Related papers: Jost B\"urgi's Method for Calculating Sines

Placing regular 4n-sided polygons correctly, equations between sine and sums of sines show up -- exactly these equations are used in B\"urgi's method to approximate sines.

There is little known about the methods used by the ancient Babylonians and Egyptians to arrive at their recorded estimates of the value of Pi. A surprisingly accurate estimate of Pi was recently revealed coded within a verse in the book of…

Ratios and coefficients are used to simplify calculations. For geometric usage these values also called function values. Like in Egypt also in Babylon such a value system can be shown. The reconstructed calculation sequence, of the Plimpton…

We discuss the main stages of development of the error calculation since the beginning of XIX-th century by insisting on what prefigures the use of Dirichlet forms and emphasizing the mathematical properties that make the use of Dirichlet…

Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern…

In this paper, an optimized version of classical Bombelli's algorithm for computing integer square roots is presented. In particular, floating-point arithmetic is used to compute the initial guess of each digit of the root, following…

Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that…

By "solving a triangle", one refers to determining the three sidelengths and the three angles, based on given information.Depending on the specific information, one or more triangles may satisfy the requirements of the given information.In…

Logic is the science of correct inferences and a logical system is a tool to prove assertions in a certain logic in a correct way. There are many logical systems, and many ways of formalizing them, e.g., using natural deduction or sequent…

A new method is presented for finding the derivative of the sine and cosine using the discoveries of Leibniz in calculus between the years 1675 and 1677, namely the derivative of the product and the quotient of two functions as well as the…

An investigation of the comparative efficiency of the different methods in which {\pi} is cal- culated. This thesis will compare and contrast five different methods in calculating {\pi} by first deriving the various proofs to each method…

In his famous work, "Measurement of a Circle," Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern…

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The…

In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove…

A new musical scale devised by the author, based on natural logarithms, is described. Most of the logarithmic pitches bear no correspondence to the twelve tones of the ancient tuning system attributed to Pythagoras, based on ratios of whole…

Giovanni Battista Benedetti (1530--1590) derived two constructions of ovals given their minor and major axes. These were published in 1585 and seem to be the first solution to this problem. Therefore, the generally accepted view that ``the…

This article discusses the reasons for the choice of the sexagesimal system by ancient Sumerians. It is shown that Sumerians chose this specific numeral system based on logical and practical reasons which enabled them to deal with big…

Multiplication and exponentiation can be defined by equations in which one of the operands is written as the sum of powers of two. When these powers are non-negative integers, the operand is integer; without this restriction it is a…

The Fibonacci numbers are familiar to all of us. They appear unexpectedly often in mathematics, so much there is an entire journal and a sequence of conferences dedicated to their study. However, there is also another sequence of numbers…

Ancient astronomers faced the problem of dealing with arcs and angles in their observations and predictions without the help of modern trigonometry. The usual method to deal with such problems was the Menelaus Theorem, explicitly discussed…