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Claudian Epigram on the Heavenly Sphere of Archimedes

Recently I discovered on Internet Archive Charles Hutton’s “A mathematical and philosophical dictionary”. Hutton’s mathematical dictionary is one of the earliest dictionaries that focused on the mathematical sciences. In this post I just want to transcribe an epigram written by Claudian about a sphere of glass created by Archimedes that showed the motions of the… Continue reading Claudian Epigram on the Heavenly Sphere of Archimedes

Lill’s Method and the Derivative of tan(θ) and the Powers of tan(θ)

Lill’s method can be used to obtain the derivatives of polynomial equations in a graphical manner. The method to obtain the derivatives is described in the paper “NOTE ON LILL’S METHOD OF SOLUTION OF NUMERICAL EQUATIONS” by B. MEULENBELD. The paper complicates the matter by having Lill representations of polynomials that don’t have perpendicular segments.… Continue reading Lill’s Method and the Derivative of tan(θ) and the Powers of tan(θ)

Quadratics with Complex Roots, Lill’s Circle and the Hyperbola

I already have a few papers that show how Lill’s method can be used to solve quadratic equations. In my paper “Lill’s Method and Graphical Solutions to Quadratic Equations” I showed 2 methods that can solve quadratics with real roots and 1 method for solving quadratics with complex roots. That paper also mentions that the… Continue reading Quadratics with Complex Roots, Lill’s Circle and the Hyperbola

A Note on Whittaker’s Root Series Formula

If you want to learn about Whittaker’s Root Series Formula see my post “Whittaker’s Root Series: Going Transcendental”. That post has useful links to introductory material on the topic. You can also see my OEIS sequences, since a lot of them were obtained using Whittaker’s formula. Whittaker’s Root Series Formula is obtained using the coefficients… Continue reading A Note on Whittaker’s Root Series Formula

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Littlewood Polynomials of Degree n with Closed Lill Paths

In this post I attempt to answer this question: How many Littlewood polynomials of degree n have a closed Lill path? A Littlewood polynomial is a polynomial that has all its coefficients equal to 1 or -1. Littlewood polynomials seem to be studied for their properties related to autocorrelation. A polynomial has a closed Lill… Continue reading Littlewood Polynomials of Degree n with Closed Lill Paths

The Tridecagon, Hyperbola and Lill’s Method

The regular tridecagon is another regular polygon that cannot be constructed using a compass and straightedge. In this post I want to show how the tridecagon can be constructed using the intersection of a circle and a hyperbola. In my previous posts “The Heptagon, Hyperbola and Lill’s Circle” and “The Nonagon, Hyperbola and Lill’s Method”… Continue reading The Tridecagon, Hyperbola and Lill’s Method

Adding and Subtracting the First n Prime Numbers to Get the Next Prime Number

In this post I want to present the following conjecture related to odd-indexed prime numbers (see OEIS sequence A031368): All the odd-indexed prime numbers larger than 2 can be obtained from all the previous prime numbers using addition and subtraction. The conjecture was inspired by Curiosa 67 from Scripta Mathematica ( page 159, Volume 7,… Continue reading Adding and Subtracting the First n Prime Numbers to Get the Next Prime Number