Discovering the same things in two such different ways: Indian and Western Calculus

David Mumford, Brown University October 22, 2007 Swarthmore College

Outline

1.

2.

3.

4.

5.

6.

Setting the stage – Babylon, place value, “Pythagoras’s” theorem over the world Euclid, the Bourbaki of the ancient world, and the pros and cons of his influence Negative numbers in the East and West Archimedes and integration Indian calculus from Aryabhata to Madhava and his school Western calculus from Oresme to Newton

Babylonian math tablets, c.1800 BCE Sulbasutras (c.800-c.200 BCE) Buddha (563-483) HELLENISTIC (YAVANA) CULTURE IN NW INDIA 326 BCE--c.100 CE Eudoxus (408-355 BCE) Euclid (325-265) Archimedes (287-212) Ptolemy (85-165 CE)

Timeline

Aryabhata (476-550) Brahmagupta (598-670) Bhaskara II (1114-1185) Madhava (1350-1425) Oresme (1323-1382) Viete (1540-1603) Newton(1643-1727) present

I. “Pythagoras’s” theorem was known to the Babylonians in 1800 BCE

2 з 1 + 24 60 + 51 3600 + 10 216000 цч » 0.0000006

Other tablets contain systematic lists of ‘Pythagorean’

k

+ =

m

value’ and sexagesimal ‘decimals’.

The Indians and Chinese both knew Pythagoras’s theorem

In India, the

Sulbasutras

(Rules of the Cord)

,

c.800 BCE: “

The cord equal to the diagonal of an oblong makes the area that both the length and width separately make”.

In China, the

Nine Chapters on the Mathematical Art

, c.100 BCE, has the ‘Gougu’ theorem:

II. Euclid was the Bourbaki of the ancient world. He avoids negatives, decimals. A

ratio

is defined as an equivalence class of pairs

(n,m)

( á la Dedekind). Here is his proof that ‘+’ between ratios is well-defined! This is, in fact, non trivial and occupies all of Book V.

III. Real and negative numbers in the West had to fight Euclid to be accepted

Stevin (1585) finally rediscovered Babylonian ideas, writing

p p

» » 3 0 1 1 4 2 1 3 6 3,1 4 1 6

iv

or

but Stifel objected that they were always inexact, ‘

they flee away perpetually’,

unlike proper Euclidean constructions.

Fermat and Descartes’ coordinates were only in the positive quadrant.

Even in the 19 th century, Augustus DeMorgan could say “

These creations of algebra (negative numbers) retain their existence in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory”

In China, negative numbers were accepted as a matter of course.

The

Nine Chapters

explain gaussian elimination for the solution of simultaneous linear equations, using an array of rods – red for positive, black for negative.

In India, negative numbers were also accepted as a matter of course.

Brahmagupta

gives all the rules, including (neg)x(neg)=(pos).

Bhaskara II

uses the negative side of the number line:

IV. Archimedes’

computation of the area of a sphere reduces to a key step which is, in effect, the calculation of the integral:

q

0 т

j d j

His proof is by an ingenious use of similar triangles:

V. Aryabhata (476-550 CE)

• Like earlier mathematicians, he wrote in highly compressed Sanskrit verse, intended to be memorized • Astronomy demanded tables of sines (‘half-chords’). To put numbers in verse, a system of number words was used: – Earth/moon/Vishnu  – Eye/twin/hand  2 – (sacrifical) fire  3 – Veda  4 – Limb (of the vedas)  1 6 • He computed to the nearest integer:

S k

=

R

sin(

k p

/ 48), 1 Ј

k

Ј 23,

R

= 3438 = but these were pretty hard to memorize #arcminutes in a radian (‘segmented half chords’) D

S k

=

S k

-

S k

1 225, 224, 222, 219, 215, 210, 205,199,191,183,174,164,154,143,131,119,106, 93, 79, 65, 51, 37, 22, 7

RADIUS (makes circum.

