Saros 145/22: The Great American Eclipse

Next Monday, the shadow of the Moon will bring a two-minute spell of darkness as it sweeps across the United States along a path from Oregon to South Carolina. The eclipse is one of a series known as Saros 145. [TM121 or search for “thatsmaths” at irishtimes.com].

Saros-145

Saros series 145 recurring every 18 years, 10 days and 8 hours.
[Image from www.GreatAmericanEclipse.com ]

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Fractions of Fractions of Fractions

Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number {x} can be expanded as a continued fraction:

\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]

where all {a_n} are integers, all positive except perhaps {a_0}. If {a_n=1} we add it to {a_{n-1}}; then the expansion is unique.

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It’s as Easy as Pi

Pi-SymbolEvery circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times  [see TM120 or search for “thatsmaths” at irishtimes.com].

The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).

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Who First Proved that C / D is Constant?

Every circle has the property that the distance around it is just over three times the distance across. This has been “common knowledge” since the earliest times. But mathematicians do not trust common knowledge; they demand proof. Who was first to prove that all circles are similar, in the sense that the ratio of circumference C to diameter D has the same value for all?

Circle-Area-Triagles

Slicing a disk to estimate pi (Image Wikimedia).

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Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

Quadrivium-Book

Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

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Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.

Glider-Loop-20

Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison]

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Robert Murphy, a “Brilliant Meteor”

A brilliant meteor that flared intensely but all too briefly”; this was how Des MacHale described the Cork-born mathematician Robert Murphy in his biography of George Boole, first professor of mathematics in Cork. Murphy was a strong influence on Boole, who quoted liberally from his publications [see TM118 or search for “thatsmaths” at irishtimes.com].

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