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Czarnek: The European Union as a peace formula is a gigantic success, it must be fought for

Czarnek: The European Union as a peace formula is a gigantic success, it must be fought for

by admin | Sep 16, 2023 | News

– As long as Marxism did not get hold of the European Union, it functioned as it should,” said the Minister for Education and Science, Przemysław Czarnek, at the opening ceremony of the College of Legal Studies at the Nicolaus Copernicus University in...
“Is the federalisation of the European Union a threat to Poland and Europe?”. Day III of the conference [TRANSMISSION]

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

by admin | Sep 15, 2023 | News

We invite you to watch the broadcast of the third day of the conference devoted to various aspects of the European Community’s federalisation process and their impact on the stability and development of the European Union. How does the direction of change in the...
“Is the federalisation of the European Union a threat to Poland and Europe?”. Day 2 of the conference [TRANSMISSION]

“Is the federalisation of the European Union a threat to Poland and Europe?”. Day 2 of the conference [TRANSMISSION]

by admin | Sep 14, 2023 | News

We invite you to watch the broadcast of the second day of the conference devoted to various aspects of the European Community’s federalisation process and their impact on the stability and development of the European Union. How does the direction of change in...
Opening of the College of Legal Sciences at the Nicolaus Copernicus University. “We want to restore Copernicus to its rightful place in the world”

Opening of the College of Legal Sciences at the Nicolaus Copernicus University. “We want to restore Copernicus to its rightful place in the world”

by admin | Sep 13, 2023 | News

– The thoughts that went through Nicolaus Copernicus’ head and were put down on paper are immortal. They are still a determinant of how we think about areas of the Universe that he did not even think about, that he could not have imagined. – said...
Opening of the College of Legal Sciences of the Nicolaus Copernicus University [TRANSMISSION]

Opening of the College of Legal Sciences of the Nicolaus Copernicus University [TRANSMISSION]

by admin | Sep 13, 2023 | News

At 7.15 p.m., the opening of the College of Legal Studies of the Nicolaus Copernicus University will take place at the premises of the Regional Branch of the National Bank of Poland in Lublin, at 6 Fryderyka Chopina Street. The ceremony will be attended by 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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