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“Is the federalisation of the European Union a threat to Poland and Europe?”. Scientific conference in Lublin [TRANSMISSION]

“Is the federalisation of the European Union a threat to Poland and Europe?”. Scientific conference in Lublin [TRANSMISSION]

by admin | Sep 13, 2023 | News

How does the direction of change in the EU affect Poland and the entire continent? Is Community federalisation an opportunity or a threat to Warsaw’s position in Europe? A scientific conference has begun in Lublin, during which Polish and foreign experts will...
Cooperation of the Copernicus Academy with the International Police Association (IPA)

Cooperation of the Copernicus Academy with the International Police Association (IPA)

by admin | Sep 12, 2023 | News

On 12 September, the Secretary General of the Copernicus Academy, Prof. Dr. Krzysztof Górski, and the Secretary General and Director of the Copernicus Academy Office, Dr. Witold Mazurek, signed a cooperation agreement with the International Police Association. The...
Scholarships for Ukrainian nationals in cooperation with the Republic of China in Taiwan. Announcement of the Competition

Scholarships for Ukrainian nationals in cooperation with the Republic of China in Taiwan. Announcement of the Competition

by admin | Sep 6, 2023 | News

Announcement by the Director of the Nicolaus Copernicus Research Centre of 6 September 2023 on the competition for funding scholarships for Ukrainian citizens Pursuant to Article 3(1) par. 1, 2 and pt. 7 of the Act of 28 April 2022 on the Copernican Academy (Journal...
Nicolaus Copernicus Grants. Information on applications submitted

Nicolaus Copernicus Grants. Information on applications submitted

by admin | Sep 5, 2023 | News

We would like to kindly inform you that within the deadline and form specified in the Announcement of the Director of the Nicolaus Copernicus Research Centre about the competition for the funding of Nicolaus Copernicus Grants, the Research Centre received 19...
‘Our mission is to inspire young minds’. Acknowledgement of Copernicus Academy

‘Our mission is to inspire young minds’. Acknowledgement of Copernicus Academy

by admin | Sep 4, 2023 | News

The 16th International Olympiad in Astronomy and Astrophysics, of which the Copernicus Academy was an official partner, came to an end on 20 August. “Thanks to your support, the Olympiad has not only become a platform for the exchange of knowledge 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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