Motivation etc.

• So Jesuits in Kochi had ample opportunity to collect and translate Indian math and astronomy texts.
• But why were they interested in doing so?

Motivation: Navigational problem

• Europe then was very poor, all dreams of wealth (whether piracy or conquest) were overseas.
• Accurate navigation was needed to bring that wealth home.
• But Vasco and Columbus were zero in navigation as we saw.
• Europeans had 3 key navigational problems.
• Latitude, loxodromes, longitude at sea.

Latitude

• Vasco did not know how to determine latitude at sea,
• carried back Indian navigational instrument, kamāl
• to have it "graduated in inches" 😄
• (Can't be done: not a linear scale it uses a harmonic scale.)

• Kamāl tells latitude by the pole star.
• Since pole star=kau (Arabic-Malayalam) =teeth, and the string of the instrument is held between the teeth
• Vasco pompously and foolishly recorded "the "pilot" was telling the distance by his teeth".

• Despite stealing kamal, there still remained a problem of telling latitude in daytime:
• by measuring angular elevation of sun.
• That requires an accurate calendar which correctly determines the date of equinox,
• as explained in my Rajju Ganita geometry text for class 9.

• More precisely, to determine latitude of a place
• from observation of solar altitude at noon
• one needs knowledge of the declination.
• Hence, the "Noorie" tables.

• Q. What did Europeans do before the "Noorie tables"?
• A. After stealing the kamal, they could determine latitude but only in the Northern hemisphere (well above the south Africa cape)
• and only at night.
• A. They had no clear way to determine latitude at sea in daytime or in the southern hemisphere.

• My 9th std. text on geometry explains how to determine the altitude of the sun at noon by observing its shadow.
• provided one can calculate the arctangent.
• Or by directly looking at the sun
• to measure the angular elevation the way the kamal is used to measure the angular elevation of the pole star.
• (But you damage your eyes.)

• My 9th std. text also explains the relation between solar altitude, latitude, and declination.
• To understand the relation between zenith and latitude, you can also see this figure.
• For more details see my 9th std. text.

• But how is declination to be determined?
• Whichever way one does it, linear interpolation or "Noorie tables"
• one needs to know the date of equinox.

• All ancient civilizations (except backward West) knew the date of equinox.
• The Maya demonstrated it most beautifully in the Chichen Itza pyramid.
• Indian and Egyptians knew about it.
• It is only the West which constantly boasted of supremacy
• which had an inferior Roman-Christian religious calendar with the wrong date of equinox (hence Easter).

• Hence, Christian priests were interested to steal calendrical knowledge
• for the Gregorian calendar reform of Christian calendar in 1582.
• They refused to acknowledge Indian sources because they always condemned knowledge from non-Christian (="heretical") sources
• and that too to correct their religious (AD-BC) calendar used to determine the date of Easter.
• (Easter, or the miraculous resurrection of Jesus after death on the cross, not Christmas, or his birth, was the main Christian festival until 2-3 centuries ago.)

Loxodromes

• Europeans navigated by plane charts, i.e., they tried to represent the spherical earth on a plane.
• They tried to travel in a fixed direction
• such as that set by a fixed direction of the compass needle
• or a straight line joining two distant stars (called a rhumb line).
• On their understanding of geometry (quadrivium) they expected this will result in straight line motion.

• This is a false expectation.
• On the curved surface of the earth
• motion in a fixed direction results in a curved line
• called loxodrome (loxos=curve, dromos = line).

• More precisely, moving in any fixed non-cardinal direction
• results in a logarithmic spiral called a loxodrome = curved line
• which spirals towards the poles.
• Escher's painting visualizing loxodromes.

• What European navigators wanted was a chart on which if they moved in a fixed direction
• they would trace a straight line path.
• This required the so-called Mercator projection
• As usual there was little or no novelty added by Europeans:
• This projection was known in Dunhuang Chinese star maps from 10th c. as Needham points out. (Needham, Shorter, p. 123)

Fundamental theorem of calculus a century before calculus!

• As Struik points out, calculating loxodromes = fundamental theorem of calculus (\(\int f' = f\))
• given a tangent (\(f'\)) to the curve (\(f\)) at any point to calculate the curve.
• No one understood or used it till Wright explained it.

Mercator projection

• The projection maps the sphere to a plane chart in such a way that straight line course
• results in a straight line on the chart (loxodromes are straight lines).
• The Mercator projection does that. (This is nothing but the usual "map of the world".
• It is a conformal projection, i.e., angles are preserved so a course can be set by a fixed direction of the compass needle.

• The mathematical formulae are these.
• The area expansion ("Greenland effect") may be explained
• to map loxodromes to straight lines it distorts areas (not a political conspiracy).
• Hence the importance of tables of secants to construct this Mercator chart
• so critical to Western methods of navigation.

• 16th century navigational manuals were full of such tables.
• E.g., Simon Stevin's tables of 1590, derived from Aryabhata via Arabs (al Khwarizmi)
• How do we know?

• Since Stevin cites al Khwarizmi who literally quotes Aryabhata's value of π
• Aryabhata's verse is prolix since he adheres to the poetic metre in his Sanskrit verse.
• with laghu as 1 matra and guru as 2, he used the Arya metre of 4 lines of 12, 18, 12, 15 matras, respectively.
• Adherence to the metre (not needed for a prose depiction) shows al Khwarizmi did not quite understand it.

• Mercator's source unknown since he was arrested by the Inquisition
• and tortured for dabbling in heretical texts.

• A few years later (1607) Clavius published trigonometric tables with precision of nine decimal places.
• Clavius as the Jesuit general in charge of the Gregorian calendar reform
• was the natural recipient of Indian math and astromomy texts stolen by Jesuits.
• His trigo tables very similar to those of "Kerala school" (even use Rsine).

