Mathematical Instruments
Book VIII. Ch. II.

# Of the Construction and Uses of the Declinatory.

Instrument is made of a very even Plate of Brass or dry Wood, in figure of a Rectangle, about one Foot in Length, and seven or eight Inches in Breadth. We draw the Diameter of a Semi-Circle upon it parallel to one of the longest Sides of this Plate, viz. parallel to AB, and we divide this Semi-Circle into two Quadrants, containing 90 Degrees each, which we divide sometimes into half Degrees, the Degrees being both ways numbered from the Point H, as may be seen in the Figure of the Instrument. When this is done, we add an Index I to the said Plate, which turns about the Center G, by means of a turned headed Rivet. On the fiducial Line of this Index we screw a Compass, with the North-Side towards the Center G, and likewise sometimes a small horizontal Dial, whose Hour-Line of 12 turns to the Center G. I shall say no more as to the Construction of this Instrument, it being easy to understand, from what has been said elsewhere in this Treatise.

## The Use of this Instrument in taking the Declinations of Planes.

A Plane is said to decline, when it does not face directly one of the Cardinal Parts of the World, which are North, South, East, and West; and the Declination thereof is measured by an Arc of the Horizon comprehended between the Prime Vertical, and the vertical Circle parallel to the said Plane, if it be vertical, viz. perpendicular to the Horizon; for if a Plane be inclined, it can be parallel to no vertical Circle. And in this Case, the Arc of the Horizon comprehended between the Prime Vertical, and that vertical Circle that is parallel to the Base of the inclined Plane, or else the Arc of the Horizon computed between the Meridian of the Place, and the vertical Circle perpendicular to the Plane, is the Plane’s Declination.

There are no Planes, unless vertical or inclined ones, that can decline; for a horizontal Plane cannot be said to decline, because the upper Surface thereof directly faces the Zenith, and it’s Plane turns towards all the four Cardinal Parts of the World indifferently.

Now, in order to find the Declination of a Plane, whether vertical or inclined, you must draw first a level Line thereon, that is, a Line parallel to the Horizon, and lay the Side AB of the Instrument along this Line: then you must turn the Index and Compass ’till the Needle fixes itself directly over the Line of the Declination or Variation thereof on the Bottom of the Box. This being done, the Degrees of the Semi Circle cut by the fiducial Line of the Index gives the Plane’s Declination towards that Coast shewn by the writing graved upon the Instrument. If, for example, the Index be found fixed upon the 45th Degree, between H and B, and the end of the Needle reflecting the North be directly over the Point S of its Line of Declination; in this Case, the Plane declines 45 Deg. South-westwardIy: but if in the same Situation of the Declinatory, the opposite end or the Needle, reflecting the South, should have fixed itself over the Point S of the said Line of Declination, then the Plane would have declined 45 Deg. North-eastwardly.

Again, If the Index be round between A and H, and the North-end of the Needle over the Point S of its Line of Declination, then the Declination or the Plane will be South-eastwardly; but if in this Situation of the Index, the South-end of the Needle fixes itself over the said Point S, then the Plane will decline North-westwardly.

If the Sun shines upon the Wall or Plane whose Declination is sought, and the time of the Day be known exactly by some good Dial, as the Astronomic Ring Dial, we may find the Declination of the Wall or Plane by means of a small horizontal Dial fastened on the Index, which must be turned ’till the Style of the Dial shews the exact Time of the Day; and then the Degrees of one of the Quadrants cut by the Fiducial Line of the Index, will be the Wall or Plane’s Declination: and by this means may be avoided the Errors caused by the Compass, as well on account of the Variation of the Needle, as because of Iron concealed near the Compass.

