Copyright © University of Cambridge. All rights reserved.
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.
8. The Arabs collect knowledge from the known world
The Arab civilisation traditionally marks its beginning from
the year 622 CE the date when Muhammad, threatened with
assassination, fled from Mecca to Medina where Muhammad and his
followers found safety and respect. Over a century later, the Arabs
had established themselves as a powerful unified force across large
parts of the Middle East and The Caliph Abu Ja'far AlMansour moved from
Damascus to establish the city of Baghdad during the years 762 to
766. AlMansour sent his emissaries to search for and collect
knowledge. From China, they learnt how to produce paper, and using
this new skill they started a programme of translation of texts on
mathematics, astronomy, science and philosophy into Arabic. This
work was continued by his successors, Caliphs Mohammad AlMahdi and Haroun AlRasheed. The quest for
knowledge became a lasting and significant part of Arab
culture.
AlMansour had founded a scientific academy that became called
'The House of Wisdom'. This academy attracted scholars from many
different countries and religions to Baghdad to work together and
establish the traditions of Arabic science that were to continue
well into the Middle Ages. Some of this work was later translated
into Latin by Mediaeval scholars and passed on into Europe. The
dominance of Baghdad and the influence of the Arab World was to
last for the next 500 years.
The scholars in the House of Wisdom came from many cultures
and translated the works of Egyptian, Babylonian, Greek, Indian and
Chinese astronomers and mathematicians. The Mathematical Treatise
of Ptolemy was one of the first to be translated from the Greek
into Arabic by Ishaq ben
Hunayn (830910). It was admired for its extensive content
and became known in Arabic as AlMegiste (the Great Book). The
name 'Almagest' has
continued to this day and it is recognized as both the great
synthesis and the culmination of mathematical astronomy of the
ancient Greek world. It was translated into Arabic at least five
times and constituted the basis of the mathematical astronomy
carried out in the Islamic world.
9. India: The Sine, Cosine and Versine
Greek astronomy began to be known in India during the period
300400 CE. However, Indian astronomers had long been using
planetary data and calculation methods from the Babylonians, and
even though it was well after Ptolemy had written the Almagest, 4th
century Indian astronomers did not entirely take over Greek
planetary theory. Ancient works like the
Pancasiddhantica (now lost)
that had been transmitted through the version by Vrahamihira [
See Part 1 section
3] and Aryabhata's
Aryabhatiya (499 CE)
demonstrated that Indian scholars had their own ways of dealing
with astronomical problems and that they had great skill in
calculation.[See Note 1 below]
Even in the oldest Indian texts, the Chord [to remind yourself
about Chords see the section on Claudius Ptolemy in the
previous article]
is not used, and instead there appear some very early versions of
trigonometric tables using Sines. However, the Indian astronomers
divided the $90^\circ$ arc into $24$ sections, thus obtaining
values of Sines for every $3^\circ45'$ of arc.
In this diagram, $SB$ is the arc for the angle $\theta$ and
$AS$ is the jiya. So the
relation between the jiya
and our sine is:
$$ jiya (\theta)=R\sin\theta$$
where $R$ is the radius of the circle.
Many Indian Sine tables use $R = 3438$ which is the result if
the circumference of the circle is $360 \times 60$ or $21,600$
minutes. [See Note 2 below]
By the 5th century, two other functions had been defined and
used. The length $EA$ was called the kotijya (our cosine), and AB was
called the utkramajya
(our versine). This was sometimes called the sama meaning an 'arrow', or
sagitta in Latin
The versine function for a circle radius $R$ is: $\mbox{vers
}\theta = R  \cos\theta $ [See Note 3 below]
In Aryabhata's work, he uses $R = 3438$ and took this value to
calculate his table of Sines. This became the standard for later
works. Comparison with Varhamihira's Sines (in sexagesimal numbers)
and Hipparchs' table (in lengths of chords) suggests a possible
transmission of at least some of the Greek works to the Hindus.
However, we have no way of knowing this for certain, and it is
quite possible that the Hindus calculated their values
independently.
The 'Great Work' (the Mahabhaskariya) of Bhaskara I
was written in about 600 CE. He produced a remarkable method for
approximating values for the Sines, by using the ratio of two
