# The Third Book ofThe Use of Lines of Numbers, Sines and Tangents, for the drawing of Hour-lines on all sorts of Planes.

There are ten several sorts of Planes, which take their denomination from those Great Circles to which they are Parallels, and may sufficiently for our use be represented in this one Fundamental Diagram, and be known by their Horizontal and Perpendicular Lines, of such as know the Latitude of the Place, and the Circles of the Sphere.

1. An Horizontal Plane parallel to the Horizon, here represented by the outward Circle ESWN.
2. A vertical Plane, parallel to the prime Vertical Circle, which passeth through the Zenith, and the Points of East and West in the Horizon, and is right to the Horizon and the Meridians; that is, maketh right Angles with them both. This is represented by EZW.
3. A Polar Plane parallel to the Circle of the Hour of 6, which passeth through the Pole, and the Points East and West, being right to the Equinoctial and the Meridian, but inclining to the Horizon, with an Angle equal to the Latitude. This is here represented by EPW.
4. An Equinoctial Plane parallel to the Equinoctial, which passeth through the Points East and West, being right to the Meridian, but inclining to the Horizon, with an Angle equal to the Complement of the latitude. This is represented by EAW.
5. A Vertical Plane inclining to the Horizon, parallel to any Great Circle, which passeth through the Points of East and West, being right to the Meridian, but inclining to the Horizon, and yet not passing through the Pole, nor parallel to the Equinoctial. This is here represented either by EIW, or EYW, or ELW.
6. A Meridian Plane parallel to the Meridian, the Circle of the Hour of 12, which passeth through the Zenith, the Pole, and the Points of the South and North, being right to the Horizon, and the prime Vertical. This is here represented by SZN.
7. A Meridian Plane inclining to the Horizon, parallel to any Great Circle, which passeth through the Points of South and North, being right to the prime vertical, but inclining to the Horizon. This is here represented by SGN.
8. A Vertical Declining Plane, parallel to any Great Circle, which passeth through the Zenith, being right to the Horizon, but inclining to the Meridian. This is represented by BZD.
9. A Polar Declining Plane, parallel to nay Great Circle, which passeth through the Pole, being right to the Equinoctial, but inclining to the Meridian. This is here represented by HPQ.
10. A Declining Inclining Plane, parallel to any Great Circle, which is right to none of the former Circles, but declining from the prime Vertical, and inclining both to the Horizon and the Meridian, and all the Hour-circles. This may be represented either by BMD, or BFD, ro BKD, or any such Great Circle, which passeth neither through the Sout and North, nor East and West points, nor through the Zenith, nor the Pole.

Each of these Planes (except the Horizontal) hath two Faces, whereon Hour-lines may be drawn, and so there are nineteen Planes in all. The Meridian Plane hath one Face to the East, and another to the West: The other Vertical Planes have one to the South, and another to the North, and the rest one to the Zenith, and another to the Nadir: but what is said of the one, may be understood of the other.

## To describe the Fundamental Diagram.

The Description of this Diagram is set down at large in the Use of my Sector, Chap. 3. But for this purpose it may suffice, if it have the Vertical Circle, the Hour-circle, the Equator, and the Tropicks first drawn in it, other Circles may be supplied afterwards, as we shall have use of them: And those may be readily drawn in this manner.

Let the outward Circle representing the Horizon be drawn, and divided into four equal parts with SN the Meridian and EW the Vertical, and each fourth part into 90 gr. That done, lay a Ruler to the Point S, and each Degree in the Quadrant EN, and note the Intersections where the Ruler crosseth the Vertical, so shall the Semidiameter EC be divided into other 90 gr. and from thence the other Semidiameter may be divided in the same sort. These may be numbered with 10, 20, 30, &c. from E toward C, and for variety with 10, 20, 30, &c. from C toward W. But for the Meridian, the South part would be best numbered according to the Declination from the Equator, and the North part according to the distance from the Pole.

Then with respect unto the Latitude, which here we suppose to be 51 gr. 30 m. open the Compasses unto 38 gr. 30 m. from C toward W, and prick them down in the Meridian from C unto P, so this point P shall represent the Pole of the World, and through it must be drawn all the Hour-circles.

Having three points E, P, W, find their Center, which will fall in the Meridian a little without the point S, and draw them into a Circle EPW, which will be the Circle of the Hour of 6.

Through this Center of the Hour of 6, draw an occult Line at length parallel to EW, so this Line shall contain the Centers of all other Hour-circles. Where the Circle of the Hour of 6 crosseth this occult Line, there will be the Centers of the Hour-circles of 9 and 3. The distance between these Centers of 9 and 3, will be equal to the Semidiameters of the Hour-circles of 10 and 2: and where these two Circles of 10 and 2 shall cross this occult Line, there will be the Centers for the Hour-circles of 11 and 7, and 5 and 1. Again, divide the distance between the Centers of 10 and 2 into three equal parts, so the feet of the Compasses will rest in two points; the one is the Center of the Hour-circle of 8, and the other the Center of the Hour-circle of 4; and the extent of the Compasses to one of these third parts shall be the true Semidiameter of these Circles, if there be no error committed in finding the other Centers.

