Time has played a central role in mathematics from its very beginnings, yet it remains one of the most mysterious aspects of the world in which we live. The beginnings of civilisation on Earth required a knowledge of the seasons, and the mysteries surrounding the length of the year, the length of the day and the length of the month began to be studied. All the world religions gave time a central role, be it in astrology, stories of creation, cyclical world histories, notions of eternity, etc. Philosophers have tried to come to grips with the concept; some have argued that time is a basic property of the universe while others have argued that it is an illusion or a property of the human mind and not of the world. A huge effort has been put into making devices to measure time with ever increasing accuracy from the beginnings of recorded history to the present day.
Quantum mechanics and relativity theory in the 20th century have shown the complexities, and sometime apparent paradoxes, in the notion of time. Yet basic mathematics takes time as understood and develops the calculus around a particle whose position at time t is given by x(t), its velocity is dx/dt , the derivative of x(t) with respect to time, and its acceleration is the second derivative. This requires time to be continuous and a time interval to always be divisible, yet quantum theory tells us that time is quantised and quite unlike mathematical time which forms the basis of applied mathematics. We shall look at the fascinating 20th century developments in understanding time in the article A history of time: 20th century time. In this article we examine how ideas about time developed, culminating in Newton's universal absolute mathematical time.
Of course the very title of these articles: "A history of time" is confusing. The idea of "history" already contains the idea of "time". But let us make it clear that what we intend to investigate in these two articles is a history of how ideas about time have developed and, as always in this archive, we emphasise the mathematical aspects.
Mathematics almost certainly began through the study of time, particularly the need to record sequences of events. An understanding of the seasons is vital for the successful growing of crops. When should crops be planted? When would the rains come? When would rivers flood? When should one harvest the crops? The natural timekeepers in the sky are the daily passage of the sun and the monthly phases of the moon. The fact that knowing the length of a year was vitally important, yet much less visible from the timekeepers in the sky, led to calculation. It was also necessary to count days and months and this gave rise to calendars. The earliest evidence of timekeeping goes back around 20000 years; evidence from markings made on sticks and bones in Europe around this time are thought to be records of days between successive new moons. Many ancient calendars were created but as an example let us look briefly at an Egyptian one from around 4500 BC.
It was important for the Egyptians to know when the Nile would flood and so this played a large role in the way their calendar developed starting from an early version around 4500 BC which was based on months. From 4236 BC the beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and by 2776 BC it was known to this degree of accuracy. A civil calendar of 365 days was created for recording dates. Later a more accurate value of 3651/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes such as the sowing of crops, harvesting crops etc. being based on the lunar month.
Dividing the year into months was natural, yet complicated since there were not an integral number of months in a year. Similarly dividing the month into days was complicated for the same reason. A day was a long period of time and there was clearly a need for dividing the day, but it was less obvious how this might be done. In around 3000 BC the Sumerians divided the day into 12 periods, and divided each of these periods into 30 parts. The Babylonian civilisation, which grew up around 1000 years later, in the same area of present day Iraq as that of the earlier Sumerians, divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. It is their division of the day which gives us the widely used modern units of time. We should note, of course, that these modern units, although deriving from the Babylonian versions, are today not defined from astronomical data. We should also note that many early units of time varied throughout the year as the length of the day and night varied with the seasons.
Now units of time require some way of measurement and, not surprisingly, because of their astronomical definitions the early devices to measure time used the sun. From around 3500 BC the gnomon was used, consisting of a vertical stick or thin monument whose shadow indicated the time of day. Later, by about 1500 BC, the sundial was in use. The problem with the sundial was that the sun took a different path through the sky throughout the year. To ensure that the sundial registered roughly the correct time all the year round the gnomon had to be set at exactly the correct angle. The sundial developed into a more accurate instrument with the introduction of the hemispherical sundial around 300 BC. By the time the Roman architect Vitruvius wrote De architectura shortly before 27 BC, he was able to describe 13 different designs of sundial in Book 9 of his work.
Of course the sun could not be used to tell the time at night and clepsydras or water clocks were in use in Egypt by 1500 BC. Water ran out through a hole in the bottom of a vessel, the inside of which had lines to indicate the passage of time. Early versions did not allow for the fact that the water ran out more slowly as the pressure dropped. Sand was also used in the still familiar hour glass where sand trickles from a container, taking a set length of time to run out.
