Chapter 8 - Ancient and Medieval India
Chapter 8 - Ancient and Medieval India
Chapter 8 - Ancient and Medieval India
MEDIEVAL INDIA
CHAPTER 8
RENYLENE JHESS
ALDREAN
NHEL MATHEW
JAKE
DANIEL SHARRAH
KENNETH MAE
TABLE OF CONTENTS
Introduction to Mathematics in India
Calculations
Geometry
Equation Solving
Indeterminate Analysis
Combinatorics
Trigonometry
Transmission To and From India
I.
Introduction to
Mathematics
in INDIA
Introduction to Mathematics in India
Monarchical States (8th
Harappan Civilization
century)
-Arose in India on the banks -Headed by the King and priests
of the Indus River on 3rd (brahmins).
millennium BCE -fortification,
-no direct evidence of its administrative centralization, and large-
mathematics scale irrigation.
Brahmasphutasiddhanta Khandakhandyaka
- a more practical text.
-a theoretical treatise -first to give rules to compute with zero.
-sequence of transmissions
composed the earliest Sanskrit
textbook entirely devoted to
mathematics, rather than having
timeline from teacher to pupil resulting
from writing of proofs of
development of infinite series,
-established the Moslem trigonometric functions
mathematics as an adjunct to Sultanate of Delhi in 1206
astronomy. -their empire last over 300
. years
Bhaskara II Vijayanagara
-An empire in the south where
-created the Lilavati and
Bijaganita on arithmetic and the mathematical school of
Madhava established
algebra respectively.
Everyone must ponder
these!
● Through the various invasion and new kingdoms, mathematics was used to solve
practical problems like calendrical questions.
● But those creative mathematicians went beyond of these and develop new areas of
mathematics. We consider in this chapter the Indian number systems and methods of
calculations, then the geometry of the Sulbasutras and later, the next algebraic methods
developed in the medieval period to solve equations (including the so-called Pell
equation), next the beginning of combinatorics, and then the development of
trigonometry and associated techniques.
● We conclude with a study of the development of power series in south India during the
fourteenth and fifteenth centuries.
II.
CALCULATIONS
THE DEVELOPMENT OF OUR
MODERN NUMERALS
● Symbol for the first nine numbers of our number system
have their origin in the Brahmi system of writing of
India, which dates back to at least the time of King
Ashoka (Mid-Third Century BCE).
● These numbers appear in various decrees of the king
inscribed on pillars throughout India.
These numerals were used all the way up to the fourth century CE, with variations
through time and geographic location.
From the fourth century on, the new number system was form and it is the Gupta
numerals.
The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were
spread throughout that empire as they conquered lands during the 4th – 6th century.
Āryabhata lists names for the various powers of 10 in his
Āryabhatīya:“dasa[ten],sata[hundred], sahasra[thousand],ayuta[ten
thousand],and niyuta[hundred thousand.]
Eka [1],Dasa [10], Sahasra [10²],…[up to 10¹⁸] (Samhitās 1500 B.C.)
Around the year 600, the Indians evidently dropped the symbols for
numbers higher than 9 and began to use their symbols for 1 through 9 in our
place value arrangement.
These shifts were not found in India, it is in the work of Severus Sebokht
remark that the Hindus have a valuable method of calculation “done by
means of nine signs.”
Severus only wrote nine signs and there is no mention of a sign of zero. However, in the Bakhshālī
manuscript, a mathematical manuscript that was discovered in 1881 in the village of Bakhshālī in
northern India, the number are written using the place value system and with a dot to represent zero.
The Bakhshālī manuscript dates from the 7th century. In this same period, other
works of Indians were generally written in a quasi-place value system to
accommodate the poetic nature of the documents.
The example for these is the work of Mahāvīra, certain words in his work stands for:
moon(1), eye(2), fire(3), and sky(0). Then the word fire-sky-moon-eye would stand
for 2103 and moon-eye-sky-fire for 3021. Note, the place value begins on the left.
The Gupta numerals eventually evolved into another form of numerals called the
Nagari numerals, and these continued to evolved until 11th century, at which they
looked like this:
III.
