John B Conway

Mathematics Department

University of Tennessee

I am a mathematician.  Relax.  I am not going to do much if any mathematics.   This lecture is about something most of us take for granted and seldom ever think about.  Time!

When is the last time you thought about time?  In some ways the millennium has made us all a bit more aware of time and I am no exception.  But my interest in time started several years ago.  I have a lot of curiosity --- about people, about nature, about mathematics, about history, about a lot of things.  One day, about three years ago, I made an observation while walking to have lunch in the University Center.  Though there are at least two systems of measurement for lengths and volumes (metric and English), there is only one system for measuring time.  Why is that?  Then I asked, ``Why are there 24 hours in a day?''  Why not 36 or 48 or 60?  Many additional questions followed.

This started me on a process of inquiry.  I put these questions to my friends, and no one gave me satisfactory answers.  I began reading.  I surfed the web. I thought a lot about what I read --- a good pastime while you are walking somewhere or waiting for an appointment.   I discovered many things about time and came to regard it as a very interesting subject.  I have sort of answered my original questions.  At least I have answered them to my satisfaction.  But I also asked myself many other questions along the way, some of which I have yet to answer.  In this lecture I'll share with you some things I learned, maybe raise a few questions you never thought about, and, I hope, provide you with a few insights into time and human nature.

Time has many faces.

I am going to concentrate on what I called Clock and Calendar time as embodied in clocks and calendars.  In other words, how do we measure time?  These are the topics that got me started and continue to interest me.


I am going to start with measuring Calendar time.  That is how history started, so why not us also.  For all of the early history of civilization there was no need for an accurate measurement of hours and minutes.  Farmers well into the middle ages awoke with the sun and the times morning, noon, afternoon, and night were sufficient for daily life.  But even prehistoric man had some need to get a calendar.

Why did prehistoric man need a calendar?  There was an advantage to knowing when the seasons began so that crops could be planted.  When will winter arrive?  When should we start our migration south?  When should the goat be sacrificed to ensure the success of the planting, or harvesting, or migration?  The Egyptians wanted to try to predict the flooding of the Nile.

Why do we have calendars?  Our society is more complex.  We need the calendar for some of the same reasons, but we have additional ones as well.  Commerce needs a calendar to plan deliveries and production.  We also want to record events.  We want to schedule events.

The history of the calendar has been propelled by such considerations and the state of the technology and science.  One of the things I discovered was the enormous influence on the making of the calendar by religions.

There are many calendars through the history of the world.  The most prevalent today is the Gregorian calendar, named after Pope Gregory XIII.  Though other calendars remain in use, this is the calendar that governs trade and commerce and is the common measure of the year.  So the answer to one of my original questions is that there are several ways to measure calendar time.

Other calendars that are in use are:

Other calendars have existed but are no longer used.  Many vanished civilizations had a calendar, such as the Mayans.  The Romans had a calendar that was a nightmare to figure out.  Another calendar of more recent vintage is the French calendar.  It was invented during the French Revolution in 1793 and was based on the decimal system.  It was abolished in January 1806 by Napoleon.

Suppose you lived in a world without a calendar.  How would you go about making one?  Why have 12 months?

The reason for 12 months is the 12 signs of the zodiac and the roughly 12 cycles of the moon in a year.

Why the varying number of days in a month?   Why are the months all around 30 days?  Why are there 7 days in a week?

Most calendars are based either on the moon, the sun, or a combination of the two.  The difficulty with making a calendar is the complete lack of cooperation by nature and religion.  Let's look at nature first.

The moon makes a complete orbit of the earth, on average, every 29.53059 days.

The earth makes a complete orbit of the sun, on average, every 365.2422 days.

The reason for the words ``on average'' here is that no two revolutions are the same.  This is about as contrary as you can get.

Given a certain attachment to the moon and its effect on tides (and a variety of beliefs of the ancients that it had even greater influences on human behavior -- the word lunatic has a common origin with the word lunar), 30 days seems a convenient number for some subdivision of the year.

