In the days before quartz crystal clocks one of my jobs as third mate of an ocean going tramp was time officer. Whenever the old man decided to flog the clocks for a new time zone I had to go around and alter the hands of every clock in the ship except the chronometer, which was kept in a locked cabinet on the bridge. Only the captain and I had a key. The chronometer was my responsibility. I had to wind it every day with seven turns of the key after receiving a radio time signal at noon Greenwich Mean Time. I recorded any difference from GMT in a book called the rate book and analysed the rate at which it was gaining or losing time. The rate could vary depending on variations in temperature as the ship steamed northwards or southwards.

The chronometer was the most important instrument on board. Without the chronometer, shipwreck was a real possibility. It was the only means of determining the ship’s longitude. For many centuries sailors had been getting wrecked because they didn’t know where they were, with only Sun, Moon and stars to guide them. The problem came to a climax in 1707 when four British warships were wrecked on the west coast of England with the loss of about 2000 men, one of the worst maritime disasters in history. The British government were so enlivened by this event that they offered a huge prize for anyone who could solve the problem of longitude at sea. An amateur clock-maker named John Harrison won the prize but he had to fight to actually collect the prize money, and then only got part of it. Dava Sobel’s book *Longitude*, tells the story.

Harrison’s solution to the problem of longitude served well for a couple of centuries but in the 20th century new technologies began to appear and nowhere more so than at sea. I was born in the century that saw more change in technology and our knowledge of the universe than in all history. From wind power to nuclear, from flag signals to radio, from guesswork navigation to radar and GPS. The first landing on the Moon took place on my watch. I remember listening to the broadcast of those famous words ‘One step for a man but a giant leap for mankind.

New technology forces us to change the way we think, to look to the future instead of the past. Einstein’s revelation of the speed of light’s relativity challenged scientific beliefs thousands of years old. Why is it so difficult to accurately measure time and place despite the new technology? Professor Wade Allison of Keble College, Oxford, writing in *Navigation News* , observed that many of the laws of physics actually conspire to give as little meaning as possible to the concept of absolute position and time.They make a virtue of confounding the aspiring navigator.

The famous 19th century British scientist Lord Kelvin, who made many contributions to nautical science, once addressed his students with the advice: “If you can’t express your knowledge in numbers then it is of a meagre and unsatisfactory kind.”

Measurement of anything requires numbers. Numbers are the mathematical language of science and time. Numbers must evolve to meet the needs of new discoveries, new threats like climate change or pandemic plagues and new technology. The language of Celestial navigation is non-Euclidean, which is different from the geometry you were taught in school. In Flat Earth geometry, or FEG, the angles of a triangle add up to 180 degrees but a spherical triangle may have angles adding up to 540 degrees. There are many different ways we can measure the world we live in.

## Numbers ain’t numbers

Fibonacci numbers derive from a puzzle posed by Leonardo Fibonacci, as he is now known, in 12th century Italy. It is paraphrased this way: A pair of rabbits is confined in a large pen. If each pair produces another pair every month and they begin breeding when they are one month old. How many pairs will be in the pen after a year? There is no evidence that Leonardo conducted this experiment and the puzzle has nothing to do with rabbits or the Moon. It is a thought bubble known to Indian mathematicians centuries before Leonardo’s time. He accompanied his father, who travelled North Africa in his work as a customs official, and encountered Middle Eastern traders and the Hindu-Arabic number system, very different from the Roman. Leonardo was famous in his own time for his prodigious output of mathematical mind- teasers and was responsible for introducing the Arabic number system into Europe through his book Liber . Abacci.

. The solution to the problem is quite simple. After one month the original pair of rabbits gives birth to another pair so there are two pairs. One month later both pairs produce another pair, making three…and so on. This procedure continues to infinity. (Leonardo’s rabbits are immortal.) and the outcome is a string of numbers in which each number is the sum of the previous two numbers

1, 2,3,5,8,13,21 34….

Whatever its validity, the Fibonacci Sequence, as it is now known, has fascinated mathematicians and others ever since because of its abundance in describing aspects of the real world. Four centuries after Leonardo’s time the famous astronomer Johannes Kepler independently discovered the sequence and its property that it converges to the Golden Ratio, studied by Greek mathematicians like Euclid and Pythagoras. The Golden Ratio is defined by the division of a line in extreme and mean ratio so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller.

The sequence has even found its way on to the stock market. An American accountant, Ralph Elliott, lived through the roaring stock market of the 1920s and the subsequent crash followed by the great depression. Suffering from a serious illness, he occupied his time studying and analysing the plunging stock prices of the Dow Jones industrial Average. The product of his investigation was a book entitled *The Wave Principle, in which *h e described the market’s behaviour as a series of five waves with different amplitudes. Three major waves were often followed by a set of minor waves and then a set of downward waves. He accredited this pattern to the fluctuating moods of investors and speculators, characterised as bulls and bears. A less charitable view might ascribe it to fear and greed, supporting a whole industry. At any rate it puts a human face on the subject.

