a little black patch; and then for that day and that night you may perform any problem by that planet in the same manner as you did by a fixed star.

But if you would be very exact you must not only seek the planet's place in the sign for that day, which is its longitude, but you must seek its latitude also in the ephemeris (which indeed in the superior planets, Jupiter, Saturn, Mars, alters but very little for whole months together) and thus set your mark in that point of latitude, or distance from its supposed place in the ecliptic, whether northward or southward, and then go to work your problem by this mark.

I shall give but one instance, which will sufficiently direct to solve all others of the same kind that relate to the planets. On the 3d of April 1723, I find by an ephemeris that the sun is about the end of the 23d degree of aries, Jupiter enters the 8th degree of capricorn, and (if I would be very exact,) I observe also that the latitude of Jupiter that day is 15 minutes or a quarter of a degree to the north: There I make a mark or put on a small black patch on the globe to stand for Jupiter. Then having rectified the globe for the latitude v. c. of London, and for the sun's place, April the 3d, I turn the mark which I made for Jupiter to the eastern edge of the horizon, and I find Jupiter will rise near the south east at a little past one in the morning: He will come to the meridian at a very little past five: He will set near the south-west, about nine in the morning.

Then if I rectify the globe for the zenith, the quadrant of altitude being brought down to the horizon, will tell you what is his altitude, and what his azimuth, at any given hour of the morning by the help of the dial and index. Or his altitude or azimuth being given you may find what it is o'clock.

By this means you may find the hour when the moon will rise and set, together with her southing, or the time of her coming. to the meridian. But let it be noted, that the moon changes her place in the zodiac so swiftly, that she moves through 13 degrees of one sign every day, or thereabout; and therefore you cannot find the precise hour and minute of her rising, setting, southing, &c. upon the globe without much more trouble than most of the other planets will give you, which change their places in the zodiac much more slowly.

Problem XXXVIII. "The day and hour of a solar eclipse being known to find all those places in which that eclipse will be

visible."

By the 13th problem find out at what place the sun is vertical at that hour of the day. Bring that place to the pole or vertical point of the wooden horizon, that is, rectify the globe for the latitude of that place; then the globe being in that situation, ubserve what places are in the upper hemisphere; for if it be a large eclipse the sun will be visibly eclipsed in most of them.

Problem XXXIX. "The day and hour of a lunar eclipse being known, to find by the globe all those places in which the same will be visible."

By problem XIII. find as before at what place the sun is vertical at that hour; then by problem IV. find the antipodes of that place Rectify the globe for the latitude of those antipodes ; thus they will be in the zenith, or in the pole of the horizon; then observe as before what places are in the upper hemisphere of the globe, for in the most of those places the moon will be visibly eclipsed. The reason of rectifying the globe for the antipodes in this problem, is because the moon must be directly opposite to the sun whensoever she is eclipsed.

SECT. XX.-Problems relating to Geography and Astronomy, to be performed by the Use of the plain Scale and Compasses.

IT is supposed that the reader is already acquainted with some of the first and easiest principles of geometry, before he can read with understanding this or any other treatise of astronomy or geography; and it is presumed also that he knows what is a chord, a tangent and a sine, and how to make and to measure an angle either by a line or scale of chords, or sines or tangents, in order to practise the problems of this last section; though a very slight knowledge of these things is sufficient for this purpose. Because several of the following problems will depend upon the altitude, or azimuth of the sun, and in order to obtain these, we sometimes use a pin or needle fixed perpendicularly, on an upright or horizontal plane; therefore the first problem I propose shall be this, (viz.)

Problem I. "How to fix a needle perpendicular on a plane, or to raise a perpendicular style or pointer in order to make observations of a shadow."

Note, Any thing fixed or set up to cast a shadow is called a style.

One way to perform this, is by having at hand a joiner's square, and while one edge of it is laid flat to the plane, the other edge of it standing up will shew when a needle or style is fixed on that plane perpendicularly, if it be applied to the side of the needle.

Note, If you have a little square made of box or any hard wood, one leg being fix, and the other eight or nine inches long, one inch or 1 broad, and an inch thick, with a thread and plummet hanging from the end of one leg, down toward the place where the other leg is joined, as in figure xiv. and a large hole for the plummet to play in it: it will be of use not only to shew you how to erect a needle truly perpendicular; but it will also discover whether any plane be truly smooth, and be horizontal or level, as well as whether any upright plane be exactly perpendicular to the

horizon. Such a square will also be very useful in the practice of any geometrical problems, by drawing one line perpendicular to another with the greatest ease,

Another way to fix a needle perpendicular to any plane, is this; describe a circle as, a, o, d, h, in figure xv. Fix a needle s p in the centre p, then measure from several opposite parts of it, as, a, o, d, h, to the tip of the needle s, and fasten the needle so as that the tips, shall be at equal distance from all those points, then it is truly perpendicular.

Note here, That in most of these practices where a perpendicular needle is required, the same end may be attained by a needle or wire strait or crooked, which may be called a style, set up sloping at random, as in figure XVI. without the trouble of fixing it perpendicular, if you do but find the point p on the plane, which lies perpendicularly under the tip of the style s, and this may be found by applying the edge of the square, described figure XIV. to the tip of the style: Though there are other ways to find this perpendicular point for nice practices in dialing by shadows, which require great exactness.

