Transcript Slide 1

Ideas of Mathematical Proof in
Classical India:
A Reading of the
Aryabhatiyabhashya
of Nilakantha Somayajin
Why Do This??
The Five Sheaths of the Human
Person (Taittiriya Upanishad)
Layer 1: Annamayakosha
Now a man here is
formed from the
essence of food. This
here is his head.
This is his right wing;
this is his left wing.
This is his torso, and
this is his tail on
which he rests.
Ashtanga First Series:
Forward Bends
First Series:
Bend more …
…One Drink Too Many.
Layer 2: Pranamayakosha
• Different from and lying within this self formed
from the essence of food is the self consisting of
life-breath … . The head is simply the outbreath, the right wing is the inter-breath, the left
wing is the in-breath; the torso is space, and the
tail on which it rests is the earth.
• “Life-breath – gods breathe along with it, as do
men and beasts …”
Second Series:
bend back …
… and waaay forwards …
… and ???
Layer 3: Manomayakosha
Manomayakosha
• Different from and lying within this self consisting
of life-breath is the self consisting of mind, which
suffuses this other self completely … . The head
is simply the Yajus formulas, the right wing is the
Rig verses; the left wing is the Saman chants;
the torso is the rule of substitution, and the tail is
the Atharva-Angirases.
• “Before they reach it, words turn back, together
with the mind…”
Third Series:
Balance
Layer 4: Vijnanamayakosha
• Different from and lying within this self consisting
of mind is the self consisting of understanding …
The head is simply faith; the right wing is the
truth; the left wing is the real; the torso is the
performance, and the tail on which it rests is the
celebration.
• “It’s understanding that conducts the sacrifice;
it’s understanding that performs the rites. It’s
understanding that all the gods venerate as the
foremost brahman.”
Fourth Series
Fourth Series
Layer 5: Anandamayakosha
• Different from and lying within this self
consisting of understanding is the self
consisting of bliss, which suffuses this
other self completely … Of this self, the
head is simply the pleasure; the right wing
is the delight; the left wing is the thrill; the
torso is the bliss, and the tail on which it
rests is the brahman.
Seated Practices:
Pranayama, Meditation
Uttaravedi Agnicayana
Vedic Ritual Reenactment
Kerala, South India, 1975
(courtesy IGNCA)
The Sacrificial Enclosure
Kalpa Computations
Soma (Ephedra Dystachia)
Agni Manthana
Agni Manthin
Agni Manthin
Verbal root manth:
1. To churn
Agni Manthin
Verbal root manth:
1. To churn
2. To agitate, disturb
Agni Manthin
Verbal root manth:
1. To churn
2. To agitate, disturb
3. To apply friction to
any part of a body in
order to produce
offspring
Churning the Ocean
Rahu and Ketu
Celestial Sphere
(Gola)
Rashis and Nakshatras
Two Main Applications of
Mathematics in India
• Kalpa (layout of
sacrificial area,
construction of ritual
devices)
• Jyotisha (“star
science”, began as
timing of Vedic rituals,
eventually turned to
astronomy/astrology)
Aryabhata
• Born 476 CE
• Astronomer,
mathematician
• Author of the
Aryabhatiya
• “taught by Brahma”
• Rotating earth
Nilakantha Somayajin
• Nilakantha = “bluethroated”
• Somayajin = “somasacrificer”
Nilakantha
•
•
•
•
1444 – 1545 CE
Kerala, South India
Namputiri Brahmin
Learned in Nyaya logic, Vedanta
philosophy
• Investigations in the scientific method,
“science and religion” issues
Jyotirmimamsa
This is what some think:
“Pleased by feats of asceticism, Brahma taught to
Aryabhata [the characteristics] of the planets ..
Because of Brahma’s omniscience, freedom
from passion, etc., how can there be criticism of
Aryabhata?”
Stupid! It is not thus. The favor of a deity causes
mental clarity only. Neither Brahma nor the Sungod taught it – only Aryabhata.
The Aryabhatiyabhashya
• Composed “late in his
life;”
• Written on multiple
levels;
• Commentary on the
Ganitapada alone is
180 pages;
• Detailed proofs, but
no diagrams!
Typical Commentarial Tasks
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•
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•
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Grammatical analysis of the current verse
Argue why the rule is best stated this way
Show that the rule is useful
Show that the verse is pretty poetry
Justify its placement in the sequence of verses
Prove the rule
Never, ever embarrass the Teacher (even if
he’s wrong)!
Aryabhatiya 7
The area of a circle is just half its
circumference multiplied by the radius.
That [area], multiplied by its own square
root, is exactly the volume of a sphere.
Nila comments:
How then is a circle made? Scratching it
out using just the tip of a paint brush with
some heated lamp-black – let that be done
by skilled artists only! At least that’s how
it’s commonly attempted. But in that case
one doesn’t get an even circumference,
due to the lack of [constant] extension.
How to get the area formula:
The portions of the area of a circular figure
all have the form of needles, because, all
around [the center], the portions are
wheel-spokes, rounded and separated at
the ends. Because of their infinity, you can
make [the wide ends] as small as you like.
Their needle-form is due to the fact that all
of the cut-up portions touch the center [of
the circle].
Dissect the Circle, Rearrange
When all of them are placed together, tipto-end in pairs, “rectangle-ness” occurs.
You Get This:
Do Same Thing,
With More Slices
In the Limit …
half the circumference
radius
But what about that volume?
• Aryabhata is saying:
V  r
3
But Really
4
3
V   r
3
Aryabhata 17(ab)
The square of the arm
plus the square of the
upright is the square
of the diagonal.
diag onal d
upright u
arm a
a u  d
2
2
2
Connection to Areas
5 X 5 = 25
3X3=9
5
3
4
4 X 4 = 16
Our Strategy:
Slice, Dice and Rearrange
u  a  (a  u)  2au
2
2
2
a- u
a- u
a
a
u
a- u
a
u
a
u
u
(a  u)  2au  d
2
d
a
2
u
a- u
a- u
(a  u)  2au  d
2
d
u
2
(a  u)  2au  d
2
d
d
d
d
2
a  u  (a  u )  2au,
2
2
2
(a  u )  2au  d ,
So
2
2
a u  d .
2
2
2