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.
Published December 2010,February 2010,December 2011,February 2011.
This is the third of three articles on
the History of Trigonometry.
Part 2 (Sections 5 - 7) can be
found here. Part 1
(Sections 1 - 4) can be found here.
"For no one can bypass the science of
triangles and reach a satisfying knowledge of the stars .... You,
who wish to study great and wonderful things, who wonder about the
movement of the stars, must read these theorems about triangles.
Knowing these ideas will open the door to all of astronomy and to
certain geometric problems. For although certain figures must be
transformed into triangles to be solved, the remaining questions of
astronomy require these books." [See Note 10 below]
The first book gives the basic definitions of quantity, ratio,
equality, circles, arcs, chords and the sine function. Next come a
list of axioms he will assume, and then $33$ theorems for right,
isosceles and scalene triangles. The formula for the area of a
triangle is given followed by the sine rule giving examples of its
application. Books III to V cover the all-important theory of
spherical trigonometry. The whole book is organised in the style of
Euclid with propositions and theorems set out in a logical
hierarchical manner. This work, published in 1533 was of great
value to Copernicus.
Regiomontanus also built the first astronomical observatory in
Germany at Nuremburg with a workshop where he built astronomical
instruments. He also took observations on a comet in 1472 that were
accurate enough to allow it to be identified as Halley's Comet that
reappeared 210 years later.
Regiomontanus died during an outbreak of plague in Rome in
Copernicus wrote a brief outline of his proposed system called
the Commentariolus that he
circulated to friends somewhere between 1510 and 1514. By this time
he had used observations of the planet Mercury and the Alfonsine
Tables to convince himself that he could explain the motion of the
Earth as one of the planets. The manuscript of Copernicus' work has
survived and it is thought that by the 1530s most of his work had
been completed, but he delayed publishing the book.
His student, Rheticus read the manuscript and made a summary of
Copernicus' theory and published it as the Narratio Prima (the First
Account) in 1540. Since it seemed that the Narratio had been well accepted
by colleagues, Copernicus was persuaded to publish more of his main
work, and in 1542 he published a section on his spherical
trigonometry as De lateribus et
angulis traingulorum (On the sides and angles of triangles).
Further persuaded by Rheticus and others, he finally agreed to
publish the whole work, De
Revolutionibus Orbium Coelestium (The Revolutions of the
Heavenly Spheres) and dedicated it to Pope Paul III. It appeared
just before Copernicus' death in 1543. [See Note 11 below]
However, soon after Rheticus' Opus Palatinum was published,
serious inaccuracies were found in the tangent and secant tables at
the ends near $1^\circ$ and $90^\circ$. Pitiscus was commissioned
to correct these errors and obtained a manuscript copy of Rheticus'
work. Many of the results were recalculated and new pages were
printed incorporating the corrections. Eventually, Pitiscus
published a new work in 1613 incorporating that of Rheticus with a
table of sines calculated to fifteen decimal places entitled the
By the beginning of the seventeenth century, the science of
trigonometry had become a sophisticated technique used in
calculating more and more accurate tables for use in astronomy and
navigation, and had been instrumental in fundamentally changing
man's concept of his world.
1. See Part
1 section 3 on the Sulbasutras.
2. See Note 4 in Part1.
The use of the capital S in Sine is to show that the radius of
the circle used is not unity, or the same as $\sin\theta$ in our
system, but could be an arbitrary length R. This means that
Sin$\theta$ is equal to R sin$\theta$ . In the Indian texts,
different astronomers took different values for R, and in most
cases the value had to be deduced from the context.
3. The advantage of the 'versine' (or reversed sine) is that
it's value is always positive and so its logarithm is defined
everywhere (except at $0^\circ$ and $180^\circ$). A positive
logarithm was necessary when calculations had to be done using
tables. The most important use was in navigation, for calculating
the distance between two points on a sphere. The perpendicular
distance from the mid point of a chord to a curve is still used as
a measure of 'deviation from straightness', for example, by railway
engineers. It is used also in optics for measuring the curvature of
lenses and mirrors, where he versine is sometimes called the sagitta from the Latin for
4. Compare the sine curve from $0^\circ$ to $180^\circ$ with $y
= -a(x- \pi/2)^2 + c$. By adjusting the values of $a$ and $c$, it
is possible to produce a curve of 'best fit' inside the sine curve.
You can obtain a remarkably good fit for $0 < x < \pi$.
5. The Hindu word jiya
for the sine was adopted by the Arabs who called the sine jiba. Eventually jiba became jaib and this word actually
meant a 'fold'. When Europeans translated the Arabic works into
Latin they translated jaib
into the word sinus
meaning a fold in Latin. In his Practica Geometriae (1220)
Fibonacci uses the term sinus rectus arcus which soon encouraged
the universal use of the word sine.