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Liu Hui
A mathematician of third-century Cao Wei China. Little is known about his life, but his commentary on the Nine Chapters on the Mathematical Art, dated around 263, and the Sea Island Mathematical Manual survive. He added explanations of why many procedures work, treating systems of linear equations with positive and negative numbers, areas and volumes, and right-triangle relations. By repeatedly increasing the number of sides of an inscribed regular polygon, he obtained π ≈ 3.1416 and examined the accuracy of approximations. The nine problems of the Sea Island Mathematical Manual use observations from two positions and right-triangle relations to determine inaccessible heights and distances.
Madhava of Sangamagrama
A mathematician of late-fourteenth-century Kerala who studied infinite series for π and trigonometric functions. Madhava’s mathematical writings are lost, but later Kerala authors including Nilakantha and Jyesthadeva report results attributed to him. They include the π/4 series now often called the Gregory-Leibniz series, series corresponding to sin, cos, and arctan, an approximation of π to eleven decimal places, and correction terms that improve convergence. These results predate comparable European rediscoveries by more than two centuries. Direct transmission from Kerala to Europe has been proposed, but the surviving evidence does not establish a specific route. Madhava’s work is therefore most safely understood as an achievement of an independently developed Kerala mathematical tradition.
Nicolaus Mercator
A seventeenth-century mathematician who connected the computation of logarithm tables with infinite series. Nicolaus Mercator was not Gerardus Mercator, the cartographer associated with the map projection. In his 1668 Logarithmotechnia, Nicolaus published the expansion now called the Mercator series, ln(1+x)=x−x²/2+x³/3−…. It converges for −1<x≤1, but is very slow near x=1, so it did not replace every logarithm table “in a few lines.” A practical algorithm must first move the input closer to zero with identities and then control truncation error. Newton was also studying binomial series and term-by-term integration more generally around this period; priority in early calculus and series cannot be reduced to one publication date. Mercator’s example is a good entrance to why a formula’s interval of convergence and efficiency matter as much as memorizing it.
Leonhard Euler
An exceptionally prolific author across analysis, number theory, mechanics, astronomy, and early graph problems. He introduced or popularized major notation and continued working by dictation with family and assistants after losing most of his sight.
Joseph Fourier
A mathematician and administrator who joined the French expedition to Egypt and later developed a theory of heat in Grenoble. His 1807 memoir used trigonometric expansions broadly, prompting objections about derivation, generality, and rigor. A revised memoir won the 1811 academy prize, and the work appeared as The Analytical Theory of Heat in 1822.
Srinivasa Ramanujan
A mathematician who rebuilt number theory in his own language under severe educational constraints. Born in Erode and raised in Kumbakonam, Ramanujan used G. S. Carr's Synopsis to reconstruct results for himself and filled notebooks with thousands of formulas. His 1913 letter from Madras led to collaboration with G. H. Hardy at Cambridge and major work on partitions, infinite series, and modular forms. His health deteriorated in Britain; he returned to India in 1919 and died at 32. Contemporary diagnoses and modern reassessments differ, so a single certain cause should not be imposed. His notebooks remain active objects of proof and discovery.
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