The Hour-circles being thus drawn, take 51 gr. 30 m. from C toward W, and prick them down in the South part of the Meridian from C unto A, and bring the third point EAW into the Circle, this Circle so drawn shall represent the Equator.

The Tropick of ♋ is 23 gr. 30 m. above the Equator, and 66 gr. 30 m. distant from the Pole, and so in this Latitude it will cross the South part of the Meridian at 28 gr from the Zenith, and the North part of the Meridian at 15 gr. below the Horizon. Take therefore 28 gr. from C toward W, and prick them down in the Meridian from C unto L, so have you the South Intersection. Then lay a Ruler to the Point E, and 15 gr. in the Quadrant NE, numbered from N toward E, and note where it crosseth the Meridian, so shall you have the North Intersection. The half way between these two Intersections will fall in the Meridian at the point aaaa, and the Circle drawn on the Center a, and Semidiameter aL, shall represent the Tropick of ♋, and here cross the Horizon before 4 in the morning, and after 8 in the evening, about 40 gr. Northward from E and W, according to the Rising and Setting of the Sun at his entrance into ♋.

The Tropick of ♑ is 23 gr. 30 m. below the Equator, and 113 gr. 30 m. distant from the Noth Pole, so that in this latitude it crosseth the South part of the Meridian at 75 gr. from the Zenith, and the North part of the Meridian at 62 gr. below the Horizon. Take therefor 75 gr. from C toward W, and prick them down in the Meridian from C unto T, so have you the South Intersection; then lay a Ruler to the point E, and 62 gr. in the Quadrant NE numbered from N toward E, and note where it crosseth the Meridian, so shall you have the North Intersection. The half way betrween these two Intersections shall be the Center whereon you may describe the Tropick of ♑, and this Tropick will cross the Horizon after 8 in the Morning, and before 4 in the Evening, about 40 gr. Southward from E and W, according to the rising and setting of the Sun at his entrance into ♑.

## To find the Inclination of any Plane.

For the distinguishing of these Planes, we may find whether they be Horizontal, or Vertical, or inclining to the Horizon, and how much they incline, either by the usual Inclinatory Quadrant, or by fitting a Thred and Plummet unto the Sector.

For let the Sector be opened to a Right Angle, the Lines of Sines to an Angle of 90 gr. inward edges of the Sector to 90 gr. and let a Thred and Plummet be hanged upon a Line parallel to the edges of one of the legs, so that Leg shall be vertical and the other leg parallel to the Horizon.

If the Plane seem to be vertical (like the Wall of an upright Building) you may try it by holding the Sector, so that the Thred may fall upon his Plummet-line: For then if the vertical edge of the Sector shall lie close to the Plane, the Plane is erect, and therefore said to be vertical; and if you draw a Line by that edge of the Sector, it shall be a Vertical Line.

If the Plane seem to be level with the Horizon, you may try it by setting the Horizontal Leg of the Sector to the Plane, and holding the other Leg upright; For then if the Thred shall fall on his Plummet-line, which way forever you turn the Sector, it is an Horizontal Plane.

If the one end of the Plane be higher than the other, and yet not vertical, it is an inclining Plane, and you may find the Inclination in this manner.

First hold the Vertical Leg of the Sector upright, and turn the Horizontal Leg about, until it lie close with the Plane; and the Thred fall on his Plummet-line; so the Line drawn by the edge of that Horizontal Leg shall be an Horizontal Line.

Suppose the Plane BGED, and that BD were thus found to be the Horizontal Line upon the Plane, then may you cross the Horizontal Line at Right Angles with a Perpendicular CF: that done, if you set one of the Legs of the Sector upon the perpendicular Line CF, and make the other Leg with a Thred and Plummet to become Vertical, you shall have the Angle between the Vertical Line and the Perpendicular on the Plane, as before in the Use of the Sector, pag. 62. And the Complement of this Angle is the Inclination of the Plane to the Horizon.

## To find the Declination of a Plane.

The Declination of a Plane is always reckoned in the Horizon between the Line of East and West, and the Horizontal Line upon the Plane. As in the Fundamental Diagram, the prime Vertical Line (which is the Line of East and West) is ECW; if the Horizontal Line of the Plane proposed shall be BCD, the Angle of Declination is ECB.

But because a Plane may decline divers ways, that we may the better distinguish them, we consider three Lines belonging to every Plane: The first is the Horizontal Line; the second the perpendicular Line, crossing the Horizontal Line at Right Angles; the third, the Axis of the Planet crossing both the Horizontal Line, and his Perpendicular, and the Plane it self at Right Angles.

The Perpendicular Line doth help to find the Inclination of the Plane, as before, the Horizontal, to find the Declination; the Axis, to give denomination unto the Plane.

