The position of the Sun in the sky depends on where we are and on the time of day. Due to the Earth's rotation, the Sun appears to cross the celestial sphere each day along a path called the ecliptic. The observer's position on Earth is given by the geographic latitude and longitude. The path … Continue reading Where is the Sun?
Tag: Spherical Trigonometry
A Model for Elliptic Geometry
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 … Continue reading A Model for Elliptic Geometry
Ireland’s Mapping Grid in Harmony with GPS
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]. Mapping the Globe Positions on the globe are given by … Continue reading Ireland’s Mapping Grid in Harmony with GPS
Trigonometric Comfort Blankets on Hilltops
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]. The pillars are about … Continue reading Trigonometric Comfort Blankets on Hilltops
Mercator’s Marvellous Map
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 … Continue reading Mercator’s Marvellous Map
Fermat’s Christmas Theorem
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. Girard also showed how the area of a spherical triangle depends on its interior angles. If the angles of a triangle on the unit sphere … Continue reading Fermat’s Christmas Theorem
Pythagoras goes Global
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. … Continue reading Pythagoras goes Global
Shackleton’s spectacular boat-trip
A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives. Endurance For eight months, Ernest Shackleton's expedition ship Endurance was carried along, ice-bound, until it was finally … Continue reading Shackleton’s spectacular boat-trip
Carving up the Globe
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. The problem of defining a uniform distribution of points on the sphere has challenged mathematicians for centuries. The vertices of the five Platonic solids achieve this but, in … Continue reading Carving up the Globe
Analemmatic Sundials
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, … Continue reading Analemmatic Sundials
Napier’s Nifty Rules
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 … Continue reading Napier’s Nifty Rules
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