# Presentation: 158474

## Topics in Math & Science:

***For Ira Myers, who was brilliant! ***
***Edited: Ruth Petersen***

***Topics in Math & Science:***
***HAVE YOU LOOKED AT YOUR CALENDAR LATELY?***

**Notes:**
This talk is respectfully dedicated to the late Dr. Ira Myers, a physicist at NASA Lewis Research Center (now renamed the John H. Glenn Research Center in honor of another truly great person), and possibly the most brilliant individual I have ever personally met. It began as a casual conversation between us many years ago, in which he demonstrated these same calendar calculations right off the top of his head! He customarily did calculations of this type spontaneously and in all areas of physics and mathematics, demonstrating a superb memory for facts and numbers, and a remarkable agility in applying them to achieving the ends of a mathematical physicist.
Thank you, Ira, for this gift which I now venture to share with students across the country. Respectfully, J.C.K.

## Preliminary Activities

***Find a dictionary with good word origins (etymologies: I recommend ******Merriam-Webster******) and look up the ******day****** and ******month****** names.Think of yourself as an archaeologist of ideas! You will be amazed at the amount of history you will encounter. Be prepared to discuss your results!***
***Look up the word ******MARS******, and read both the definition and the etymology. What do you learn?***

***.***

**Notes:**
What is an etymology, and how does it differ from a definition?
What was Aristotle’s concept of ‘definition’?
What can word etymologies teach us?

## Look up the word MARCH. What connection do you find between Mars and March?

***Assume that you have a calendar of 365 days per year. How much will the calendar get out of step with real time if the tropical* year is actually 365.25 days long? (Make your estimate in days per year).***
***How much will your calendar get out of step in one century??***

**[******The length of the year used by astronomers, measured according to the sun’s motion. We now know that one tropical year is 365.2422 days.*****]**

***.***

**Notes:**
The calendar is a marvelous collection of historical facts and mythological references. One has only to take the time to investigate this most commonplace of all modern artifacts to discover the richness of its content! I challenge anyone who reads this to take up your calendar, your dictionary, and your calculator and have at it! You will be absolutely amazed at what you can learn!

## How would you fix the problem?

***How would you fix the problem?***
***How many days per year (on the average) would your new calendar have?***
***Now use the actual tropical year of 365.2422 days. How much will your new calendar get out of step with real time?***
***Can you devise another “fix”?***
***Can you predict the accuracy of THIS new calendar?***
***Finally: ******HAVE A CALENDAR CLOSE BY TO ANSWER SOME ADDITIONAL IN-CLASS QUESTIONS!***

***.***

**Notes:**
These notes are a start into the history and mathematics of one of the most amazing, and yet taken-for-granted, timepieces ever invented and used by the human mind.

## Have you ever taken a careful look at your pocket calendar? You might be amazed just how much history, mythology, mathematics, and astronomy you would find there!

***Have you ever taken a careful look at your pocket calendar? You might be amazed just how much history, mythology, mathematics, and astronomy you would find there!***

***.***

**Notes:**
Have you ever taken a careful look at your pocket calendar? You might be amazed just how much history, mythology, mathematics, and astronomy you would find there!

## You should have already discovered that:

***You should have already discovered that:***

***Please use the results of your preliminary activities to tell us what you have learned about some of the day and month names and their sources.***

***The day names derive from ancient Norse and Teutonic sources.***

***The month names derive from Ancient Rome!***

***.***

**Notes:**
Sunday: Sunnandaeg – Sun’s day
Monday: Monandaeg – Moon’s day
Tuesday: Tiwesdaeg – Tiu’s day (for Tiu, ancient Germanic god of war)
Wednesday: Wodnesdaeg – Odin’s day (for Odin, supreme god and creator in Norse mythology)
Thursday: Thunresdaeg – Thor’s day (for Thor, Norse god of thunder, weather, and crops)
Friday: Frigedaeg – Frigga’s day (for Frigga, Odin’s wife, and goddess of marriage and the hearth)
Saturday: Saeterndaeg – Saturn’s day (for Saturn, Roman god of agriculture)
OK – now YOU do the month names!!! 

## Next, please use your results to tell us the role Mars plays in our calendar.

***.***

**Notes:**
Mars was the Roman god of war. You have only to anthropomorphize (see next chart) modern warfare to realize just what a powerful and fickle deity Mars must have been! In order to propitiate him, the ancient Romans performed numerous rituals, offered sacrifices, and even gave him star billing on their calendar. Our month name, “March,” derives, etymologically, from, “Mars.” March was the first, not the third, month of the ancient Roman calendar.

