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.
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.
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.
Fig. II.
Earth’s Semi-Disc.
370. Make a Scale of any convenient length (six inches at least) as AC, and divide it into as many equal parts as the semi-diameter of the Earth’s Disc contains minutes, which in this construction of the Eclipse for London in April 1764, is 54 minutes and 57 seconds; but as it wants only 3ʺ of 55ʹ the Scale may be divided into 55 equal parts, as in the Figure. Then, with the whole length of the Scale as a Radius, setting one foot of your Compasses in C as a center, describe the Semi-circle AMB for the northern Hemisphere or Semi-disc of the Earth, as seen from the Sun at that time. Had the Place for which the Construction is made been in South Latitude, this Semi-circle would have been the Southern Hemisphere of the Earth’s Disc.
Axis of the Ecliptic.
371. Upon the center C raise the straight line CH for the Axis of the Ecliptic, perpendicular to ACB.
North Pole of the Earth.