So suppose you decide to base your calendar on the moon.  It is quite natural to say that one month has 30 days.  (Also the numbers 29 and 31 cannot be divided by any other number, so they have distinct disadvantages.  Thus twelve months equals 360 days.  You rather quickly get out of sync with the moon as well as out of sync with the rotation about the sun.  But if you are a lunar calendarist, you don't care about the sun.

The Islamic calendar is lunar based as were many early, pre-christian calendars.  The moon is easier to observe than the sun, so it offers some advantage.  It also seems to be the method of calendaring preferred by more agrarian cultures.  Of course to maintain coordination with the moon you have to have about as many months with 29 days as with 30 so that every month starts around the beginning of a full moon.

The Islamic year is about 11 days shorter than our calendar.   In fact,

365-(6 times 30 + 6 times 29) = 11

So the year number of the Islamic calendar is gaining on our calendar.  In the Islamic calendar, years are counted since the Hijra, that is, Mohammed's flight to Medina, which is assumed to have taken place 16 July C.E. 622 (Julian calendar).  We are presently in the Islamic year 1421 (I think).  In our year 20,874 the Islamic calendar will also be in year 20,874.  Let's hope that Islamic and western cucltures come together in mutual understanding sooner than the calendars match their years.

The Jewish calendar is based on both the moon and the sun.  Non-leap years have 12 months of either 29 or 30 days.  In leap years they add a thirteenth month.  An ordinary (non-leap) year has 353, 354, or 355 days.  A leap year has 383, 384, or 385 days.

 All these differences exist to keep bringing the calendar in line with the phases of the moon and the position of the earth in its orbit.

The Romans also had a calendar.  By all accounts it was a mess.  The Romans were lousy scientists --- what do you expect from a culture that had such an absurd numbering system.  (Try doing arithmetic in Roman numerals.)  Their calendar seems to have undergone frequent ad hoc changes to try to align it with the sun and moon.

Then along came Julius Caesar.  In his stay in Egypt, between courting Cleopatra and defeating armies, he discovered Egyptian science.  The Egyptians believed that the year was 365.25 days long.  More accurately, they said the year was 365 and 1/4 days long.  The Egyptians certainly did not have decimals.  They dealt with fractions, but their only concept of fractions had a 1 in the numerator.  Also their instruments lack the accuracy to have arrived at the 365.2422 day length of the year.

Back in the good old days of the Roman Empire, whatever Caesar said was what happened.  In 45 BC Caesar declares a new calendar.  This is the Julian calendar.  It has 12 months with the days distributed as we have them now and ever fourth year there should be a leap year with one extra day.  The trouble here is that a year is not exactly 365.25 days but 365.2422 days.

By the way, how is this measured?  What is used here is technically known as the solar year and is defined as the time between two vernal equinoxes.  (The vernal equinox is the time when the center of the sun crosses the equator from north to south.  It is usually around March 21, the beginning of spring, and at that time the length of the day equals the length of night.  This is in contrast to the autumnal equinox when the center of the sun crosses the equator from south to north, around September 22, the beginning of autumn.)

If nature hasn't confused you enough yet, there is another way that is used to measure the year.  The sidereal year is the time it takes the earth to make one complete orbit around the sun.  This is 365.25636 days, a little longer than the solar year.  The sidereal year is used by astronomers and sailors, where the position of the earth relative to the stars is paramount.  I believe that all solar based calendars use the solar year.

Back to Caesar and his Julian calendar.  Let's do a calculation.

Julian year
365.2500 days
Solar year
365.2422 days
Difference =
.0078 days
11.232 minutes

Therefore the Julian calendar gets one day ahead of the solar year every 125 years.  With time and the development of more accurate scientific instruments, this discrepancy was discovered.  In 1267 an English monk, Roger Bacon, observed that this had resulted in the then present year being 9 days ahead of the true calendar.  Bacon was teaching in Paris and wrote to Pope Clement IV alerting him to this fact.  Bacon called for reform, the need for which he felt overwhelming because it meant that Easter was being celebrated on the wrong day of the year.