## 1. The ancients understood the sky as being populated by various gods.

***1. The ancients understood the sky as being populated by various gods.***

***2. These gods were eternal, like the fixed stars.***

***3. But the planets were mysterious wanderers whose changing positions were considered important to events on earth.***

***4. Mars the war god, was particularly to be feared . . .and respected . . .***

***.***

**Notes:**
To understand how the ancients thought, try to express puzzling (and not-so-puzzling) phenomena strictly in terms of human characteristics rather than using the mathematical abstractions which we have available to us today. Remember, the ancients did not have these abstractions: they had only themselves and their societies as models with which to explore the unknown! We call this process, anthropomorphization (from the Greek anthropos, human and morphos, form).

## From your word studies, what can you say about Mars’ role in the ancient Roman Calendar? 

***From your word studies, what can you say about Mars’ role in the ancient Roman Calendar? ***

***What can you say about the number of days in the ancient Roman calendar?***

**Notes:**
Since military campaigns were most likely launched in Spring to optimize the use of the warm seasons (summer and early fall), March was assigned to the spring. Since the ancient Romans did not have accurate means for making astronomical measurements, they were not aware that the solar year was not an exact number of days in length; hence, their calendar had 365 days in each year.

## .

***.***

**Notes:**
Let’s review!

## If you begin the calendar with March, then what numerical positions do the last four months of our calendar occupy?

***What have you learned from you etymology studies about the names of ******these ******months in our calendar?***

***Do you see a connection, or are you puzzled?***

***.***

**Notes:**
March	- 1				August	- 6
April	- 2				September	- 7
May	- 3				October	- 8
June	- 4				November	- 9
 July	- 5				December	- 10

## September – Latin, Septem, 7

***October – Latin, Octa, 8***

***November – Latin, Novem, 9***

***December – Latin, Decem, 10***

***But ... these months now occupy positions 9, 10, 11, & 12 on OUR calendar! How did this happen?***

***.***

**Notes:**
Our present day calendar actually contains etymological artifacts extending all the way back in time to the Ancient Romans! It also preserves for us clues to changes made in the calendar reforms of antiquity!

## The calendar gradually fell out of step with days, months, and seasons as determined by the actual position of the sun until, by the time of Julius Caesar, it was badly in need of reform.

***The calendar gradually fell out of step with days, months, and seasons as determined by the actual position of the sun until, by the time of Julius Caesar, it was badly in need of reform.***

***.***

**Notes:**
Because the tropical year is not exactly 365 days, but is just a little longer, the ancient Roman calendar drifted with each passing year. Since the calendar year turned over 0.2422 days before the solar year, it actually fell behind times and seasons.

## Date = 46 B. C.

***Date = 46 B. C.***

***With the help of Egyptian astronomers, Julius Caesar reformed the calendar to have one year out of every four possess an extra day - leap year!****** ***

***He also added two more months ***_***IN FRONT OF***_*** March... ***

***...******& September went from being #7 to #9, etc.***

***.***

**Notes:**
One wonders, were there no astronomers in the Roman Empire worthy of doing the calculations that the great Caesar needed? Evidently not – at least in Caesar’s opinion! By the way, Julius Caesar named a month for himself. Do you know which one it is? So did Caesar Augustus. His should be intuitively obvious!  

## So, with 3 years of 365 days each, and 1 year of 366, Julius Caesar REDEFINED the tropical year.

***From your preliminary activity calculations, how long is this new *****“*****Julian” year?***

***.***

**Notes:**
OK: Time for some number crunching!

## .

***.***

**Notes:**
The weighted average is as important a concept, here, as is the 365.25 days in Julius Caesar’s tropical year. Weighted averages are used throughout statistics and in physics: particularly, in gas dynamics and quantum theory.

## Please Note: When a = b = c = ... m = 1, then:

_***Please Note***_***: ******When *****a = b = c = ... m = 1, *****then*****:**
******x******** = (x****1**** + x****2**** + ... + x****n****)/n**

**Notes:**
The weighted average is a more general case of the usual concept average. The usual concept is shown on the bottom of the slide: n variables are added and the result is then divided by n. The weighted average is shown in the definition above: n variables, each with respective weights a, b, ..., m, are added and the result is then divided by the sum of the weights. If you wish, the variable x1 may be thought of as being repeated a times, the variable x2 as being repeated b times, and so on.

## Problem: Julius Caesar’s calendar, on the other hand, was too long by

***365.25 - 365.2422 = 0.0078 days.***
***Estimate the accuracy of ***_***this***_*** calendar.***

***.***

**Notes:**
Caesar’s calendar drifted, too, because, while his estimate of the tropical year was closer to the truth, it was still not exact. Unlike the ancient Roman calendar, Caesar’s calendar year turned over 0.0078 days after the solar year; so it gained on the solar year rather than falling behind. We shall see that the next calendar reform actually had to eliminate some days in the reform year to bring timekeeping back on track!

## 128

***(0.0078 days/year) = 1 day/_____ years***

***128***

***.***

**Notes:**
The concept of simple proportion is used to do this calculation:
0.0078 days:1 year::1 day:x years
  X = 128.2 years ~ 128 years
Caesar’s calendar gained one day every 128 years (approximately).

