The Third Book of the Sector Containing the Use of the particular Lines.
Having shewed some use of the Lines an the flat sides of the Sector, there remain only those on the edges. And here one half of the outward edge is divided into inches, and numbered according to their distance from the ends of the Sector.. As in the Sector of fourteen inches long, where we find 1 and 13, it sheweth that division to be 1 inch from the nearer end, and 13 inches from the farther end of the Sector.
The other half containeth a Line of lesser Tangents, to which the Gnomon is Radius; They are here continued to 75 gr. And if there be need to produce them farther, Take 45 d, out of the number of degrees required, and double the remainder: so the Tangent and Secant of this double remainder being added, shall make up the Tangent of the degrees required.
As if AB being the Radius, and BC the Tangent Line, it were required to find the Tangent of 75 gr. If we take 45 gr out of 75 gr. the remainder is 30 gr. and the double 60 gr. whose Tangent is BD, and the Secant AD: if then we add AD to BD, it maketh BC, the Tangent of 75 gr. which was required. In like sort, the secant of 61 gr. added to the Tangent of 61 gr. giveth the Tangent of 75 gr. 30 m. and the Secant of 62 gr. added to the Tangent of 62 gr. giveth the Tangent of 76 gr. and so in the rest. The use of this Line may be,
Hold the Sector so as the Tangent BC, may be Vertical, and the Gnomon BA, parallel to the Horizon; then turn the Gnomon toward the Sun, so that it may cat a shadow upon the Tangent, and the end of the shadow shall shew the Altitude of the Sun. So if the end of the Gnomon at A, do give a shadow unto H, it sheweth that the Altitude is 38 gr ½, if unto D, then 60 gr. and so the rest.
There is another use of this Tangent Line, for drawing of the hour Lines upon an ordinary Plane, whereof I will set down these Propositions.
First draw a right Line AC for the Horizon, and the Equator, and cross it at the Point A, about the middle of the Line, with AB another right Line, which may serve for the meridian, and the hour of 12; then take out 15 gr. out of the Tangents, and prick them down in the Equator on both sides from 12: so the one Point shall serve for the hour of 11, and the other for the hour of 1. Again, take out the Tangent of 30 gr., and prick it down in the Equator on both sides from 12: so the one of these Points shall serve for the hour of 10, and the other for the hour of 2. In like manner may you prick down the Tangent of 45 gr. for the hours of 9 and 3, and the tangent of 60 gr. for the hours of 8 and 4, and the Tangent of 75 gr. for the hours of 7 and 5.
Or if any please to set down the parts of an hour, he may allow 7 gr. 30 m. for every half hour, and 3 gr. 45 m. for every quarter. This done, you are to consider the Latitude of the place, and the quality of the Plane. So the Secant of the Latitude shall be the Semidiameter in a Vertical Plane, and the secant of the Complement of the Latitude in an Horizontal Plane.
For example, about London, the latitude is 51 gr. 30 m. and let the Plane be Vertical. If you take AV, the Secant of 51 gr. 30 m. out of the Sector, and prick if down in the Meridian Line from A to V, the Point V shall be the Center and if you draw right Lines from V unto 11, and 10, and the rest of the hour Points, they shall be the hour Lines required.
But if the Plane be Horizontal, then you are to take out AH the Secant of 38 gr. 30 m. for the Semidiameter, and prick it down in the Meridian Line from A unto H; so the right Lines drawn from the Center H unto the hour Points, shall be the hour Lines required; only the hour of 6 is wanting, and that must always be drawn Parallel to the Equator, through the Center V in a Vertical, through the Center H, in an Horizontal Plane.
This being done, if you set the Lines AH, HV, to a right Angle (HAV) the right Line HV the Base of the Triangle shall be the Axis of the style for either Plane.
In a Polar Plane the Equator may be also the same with the Horizontal Line, and the Hour Points may be pricked on as before, but the hour Lines must be drawn Parallel to the Meridian.
In the Meridian Plane, the Equator will cut the Horizontal Line with an Angle equal to the Complement of the latitude of the place; then may you make choice of the Point A, and there cross the Equator with a right Line, which may serve for the hour of 6: so the Tangent of 15 gr. being pricked down in the Equator on both sides from 6, shall serve for the hours of 5 and 7; and the Tangent of 30 gr. for the hours of 8 and 4, and the Tangent of 45 gr. for the hours of 3 and 9; and the Tangent of 60 gr. for the hours of 2 and 10; and the Tangent of 75 gr. for the hours of 1 and 11. And if you draw right Line through the hour Points, crossing the Equator at right Angles, they shall be the hour Lines required.
The Substilar will be the same with the hour of 12 in the Polar Plane, and with the hour of 6 in the Meridian Plane: the Axes of the stile may be Parallel to the Substilar in either Plane according to the distance of the third hour from the Substilar.
 First, draw AV the Meridian, and AE the Horizontal Line, crossing one the other at right Angles in the point A.
 Then take out AV, the Secant of the Latitude of the place, which you may suppose to be 51 gr. 30 m. and prick it down in the Meridian Line from A unto V.