GEOMETRY
Pythagorean Theorem:
● The areas of the squares produced separately by the
length and the width of a rectangle together is equal to
the area of the square produced by the diagonal. This is
observed in rectangles having sides 3 and 4, 12 and 5, 15
and 8, 7 and 24, 12 and 35, 15 and 36.
c2 = a2 + b2
For example:
24 Given:
a=7 ; b=24 ; c=?
To check:
7 Solution: c2 = a2 + b2
? c2 = a2 + b2 252 = 72 + 242
c2 = 72 + 242 625= 49 + 576
c2 = 49 + 576 625 = 625
c2 = 625
√ c2 = √ 625
c = 25
Aryabhata too presented some geometric results:
STANZA II, 16 The upright side is the distance between the tips of the two shadows
multiplied by a shadow divided by the decrease. That upright side multiplied by the
gnomon, divided by its shadow, becomes the base.
The shadows of two equal gnomons (of height 12 arigulas) are observed to be
respectively 10 and 16 angulas and the distance between the tips of the
shadows is seen as 30…
𝑑 𝑠1 𝑢𝑔
𝑢= 𝑎𝑛𝑑 h=
𝑠2 − 𝑠1 𝑠1
Note that this problem is very similar both in form and solution method to problem I in the
Chinese Sea Island Mathematical Manual.
STANZA II, 17 ... In a circle,
the product of both arrows is
the square of the half-chord,
certainly for two bow fields.
Many other geometric formulas, some exact,
some stated as exact but only approximate, and
some stated explicitly as approximate, occur in
various Indian mathematical texts. But we will
conclude this section with two remarkable
results of Brahmagupta dealing with cyclic
quadrilaterals (quadrilaterals inscribed in
circles), given in chapter 12 of the
Brähmasphutasiddhānta.
The first is the following:
The accurate area of a cyclic quadrilateral is the square
root of the product of the halves of the sums of the sides
diminished by each side of the quadrilateral.
Aryabhata
in dealing with arithmetic progressions in two
stanzas of his Aryabhari ya, provided what
amounts to the quadratic formula in a special
case:
Stanza II – 19
Stanza II
– 20
•Aryabhata did not explicitly present here a general procedure for solving quadratic
equations
Brahmagupta
Born 598 (possibly) Ujjain, India
Died 670 India
Brahmagupta was an Indian mathematician and
astronomer. He is the author of two early works on
mathematics and astronomy
a century and a quarter later, did so for the equation
ax² + bx= c
Example:
x2 - 10x = -9:
given: a=1 b= -10 c=-9 x=
x= x=
x= x=
x= x=9
Several hundred years later, Bhāskara II did deal with multiple roots, at least when both
are positive. His basic technique for solving quadratic equations was that of completing
the square. Namely, he added an appropriate number to both sides of ax² + bx = c so that
the left side becomes a perfect square: (rx-S)²=d. He then solved the equation rx-S=√d for
x.
But, he noted, "if the root of the absolute side of the equation is less than the number,
having the negative sign, comprised in the root of the side involving the unknown, then
putting it negative or positive, a twofold value is to be found of the unknown quantity. In
other words, if √d<S, then there are two values for x, namely, (S + √d)/rand ( S-√d)/r.
Bhāskara did, however, hedge his bets. As he says, "this [holds] in some cases." We
consider two examples to see what he meant.
Example:
1 2
( 𝑥) + 12 = 𝑥
8
1 2
(( 8 𝑥) + 12 = 𝑥) 64
𝑥 2 + 768 = 64𝑥
𝑥 2 − 64𝑥 = −768
Linear Congruence
Āryabhatta’s work:
A simple Diophantine equation would be ax + by = c. In this equation a, b and c
are given integers; and x and y unknown integers. Aryabhatta is the earliest
known work which examines integer solutions to Diophantine equations of the
form by = ax + c and by = ax – c.
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer. He is the author of two early works on
mathematics and astronomy: the Brāhmasphuṭasiddhānta
Brahmagupta works:
Known for: Zero; Modern number system.
