|
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)
|
|
|
(0 vote)