But take notice here, that if you use this method of a style, set up sloping at random as in figure xvI. then with your compasses you must measure the distance from the tip of the style s to the point p, and that distance must be counted and used as the length of the perpendicular style sp in figure xv. wheresoever you have occasion to know or use the length of it. Observe also, that if the tip of your style (whether strait or crooked) be more than three or four inches high from the plane, you will scarce be able to mark the point of shadow exactly, because of the penumbra or faint shadow which leaves the point or edge of a shadow undetermined.

On a horizontal or level plane you must use a much shorter style when the sun is low, or in winter, because the shadow is long; but in the longest days in summer a four inch style is sufficient, though the shadow at that season be very short all the middle hours of the day. From the tip of the style to the tip of the shadow should never be above six inches distance.

After all, if you have frequent occasion for a perpendicular style to observe a shadow by it, I know nothing easier than to get a small prism of wood, or ivory, or rather of brass, such as is described figure XVII. Let the base be a right angled triangle a c: the line B C an inch: A B two inches: and let the height of the prism, viz. A D or c E be three inches (or near four inches if you please.) By this means you obtain three perpendicular styles of different lengths, according as you want the shadow to be either longer or shorter in summer or in winter. If you set it upon the square side A B D O, your perpendicular style will be B c or o E: if it be в o, then c is the tip of the style, and в marks the

1

point on the plane. If you set it on the square side B co E as it stands in the figure, then A B, or D o is your perpendicular style. Or if you set it on its triangular base A B C, then either a D, or BO OF CE will be your perpendicular style.

This little plain prism has these great advantages in it, viz. That you can set it up in a moment on a perfectly smooth plane, and you are sure it is perpendicular to the plane; and then if you require it to stand there any time, and it should happen to be moved, if you have but fixed and marked its place by the lower edges on the plane, and remember which edge you designed for the style, you may set it exactly in the same position again.

Problem II. "How to take the altitude of the sun by a needle fixed on a horizontal plane, or by any perpendicular style."

In all these practices be sure that your plane be truly level or horizontal, which you cannot well know without some such, instrument as I have described before, figure XIV. which serves instead of a level.

You must apply this instrument or square not only to one, part, but to every part of the plane, wheresoever you can imagine the shadow will fall, to see if it be precisely horizontal or level; for a very small variation from the level will cause a great difference in the length and in the point of shadow; and upon this account there are few window-stools, or any boards or posts fixed by the common work of carpenters sufficiently level for a just observation in astronomy or dialling.

Fix your perpendicular style p s, as in figure xvIII. observe the point of shadow cast from the tip of the styles; draw P c ; then take the height of the style Ps in your compasses; set it perpendicular on PC; draw the line s c on the plane, and the angle c is the sun's altitude, viz. 35 degrees.

Here it is evident that if you suppose c the centre and c p to be the radius, then p s is the tangent of the altitude 35 degrees; for it measures the angle c or the arch P A.. But if you make s the centre, and suppose s P to be the radius of a circle, cp is the tangent of the coaltitude of the sun, viz. 55 degrees, for it is that tangent which measures the angles s or the arch PE

Hence it will follow that if you fix a perpendicular needle, pointer or style, on any horizontal plane, and divide a line, as Pc, according to the scale of tangents, whose radius shall be Ps, beginning at P towards c, and make this line of tangents moveable round the centre P, the shadow of the style will shew you the coaltitude of the sun at any time on that moveable scale of tangents.

Or if the scale of tangents P c be divided on the immoveable horizontal plane itself, and you describe concentric circles on the centre P through every degree of that scale, the shadow of the tip of the style will shew the coaltitude among those circles; for they will exactly represent the parallels of altitude in the heavens.

Note, This is described thus particularly rather for demonstration than use, because when the sun is low the shadow p c will be extended many feet or yards.

Problem III. "To take the altitude of the sun by a style on a perpendicular or upright plane.”

Fix your style A в perpendicular to a flat board as figure XIX. raise your board exactly upright, and turn it to the sun; so that the shadow of the style A D may be cast downward directly perpendicular from the centre a in the line a Q. Then take the length of the style A B in your compasses, and set it on the board at right angles to the line of shadow, from A to B ; draw the line BD; and the angle A D B shall be the sun's coaltitude, (or zenith distance as it is sometimes called) viz. 55 degrees; the tangent of which is A B to the radius DA, and the angle A B D (which is the compliment of it) or 35d. shall be the sun's altitude; the tangent of which is A D to the radius B A.

Or to make this more evident, draw the obscure line D o parallel to ▲ B, i. e. horizontal, and the angle B D O will plainly appear to be the angle of the sun's altitude 35 degrees.

Hence it will follow, that if the line A D be prolonged to a and divided according to the degrees of the scale of tangents, this board or instrument will be always ready to shew the sun's altitude on that scale, the shadow of the style A B turned directly to the sun, when the board is held up and made to stand perpendicular to the horizon.

N. B. This is the foundation of those dials which are made on moveable columns, or on walking canes, which shew the hour of the day by the different altitudes of the sun in the various seasons of the year.

Note, There are several other ways to find the altitude of the sun by a moveable or immoveable upright plane, and a perpendicular style fixed on it. But none of those ways of taking an altitude by the point or end of the shadow are the most commodious and exact for common use; I have chiefly mentioned them to lead the learner into a more familiar and perfect acquaintance with the nature and reason of these operations.

If no regular instrument be at hand to take the sun's altitude, I prefer the following method above any others.

Problem IV. "To find the sun's or any star's altitude by the plain board, thread and plummet."