# Some Facts and Figures.

Antipodal.   The Earth's surface is approximately four fifths covered by ocean. It is thus obvious that many parts of the ocean will be antipodal to ocean.

(Two points on the Earth's surface are said to be antipodal to each other if the straight line joining them goes through the centre of the Earth.)

Are there any areas of land which are exactly antipodal to land?

I sometimes used to wonder that question, just out of curiosity. It is sometimes said that Australia and New Zealand are opposite to Britain, but that is at best an approximation - and in the case of Australia, not even a particularly close approximation at that. Both are closer to the equator the Britain is, and are west of the 180 degree line of longitude.

Two points on the Earth are antipodal if one has (a) a South lattitude equal to the other's North lattitude, and (b) a longtitude which differs by 180 degrees. If one is measuring longitude on the map using 0 to 360 degrees, this calculation is easy - just add or take 180 degrees from the longitude of one to get that of the other. If as is more usual, the map indicates longitude from 180 degrees West to 180 degrees East of Greenwich, then one needs to regard the degrees of longitude as positive or negative depending which side of the Greenwich meridian it is.

It turns out that part of New Zealand is antipodal to parts of Spain and Portugal.

Parts of South East Asia, including China and Mongolia, are antipodal to South America.

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Elliptical Orbits, and Bank Notes.   It has been brought to my attention, that some bank notes issued by the Bank of England in the past, contained a picture of the English scientist Isaac Newton, and also a diagram of planets orbiting the Sun in elliptical orbits. However, whilst both Newton and Kepler realised that a planet orbits the Sun in a ellipse with the Sun at one focus (Kepler's First Law), the artist designing the bank notes appears to have believed that these scientists were careless in drawing the Sun off centre, and thus "corrected" the situation by drawing the Sun at the centre of the ellipse, on the bank notes.

Newton used Kepler's laws on planetary motion to formulate his laws of gravity, including his inverse-square law – which states that the force of gravity between two masses is an attractive force whose magnitude is inversely proportional to the square of the distance between them.

As Newton and others were aware at the time, there are two possible types of attractive force law which give rise to elliptical orbits - Hooke's direct first-power law (attractive force directly proportional to the first power of distance - sometimes known as the harmonic-oscilator force law), and Newton's inverse-square law (attractice force inversely proportional to the square of the distance). The difference between these types of orbit is that in the case of Hooke's Law the central force of attraction would be at the centre of the ellipse, whereas in the case of Newton's inverse-square law the central force (in this case the Sun) is at one focus of the ellipse.

Hooke's harmonic-oscilator force law does not apply to gravitation, although it does apply to certain other forces, including (appoximately) certain types of elastic force. Hooke's Law applies to a first approximation to pendulum motion provided that the pendulum only traverses small arcs (such that the sin of the angle is appoximately equal to the angle itself measured in radians).

Newton's inverse-square law applies to gravitation, and also to electro-static forces (whether attractive or repulsive).

Perhaps the artist on the bank notes confused Newton's Law with Hooke's Law!

(As an aside, be it noted that should an attractive force tail off with distance according to an inverse-cube law, or even more sharply, then stable orbits are not possible; the orbiting body would end up either escaping from the system, or spiraling in towards the central source of attraction.)

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Isaac Newton.   Isaac Newton (4 Jan 1643 - 31 Mar 1727, new-style dating) (25 Dec 1642 - 20 Mar 1727, old-style dating) was one of the most famous of English Scientists, being credited among other things, with his laws of motion and of gravity, and for his book Philosophić Naturalis Principia Mathematica, published in 1687. He formulated the concepts of conservation of momentum and angular momentum, and showed experimentally that white light can be split by a prism into the different colours. He invented the reflecting telescope, and shares the credit with Gottfried Leibniz for inventing calculus. He did research into the relationship between orbital configurations and how the (attractive) force varies with distance from the central object. He worked out from Kepler's laws that the gravitational attractive force follows an inverse-square law in relation to distance.

Though not so well known, Newton also devoted much time to study of the Bible. He believed in a monotheistic God as the masterfull creator. He wrote a number of religious tracts, many of which were not published during his lifetime. He disputed the existence of the Trinity. Political climate in Newton's time was hostile to non-trinitarian views, and such writers were persecuted.

Up to Newton's time most Fellows at Oxford and Cambridge were expected to be ordained priests of the Anglican Church. However Lucas' will stipulated that the holder of the Lucasian Professor of Mathematics should not be active in the Church. Newton, not wishing to take holy orders, referred to Lucas' stipulation in an appeal to King Charles II, who then excused holders of this professorship from the requirement to take orders.

Newton was twice a member of Parliament.

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About Blue Moons.   There is a phrase Once in a blue moon, which is sometimes used in a rather-vague way to mean "not very often".

What, however, does the phrase actually mean? Or does "a blue moon", in fact, have any definite meaning?

Although, during a lunar eclipse, the moon does actually darken, and become some unusual colour (depending on the Earth's atmospheric conditions at the time), the phrase Blue Moon, is currently used frequently, to refer to a second Full Moon in one calendar month. The term has had some variation in meaning though, even over the past century.

Since the lunar month is around 29.5 days, and all calendar months on the modern calendar, with the exception of February, exceed this, it follows that sometimes, though not very often, a month with 31 days, or more rarely, 30 days, can have a Full Moon near the beginning of the month, then a second Full Moon near the end. It is this second Full Moon which is sometimes referred to as the Blue Moon.

Because February has less than 29.5 days, it will sometimes, though not very often, not have a Full Moon. When February does not have a Full Moon, January and March will each have two Full Moons (occasionally it will be January and April (or rarely May), or December and March), leading to what some refer to as a Double Blue Moon.

As of 2020, the last Double Blue Moons (assuming times are calculated using Greenwich Mean Time) were in January and March of 2018, and the ones before that January and March of 1999. In these years, February did not have a Full Moon.

Because, astronomically speaking, the Full Moon occurs at a particular moment in time, it might occur because of time zones, that the date of a particular Full Moon, measured according to local time, will vary depending on where one is on the Earth's surface. Because of this, on some occasions, certain places on the Earth may experience a Blue Moon in a particular month, whereas others do not.

Countries near the Greenwich meridian experienced a Blue Moon in December 2009. Parts of east Asia, Australia and New Zealand, however, instead experienced a Blue Moon in January 2010. Some of these areas also experienced a second Blue Moon in March 2010, thus had a Double Blue Moon.