Why are there 24 hours in a day? We inherited that number from ancient Babylon, a major centre for astronomy, mathematics and philosophy forming the foundation of our civilisation today. There is no doubt that our number system derives from the fact that humans have 10 fingers and toes. Ten is the radix of the denary number system used almost universally in markets but why do we buy eggs by the dozen? Why didn’t the Babylonians divide the day into 20 hours? The base of any number system is called its radix and you can create a number system with any radix you please.
There is no such thing as a straight line on the surface of a sphere. We have geodesics, including meridians of longitude, which are great circles with their centre at the centre of the sphere.
- We have parallels of latitude also called almucantars.
- We have rhumb lines or loxodromes representing lines of constant direction, which is the track followed by a ship or aircraft steering a compass course, or some share price on a constant trajectory..
- The angles of a triangle add up to anywhere between 180 degrees and 540 degrees or six right angles, not 2.
- There is no such thing as a square on the surface of a sphere; no figure with four right angles and four equal sides. The nearest thing to a square is a figure straddling the equator and bounded by two meridians and two geodesics.
- .The geodesic is the shortest distance between two points.
- Almucantars define the latitude of points on the sphere.
The circumference of an almucantar is the cosine of its latitude. For example, Brisbane is in south latitude 27 degrees 28 minutes and its almucantar is 19,336 nautical miles. Sydney lies in 33 degrees 52 minutes and its almucantar is 18,119 nautical miles. Melbourne lies in 37 degrees 45 minutes and its almucantar is 17,008 nautical miles.
In agreement with the Galilean principle of relativity and setting aside Tectonic shifts on geological time scales, I postulate that Brisbane, Sydney and Melbourne are more or less stationary relative to one another, but if the world rotates once every 24 hours Brisbane is travelling at 805.6 knots, Sydney at 755 knots and Melbourne at 711.6 knots. Brisbane is travelling at 94 knots relative to Melbourne.
That’s absurd, is it not? The word ‘absurd’ means inconsistent with the axioms. There is something absurd about our axioms of timekeeping inherited from antiquity. We can resolve this issue by observing that the confusion over the missing day upon Victoria’s return home is nothing more than confusion of two different regimes of time, namely sidereal time measured by the stars and mean solar time derived from the apparent motion of the Sun. Navigators and astronomers employ both in the course of their work but landlubbers generally confine themselves to mean solar time. As the name implies, mean solar time is an average of the Sun’s rather erratic travels through the sky. Mean solar time is calculated from two imaginary bodies ; the Mean Sun travelling at an average speed in the equinoctial and the Dynamical Mean Sun travelling at an average speed in the ecliptic . I shall return to this matter in due course.
Representing the sphere on a flat sheet of paper is almost an industry in its own right. There are many different map projections tailored to illustrate different properties of the sphere. The most important for navigators is the Mercator projection, which is also the most common map of the world in school atlases. It was devised in the 16th century by a Flemish mathematician Gerhard Kremer who became known as Gerardus Mercator. It features the loxodrome, which is why it is important for navigators. Another name for loxodrome is Golden Spiral, which is related to the set of Fibonacci numbers devised or discovered in the 13th century by a mathematician known these days as Leonardo of Pisa. He wrote a book called Liber Abaci, or Book of Calculations with articles on currency exchanges, weights and measures, calculating profits and other matters of interest to traders.
Through his experience in the middle East he realised the Arabic or Hindu system of numbers was superior to clumsy Roman numerals for calculations. He played a big role introducing Arabic numerals into Europe. Fibonacci as he is now known published many books but is remembered for his rabbit problem, paraphrased as follows.
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?
In thrall to the contemporary Julian Calendar, Leonardo is careless in his treatment of time, There is no mention of observations of the Moon’s altitude and azimuth. He was not a navigator or astronomer. He assumes 12 months in a year, an error that misleads people to this day. Fibonacci numbers have nothing to do with the Moon or with rabbits but Leonardo’s puzzle has intrigued mathematicians and others for centuries. As he explained:The first pair produces issue in the first month so in this month there will be two pairs. Of these, one pair gives birth in the following month and in the second month there will be three pairs. Of these, two pairs will produce issue in the following month…
The set of Fibonacci numbers emerges as the sequence 1, 1, 2, 3, 5, 8, 13, 21,…. in which each number is the sum of the previous two numbers. This still has nothing to do with the Moon or with rabbits but 400 years after Fibonacci’s time Johannes Kepler, the astronomer who discovered that the planets orbit the Sun in ellipses, found the same sequence independentally of Leonardo’s work. He commented that the numbers revealed the seminal faculty, or biological marvel of procreation. Kepler proved the ratio of consecutive numbers approached 1.618 or phi, as a limit. Phi is a number well known to ancient Greek mathematicians as the Golden Mean or Divine Proportion and the sequence is called the Golden Spiral.
Navigators know the Golden Spiral as a loxodrome. In spherical trigonometry it is the track followed by a ship or aircraft steering a compass course. It is a ‘straight line on a Mercator chart. If a loxodrome is extended indefinitely it ends up at the N or S pole but we still have no involvement with rabbits.