quadratic functions. This was based entirely on comparing the
results of his calculations with earlier values. [See Note 4 below
]
However, since these tables only gave values for every
$3^\circ 45'$, there was cnsiderable room for improvement. It is
curious that since Ptolemy's table of chords enabled him to find
values equivalent to Sines from $\frac{1}{4}^\circ$ to $90^\circ$
that the Indian scholars did not go further at this stage. Later,
Brahmagupta (598670) produced an ingenious method based on second
order differences to obtain the Sine of any angle from an initial
set of only six values from $0^\circ$ in $15^\circ$ intervals to
$90^\circ$.
10. Trigonometry in the Arab Civilisation
The introduction and development of trigonometry into an
independent science in the Arab civilisation took, in all, some 400
years. In the early 770s Indian astronomical works reached the
Caliph AlMansur in
Baghdad, and were translated as the Zij alSindhind, and this
introduced Indian calculation methods into Islam.
Famous for his algebra book,
Abu Ja'far Muhammad ibn Musa
alKhwarizmi (see
The Development of
Algebra Part 1) had also written a book on Indian methods of
calculation (
alhisab
alhindi) and he produced an improved version of the
Zij alSindhind. AlKhwarizmi's
version of
Zij used Sines
and Versines, and developed procedures for tangents and cotangents
to solve astronomical problems. AlKhwarizmi's
Zij was copied many times and
versions of it were used for a long time.
Many works in Greek, Sanskrit, and Syriac were brought by
scholars to AlMansur's House of Wisdom and translated. Among these
were the works of Euclid, Archimedes Apollonius and of course,
Ptolemy. The Arabs now had two competing versions of astronomy, and
soon the Almagest prevailed.
The Indian use of the sine and its related functions were much
easier to apply in calculations, and the sexagesimal system from
the Babylonians continued to be used, so apart from these two
changes, the early Arabic versions of the Almagest remained
faithful to Ptolemy. [See Note 5 below]
Abu alWafa
alBuzjani (Abul Wafa 940998) made important contributions
to both geometry and arithmetic and was the first to study
trigonometric identities systematically. The study of identities
was important because by establishing relationships between sums
and differences, and fractions and multiples of angles, more
efficient astronomical calculations could be conducted and more
accurate tables could be established.
The sine, versine and cosine had been developed in the context
of astronomical problems, whereas the tangent and cotangent were
developed from the study of shadows of the gnomon. In his Almagest, Abul Wafa brought them
together and established the relations between the six fundamental
trigonometric functions for the first time. He also used $R = 1$
for the radius of the basic circle.
From these relations Abul Wafa was able to demonstrate a
number of new identities using these new functions:
$$sec^2\theta = 1 + tan^2\theta \mbox{. . . . . . . }cosec^2
\theta=1+cot^2 \theta$$
Abul Wafa also devised methods for calculating trigonometric
tables by an improved differencing technique to obtain values that
were accurate to $5$ sexagesimal ($8$ decimal) places.
Greek astronomers had long since introduced a model of the
universe with the stars on the inside of a vast sphere. They had
also worked with spherical triangles, but Abul Wafa was the first
Arab astronomer to develop ways of measuring the distance between
stars using his new system of trigonometric functions including the
versine.
In the diagram above, the blue triangle with sides $a$, $b$, and
$c$ represents the distances between stars on the inside of a
sphere. The apex where the three angles $\alpha$, $\beta$, and
$\gamma$ are marked, is the position of the observer. The blue
curves are Great Circles on the sphere, and by measuring the
angles, finding more accurate values for their functions, and
assuming a value for $R$ the radius of the sphere, it became
possible to find the greatcircle distances between the stars.
By an ingenious application of Menelaos' Theorem [See
History of
Trigonometry Part 2] using special cases of great circles with
two right angles, Abul Wafa showed how the theorem could be applied
in spherical triangles. This was a considerable advance in
Spherical Trigonometry that enabled the calculation of the correct
direction for prayer (the quibla) and was to have important
applications in Navigation and Cartography.
The Abul Wafa crater of the Moon is named in recognition of
his work in astronomy.

