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Copernican Talks. Prof Stys: medicine is a global science and the whole world benefits

Copernican Talks. Prof Stys: medicine is a global science and the whole world benefits

by admin | Sep 22, 2023 | News

Scientists will collaborate regardless of borders, emphasises Professor of Medicine at Sanford School of Medicine (University of South Dakota) Adam T. Stys, explaining how the Copernicus Academy Medical Chamber works. Prof Adam T. Stys of the Sanford Cardiovascular...
2nd Świętokrzyskie Development Forum. “This is an important event for the entire region”

2nd Świętokrzyskie Development Forum. “This is an important event for the entire region”

by admin | Sep 22, 2023 | News

In order to use the region’s unique capital and ensure its multidirectional development, it is necessary to cooperate with the world of science, emphasised in a letter to the organisers and participants of the Świętokrzyskie Development Forum:...
Cooperation between science and business. Swietokrzyskie Development Forum launches on Friday

Cooperation between science and business. Swietokrzyskie Development Forum launches on Friday

by admin | Sep 21, 2023 | News

On 22 September, Kielce will host the second Świętokrzyskie Development Forum: Education-Science-Development. Copernicus Academy is the strategic partner of the event. Talks about cooperation between research centres and businesses, an opportunity to exchange good...
New prospects for cooperation. Talks at the Copernicus Academy with a delegation from Taiwan

New prospects for cooperation. Talks at the Copernicus Academy with a delegation from Taiwan

by admin | Sep 21, 2023 | News

A meeting was held at the Copernicus Academy between representatives of the Taiwanese Ministry of Education and the authorities of Taiwanese higher education institutions and representatives of the largest Polish universities, including the University of Wrocław, the...
Prof. Górski: We want the Academy to be a living monument to Nicolaus Copernicus

Prof. Górski: We want the Academy to be a living monument to Nicolaus Copernicus

by admin | Sep 18, 2023 | News

We want to restore Copernicus to its rightful place in the world,” said the Secretary General of the Copernican Academy, Prof. Dr. Krzysztof Górski, at the opening ceremony of the College of Legal Sciences at the Nicolaus Copernicus University in Lublin. Opening...
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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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