• Thus Nilakantha says in his Aryabhatiya-bhashya that
• Madhava of Samgamgrama gave a value of the circumference for a radius of nava-nikharva (\(9\times 10^{11}\))
• And, the Karanapaddhati gives the value of the circumference for a radius of 10,000,000,000.
• The same value of radius is used by Clavius as clear from the very title of his book.

Longitude

• Longitude calculation in Indian tradition requires, as Brahmagupta says, knowledge of the radius of the earth \(R_E\)
• easily calculated from elementary trigonometry
• by measuring the height of a hill h and the angle of dip from it θ.
• But mathematically backward Europeans could not do it (even after stealing trigo values in 16th c.)

• Therefore, the European navigational problem of determining longitude at sea persisted until 18th c.
• Many European governments offered large prizes for its solution,
• the last being the British longitude prize legislated by an act of British parliament 1711-12.
• Only half of it given away in 1762
• because the Board of Longitude was only half sure that the prize had been won!

• The chronometer for which the prize was given became a reliable instrument only by the mid-19th century.

• Why did the Europeans have such a big problem with navigation?
• For many reasons. First, because they were superstitious
• and the Bible said the earth is flat
• unlike Indians who always knew the earth is round
• hence its very Sanskrit name is भूगोल

• Āryabhaṭa said the earth is round like a kadamba flower
• Lalla explained that is inferred from the fact that far off trees cannot be seen.
• All Indian ganita texts contain estimates of the earth radius
• whose high accuracy was confirmed by al Beruni's estimate in Arabic miles
• (see my book on Indian calendar).

• Getting over Bible superstitions, Columbus, to get funds for his project of traveling West to go East,
• deliberately underestimated the size of the earth by 40%
• this led to navigational disasters
• and the Portuguese law banning the carrying of globes aboard ships.

• Further, the fact is that though Europeans stole trigonometric values
• they did not know how to use them to measure the radius of the earth.

• Today Internet is full of bunkum stories of Eratosthenes
• a Greek who supposedly measured the size of the earth.
• Nobody gives an actual primary source? Why not? (Bcoz source from 19th c.)

• Picard's 17th c. measurement was preceded even by al Mamun's 9th c. measurement by 800 years as usual.
• However, important point is that this is all in retrospect:
• in prospect no navigator was willing to trust his life to Picard
• therefore European problem of determining longitude at sea persisted.

• Further, as we will see in next lecture
• Europeans were very backward in math
• even in elementary primary-school arithmetic
• until the end 19th c., and even into the 20th c. algebra school-text of Hall and Knight which I was forced to study.

Indian method of determining longitude

• Many methods.
• A simple method was this:
• If one knows (1) the latitudes of two towns,(2) the distance between them and (3) the radius of the earth,
• one can calculate the longitude difference in yojanas
• by solving the longitude triangle as described by Bhaskara 1 (Laghu Bhaskariya 1.32, p. 11)

• This was described as a gross method since it neglected the curvature of the earth
• according to भटस्य शिष्या: (Bhaskara 1, Mahabhaskariya, 2.5
• (Note how only the term भट (NOT भट्ट)is used for Āryabhaṭa indicating he was lower caste).

• However, Europeans kept applying this gross method for centuries
• Further, they measured distance at sea by the crude method of "heaving the log" and using a log-book
• to record the ship's speed in knots (on the rope tied to the log).
• This, was a notoriously unreliable method of speed determination.

• Of course, it still needed accurate trigonometric values to solve the triangle
• from course angle and distance travelled.
• Naturally, they grabbed knowledge from every available source.
• There is ample circumstantial evidence they did so.

Circumstantial evidence

• Long list discussed in Cultural foundations of Math
• meridian of Greenwich a copy of Meridian of Ujjaini called yamayottari
• identical infinite series
  • "Newton's" sine series (Newton claimed sine series, and only rigor in calculus)
  • "Leibniz" series for π,
• Fermat's challenge problem,
• "Pascal's" triangle for binomial coefficients \(̱̱\binom{n}{r}\) called khanda meru
• in theory of permutations and combinations first developed in India for the Vedic metre (see my article Probability in Ancient India)

• Aryabhata's recursive method = "Euler" method (will cover this in detail, later)
• Nilkantha's planetary model= Tychonic model (see my book on Indian calendar) (Tycho as Royal astronomer to the Holy Roman empire was a natural recipient of stolen Indian texts)
• Madhava's sine values = Clavius' trigonometric values (which were an interpolated version)
• Brahmagupta-Vateshvar="Stirling's" formula

Documentary evidence

• Primary sources [Ricci's letter. (Ricci was Clavius' favourite student and biographer.)
• He was naturally searching for Indian calendrical texts in Kochi
• since this was right before the Gregorian calendar reform of 1582
• authored by Clavius.

• But in those days of the Inquisition
• it was impossible for the padres to admit that the reform of the Christian RELIGIOUS calendar was based on superior HERETICAL knowledge.
• (Note it was the Christian religious calendar,
• officially adopted at the first Nicene council to fix the date of Easter
• NOT the Julian calendar)

• which did not have any AD/BC as its starting point
• which starting point was fixed in the 6th c.
• in relation to the date of Easter.

• As usual the chief padre, the pope lied about the 1582 calendar reform telling the story that it was
• based on the "Alfonsine tables" from centuries ago
• and due to one Alyosisus Lilius a shadowy figure trained as a physician (not astronomer)
• But the story "established" that the knowledge had a valid Christian lineage.
• Why, then, was Ricci searching for knowledge of the calendar among heretics?😄