Part One. Chapter I.

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bounded original: "minantur," likely completing "terminantur" from the previous page, they considered it to be spherical. In that sphere, they contemplated either the stars truly existing there, or the projected images of lower bodies and their movements. They wanted to measure the distances of the visible places in that same sphere. Or, to put it better, they wanted to measure the visual angles sent out from here. They treated these angles as if they were projected from the center of that spherical surface. For they considered the Earth to be like a point in relation to such a surface. The measure of an angle is any arc described around the vertex original: "puncto," meaning the point of the angle as its center. Therefore, to obtain the measure of these angles, they chose to measure the circumferences of circles passing through the center of that surface instead of the angles themselves. They called these "great circles" The largest possible circle that can be drawn on a sphere, sharing the same center as the sphere, like the equator on Earth.. When they realized that triangles, or other spherical figures that can be broken down into triangles, were formed by the intersection of several great circles passing through the center of the sphere, a new necessity arose. They needed to establish a doctrine for this type of triangle. The sides of these triangles are always the circumferences of great circles. The angles correspond to the circumferences subtended by them on a great circle whose pole is the point where the angle is formed, as will be explained later. These could rightly be called spherical triangles A figure on the surface of a sphere bounded by three arcs of great circles.. After considering these things carefully, they realized that the investigation of celestial matters could be done safely by measuring both plane and spherical triangles. They recognized that they first had to master the art of these measurements.

When they began examining these triangles, they did not think much about measuring the area or the space within them. They thought areas were of little use in celestial studies. Instead, their only concern was to measure the sides and angles. They saw that in spherical geometry, the measurement of angles could be correctly exchanged for the measurement of the circumferences on which they stand. Therefore, they saw that only circumferences of circles would appear in spherical triangles. In plane triangles, however, they knew a circle could be drawn around any triangle. In such a circle, the circumferences across from the angles relate to the angles themselves. Because of this, they likewise exchanged the measurement of angles for circumferences in plane triangles. Thus, in both types of triangles, they found they only needed to measure circumferences and sides, or the straight lines stretched across them. They finally realized the whole matter could be reduced to this single problem: namely, the division of the cir-