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A Jubilee for the Heart of Polish Observational Astronomy: The 200th Anniversary of the University of Warsaw Astronomical Observatory.

A Jubilee for the Heart of Polish Observational Astronomy: The 200th Anniversary of the University of Warsaw Astronomical Observatory.

by admin | Sep 5, 2025 | News

The University of Warsaw Astronomical Observatory marks its bicentennial this year. Its history dates back to 1816, when the Chair of Astronomy was established, and thanks to the efforts of Franciszek Armiński, the observatory building on Aleje Ujazdowskie opened in...
Jubilee of the heart of Polish Observational Astronomy:Celebrating 200 Years of the Astronomical Observatory of the University of Warsaw”

Jubilee of the heart of Polish Observational Astronomy:Celebrating 200 Years of the Astronomical Observatory of the University of Warsaw”

by admin | Sep 5, 2025 | Bez kategorii

This year, the Astronomical Observatory of the University of Warsaw celebrates its 200th anniversary. The history of this distinguished institution dates back to 1816, when the Department of Astronomy was founded. Thanks to the efforts of Franciszek Armiński, the...
Presentation of Awards to Schools of the Laureates of the Nicolaus Copernicus Academy Competition “Nicolaus Copernicus – Father of the Modern Economy” 

Presentation of Awards to Schools of the Laureates of the Nicolaus Copernicus Academy Competition “Nicolaus Copernicus – Father of the Modern Economy” 

by admin | Jul 4, 2025 | News

On June 26, 2025, representatives of the Nicolaus Copernicus Academy – Paweł Derecki and Anna Ostrowska – presented awards to six schools whose students became laureates of the nationwide competition “Nicolaus Copernicus – Father of the Modern...
Results of the Competition: “Nicolaus Copernicus – Father of the Modern Economy”

Results of the Competition: “Nicolaus Copernicus – Father of the Modern Economy”

by admin | Jun 25, 2025 | News

We present the list of schools that have been named winners of the competition “Nicolaus Copernicus – Father of the Modern Economy.” All student teams demonstrated not only creativity but also an interesting and mature perspective on the legacy of Nicolaus...
Visit of the Nicolaus Copernicus Academy Delegation to the Astronomical Observatory in Padua

Visit of the Nicolaus Copernicus Academy Delegation to the Astronomical Observatory in Padua

by admin | Jun 20, 2025 | News

On June 12, 2025, a delegation from the Nicolaus Copernicus Academy, led by Prof. Krzysztof M. Górski, visited the astronomical observatory in Padua, which is part of the National Institute for Astrophysics (INAF – Istituto Nazionale di Astrofisica). The host was...
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Cudzysłów

Kopernikańskie twierdzenia cosinusów dla trójkątów sferycznych

Rozdział XIV

Twierdzenia III i XII

Cudzysłów

Rozważamy trzy punkty A, B, i C na sferze o promieniu R. Jeżeli połączymy je łukami (wzdłuż kół wielkich) to otrzymamy trójkąt sferyczny ABC.
Rozważamy trójkąty, które są prostokątne i mają boki krótsze niż półokrąg, jak na rysunku.

Cudzysłów

III Twierdzenie Kopernika:

W prostokątnym trójkącie sferycznym ABC na sferze o promieniu R (gdzie kąt C jest prosty) zachodzą następujące proporcje pomiędzy długościami boków:

AB / BC = R / BC

Czyli że stosunek przeciwprostokątnej do jednej z przyprostokątnych jest równy stosunkowi promienia do drugiej przyprostokątnej. Znaczy to, że jeżeli znamy dwa boki, to możemy znaleźć trzeci.

[Kopernik, Mikołaj (1473-1543), “Mikołaja Kopernika Toruńczyka O obrotach ciał niebieskich ksiąg sześć”,  Kujawsko-Pomorska Biblioteka cyfrowa, UMK, 1854, Rozdział XIII, str.63-64]

Cudzysłów

Czyli Kopernik dowiódł szczególną wersję twierdzenia cosinusów dla trójkątów sferycznych.  Teraz możemy  założyć że promień sfery R =1. Mierzymy w radianach długość boku (łuku) leżącego naprzeciwko danego kąta jako łuku na sferze (od środka sfery) i mamy:

AB = c         BC = a         AC = b
(tutaj a, b, c są miarami kątów AOB, BOC, AOC w radianach)
Jeżeli kąt C jest prosty, możemy zapisać twierdzenie Kopernika jako
cos c / cos b = cos a

[Kopernik, Mikołaj (1473-1543), “Mikołaja Kopernika Toruńczyka O obrotach ciał niebieskich ksiąg sześć”,  Kujawsko-Pomorska Biblioteka cyfrowa, UMK, 1854, Rozdział XIII, str.63-64]

553 rocznica urodzin Mikołaja Kopernika

Zobacz więcej
Cudzysłów

Mamy też ogólniejsze twierdzenie cosinusów dla trójkątów sferycznych na sferze o promieniu R =1, gdzie kąty α, β, γ są kątami sferycznymi trójkąta ABC.

XII Twierdzenie Kopernika:
cos c = cos a   cos b + sin a   sin b   cos γ

Znaczy to, że jeżeli znamy dwa boki i przynajmniej jeden kąt, to możemy znaleźć trzeci, co jest sferyczną wersją twierdzenia cosinusów dla wszystkich trójkątów płaskich.

[Kopernik, Mikołaj (1473-1543), “Mikołaja Kopernika Toruńczyka O obrotach ciał niebieskich ksiąg sześć”,  Kujawsko-Pomorska Biblioteka cyfrowa, UMK, 1854, Rozdział XIII, str.63-64]

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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 triangle ABC.
We consider right-angled spherical triangles with sides shorter than a semicircle, as shown in the figure.
Cudzysłów
Copernicus’ Third Theorem:

In a right-angled spherical triangle ABC on a sphere of radius R (where angle Cis 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.

[Citation: 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

[Citation: 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.
 

[Citation: Copernicus, Nicolaus (1473–1543), De revolutionibus orbium coelestium, Kujawsko-Pomorska Digital Library, UMK, 1854, Chapter XIII, p. 73]

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