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Visit of Copernicus Academy representatives to Washington (8-10.11.2023)

Visit of Copernicus Academy representatives to Washington (8-10.11.2023)

by admin | Nov 15, 2023 | News

Participants of the opening of the exhibition dedicated to John Paul II From 8 to 10 November 2023, representatives of the Copernican Academy, including the Secretary General, Professor Christopher Gorski, were in Washington DC for a business visit. On 8 November,...
“Wyborcza.pl” – At least seven lies in one article about the Academy

“Wyborcza.pl” – At least seven lies in one article about the Academy

by admin | Nov 15, 2023 | News

The Copernicus Academy – in accordance with current press law – is demanding a correction from wyborcza.pl. She has formally approached the website’s editor-in-chief Jarosław Kurski on this matter. The article in question is by Alicja Gardulska:...
“Copernicus’ insights are still relevant today and provide an inspiring point of reference for economic researchers.” – interview with Prof. Krzysztof M. Gorski, Secretary General of the AK

“Copernicus’ insights are still relevant today and provide an inspiring point of reference for economic researchers.” – interview with Prof. Krzysztof M. Gorski, Secretary General of the AK

by admin | Nov 9, 2023 | News

Professor, this year marks the 550th anniversary of the birth of Nicolaus Copernicus. Why are competitions in economics one of the elements of the celebrations? Prof. Krzysztof M. Górski, Secretary General of the Copernican Academy: Yes, our competitions are held in...
‘Copernican Academy expands its spectrum of activities’. Christian Theological Academy new AK partner

‘Copernican Academy expands its spectrum of activities’. Christian Theological Academy new AK partner

by admin | Oct 25, 2023 | News

– I am glad that the Copernican Academy is expanding its spectrum of activities and, in the sphere of cooperation with theological academies, is going beyond the Roman Catholic Church,’ stressed Archbishop Prof. Jerzy Pańkowski, Rector of the Christian...
Archbishop Pankowski: Theological teachings frame our behaviour and values

Archbishop Pankowski: Theological teachings frame our behaviour and values

by admin | Oct 25, 2023 | News

Attempts to oust theological sciences from universities may be related to the fact that theology and religious studies constitute a kind of framework of our behaviour and values, beyond which we cannot go, and therefore such attempts have been made, are repeated and...
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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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