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Copernican Talks. Prof. Gorski: We have ambitious plans, we want to create a new type of scientific institution

Copernican Talks. Prof. Gorski: We have ambitious plans, we want to create a new type of scientific institution

by admin | May 31, 2023 | News

At the Copernicus Academy, we want to create a new type of institution that will enable effective cooperation between Polish and foreign scientists,” stresses the Secretary General of the Academy, Prof. Krzysztof Górski. On 24 May, the Secretary General of the...
Opening of the College of Philosophy and Theology at Nicolaus Copernicus University

Opening of the College of Philosophy and Theology at Nicolaus Copernicus University

by admin | May 25, 2023 | News

On 25 May 2023, the inauguration of the College of Philosophy and Theology of the SGMK will take place at 20 Basztowa Street in Kraków. The event will be attended by representatives of the highest authorities of the Ministry of Education and Science, the Copernicus...
Opening of the College of Philosophy and Theology. Archbishop Jędraszewski: There is an interplay between reason and truth

Opening of the College of Philosophy and Theology. Archbishop Jędraszewski: There is an interplay between reason and truth

by admin | May 25, 2023 | News

‘I am deeply convinced that the opening of the College represents a great opportunity for the encounter between reason and faith, philosophy and theology to find answers to the anxieties of the modern world,’ said Krakow Metropolitan Archbishop Marek...
Czarnek: Colleges to educate scientific elite in Copernican areas

Czarnek: Colleges to educate scientific elite in Copernican areas

by admin | May 24, 2023 | News

– The College of Economics and Management, which opened today, but also the other colleges opening within Nicolaus Copernicus University, are intended above all to provide PhDs and research at the highest world level. In cooperation with scientists from all over...
Opening of the College of Economic Sciences and Management. Czarnek: Theory has become fact

Opening of the College of Economic Sciences and Management. Czarnek: Theory has become fact

by admin | May 24, 2023 | News

– The Copernican Academy is an example of a new dimension of cooperation at the highest level, of the most eminent representatives of the world of science in Copernican fields,” explained Minister Przemysław Czarnek. The opening ceremony of the College of...
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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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