Science and Mathematics Correspondence
David McNaughton

1. The Keralite calendar drifts through the seasons.

2. Astrology - and statistical significance tests.

3. Cycles and resonances in the lunar orbit.

4. Russell's paradox: (self-including sets).

5. When can we see an 'upside-down' crescent?

6. Is it feasible to make rules for calendars thousands of years ahead?

7. The evidence for large variations in lunation-values.

8. Attempt to relate the Velocity of Light to Earth/moon parameters.

9. Visualising Complex Number relationships requires four dimensions.

10. Eclipse predictions in earlier centuries.

11. Significance of Delta-T, whose future "growth" may be less than expected.

1. To "Gulf News" Tabloid (Dubai, UAE) - 13th December 1993:

I enjoyed reading about the Keralite Calendar in your 30th October issue. It is probably the only system in the world which has the occasional 32-day 'month' - (during June/July, because that is when the sun migrates more slowly, and because a new Keralite month starts whenever the sun crosses a stellar zodiacal boundary).

However, you did not mention its main disadvantage - that its dates do not remain constant with respect to the seasons. Instead, there is a gradual drift due to precession of Earth's spin-axis.

To illustrate, Spring Equinox (around 21st March) presently occurs when the sun is close to the 'circlet' asterism in constellation Pisces. In 6500 years' time, however, this is where the sun will be at northern Summer Solstice.

By then, the seasons will therefore occupy much earlier positions in the Keralite Calendar, than they do now.

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2. From Meena Narayan (Dubai, UAE) - 1994:

I would appreciate any comments you might care to make regarding astrology.

Reply (quoted in "Khaleej Times" Weekend, 20th May 1994):

Science is not capable of adequately explaining everything in the universe (particularly regarding the human mind and soul), so it is possible that certain astrological claims do contain elements of truth. However, most horoscopes are so vague that we could almost choose any zodiacal sign at random, and apply its 'predictions' to our daily lives.

To assess whether there really is something in it, a survey should be carried out of perhaps a million car drivers categorised by birth sign - showing how many traffic violations they incur in (say) a ten-year period. If there then appeared to be a bias favouring or denouncing particular zodiacal signs, then a standard statistical procedure (known as the 'Contingency Table Chi-Squared Test') could easily be applied to the data. That calculates the likelihood of the bias being merely an accidental outcome - a result of pure chance. Like repeatedly throwing a dice and obtaining six every time, that likelihood would never become zero, but it would be impressive to state (for example) that the odds were 100 million to one against there being "no connection between birth-sign and driver safety".

Every astrological "guideline", without exception, should be scrutinised in that manner and labelled with a "probability level". Only then can people decide whether or not to trust them.

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3. To Roger Sinnott (Sky & Telescope, USA) and Ala'a Jawad (Kuwait) - 1994:

It was stimulating to read about cycles in lunar orbital times in Sky & Telescope (November 1993, pages 76-77).

The average length of the 'medium' cycle is really 3232.9 days, (not 3307 as stated in the article). It is equal to the period of revolution of the axes of the lunar orbit, i.e. 8.851 years: see Jean Meeus's "Les durées extremes de la lunaison", 1'Astronomie (Société de France) volume 102, pages 288-289 (July-August 1988).

Individual waves in that medium cycle do not form a "spectrum" of sizes: instead only four values are found - namely 3307.57 days, 3277.76 days, and (less frequently) 2893.99 days - and also (on very rare occasions) 3691.35 days. Variations around those key values are mostly ±0.01 day; they seem to be guided or restricted by resonances between synodic, anomalistic and possibly other types of 'month'. Those first two wave-intervals usually occur successively, thereby spanning a saros.

The 'short' lunation cycle averages 411.8 days, not 413. Individual waves often do cover 14 lunar months (413.43 days), but are subjected to a gradual and continuous retuning process.

As Ala'a Jawad states, long-term maxima are indeed sometimes 177 years apart, but occasionally they seem to recur after 186 (or even just 62) years. It may be significant that both 177 and 62 are approximate multiples of 8.851, such that individual wave-values are determined by resonance, but oscillate around the long-period mean (just as with the medium cycle, as described above).

Note added subsequently:
In chapter 4 of his "More Mathematical Astronomy Morsels" (Willmann-Bell, Inc., 2002), Jean Meeus shows that the average interval between long-term maxima is 184 years.