He gave somewhat clearer explanations. He just presented an algorithm. Brahmagupta’s description
of his method of 𝑘𝑢𝑡𝑡𝑎𝑘𝑎 or “pulverizer,” taken from chapter 18 of his text, with an example he
used: N ≡ 10 (mod 137) and N ≡ 0 (mod 60).This problem can be rewritten as the single equation
137x + 10 = 60y.
Divide 137 by 60 and continue by dividing the residues
Brahmagupta
In other words, apply the Euclidean algorithm until the final nonzero remainder is reached:
137 = 2 . 60 + 17
60 = 3 . 17 + 9
17 = 1 . 9 + 8
9=1.8+1
The final remainder is 1. Multiply that by some number v so that 1 . v ± 10 is exactly divisible by the last divisor,
in this case 8. Brahmagupta explained that one uses the + when there are an even number of quotients and the −
when there are an odd number.
Here, because 0 is one of the quotients, the last equation becomes 1v − 10 = 8w. Choose v = 18 and w = 1. The
new column of numbers is then;
0
2
3
1
1
18
1
Brahmagupta
Multiply 18 by 1 and add 1 to get 19. Then replace the term “above,” namely, 1,
by 19, and remove the last term. Continue in this way (as in the table below)
until there are only two terms.
0 0 0 0 0 130
2 2 2 2 297 297
3 3 3 130 130
1 1 37 37
1 19 19
18 18
1
The top term, is 130. So x = 130, y = 297, is a solution to the original equation.
The Pell Equation
The ability to solve systems of pairs of linear congruences turned out to
be important in the solution of another type of indeterminate equation,
the quadratic equation of the form Dx2±b=y2
Today, the special case where b = 1 is usually referred to as Pell’s
equation. But although there are indications that the Greeks could solve
a few of these equations, the general case, first developed in India, was
undoubtedly the high point of medieval Indian algebra.
Brahmagupta
Brahmagupta
He gave the first explanation of the method of solving these problems. And, as in the case of the 𝑘𝑢𝑡𝑡𝑎𝑘𝑎, he
introduced rules for dealing with equations of this type, in conjunction with examples.
This equation, 92x2+1=y2 will be solved here in considerably less than a year.
The product of the first pair, multiplied by the multiplier, with the product of the last pair, is the
last root.
The sum of the thunderbolt products [cross multiplication] is the [new] first root. The additive is
equal to the product of the additives.
Bhaskara
Bhaskara’s goal in his Lilavati was to show how any equation of the form
Dx2±b=y2 can be solved in integers. He began by recapitulating Brahmagupta’s
procedure. In particular, he emphasized that once one had found one solution
pair, indefinitely many others could be found by composition.
More importantly, however, he discussed the so-called cyclic method. The basic
idea is that by continued appropriate choices of solution pairs for various
additives by use of the kuttaka method, one eventually reaches one that has the
desired additive 1.
Bhaskara
Bhaskara’s rule for the general case Dx2±b=y2 and follow its use in one of his examples,
67x2+1=y2
Making the smaller and larger roots and the additive into the dividend, the additive, and the
divisor,the multiplier is to be imagined
When the square of the multiplier is subtracted from the “nature” or is diminished by the “nature”so
that the remainder is small, that divided by the additive is the new additive. It is reversed if the
square of the multiplier is subtracted from the “nature.” The quotient of the multiplier is the smaller
square root; from that is found the greatest root.
Then it is done repeatedly, leaving aside the previous square roots and additives. They call this the
chakrav¯ala (circle). Thus there are two integer square roots increased by four, two or one. The
supposition for the sake of an additive one is from the roots with four and two as additives.
VI.
COMBINATORICS
Combinatorical rules appear in India, although again
without any proofs or justifications.
For example, the medical treatise of Susruta, perhaps
written in the sixth century BCE, states that 63
combinations can be made out of six different tastes—
bitter, sour, salty, astringent, sweet, hot—by taking
them one at a time, two at a time,three at a time, and so
on.
In other words, there are 6 single tastes, 15 combinations
of two, 20 combinations of three, and so forth. Other
works from the same general time period include similar
calculations dealing with such topics as philosophical
categories and senses. In all these examples, however,
the numbers are small enough that simple enumeration is
sufficient to produce the answers.