For example: In a Vertical Plane in the Fundamental Diagram, represented by FZW, the Horizontal Line is ECW, the same with the Line of East and West, and therefore no Declination. The perpendicular crossing it is CZ, the same with the Vertical Line, drawn from the Center to the Zenith, right unto the Horizon, and there is no Inclination. The Axis of the Plane is SCN, the same with the Meridian Line, drawn from the South to the North, and accordingly gives the denomination to the Plane. For the Plane having two Faces, and the Axis two Poles, S and N, the Pole S falling directly into the South, doth cause that Face to which it is next, to be called the South Face; and the other Pole at N, pointing into the North, doth give the denomination to the other Face, and make it to be called the North Face of this Plane.

In like manner, in the Declining Inclining Plane in the Fundamental Diagram, represented by BFD, the Horizontal Line is BCD, which crosseth the prime Vertical Line ECW, and therefore it is called a Declining Plane, according to the Angle of Declination ECB or WCD. The Perpendicular to this Horizontal Line is CF, where the point F falleth in the Plane QZH perpendicular to the Plane proposed, between the Zenith and the North part of the Horizon; and therefore it is called a Plane inclining to the Northward, according to the ARK FQ, or the Angle FCQ. The Axis of the Plane is here represented by the Line CK, where the Pole K is 90 gr. distant from the Plane, and so is as much above the Horizon H, and the other Pole as much below the Horizon at Q, as the Plane at F is distant from the Zenith: And this Pole K here falling between the Meridian and the prime Vertical Circle into the Southwest part of the World, this upper Face of the Plane is therefore called the South-west Face, and the lower the North-east Face of the Plane.

The declination from the prime Vertical may be found by the Needle in the usual Inclinatory Quadrant, or rather by comparing the Horizontal Line drawn upon the Plane, with the Azimuth of the Sun, and the Meridian Line, in such sort as before we found the Variation of the Magnetical Needle. For take any Board that hath one side straight, and draw as in the last Diagram the Line HO parallel to that side, and the Line ZM perpendicular unto it, and on the Center Z make a Semicircle HMO: this done, hold the Board to the Plane, so as HO may be parallel to BD the Horizontal Line on the Plane, and the Board parallel to the Horizon: then the Sun shining upon it, hold out a Thred and Plummet, so as the Thred being Vertical, the shadow of the Sun may fall on the Center Z; and draw the Line of Shadow AZ, representing the common Section which the Azimuth of the Sun makes with the Plane of the Horizon, and let another take the Altitude of the Sun at the same instant; so by resolving a Triangle, as I shewed before, you may find what Azimuth the Sun was in when he gave shadow upon AZ.

Suppose the Azimuth to be 72 gr. 52 m. from the North to the Westward, and therefore 17 gr. 8 m. from the West, we may allow these 17 gr. 8 m. from A unto V, and draw the Line ZV, and so we have the true West point of the prime Vertical Line: then allowing 90 gr. from V unto S, we have the South point of the Meridian Line ZS, and the Angle HZV shall give the Declination of the Plane from the Vertical, and the Angle OZS the Declination of the Plane from the Meridian.

Or we may take out only the Angle AZH, which the Line of Shadow makes with the Horizontal Line of the Plane, and compare it with the Angle AZV, which the Line of Shadow makes with the prime Vertical. And so here, if AZV the Suns Azimuth shall be 17 gr. 8 m. past the West, and yet the Line of Shadow AZ 7 gr. 12 m. short of the Plane, the Declination of the Plane shall be 24 gr. 20 m. as may appear by the site of the Plane and the Circles.

If the Altitude of the Sun be taken at such time as the Shadow of the Thread falleth on BD or HO, and then a Triangle resolved, the Declination of the Plane will be such as the Azimuth of the Sun from the Prime Vertical.

If at such a time the Shadow falleth on MZ, the Declination will be such as the Azimuth of the Sun from the Meridian.

If it be a fair Summers day, you may first find what Altitude the Sun will have when it commeth to the due East or West, and then expect until he come to that Altitude, so the Declination of the Plane shall be such as the Angle contained between the Line HO and the Line of the Shadow.

Having distinguished the Planes, the next care will be for the placing of the Style, and the drawing of the Hour-lines.

The Style will be as the Axis of the World, sometimes parallel to the Plane, sometimes perpendicular, sometimes cut the Plane with Oblique Angles.

The Hour-lines will be either parallel one to the other, or meet in a Center with equal Angles, or meet with unequal Angles. If the Style be perpendicular to the Plane, the Angles at the Center will be equal; and this falls out only in the South and North Face of the Equinoctial Plane. If the Style be parallel to the Plane, the Hour-lines will be also parallel one to another; and this falls out in all Polar Planes, as in the East and West Meridian Planes parallel to the Circle of the Hour of 12, in the upper and lower direct Polars, parallel to the Circle of the Hour of 6, and in the upper and lower declini9ng Polars, which are parallel to any of the other Hour-circles.

But in the Horizontal and all other Planes, the Style will cut the Plane with an acute Angle, nad the Hour-lines will meet at the root of the Style, and there make unequal Angles.

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