Nothing happened until In I Pope Gregory XIII issued a papal bull adopting a new calendar that resulted from a commission he had set up to attack this problem.  The commission, led by the Bavarian mathematician Christopher Clavius and the Italian doctor Aloysius Lilius, proposed the calendar we use today.  There is a leap year every 4 years, like the Julian calendar, except in this calendar there is no leap year at the beginning of each century that is not divisible by 400 -- like 1700, 1800, and 1900.

Everyone did not adopt the Gregorian calendar.  Here is the time table of acceptance of the Gregorian calendar.

1582 Italy, Spain, Portugal, Luxembourg, France, Belgium, German Catholic States, Catholic Netherlands, Poland
1583 Austria
1584 Catholic Switzerland
1587 Hungary
1600 Scotland
1700 Protestant Netherlands, Denmark, German Protestant States
1752 Britain and Empire (including American colonies), Quakers
1753 Sweden
1812 Rest of Switzerland
1867 Alaska
1873 Japan
1875 Egypt
1912 China, Albania
1917 Turkey
1918 Russia
1919 Yugoslavia, Romania
1923 Greece
1924 Eastern Orthodox Church in Romania, Yugoslavia, and Greece

Indeed, to this day the Russian Orthodox Church uses the Julian calendar.  There was a human dynamic at play in this that involved both religion and politics.  There was no world organization, no entity, in existence with any international influence other than the Roman Catholic Church.  No one could have carried out calendar reform on an international scale other than the Roman Church.  On the other hand, the fact that it was proposed by the Roman Church guaranteed that many countries would immediately be opposed to the reform.

How was the change carried out?  In 1582 the Julian calendar was 10 days behind the Gregorian.  It was decided that on October 5, 1582 (Julian) they would suddenly switch the date to October 15 and begin using the new calendar.  (I don't know why October 5 was chosen, but I suspect it was connected to religious observance.  Perhaps that 10 day period had the fewest religious observances of consequence.)  There were many problems.  Suppose you had a loan and were paying interest.  How was you interest calculated that month?  How did they calculate rent for the month of October?  When is your birthday?

There have been several efforts to reform the calendar even further.  How about having a calendar when every February 8 is a Tuesday?  One possibility is to observe that

7 days per week $\times$ 52 weeks = 364 days

So one suggestion is that there be a blank day at the end of the year and January 1 always starts on a Sunday.  In leap years, have two blank days.  This was proposed by many, was debated rather strenuously in the League of Nations in the 1920s.  Alas, nothing came of it.  One point of opposition:  several religions got upset about violating the prescription to worship every seventh day.  An occasional blank day in effect creates an occasional eight day week and throws out the 7 day cycle of worship.

There is a flaw in the Gregorian calendar that is more serious than the inconvenience of shifting days on the calendar.  The Gregorian calendar overstates the length of the year by about 26 seconds.  This means that in about the year 4916 the calendar will be a full day ahead of the solar year and we will have to skip having leap year that year.

Question:  Will the world be sufficiently organized and harmonious to be able to come to the decision to drop February 29 in 4916?


Let's now turn our attention to clocks.  I am not going to spend much time on the history of the clock.  In fact the history of the clock is complicated.

What are some instances when we have to measure time accurately?

What are some instances when we have to measure time with some accuracy?
  • When we try to determine how many minutes we have remaining in an exam.
  • When we try to make a soft boiled egg.
  • What are some instances when we are rather casual about measuring time accurately. At other times, we are completely negligent about measuring time.  Anyone who has spent a couple of weeks at the seaside or motoring through the French countryside has probably asked, ``What day is it?''

    How did civilization get started in the quest for accuracy in time?  There were a number of instances in ancient time when some accuracy in time was required, but few if any where great accuracy was needed.  This remained the case until the age of the great sea explorations started.  The prime motivation for gaining accuracy in clocks was the desire to be able to judge longitude.  Like many scientific advances, there was a pressing problem that motivated the search for more accurate time --- how to navigate the seas safely.

