Discovering the same things in two such different ways: Indian

and Western CalculusDavid Mumford, Brown University

October 22, 2007Swarthmore College

Outline

1. Setting the stage – Babylon, place value, “Pythagoras’s” theorem over the world

2. Euclid, the Bourbaki of the ancient world, and the pros and cons of his influence

3. Negative numbers in the East and West

4. Archimedes and integration

5. Indian calculus from Aryabhata to Madhava and his school

6. Western calculus from Oresme to Newton

Timeline

Babylonian math tablets, c.1800 BCE

Buddha (563-483)Eudoxus (408-355 BCE)Euclid (325-265)Archimedes (287-212)Ptolemy (85-165 CE)

Aryabhata (476-550)Brahmagupta (598-670)

Oresme (1323-1382)Viete (1540-1603)Newton(1643-1727)

Bhaskara II (1114-1185)Madhava (1350-1425)

Sulbasutras (c.800-c.200 BCE)

HELLENISTIC (YAVANA) CULTURE IN NW INDIA 326 BCE--c.100 CE

present

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

24 51 102 1 0.0000006

60 3600 216000

æ ö÷ç- + + + »÷ç ÷çè ø

Other tablets contain systematic lists of ‘Pythagorean’ triples . Note that they are using ‘place value’ and sexagesimal ‘decimals’.

2 2 2k m+ =

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 acceptedStevin (1585) finally rediscovered Babylonian ideas, writing

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 19th 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”

3 0 1 1 4 2 1 3 6 or3,1 4 1 6iv

pp»

¢¢¢¢¢¢»

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:

His proof is by an ingenious use of similar triangles:

0

sin( ) 1 cos( )dq

j j q= -ò

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 1– Eye/twin/hand 2– (sacrifical) fire 3– Veda 4– Limb (of the vedas) 6

• He computed to the nearest integer:

but these were pretty hard to memorize• So they memorized the first differences

(‘segmented half chords’)

sin( / 48), 1 23, 3438 #arcminutes in a radiankS R k k Rp= £ £ = =

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

1k k kS S S -D = -

RADIUS(makes circum.360*60)

ANGLEin degrees

RAD*SIN (HALF

CHORD)DELTA

RAD*SIN

DELTAOF DELTA

OF RAD*SIN

Aryabhata'srule

3438 3.75 225 225 1 1

7.5 449 224 2 2

11.25 671 222 3 3

15 890 219 4 4

18.75 1105 215 4 5

22.5 1316 211 6 6

26.25 1521 205 7 6

30 1719 198 7 7

33.75 1910 191 8 8

37.5 2093 183 9 9

41.25 2267 174 10 10

45 2431 164 10 10

48.75 2585 154 11 11

52.5 2728 143 12 12

56.25 2859 131 13 12

60 2977 118 12 13

63.75 3083 106 13 13

67.5 3176 93 13 14

71.25 3256 80 15 14

75 3321 65 14 14

78.75 3372 51 14 14

82.5 3409 37 15 15

86.25 3431 22 15 15

90 3438 7

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:

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

• 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).

1 2 1 1 1( ).( )k k kS S S S S S- -D - D = D - D2 .k kS C SD =-

sin sin¢¢=-

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( ) sin( ) vert.side hor.sideof small = of large cos( )

(chord of angle 2 ) hypot hypot

cos( ) cos( ) hor.side vert.sideof small = of large sin( )

(chord of angle 2 ) hypot hypot

R

R

q q q qq

q

q q q qq

q

+D - - D= =

D

+D - - D= =

D

Madhava – the power series for arctan

( )( )

1

1

1

1

1

1

2

1

2

If angle , then arc arc

.

using similar triangles

1 1

1

m

m k k

k k

k k

k

k k

k k

k

POPPP PP

S QR P

OPPP

OP OP

PP PP

dx x

q

-

-

-

-

-

-

==»

=

=

» × +

» +

ååå

å

åò

Then he sums nk from which he integrates xk gets, e.g.1 1 1

14 3 5 7

p= - + - +

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)

22nn=å

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 14th century, Madhava uses power series, Fund. Theorem.

• Libraries, monasteries and universities

• Decimals and negatives accepted very slowly, suspect even in 19th cent.

• Dominant influence of Euclid’s proofs by contradiction

• From Oresme, key idea of fcns. of time, plotted in space (but not sine!)

• In 17th century, Newton bases his work on power series, Fund. Theorem.