In the ninth century, Mahavira gave an
explicit algorithm for calculating the
number of combinations:
The rule regarding the possible varieties
of combinations among given things:
Beginning with one and increasing by
one, let the numbers going up to the
given number of things be written
down in regular order and in the
inverse order (respectively) in an
upper and a lower horizontal row.
If the product of one, two, three, or more of the
numbers in the upper row taken from right to left be
divided by the corresponding product of one, two,
three, or more of the numbers in the lower row,also
taken from right to left, the quantity required in each
such case of combination is obtained as the result.
C(n,r)=n(n-1)(n-2)...(n-r+1)/r!
COMBINATIONS
The number of ways of choosing r elements
from S (order does not matter).
S={A,B,C}
e.g., AB , AC, CB
The number of r-combinations C(n,r) of a set
with n=|S| elements is
C(n,r)= n!/r!(n-r)!
Combination Example
How many ways if 3 balls can be selected of 5 balls.
The number of balls doesn’t matter
C(n,r)= n!/r!(n-r)!
C(5,3)= 5!/3!(5-3)!
5×4×3×2×1/ 3×2×1×2
P(10,3) = 10.9.8=720,
So there is a 1 in 720 chance
That you’ll survive
VII.
TRIGONOMETRY
1.) TRIGONOMETRY
Derived from Greek trigōnon, “triangle” and
metron, “measure”.
Began as a branch of geometry and was
utilized extensively by early Greek
mathematicians to determine unknown
distances.
Branch of mathematics that studies
relationships between side lengths and angles
of triangles.
2.) HIPPARCHUS OF NICAEA (162-127 BCE)
Founder of Trigonometry
Greek Astronomer
Mathematician
Found a relationship between the lunar
and solar distances that enabled him to
calculate that the Moon’s mean distance
from Earth is approximately 63 times
Earth’s radius.
3.) PTOLEMY OF ALEXANDRIA (c.90 –
168 AD)
Greek astronomer and
mathematician, worked out
the ideas of Hipparchus.
He developed the Ptolemaic
Theory of Astronomy.
Wrote the book “The
Almagest” around 150 AD.
4.) CONSTRUCTION OF SINE TABLES
Golden Age Indian
Mathematician made
fundamental advances in the
Theory of Trigonometry.
Indian astronomers used
trigonometry tables to
estimates the relative
distance of the Earth to the
sun and moon.
5.) PAITAMAHASIDDHANTA
Earliest known Indian work
containing trigonometry, written in
the early fifth century.
Contain a table of "half-chords", a
Sanskrit term jya-ardha.
6.) ARYABHATA (476-550 AD)
His work Aryabhatiya contains the
earliest surviving tables of sine
values and versine values, in
3.75° intervals from 0° to 90°, to an
accuracy of 4 decimal places.
He used this to estimate the
circumference of Earth, arriving at
a figure of 24,835 miles, only 70
miles off its true value.
7.) STANZA II,12
By what number the second Sine is less than the first
Sine and by the quotient obtained by dividing the sum of
the preceding Sines by the first sine, by the sum of these
two quantities the following fires are less than the first
Sine.
Indian Mathematicians use the following words:
jya - sine
kojya - cosine
utkrama-jya - versine
otkram jya - inverse sine
8.) ETYMOLOGY OF SINE
The English word “sine” comes
from a series of mistranslations of
the Sanskrit jya-ardha (chord-
half).
Aryabhata frequently abbreviated
this term as jya or jiva.
9.) The “first sine” S₁ in Indian trigonometry always
means the Sine of an arc 3 ¾° = 3°45`, and this Sine, in a
circle of radius 3438.
The formula of nth Sine Sn (the Sine of n×3°45):
STANZA I,10
The twenty-four Sine [differences] reckoned in minutes of arc are
225,224,222,219,215,210,205,199,191,183,174,164,154,143,131,119,
106,93,79,65,51,37,22,7.33
10.) SURYA SIDDHANTA
By unknown authors and from
around 400 CE,
It refers to eight part of the
minutes of a sign.
First real use of sine, cosine,
inverse sine, tangent and
secants.
11.) VARĀHAMIHIRA (C. 505- 587)
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