f. 16r The addition of Johannes Regiomontanus. 'When the height of altitude, etc'. Albategnius in his many demonstrations resorts to straight lines and concludes many things from the similarity of triangles. That method seems easy to the intellect, unless the intersections of the planes cause lack of clarity. On account of this, therefore, let us suppose the meridian circle ABGD to be cut by the circle of the horizon along the line BD and by the circle of the equinoctial [i.e. celestial equator] along the line FR; by the parallel of the Sun along the line LP; and let the axis of the World in that meridian be the line ZY. Now let me place the Sun on the horizon [i.e. K] so that I may know the amplitude of the arising [amplitudinem orientis], which they call 'the latitude of sunrise' [latitudinem ortus]. Since, however, the parallel of the Sun cuts the meridian orthogonally, and likewise the horizon cuts the meridian orthogonally, the section of the horizon will also be orthogonal to the parallel at line KD. Thus KD is the versine [sinus versus] of that arc of the horizon which is between the centre of the Sun and the meridian to the north, whence EK will be the sine [sinus rectus] of the complement of the quarter circle, that is to say, of the latitude of sunrise. Then, on account of the similarity of the triangles NEK and XED, you will find that the sine EK of the latitude of sunrise is known. Next, let us imagine the meridian circle to be cut by the almicantarath circle [1] [along] the line through Q, which will cut LP at the point O, from which is dropped to the horizon the perpendicular OH. However, that will be equal to the sine of the altitude of the Sun in this position, which is easily determined. The two triangles OHK and XED will therefore be similar, for the angles X and H are right-angled. However, the external angle K, and the internal angle D are equal on account of the lines NK and XD being parallel; so the ratio EX to XD becomes the same as OH to HK; but EX is the sine of the complement of the latitude of the region, XD the sine of the latitude of the region, and OH the sine of the altitude of the Sun. So the line HK is made known; from which, with the line EK subtracted, that is to say the sine of the latitude of sunrise, the remainder EH is known, which is equal to OS. OS, however, is the sine of the arc in the almicantarath circle, which is the arc between the height of the altitude of the Sun and the height lacking in declination [zenith declinatione carens, i.e. east] which is already known by the parts of which the semidiameter of the great circle in the Sphere is taken to be the whole sine [sinus totus][2]. It remains for it to be made known in such parts as if the semidiameter of the almicantarath circle, that is the line MS, is the whole sine. MS, however, is known in terms of the great parts [i.e. the parts of the great circle] and the line OS in the same parts, for the line MS is the sine of the complement of the altitude of the Sun. Wherefore taking the line MS as the whole sine, the line OS will be known in parts of the same sine. From this, you will be able to do it unaided when the Sun is in the middle of the southern zodiac. End of addition. [1] A circle on the celestial sphere parallel to the horizon, typically one of a series that cut the meridian at equal angular separations; a parallel of altitude [OED]. [2] Because sines were constructed on the basis of the radius of a circle, the whole sine is identified with that radius. So when Regiomontanus constructs a triangle whose hypotenuse is a radius, the length of the line sought is equal to the sine of the angle it subtends.