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When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home.

Mathematics and the Imagination, New York: Simon and Schuster, 1940.

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# Servois' 1813 Perpetual Calendar, with an English Translation

## Servois' Perpetual Calendar

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' article on the perpetual calendar, Gergonne attached the following footnote:

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 [p. 84].

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

1. To determine which day of the week corresponds to a certain day of a designated month of a given year.
2. To determine which days of a designated month, in a given year, correspond to a certain day of the week.
3. To determine which are the months of a designated year, in which a certain day of the week corresponds to a certain date.
4. 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.

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Petrilli, Jr., Salvatore J., "Servois' 1813 Perpetual Calendar, with an English Translation," Loci (June 2012), DOI: 10.4169/loci003884