相关论文: Aryabhata and the Construction of the First Trigon…
We discuss the work of a brilliant line of Mathematicians who lived in central Kerala and starting with its founder Madhava (1350 CE) developed what can best be described as Calculus and applied it to a class of trigonometric functions. We…
It is shown that with appropriate boundary conditions, a real function satisfying the differential equation $f'(x) = f(x+a)$ has all known properties of the sine function. A number of elementary derivations are presented including proofs…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
We investigate the mathematics behind 1500 year old root extraction methods presented by Aryabhata in his famous mathematical treatise "Aryabhatiya". Also, we look at their computational complexity.
The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics…
This paper presents the principle of relativity of motion as described in Aryabhata's text Aryabhatiya. This principle is likely to have been instrumental in the faming of Aryabhata's theory that the earth rotated on its axis and,…
The first isodynamic point of a triangle is one of many notable points associated with a triangle. It is named X(15) in the Encyclopedia of Triangle Centers. This paper surveys known results about this point and gives additional properties…
This paper proposes a trigonometric interpolation-based approach (TIBA) to approximate solutions of mixed boundary value problems of second-order ODEs. TIBA leverages the analytic attractiveness of a trigonometric polynomial to reformulate…
This paper is a sequel to arXiv:2511.01024 (Base 1), where an axiomatic framework for angles and the foundations of difference-angle geometry were introduced. In difference-angle geometry, where the difference of slopes of lines is treated…
An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This…
Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly)…
The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. We briefly recall why this introduction was necessary. We then show that…
Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie…
For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha,$ $ \cdots, (N-1) \alpha$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one…
This paper presents certains aspects of the mathematics of Aryabhata that are of interest to the cryptography community.
Only very recently a trigonometric proof of the Pythagoras theorem was given by Zimba \cite{1}, many authors thought this was not possible. In this note we give other trigonometric proofs of Pythagoras theorem by establishing,…
Comparison of answers offered by a computer algebra system (CAS) with answers derived by a student without a CAS is relevant, for instance, in the context of computer-aided assessment (CAA). The issues of identity, equivalence and…
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC . We wish to…
How were surfaces evaluated before the invention of the sexagesimal place value notation in Mesopotamia? This chapter examines a group of five tablets containing tables for surfaces of squares and rectangles dated to the Early Dynastic…
We perform a canonical analysis of the bimetric theory in the metric formulation, computing the constraints and their algebra explicitly. In particular, we compute a secondary constraint, that has been argued to exist earlier, and show that…