Abu alRayhan Muhammad ibn Ahmad AlBiruni (9731050) was an
outstanding scholar reputed to have written over 100 treatises on
astronomy, science, mathematics, geography, history, geodesy and
philosophy. Only about twenty of these works now survive, and only
about a dozen of these have been published.
AlBiruni's treatise entitled Maqalid 'ilm alhay'a (Keys to
the Science of Astronomy) ran to over one thousand pages and
contained extensive developments in on trigonometry. Among many
theorems, he produced a demonstration of the tangent formula, shown
below.

From the diagram, $O$ is the centre of the semicircle, and
$AED$ a rightangled triangle with a perpendicular from $E$ to
$C$.
Consequently, triangles $AEC$ and $EDC$ are similar.
Angle $EOD$ is twice angle $EAD$, and angles $EAC$ and $DEC$
are equal.
If the radius of the circle $R =1$, then $EC = \sin \theta$
and $OC = \cos\theta$
$$\mbox{So
}\tan\left(\frac{\theta}{2}\right)=\frac{EC}{AC}=\frac{\sin\theta}{1+\cos\theta}
\mbox{ . . . and . . . }
\tan\left(\frac{\theta}{2}\right)=\frac{DC}{EC}=\frac{1\cos\theta}{\sin\theta}$$
From which he derived the half angle and multiple angle
formulae. [See Note 6 below]
While many new aspects of trigonometry were being discovered,
the chord, sine, versine and cosine were developed in the
investigation of astronomical problems, and conceived of as
properties of angles at the centre of the heavenly sphere. In
contrast, tangent and cotangent properties were derived from the
measurement of shadows of a gnomon and the problems of telling the
time.
In his
Demarcation of the
Coordinates of Cities he used spherical triangles for
finding the coordinates of cities and other places to establish
local meridian (the
quibla) and thereby finding the
correct direction of Mecca, and in his
Exhaustive Treatise on Shadows
he showed how to use gnomons [See
A Brief History of
Time Measurement] for finding the time of day.
Abu Muhammad Jabir ibn
Aflah (Jabir ibn Aflah c1100  c1160) probably worked in
Seville during the first part of the 12th century. His work is seen
as significant in passing on knowledge to Europe. Jabir ibn Aflah
was considered a vigorous critic of Ptolemy's astronomy. His
treatise helped to spread trigonometry in Europe in the 13th
century, and his theorems were used by the astronomers who compiled
the influential Libro del
Cuadrante Sennero (Book of the Sine Quadrant) under the
patronage of King Alfonso X the Wise of Castille (12211284).
A result of this project was the creation of much more
accurate astronomical tables for calculating the position of the
Sun, Moon and Planets, relative to the fixed stars, called the
Alfonsine Tables made in Toledo somewhere between 1252 and 1270.
These were the tables Columbus used to sail to the New World, and
they remained the most accurate tables until the 16th
century.
By the end of the 10th century trigonometry occupied an
important place in astronomy texts with chapters on sines and
chords, shadows (tangents and cotangents) and the formulae for
spherical calculations. There was also considerable interest in the
resolution of plane triangles. But a completely new type of work by
Nasir alDin alTusi
(AlTusi 12011274) entitled Kashf alqina 'an asrar shakl
alqatta (Treatise on the Secrets of the Sector Figure), was
the first treatment of trigonometry in its own right, as a complete
subject apart from Astronomy. The work contained a systematic
discussion on the application of proportional reasoning to solving
plane and spherical triangles, and a thorough treatment of the
formulae for solving triangles and trigonometric identities.
AlTusi originally wrote in Persian, but later wrote an Arabic
version. The only surviving Persian version of his work is in the
Bodleian Library in Oxford.
This was a collection and major improvement on earlier
knowledge. Books I, II and IV contain parts of the Elements, the
Almagest and a number of other Greek sources. Book III deals with
the basic geometry for spherical triangles and the resolution of
plane triangles using the sine theorem:
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Book V contains the principal chapters on trigonometry dealing
with rightangled triangles and the six fundamental relations
equivalent to those we use today; sine, cosine, tangent, cotangent,
secant and cosecant. He provided many new proofs and showed how
they could be used to solve many problems more easily.

