The branch of
mathematics that is related to the study of the
triangle is called Trigonometry. A triangle is
a close region that is constructed with the help
of three straight lines that finally form its
structure. Trigonometry is associated with the
study of the relationships that are found between
the angles and the sides of the triangle.
Trigonometry has got the pride to be one of the
most ancient subjects that were extremely famous
all over the world and scholars from all over
the world studied those ancient subjects. (Brown,
1990)
The invention of trigonometry is associated with
the geometric school of Alexandria that was well-known
for the studies of astronomy. Quantitative geometry
is necessary for quantitative astronomy. Quantitative
geometry is the geometry of globe. The trigonometry
was invented by the astronomers. Those astronomers
had applied the trigonometry to the study of sky.
Hypparchus of Rhodes (II century. B.C.) is called
the founder of trigonometry although spherical
geometry had been discussed by Eudoxus of Cnidos
and Euclid from Alexandria. Theodosius of Tripoli
and Menelaus of Alexandria had contributed fundamentally
towards spherical trigonometry. Both of them had
written under the title ‘Sphaerica’.
But Ptolemy Claude is known to provide the biggest
information about trignometrical methods from
Alexandria. (Foerster, 1990)
The oldest known work on trigonometric tables
is the Syntaxis Mathematica written by Ptolemy
of Alexandria about 140 AD. We know almost nothing
about Ptolemy himself, but his work (of thirteen
books) surpassed all previous similar works and
became known as the greatest, "almagest,"
which we call it to this day. In the Almagest,
Ptolemy describes (and proves) a method for deriving
a table of chords subtended by arcs of 1 degree
to 180 degrees. This approximates today's table
of sines (and cosines). Ptolemy extended the work
of Hipparchus (died 125 B.C.) and followed his
method. (Brown, 1990)He started with the circle
divided into 360 parts and a diameter of 120 units.
Using basic constructions for the sides of regular
pentagon, hexagon, and decagon, he computed the
chords for arcs of 36 degrees, 60 degrees, and
72 degrees. He also computed chords of supplement
arcs, proving the theorem which bears his name
and which allows him to compute chords of difference
of arcs. He computed chords of half arcs and eventually
found the chord for 1/2 degrees; from this one
can compute his table. In this process, Ptolemy
uses several of the relationships we will develop
below involving chords. (The sine function, as
we know it, was devised in the sixth century by
Arabs). (Toomer, 1973)
Alexandrian trigonometry uses chords while modern
trigonometry uses sine. The semi circumference
was divided into degrees, 180 equal parts in the
Babylonian tradition and the diameter was divided
into 120 equal parts. As a result, goniometer
was obtained. Goniometer consists of two parts,
the round part and the flat part. Arcs are measured
with the help of the round part and the relative
cords are measured with the help of the flat part.
The arc is measured in degrees, which is a unit
that measures the circumference 360. the cord
is measured in units so that the radius measures
60. (Foerster, 1990)
The Theorem of Ptolemy
According to the theorem of Ptolemy, if a quadrilateral
is inscribed in a circle, the sum of the products
of the two opposed sides is equal to the product
of the multiplication of the diagonals.
According to the quadrilateral PQRS, we have the
following formulae.
PR x QS = PQ x RS + PS x QR.
The theorem of Ptolemy is as follows:
c2 (a) = 60 c2 (2 a) / 120 + c (180 – 2
a) (Toomer, 1996)
Work of Menelaus
Menelaus of Alexandria had lived before Ptolemy
because Ptolemy had mentioned Menelaus in his
work. Menelaus had written many books such as
‘The Book of Spherical Propositions’,
three books on the ‘Elements of Geometry’
that were edited by Thabit ibn Ourra and ‘The
Book on the Triangle’. The translation of
some of these books are found in Arabic. Among
many books, only ‘Sphaerica’ is still
known. (Aintabi, 1971) The knowledge about spherical
triangles and the applications of such triangles
to astronomy is provided in this book. Menelaus
was the first mathematician that give the definition
of a spherical triangle. He had used arcs of great
circles in his Book I of Sphaerica. Before that
time, arcs of parallel circles on the sphere had
been used. This innovation was found to be a turning
point in the formation and development of spherical
trigonometry. The Book II is about the application
of spherical geometry to astronomy. The proof
that Menelaus had given in this book are far better
than the proofs given by Theodosius in his Sphaerica.
