Marx and mathematics-3:
The European navigational problem and the dissemination of the Indian calculus in Europe

C. K. Raju

Why exactly were Europeans so keen to “discover” this Indian knowledge of calculus?

Here, I will list only the economic reasons. The then key European means of producing wealth was through overseas “trade”, mostly just piracy and loot. But whether trade or piracy, transporting the wealth back required a reliable means of navigation which Europeans then lacked. And (celestial) navigation needed precise trigonometric values.1 The most precise trigonometric values in the world, then, were derived in India (accurate to the sexagesimal thirds or about nine decimal places) using the Indian infinite series (and its sophisticated enhancements). This is exactly what was later falsely claimed to have been “discovered” by Newton and Leibniz on the doctrine of Christian discovery that the first Christian to sight a piece of land or knowledge becomes its “discoverer”.

The European problem of navigation
The precise trigonometric values, derived in India using calculus and its infinite series, were needed to determine one’s position at sea, or latitude, longitude and loxodromes. (Indians used only the first two.)

Loxodromes. The problem of loxodromes was peculiar to the inferior European method of navigation using plane charts. The problem is that a plane chart does not correctly represent the curved surface of the earth. A straight line course set by a compass or stellar rhumb line results in a ship moving on the curved surface of the earth in a spiral, a curved line—a loxodrome. Indians (such as the 7th c. Bhaskar) knew and explicitly stated that though lines of latitude and longitude meet at right angles, the “Pythagorean theorem” does not apply to (spherical) triangles drawn on the curved surface or earth.2

Therefore, for Europeans, a key navigational problem in the 16th c. was to devise a plane map so that a straight line course would result in a straight line on the map. The solution to this problem was the Mercator chart, or common map of the world, which involves a projection today called the Mercator projection, with its characteristic feature that it distorts areas. Precise trigonometric values (“tables of secants”, or reciprocal of cosines) were needed to calculate the so-called Mercator projection. Though Needham has documented that the Mercator projection was available in Chinese star maps, from long before Mercator, there is no documentation for the source of Mercator’s precise trigonometric values, naturally not because Mercator was arrested by the Inquisition, a fact rarely revealed.

Anyway, 16th c. European navigational theorists like Simon Stevin hence state the earlier version of these trigonometric values known to Europeans (accurate to the first sexagesimal minute, or about 5 decimal places). These values were quite clearly obtained from Aryabhata via al Khawarizmi because of the peculiar poetical way of stating the value of π used by Aryabhata (Ganita 10), for the sake of the metre, which roundabout way is repeated verbatim by both al Khwarizmi and the 16th c. Stevin.

Longitude. Determining longitude at sea, of course, was a major European problem which persisted until the 18th c., as many European governments acknowledged and offered huge prizes for, from the 15th c. onward. This was again an exclusively European problem due to bad European mathematics. As the 7th c. Brahmagupta3 caustically remarks ignorance of the earth’s radius makes longitude determination futile. European with their inferior Roman arithmetic (which had no systematic notation for fractions) could not determine the radius of the earth accurately, hence had problems with determining longitude. These problems persisted well into the mid-18th c., at least, regardless of retrospective stories of how Europeans managed to belatedly determine the radius towards the end-17th c., because those determinations had no credibility in prospect with sailors whose lives depended on it.

Latitude and calendar reform. But we need to note what heroic accounts of European navigation miss: that 16th c. Europe had problems even with the determination of latitude. Vasco carried back the Indian instrument of navigation, the kamal, used by the Indian navigator who brought him from Africa to India (so he could “discover” India). This instrument was also called “rapalagai” or “night instrument” (in Malayalam), and could determine latitude at night through observations of the pole star.4 It is just a precise instrument for angle measurement, and can, in principle, be used to measure the solar altitude at noon.

But, to be able to determine latitude in daytime (e.g. from a measurement of solar altitude at noon) one needs an accurate calendar, which gives the date of equinox accurately. This knowledge was then missing in Europe. The then-prevalent European (Julian) calendar (adopted as the Christian calendar at the first Council of Nicea), was way off the mark by eleven whole days. This was coarsely corrected by the 1582 Gregorian reform of the calendar. (Coarse because, Europeans still lacked fractions, and used a rudimentary system of leap years instead of stating the duration of the tropical year as a precise fraction. This crude method of leap years gets the [tropical] year right only on a thousand year average, not from year to year, as needed.)

The claim that the calendar reform was purely a “religious” matter is not credible, since attempts to correct the slip in the date of Easter were first started by pope Hilarius, in the 5th c. The Gregorian reform was directly related to the huge practical European needs of accurate navigation in the 16th c. Hence, after first rejecting it as a papal plot (Newton thought he was born in Christmas day), Protestant Britain adopted the papal solution in 1752, and Soviet Union in 1918. (Hence, the “Great October revolution” comes in November.)

Secondly, for the same economic reason, there was a need for accurate astronomical models. Both accurate trigonometric values, and accurate astronomical models (and an accurate calendar, correctly stating the equinox and the duration of the year) were found in India, and sought by key Jesuits such as Matteo Ricci,5 a dear student of Clavius who actually authored the Gregorian reform.

