84) by Mr. Servois engaged me to review, in the

Correspondance astronomique et géographiqueof Mr. the Baron Zach (August 1800), the article of Mr. Gauss which gave him the idea, and where this illustrious geometer learned to find, without epoch, golden number, or dominical letter, the date of the Feast of Easter, for any year and present thusly, in two pages, all of the theory of the calendar, whether Julian or Gregorian [Français 1814, p. 273].

Interestingly, Français extended this compliment two months after Servois publically stated in the pages of the *Annales* that “I had long thought of calling the ideas of Messrs. Argand and Français on complex numbers by the odious qualifications of *useless* and *erroneous *..." [Servois 1814a, p. 228]. If Français had read Servois’ harsh criticism of his work before he submitted this letter, then we may conclude that Français was definitely a thick-skinned gentleman.

In summary, François-Joseph Servois was a man with a variety of mathematical talents ranging from geometry to calculus, who in 1813 added the workings of the calendar to his mathematical repertoire. Servois' calendar may not be considered ground breaking in the field of mathematics; however, Bradley may have said it best when he wrote that the calendar is “clever, and efficiently manages a large quantity of information in a small space ....” [Bradley 2002, p. 11].

This article provides teachers of mathematics with an original source to use with their students, which, unlike much source material, does not require advanced mathematical understanding. The article could be used as the foundation for many classroom activities in a number of different mathematics courses, as well as student-centered research projects.

For instance, the material in this paper could be used to create an activity in a high school or liberal arts mathematics course. There are many websites which provide easy instructions for creating perpetual calendars, such as Robinson [2002]. Teachers can create their own perpetual calendars and have students discover concepts, such as corresponding months and the behavior of leap years. Additionally, teachers can have students create their own perpetual calendars. Teachers could encourage multiculturalism through this project by allowing students to create perpetual calendars for different cultures or religions; for example, one based on the Hebrew calendar.

Additionally, the material in this paper could also be incorporated into a history of mathematics course. If possible, it is always better to learn about the history of mathematics from a mathematician's own words. Servois' “Calendrier perpétuel” [1813] can easily be used for such a purpose. Students in a history of mathematics course could investigate the mathematical advances occurring in France around this time, other mathematical works of Servois, or the history of the calculation of the date of Easter Sunday.

Furthermore, Servois' 1813 paper provides students with an opportunity to conduct research on the history of mathematics. Open questions include, for instance:

- Who was the Baron de Zach? Besides being an editor of several mathematical journals, did he make any contributions to the field of mathematics? Research on this topic should begin with Vargha [2005].
- Within Servois' paper, Gergonne mentioned the Persian intercalation. What other intercalations exist? How do they differ from the Gregorian intercalation? Readers interested in this question can begin their research by examining [Richards 1998, especially pp. 89, 93-94, 100, 179, 207, 214, 232, 289, 297, 310, 318].

**Perpetual Calendar**^{1}

By Mr. Servois, professor of the artillery schools^{*}

[84]^{2} The calendar, whose uses I shall explain, may be used to solve this general question, which includes four particulars: *Of *[85]* these four things: if any of these three are given: a year of the common era, the name of a month of that year, a day of the month, and the name of the day of the week which corresponds to this day, then what is the fourth?*

I shall provide some examples, which are always much more clear than explanations, and will demonstrate the advantages that can be drawn from this little calendar (*See the plate* [Figure 9]).

*PROBLEM I. Determine which day of the week corresponds to a certain day of a designated month of a given year?*

Example. *We wish to know, which day of the week corresponds to January 28, 1821?*

In the table, look for the column that includes the number 21 with which the year terminates; you will find that this is the first on the left. In the same column look for the word *January,* which you will find at the top, followed by *October*. Then, move horizontally along the first line until you find yourself vertically above the last of the medallions at the bottom, which contains only the date given 28. You will find the word *Sunday* on the circle at which you will stop, and will tell you that January 28, 1821 will be a Sunday.

** **** **

**Figure 9.** Servois' perpetual calendar appeared on page 92 of the *Annales.*

*Remark.* If the year is a leap year, that is to say, if the number formed by its last two digits on the right is a multiple of 4, then during the first two months, *January* and *February*, we must make use of the column that immediately precedes, on the left, the column that contains the indication, and use the last one if this column is the first one. This is a general remark.