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4. To Dr A.W. Moore (Oxford University, UK) - 1995:

It was interesting to read your article on infinity in the April 1995 issue of Scientific American.

My belief regarding Russell's paradox is that its cyclic "self-contradiction" is merely a situation which has not been defined properly; i.e. there is confusion due to wording which at first glance looks sensible, but which in reality is incomplete due to a subtle twist. Thus, maybe it is simply an illustration of how language and mathematical logic can be made to diverge. That explanation does seem valid when we concentrate just on one of its real-life examples: i.e. "Does the village barber shave himself or not?"

Perhaps too, Russell's paradox is a compact manifestation of Gödel's theorem - that it is possible to set up a system containing fundamental axioms, only to discover later that we can deduce contradictory conclusions within that system (until of course we add the extra axiom "no set can be a member of itself").

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5. From Maya Menzes (Dubai, UAE) - 1997:

Is it ever possible to see an upside-down crescent from some part of the Earth - say the North or South Pole?


That is certainly never possible during the night (from anywhere) - nor around dusk or dawn. And a very young crescent will never appear upside-down (meaning a New Moon which is less than about 1½ days old).

To become visible during twilight, a thin lunar crescent needs to be higher than the sun: (at dusk, for example, the sun needs to set first to allow the western sky to darken sufficiently). With that configuration, the horns of the crescent will point upwards.

However, if the moon is between about three and four days old, then it is sometimes possible for it to appear as an inverted crescent - but during broad daylight. An hour or two before noon, for example, the upside-down moon would then be somewhere above the eastern horizon (and lower than the sun). In order to be visible it needs to be sufficiently far away from the solar glare. It also helps if the sky is clean and a pure blue colour; (dust-haze, on the other hand, gives the sky a lighter shade or perhaps even a cream colour - in which case the crescent will not be prominent). Thus, this phenomenon will be witnessed more often from very high altitude locations.

Similarly, an old moon which is about three or four days short of conjunction might look 'upside down' just after midday - in the western sky.

Incidentally, it is comparatively easy to photograph an upside-down crescent-shaped Earth from the lunar surface - because there is no atmosphere there to scatter the sunlight all round the sky; (the nearby sun can of course be shielded from view).

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6. To MINDSPORT, “Khaleej Times” Weekend (Dubai, UAE) - 19th March 1999:

It is not possible to devise a rigid rule for leap-year corrections which tries to go ahead several thousand years, because the lengths of our day and of our year will not always remain the same as they are now.

Two thousand years ago, Earth’s year was about 10 seconds longer than it is at present. (Variations are induced by gravitational interaction with other planets, and may be calculated using the laws of celestial dynamics, modified by relativity theory).

Around AD 1, our day-length was 0.03 (or perhaps 0.04) seconds shorter than it is today. Tidal friction causes Earth’s rotation to slow down (but not at a constant rate: irregular fluctuations are induced, for example, by volcanic and tectonic convulsions). We can deduce these figures approximately by studying occurrences of ancient eclipses.

Professor Selvan is right that present values imply an accumulated error in the Gregorian calendar of just over one day every 4000 years. However, if we select the parameters prevailing 2000 years ago, then the calculated error is only about a quarter of a day during 4000 years.

It is not worth trying to calculate what the figures might become in AD 4000 and AD 8000, as their day-lengths cannot be computed accurately.

[A similar point was made - regarding Islamic calendars - in a letter to Hamdard Islamicus (Karachi) XVIII(4), pages 123-125 (Winter 1995)].

Note added subsequently:
In chapter 63 of his "More Mathematical Astronomy Morsels" (Willmann-Bell, Inc., 2002), Jean Meeus discusses this question in great detail. Strictly speaking, the Gregorian calendar aims to keep the date of (northern) Spring Equinox close to 21st March: this is really how it should be defined and treated. We get different results if we work instead with dates of the September Equinox, or with one of the solstices.

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7. From Mostafa Afifi (New Jersey, USA) - 2002:

I am not convinced that the lunar orbit changes as drastically as astronomers claim, i.e. with a lunation fluctuating between 29 days 6½ hours - and 29days 20 hours. I believe that this period cannot vary much from its mean value of 29.53 days. What evidence is there to support the scientists?