Menelaus’s theorem is found in the Book
3. spherical trigonometry is found in this book.
(Schmidt, 1955)
A sphercial triangle version was produced by
Menelaus as he proposed and proved that theorem.
Although ‘Sphaerica’ had been translated
into Arabic, but none of the translations was
found to be the exact one. Proclus had pointed
out some geometrical result of Menelaus. That
result was not found in his written book. Menelaus
had proved a theorem in the Euclid’s ‘Elements’.
The Archytas’s solution for the problem
of duplicating the cube was found in Menelaus’s
book ‘Elements of Geometry’. (Tannery,
1883)
Development of Trigonometry by Arabs and Indians
Mathematical sciences continued to develop during
the Roman period although Romans did not play
any role in the development of mathematical sciences
but they did not hinder its progress. The Arabs
were the natural successors of the Greek geometers.
The Arabs had faced different traditions and they
had assimilated most of them very quickly. The
Arabs were standing on such cross roads that had
a variety of mathematical traditions. At one side,
the Babylonian and Egyptian cultures were merging
with the classic Greek geometry and on the other
side; they were facing the innovations of Indian
mathematicians. The Arab influence encouraged
some fundamental discoveries that include both
on paper and technological to reach the west because
those fundamental discoveries were very crucial
in the development of science and in the diffusion
of culture. Those discoveries include both positional
and scientific notations such as the use of numeric
characters that were called Arab. (Blitzer, 2003)
Abu'l-Wafa had also contributed a lot in the field
of mathematics. In his time, the arithmetic books
were written in two types. One type was the use
of Indian symbols and the other type was the use
of finger-reckoning. Abu’l-Wafa had written
a book for practical use, ‘A book on those
geometric constructions which are necessary for
a craftsman’.
There were thirteen chapters in that book and
the topics covered in that book were the construction
of right angles, construction of parabolas, inscribing
of various polygons in given polygons, approximate
angle trisections etc. Abu’l-Wafa was the
first who used the tan function. He was also the
first who compiled the tables of sines and tangents
at 15’ intervals. This work is written down
in ‘Theories of the Moon’ as an investigation
to find out the orbit of the Moon. Besides, he
also introduced cosec and sec function and worked
on the relationship among the six trigonometric
lines that were associated with an arc. He introduced
a new method for the calculation of sine tables.
The trigonometric tables that he had designed
were accurate to 8 decimal places. (Saidan, 1974)
Indian mathematicians have outstanding contributions
in the development of mathematics specially trigonometry.
Indians had invented the beautiful system of numbers
that is the foundation of much of mathematical
development. In this system, a set of ten symbols
was used and each symbol was associated with an
absolute value and a place value. This inventions
may seem very simple nowadays but all the calculations
are based on this system because this was the
foundation of the arithmetic. India is responsible
for the innovation that was related to Alexandrine
trigonometry. That innovation was the use of sine
instead of the use of chord. Work was done on
the implementation of sine and the first table
of sine was developed that was known as Surya
Siddhanta. This table was developed around IV
or V centuries ago. That table is very important
because it contains that calculus of sine of the
multiple of 3o 45’, until 90o. Indian astronomers
are also responsible for the addition of the cosine
to the sine, the cotangent and the tangent. During
the eighth century A.D., the translation of the
sine table into Arab took place. The Arab astronomers
were very genius and they put their efforts in
the field of circular functions and then they
realized that those circular functions need some
changes as well as improvements. (Foerster, 1990)
The sine of the complementary arc was known as
cosine: cos a = sin (900 – a)
As the cosine were to find directly in the tables
of sine so there was not any need of developing
the table of cosine. Gnomonics is called the science
of sundials and the cotangent and the tangent
were related to this science. The hypotenuse of
the triangles that contain gnomon and its shades
represented the cosecant and the secant. So it
can be said that the construction vertical and
horizontal sundials is connected with the cotangent
(and cosecant) and tangent (and secant) respectively.
Among tangent, cotangent, secant and cosecant,
only the table of tangent has been formed because
it was realized that the cotangent is complementary
tangent just the same as the cosine. The original
term for tangent was zill, that is umbra recta
in Latin and the original term for cotangent was
zill makus, which is umbra versa in Latin. T.
