ÐÓ°ÉÔ­´´

Q: Interested readers were asked to write in and explain if (and why)
the published explanations to the ‘Moon’s up’ question (4 June) – one employing
spherical geometry and one employing plane geometry – were essentially the
same. Readers were also asked to explain one more thing – if rays of light
from the Sun are parallel, why is it that sunbeams penetrating a hole in
the clouds spread out in the fan-shaped pattern so often depicted by Turner
and other painters of seascapes? Another coincidence, an illusion, or something
else?

* * *

A: The answers to the problem of the apparently wrongly illuminated
Moon are not as incompatible as they appear. Two of the answers, presented
in terms of plane geometry, had the correct idea of a distant and relatively
large Sun emitting virtually parallel rays. This can be superimposed on
the spherical geometry of the other answer, where use of a point source
Sun clarifies the diagrams. So the virtually parallel rays (of the plane
geometry rep-lies) are following great circles (as per the spherical geometry
reply) – the shortest distance between two points in spherical geometry.
Spherical geometry is used to cope with the illusions inherent in observing
what appears to be a dome (the sky) from the surface of a spinning sphere
(the Earth). Actual bending of light is not implied, any more than when
a straight pole casts a curved shadow on a hump in the ground.

John Fishwick The Nautical Campus Astronomical Society, Fleetwood, Lancashire

* * *

A: You can use a broom handle to illustrate the Moon’s illumination.
If you hold the broom so it appears to pass through the middle of both Sun
and Moon, you will find the illuminated part of the Moon is symmetrical
above and below this line. That the illuminated side seems not to point
directly at the Sun is an optical illusion. We perceive the sky as a dome
and mentally trace out curved lines across it rather than straight ones.
The handle forces you to imagine the straight line followed by the light
rays.

Stuart Arnold Gauting, Germany

* * *

A: The diverging sunbeam phenomenon is most often seen in early morning
or evening, not when the Sun is high in the sky, and therein lies the clue.
The beams of light are not descending perpendicular to the Earth’s surface,
but taking a shallow angle in relation to the viewer. This gives a perspective
effect, as with railway lines that converge at a point in the distance.
The rays of light similarly seem to converge to a point just above the clouds.
In this case though, there are no railway sleepers getting ever closer with
distance, to help the eye.

B. Fisher Hitchin, Hertfordshire

Roman knows

Q: How did the Romans do basic arithmetic using their system of numerals?
Using our Arabic number system we find it easy to place two numbers, one
above the other, and add or subtract. How can this be done using a number
such as MCMXCIV (1994)?

* * *

A: Basic arithmetic by writing down the numbers is effectively impossible
using Roman numerals. This is why we use Arabic numerals even though countries
which accepted Christianity from Rome adapted the Roman alphabet for writing
their languages. The ancient Romans did not have a problem with basic arithmetic
because they used the abacus as a calculating tool. We should not forget
that our procedures for basic arithmetic using pencil and paper are also
just tools for calculating. The memorising of rules and simple arithmetical
calculations for use in our tools is analogous to learning which beads to
move on the abacus.

These days the pocket calculator is the ultimate tool for basic arithmetic
and I am sure it is possible to make one that operates in Roman numerals.
If the ancient Romans had invented the pocket calculator we would probably
still be using Roman numerals. After all, when the computer arrived we
did not change to the binary system for everyday arithmetic.

M. W. Kawecki Dhahran, Saudi Arabia

* * *

A: The Romans used the abacus. The Greek word abax refers to a board
covered with sand and in its simplest form the Roman abacus was scratched
on the ground. It had columns for units, tens, hundreds and so on, the numbers
being represented by marks in the sand, or by pebbles. They also used a
frame with beads like a simpler version of the modern Chinese abacus. A
more complicated grid of squares was used for multiplication, with the two
numbers written along adjacent sides and the units, tens, etc. multiplied
along the diagonals and added up on the opposite sides. This could not be
used for division, for which the Romans had to break the sum down into steps
similar to those we use for long division today. It may seem strange that
they did not develop the use of tens and units into a proper positional
numbering system. However, the crucial development in bringing about the
system we use today was the invention of a symbol for zero by the Indian
astronomer Aryabhata around 1500 years ago.

Ralph Hancock London

A copy of Ralph Hancock’s article from the Open University magazine
M500 on the multiplying abacus is available by post, free of charge. Please
fax or write to New ÐÓ°ÉÔ­´´ at the address below – Ed

* * *

A: Roman numerals themselves were used only to record numbers. Their
great strength is that they are difficult to falsify and so they remained
in use for business and administration long after decimal notation was in
use for calculation.

Robert Fairthorne Farnborough, Hampshire

Canned gas

Q: We often hear that Freons in aerosols and refrigerators can be damaging
when they are released. Before these gases were utilised in containers,
where did they exist?

* * *

A: Freons are entirely synthetic. Prior to their use in aerosols and
refrigerators they did not exist. So the only action required to alleviate
the problems caused by their release is to stop producing them.

David Stocker Dubendorf, Switzerland

* * *

A: Freons are synthetic compounds based on paraffin hyd-rocarbons with
most or all of the hydrogen atoms substituted by fluorine and chlorine.
Freon is a trade name, and together with Arctons they go under the more
general name of halons. They were originally chemical curiosities but their
inertness and ease of change between gaseous and liquid phases made them
ideal for a variety of uses such as extinguishing fires and aerosol propellants.

R. V. Foster Maidenhead, Berkshire

This week’s questions

Mirror image: Why is an image in a mirror inverted left to right but
not top to bottom?

Kishor Bhagwati Lausanne, Switzerland

* * *

Aromatic pees: What is the aromatic compound excreted in urine after
eating asparagus?

Barry Timms South Dakota, US

* * *

Eggstraordinary: Why does a very fresh egg take appreciably longer to
cook than one that is a day or more old?

Richard Moore Ashurst Wood, West Sussex