360*60)

3438

ANGLE in degrees

3.75

7.5

11.25

15 18.75

22.5

26.25

30 33.75

37.5

41.25

45 48.75

52.5

56.25

60 63.75

67.5

71.25

75 78.75

82.5

86.25

90

RAD*SIN (HALF CHORD)

225 449

DELTA RAD*SIN

225 224

DELTA OF DELTA OF RAD*SIN

1 2

Aryabha ta's rule

1 2 671 890 1105 1316 1521 1719 222 219 215 211 205 198 3 4 4 6 7 7 3 4 5 6 6 7 1910 2093 2267 2431 2585 2728 2859 2977 3083 3176 3256 3321 3372 3409 3431 3438 191 183 174 164 154 143 131 118 106 93 80 65 51 37 22 7 8 9 10 10 11 12 13 12 13 13 15 14 14 15 15 8 9 10 10 11 12 12 13 13 14 14 14 14 15 15

Aryabhata (cont.)

• Again he saw a pattern (book 2, verse 12):

The segmented second half-(chord) is smaller than the first half-chord of a (unit) arc by certain (amounts). The remaining (segmented) half-(chords) are (successively) smaller by those (amounts) and by fractions of the first half-chord accumulated.

• Bhaskara I, at least, interpreted this by the true formula: D

S

- D

S

= D

S

- D

S

).(

S S

)

k k

differential equation 1 • This is just a finite difference form of the sin

ўў= -

2 sin 1

k k

= 1 1

C S k

• Arguably Hooke first discovered this in the West in 1676 and stated it just as enigmatically by an anagram!

‘ceiiinosssttuv’

(

Ut tensio, sic vis

, or

As the extension, so the force

).

Madhava – rigorous derivation of the first derivatives of sin and cos

The two shaded triangles are similar. Equating the ratios of their sides to their hypotenuse: sin(

q

+ D

q

) sin(

q

- D

q

) = (chord of angle 2 D

q

)

R

vert.side

of small = hypot hor.side

of large hypot = cos(

q

+ D

q

) cos(

q

- D

q

) = (chord of angle 2 D

q

)

R

hor.side

of small = hypot vert.side

of large hypot =

Madhava – the power series for arctan

If

q

arc =

PP m

angle

POP m

, then = » е е arc

P P k S Q k k

1 1

R P k k

= е 1

OP k

1 = е

k

1 using similar triangles

k

.

k

»

PP

1 Ч е +

PP k

2 ) 1 » т

dx

( 1 +

x

2 ) Then he sums

n k p

4 from which he integrates

x k

1 3 + 1 5 1 7 + gets, e.g.

VI. Nicole Oresme (1323-1382)

De configurationibus qualitatum et motuum

“The quantity of any linear quality is to be imagined by a surface whose length or base is a line protracted in a subject of this kind and whose breadth or altitude is designated by a line erected perpendicularly on the aforesaid base. And I understand by “linear quality” the quality of some line in the subject informed with a quality.

That the quantity of such a linear quality can be imagined by a surface of this sort is obvious, since one can give a surface equal to the quality in length or extension and which would have an altitude similar to the intensity of the quality. But it is apparent that we ought to imagine a quality in this way in order to recognize its disposition more easily, for its uniformity and its difformity are examined more quickly, more easily and more clearly when something similar to it is described in a sensible figure. …” (I.iv) His qualities include:

distance, velocity, temperature, pain, grace

. His subjects (our domains) include:

1,2 and 3D objects and intervals of time

Oresme (cont)

He talks about the area of the graph (‘ the total quantity of the quality’ ) and, in his examples, makes clear that the area of the graph of the velocity of change of a quality is the total change. ‘The Fundamental Theorem’ – of course no equations!

“If some mobile were moved with a certain velocity in the first proportional part of some period of time, and in the second part it were moved twice as rapidly and in the third three times as fast … the mobile in the whole hour would traverse precisely four times what it traversed in the first half hour.” (III.viii) е

n

2

n

= 2

The East and West compared

• Strong oral family-based transmission • Decimals and negative numbers ubiquitous from early CE times • Heuristic arguments by construction, no use of excluded middle • Calculus arose from finite differences, analysis of sine fcn., the circle/sphere • In 14 th century, Madhava uses power series, Fund. Theorem.

• Libraries, monasteries and universities • Decimals and negatives accepted

very

slowly, suspect even in 19 th cent.

• Dominant influence of Euclid’s proofs by contradiction • From Oresme, key idea of fcns. of time, plotted in space (but not sine!) • In 17 th century, Newton bases his work on power series, Fund. Theorem.