Thus, for example, if we wanted to find the day of the week that corresponds to January 28, 1824, then because 24, which belongs in the 5th column, is a multiple of 4, and because January is one of the first two months, we must make use of the 4th column. There we find *January* followed by *October* in the fourth circle down. Then moving horizontally to the right until the last [86] column, below which the day 28 is found, the word *Wednesday* that we find in the circle upon which we stopped, tells us that January 28, 1824 must be a Wednesday.

*PROBLEM II. To determine which days of a designated month, in a given year, correspond to a certain day of the week?*

*Example.* We wish to know, what are the days in February which will be Sundays in the year 1836?

Because 36 is in the 6th column and is a multiple of 4, and because *February* is one of the first two months, I am using the 5th column. I look there for the word *February* that is at the top, followed by *March* and *November*, and I move horizontally until the word *Sunday*, which belongs to the last column. Otherwise, I look for the word *Sunday* in the 5th column, and I again move horizontally until I meet the word *February*. I fall once again on the last column and I read at the bottom that the Sundays in February 1836 will be 7, 14, and 21.

*PROBLEM III. To determine which are the months of a designated year, in which a certain day of the week corresponds to a certain date?*

*Example.* We wish to know, what are the months of the year 1825 that will begin on a Monday?

25 is found to belong to the 6th column in which I look for the word *Monday*. I move horizontally to the left, starting from this word, until I meet the first column, under which I find the day numbered 1, and I read in this circle that there is but the single month of August in the year 1825 that should begin on Monday.

If it were a matter of the year 1828, which is a leap your, we will first find the word *Monday* in the 2nd column, which immediately precedes the one that includes the number 28. Then moving to the left horizontally until the first column, below which lies the day numbered 1, we first find the months of April and July, which we reject, because they fall beyond the first two, and we had used the column that precedes the year. Then taking [87] the word *Monday* in the third column and moving horizontally to the first one, we meet the months of *September* and *December*, both of which we admit, because they fall after the first two, which are consequently the only ones of the year 1828 which begin on a Monday.

*PROBLEM IV. To determine which are the years in which a certain day of the week coincides with a given date of a designated month?*

*Example.* We wish to know in which years April 1st will be a Sunday?

The number 1 is found at the bottom of the first column and *April* is found in the lowest circle in this column, which also contains the word *Sunday*. We then conclude that the years that have April 1st on a Sunday are: 1804, 1810, 1821, 1827, 1832, 1838, 1849, 1855, 1860, 1866, 1877, 1883, 1888, 1894, etc.

Suppose it were a matter of one of the first two months of the year -- for example, if we wished to know which were the years in which the 7th of February is a Saturday. The number 7 is found in the last column, where the month *February* is in the 3rd circle. Then moving horizontally to the left, until the one that contains the word *Saturday*, we find that it is in the fourth column. However, we must reject all the leap years in this column and substitute the asterisks that we find there with the leap years of the next column, which gives 1801, 1807, 1818, 1824, 1829, 1835, 1846, 1852, 1857, 1863, 1874, 1880, 1885, 1891, etc.

*Remark.* This calendar is only prepared for the current century, but we will make it truly perpetual by a simple transposition of the numbers that express the years, from one column to another, in such a way that the number 00 is found in the 7th column, in the 5th, in the 3rd, or in the first column depending on whether the number to the left of the last two digits gives a remainder of 0, 1, 2, or 3 when divided by four, so that with only four tables, [88] we have a calendar that can be used for all centuries, past and future, at least until the error, presently negligible, does not become, by the accumulation of the centuries, large enough to require a new reform.^{**}

Mr. Gauss^{3} has given in the 2nd volume for 1802 of the excellent journal *Astronomico-Géographique*^{4} by the Baron de Zach,^{5} a method for calculating the epoch of Easter for each year. From this I deduced the following table,^{6} which would be easy to extend and which, for every year of the nineteenth century, gives the epoch of the full moon of March.

[89] The decades are to the left of the table and the units are above, as in tables of logarithms. Dates less than 20 belong to the month of April and the others to the month of March. The law of this table is very simple: by writing every one of the 30 days from March 21st to April 19th inclusive in a circle, these dates taken nineteen at a time, in the direct order, form the horizontal rows, and taken alternately nine at a time and ten at a time give the vertical columns.