(i) The strong gravity of the sun is constantly disturbing the moon's motion round our planet, and its (highly complex) effects may be calculated accurately according to well-established laws and equations of celestial dynamics (and relativity theory). The computations agree extremely well with actual observations. Nowadays, the moon's position may be measured to within a few metres using laser reflectors placed on its surface by astronauts.
   The fastest orbits occur when New Moon falls near perigee. When New Moon falls near apogee, on the other hand, the balance of forces changes, and a lunar circuit is significantly slower. Jean Meeus has shown that extremely long lunations occur when a New Moon at apogee is also passing through one of its nodes - particularly if that coincides with the time when Earth is near perihelion.

(ii) There are plenty of examples of two (partial) solar eclipses which are separated by only 29 days 7 hours. (A new lunation always commences during a solar eclipse).

(iii) It is not difficult to compare clock-times, in successive months, when the moon draws alongside and "overtakes" a particular star. In that manner, even ancient Greek and mediaeval Arab astronomers managed to notice significant differences in the duration of a sidereal month. Alternatively, you could work with intervals between occasions when exactly half the lunar disc is illuminated, perhaps utilising the cross-hairs in an eyepiece to achieve great accuracy.

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8. From Suleiman Segura (Spain) - June 2004:

What do you think about the article by Dr Mansour Hassab-Elnaby at - which attempts to derive the Speed of Light from movements of the Earth and moon within our solar system?


Many of the key values in the calculations do not remain the same with the passage of time. These include the length of our day and of our year. Earth's rate of spin is gradually slowing down (so its days are becoming longer), partly due to tidal and atmospheric friction. Furthermore the modifications tend to be irregular and unpredictable, because they are influenced by volcanic and tectonic convulsions.

The time-span comprising Earth's 'year' does vary according to our planet's orbital eccentricity, which is presently decreasing. Unfortunately, as soon as we try and examine its behaviour over a period of a few hundred thousand years, we find that the fluctuations cannot easily be expressed mathematically. In addition, it is necessary to keep track of changes in the rate of precession of Earth's spin-axis. It seems too that Dr Hassab-Elnaby utilises the anomalistic year in his computations, but he does not appear to consider the slow migration of Earth's perihelion-point.

Thirdly, the time taken for the moon to orbit our planet is not constant. Even if we try and define an 'average' month, we encounter the problem of the moon's long-term deceleration. To be more specific, the mean interval between successive New Moons is 29.530589 days, but a thousand years ago it was 29.530587 days. There has of course been a similar alteration in the length of a Sidereal Month.

Furthermore, the distance of the moon from the Earth is increasing very slowly but surely.

Therefore, if we adopt parameters prevailing several thousand years ago, and reexamine the arithmetic portion of that article - then we will arrive at a different answer for the speed of light. Obviously, the discrepancy will be worse if we care to go back (or forward) a million years, say.

A useful book throwing more light on this business - is "More Mathematical Astronomy Morsels" by Jean Meeus (Willmann-Bell, Inc., 2002). Chapter 33 discusses the eccentricity of Earth's orbit. Its chapter 6 is also illuminating - demonstrating that extreme low lunar perigee values in 3000 to 4000 years' time, will not be as close as they were 3000 years ago.

Chapter 4 is interesting for its treatment of cycles and trends in lunation-values. Even though this quantity does not feature directly in Dr Hassab-Elnaby's article, it is interesting that lunar months in excess of 30 days are possible if we are prepared to look back (or forward) a million years or so - whenever Earth's orbital eccentricity becames abnormally high (page 30). That will probably have an effect on the length of a Sidereal Month too, because gravitational perturbations by the sun will be that much stronger at perihelion.

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9. From J. G. W. (USA) - January 2005:

I have been thinking about one of the uses of complex numbers - whereby they often supply 'missing' roots of polynomial equations. However, I suspect that, in order to appreciate the full spectrum of all possible interrelationships involved there, it would be necessary to visualise them in three, or maybe four dimensions - but I have not yet managed to think this through mathematically.


Yes, you are right: it does provide a stimulating exercise in four-dimensional thinking. We need not bother with an example containing 'missing roots'; instead, let me illustrate with what is a very simple 'polynomial' - namely
y = x:squared.
If we allow "x" to take on complex values (which are a mixture of "real" and "imaginary"), then "y", too, will usually end up as a complex number. In that scenario, in order to cover all types and possibilities of variation, we would therefore need two "x:axes" and two "y:axes" - thus requiring four dimensions for a complete and proper appreciation.

