Page 21 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes
and a negative fallout of this irrational influence was the rejection of scientific theories that
were far ahead of their time. One of the greatest scientists of the Gupta period—Aryabhatta
(born in 476 AD, Kusumpura, Bihar)—provided a systematic treatment of the position of the
planets in space. He correctly posited the axial rotation of the earth, and inferred correctly
that the orbits of the planets were ellipses. He also correctly deduced that the moon and the
planets shined by reflected sunlight and provided a valid explanation for the solar and lunar
eclipses rejecting the superstitions and mythical belief systems surrounding the phenomenon.
Although Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of science,
Nizamabad, Andhra) recognized his genius and the tremendous value of his scientific
contributions, some later astronomers continued to believe in a static earth and rejected his
rational explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a profound
influence on the astronomers and mathematicians who followed him, particularly on those
from the Asmaka school.
Mathematics played a vital role in Aryabhatta’s revolutionary understanding of the solar
system. His calculations on pi, the circumferance of the earth (62832 miles) and the length
of the solar year (within about 13 minutes of the modern calculation) were remarkably close
approximations. In making such calculations, Aryabhatta had to solve several mathematical
problems that had not been addressed before including problems in algebra (beej-ganit) and
trigonometry (trikonmiti).

Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as
the longitudes of the planets; conjunctions of the planets with each other and with bright stars;
risings and settings of the planets; and the lunar crescent. Again, these studies required still
more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided
by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst
his most important contributions was his formula for calculating the sine function which was
99% accurate. He also did pioneering work on indeterminate equations and considered for the
first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

Another important astronomer/mathematician was Varahamihira (6th C, Ujjain) who
compiled previously written texts on astronomy and made important additions to Aryabhatta’s
trigonometric formulas. His works on permutations and combinations complemented what
had been previously achieved by Jain mathematicians and provided a method of calculation
of nCr that closely resembles the much more recent Pascal’s Triangle. In the 7th century,
Brahmagupta did important work in enumerating the basic principles of algebra. In addition
to listing the algebraic properties of zero, he also listed the algebraic properties of negative
numbers. His work on solutions to quadratic indeterminate equations anticipated the work
of Euler and Lagrange.
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged
to introduce the concept of infinitesimals—i.e. tatkalika gati to designate the infinitesimal, or
near-instantaneous motion of the moon—and express it in the form of a basic differential
equation. Aryabhatta’s equations were elaborated on by Manjula (10th C) and Bhaskaracharya
(12th C) who derived the differential of the sine function. Later mathematicians used their
intuitive understanding of integration in deriving the areas of curved surfaces and the volumes
enclosed by them.

Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation of trigonometric
tables and measurement units. Yativrsabha’s work Tiloyapannatti (6th C) gives various units
for measuring distances and time and also describes the system of infinite time measures.
Contd...

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