Review of Henry Thomas Colebrooke's 1817 Translation of Sanscrit Works on Division by Zero

Abstract

We review the introduction of the number zero and of total arithmetical operations of addition, subtraction, multiplication, and division in various Sanscrit works from 628 CE to 1621 CE, translated into English in the book "ALGEBRA WITH ARITHMETIC AND MENSURATION, FROM THE SANSCRIT OF BRAHMEGUPTA AND BHASCARA" by Henry Thomas Colebrooke, 1817 CE. We note the existence of intermediate steps in the development of zero from a placeholder into a number. We introduce the concepts of a paraconsistent arithmetic that is uniquely determined but which contradicts its axioms, and of a paralogical arithmetic that is uniquely determined but which is non-logical. We find that Brahmegupta, 628 CE, described two total paraconsistent arithmetics of fractions that contain the lexical operations of real arithmetic. Bhascara, 1150 CE, used only one of Brahmegupta's fractional arithmetics, but we generously credit Bhascara with making this arithmetic consistent. This manoeuvrer makes both of Brahmegupta's total fractional arithmetics consistent. However, Bhascara also introduced an inchoate arithmetic of infinitesimal tuples, which reintroduced Brahmegupta's inconsistency. We suggest that deterministic arithmetics are useful, despite any logical shortcomings. We illustrate this with the IEEE 754 Standard for Floating-Point Arithmetic, which is useful because it succeeds in its aim of specifying a deterministic arithmetic, despite the fact that this arithmetic is non-logical. We conclude that Brahmegupta and Bhascara produced a more logically sound total arithmetic than the IEEE standards committee. We recommend that computer arithmetic is founded on transreal arithmetic, which surpasses the Sanscrit arithmetics in its application to calculus and mathematical physics.