General Uranometric Directory (Directorium Generale Uranometricum)

understand, which I judge will bring no small pleasure and utility to the minds of students. This will cause those who usually abandon their studies halfway through, discouraged by the extreme difficulty of the subjects, to apply themselves to this study with a more eager spirit.

Therefore, the ancient Hipparchus Hipparchus of Nicaea (c. 190 to 120 BCE) was a Greek astronomer and the father of trigonometry. and other famous mathematicians after him, including Menelaus original: "Mileus Romanus." Menelaus of Alexandria (c. 70 to 140 CE) was a mathematician and astronomer who wrote about spherical geometry. and especially Ptolemy, devoted all their strength to contemplating celestial things. They noted the movements of both the wandering stars planets and the fixed stars with great skill and diligent observation. They noticed that these movements were not regular but followed an uncertain and wandering course, extending both in length and toward the sides of the world This refers to movements in longitude and latitude on the celestial sphere.. In such a diverse variety of moving things that seem to wander without a fixed law, even the most skilled men could not establish anything certain.

Indeed, they might have thought themselves like the ignorant masses. The common people often ask foolish questions about the "ladders" astronomers use to climb the walls of heaven. These people are often as skeptical as they are foolish regarding the wonderful things they hear from astronomers. Perhaps these mathematicians would have thought they were dreaming about such distant bodies rather than understanding any truth, had they not thought of constructing ladders for themselves. These ladders would allow them to ascend to the highest parts of the universe.

They realized that the only way to grasp the progress, stations, approaches, and retreats of bodies at such a great distance was through the help of their eyes. They saw that we can only project visual lines and angles from this small globe of Earth into the Heaven. From straight lines and angles, only triangles or figures that can be broken down into triangles are constantly formed. Therefore, they found the core of this matter: if they discovered a certain rule for measuring triangles, both their sides and their angles, then they would have built a most suitable ladder for themselves to climb to the heavens.

This ladder is invisible to the "blind" common people but visible to those upon whom the light of the mathematical sciences has shone. It is truly a wonder. Led by this necessity, they invented the art of measuring triangles with straight sides (which are usually called plane triangles), as was allowed at the beginning of the matter, which I will soon explain. However, when the visual rays of the observer reach the extreme surface of the world...