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Science, faith, inspiration: opening of the exhibition ‘Nicolaus Copernicus – life and work’

Science, faith, inspiration: opening of the exhibition ‘Nicolaus Copernicus – life and work’

by admin | May 23, 2023 | News

On 23 May 2023 at 11:30 a.m., the opening ceremony of the exhibition ‘Nicolaus Copernicus – life and work’ will take place at the building of the District Branch of the National Bank of Poland at 20 Basztowa Street in Kraków. It is co-organised by...
Opening of the College of Economics and Management of Nicolaus Copernicus University

Opening of the College of Economics and Management of Nicolaus Copernicus University

by admin | May 23, 2023 | News

On 24 May 2023, the College of Economic and Management Sciences of the Nicolaus Copernicus University (SGMK) will hold its official opening ceremony at the premises of the Nicolaus Copernicus University at 47a Nowogrodzka Street in Warsaw. The event will be attended...
The exhibition “Nicolaus Copernicus life and work” launches into the world on

The exhibition “Nicolaus Copernicus life and work” launches into the world on

by admin | May 23, 2023 | News

On 23 May 2023, the opening ceremony of the exhibition ‘Nicolaus Copernicus – life and work’ prepared by the Ministry of Foreign Affairs took place at the headquarters of the district branch of the National Bank of Poland in Krakow. In the following...
Prof. Gorski to Poles in Lithuania: Copernicus’ life is an illustration of the four cardinal virtues

Prof. Gorski to Poles in Lithuania: Copernicus’ life is an illustration of the four cardinal virtues

by admin | May 18, 2023 | News

In Vilnius, Polish young people, teachers and Polish scientists active in Lithuania took part in a panel organised by the Ministry of Education and Science. A short and moving lecture was also given by Professor Krzysztof Górski – Secretary General of the...
Prof. Grzegorz Górski met with the chairman of the Italian commission for culture, education, science and sport, Roberto Marti

Prof. Grzegorz Górski met with the chairman of the Italian commission for culture, education, science and sport, Roberto Marti

by admin | May 18, 2023 | News

On 17 May 2023, the coordinator of the expert department of the Nicolaus Copernicus Research Centre, Prof. Grzegorz Górski, held a conversation in Rome with the President of the Committee on Culture and Heritage, Education, Research and Sport of the Senate of the...
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Cudzysłów
Copernican Cosine Theorems for Spherical Triangles

Chapter XIV

Theorems III and XII

Cudzysłów
We consider three points A, B, and C on a sphere of radius R. If we connect them by arcs (along great circles), we obtain a spherical triangleABC.
We consider right-angled spherical triangles with sides shorter than a semicircle, as shown in the
figure.
Cudzysłów

III Copernicus’ /b>Third Theorem<:

In a right-angled spherical triangle ABC on a sphere of radius R (where angle C is a right angle), the following proportion between the sides holds:

AB / BC = R / BC

That is, the ratio of the hypotenuse to one leg equals the ratio of the radius to the adjacent leg.
This means that if we know two sides, we can determine the third.

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”,  Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]

Cudzysłów
In other words, Copernicus proved a special case of the spherical law of cosines.  Now, we may assume that the sphere has radius R =1.

We measure in radians the length of the side (arc) opposite to a given angle as an arc on the sphere (from the sphere’s center).
We have:

AB = c         BC = a         AC = b
(here a, b, c are the measures of angles AOB, BOC, AOC in radians)

If angle C is right, we can write Copernicus’ theorem as:

cos c / cos b = cos a

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”,  Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]

Cudzysłów
Theorem of Pythagoras for spherical triangles

That is, if we know two sides, we can find the third.This is the spherical version of the Pythagorean theorem, which can be written as:

cos c = cos b   cos a

Cudzysłów
We also have the general spherical law of cosines for triangles on a sphere of radius R=1, where α, β, γ are the spherical angles of triangle ABC.

Copernicus’ Twelfth Theorem:
cos c = cos a   cos b + sin a   sin b   cos γ

That is, if we know two sides and at least one angle, we can determine the third, which is the spherical version of the law of cosines for all plane triangles.

 

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”,  Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]

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