The techniques can be applied to almost any market , large or small over time scales long or short. Timing is of course vital in such an environment.

Modern traders have access to software based on the principles of Fibonacci numbers, now described as Technical analysis. It has been found that certain numbers in the Fibonacci sequence, on the balance of probabilities, do tend to appear in patterns indicating the possibility of profit for an acute trader.

Leonardo of Pisa was a famous mathematician of his day. He was largely responsible for introducing Arabic or Hindu numbers into Europe. Much more versatile than Roman numerals, They gained rapid acceptance among traders. Multiplication and division were greatly simplified by the use of zero (a new idea to Romans) and a system whereby the value of an individual number, or digit, depends on its position in a string of other numbers. For example, in the denary or decimal number system to base 10, 5672 means 2+7×10+6×102^+5×103^. That is very different if the digits are rearranged:2675.

## radix

In the digital number system the base or radix is the number of unique digits, including the digit zero, used to represent numbers in a positional system. The most common system is the decimal or denary with radix 10 because humans have 10 fingers and toes. A number system for spiders would have 11 digits and the octopus number system has 7 digits, the number of days in a week. In the binary number system, 100 to base 2, 100 means four in the denary system. (No ones, no twos and one 2 squared= four

Good mathematics is silent music and Fibonacci numbers are the orchestra. The performance begins with the Golden Ratio, which has the uninspiring definition : A straight line is divided in extreme and mean ratio when the whole line is to the greater segment as the greater is to the lesser. I don’t want you to get too exited over that but I will mention that the Golden ratio is also known as a rhumb line or loxodrome. If you are a sailor you will know that a rhumb line is a ‘straight line’ on a nautical chart drawn on the Mercator projection. You will be familiar with the Mercator projection because it is the most common representation of the world in school atlases.

Gerardus Mercator’s depiction of the surface of a sphere on a flat sheet of paper in 1569 was a prodigious mathematical achievement ranking with invention of the chronometer in maritime history. For navigators, the chart’s most important feature is the rhumb line, which is the line followed by a ship steering a constant compass direction. To achieve that, the scales of distance and area are distorted. Meridians of longitude, which mark the scale of time, converge to the north and south poles but are depicted as vertical parallel straight lines. To compensate, the spacing of parallels of latitude increases northward and southward. The time scale is grossly extended and land features appear greatly enlarged. The Mercator chart is not suitable for use in high polar latitudes. The polar stereographic projection is often used instead.

Compare that with the Cartesian plane universally used by practitioners of FEG and question the depiction of time scales on graphs covered with little squares. It doesn’t work. There is no such thing as a square on the surface of a sphere and the only ‘straight line’ is the sea horizon. I have been staring at it for most of my life and have never seen any other ‘straight line than the sea horizon. I am not talking about lines drawn on a sheet of paper but observations of the real world. The Mercator projection achieves a kind of reality by distorting the elements of Flat Earth geometry. Why is the sea horizon a ‘straight line’ if the Earth is nearly spherical? In order to extend their view, sailing ship sailors would keep a lookout from the crow’s nest at the top of the mast. Then they might begin to see the curvature of the Earth.

Nowadays we have radar to keep a lookout and a variety of radars for different purposes. A radar transmits pulses of high frequency radiation and calculates the distance from dangers or targets by measuring the time taken for the radar pulse to bounce off a target and return. So-called over- the- horizon- radars can measure very great distances well out of sight from the ship. The pulse does not travel in a ‘straight line’ but follows the curvature the Earth.

It was almost a spiritual event for me to climb out of my bu Earth.before dawn, stagger on deck with a sextant, take sights of my favourite stars and then do a bit of arithmetic, draw lines on a chart and announce with confidence ‘I am here. No one has ever been at this timeplace before and never will be again. I am unique in world history.” I suppose I should share that with my shipmates, but I did the work and saved them all from shipwreck. A man can be proud of that. The advent of the GPS was a sad event for me. It took all the fun out of navigation. Navigators no longer have the satisfaction of arriving at their destination bang on the nose after an ocean passage. Nowadays we just push buttons to read off latitude and longitude. The GPS has also rendered 2,500 years of geometry obsolete. I will say that again in case you missed it. The GPS has rendered 2,500 years of geometry obsolete.

My experience from those days taught me the difference between a chronometer and a clock. A chronometer is an instrument for measuring time. A clock and its cousin the calendar, is a tool for regulating human society according to the whims of priests and politicians. Aboard ship the captain fills that despotic role.

## The calendar’s missing day

One day in September 1522, 18 emaciated men staggered ashore at Seville after a three-year voyage; survivors of a fleet of five ships and 270 men. The local populace were more astounded than those like me listening to Neil Armstrong on radio in 1969

One puzzling aspect of their voyage was that all their diaries and the ship’s logbook were one day in error. According to them it was Saturday but the locals knew it was Friday. The importance of this lay in the fact that they may have committed the sin of eating meat on a Friday. No one could know how many Hail Maries they would have to say to absolve themselves of sin.