There was high religious significance in time measurement in these early civilisations. Of course the importance of successful crop management to the survival of a civilisation meant that time gained a religious significance, and the astronomical way that time was measured emphasised this. Also religious observance required certain events to be carried out at specific times, and detailed knowledge of a calendar was necessary. An example of what we mean is illustrated by the way the Christian festival of Easter is still calculated. It occurs on the first Sunday after the first full moon that occurs on or after the vernal equinox. A more sophisticated weaving of time into religions took place with Pythagoras and Buddha, around 500 BC, each claiming a cyclical nature of time with humans being reborn during the flow of time. Both could recall memories of previous existences on Earth and indeed this idea is a very natural one given the cyclical nature of time as observed in the seasons and years. Some religions such as Judaism and Christianity are based on a creation story where time begins in the act of creation. In these the creator is often considered to be outside of time, a concept which is hard to understand. Let us examine some early, yet significant, contributions to the understanding of the concept of time.
Zeno of Elea, around 450 BC, gave a number of paradoxes which indicated puzzling aspects of time. In the paradox 'The Arrow' Zeno argues, in Aristotle's words:-
If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.
The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-
... for time is not composed of indivisible 'nows', no more than is any other magnitude.
For Aristotle to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. On the other hand if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Really one feels that Aristotle had dismissed Zeno's subtle paradox far too readily. The paradox is brilliant with Zeno putting his finger on a genuinely deep puzzle.
Let us look at how Plato and Aristotle viewed time. Plato argued that time was created when the creator fashioned the world from existing material, giving form to primitive matter. Plato argues in the Timaeus that the creator:-
... sought to make the universe eternal, so far as might be. Now the nature of the ideal being was everlasting, but to bestow this attribute in its fullness upon a creature was impossible. Wherefore he resolved to have a moving image of eternity, and when he set in order the heavens, he made this image eternal but moving, according to number, while eternity itself rests upon unity; and this image we call Time.
According to Plato then, time was created at the same instant as the heavens, see . Aristotle, however, argues against Plato's idea that time was created. His ideas relate time to motion. In a sense this is reasonable since to Aristotle time was measured by the motions of the heavenly bodies so a period of time was represented by the movement of the sun across the sky. Other ways of telling time such as the water clock and the hour glass also identified time with movement, in these cases movement of water or sand. There is an argument, claims Aristotle, to say that time does not exist, for the past no longer exists and the future does not yet exist. Having looked at this argument, he rejects it and defines time as motion which can be enumerated. From this we see why he argues against Zeno's arrow paradox. For Zeno, the arrow cannot move for it does not move in the instant. However, for Aristotle time itself is motion, the flow of time is the motion of the arrow or of the sun and moon across the sky.
St Augustine, around the end of the 4th century AD, was responsible for bringing much of Plato's philosophy into Christianity. Plato's version of the creation does not quite fit the Genesis account since in that God creates the world from nothing, while for Plato the world was created by bringing order to primitive matter. However, St Augustine agrees with Plato that time begins with the creation. He answers the question of why the world was not created sooner by stating clearly that there is no 'sooner'. However, the concept of time was still a puzzle to St Augustine:-
What then is time? If no one asks of me, I know; if I wish to explain to him who asks, I know not.
Like Aristotle, St Augustine questions whether the past or future really exist. Surely only the present actually exists and this is instantaneous, only measured by its passing. Yet, like Aristotle, St Augustine says how can it be that past and future time do not exist. He tried to answer the apparent contradiction by claiming that past time can only be thought of as past if one is thinking of it in the present. He identifies three times:-
... a present of things past, a present of things present, and a present of things future. ... The present of things past is memory, the present of things present is sight, and the present of things future is expectation.
St Augustine had reached conclusions that time does not exist without an intelligent being who is able to think in the present about things past, present and future. However he was not really happy with his own ideas and prayed for enlightenment:-
My soul yearns to know this most entangled enigma. I confess to Thee, O Lord, that I am as yet ignorant what time is.
Certainly St Augustine was right to feel that his ideas are less than satisfactory, yet that said, he had thought more deeply about time than anyone seems to have done before him including the greatest of the Greek philosophers, and more deeply than anyone else seems to have done during the following one thousand years. If his ideas are less than satisfactory, at least St Augustine has appreciated for the first time what a complex and puzzling concept time is.
There was some progress in clocks to measure periods of time going in the period when St Augustine was contemplating the puzzle. The developments were not, however, in new types of clock, merely in improved designs of sundials and water clocks. Mechanical devices were added to water clocks to strike bells, move hands on a dial, open doors to display figures like the modern cuckoo clock but these did nothing to improve the basic time keeping. There was much interest in clocks, however, and in the first century BC, the Tower of the Winds was constructed in Athens. This had both sundials, and water clocks which drove mechanical devices to display the hour on a 24 hour scale. It had other features relating to time such as displaying the season, and various astrological dates.
