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For example, given arc fbc, its first right sine was called ch. This is because it is common to both arcs fbc and cd, which together complete the semi-circle circumference. However, the second right sine, or sine of the complement Original: "sinus complementi". This is the origin of the modern term "cosine"., was called cg. This is the sine of arc cb, which is the amount by which arc cf exceeds a quadrant.

The part of the diameter caught between any right sine and the end of that same arc on that diameter (such as gb or hd) they called the sagitta Latin for 'arrow'. In geometry, it is the distance from the center of an arc to the center of its chord, resembling an arrow on a bow. or the versed sine of that arc. Thus, gb is the versed sine of arc bc, and hd is that of arc cd. Likewise, hf is the versed sine of arc fbc. The radius of the larger quadrant was called the total sine The "total sine" is the value of the radius. In early trigonometry, sines were expressed as whole numbers based on a large radius rather than decimals between 0 and 1..

A geometric diagram displays a circle with a horizontal diameter FD and a center E. A vertical radius EB points to the top of the circle. Points A and C sit on the upper circumference, joined by a horizontal chord AC that crosses the vertical radius at point G. From point C, a vertical line CH drops down to the diameter FD, meeting it at point H. Radii EA and EC connect the center to the circumference.

We know now what the Saracens Cavalieri refers here to medieval Arabic mathematicians who preserved and expanded upon Greek trigonometry. meant by the first and second right sine (also called the sine of the complement), the sagitta, and the versed sine. This applies whether the arcs are smaller or larger than a quadrant. We also know the total sine. Learning these names is very necessary for what will be said later, and it is vital they are correctly understood. With these names ready, they constructed a Table of Sines very easily, as these were the halves of the chords provided by the Ancients.

Next came Johannes Regiomontanus Also known as Jo. de Monteregio (1436–1476), an influential German mathematician who helped transition European mathematics into the Renaissance.. While he kept these same names, he saw that the work could be made more convenient. To avoid working with fractions, he realized that the semi-diameter or radius of a circle (such as ed) should not be divided into 60 parts, as the Ancients and Saracens had done. Instead, he understood it as being divided into 6,000,000 small parts. Eventually, noticing that a unit based on powers of ten provided a better shortcut, he preferred the radius ed to be cut into 10,000,000 parts. Thus he built a table showing how many of these small parts correspond to the right sine of any given