Using this table, if we want to know the epoch of the full moon of March for the year 1854, we will find on the spot that it is April 12th. On the other hand, if we want to know in what years the full moon will fall on April 4th, we will find that this will occur in the years 1814, 1833, 1852, 1871, and 1890.

Also, as Easter is set as the Sunday that immediately follows the full moon of March, it is easy, using the combination of this little table with our calendar, to determine the epoch of Easter for each year and reciprocally to assign the years in which this feast falls on a designated epoch.

For example, if we want to know the epoch of Easter for the year 1852, because we have found that for this year, the full moon of March comes on April 4th and, as we have found elsewhere from the calendar, that April 4th is a Sunday, we then conclude that in 1852 Easter falls on April 11th.

[90] Conversely, if we ask in which years Easter will fall on April 1st, we have already seen that this day was a Sunday only in 1804, 1810, 1821, 1827, 1832, 1838, 1849, 1855, 1860, 1866, 1877, 1883, 1888, etc. On the other hand, in order that Easter falls on April 1st, the full moon of March must occur from March 26th to April 1st, inclusive, which occurs only in the years 1801, 1804, 1809, 1812, 1817, 1820, 1823, 1828, 1831, 1836, 1839, 1842, 1847, 1850, 1855, 1858, 1861, 1866, 1869, 1874, 1877, 1880, 1885, 1888, 1893, 1896, 1899, etc. Therefore, Easter will occur on April 1st only in the years 1804, 1855, 1866, 1877, 1888, etc.

* It was only through the plea of the editor of the *Annales* that Mr. Servois who had sent him this ingenious calendar, without attaching the least importance to it, was willing to permit it to appear in this journal, in which we think that it would not at all be out of place.

** I do not know if it has yet been remarked that the Persian intercalation, I mean that of 8 days over 33 years, slightly more accurate than the Gregorian intercalation, could be brought back in a completely remarkable way because of its precision and uniformity. To do this would require adding one day every four years, to delete it every century, to restore it every four centuries, to delete it every ten thousand years, to restore it every forty thousand years, and so on. Indeed, this would give the length of the mean year as, \[365^j + \frac{1}{4} - \frac{1}{100} + \frac{1}{400} - \frac{1}{10000} + \frac{1}{40000} - \ldots\] or \[365^j + \left(\frac{1}{4} + \frac{1}{400} + \frac{1}{4000} + \ldots \right) - \left(\frac{1}{100} +\frac{1}{10000} + \ldots \right)\] or \[365^j + \frac{25}{99} - \frac{1}{99} = 365^j + \frac{24}{99} = 365^j + \frac{8}{33}.\] J.D.G.^{7}

*Calendrier perpétuel*, an article in*Annales des Mathématiques pures et appliquées***4**(1813-1814), pp. 84-90. Some citations might begin with the word*Chronologie*because the headline in Gergonne's*Annales*is the editorial category to which the article was assigned.- Numbers in square brackets represent the original page numbers of the article in Gergonne's
*Annales*. - Johann Carl Friedrich Gauss (1777-1855).
- Servois stated that Gauss' article [1800] appeared in the
*Astronomico-Géographique;*however, this paper actually appeared in the*Monatliche Correspondenz*. Around 1813, the*Monatliche Correspondenz*was no longer going to put out publications and there were discussions of renaming the journal to*Astronomico-Géographique;*however, this never occurred. Interestingly, Zach started a new journal in 1818 named*Astronomico-Géographique*. Also, there appears to be a typographical error on Servois' part. Servois claimed that the paper appeared in the 2nd volume of the journal in 1802; however, this paper appeared in 1800. - Baron Franz Xaver von Zach (1754-1832).
- Image used courtesy of the Science, Industry and Business Library of the New York Public Library.
- In this footnote, Gergonne used the superscript \(j\) to symbolize the word
*jours,*which means days.

**Acknowledgments**

The author expresses deep gratitude to Robert E. Bradley, professor of mathematics at Adelphi University, for graciously dedicating so much of his time, patience, and knowledge in editing my translation of Servois' “Calendrier perpétuel” and providing general editorial comments for the entire manuscript.

Additionally, the author is extremely grateful to the referees for their many helpful suggestions and corrections.

**About the Author**

Salvatore J. Petrilli, Jr. is an assistant professor at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received the Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His research interests include history of mathematics and mathematics education.

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