Mathematical Instruments
Book VIII. Ch. V.

# Of the Construction and Uses of Portable Dials.

## Of the Construction of a Globe.

This Figure represents a Globe, whereon are drawn the Meridians or Hour-Circles. There are divers Sizes of them; the great ones are set up in Gardens, and are of Stone or Wood well painted, and the small ones are made of Brass, having Compasses belonging to them, and may be reckoned among the Number of Portable Dials.

The manner of turning round Balls of any Matter is well known, but if a large Stone-Ball is to be made, that cannot be turned because of it’s Weight: first, you must nighly form it with a Chissel, and then take a wooden or brass Semi-Circle of the same Diameter as you design your Ball. This being done, turn the Semi-Circle about the Ball, and take away all the Superfluities with a Raspe, until the Semi-Circle every where and way just touches the Superficies thereof; afterwards make it smooth with a Pumice-Stone or Sea-Dog Fish’s Skin, &c.

The Globe being well rounded and made smooth, you must take the Diameter thereof with a Pair of Spheric Compasses, viz. such whose Points are crooked, which suppose the right Line AB; this Line is divided into two equal Parts in E by the vertical Line ZN, the upper Point whereof Z, represents the Zenith, and the lower one N, the Nadir. Now set one Point of the Spheric Compasses in E, and extend the other to A, and draw the Meridian Circle AZBN; likewise setting one Foot of your Compasses in Z, with the last Opening describe the Circle AEB, representing the Horizon; and from the Point B to C count 49 Deg. the Elevation of the Pole on the Meridian, and setting one Foot of your Compasses in the Point C, representing the North Pole, extend the other to 41 Deg. on the Meridian below the Point B, and draw the Equinoctial Circle; likewise setting one Foot of your Compasses, opened to the same Distance as before, upon the Point in the Meridian cut by the Equinoctial, you may draw the Hour-Circle of 6 passing thro’ the Poles C and D. By this means the Equinoctial shall be divided into four equal Parts by the Meridian and Hour-Circle of 6; and if each of these four Parts be divided into six equal Parts, for the 24 Hours of a Natural Day, and about the Points of Division as Centers, with the extent of a Quadrant of the Globe, Circles be described; these will all pass thro’ the Poles of the World C and D, and are the Hour-Circles. If you have a mind to have the half Hours or Quarters, each of the Divisions on the Equinoctial must be divided into 2 or 4 equal Parts. The Hour-Circles are numbered round the Equinoctial both above and below it, as appears per Figure.

If the Parallels of the Signs are to be drawn upon the Globe, you must count Upon the Meridian both ways from the Equinoctial, the Declination for every Sign, according to the Table expressed; as, for Example, For the two Tropicks you must count 23 Deg. 30 Min. from the Equinoctial, and about the Poles C and D, draw Circles on the Globe. Note, The two Polar Circles must be drawn at 23 Deg. 30 Min. from the Poles, or 66 Deg. 30 Min. from the Equinoctial.

The Globe thus ordered must be placed upon a Pedestal proportionable to the bigness thereof in a Hole made in the Nadir N, distant from the Pole the Complement of it’s Elevation (viz. 41 Deg.) and fixed in a Garden, or elsewhere, well exposed to the Sun, so as to be conformable to the Sphere of the World.

But if it be a small Portable Globe, we place a little Compass upon the Pedestal thereof, that so the Globe may be set North and South when the Hour of the Day is to be shewn thereby, which is shewn thereon without a Style, by the Shadow of the same Globe: for the Shadow or Light thereon always occupies one half of the Globe’s Convexity, when the Sun shines upon it; and so the Extremity of the Shadow or Light shews the Hour in two opposite Places. If, moreover, the different Countries on the Earth’s Supersicies, as likewise the principal Cities, are laid down upon the Globe according to their true Latitudes and Longitudes, you may discover any Moment the Sun shines upon the same, by the illuminated part thereof, what Places of the Earth the Sun mines upon, and what Places are in Darkness. The Extremity of the Shadow shews likewise what Places the Sun is rising or setting at; and what Places have long Days, and what have short Nights: you may likewise distinguish thereon the Places towards the Poles that have perpetual Night and Day. All this is easy to be understood by those who are acquainted with the Nature of the Sphere. Note, This Dial is the most natural of all others, because it resembles the Earth itself, and the Sun mines thereon as he does on the Earth.

