Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at irishtimes.com].

### Fourier’s Wonderful Idea – II

Published April 5, 2018 Irish Times Leave a CommentTags: Analysis, History

**Solving PDEs by a Roundabout Route**

Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].

### Fourier’s Wonderful Idea – I

Published March 29, 2018 Occasional Leave a CommentTags: Analysis, History

**Breaking Complex Objects into Simple Pieces**

“In a memorable session of the French Academy on the

21st of December 1807, the mathematician and engineer

Joseph Fourier announced a thesis which inaugurated a

new chapter in the history of mathematics. The claim of

Fourier appeared to the older members of the Academy,

including the great analyst Lagrange, entirely incredible.”

** Introduction **

The above words open the *Discourse on Fourier Series*, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

“It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002).

Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

### Cubic Skulduggery & Intrigue

Published March 15, 2018 Irish Times Leave a CommentTags: Algebra, History

Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call *x *enters in the form of powers, that is, *x* multiplied repeatedly by itself. When only *x* appears, we have a linear equation. If *x*-squared enters, we have a quadratic. The third power of *x* yields a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at irishtimes.com].

### Subtract 0 and divide by 1

Published March 8, 2018 Occasional Leave a CommentTags: Analysis, History

We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that

This is an essential step in deriving an expression for the derivative of .