A geometric diagram illustrating trigonometric principles is overlaid on the text. It depicts a circle centered at point e with a horizontal diameter fd and a vertical radius eb forming a quadrant bd. A point c is marked on the arc bd. From c, a line ca is drawn parallel to fd, intersecting the radius eb at point g and the opposite arc bf at point a. Another line ch is drawn from c parallel to eb, meeting the diameter fd at point h. Lines ec and ea are radii connecting the center to the arc.

CHAPTER TWO.

How later scholars reduced the astronomical art of calculation into a better and easier form than the ancients, regarding what pertains to the construction of tables.

When, therefore, later scholars examined the aforementioned form of calculation used by the Ancients, the Saracens Cavalieri uses this term to refer to medieval Arabic mathematicians, such as al-Battani, who were instrumental in transmitting and improving Greek mathematics. are considered to have been the first of all who realized that the work could be completed more quickly by using smaller numbers. They noticed that the proportion between the halves of chords is the same as the proportion between the whole chords. Therefore, they abandoned the Ptolemaic and ancient way of speaking and devised new terms in their own language, as will now become manifest in the accompanying diagram.

Let there be, therefore, a circle, bfd, whose center is e, and diameter fd, upon which the radius be stands perpendicularly. Once any point c is taken on the arc bd of the quadrant A quarter of a circle or its circumference, measuring 90 degrees., let the line ca be drawn through it parallel to fd, intersecting be at g, and the arc bf at a. From that same point c, let ch be drawn parallel to be, falling upon the diameter at h, to which it will also be perpendicular. Finally, let the radii ea and ec be drawn.

The Ancients used to call ac the chord A straight line segment whose endpoints both lie on a circular arc. of the arc abc (which some later called the subtended line, and others the inscribed line). This chord is divided into two equal parts by the radius be at point g. Because of this, these later scholars took only the half of that same chord ac, which is cg. They called this the first right sine original: "Sinus rectus primus" of the arc cb, or of the angle ceb which it subtends at the center.

In the same way, ch, which is the half of the chord of double the arc cd, was called the first right sine of the arc cd, or of the angle ced. Since the radius eb completes a quadrant when added to the arc cd (specifically, the arc bd), the line gc was not only called the first right sine of the arc bc by them, but also the second right sine, or the sine of the complement original: "Sinus complementi" of the arc cd, or of the angle ced.

From this, it follows that ch was not only called the first right sine of the arc cd, or of the angle ced, but also the second right sine, or the sine of the complement of the arc bc. This is because the arc cd, when added to the arc cb, completes the quadrant bd. This terminology holds as long as the chosen arcs are smaller than a quadrant. If they should exceed a quadrant, as...