## Archive for November, 2016

### Taylor Expansions from India FIg. 1: Brook Taylor (1685-1731). Image from NPG.

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for ${\sin x}$, ${\cos x}$ and ${\arctan x}$ were known to mathematicians in India about 400 years before Taylor’s time.
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### Marvellous Merchiston’s Logarithms

Log tables, invaluable in science, industry and commerce for 350 years, have been consigned to the scrap heap. But logarithms remain at the core of science, as a wide range of physical phenomena follow logarithmic laws  [TM103 or search for “thatsmaths” at irishtimes.com]. Android app RealCalc with natural and common log buttons indicated.

### Which is larger, e^pi or pi^e?

Which is greater, ${x^y}$ or ${y^x}$? Of course, it depends on the values of x and y. We might consider a particular case: Is ${e^\pi > \pi^e}$ or ${\pi^e > e^\pi}$? Contour plot of x^y – y^x, positive in the yellow regions, negative in the blue ones.

### A New Window on the World

The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. Galileo put it this way: the book of nature is written in the language of mathematics [TM102, or search for “thatsmaths” at irishtimes.com]. Visualization of gravitational waves. Image credit MPI/Gravitational Physics/ITP Frankfurt/ZI Berlin.