Chapter 3
Elsewhere: the parts most textbooks skip
Calculus is usually taught as a European relay race. It was not. Three other traditions got a long way, and in two cases they got further, earlier, than anything in Europe until the 1600s.
China: the principle Cavalieri gets the credit for
Liu Hui (刘徽, third century CE) wrote a commentary on the Nine Chapters on the Mathematical Art in 263 CE that includes a circle-cutting algorithm: inscribe a polygon, keep doubling the sides, and squeeze π. His instruction, "divide the circle until it cannot be divided any more," is either a limit or a contradiction depending on how charitable you are, and he knew it (Source 20, Guo).
Then Liu Hui published his own failure. He rejected the sphere formula in the Nine Chapters and built the right object to attack instead: the mouhe fanggai, the "box-lid" solid common to two cylinders inscribed in a cube at right angles. He saw correctly that the sphere is to that solid as π is to 4. Then he could not compute the box-lid's volume, said so explicitly in print, and left the problem open, closing with a short verse titled "The geometer's frustration":
Though the diminution increases, / It doesn't quite fit... I dare to let the doubtful points stand, / Waiting for one who can expound them. (Source 23, Wagner, Donald Wagner's translation, pp. 59 to 79)
Two and a half centuries later, somebody answered. Zu Gengzhi, son of Zu Chongzhi, finished the problem using the principle now stated as: if two solids have equal cross-sectional areas at every height, they have equal volumes. In the original:
疊棋成立積 / 緣幂勢既同 / 則積不容異 "If blocks are piled up to form volumes, / And corresponding areas are equal, / Then the volumes cannot be unequal." (Source 23, Wagner, Donald Wagner's translation, pp. 59 to 79)
That is Cavalieri's principle, about eleven hundred years before Cavalieri. The chain runs Liu Hui (263) to Zu Chongzhi (429 to 501) to Zu Gengzhi (c. 450 to 520) to Li Chunfeng (602 to 670), who wrote it down so it survived (Source 23, Wagner).
Zu Chongzhi also gave bounds on π that stood for roughly nine hundred years:
Read this equation in words
Pi lies between three point one four one five nine two six and three point one four one five nine two seven. Pi is approximately three fifty-five over one thirteen, which is three point one four one five nine two nine two, and so on. The difference between that fraction and pi is two point six six eight times ten to the negative seventh.
Both bounds check out, and 355/113 agrees with π to six decimal places (Appendix E.2). It is also the best rational approximation to π with a denominator under about sixteen thousand, and it is trivially memorable: write 113355, split it down the middle, put the big half over the small half.
The ending is bleak and useful. Zu Chongzhi's book was dropped from the imperial curriculum for being too hard, and then the only copy rotted. A civilization lost its best mathematics because of an examination-board decision (Source 21, O'Connor).
The vocabulary is worth a minute of class time on its own. Chinese solid geometry names shapes after things you could point at in a field: a prism is a "moat-wall", a pyramid is a "male horse", a tetrahedron is a "turtle's foreleg" (Source 20, Guo). Compare the Greek habit of naming after abstractions.
India: instantaneous velocity, because of the Moon
Bhāskara II (1114 to 1185) needed instantaneous velocity for a religious reason. The Moon moves too fast for a daily average to predict when a tithi, a lunar day, will end. So in the Siddhānta Śiromaṇi (c. 1150) he calls daily average motion "only a rough or approximate rate", introduces tatkālika gati, instantaneous motion, because the Moon "varies from moment to moment", and uses the Rcosine as the rate of change of the Rsine (Source 50, Ramasubramanian, pp. 20 to 23).
The derivative enters Indian mathematics for the same reason it enters Newton's: somebody needed to know where a moving body is right now, and an average was not good enough.
He also states, in Golādhyāya 4.39, that where the velocity correction vanishes the equation of the center is extremal (Source 50, Ramasubramanian). This is often described as Rolle's theorem. It is not. It is Fermat's rule for a maximum, and the research file corrects that specifically.
Kerala: infinite series, two centuries before Newton
Mādhava of Saṅgamagrāma (c. 1340 to c. 1425) is the most important mathematician in this chapter and left no surviving mathematical book. We know him the way we know Socrates: through students, and students of students, who quote his verses and say "as Mādhava said" (Source 55, O'Connor). The lineage is named and traceable: Mādhava, Parameśvara, Dāmodara, Nīlakaṇṭha, with Jyeṣṭhadeva also a student of Dāmodara.