AlTusi invented a new geometrical technique now called the
'AlTusi couple' that generated linear motion from the sum of two
circular motions. He used this technique to replace the equant used
by Ptolemy, and this device was later used by Copernicus in his
heliocentric model of the universe. 
AlTusi was one of the greatest scientists of Mediaeval Islam
and responsible for some 150 works ranging from astronomy,
mathematics and science to philosophy and poetry. 

11. Arab Science and Technology Reaches Europe
The Arab astronomers had learnt much from India, and there was
contact with the Chinese along the Silk Road and through the sea
routes, so that Arab trading posts were established in India and in
China. Through these contacts Indian Buddhism spread into China and
was well established by the 3rd century BCE, probably later
carrying with it some of the calculation techniques of Indian
astronomy. However few, if any, technological innovations seemed to
have passed from China to India or Arabia.
By 790 CE, the Arab empire had reached its furthest expansion
in Europe, conquering most of the Iberian peninsula, an area called
AlAndalus by the Arabs.
[See Note 7 below]
At this time many religions and races coexisted in Iberia,
each contributing to the culture. The Muslim religion was generally
very tolerant towards others, and literacy in Islamic Iberia was
more widespread than any other country in Western Europe. By the
10th century Cordoba was said to have equally good libraries and
educational establishments as Baghdad, and the cities of Cordoba
and Toledo became centres of a flourishing translation
business.
Between 1095 and 1291 a series of religiously inspired
military Crusades were waged by the Christians of Europe against
the Arab Empire. The principal reason was the restoration of
Christian control over the Holy Land, but there were also many
other political and economic reasons. [See Note 8 below]
In all this turmoil and conflict there were periods of calm
and centres of stability, where scholars of all cultures were able
to meet and knowledge was developed, translated and transmitted
into Western Europe. The three principal routes through which Greek
and Arab science became known were Constantinople (now Istanbul)
Sicily and Spain. Greek texts became known to European monks and
scholars who travelled with the armies through Constantinople on
their way South to the Holy Land. These people learnt Greek and
were able to translate the classical works into Latin. From Sicily,
Arabs traded with Italy, and translation took place there, but
probably the major route by which Arabic science reached Europe was
from the translation houses of Toledo and Cordoba, across the
Pyrenees into southwestern France.
During the twelfth and thirteenth century hundreds of works
from Arabic, Greek and Hebrew sources were translated into Latin
and the new knowledge was gradually disseminated across Christian
Europe.
Geometrical knowledge in early Mediaeval Europe was a very
practical subject. It dealt with areas, heights, volumes and
calculations with fractions for measuring fields and the building
of large manors, churches, castles and cathedrals.
Hugh of St. Victor (10781141) in his Practica Geometriae divides the
material into Theorica (what is known and practised by a teacher)
and Practica (what is done by a builder or mason). Theoretical
geometry in the Euclidean sense was virtually unknown until the
first translations of Euclid appeared in the West.
The astrolabe was commonly used to measure heights by using
its 'medicline' (a sighting instrument fixed at the centre of the
circle) and the shadow square engraved in the centre of the
instrument, and then comparing the similar triangles. The
horizontal distance from the centre of the astrolabe to the edge of
the square was marked with twelve equal divisions.
This system was in use well into the 16th century as seen in
the illustration below:
This is from Thomas Digges' Pantometria of 1571. The same
system is still used, but the square in the quadrant is marked with
six divisions.
A popular twelfth century text, the Artis Cuiuslibet Consummatio
shows the gradual insertion of more technical knowledge, where the
measuring of heights (altimetry) was much more related to
astronomy, showing how to construct gnomons and shadow squares.
Gradually the translations made on the continent of Europe came to
England.
Richard of Wallingford (12921336)
After entering Oxford University in about 1308, Richard
entered monastic life at St Alban's in 1316. After his ordination
as a priest, his Abbot sent him back to Oxford where he studied for
nine years. In 1327 he became Abbot of St Albans.