Progress in timekeeping in Europe was non-existent from around 500 AD to 1300 AD, but in other countries progress did continue with mechanical clocks being introduced in China. However the invention of the verge escapement in Europe in the 14th century led to a revolution in mechanical clocks. The verge escapement worked by having a wheel with cogs which was prevented from spinning by a pair of metal leaves which moved up and down to allow the cog wheel to move forward one cog at a time. The leaves were attached to a foliot, a weighted crossbar, on which the small weights could be moved back and forward to adjust the rate at which the bar oscillated. The whole of the mechanism was powered by heavy weights which drove the cog wheel. Such clocks were more accurate than any of the earlier ways of measuring time, but they were very difficult to adjust to obtain that accuracy. The speed which the clock ran at was still completely dependent on the power applied by the weights, and by the amount of friction in the mechanism.
An early example of such a mechanical clock was the one constructed in Strasbourg between 1352 and 1354. The Clock of the Three Kings was built in Strasbourg Cathedral and stood twelve metres high. Clocks at this time needed to be big so that the weights had a long drop, otherwise the weights required to be wound up frequently, but the Strasbourg clock was more than just a clock for it related time to all its astronomical origins. In addition, the clock was a work of art in terms of both the decoration and the novelty of the automata. The lowest portion of the clock consisted of a calendar, above which was an astrolabe, while above that again there was a statue of the Virgin and Child. Every hour figures of the Magi appeared and bowed before the Virgin and Child while chimes rang out and an automaton cock crowed and flapped its wings.
Time had gained a certain status which it had not enjoyed earlier. Mechanical clocks were an important status symbol but towns only required their own local times, and it would be another 500 years before the advent of the train made standard time zones a necessity.
We will return to the Strasbourg astronomical clock in a moment, but first let us consider the 14th century work De proportionibus proportionum by Oresme. In this Oresme discussed whether the celestial motions are commensurable or, expressed another way, is there a basic time interval so that the day, month, and year are all exact integer multiples of it. On the one hand Oresme says one might expect that a creator of the universe would have arranged things so that this is so. He ends his discussion, however, by coming down on the side that no two celestial motions are commensurable. It is a fascinating discussion which essentially asks if time as measured by the sun and the moon is the "same" time.
The Strasbourg astronomical clock ran for about 150 years before its mechanism failed. A decision to replace the clock was made and work began in the cathedral on a new clock in 1547. Chrétien Herlin, an astronomer and professor of mathematics at Strasbourg Academy, was in charge of the project with two assistants who were also mathematicians. When the project began the cathedral was a Protestant one, since the Reformation had swept through Germany about twenty years earlier. However the cathedral returned to be a Roman Catholic one soon after construction of the new clock began, the project was put on hold and only restarted in 1571 when the cathedral was again a Protestant church. Herlin's successor as professor of mathematics at Strasbourg Academy was Conrad Dasypodius and he was now put in charge.
Both a mechanical and artistic triumph, this clock illustrated clearly how time was thought of in the 16th century. Time was astronomical, founded on the movements of the celestial bodies, and this is represented by there being a celestial globe with 48 constellations and 1022 stars. The movements of the sun, moon and five planets were shown. Eclipses of the sun and moon and phases of the moon were also shown, and there was an astrolabe which was designed on Ptolemy's version of the universe.
During the 16th century the solution of problems relating to time became of utmost importance because of its relation to finding the longitude. In an age of exploration on a world scale, determining position became a crucial problem and much effort was put into its solution. The realisation that an absolute time standard for the world would allow the calculation of the longitude of any position by comparison with local time was a major driving force in efforts to devise accurate clocks. It also led to a clear distinction in people's minds between an absolute time and a local time. Gemma Frisius wrote in 1530:-
... while we are on our journey we should see to it that our clock never stops. When we have completed a journey of 15 or 20 miles, it may please us to learn the difference of longitude between where we have reached and our place of departure. We must wait until the hand of our clock exactly touches the point of an hour and at the same moment by means of an astrolabe ... we must find out the time of the place we now find ourselves.
In the 17th century Galileo discovered a 'clock' in the sky which recorded 'absolute time', namely the times of the eclipses of Jupiter's moons. Theoretically this provided a solution to the longitude problem, but in practice observing the eclipses of Jupiter's moons from the deck of a ship was essentially impossible. Several large prizes were offered for a solution to the problem of determining longitude and Galileo tried the persuade the Spanish Court in 1616 that he could determine absolute time using Jupiter's moons and, after failing to convince them, tried to persuade Holland of his method when they offered a large prize in 1636.
This was not the only contribution Galileo made to the study of time. Long before his discovery of Jupiter's moons he discovered the fundamental property of the pendulum in 1583. While attending services in Pisa cathedral be noticed that a swinging lamp in the cathedral took the same time to swing irrespective of how large the displacement. Of course one might reasonably ask how he discovered this since in Galileo's time there was no device to accurately measure small intervals of time. In fact Galileo used the biological clock built into his body, for he used his own pulse to compare the time taken for the pendulum to swing. Galileo does not seem to have realised that his discovery might be used to design an accurate clock until many years later, but around 1640 he did design the first pendulum clock. Galileo died in early 1642 but the significance of his clock design was certainly realised by his son who tried to make a clock to Galileo's design, but failed.
