368. Having collected these Elements or Requisites, the following part of the work may be very much facilitated by means of a good Sector, with the use of which the reader should be so well acquainted, as to know how to open it to any given Radius, as far as it will go; and to take off the Chord or Sine of any Arc of that Radius. This is done by first taking the extent of the given Radius in your Compasses, and then opening the Sector so as the distance cross-wise between the ends of the lines of Sines or Chords at _S_ or _C_, from Leg to Leg of the Sector, may be equal to that extent; then, without altering the Sector, take the Sine or Chord of the given Arc with your Compasses extended cross-wise from Leg to Leg of the Sector in these lines. But if the operator has not a Sector, he must construct these lines to such different lengths as he wants them in the projection. And lest this Treatise should fall into the hands of any person who would wish to project the Figure of a solar or lunar Eclipse, and has not a Sector to do it by, we shall shew how he may make a line of Sines or Chords to any Radius. [Sidenote: Fig. II. How to make a line of Chords. Pl. XII.] 369. Draw the right line _BCA_ at pleasure; and upon _C_ as a Center, with the distance _CA_ or _CB_ as a Radius, describe the Semi-circle _BDA_; and from the Center _C_ draw _AC_ perpendicular to _BCA_. Then divide the Quadrants _AD_ and _BD_ each into 90 equal parts or degrees, and join the right line _AD_ for the Chord of the Quadrant _AD_. This done, setting one foot of the Compasses in _A_, extend the other to the different divisions of the Quadrant _AD_; and so transfer them to the right line _AD_ as in the Figure, and you have a line of Chords _AD_ to the Radius _CA_. _N. B._ 60 Degrees on the Line of Chords is always equal to the Radius of the Circle it is made from; as is evident by the Figure, where the Arch _E_, whose Center is _A_, drawn from 60 on the Quadrant _AD_, cuts the Chord line in 60 degrees, and terminates in the Center _C_. [Sidenote: And of Sines.] Then, from the divisions or degrees of the Quadrant _BD_, draw lines parallel to _CD_, which will fall perpendicularly on the Radius _BC_, dividing it into a line of Sines; and it will be near enough for the present purpose, to have them to every fifth Degree, as in the Figure. And thus the young _Tyro_ may supply himself with Chords and Sines, if he has not a Sector. But as the Sector greatly shortens the work, we shall describe the projection as done by it, so far as Signs and Chords are required. [Sidenote: Fig. II. Earth’s Semi-Disc.]