Science

Eratosthenes Measures the Earth

Eratosthenes Measures the Earth
Eratosthenes Measures the Earth

Like Aristotle[1] before him and in keeping with virtually all educated opinion in ancient Greece[2], Eratosthenes (c. 276 BC - c. 194 BC) assumed the earth was spherical. He set out to measure its size using a sound method that has stood the test of time. His results were good (about \( 5.4\% \) too high) and no one did better for over a thousand years. His near correct result was ignored well into the modern era, including by Columbus. Snellius and Picard finally nailed down the earth's circumference in the seventeenth century using Eratosthenes' experimental design and taking advantage of advances in mathematics and instrumentation that had accrued over the intervening \( 1900 \) years.

Newton and Kepler's Laws

Issac Newton 1689
Issac Newton in 1689, at the height of his powers (age 46).

Nature and Nature's Laws lay hid in Night:
God said, 'Let Newton be!' and all was Light.

- Alexander Pope

Kepler's laws of planetary motion are:

1. The orbit of a planet is an ellipse with the Sun at one focus.
2. A line from a planet to the Sun sweeps out equal areas in equal times.
3. The square of the time of revolution of a planet is proportional to the cube of the transverse axis of its elliptical orbit, with the same constant of proportionality for all planets.

Gauss and the Fast Fourier Transform

Gauss Stamp

The Fast Fourier Transform (FFT) is a modern algorithm to compute the Fourier coefficients of a finite sequence. Fourier will forever be known by his assertion in 1807 that any function could be expressed as a linear combination of sines and cosines, its Fourier series. "Any" was a little ambitious, counter-examples coming to the fore in due time. A fair amount of mathematics from that time to this has been devoted to refining Fourier's insight and studying trigonometric series, a subject that led Georg Cantor to founding set theory. Piecewise smoothness is sufficient for pointwise convergence on \( [-\pi, \pi] \):

\[ f(x) = {a_0 \over 2} + \sum_{j=1}^\infty \left( a_j \cos jx + b_j \sin jx \right), \]

Calendar

August 2013 calendar

My original calendar program was on the Casio fx-3600p calculator in 1980 or so - my first programming venue and exercised partly in spare moments when driving truck out on the route; a precursor mobile device you could say. My buddy Dave got me started. I might have scarred myself permanently though. The transition from math to software engineering is always tricky, considering that there are many commonalities, but just as important differences to snag the self-taught and perhaps obstinate and all-too-confident mathematician (perish the thought). The 3600 had this bizarre little macro language providing for a trade-off between memory and program size. You could have (say) fifty memory locations and 400 instructions or twenty locations and 600 instructions. They're really variables of course, but the memory locations were designated K0 through K19 or something.

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