# CHAP. IV.To draw the Hour-lines in an Horizontal Plane.

An Horizontal Plane is that which is parallel to the Horizon, represented in the Fundamental Diagram by the outward Circle ESWN, in which the Diameter SN drawn from the South to the North, may go both for the Meridian Line and the Meridian Circle, Z for the Zenith, P for the Pole of the World, and the Circles drawn through P for the Hour-circles of 1, 2, 3, 4, &c. as they are numbered from the Meridian.

These are equal at the Pole, and at the Equator, but unequally distant at the Horizon; the distance between the Meridian and the first Hour being not full 12 gr. the distance between the fifth and sixth Hour about 18 gr. which inequality being observed, if you suppose Right Lines drawn from the Center C to the Intersections of these Hour-circles with the Horizon, the Line so drawn shall be the Hour-lines here inquired. And then, if you can, imagine a Line drawn from the center C, toward P the Pole of the World, and raised above the Meridian Line CN, so as the Angle PCN may be equal to the Latitude of the Place, this Right Line CP shall be the Axis of the Style. And so you have both Style nad Hour-lines ready drawn to your hand. But more particularly to our purpose.

These Hour-lines considered, with the Meridian and the Horizon, do make divers Triangles, PN1, PN2, PN3, in which we have known, first, the Right Angle at N, the North Intersection of the Meridian and the Horizon; secondly, the Side PN, the Ark of the Meridian between the Pole and the Horizon, which is always equal to the Latitude of the Place; thirdly, the Angles at the Pole, made by the Meridian and the Hour-circles, the Angle NP1 being 15 gr. NP2 30 gr. each Hour 15 gr. more than the other, each half Hour 7 gr. 30 m. each quarter 3 gr. 45 m. as in the second Column of this table. And these three being known, we may find the Arcs of the Horizon between the Meridian and the Hour-circles N1, N2, N3, &c. For,

• As the Sine of 90 gr. is to the Sine of the Latitude:
• So the Tangent of the Hour, to the Tangent of the Hour-line from the Meridian.
 Latit. 51 30 Ho. Ang. Po. Arc. Po. Gr. M. Gr. M 12 0 0 0 0 3 45 2 56 7 30 5 52 11 15 8 51 1 15 0 11 50 18 45 14 52 22 30 17 57 26 15 21 6 2 30 0 24 20 33 45 27 36 37 30 31 9 41 15 34 28 3 45 0 38 3 48 45 41 45 52 30 45 34 56 15 49 30 4 60 0 53 35 63 45 57 47 67 30 62 6 71 15 66 33 5 75 0 71 6 78 45 75 45 82 30 80 25 86 15 85 13 6 90 0 90 0

Extend the Compasses from the Sine of 90 gr. to the Sine of the Latitude; so the same Extent shall reach from the Tangent of the Hour, to the Tangent of the Hour-line from the Meridian. Thus the Latitude being 51 gr. 30 m. I extend the Compasses from the Sine of 90 gr. to the Sine of 51 gr. 30 m. and find the same extent to reach from the Tangent of 3 gr 45 m. unto the Tangent of 2 gr. 56 m. for the distance of the first quarter from the Meridian; and from the Tangent of 7 gr. 30 m. unto the Tangent of 5 gr. 52 m. for the half Hour; and from the Tangent of 11 gr. 15 m. to the Tangent of 8 gr. 51 m. for the third quarter; and from the Tangent of 15 gr. 0 m. unto 11 gr. 50 m. for the first Hour; And so the rest, as in the third Column of this Table, under the Title of the Arks of the Plane.

Only when I come to set one Foot of the Compasses to 48 gr. 45 m. for the finding of a quarter past 3, the other Foot will fall out of the Line, and then I may either take out so much as is out of the Line beyond 45 gr. and turn it back into the Line, and it will reach from 45 gr. to 41 gr. 45 m. or I may use cross work, extending the Compasses from the Sine of 90 gr. to the Tangent of 48 gr. 45 m. so the same extent will reach from the Sine of 51 gr. 30 m. to the Tangent of 41 gr. 45 m. And such is the distance of the Line of 3 Hours ¼ from the Meridian.

This done, I come to the Plane, and there according as the Lines do fall in the Fundamental Diagram,

1. I draw the Right Line SN, serving for the meridian, the Hour of 12, and the Substylar.
2. In this Meridian I make choice of a center at C, and there describe an occult Circle representing the Horizon.
3. I find a Chord of 11 gr. 50 m. and inscribe it into this Circle on either side of the Meridian, for the Hours of 11 and 1; in like manner, a Chord of 24 gr. 20 m. for the Hours of 10 nad 2; and a Chord of 38 gr. 3 m. for the Hours of 9 1nd 3: And so for the rest of the Hours, their Halves, and Quarters.
4. I Draw Right Lines through the center, and the Terms of these Chords, and these Lines so drawn are the Hour-lines required.

The Line belonging to the Hour of 6 will be perpendicular to the Meridian, and the Hour-lines before 6 in the Morning, or after 6 in the Evening, may be supplied by continuing their opposite Hour-lines beyond the Center, as the Hour-line of 7 in the Morning continued, will be the Hour-line of 7 in the Evening: And so the rest.

Lastly, I set up the Style over the meridian, so as it may cut the Plane in the Center, and there make an Angle with the meridian equal to the latitude of the Place; so it shall represent the Axis of the World, and be truly placed for casting of the Shadow upon the Hour-lines in an Horizontal Plane.

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