However, most of us are simply not capable of visualising more than three dimensions. So, to illustrate - and to try and gain at least some idea - let's see what might be managed with a series of 'partial pictures'. Essentially, this consists of examining [3-D] 'snapshots' of specified sections of that complex quadratic function. This involves taking three-dimensional slices through a four-dimensional universe - so we should really call them "hyperslices", or perhaps "hypersurfaces"- because they embrace more than two dimensions. [This exercise may be compared with the simpler, more familiar one of examining a two-dimensional surface within our ordinary 3-space: for example, think of a plane surface produced and exposed by a straight cut or slice through a solid object].

To obtain our first 3-D hyperslice, we restrict ourselves to real (=everyday) and to imaginary "x:numbers". (This means excluding complex [i.e. mixed] values of x). The above expression 'x:squared' will then produce only real values of y ... so in this partial representation, just one "y:axis" will suffice, and we can proceed by plotting points and joining them up into line-graphs. These lines show where this particular 3-D hyperslice intersects the 4-D function "y=x:squared".

The real values of "x" will of course produce the parabola which is familiar to mathematicians ... while the imaginary values will yield another parabola, inverted with respect to the first, but of identical shape. The two parabolas are mutually perpendicular, and touch at the 'origin' (zero-point) - whilst lying on opposite sides of the "y equals zero" plane. In this particular hyperslice or snapshot, only one "y:axis" appears - i.e., the axis labelled with real values of y.

A totally different (second) 'snapshot' is also feasible - constructed around the other "y:axis" [i.e. the one containing all imaginary values of "y"]. In this second 3-D picture, consider just those x:values which are "multiples of 1+i". Squaring them yields only imaginary values of "y" [i.e. without any real component] - so all associated points may easily be plotted in this new three-dimensional hyperdiagram. Afterwards, we can put in more points - corresponding to x:values which are "multiples of 1-i". After completing both these graphs, we (again) end up with two mutually inverted and perpendicular parabolas - shaped just like those in the first hyperslice. However, unlike those other parabolas, these new ones are not coplanar with either of the two "x:axes" - (instead, they have been rotated through 45 degrees).

It is also feasible to examine x:values which are "multiples of 2+i". It is easier to visualise this third 'snapshot' if both "y:axes" are included; there is then room for one additional axis, which could be the line containing all these x:points.

More snapshots or hyperslices may be obtained by generalising, feeding in x:values which are "multiples of a+bi"; (some algebraic manipulation is necessary, but it is not difficult). And we always find that the local, individual graphs relating y with x are parabolas - shaped exactly like the first ones.

Now we can attempt to visualise the complete four-dimensional (complex-number) function "y=x:squared". All parabolas must presumably join together - with one placed behind the other. The result may be compared with a three-dimensional parabolic trough [... or tunnel - depending which way up it is]. The 'wall' of that trough is a two-dimensional surface winding its way through 4-D space. Each individual parabola is merely a slice across the trough. The two mutually inverted parabolas in the first 'snapshot' must form part of that same extended surface, which manages to turn through 90 degrees on its way from the top one to the lower one. (Note the suggestion of 45-degree rotation between the first and second 'snapshots' above). With the extra freedom offered by 4-D space, it is evidently easy for a two-dimensional surface to twist itself round like that without becoming too confused and 'mixed up', although it does require some effort to try and imagine the result - (just as the inhabitants of "Flatland" would have had difficulty picturing a Möbius Strip, with its characteristic twist).

Every parabola is of course anchored to the origin (zero-point), so the following representation may be helpful:
Start with an ordinary "y = x:squared" parabola in a horizontal plane - plotted using real values of "x" and "y" . Then slowly spin it round a vertical axis pierced through the origin. At the same time, rotate the parabola round its own axis (which is also the "y:axis"), such that it executes one revolution while the horizontal plane completes two. I believe that the convoluted surface traced out by that parabola (in 3-D space), will then be similar to the graph, in four dimensions, of all complex values of "y=x:squared".

Of course, we could have tried to examine an even simpler equation in four dimensions - such as "y=x". However, the above example probably enjoys an advantage in that the parabola provides a very distinctive feature on which we can focus our attention.

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10. From Dr Tahir Ijaz (California) - December 2005:

How accurate and reliable were eclipse predictions in the 19th century? I am thinking in particular about the Ahmadi phenomena of March/April 1894. I believe that Mirza Ghulam Ahmad was completely unaware of those imminent eclipses when he claimed to be the Mahdi.