Richard's early work was a series of instructions (canons) for
the use of astronomical tables that had been drawn up by John
Maudith, the Merton College Astronomer. Later he wrote an important
work, the Quadripartitum, on the fundamentals of trigonometry
needed for the solution of problems of spherical astronomy. The
first part of this work is a theory of trigonometrical identities,
and was regarded as a basis for the calculation of sines, cosines,
chords and versed sines. The next two parts of the Quadripartitum
dealt with a systematic and rigorous exposition of Menelaos'
theorem. The work ends with an application of these principles to
astronomy. The main sources of the work appear to be Ptolemy's
Almagest, and Thabit ibn Qurra (826901 CE). 
The Quadripartitum
was probably the first comprehensive mediaeval treatise on
trigonometry to have been written in Europe, at least outside Spain
and Islam. When Richard was abbot of St. Albans, he revised the
work, taking into account the Flores of Jabir ibn Afla.
In 1326 to 1327 Richard also designed a calculation device,
called an
equatorium, a
complex geared astrolabe with four faces. He described how this
could be used to calculate lunar, solar and planetary longitudes
and thereby predict eclipses in his
Tractatus Albionis. It is
possible that this led to his design for an astronomical clock
described in his
Tractatus
Horologii Asronomici, (Treatise on the Astronomical Clock)
of 1327, which was the most
complex clock
mechanism known at the time. The mechanism comprised a rotating
star map that modeled the lunar eclipse and planets by gearing,
presented as a geocentric model. It appeared at a transitional
period in clock design, just before the advent of the escapement.
This makes it one of the first true clocks, and certainly one of
first self powered models of the heavens. Unfortunately it was
destroyed during Henry VIII's reformation at the dissolution of St
Albans Monastery in 1539.
Georg von Peuerbach (14231461)
Peuerbach's work helped to pave the way for the Copernican
conception of the world system; he created a new theory of the
planets, made better calculations for eclipses and movements of the
planets and introduced the use of the sine into his
trigonometry.
His early work, Tabulae Eclipsium circulated in manuscript was
not published until 1514, contained tables of his eclipse
calculations that were based on the Alfonsine Tables. He calculated
sines for every minute of arc for a radius of 600,000 units and he
introduced the HinduArabic numerals in his tables. [See Note 9
below] 

Peuerbach's Theoricae Novae
Planetarum, (New Theories of the Planets) was composed about
1454 was published in 1473 by Regiomontanus' printing press in
Nuremburg. While the book was involved in attempting a technical
resolution of the theories of Eudoxus and Ptolemy, Peuerbach
claimed that the movement of the planets was determined by the Sun,
and this has been seen as a step towards the Copernican theory.
This book was read by Copernicus, Galileo and Kepler and became the
standard astronomical text well into the seventeenth century.
In 1460 he began working on a new translation of Ptolemy's
Almagest, but he had only completed six of the projected thirteen
books before died in 1461.
Johannes Muller von Konigsberg or Regiomontanus
(14361476)

Regiomontanus had become a pupil of Peuerbach at the University
of Vienna in 1450. Later, he undertook with Peuerbach to correct
the errors found in the Alfonsine Tables. He had a printing press
where he produced tables of sines and tangents and continued
Puerbach's innovation of using HinduArabic numerals. 
As promised, he finished Peuerbach's Epitome of the Almagest,
which he completed in 1462 and was printed in Venice. The Epitome
was not just a translation, it added new observations, revised
calculations and made critical comments about Ptolemy's work. 