The gnomonic projection displays geodesics as ‘straight lines’ and is used when plotting the shortest course between two points. While following this ‘straight line’ the ship or aircraft is altering course constantly, so what do you mean by ‘straight line’? Call me stupid but I don’t understand ‘straight line.’ It is not a geodesic, not an almucantar and not a loxodrome.
The stereographic projection is often used to map polar regions where the Mercator projection becomes unusable because of its expanding scale of latitude and shrinking scale of longitude, which is to say, of time. Since there are 24 hours in a day and 360 degrees in the Earth’s circumference then one hour of time is 15 degrees of longitude, positive in eastwards directions and negative westwards. The loxodrome/ rhumb line is also known as the equiangular spiral.
To maintain the desirable feature of a loxodrome as a ‘straight’ line on a Mercator chart, meridians, which converge on the poles in the real world, are represented as vertical ‘straight’lines disappearing to infinity on the chart. The scale of longitude, which is the distance between charted meridians, is uniform. Fifteen degrees of longitude is one hour of time but the scale of distance varies with latitude across the globe. Very high latitudes cannot be represented on a Mercator chart. The time scale is extended to Eternity, which mathematicians call asymptotes, represented as hyperbolae.
The tools of Flat Earth Geometry are straight-edge rule, pencil, compass for drawing circles and computer or pocket calculator. In addition to FEG tools, the navigator’s tools before mid 20th century were chronometer, sextant, nautical almanac, mathematical tables, magnetic compass and a vigilant lookout for dangers. In the 21st century the sextant and chronometer have been replaced by the GPS, the magnetic compass by the gyrocompass and a lookout for dangers. has in some cases been replaced by radar. Robot ships will one day eliminate humans. I retired as a navigator before becoming redundant. The mariners compass, probably invented by the Chinese about a thousand years ago, differs from the FEG representation of a circle. The mariners compass divides the circle into 32 points and 128 quarter points. It is a 32 bit byte in the binary number system =110000, and should be familiar to anyone using a computer. A compass point is 11 degrees 15 minutes and a quarter point is 2 degrees 48 minutes 45 seconds of arc. We inherited that number system from ancient Babylon so the Chinese appear to have anticipated the digital revolution of the computer age by simplifying numbers down to compass points. What is a number anyway? It is the language of time, calendars and clocks. You can’t escape from numbers this side of death.
Computer nerds invented the hexadecimal number system to base 16, which divides the circle into 64 points = a 64 bit byte. A point in the mariners compass is an angle of 11 degrees 15 minutes of arc. A point in the hexadecimal system is an angle of 5 degrees 37 minutes and 30 seconds so one Hex=2 compass points. There was no requirement for the hexadecimal number system. The Mariner’s compass says it all. HEX is even more complicated than the Babylonian system, incorrectly called sexagesimal. The base of a number system is called the radix and you can have any radix you please. In any number system with radix R the number of digits is R minus 1. If the Babylonian system were truly sexagesimal it would require 59 digits. A number system for spiders has 11 digits and the octopus number system has 7 digits, the number of days in a week and the number of planets known to the ancients: Mercury, Venus, Earth, Mars, Jupiter and Saturn. The octopus number system has been adopted for the calendar by most if not all religions around the world. Maybe God is an octopus.
The human animal has 10 fingers and toes and it’s no surprise that they invented a number system to base 10 with nine digits, called the denary number system. Various civilisations of Central America and the Inuit Eskimo people adopted radix 20, counting on their toes as well as their fingers. They were all feggers. The Mayan vigesimal number system has 19 digits which is the number of years in the Metonic cycle extensively used in calendars of ancient Babylon. To this day the Metonic cycle plays a role in the Catholic church’s calculation of the date of Easter so it could be said the Gregorian calendar is also vigesimal.
If you are contemplating buying a computer you will need to familiarise yourself with the jargon of bits and bytes, megabits, kilobytes, RAM, MBPS or megabits per second, which is the ‘speed’ of a computer. The prefixes kilo, hecto, deka, mega, giga …are those of the International Standard of units, SI. They have precise meanings representing powers of 10, 100, 1,000, 1,000,000 1,000,000,000. Computer engineers put their own twist on that and kilo in computer talk means 1024 bytes because 2 ^10 creates engineering difficulties. They claim 1024 is near enough to 1,000. If a navigator did that kind of thing he would soon be shipwrecked.
You need to clarify whether the salesperson in the computer shop is talking about powers of 2, powers of 10 or the arbitrary computer kilo, 1024, which is just sloppy arithmetic. Let’s get it straight. Kilo in the International system of units (SI) means 1,000, not 1024. When the salesman in the computer shop extolls the virtues of that shiny computer with a speed of 50 megabits per second ask him to explain what he means. Ask him why he thinks a kilobyte is 1024 bytes. Please let me know his answer. *************************
The dark oceans were the womb of life: from the protecting oceans life emerged. We still bear in our bodies-in our blood, in the salty bitterness of our tears-the marks of this remote past. Retracing the past, man, the present dominator of the emerged earth, is now returning to the ocean depths. His penetration of the deep could mark the beginning of the end for man , and indeed for life as we know it on this earth: it could also be a unique opportunity to lay solid foundations for a peaceful and increasingly prosperous future for all peoples.
to the UN General Assembly, 1967