For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.
Posts Tagged 'Spherical Trigonometry'
A Model for Elliptic Geometry
Published May 13, 2021 Occasional Leave a CommentTags: Geometry, Spherical Trigonometry
Ireland’s Mapping Grid in Harmony with GPS
Published November 19, 2020 Irish Times Leave a CommentTags: Geophysics, Maps, Spherical Trigonometry
The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [TM199 or search for “thatsmaths” at irishtimes.com].
Transverse Mercator projection with central meridian at Greenwich.
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Trigonometric Comfort Blankets on Hilltops
Published July 5, 2018 Irish Times Leave a CommentTags: Geophysics, Maps, Spherical Trigonometry
On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country [see TM142, or search for “thatsmaths” at irishtimes.com].
Trig pillar on summit of Croaghan Moira, Wicklow [Image from https://mountainviews.ie/%5D.
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Mercator’s Marvellous Map
Published May 21, 2015 Irish Times Leave a CommentTags: Geometry, Geophysics, Spherical Trigonometry
Try to wrap a football in aluminium foil and you will discover that you have to crumple up the foil to make it fit snugly to the ball. In the same way, it is impossible to represent the curved surface of the Earth on a flat plane without some distortion. [See this week’s That’s Maths column (TM068): search for “thatsmaths” at irishtimes.com].
![Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].](https://thatsmaths.files.wordpress.com/2015/05/mercator-projection-75deg.jpg)
Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].
Fermat’s Christmas Theorem
Published December 25, 2014 Occasional Leave a CommentTags: Arithmetic, Number Theory, Pi, Primes, Spherical Trigonometry
Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations ‘sin’, ‘cos’ and ‘tan’ for the trigonometric functions.
Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)
Pythagoras goes Global
Published April 11, 2013 Occasional Leave a CommentTags: Pythagoras, Spherical Trigonometry
Spherical trigonometry has all the qualities we expect of the best mathematics: it is beautiful, useful and fun. It played an enormously important role in science for thousands of years. It was crucial for astronomy, and essential for global navigation. Yet, it has fallen out of fashion, and is almost completely ignored in modern education.
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Carving up the Globe
Published October 18, 2012 Irish Times Leave a CommentTags: Algorithms, Geometry, Numerical Analysis, Spherical Trigonometry
This week, That’s Maths (TM007) describes various ways of dividing up the sphere. This is an important problem in geometry, biology, chemistry, astronomy, meteorology and climate modelling. Continue reading ‘Carving up the Globe’
Analemmatic Sundials
Published August 16, 2012 Irish Times Leave a CommentTags: Spherical Trigonometry, Time measurement
This week’s That’s Maths article, TM003, describes the analemmatic sundial on the East Pier in Dun Laoghaire.
An article in Plus Magazine, by Chris Sangwin and Chris Budd, gives a description of the theory of these sundials and instructions on how to build one.
A script to design an analemmatic sundial, written by Alexander R. Pruss, is available here. To run it, just enter the width of the sundial, the latitude and longitude, and the timezone. The script will generate all the required dimensions.
Here is a technical article, The Equation of Time and the Analemma (PDF), submitted to the Bulletin of the Irish Mathematical Society.
Napier’s Nifty Rules
Published August 12, 2012 Occasional Leave a CommentTags: Spherical Trigonometry
Spherical trigonometry is not in vogue. A century ago, a Tripos student might resolve a half-dozen spherical triangles before breakfast. Today, even the basics of the subject are unknown to many students of mathematics. That is a pity, because there are many elegant and surprising results in spherical trigonometry. For example, two spherical triangles that are similar – having corresponding angles equal – have the same area. This contrasts sharply with the situation for plane geometry. Continue reading ‘Napier’s Nifty Rules’