Dissemination of calculus in Europe
In the 16th c., the natural first European recipient of the translated mathematical and astronomy texts from India was the Jesuit general Christoph Clavius (who published Indian trigonometric values with the same accuracy to the tenth decimal place, in his own name in 1607, without even whispering its “heretical” or non-Christian source). Another key recipient was Tycho Brahe, Kepler’s master, and the Astronomer Royal to the Holy Roman empire, whose astronomical model is a carbon copy of Nilakantha’s planetary model. Once again, in those days of the Inquisition, Tycho kept his “heretical” sources a secret even from his assistant Kepler. Though Ricci’s master, Clavius, was also the author of the Gregorian reform, the pope was loathe to acknowledge non-Christian knowledge as the source of this religious reform, and falsely attributed the knowledge to some (unknown) old European texts to avoid the charge of relying on heathen texts, for the key aspect of Nicene Christianity, the date of Easter, determined by the Christian calendar.

Much later, Galileo, who had access to the Jesuit archives, refused to dabble in infinite series (denounced also by Descartes), and passed on credit/discredit for this knowledge to his student Cavalieri.

That others such as Pascal and Fermat had access to translated Indian texts, after nearly a century of their arrival in Europe, is clear from the strong circumstantial evidence of Fermat’s challenge problem,6 which was so hard that no European mathematician could then solve it. But this is a solved exercise in Bhaskar II’s text.7 Indeed, Fermat copies Bhaskar who himself poses it as a sort of challenge: “Declare it friend if the method [of solution] be spread over your mind like a creeper”. The problem was so hard because the numbers involved in it are very large, 226153980, and 1766319049. Hence, the claim of a “coincidence” between Bhaskar and Fermat is simply not credible, even by the shamelessly brazen standards of Western history in claiming “independent rediscovery” (with credit given ALWAYS to the later-day “discoverer”!). A later day-commentary on Bhaskar’s work, Kriyakramkari, was the probable source for Fermat, since it too contains the sine values (which Europeans greatly desired), and translated and took to Europe.

Likewise, things like “Stirling’s” formula (for quadratic interpolation) was known to Vateshwara, long before the Aryabhata school in Kerala became prominent. Later, Euler became famous by claiming credit for solving Fermat’s challenge problem, and it is to be noted that Euler had access to Indian texts and studied them, and wrote an article on the Indian calendar in 1700.8 The famous “Euler’s” method of numerically solving differential equations too is just the method used by Aryabhata to derive his sine values.9

In short, semi-digested knowledge of the calculus and other Indian mathematical texts was widely circulating in Europe from long before Newton and during his time. But, of course, Westerners just mule-headedly keep claiming “discovery” or “independent rediscovery” of all knowledge in the world and attribute it to Archimedes and Copernicus etc. to this day. And the colonially educated are their bhakts who will believe it all on faith and persistently refuse to check the facts, but instead side with Western authority as propagated by Wikipedia stories.

Could Marx have known about the true origin of calculus? He could have, in principle, though he probably did not. Knowledge of the infinite series in various Indian texts was already publicised in Britain in 1832 when Marx was only 14 years old, and published in 1835 by Whish.10

To conclude: Europeans stole the calculus from India. They desperately needed its related precise trigonometric values for navigation related to their key source of (overseas) wealth. In the process of correcting their navigational technique, they also needed to carry out a religious reform of the Christian calendar (1582), and were loathe to acknowledge that it was based on superior non-Christian knowledge. Hence, as traditional in the West, they simply wrote a brazenly false history of it, that calculus was “discovered” by Newton and Leibniz.

But, granting this corrected history of calculus, how exactly does it affect Marx’s understanding of calculus?

(To be continued)

1  See references cited in part 2 to my book, articles for the Springer Encyclopedia, and MIT talk. The relevant verse may also be found online at, its translation is given at, and the accuracy of the resulting sine values is explained at Also see e,g the summary in Ghadar Jari Hai,
2  Laghu Bhaskariya 1.27. For further elaboration, see Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE, 224–25 chp. 4, “Time, latitude, longitude and the globe.”
3  ब्राह्मस्फुटसिद्धांत, chapter 11, तन्त्रपरीक्षाध्याय, verses 15-16
4  Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE, chp. 5, “Kamal or Rapalagai”{.
5  Matteo Ricci, 1581, Goa, 38 (I) ff 129r–130v (Jesuit archives, Vatican); corrected and reproduced in Documenta Indica, XII, 472–477 (p. 474).
6  Fermat, Feb 1657, Ouvres, p. 332 et seq.
7  Beejaganita, 87.
8  For references to all the original sources see Cultural Foundations of Mathematics, cited above,
9  Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE, chp. 3, “Infinite series and pi.”
10  Whish, Charles M, “On the Hindu Quadrature of the Circle and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati and Sadratnamala”,” Trans. R. Asiatic Soc. Gr. Britain and Ireland 3 (1835): 509–523.

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Sep 4, 2020

Prof. C. K. Raju

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