A major Chinese mathematician of the 19th century, Li Shanlan escaped to Shanghai in 1852 during the Taiping Rebellion — “one of the bloodiest wars in human history”. With the help of two Protestant missionaries, namely Joseph Edkins and Alexander Wylie, Li began to translate major works of Western mathematics into Chinese for the first time — from ancient Euclid to modern algebra.

This was an echo of another collaboration undertaken in the early 1600s by the Jesuit missionary Matteo Ricci and Xu Guangqi, who had opined to his Ming dynasty rulers that native mathematics was in total decline (like “tattered sandals”) and had to be discarded as such. Some scholars maintain that Ricci had more to gain from this enterprise than Xu, which also exposed Chinese philosophy, including the I Ching, to Europe. “I became increasingly skeptical of Xu’s claims,” writes historian Roger Hart in his book The Chinese Roots of Linear Algebra (2010), deepening the mystery around this period.

Like most of early mathematics, linear algebra emerged from immediate, material needs of human culture and commerce. Suppose a Mesopotamian farmer in 2000 BC who only knows that two fields A and B together produce eight tonnes of grain per season, but when B doubled production, the total yield became 10 tonnes. So how much does each field produce? In algebra one would write this as two ‘linear equations’: (a + b = 8) and ( a +2b = 10). If you subtract the first equation from the second, one variable (or ‘unknown’) disappears and we get the answer b = 2, which, when plugged back in, gives us a = 6.

These equations are called linear because they represent lines in two-dimensional space, which intersect in a point — viz the solution (a= 6, b= 2). If we had three variables (a, b, c) in the above, the equation could represent a two-dimensional plane — like an infinite sheet of white paper. Now one can imagine how two lines intersect in a single point, but also how two planes intersect in a line.

Subtracting one equation from another to reduce the number of unknowns later became known as ‘Gaussian elimination’. The first algorithm to do this kind of procedure mechanically was for Charles Babbage’s analytical engine, written by Lady Ada Lovelace (for 10 equations with 10 unknowns).

Allegedly the ‘first computer programmer’ in history, Lovelace was also a victim of uterine cancer; her excruciating descent and death in 1852 eased only by frequent doses of laudanum (a tincture of opium), and other prescribed narcotics. Li Shanlan’s escape took place in the same year between the two Opium Wars (1839-1860), when the British empire was struggling to protect its most important source of revenue — the opium which came to Canton all the way upstream from Ghazipore and Patna in central India.

One of the works Li translated was an algebra textbook by Augustus De Morgan. Lovelace was a student of the latter, along with JJ Sylvester, who, in a macabre turn of events, would coin the modern name for arrays of numbers that represented ‘nxn’ linear equations — matrix ( which is Latin for womb or uterus).

Speaking of the Ricci-Guangqi project, Hart continues, “General solutions to systems of n equations in n unknowns had not been known in Europe at the time, and their importance was not lost on the Jesuits and their collaborators. For they copied, without attribution, one by one, linear algebra problems from the very Chinese mathematical treatises they had denounced as vulgar and corrupt. They then included those problems in a work titled Guide To Calculation ( Tong wen suan zhi , 1613), which they presented as a translation of German Jesuit Christopher Clavius’s Epitome arithmeticae practicae (1583). Chinese readers of this “translation” had no way of knowing…”

Every matrix has a certain telltale signature known as the determinant which can be calculated from its table of values. It tells us whether the n equations in n unknowns have a solution. So if there are eight equations in three unknowns, perhaps they might intersect in three-dimensional space to enclose a perfect cube. The determinant will tell us the volume of that cube. By multiplying another matrix to such a matrix, one can transform the cube into some other shape — expanding and twisting it, or contracting it.

The book Alice’s Adventures In Wonderland (1865) alludes to this often, such as when the hookah-smoking Caterpillar offers her a mushroom which can restore Alice to her proper size (perhaps an allegory for an “invertible matrix”). We know this because the mathematician Charles Dodgson (aka Lewis Carroll) was no stranger to linear algebra. PG Harper and DL Weaire, in their Introduction To Physical Mathematics (1985), report that: “...when Queen Victoria, an admirer of Alice, requested a personal copy of his next book, the author sent her The Theory Of Determinants. Presumably, she was not amused.”

Rohit Guptaexplores the history of science as Compasswallah;@fadesingh

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