How to limit and draw the Ellipsis.

Having the substylar or proper Meridian drawn upon the plain, in its true position, and also knowing the elevation of the Pole above the plain, you may, upon the substylar line, set out your shorter diameter, and on a line drawn perpendicular thereto, you may set off the longer diameter, in this pro∣portion: Let the longer Diameter be as the Radius, the shorter as the Sine of the plains Latitude taken to the same Radius. Or, the longer semidiameter may be the Radius, and the shorter semidiameter must then be the Sine of the elevation of the Pole above the plain.

Upon these two extream diameters thus limited, may the Ellipsis be described with Elliptical Compasses. Or other∣wise it may be done by two Circles, the one circumscribed, the other inscribed, both divided into like parts, and so points found through which the Ellipsis is to passe, as in the for∣mer direct plains was prescribed, and through those points the Ellipsis may, with a stedfast hand be described. In right Horizons or Polar plains there will be no Ellipsis at all, but it vanisheth into a streight line only.

How to divide the Ellipsis into such requisite parts as the plain shall require.

If you should put in the houres proper to the plain there will be no difficulty, for then the substilar being taken for the line of 12, the houres must be drawn as before was shew∣ed in direct plains. For in this case the Dial is Horizontal, shewing the houre of the day to all places that lie under that Meridian under which the plain it selfe lyeth▪ But

Page  24If you would put in the houres of the place (as most fre∣quently is desired) then you are to work (for the division of the Ellipsis) either by calculation or by protraction. If you calculate for the Ellipsis, you must first forme all the angles at the Pole as the manner is in these kindes of Dials: Then you are to invert the terms of proportion from the common way of calculating ordinary Dials, and say thus:

As the Sine of the Poles elevation above the plain, Is to the Radius;

So is the Tangent of each angle at the Pole, To the Tangent of the angular distances of all the houres from the substylar line of the plain.

Page  25Supposing therefore that you had your Ellipsis before described, and the substilar line set off from the plains Ver∣tical line in its true position, you may from that substylar line (and by the horary spaces found by calculation) set in every houre point into the Ellipsis, by help of any Circle described upon the center A. As you see here are two Circles, the circumscribed and the inscribed, either of which, or any other circle else, will serve to prick on the houres by. The houre points are here set upon the circumscribed Circle, and transferred into the Ellipsis by a Ruler laid to A, the common center of the Ellipsis and of the Circle. This is one way, and is best done by calculation: here you may also put in the halves, and quarters, and halfe quarters of houres, or what other parts you shall best like of.

How to limit the Ellipsis, and divide it, without Calculation.

But if you are desirous to do it rather by protraction, then must you work somwhat as formerly was done, the manner I will in briefe shew, wherein you may see the way both how to describe the Ellipsis, and also how to divide it, all under one.

1. Having prepared all necessary requisites before hand, you must first set off the substilar line in its true position from the Vertical line, which to do, I here suppose already known. So in the following figure you see A C B drawn for the substilar line, or the proper meridian of the plain.

2. Upon this plain, and upon some point of the Substy∣lar line, as at A, as upon a center, describe the Circle B M E, and quarter it, and let the Semidiameter of that Circle be counted as the Radius

Page  263. Upon one of the quarters, set off the Poles elevation as from E to F, and draw A F, and from E to A F take the least distance, as E G, and with that distance, upon the former center A, describe another lesser Circle as C H.

4. From B (where the greater Circle cuts the substylar) set B M equall to the plains difference of Longitude: and from M divide that exterior Circle into 24 equal parts, which signifie the 24 houres; and by a Ruler laid to the common center A, transfer them into the lesser Circle, as M H, 10 I, 6 K, &c.

Page  375. From every point in the great Circle, draw lines paral∣lel to the substylar, and from every point of the lesser Circle draw lines perpendicular to the substylar, or parallel to the longest diameter P A R, and so each couple of these lines shall cut each other at right angles.

6. Note where every line thus drawn from any point of the greatest Circle, meets with the other line which is drawn from the like point in the lesser Circle. For the concourses of these answerable lines are the points here required. That is, Those are the points which shew both where the Ellipsis is to be drawn, and also where it is to be divided into its re∣quisite parts.

Thus in the precedent figure: M O and H O meet in the point O, 10 L and I L meet in the point L, 6 N and K N meet in the point N, and so the rest will meet in their due places. These points shew the way where through the Ellipsis is to be drawn, and the same points shew where the houres are to be marked out. The like may be done for quarters and halfe quarters of houres, or any other division that shall be best liked of.

There are many other wayes to do the same things, but I suppose this to be most expedient.

Note: that

1. The Index in these Elliptical Dials must stand per∣pendicularly upright upon the plain, making right angles with it every way, as it was ordered to do upon the former sort of direct plains.

Page  282. The annual course of the Sun must be limited as for∣merly in the other plains, that is, making the greater semidi∣ameter as a Badius, you must finde the Co-sine of the Poles elevation above the plain. This Co-sine is to be made either as a Tangent of 45 gr. whereof you are only to use 23½: Or else it is to be made a Decimal Scale. By both these you have Tables and Rules how to compleat the Suns annual course, either in degrees of the Signes in the Zodiac, or by the dayes of the 12 moneths.

3. Either the Index must move and the Ellipsis lie still, or contrarily. Every man in this must do as his invention shall best suggest. And that motion must be made either upon the substylar line, or else parallel to it, which way so∣ever it be, it must be precisely and punctually ordered.

4. All other things concerning the time of Sun rising and setting, the Suns Amplitudes and Declinations, may here (in the same manner as before) be inserted either proper to the plain, or proper to the place, as shall be desired.

5. In right Horizons or Polar plains, the Ellipsis closeth quite up into a streight line: and so the division of it is only by the exterior Circles parts projected upon it by those lines that are drawn parallel to the substylar.

Thus far of the Elliptical Dial, as it is to be described upon declining plains whose Indexes stand perpen∣dicular to the plains, and which do not lie under the Meridian of the place but swerve from it.