Logo Akademii Kopernikańskiej
  • Academy
    • About the Nicolaus Copernicus Academy
      • General Assembly
      • Presidium of the NCA
      • Nicolaus Copernicus Research Center
      • Office of the Nicolaus Copernicus Academy
    • Chambers of the Copernican Academy
      • Chamber of Astronomy and Mathematical-Natural Sciences
      • Chamber of Medical Sciences
      • Chamber of Economics and Management
      • Chamber of Philosophy and Theology
      • Chamber of Legal Sciences
    • Copernican Awards
      • Ambassadors of the Nicolaus Copernicus Academy
    • Collaboration
      • Meet the Partners of the Nicolaus Copernicus Academy
      • Partners in Promotion of the Exhibition
  • World Copernican Congress
    • I World Copernican Congress
      • Programme
      • Establishment of the Nicolaus Copernicus Academy
    • Conference panels
      • Astronomy panel
      • Economics and Management Panel
      • Law Panel
      • Philosophy and Theology Panel
      • Medical Sciences Panel
    • Archive 1st World Copernican Congress
      • YouTube
      • Kopernik TV
      • Chronicle
      • Photo Gallery
  • Around Nicolaus Copernicus
    • Celebration of the birthday of Nicolas Copernicus:
      • 550th anniversary of birth – 1st World Copernican Congress 2023
      • 552nd Anniversary of Birth – Radio Broadcast Inspired by the Exhibition
      • 553rd anniversary of the birth of Nicolaus Copernicus
    • The works of Nicolaus Copernicus
    • Quotes
    • Exhibition
      • Competitions about Nicolaus Copernicus
    • Lectures Along Nicolaus Copernicus’ Trail
      • Lidzbark Warmiński
      • Frombork
  • Publishing House
    • Publishing House
    • Information for Authors
    • New Releases
    • Journals
    • Editorial Committee
  • News
  • Contact
  • Polski
  • Deutsch
  • Français
  • Русский
  • Português
  • Español
  • 中文 (中国)
  • Italiano
Menu
Opening of the College of Economics and Management of Nicolaus Copernicus University

Opening of the College of Economics and Management of Nicolaus Copernicus University

by admin | May 23, 2023 | News

On 24 May 2023, the College of Economic and Management Sciences of the Nicolaus Copernicus University (SGMK) will hold its official opening ceremony at the premises of the Nicolaus Copernicus University at 47a Nowogrodzka Street in Warsaw. The event will be attended...
The exhibition “Nicolaus Copernicus life and work” launches into the world on

The exhibition “Nicolaus Copernicus life and work” launches into the world on

by admin | May 23, 2023 | News

On 23 May 2023, the opening ceremony of the exhibition ‘Nicolaus Copernicus – life and work’ prepared by the Ministry of Foreign Affairs took place at the headquarters of the district branch of the National Bank of Poland in Krakow. In the following...
Prof. Gorski to Poles in Lithuania: Copernicus’ life is an illustration of the four cardinal virtues

Prof. Gorski to Poles in Lithuania: Copernicus’ life is an illustration of the four cardinal virtues

by admin | May 18, 2023 | News

In Vilnius, Polish young people, teachers and Polish scientists active in Lithuania took part in a panel organised by the Ministry of Education and Science. A short and moving lecture was also given by Professor Krzysztof Górski – Secretary General of the...
Prof. Grzegorz Górski met with the chairman of the Italian commission for culture, education, science and sport, Roberto Marti

Prof. Grzegorz Górski met with the chairman of the Italian commission for culture, education, science and sport, Roberto Marti

by admin | May 18, 2023 | News

On 17 May 2023, the coordinator of the expert department of the Nicolaus Copernicus Research Centre, Prof. Grzegorz Górski, held a conversation in Rome with the President of the Committee on Culture and Heritage, Education, Research and Sport of the Senate of the...
Meeting between Prof. Gorski and the Vice-President of the Senate of the Italian Republic

Meeting between Prof. Gorski and the Vice-President of the Senate of the Italian Republic

by admin | May 18, 2023 | News

On 17 May 2023, a representative of the Copernicus Academy, Prof. Grzegorz Górski, met in Rome with the Vice-President of the Senate of the Italian Republic, Gian Mark Centinaio. Prof. Górski’s visit to Italy is aimed at promoting Polish science and culture...
Prof. Grzegorz Górski and Italian politician Simona Baldassarre discussed cooperation at a meeting in Rome

Prof. Grzegorz Górski and Italian politician Simona Baldassarre discussed cooperation at a meeting in Rome

by admin | May 18, 2023 | News

On 17 May 2003, the coordinator of the expert department of the Nicolaus Copernicus Research Centre, Professor Grzegorz Górski, met the Italian politician Simona Baldassarre. The purpose of the meeting was to discuss the possibilities of cooperation between the...
« Older Entries
Next Entries »

Recent Posts

  • Women of Extremes. On the Scientists Working in ExtremeEnvironmental Conditions
  • The Nicolaus Copernicus Academy Joins Our Member’s Family in Prayer
  • People of Science. Algorithms of Celestial Mapping
  • Happy Easter
  • “Women in Science” – Part Two of the Interview with Dr Graça R

Recent Comments

No comments to show.
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]

Akademia Kopernikanska
Do góry
Polityka prywatności
Polityka Cookies
Konkursy
Deklaracja Dostępności
Kontakt
biuro@akademiakopernikańska.gov.pl
office@nca.gov.pl
+48 782 950 350
Kontakt dla mediów
media@akademiakopernikanska.pl
+48 782 950 050
Social Media
  • Follow
  • Follow
  • Follow
  • Follow
  • Follow
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]

Akademia Kopernikanska
Up
Privacy Policy
Cookies Policy
Contests
Accessibility Declaration
Contact
biuro@akademiakopernikańska.gov.pl
office@nca.gov.pl
+48 782 950 350
Contact for media
media@akademiakopernikanska.pl
+48 782 950 050
Social Media
  • Follow
  • Follow
  • Follow
  • Follow
  • Follow
This site uses cookies to provide services at the highest level. By continuing to use the website, you agree to their use in accordance with the privacy policy.