Realising that there was a need for a systematic account of
trigonometry, Regiomontanus began his major work, the De Triangulis Omnimodis
(Concerning Triangles of Every Kind) 1464. In his preface to the
Reader he says,
"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 allimportant 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
1476.
12. The Final Chapter: Trigonometry Changes the World
System
Nicolaus Copernicus (1473  1543)
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]
Georg Joachim von Lauchen called Rheticus (15141574)

Rheticus had facilitated the publication of Copernicus' work,
and had clearly understood the basic principles of the new
planetary theory.
In 1551, with the help of six assistants, Rheticus
recalculated and produced the Opus Palatinum de Triangulis (Canon
of the Science of Triangles) which became the first publication of
tables of all six trigonometric functions. This was intended to be
an introduction to his greatest work, The Science of
Triangles.
When he died his work was still unfinished, but like
Copernicus, Rheticus acquired a student, Valentinus Otho who
supervised the calculation (by hand) of some one hundred thousand
ratios to at least ten decimal places filling some 1,500 pages.
This was finally completed in 1596. These tables were accurate
enough to be used as the basis for astronomical calculations up to
the early 20th century.

Bartholomaeus Pitiscus (1561  1613)

The term trigonometry
is due to Pitiscus and as first appeared in his Trigonometria: sive de solutione
triangulorum tractatus brevis et perspicuus, published in
1595. A revised version in1600 was the Canon triangularum sive tabulae sinuum,
tangentium et secantium ad partes radii 100000 (A Canon of
triangles, or tables of sines tangents and secants with a radius of
100,000 parts.) The book shows how to construct sine and other
tables, and presents a number of theorems on plane and spherical
trigonometry with their proofs. [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
Thesaurus
Mathematicus.
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.
Reflection
See the notes to this article to read some thoughts on the
value of teachig the history of mathematics.
References
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to the Stars. N.Y and Chichester Wiley
Skywatching in three ancient cultures: Megalithic Astronomy,
the Maya and the Inca. The first chapter (almost a third of the
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With clear diagrams and explanations this give a fascinating
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A fascinating description of one person's 'detective quest' to
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his amazing device.
Web Links
The Muslim Heritage
Site is very interesting. It is a valuable source of
information on "1,000 years of missing history from 600 to 1600."
You can find a large number of biographies of Muslim Scholars of
the Past, a time Line of Events, and much more.
http://www.muslimheritage.com
The Wallingford Clock
http://www.wallingfordclock.talktalk.net
Here is the general description and explanation of the various
mechanisms of the reconstruction of the clock. Some parts of this
site may not work.
There are many specialised websites where you can obtain
information. If you 'Google' your enquiry and use the Wikipedia
option you can usually obtain reasonably reliable results and ideas
for more searches if you need to look any further.
Notes
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
arrow.
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.
6. In the diagram, if $OB = t$ and $R = 1$, then in triangle
$ABO, AO^2 + OB^2 = 1 + t ^2$. From which we get the familiar
parametric formulae.
$$\cos\theta=\frac{1t^2}{1+t^2}\mbox{ etc.}$$
7. The maximum area occupied by the Arabs was most of Spain
and Portugal with the exception of the kingdom of Asturias in the
North, and a part of France now called the Languedoc.
8. The Holy Land with its capital Jerusalem, consisted roughly
of what is now Israel and Palestine. There were many other reasons
for the Crusades; the loss of power and territory of the older
Christian Empires, the growing problem of the slave trade run by
Arabs, and by taking part in these campaigns, some Christian
kingdoms thought they could gain political advantage over their
rivals.
9. The reason for using such large numbers for the radius is
much the same as in the past, the arithmetic was much easier using
a large unit like 100,000 where the small parts (in this case
minutes of arc) could be managed in integral parts rather than
fractions. Similar considerations were applied by Napier, Briggs
and Burgi in their invention of logarithms.
10. This quotation comes from the preface to the reader in De
Triangulis, translated by Barnabas Hughes.