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‘Their research expands the boundaries of human cognition’. The Copernican Prizes were presented on

‘Their research expands the boundaries of human cognition’. The Copernican Prizes were presented on

by admin | Jun 20, 2023 | News

Professors P. James E. Peebles and Barry C. Barish became the first ever recipients of the Copernican Prizes. “This distinction in the field of science, is the realisation of the concept of appreciating the achievements of the most excellent researchers whose...
One million gold for Nobel laureates. Professors James Peebles and Barry Barish winners of the Copernican Prize

One million gold for Nobel laureates. Professors James Peebles and Barry Barish winners of the Copernican Prize

by admin | Jun 19, 2023 | News

The names of the two scientists who will receive this year’s Copernican Prizes have been announced. Professors P. James E. Peebles and Barry C. Barish are the first recipients of the award – the award gala will take place on Tuesday, 20 June at the Royal...
Half a million zlotys for the ‘Polish Nobel Prize’. First ever presentation of the Copernicus Prizes

Half a million zlotys for the ‘Polish Nobel Prize’. First ever presentation of the Copernicus Prizes

by admin | Jun 16, 2023 | News

Next Tuesday (20 June), the Royal Castle in Warsaw will host a ceremony to present the Copernican Prizes, awards for outstanding scientific achievements of groundbreaking importance and international scope. The winners will be revealed on Monday, 19 June at a press...
Taiwan donates $3.5 million to educate Ukraine’s future elites in Poland

Taiwan donates $3.5 million to educate Ukraine’s future elites in Poland

by admin | Jun 16, 2023 | News

Doctoral and postdoctoral students from Ukraine will be scholarship holders at the Nicolaus Copernicus University in Warsaw from October. This is the result of a collaboration between officials in Taipei and Warsaw, which will provide $3.5 million in scholarships for...
‘This year is special’. End of the school year in the municipality of Gzy

‘This year is special’. End of the school year in the municipality of Gzy

by admin | Jun 3, 2023 | News

On 3 June, the Secretary General of the Copernicus Academy, Professor Krzysztof Górski, and the Deputy Minister of Education, Tomasz Rzymkowski, attended the end of the school year in two schools in the municipality of Gzy. On 3 June, the Secretary General of 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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