## .

***Well, by the 16th century, ***
***the equinox had slipped again, this time  by 10 days, from March 21 to March 11.***

***Pope Gregory XIII ordered***
***that 10 days be dropped from the calendar, and that years ending in hundreds be leap years ***_***only***_*** if divisible by 400.***

***.***

**Notes:**
At this time in Western history, the Catholic Church held considerable sway over human affairs. Pope Gregory constituted a committee to study the accumulated calendar data and to do all of the necessary calculations for his calendar reform. He also laid the groundwork for the Vatican Observatory, the oldest observatory in the Western world. His ‘Rule of 400’ eliminated nine leap years out of every 1200 years. The Gregorian calendar is used throughout the Western world to this day. Other societies (Middle East and Eastern, for example) still use the Julian (or other) calendars.

## Interesting Note...

***Interesting Note...***
***History records***
***that non-Catholic countries in Europe did not accept the change immediately.***
***Great Britain did not accept the change***
***(the *****“*****New Style******” calendar)***
***until 1752.***

***We still use the Gregorian calendar in the Western World... ***

***.***

**Notes:**
The dropping of ten days from the calendar in 1582 caused a great deal of confusion (and not a little bit of superstition!) throughout Europe. For example, what would a person do whose birthday fell on one of the dropped days? During the time between 1582 and 1752, therefore, and especially in England, dates were often written with two day numbers, such as: November 23/13 1742. Hence the “old style date,” and the “new style date,” went hand in hand for almost two centuries.

## Hint: In Caesar’s calendar, ALL years divisible by 4 are leap years!

***Hint: In Caesar’s calendar, ******ALL****** years divisible by 4 are leap years!***

***Pope Gregory “reset” the calendar by eliminating 10 days and specifying that years ending in hundreds be leap years only if also divisible by 400.***
***Problem:****** What can you say about the accuracy of Pope Gregory’s calendar?***

***.***

**Notes:**
And now, folks: Time for some more number crunching!!! 

## Answer: In 1,200 years:

***A total of ******300 years****** are****** ******divisible by ******4******, ******leaving a total of ******900 years****** not divisible by ******4******.***
***(Starting with AD 100), there are****** ******12 years ending in “00”****** that are ***_***possible***_*** leap years, ******BUT...***
***Only ******3 such years out of every 12 are ***_***also***_*** divisible by 400****** *****(*****e.g.,***** {*****400, 800, 1200*****}*****, *****{*****1600, 2000, 2400 *****}*****, etc. ******Try it for yourself!!! *****);***** so, ******only 3 years ending in “00” are ***_***actual***_*** leap years.***

***.***

**Notes:**
The number 1,200 is divisible by both 4 and 400. So I chose it as a convenient number of years with which to work. You may try other numbers of years, if you wish. If you do everything correctly, you will obtain the same result as is obtained here.

## Since 12 - 3 = 9, the Gregorian calendar eliminates 9 leap years (ending in “00” ) out of every 1,200 years.

***Thus, 300 - 9 = 291 years out of every 1,200 are leap years, and 900 + 9 = 909 are regular years.***

***.***

**Notes:**
So, 9 leap years are lost while 9 non-leap years are gained in the 1,200 year group we have chosen to work with.

## And . . .

***And . . .***

***365.2425 - 365.2422 = ******0.0003 day/year****** ***

***.***

**Notes:**
There’s that weighted average again!... Finally, comparison with the modern tropical year gives a measurement of accuracy.
Well, we have gone from an accuracy of 1 day every four years, to an accuracy of one day every 128 years, to an accuracy of 1 day every 3,333.33... years in just under two millennia. That’s progress. Sometimes knowledge which is truly worth while is hard won, and obtained by a lot of effort. Think how many people it took to fulfill President Kennedy’s mandate to go to the moon within a decade! Sometimes we take too much for granted those things which are the genuine treasures of our Western cultural heritage!!!

## Finally:

**(*****0.0003 day/yr*****)****-1***** = 3,333.33...yr/day***
***Giving an accuracy of ******1 day every 3333 years******!***

***.***

**Notes:**
And this is where we stop. Yes, it is possible to reform the calendar further, given the precise astronomical measurements available to us today. But one day in more than 3,000 years seems to be accurate enough even for 21st century time keeping, so things are being left alone...
...at least for the moment!

## And that about does it!

***And that about does it!***
_***For your post-conference activity***_***---***
***Next time you look at a calendar,***
***or a clock, or a street name, or whatever, stop to ask yourself, *****“*****What’s in it?”***
***Use your dictionary and your math skills and pry into things.***
***Even in the most everyday things, you will usually find *****“*****Far more than***
***meets the eye!”***
***Ciao!***

***.***

**Notes:**
CIAO!!!

## For those interested in talking more, contact me at:

***joseph.c.kolecki@grc.nasa.gov ***

***.***

**Notes:**
Please feel free to contact me!