He had the series we call Leibniz's:
Read this equation in words
Pi over four equals one, minus one third, plus one fifth, minus one seventh, and so on.
and the series we call Newton's for sine and cosine, and the one we call Gregory's for arctangent. He also knew this series is uselessly slow, and did something about it. Running it out to ten thousand terms buys three correct decimal places (Appendix E.4):
| Terms | Value | Error |
|---|---|---|
| 10 | 3.0418396189 | 9.98 × 10⁻² |
| 100 | 3.1315929036 | 1.00 × 10⁻² |
| 1,000 | 3.1405926538 | 1.00 × 10⁻³ |
| 10,000 | 3.1414926536 | 1.00 × 10⁻⁴ |
So Mādhava built correction terms, and the Yuktibhāṣā states an exactness criterion for them, namely that the correction function should satisfy (Source 50, Ramasubramanian). That is error analysis, not guessing.
His π value is recorded as a poem. In the bhūta-saṅkhyā word-numeral system, numbers are written as words for things that come in known quantities: gods, eyes, elephants, snakes, fire, the three qualities, the Vedas, the lunar mansions. Decoded, the string is 2827433388233, and the value is:
Read this equation in words
Pi is approximately the thirteen-digit number two eight two seven four three three three eight eight two three three, over nine times ten to the eleventh, which is three point one four one five nine two six five three five nine two two two, and so on. Pi itself is three point one four one five nine two six five three five eight nine seven nine, and so on.
Eleven correct decimal places (verified, Appendix E.3). The system exists because verses are easier to memorize and harder to corrupt in copying than digits are. Indian mathematics was, for centuries, sung (Source 50, Ramasubramanian).
Nīlakaṇṭha Somayāji (born 14 June 1444; death unsettled: sources give "after 1501", c. 1550, and 1545) dates his own work by eclipses he watched: 6 March 1467, 28 July 1501 (Source 56, O'Connor). A student with a laptop and a modern ephemeris can check those, which makes a five-hundred-year-old Sanskrit text verifiable in a homework assignment.
Jyeṣṭhadeva's Yuktibhāṣā (c. 1530) is written in Malayalam prose, not Sanskrit verse, and it explains why. That is a deliberate pedagogical choice: the text was written to be understood rather than to be authoritative (Source 50, Ramasubramanian).
The one documented link between these two chapters
Here is a fact worth putting in front of students, because it is concrete where most transmission talk is speculative. The Kerala derivation of the sine series uses ibn al-Haytham's summation identity. Katz traces it precisely: in the step where the derivation passes from one line to the next, "the Indians used ibn al-Haytham's equation for the case k = 1", and they then sketch a proof of the general result themselves, showing that for any k the sum of the kth powers of the first n integers is approximately n^(k+1)/(k+1) (Source 61, Katz, p. 172).
And immediately: "Although the Indian mathematicians did not refer to ibn al-Haytham or any other predecessor" (Source 61, Katz, p. 172). The mathematics is shared. The citation is absent. Katz's own reading is that "there seems a good chance that transmission to India did occur" (Source 61, Katz, p. 174).
Baghdad to Kerala and Kerala to Europe are two different arguments, and they do not stand or fall together. They rest on two different bodies of evidence, and the first is on much firmer ground than the second (Source 61, Katz).
The transmission question, and how to teach a live dispute
In 1834 Charles Whish, a young East India Company civil servant in Malabar, read the manuscripts, realized the local astronomers had infinite series for π before Newton was born, and published it in London (the paper is dated 1834 or 1835 in different sources). Nobody cared for about a hundred and twenty years. He died at 39 (Source 51, Almeida).
Whether Kerala mathematics reached Europe through Jesuit missionaries is an open question among working historians. Dennis Almeida and George Gheverghese Joseph argue for transmission on grounds of motive, opportunity, and resemblance: the Jesuits ran a scientific network from Rome to Beijing, they were in Kerala, they were collecting calendrical material, and the series show up in Europe soon after (Source 51, Almeida). Skeptics point out that no document has been produced.