Fink (1561-1656) is responsible for the introduction
of the term tangent in 1583. While E. Gunter (1581-1626)
introduced the term cotangent in 1620. After the
introduction of those functions, it was realized
that the tables of those functions should be prepared
while it was also felt that the already existing
tables require some improvement. Initially Arab
mathematicians and then the Europeans put their
efforts in the formation of the tables as well
as in the improvement of the older one. (Blitzer,
2003)
Hindus Work on Trigonometry
Hindus were the first who actually invented the
sine of an angle. The tables of half cords were
given by Aryabhata in about 500. these tables
are now the sine tables. Then Brahmagupta in 628
produced the same table. Bhaskara in 1150 invented
the detailed method for the construction of a
table of sines. That table of sine could calculate
the sine of any angle. the approximate values
of sine could be calculated with the help of a
table given by Aryabhata. In this table, the approximate
values could be calculated at the intervals of
90 /24 = 3 45'. He used a formula to do such type
of calculation. The formula was sin (n+1) x -
sin nx in terms of sin nx and sin (n-1) x. Aryabhata
is also known for the introduction of versine
(versin = 1 - cosine) into trigonometry. Aryabhata
also gave some other rules that were used for
the summing of the first n integers, the squares
and the cubes of these integers could be determined.
(Sen, 1963) He also proposed the formula for the
areas of a triangle and the areas of a circle.
Both of the formulae are correct. Some historians
claim that the formulae proposed by aryabhata
for the volumes of a pyramid and of a sphere were
wrong. (Elfering, 1977)
Brahmagupta was famous because of his understanding
of the number system that was not found among
the mathematicians of that period. He defined
zero in the Brahmasphutasiddhant. He defined that
zero is the result of subtraction of a number
from itself. He also presented algorithm for the
calculation of square roots. He used the interpolation
formula for computing the values of sines. (Sarasvati,
1986) Bhaskaracharya is known the top most mathematician
in 12th century. He understood the number systems
and the solved such equations that was not done
by European mathematicians for several centuries.
He understood about negative numbers and zero.
Bhaskaracharya had shown interesting results on
trigonometry. The mathematicians before bhaskaracharya
did not give trigonometry any particular importance
because they thought that trigonometry is just
a tool that is used for calculation but bhaskaracharya
was found more interested in trigonometry than
in any other branch of mathematics. The interesting
results of the work of bhaskaracharya are as follows:
sin (a + b) = sin a cos b + cos a sin b
and
sin (a - b) = sin a cos b - cos a sin b. (Chaudhary
& Jha, 1990)
Development of Trigonometry in Europe
Arabs are responsible to bring the trigonometry
into the West. No significant contributions in
the field of trigonometry were found before the
fifteenth century. In the fifteenth century, attention
was again given to trigonometrical studies as
a requirement for astronomy. Tables that are more
specific in two directions are required for a
higher precision of instruments. Among those two
directions of tables are sines that have a bigger
number of decimals and angles that have smaller
intervals. George Peurbach (1423-1461) put his
attention towards the interval between the arcs.
He calculated the table of sine that contained
the intervals of 10’. Johann Muller (1436-1476)
work harder on the same field and composed such
a table in which the intervals were only of one
prime. There was a significant increase in exactitude.
(Foerster, 1990)This one was given from the dimension
of the radius of the goniometric circle.
Today this is given by the number of the decimal
figures. The values of R sin a were reported in
integer numbers, that had the range from 0 to
10000 that could correspond to four decimal figures,
when the radius was taken as R = 10000. That radius
was also called toto sine. Then the value of radius
was taken as R = 600000 in the table composed
by Peurbach. Then the value of radius was taken
first as R = 6000000 and then R = 10000000 that
could correspond to seven decimals in the table
composed by Regiomontanus. That time is considered
as the first time when the base 10 was definitely
adopted and the liberation had taken place from
the use of sexagesimal system of sine. (Blitzer,
2003)
Typography is also responsible for the development
of trigonometry. Typography is totally based on
rectilinear trigonometry and this feature of typography
is totally different from astronomy. Efforts are
done for the study of triangles and their solutions
because they are required in the topographical
survey. Regiomontanus wrote the first treatise
of trigonometry in 1464 that is called De trianglulis
omnimodis. Nicolaus Copernicus was the one who
included De revolutionibus orbium caelestium in
his work. G. J. Rheticus then shed light on Copernicus’s
work in his work. Rheticus is known for the preparation
of a monumental series of tables that contain
the six circular functions. Those functions possess
the intervals of 10” and they work for a
radius whose value is R = 10000000. (Blitzer,
2003)
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