    If you are trying to navigate the open seas, far from the sight of land and with no landmarks to guide you, you have to be able to determine with some accuracy your position on the globe.  This means finding your longitude and latitude.  These are the vertical and horizontal ``lines'' marking position on the earth.  (Which is which?  An easy way to remember is that the word ``latitude'' sounds a bit like ``ladder'' and the lines of latitude run horizontally like the rungs of a ladder.)

    Determining latitude is not a big problem.  Sailors from long ago knew how to do this with varying degrees of accuracy.  Some could have an intuitive idea of their latitude by the length of the day.  Others by the height of the sun above the horizon or the position of so called ``fixed stars.''

    Let's suppose you want to determine your latitude by finding the height of the sun.  Using a sextant or some such similar instrument, at noon you determine the angle between the sun and the horizon.  For the given day of the year, you know the angle the sun makes with the horizon at the equator (this was determined by your friendly astronomer and is contained in a book kept on board).  A comparison of these angles would then give you your distance from the equator and you can then read off your latitude.  It is a relatively starightforward exercise in geometry/trigonometry that would make a somewhat exacting exercise in a mathematics course.

    The point here is that you are perpendicular to the line made by the sun as it crosses the sky. So this simple angle measurement works.  The longitudinal lines run parallel with the equator, the direction of the sun's movement, and in the direction of the earth's rotation.  A simple angle is useless.  Over the years, shipping suffered.  Many ships were lost because they didn't know where they were:  if they thought they were far west of an island but were close to it, they might hit rocks and sink.  For safety, the ships of the day follow the same well established routes and stayed as close to land as possible.  This clogged the sea lanes and also made shipping more vulnerable to pirates who could predict where the boats would be.  Lacking a knowledge of longitude often forced navigators to take safer but far longer routes:  go due south to the Azores then turn left.  These longer trips exposed the crews to scurvy.

    Various kings (Phillip III of Spain, Louis XIV of France, Charles II of England) instituted prizes for the discovery of a reliable method of determining longitude.  Astronomical observatories were established around the world in an effort to gather the information needed to solve the problem.  It was in 1675 that Charles II of England had the Royal Observatory built in Greenwich for the purpose of perfecting astronomy and navigation.

    Various solutions were proposed by the leading minds of the time.  Galileo proposed using the the eclipses of the moons of Jupiter.  This was a big improvement on land, but not so helpful in a rolling ship or on a cloudy night.  Various other celestial solutions were also proposed, but they had severe drawbacks.  Some proposed the bizarre, like stationing boats at regular intervals across the ocean.  These boats would first cannons straight into the air, and, when the shot burst, sailors could time the difference between their sighting of the burst and the hearing of its blast, thereby determing their distance from the stationed boat.  This also had drawbacks, not the least of which was being able to anchor a ship in the middle of the ocean so its position was fixed.

    Another proposal was to use a magnetic compass and determine the angle between the true north pole (as measured against the north star) and the magnetic north pole.  With this there are theoretical difficulties -- the magnetic pole varies in intensity from place-to-place and from time-to-time.  There were also practical problems in that compasses were not so accurate and changed significantly with time.

    In 1530 the Flemish astronomer Gemma Frisius proposed using the recently invented mechanical clock to determine longitude.  The basic idea is that

    (360 degrees ) divided by (24 hours)= 15.

    So two locations one hour apart differ in longitude by 15 degrees.  Here is how good timing will help.  You leave port and have your clock set to local time.  You travel west.  One day you examine the sky to determine that the sun is directly overhead; that is, it is noon where you are.  You look at the on board clock and see that at your home port it is 2:00 PM.  This two hour difference means that you are 30 degrees west of your home port.  The difficulty here is having an accurate clock.  There is also a difficulty with cloudy days.  But at night you could use the known positions of the moon and stars at certain times to substitute for the concept of noon.