This is a dispute, not a fact. Two competent people looking at identical evidence and disagreeing is the whole lesson (Source 52, Pearce).
Before you read on: motive, opportunity, and resemblance are all present. What single thing would settle it?
The answer, and why nobody can produce it
A document. One Jesuit letter, inventory, or translation that carries a Kerala series to Europe. Nobody has one.
Notice what that means about the three arguments in favor. Motive, opportunity, and resemblance are exactly the evidence you would expect to find whether or not the transmission happened, because the Jesuits really did run a network from Rome to Beijing, they really were in Kerala, and two people can find the same series independently. Evidence that looks the same under both stories cannot tell you which story is true.
That is the difference between a case and a proof, and it is worth more than any settled result in this chapter.
One more thing belongs here, because it is about how histories get written. Carl Boyer's 1949 history dismissed Indian calculus in two pages, at the exact moment the Yuktibhāṣā was going to press in Kerala (Source 50, Ramasubramanian). Historiography has a supply-chain problem.
The Islamic world
Ibn al-Haytham (Alhazen, 965 to 1039) needed the sum of fourth powers to find the volume of a paraboloid, because a paraboloid's cross-sectional radius squared is linear in height, and squaring the area term pushes you to degree four (Source 54, Dennis; Source 61, Katz). Nobody wanted Σi⁴ for its own sake; a solid shape demanded it.
The machinery is one identity, proved by induction:
Read this equation in words
n plus one, times the sum from i equals one to n of i to the k, equals the sum from i equals one to n of i to the k plus one, plus, for each p from one to n, the sum from i equals one to p of i to the k, all added together.
He states it only for n = 4 and k = 1, 2, 3, and proves each case by induction on n, but the argument generalizes immediately (Source 61, Katz, p. 166). I checked it for k = 1 through 4 (Appendix E.13). He stopped at the fourth power because he needed nothing higher, and what he needed it for was this: rotate the parabola x = ky² about the line x = kb², slice the solid into discs, and the volume comes out at exactly eight fifteenths of the circumscribing cylinder (Source 61, Katz, p. 168). That ratio is exact, and I verified it symbolically (Appendix E.14).
Guess before you read on
He worked out the sum of fourth powers exactly, every term of it, and proved it by induction. You would write Σi⁴ ≈ n⁵/5 and throw the rest away without a second thought. Name the one thing he did not have that lets you do that.
I have a guess
A limit. Divide Σi⁴ by n⁵, let n grow, and every term except 1/5 dies, which is ∫₀¹x⁴dx. That move needs a limit concept. He did not have one, so he needed the formula exact, down to the last term (Source 54, Dennis).
If the exact formula looked like showing off, turn it around: it is the price of not having limits, and you get to skip it because somebody else paid.
The instructive part is why he needed an exact closed formula rather than a leading term. Σi⁴ ≈ n⁵/5; divide by n⁵, let n grow, and the other terms die, leaving 1/5, which is ∫₀¹x⁴dx. We only need the leading term because we have limits. He needed the exact formula because he did not. The formula is the price of not having a limit concept (Source 54, Dennis).
And his proof is a Greek-style double reductio: he brackets the paraboloid between eight fifteenths of the cylinder and eight fifteenths of the cylinder-less-its-top-slice, then makes the top slice as small as he likes (Source 61, Katz, p. 168). Six hundred and fifty years later, Fermat and Roberval do the same thing with the same inequality, and Katz shows the two arguments side by side (Source 61, Katz, pp. 163 to 164).
The formula then travels without its owner. It turns up in Morocco in the work of ibn Haydur (d. 1413) and ibn Ghazi (1437 to 1514), and in Samarkand in al-Kāshī's Calculator's Key. Katz is blunt about the limit of the evidence: "we do not know, however, how these mathematicians learned of the formula or for what purpose they used it" (Source 61, Katz, p. 169).
His biography is the best cautionary tale in the history of engineering. He promised the caliph al-Hakim he could control the Nile, traveled to Aswan, realized the project was impossible with available technology, and then, knowing what al-Hakim did to people who disappointed him, spent roughly a decade faking insanity. He used the decade to write the Optics, the book that established experiment as the arbiter of physical theory (Source 58, O'Connor). Modern optics exists because a scientist was frightened of his boss. The Aswan High Dam was later built roughly where he stood.
