Labels within the armillary sphere: Arctic circle, Pole of the zodiac, Tropic of Cancer, Zodiac, Equinoctial line, Tropic of Capricorn, Antarctic circle, Pole of the world.

SITS HIGHER UPON THE LOFTY IMAGE OF THE WORLD

original: "ANIMVS ALTIOR INCVBAT SVBLIMA GINE MVNDI." This motto suggests that the human mind or soul is elevated when it contemplates the structure of the universe.

Johannes de Monte Regio, commonly known as Regiomontanus (1436–1476), was a celebrated mathematician and astronomer who updated Ptolemy's ancient works.

Previous page translation for continuity:

Translator's note: Large gothic text from the back of the page is visible through the paper, appearing as faint shadows across the middle of the text.

12. 2. Proposition 12 of Book 2 of Euclid’s "Elements." This theorem allows astronomers to calculate distances in triangles that do not have a right angle.

Obtuse triangle

If the perpendicular line ad falls upon the extended line bc (according to Proposition 12 of Book 2), then I say that the square of the side bc In this diagram, bc is the side opposite the wide (obtuse) angle. which spans the obtuse angle is greater than the two squares of the lines ab and ac that enclose that obtuse angle. The difference is equal to twice the rectangle formed by the lines ad and bc In modern algebra, this is a geometric way of expressing the Law of Cosines: a² = b² + c² + 2bc(cos A). Since the cosine of an obtuse angle is negative, the formula adds area to compensate. / ...the power of the lines in respect to these squares... The Latin term "potens" refers to the "power" or area that a line produces when it is squared.

13. 2. Proposition 13 of Book 2 of Euclid’s "Elements." This proposition deals with "acute" triangles, where all angles are less than 90 degrees.