Peter Hansen's refined lunar theory dates from the 1860's, being further developed by Simon Newcombe in the early 1880's. In 1887, the Austrian astronomer Theodor von Oppolzer published his "Canon der Finsternisse" showing details of past and future eclipses during a period of more than 3000 years. His accuracy was really commendable. Needless to say, the computations were very time-consuming in those days.

Even before those dates, scientists used to attempt predictions - making the best of simple and crude mathematical techniques. This is confirmed by the fact that expeditions were often organised to observe solar eclipses. Patrick Moore's "Guinness Book of Astronomy" mentions a party sent to Maine from Harvard University in October 1780 (when they had to cross the battle-line during the American War of Independence!) - but unfortunately their calculations were not good enough to position themselves within the zone of totality. However, astronomers were more successful with the solar eclipse of June 1806.

Even before the 18th century, it proved possible to predict eclipses, although only in general terms. Their times were often wrong by a few hours, and (as happened in Maine in 1780), people did not usually know in advance exactly where a total solar eclipse could be witnessed - because its zone or track is comparatively narrow.

From Babylonian times, ancient astronomers knew about the 'saros cycle', and could often invoke it to forecast lunar eclipses; they sometimes had success too with solar ones. In that manner, the Greek philosopher Thales of Miletus apparently announced beforehand the solar eclipse of May 585 BC.

A 'saros series' lasts over 1000 years, consisting of solar eclipses which are 18.03 years apart. The central line (or 'track of totality') of each event tends to run slightly north or south of the earlier ones; in this manner, there is a slow migration from one pole to the other. About two weeks before or after every solar eclipse, there is of course a lunar one. At any one moment, there will be between 30 and 40 saros series running independently.

The significance of the saros is that 6585.32 days after an eclipse there will be another one, and then a further event after a similar interval - and so on until the sequence eventually leaves the Earth near the North or South Pole. In particular, the triple-saros was the key to solar eclipse prediction - because after that time they usually reappear over approximately the same portion of the globe. Thus, the Greeks gave that period a special name: an "exeligmos".

Thus, if Indian astronomers had noticed and recorded the partial solar eclipse of 30th January 1786, as well as the subsequent one on 4th March 1840 (belonging to the same exeligmos series) - then application of the ancient Greek or Babylonian rule would certainly have anticipated the appearance in the subcontinent of the Ahmadi solar eclipse of 6th April 1894. Similarly, there was a lunar eclipse visible on 14th January 1786, followed by another one exactly 19756 days later on 17th February 1840: it was therefore perfectly reasonable to expect the next member of the series to appear on 21st March 1894 - i.e., after another 19756-day interval.

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11. To Sheeba Mumtaz (Australia) - March 2011:

In the article "A Universal Islamic Calendar" (Hamdard Islamicus, January 1997 - summarized at ), the uncertainty regarding the date of 1st Ramadan 1511H could be regarded as an illustration of an attempted prediction which is not yet possible - because it depends on unknown future fluctuations in Earth's rotation-rate. That particular example (i.e., deciding whether Ramadan will commence on the 24th or on the 25th of March 2088) is of course linked to the criterion used by the article for determining the start-date of Islamic months - but similar scenarios could also arise with other criteria, including ones based on calculations of crescent sightability.

In 1996, when the above-mentioned article was composed, most Delta-T formulae suggested a 2088 value in excess of 200 seconds, in which case the Ramadan date will be 24th March. However, it now seems questionable whether Delta-T will even reach 175 seconds - so 25th March looks more likely.

You might like to look at the second diagram in the Wikipedia article at - which is accompanied by a tabulation showing years when leap seconds were added: (remember that the addition of leap seconds corresponds to the "growth" of Delta-T). Between 2000 and 2006 the slope of the graph decreased - showing that, between those years, Delta-T did not increase as quickly as it had during earlier years. In order to attain a value of 200 seconds by year 2088, we will not only have to add one leap second every year - it will also be necessary to start inserting two annual leap seconds quite soon.

A diagram on p.33 in the New Scientist of 30th January 1999 suggests the existence of a long-period oscillation in Earth's spin rate. If that oscillation really does exist, then - during the next few century or two - the year-by-year increase in Delta-T will be less than is envisaged. Stephenson and Morrison published supporting evidence in their paper "Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990": Philosophical Transactions of the Royal Society of London, Series A 351 (1995), pp. 165-202.

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