Thābit ibn Qurra (836 to 901) computed volumes using unequal partitions (Source 57, O'Connor). Every calculus student meets Riemann sums with equal-width strips. Thābit saw that unequal widths can be the right tool, a thousand years before Riemann formalized the general partition. His religious background matters historically: the House of Wisdom's translation program depended on a pagan, Syriac-speaking minority who had kept reading Greek. The transmission of Greek mathematics to Europe runs through a community that was neither Greek, nor Muslim, nor Christian (Source 57, O'Connor).
Sharaf al-Dīn al-Ṭūsī (c. 1135 to 1213) found the maximum of a cubic about 450 years before Fermat, and used the discriminant condition D ≥ 0 to ask when a solution exists rather than what it is (Source 53, O'Connor). Whether he "had the derivative" is disputed: Roshdi Rashed says yes, Jan Hogendijk says the evidence does not support it.
Before you read on: what would al-Ṭūsī have had to write down for you to say he "had" the derivative?
What the two sides are arguing about
Not the mathematics. Rashed and Hogendijk agree on what is on the page. They disagree about what counts as having a concept.
Rashed's standard is the procedure: al-Ṭūsī computes the quantity we would compute and gets the answer we would get.
Hogendijk's standard is the object: to have the derivative you have to treat it as a thing in its own right, something you can name, differentiate again, and reason about apart from the cubic that produced it.
You will cross that exact line in Derivatives 1.4, where a derivative stops being something you do to a curve at one point and becomes a function in its own right. Your course walks over the boundary these two historians are arguing about, in one section.
al-Kāshī (d. 1429) stated his error budget as "no more than the width of a horse's hair on the circumference of the universe", which is a fifteenth-century engineer's way of saying sixteen decimal places, and a much better hook for a teenager than the digits are (Source 63, Aydin). One correction for the textbooks: al-Kāshī did not invent decimal fractions. al-Uqlīdisī did, around 950 (Source 63, Aydin).
The word "sine" is a mistranslation, and we never fixed it
This is the best language-and-mathematics crossover available, and it takes three minutes to teach.
- Sanskrit ardha-jyā, "half-bowstring". A chord of a circle looks like a drawn bowstring; the arc is the bow. Shortened to jyā.
- Borrowed into Arabic as jiba, a meaningless loanword.
- Arabic script omits short vowels, so jiba is written identically to jaib, a real Arabic word meaning a fold, bay, or bosom.
- A twelfth-century translator in Toledo read the real word and rendered it into Latin as sinus.
- Edmund Gunter abbreviated it to "sin" in 1624.
(S060; the translator is either Robert of Chester or Gerard of Cremona, and sources disagree.)
Every time a student writes "sin", they are writing a Sanskrit word for archery equipment, mangled through Arabic, mistranslated into Latin, and abbreviated by a Jacobean clergyman.
How to close this chapter honestly
The temptation with a chapter like this is to swing from "Europe invented calculus" to "Europe stole calculus". Both are wrong, and the standard citation says so plainly. Katz, having laid out everything above, ends:
There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today. (Source 61, Katz, pp. 173 to 174)
And then, in the next breath, the honest caveat:
But what we do not know is whether the immediate predecessors of Newton and Leibniz, including in particular Fermat and Roberval, learned of some of the ideas of the Islamic or Indian mathematicians through sources of which we are not now aware. (Source 61, Katz, p. 174)
That is the register to write this chapter in. Islamic and Indian mathematicians had large pieces of the calculus, centuries early, and used them for the specific problems they cared about. What they did not do, on the evidence we have, is unify them. Katz's diagnosis: "There were apparently only specific cases in which these ideas were needed" (Source 61, Katz, p. 173).
One question before you go
"Sin" is a Sanskrit word that got mangled twice on the way to your calculator. What did the Sanskrit word mean?
Show the answer
Half a bowstring. Ardha-jyā, shortened to jyā, because a chord looks like a drawn bowstring and the arc is the bow. It went into Arabic as jiba, a meaningless loanword written identically to jaib, a fold or a bay, and a twelfth-century translator in Toledo read the real word and put sinus into Latin (Source 60).
That is the word you use all through Derivatives 2.2, and it is the one piece of notation in your course that means nothing at all. Nobody ever fixed it, so every "sin" you write is archery equipment.