 Because it is a declining Plane, and you may suppose it to decline 40 gr. East ward, you are to make an Angle of the Declination upon the Center A, below the Horizontal Line, and to the left hand of the Meridian Line, because the declination is Eastward, for otherwise it should have been to the right hand, if the Declination had been Westward.
 Take AH, the Secant of the Complement of the Latitude out of the Sector, and prick it down in the Line of Declination from A unto H, as you did before for the Semidiameter in the Horizontal Plane.
 Draw a Line at full length through the Point A, which must be Perpendicular unto AH, and cut the Horizontal Line according to the Angles of Declination, and it will be as the Equator in the Horizontal Plane.
 Take the hour Points out of the tangent Line in the Sector and prick them down in this Equator on both sides from the hour of 12 at A.
 Lay your Ruler, and draw right Lines through the Center H, and each of these hour Points: so have you all hour Lines of an Horizontal Plane, only the hour of 6 is wanting, and that may be drawn through H perpendicular to HA.
 Lastly, you are to observe and mark the Intersection, which these hour lines do make with AE the Horizontal Line of the Plane: and then if you draw right Lines through the Center V, and each of these Intersections, they shall be the hour Lines required.
The Line HF drawn up to the Horizon, and Parallel to the Meridian, will give the Substilar VF: The Line FG drawn perpendicular to VF, and equal to FH, will give VG, the Axis of the stile.
Having drawn a right Line for the Equator as before, and made choice of the Point a, for the hour of 12: you may at pleasure cut off two equal Lines A10 and A2. Then upon the distance between 10 and 2, make an Equilateral Triangle, and you shall have B for the Center of your Equator, and the Line AB shall give the distance from A to 9, and from A to 3. That done, take out the distance between 9 and 3, and this shall give the distance from B unto 8, and from B unto 4: again, from 4 to 11, and from 8 unto1, and also from 8 unto 7. So have you your hour Points, and if you take out the distance B1, B3, B5, &c. You may find the Points not only for the half hour, but also for the quarters.
But if it so fall out, that some of these hour Points fall out of your Plane, you may help your self by the larger tangent, both in the Vertical, and Horizontal Planes.
For if at the hour Points of 3 and 9, in the Scheme of the Horizontal and Vertical Dials, you draw occult Lines Parallel to the Meridian; the distances DC between the hour Line of 6 and the hour Points of 3 and 9, will be equal to the Semidiameter AV in a vertical, and AH in a Horizontal Plane, and if they be divided in such sort as the Line AC is divided, you shall have the Points of 4, and 5, and 7, and 8, with their halfs and quarters.
As in the Horizontal Plane, take out the Semidiameter AH, and make it a Parallel Radius by fitting it over in the Sines of 90 and 90: Then take 15 gr. Out of the larger tangent and lay them down on the Lines of Sines, where they will reach from the Center unto the Sines of 15 gr. 32 m. therefore take out the Parallel Sine of 15 gr. 32 m. and it shall give the distance from 6 unto 5, and from 6 unto 7, in your Horizontal Plane. That done, take out 30 gr out of the larger tangent, and lay them on the Sines, from the Center unto the Sines of 35 gr. 16 m. and the Parallel Sine of 35 gr. 16 m. shall give you tze distance from 6 unto 4, and from 6 unto 8, in your Horizontal Plane. The like may be done for the half hours and quarters.
So also in the Vertical declining Plane. If you first take out the Secant of the declination of the Plane, and prick it down in the Horizontal Line from A untoE, and through E draw right Lines Parallel to the Meridian, which will cut the former hour Lines of 3 and 9, or one of them in the Point C; then take out the Semidiameter AV, and prick it down in these Parallels from C unto D, and draw a right Lines from A unto C, and from V unto D, the Line VD shall be the hour of 6, and if you divide the like Line DC in the Horizontal Plane, you shall have all the hour Points required.
Or you may find the Point D, in the hour of 6, without knowledge either of H or C. For having pricked down AV in the Meridian Line, and AE in the Horizontal Line, and draw Parallels to the meridian through the Points at E, you may take the tangent of the Latitude out of the Sector, and fit it over in the Sines of 90 and 90: so the Parallel Sine of the Declination measured in the same Tangent Line, shall there shew the Complement of the Angle DVA, which the hour Line of 6 maketh with the Meridian; then having the Point D, take out the Semidiameter VA, and prick it down in those Parallels from D unto C: so shall you have the Lines DC and AC to be divided as before.
The like might be used for the hour Lines upon all other Planes. But I must not write all that may be done by the Sector. It may suffice that I have wrote something of the Use of each Line, and thereby given the ingenuous Reader occasion to think of more.
It is well known to many of you, that this Sector was thus contrived, the most part of this Book written in Latine, many Copies transcribed and dispersed more than sixteen years since. I am the last contented to give way that it come forth in English. Not that I think it worthy either of my labor, or the publick view, but partly to satisfie their importunity, who not understanding the Latine, yet were at the charge to buy the Instrument, and partly for my own ease. For as it is painful for others to transcribe my Copy, so it is troublesome for me to give satisfaction herein to all that desire it. If I find this to give you content, it shall incourage me to do the like for my Crossstaff, and some other Instruments. In the mean time bear with the Printers faults, and so I rest.
Gresham College 1 Maij 1623. E. G.
Finis