    Clocks in Frisius's day were very inaccurate.  They lost and gained two hours per day.  Various people set about to invent more accurate clocks.  Galileo had discovered that the rate of swinging of a pendulum only depends on the length of the pendulum's arm.  In 1656 the Dutch astronomer and physicist Christian Huygens invented a clock based on the regularity of the swinging of a pendulum.  This was an enormous improvement in accuracy --- on land.  With the usual rocking back and forth of a ship, the swinging of the pendulum is disturbed and the accuracy lost.  Huygens changed to a spring driven clock rather than a pendulum.   Still, the amount of accuracy needed was not achieved.  Springs are affected by motion, temperature differences, salt in the air.

    Why the need for great accuracy?  With one hour difference in time meaning 15 degrees of longitude, it takes only 4 minutes difference in time to equal one degree.  One degree is 60 nautical miles or 68 regular miles.  So a small inaccuracy in the clock over a long sea voyage would result in a significant error in determining position.

    In 1714 the English parliament passed The Longitude Act, which offered a prize of 20,000 pounds for a solution of the longitude problem.  In 1735 the English clock maker, John Harrison, completed work on a clock that he believed was sufficiently accurate to solve the longitude problem and claim the prize.

    He was right.  On a trial sea voyage form Lisbon to England the clock had lost on a couple of seconds.  What followed over the next many years is a tale of the quest for perfection by Harrison, jealousy by his contemporaries, and bureaucracy.  Harrison, who seems to have been one of the great geniuses in history, was not satisfied and continued working toward perfection.  Twenty-six years after his first clock, he produced his fourth clock.

    (For more pictures of Harrison clocks, see the display by the Royal Observatory.  The home site of the Royal Observatory at Greenwich is a good one to gather a lot of information of time and related topics.)

     It is about 5 inches in diameter and weighs about 5 pounds.  Harrison's son set sail for the West Indies with this clock aboard the ship Deptford on 18 November 1761. They arrived in Jamaica on 19 January 1762, where the watch was found to be only 5.1 seconds slow!  That's an efficiency rate of


    Prior to this efficiency rates for clocks on land were on the order of about 98%.

    Modern times have seen great strides in timing accuracy.  Recently (in the 1930s) the quartz clock has come to be.  Rather than the oscillation of a pendulum, the natural oscillation of the quartz crystal is used to regulate the movement of the clocks hands or digital display.  The quartz watch is accurate to within .001 second in 1 day.  This is an efficiency rate of


    Still more recently we have the Atomic clock, starting in the 1950s.  By the 1990s this clock was accurate to within

    one second in 1 MILLION years!!

    The Atomic clock uses the oscillation of the cesium atom.  Using atomic phenomena, an international agreement was reached that the length of a second is

    9,192,631,770 cycles of cesium 133.

    Why would you possibly want this much accuracy?  Partly this kind of accuracy is needed if you want to send space probes to distant planets, where a very small miscalculation can result in an enormous error after the craft has flown millions of miles.  It is also used for astronomical and physical research.  Such accuracy was needed, for example, to verify Einstein's theory of relativity and, in particular, the part of that theory that holds that gravity changes time.  A clock on top of Mount Everest, for instance, was predicted to run 30 millionths of a second per day faster than an identical clock at sea level. The only way to make measurements this accurate is to control a clock by the infinitesimal oscillations of the atom itself.

    But even closer to home is the Global Positioning System or GPS.  GPS has 24 satellites in orbit around the earth about twice each day.  Each satellite has on board 4 separate atomic clocks.  Precision in the time is essential for the satellite to determine its own position and thus function as designed.  What does the GPS do?

    GPS itself was born as a military tool to enable a rocket to precisely hit its targets. Today it is also used by ships, planes, and private citizens.  Many new automobiles have a GPS receiver built into the car.  Anyone can buy a hand held GPS receiver at some department stores or on the web.

    Each satellite essentially transmits the following information:

    I am satellite X
    I am in position Y
    It is time Z

    GPS picture

    You can find the exact time by contacting the US Naval Observatory, which reads out the time from its atomic clock.  You can also find the answers to a number of questions about time as furnished by the National Institue of Standards and Technology.

    Some web sites on time.

    (Some of these were linked in the text.)












    Hard copy references