The first to succeed in making a pendulum clock was Huygens in 1656, see . This invention brought a new accuracy to the measurement of time, with his early versions achieving errors of less than 1 minute a day. With a later improved design Huygens was able to build a clock accurate to within 10 seconds in a day. Hooke used the natural oscillation of a spring to control the balance of a clock and some years later Huygens also experimented with a balance wheel and spring assembly which can still be found in mechanical wrist watches.
Descartes used mathematical principles to explain the world and many, including the early members of the Royal Society in London, followed his example. This, however, provoked a reaction among many who used religion to explain the world and objected to the mechanical approach. Boyle, a great advocate for mathematical descriptions of the world, provided an answer by stating clearly that he believed in a God who could create a mechanical universe which operated with certain laws and he gave as an example the Strasbourg clock. There was a parallel, said Boyle, between the creator of the Strasbourg clock who built a mechanism which ran on its own without the intervention of the builder and the universe made by God which operated according to his laws but without his intervention. The ultimate version of the mechanical universe appeared in Newton's Principia in 1687. The whole description of Newton's laws depended on time and so Newton began by defining time as:-
... absolute, true, mathematical time, [which] of itself, and from its own nature, flows equably without relation to anything external.
This was a major new idea regarding time, see , ,  and . No longer was time determined by the universe, but rather Newton postulated an absolute clock, external to the universe, which measured time independently of the universe itself. With his ideas Newton put time into a new place in mathematics. The calculus was his theory of fluxions, relating motion to this universal flux of time. No longer could time be said to be an illusion as some ancient philosophers had suggested, for now the whole of science was being built on laws based totally on the notion of time.
Not everyone was convinced by Newton's arguments, however, and Leibniz argued against Newton's notion of absolute time using religious reasoning. He believed that God was rational and therefore required a reason for every action. So how could God choose an instant to create the universe? If there was no way to distinguish one time from another, as Newton had claimed, God was faced with an impossible choice to decide rationally on the moment of creation. Although many today may not see this argument by Leibniz as a scientific one, it can be turned into a philosophical argument and, in a modern form, would argue that time was created at the instant of creation. Leibniz used another argument too. If two things are identical in every respect then, claimed Leibniz, they are one. Newton's absolute space and time are identical everywhere at all times by their very definition so Leibniz claimed any two positions are one as are any two times.
There was another interesting consequence of Newton's description of the universe based on his precise mathematical laws, and this was fully understood by Laplace. If one knew the exact position and motion of every particle in the universe then one could calculate the future position, and also the past position, of every particle. An example would be the prediction of eclipses of the sun and moon which is possible from knowing the positions and motions of the bodies in the solar system and then using Newton's laws. Such knowledge also allow us to calculate when such eclipses occurred in the past. Laplace correctly argued that given the laws of mechanics, the complete picture of the past and future world is encapsulated in the present world. There followed a period when science aimed to calculate with ever increasing accuracy.
Newton's laws were quickly accepted because they led to correct predictions about the world. They did contain several puzzles, however, one of which was that they simply described the way that the world is, and do nothing to say why it is so. Their most puzzling aspect as far as time is concerned is that they work equally well if time runs forward or if time runs backwards. There is a complete time symmetry in the laws, yet human experience leads to the belief that time always flows forward. It was only in the middle of the 19th century that the second law of thermodynamics was proposed by Clausiusand this was the first law to lack symmetry in the direction of time, see .
Clausius read a paper to the Berlin Academy on 18 February 1850 which contained this second law of thermodynamics. He defined entropy which originally measured the amount energy in the form of work that can be extracted from a hot gas but later came to represent a more general measure of the randomness of a system. The second law of thermodynamics states that the entropy of a closed system will always increase, that is its randomness will always increase. This is illustrated by the fact that if we take a box with a membrane across the middle, one side filled with a hot gas, the other with a cold gas, then removing the membrane will result in the hot and cold gases mixing and the temperature will approach the average temperature of the two gases. One would not expect to see the reverse happen. If the box was filled with gas, one would not expect high energy molecules to move to one side of the box and low energy molecules to the other. The system is not time symmetric.
Despite the difficulties which still existed in understanding the notion of time, by the last part of the 19th century one would have to say that Newton's universal time had proved extremely effective in providing a basis for laws which had been observed to hold to a high degree of accuracy. Although we are far from understanding the notion of time today, the 20th century saw a revolution in the study of time. We may not understand time, but we know that Newton's absolute time cannot provide the answer. In the article A history of time: 20th century time we examine this revolution.
Article by: J J O'Connor and E F Robertson
MacTutor History of Mathematics