They would have sought forgiveness in the church of Santa Maria de la Victoria in Triana, a suburb of Seville. By coincidence, the church had the same name as their ship. Around this time, a Dominican monk named Tomas Torquemada was Inquisitor General of Spain, a zealous pursuer of heretics estimated to have burned about 2000 of them at the stake. There is no evidence that any of the survivors fell victim to him but they would have been anxious.

Read about the first circumnavigation

ISBN 978-0-9946390-2-8

Of course the Catholic Church could not get away with such behaviour in modern times. These days they are restricted to the sexual abuse of children. One of the most senior priests of the Catholic Church, Cardinal George Pell, was convicted of the crime. He served time in prison but eventually the case was dismissed. Nevertheless, several of his colleagues declared their defiance of the law relating to sexual abuse of children. If you want to protect your children from sexual abuse by priests the Catholic Church needs to be declared a criminal organisation like the Mafia. Its influence in the Gregorian calendar, to which we are all subject, needs to be replaced by the abundance of scientific knowledge accumulated since the first circumnavigation. The Gregorian calendar is simply out of date. Nowadays the sky is crisscrossed by satellites providing all the temporal information required.T he Catholic Church should have been declared a murderous organisation back in the time of the Spanish Inquisition. The Catholic church is not an appropriate trustee of the calendar.

The puzzle of the missing day was put down to the fact that, travelling always westwards the ship’s crew had overtaken the Sun by one day. Perhaps they could be forgiven the apparent heresy. The realisation eventually, after many years, gave rise to the International Date Line, a permanent line of demarcation between yesterday and tomorrow. So, how do we define the gap between yesterday and tomorrow that we call today? Stand astride the International Date Line with one leg in Saturday and the other in Sunday and ask yourself ‘What time zone is my penis in? Scientists these days are measuring time down to nanoseconds.

Had the *Victoria *sailed eastwards around the world they would have lost one day instead of gaining one day. Who steals the missing day in each case? Where does that fit in a calendar having nothing to do with the Catholic Church? People born on the 29th of February have a vested interest in this question, especially the majority of people in the world who are not Christians.

Nowadays we know that, depending on your point of view, it wasn’t the *Victoria* overtaking the Sun but Seville falling behind the stars.

Much of the nautical technology developed in the 20th century evolved from the work of Albert Einstein and the mathematics of James Maxwell which predicted the existence of electromagnetic waves. The speed of light is an absolute phenomenon of the universe; perhaps the only one, and Maxwell mathematically predicted its speed. Of course navigators and astronomers had been pondering the mysteries of the sky for millennia but the first real break- through could be claimed for the Danish astronomer Roemer in 1675. He made a study of the moons of Jupiter being eclipsed by their own planet. He found the eclipses varied with the regular motion of Jupiter around the Sun.

A couple of centuries later Michelson and Morley attempted to measure the speed of light from different directions in the sky at different times of year. The experiment failed to find any variation The relative speed of light is not affected by the motion of the observer.

Einstein’s genius realised that the speed of light is time itself and may be subject to dilation if the observer travels at a very high speed. Time is subject to dilation, as we all know. Doesn’t time fly when you’re having fun?

Scientists have long been troubled by the fact that the laws of physics, including Newton’s laws, work equally well in reverse, going backwards.

Another important invention of the 20^{th} century was the gyro compass. The magnetic compass was probably invented by the Chinese about a thousand years ago. It suffers from the problem that the Earth’s magnetic field is variable over time and over different regions. The magnetic compass indicates magnetic north instead of so-called true north, which is the pole star, Polaris, within about one degree of azimuth. It is also affected by ferrous material aboard ship and needs to be corrected by placing small magnets in strategic places.

The Earth’s magnetic poles are continually on the move : evidence of the turmoil of earthquakes, volcanoes and the cauldron of molten rock to remind us of the sleeping dragon below. The magnetic poles have reversed several times in geological history. Navigators even nowadays are at the mercy of various forces and a different geometry than Flat Earth Geometry (FEG) that you were taught in school.

The tools of FEG are the straight-edge, pencil , compass (the other kind of compass) and computer or pocket calculator. In addition to FEG tools, the navigator’s tools before mid-20^{th} century were chronometer, sextant, mathematical tables, magnetic compass and a vigilant lookout for dangers. In the 21^{st} century the sextant and chronometer have been replaced by GPS, the magnetic compass by the gyro compass and a vigilant look out in some cases has been replaced by radar. Robot ships are bound to proliferate, like robot aircraft and driverless cars on the road. I retired as a navigator before becoming redundant. New technology put me out of work; a common issue nowadays as time accelerates and humans are replaced by robots, but a knowledge of the heavens and the environment is even more important nowadays than in olden days as we grapple with the advance of global warming and worldwide virus epidemics.