An article which gives an account of some properties of magic squares.
When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.
This short article gives an outline of the origins of Morse code and its inventor and how the frequency of letters is reflected in the code they were given.
This is a quotation from Neugebauer (1969): "The common belief
that we gain "historical perspective" with increasing distance
seems to me utterly to misrepresent the actual situation. What we
gain is merely confidence in generalisations which we would never
dare make if we had access to the real wealth of contemporary
evidence." (Preface vi).
This warning goes for the study of all history, but particularly
the history of past science. So much has been lost of all the
material that was written, we can never really be sure of any claim
for the first appearance or the 'invention' of an idea. The more
that has recently been studied, the less we can rely on former
claims that, for example, 'Hipparchus invented trigonometry'. What
he did, we now know, was to modify a technique known to both
Egyptian and Babylonian scholars, using the data that had been
passed on to him, and in this respect, he was an important link in
a chain. It depends what you mean by 'invented' and 'trigonometry'.
Not until the Arabs used the Indian idea of 'sine' and then
developed the Greek and earlier techniques to apply them to map
making, geography, land measurement, and other sciences does
'trigonometry' become a separately established part of
The earliest conceptions.
Early man was aware of the rotation of the heavens and that this
was a recurrent phenomenon. Circular movement and repeated events
like the seasons, and natural changes like the flooding of rivers,
began to be linked to events in the sky. Further observation led to
the recognition of patterns in the stars and the regular phases of
the moon. The motion of the moon, set against the patterns of the
stars, led to the conception that different sections of the sky
were visible at different times of the year. This led to the idea
that the heavens could be split up into sections (decans of 10 degrees each in
Egypt, and in Mesopotamia, sections of 30 degrees for the zodiacal
Thus rotation in a circle, and
division of the rotation into sections was the foundation for the
idea of angle.
Does your school have a telescope? Even with a modest one, you
can see a lot. Where is your nearest planetarium? These can often
be found at, or borrowed from, your regional Science Centre.
Measuring Heights and Distances.
The Merkhet was used for measuring both horizontally and
vertically. Simple measures of heights and distances were used to
locate objects in the sky. From the practical measurement of a
height and a horizontal distance, we get the tangent ratio, and
from the measurement of the half-chord and the radius of the circle
we obtain the sine ratio.
The geometrical idea of ratio, the comparison of two like
quantities, was formally stated in Euclid Book V
(Definitions 3, and 5,) and then extended to proportion, which is the equivalence of ratios
Note that our use of the
word proportion has two senses. In the expression "A is proportional to B" or "A and
B are in proportion" we
are comparing two ratios in the Euclidean sense, whereas "X is a proportion of Y" refers to
a part-whole relationship.
Comparing ratios of quantities lies at the heart of many
practical situations. If the entities in the ratio are different
then there are many kinds of mixed ratios, miles/hour (mph),
metres/second (mps), pounds per square inch (lbs/inch$^2$) and so
on. Also we find many concepts in Physics like density (mass/unit
volume, or m/cm$^3$), the ampere (electric charge flowing per
second dQ/dt), the Galileo (metres per second$^2$ or m/s$^2$) all
measured in mixed ratios.
Once we have the possibility of comparing ratios of unlike
magnitudes (objects, substances, etc.) we are in a very powerful
Provided we can establish a relationship of equality between
pairs of ratios of unlike things, the possibility for scientific
experiment and explanation is open.
If we are able to establish a direct relationship between two
quantities so that naming one quantity precisely identifies
another, (and vice versa) then we call this a function. We say that
'y is a function of x' and use various symbols to show this. The
precise definition of function involves a very sophisticated
collection of concepts, and even the 'loose' use of the word (like
the familiar 'function machine') involves ideas that have developed
over many years. In fact, our modern definition of function did not
appear until the middle of the 20th century!
There is no evidence that ancient people thought of any
relationships between numbers, geometrical quantities or events in
this way. They certainly recognised and used relationships in their
reasoning, they realised that some relationships were regularly
repeated, but to attribute the idea of a function to them is quite