## Introduction

Despite the fact that there is a lot of science involved in today’s time and calendar management, the start of this journey was based on very flexible logic, one of the earliest Roman Calendars was based on a thirty day lunar cycle and included an eight-day market week. In reality this eight-day week did not have any scientific logic but was based on the fact that this was the time taken for processing of Goat’s cheese. The first calendar included four months of thirty days each and March, April, May and June were the months that were named based on roman mythological beings. [ CITATION bea11 \l 1033 ]

It is believed that around 738 B.C. six, more months were added by Romulus, the founder of Rome, after which Numa Pompilius the next ruler finally added January and February to complete the entire year which had 355 days in total, this calendar did not include the eight-day week however was pretty much according to the monthly lunar cycle.[ CITATION cal11 \l 1033 ] In this paper we will look at the various aspects of the development of Calendar to reach today’s stage and we will also analyze the importance of calculations and will see graphical representation of the modern astronomical studies on Calendar.

It is true that with the evolution of different civilizations, not only language and cultures but even calendars were individually developed. As of now there are various calendar systems in use today, to name a few of them we can say Hebrew, Chinese, Hindu and Ethiopian etc. however the basic concept behind all these is the periodic motion of the Sun and the Moon. [ CITATION Wilns \l 1033 ]

In order to understand this concept lets see that one year is actually equal to 365.24219878 days and a lunar month (on the basis of moon’s rotation cycle) is 29.530589 days, now if we find a ratio between these two we will realize that it comes to 12.36826, and this also the number of lunar rotations in a year. Hence we have 12 months with each month approximately equal to one lunar rotation. The oldest calendar known was actually the Babylonian Calendar and it basically consisted twelve months, and each month has 29 and 30 days alternatively making each year complete with only 354 days, again Egyptian calendar later corrected this issue and added 30 days to each month making it 360 days in a year. [ CITATION Fre05 \l 1033 ]

The discrepancy in the Egyptian calendar was soon noticed as per the total rotations by the moon and sun and then during the Pharaonic times, there were five more days added to the year making it equal to the current calendar with 365 days. There was still a discrepancy related to the leap year and it was understood that the normal year was understood to be 365.25 years which is close to the solar cycle year 365.24219878, this was studied well and in 1582 Pope Gregory XIII, created a team to work on this issue and then finally the astrologer Aloysius Lilius recommended the leap year system and the new year started getting celebrated from March 25th to January 1st of each year. [ CITATION cal11 \l 1033 ]

These methods and developments resulted in the final modern calendar however there is more science related to it and we will see that how we have come a long way by the merger of science and astrology in the development of modern calendar system.

## The relationship between Continued Fractions and Calendar System

The astrological and mathematical systems have been able to explain some remarkable theories of time management, and one of the most interesting topics of study is the expansion of continued fractions. These have really helped to make great approximations on the basis of convergent αn.

Theorem 1 Every convergent αn is a good approximation (in the sense of Deﬁnition 1) for α and conversely, every good approximation for α is one of the numbers αn for some n ≥ 1. In fact qn is the smallest integer q > qn−1 such that |qα − p| < |qn−1α − pn−| for any integer p. [ CITATION Ser95 \l 1033 ]

## Now lets see though a graphical representation that how this is helpful.

Now in order to find the approximation in relation to the numbers, we need to observe the convergent of √2 in the above graph with respect to π. We will then hold the above theorem true to find the relative values as per the above relation. [ CITATION Ser95 \l 1033 ]

It is clear that the approximations match when the value is q= 2, 5, 12 & 29, hence the theorem holds true, now with the above theorem it is easy to plot and predict the approximations related to the Astrological numbers. This was actually used in the calendar approximations by Christian Huygens, now let’s see the application of this theory with respect to the calendars and notice that how this can provide a revolutionary accuracy to the system.[ CITATION Ser95 \l 1033 ]

## Continued Fraction: The application

The modern calendar works on the concept as per which there is a cycle that spans for q years out of which p years are leap years, and q-p are non leap years, now we choose a value for q and p in such a manner that the mean of these two is closest possible to the astronomical year. If we consider the cycle of these years to be 365q+p days, then the length of each year will become 365+p/q, and in that case the value of p/q as per our knowledge will be α = 0.24219878 from the above theorem of convergence if we draw a graph to find the sequence we will realize as mentioned below: [ CITATION Wilns \l 1033 ]

P1/q1=1/4, p2/q2=7/29,p3/q3=8/33, p4/q4=31/128 and p5/q5=163/673

Now if we observe the above results closely, we will realize that the first cycle presents a single leap year in 4 years cycle which is like the Julian Calendar system, however the other approximations suggest us cycle lengths of 29,33,128 and 673 years. Now among the options discussed the 33 years cycle is the most preferred one as it gives a very precise and close number related to the current calendar. [ CITATION Wilns \l 1033 ]

## The Calendar Error

This is something which has really interested various astrologers and mathematicians since centuries; the above application of continuous fraction is definitely helpful in developing a precise and accurate however it is also capable of telling us about the approximate calendar error which takes place after few centuries due to the difference between the leap year calculation system in the Solar and Gregorian calendar format.[ CITATION Fre05 \l 1033 ]

According to the Gregorian calendar format there will be around 97 leap years in a span of 400 years, now the same as compared to the Solar year would make some 121 leap years in the span of 500 years and per the above calculation there is a difference of 26 seconds between the Gregorian Year and the Solar year which makes it a 1 days error in 3,320 years. Considering the above data if we draw a graph, plotting the two calendar options together for a time period of 100 years we will see that:[ CITATION Fre05 \l 1033 ]

The saw tooth-oscillations are the leap year insertions as per the calendar regulations during the period of time being observed, now we know that every year ending with 00 has to be an exception to leap year and in case if we avoid the same we will realize that there will be a calendar error due to which the changes will take place and the graph will look somewhat like as mentioned below.

Therefore, the graph below shows that there will be a difference of two days in just 400 years bringing a great error in the time calculation; hence there is an exception of no leap year at “00” so that the above error may be avoided and that makes the current calendar very accurate.

## Conclusion

We have come a long way with our developments related to calendars and we have definitely used the most latest science and technology to create the calendars with maximum accuracy, it is important to understand that the time calculations have to be error free as they have a lot relativity with regards to astrological science, as we have seen above and therefore regardless of the calendars used by different civilizations in ancient times, there is a common use of the modern calendar across the world in today’s scenario which makes it easy for everyone to understand and work according to the geographical time differences.

## Works Cited

beaconlearningcenter. History of the Calendar. 14 May 2011 <http://www.beaconlearningcenter.com/documents/1390_01.pdf>.

calendar-origins.com. Roman Calendar. 14 May 2011 <http://calendar-origins.com>.

Fredrick, Ricky V. “Mathematics of the Gregorian calendar.” Math. Intelligencer, Vol.7 No.1 (2005): 53-56.

Grabovsky, Yury. Modern calendar and continued fractions. London: Temple University, 2008.

Serge, Lang. Introduction to Diophantine approximations. 2nd Ed. New York: Springer-Verlag, 1995.

William, John B. Thron, Wolfgang J. Continued fractions. 1980: Addison-Wesley, Encyclopedia of Mathematics and its Applications.