When the Sun shines upon a Wall, we may find likewise the Substyle or proper Meridian by means of observing two Extremities of the Shadow of an Iron-Rod, in the manner we have above mentioned, and afterwards the Declination; or else we may draw a meridian Line upon an horizontal Plane near the Wall, which being produced to the Wall, will be a means to find the Declination thereof, as also to find the Variation of the Needle. Now the manner of drawing a Meridian Line is thus:

Draw a Circle upon some level Plane, (suppose this to be represented by the Figure M) and in the Center thereof set up a sharp Style very upright, or else fix a crooked Style in some Place, as A, in such manner, that a Line drawn from its sharp end to the Center of the said Circle be perpendicular to the Plane of the Circle; which you may do by a Square. But before you draw the Circle, it is necessary to know the Length of the Shadow of the Style, that so the Circumference of the Circle may be drawn thro’ the Extremity of the Shadow of the Style observed some time before Noon. Now the Circle being drawn, suppose the Extremity of the Shadow touches the Circumference of the Circle in the Morning at the Point G, and about as many Hours after Noon as when in the Morning you observed the Extremity of the said Shadow in G before Noon, you find the Extremity of the Shadow again to touch the Circumference of the Circle in F; then if the Arc FG be bisected in the Point C, and the Diameter BC be drawn, this Diameter will be a meridian Line.

If you have a mind to find a meridian Line when the Sun is in the equinoctial Line, there is no need of drawing a Circle, for all the Extremities of the Shadow of the Style will then be in a right Line, as ED, which is the common Section of the Equinoctial and the horizontal Plane; and so any right Line, as BC, cutting ED at right Angles, will be a meridian Line.

Thus having drawn a meridian Line, if the Hour-Line of 12 of a horizontal Dial be placed so as to coincide therewith, we may have, the Time of the Day thereby: and therefore if at the same time the Index of the Declinatory be turned so, that the small horizontal Dial fastened thereon shews the same Hour or Part, then the Degrees of the Circumference of the Instrument cut by the Index, will shew the Declination of the Wall or Plane. Or else you may produce the abovesaid meridian Line ’till it cuts the declining Plane, for then it will make two unequal Angles with the horizontal Line drawn upon the Plane, viz. an acute and obtuse Angle, which being measured with all the Exactness possible, the Difference between either of these Angles and a right Angle, will be the Declination of the Plane. For Example, if the acute Angle be 50 Deg. and consequently the obtuse one 130 Deg. then the Difference between either of them and a right Angle, will be 40 Deg. for the Declination of the Plane.

If you have a mind to find the Variation of the Needle, apply one of the Sides of the square Box of the Compass along the meridian Line drawn on the Plane; and when the Needle is at rest, observe how many Degrees the North Point thereof is distant from the Flower-de-luce of the Card; and these Degrees will be the Needle’s Declination or Variation; but this Variation will not last long, for it changes continually. Note, When the Declinations of Planes be taken with a Compass, you must have regard to the Variation of the Needle, in letting it rest over a Line shewing the Variation, which is drawn commonly on the Bottom of the Compass-Box.

## The Use of the Declinatory in taking the Inclinations of Planes.

This Instrument serves to take the Inclinations of Planes, as well as their Declinations, that is, the Angles the Planes make with the Horizon, and for this end there is a little Hole in the Center G, having a Plumb-Line fastened therein.

The 17th Figure shews the manner of taking the Declinations and Inclinations of Planes. The Plane A, of this Figure, whereon the Declinatory is applied, is a vertical Meridional undeclining Plane. The Plane B declines South North-westwardly 44 Degrees. The Plane C, is a direct West one. The Plane D, declines 45 Degrees North-westwardly. And the other Declinations are taken in the same manner, in applying the Side AB of the Declinatory to them, so that the Plane of the Semi-circle be parallel to the Horizon.

Now to measure the Angle of a Plane’s Inclination, you must apply some one of the other Sides of the Instrument to the Plane or Wall, and keeping the Plane of the Semi-circle perpendicular to the Horizon, see what Number of Degrees of the Circumference thereof the Plumb-Line plays upon, for these will be the quantity of the said Angle of Inclination.

If, for example, the Side CD be applied to the Plane E, and the Plumb-Line plays upon the Line GH, then the said Plane will be parallel to the Horizon. But if the Side CA of the Instrument being applied on the Plane F, and the Plumb Line plays, as per Figure, this Plane inclines 45 Degrees upwards. Again, If the Instrument being applied to the Plane G, and the Plumb-Line plays upon the Diameter, then this Plane is vertical. And lastly, If the Side AC, being applied on the Plane H, and the Plumb-Line plays as per Figure, then the Inclination thereof will be 45 Deg. downwards.