You may find the Hour of the Day otherwise, by means of a thin brass Semi-Circle divided into twice 90 Deg. adjusted to the Poles or Extremes of the Axis, by help of two little Ferrils. This Semi-Circle being turned about the Globe with your Hand, until it only makes a perpendicular Shadow upon the Globe, represents the Hour-Circle wherein the Sun is, and consequently shews the Hour of the Day, and also what Places of the Earth it is Noon at that Time. But in this Case the Number 12 must be set to the Meridian, and the Numbers 6 and 6 to the two Points wherein the Equinoctial cuts the Horizon: and this is the reason why we commonly place two rows of Figures along the Equinoctial. The Shadow of the two ends of the Axis, if they are continued out far enough beyond the Poles, and the Hours are figured round the Polar Circles, will likewise shew the Hour. Note, In order to make small Portable Globes universal, we adjust Quadrants underneath them, that so the Pedestal may be slid according to the Elevation of the Pole. This is easy to be understood.

## The Construction and Use of the Concave and Convex Semi-Cylinder

These Dials, which are made of different bignesses, the small ones of Brass, and the great ones of Stone or Wood, are very curious on account of their shewing the Hour of the Day without a Style. Their Exactness consists very much in being very round and even both within side and without.

The 13th Figure represents one of these Dials, set upon and fastened on it’s Pedestal, inclining to the Horizon under an Angle equal to the Elevation of the Pole, and directly facing the South: and therefore the Hour-Lines and the Edges AB, ab, serving as a Style, are all parallel between themselves, and to the Axis of the World. The whole Convex Cylinder is divided into 24 equal Parts, or twice 12 Hours, by parallel Lines; and the Concave Semi-Cylinder is divided into 6 equal Parts by Right Lines, which are the Hour-Lines from 6 in the Morning, to 6 in the Afternoon.

Now when the Sun shines upon this Dial, the Hour of the Day is shewn on the Convex side thereof, by the defect of Light, that is, by a right Line separating the Light from the Shadow. But the Hour of the Day is shewn in the Concave part of the Dial, by the Shadow of one of the Edges AB or ab; so that when the Sun in the Morning is come to the Hour-Circle of 6. the Shadow of the east Edge ab will then fall upon the other Edge AB, which is the Hour-Line of 6: and as the Sun rises higher above the Horizon, the Shadow of the said edge ab will descend and shew the Hour among the Hour-Lines. (Note, The Figures on the Top are for the Morning Hours, and those on the Bottom for the Afternoon ones.) When the Sun is come to the Meridian, he directly shines into the Dial, and then the Edges will cast no Shadow: but when the Sun has passed the Meridian, and descends westwards, the Shadow of the opposite Edge AB will shew the Hour from 12 to 6 in the Evening. If you have a mind to have the Halves and Quarters of Hours, you need but double or quadruple the Divisions.

Small Dials of this kind have Companies belonging to them, that so the Dials may be set North and South.

## The Construction and Use of the Vertical Cylinder

This is a vertical Dial drawn upon the Superficies of a Cylinder by means of a Table of the Sun’s Altitude above the Horizon at every Hour, when he enters into every 10th Degree of the Signs, according to the Latitude of the Place for which the Dial is to be drawn; and for this end the following Table is calculated for 49 Degrees of Latitude.

A Table of the Sun’s Altitudes for every Hour of the Day at his Entrance into every 10th Degree of the Signs, for the Latitude of 49 Degrees.
Hours
Signs XII

XI
I
X
II
IX
III
VIII
IV
VII
V
VI
VI
V
VII
30 ♋︎ 64° 30′ 61° 56′ 55° 19′ 46° 35′ 37° 1′ 27° 10′ 17° 30′ 8° 21′
20 10 64° 9′ 61° 33′ 55° 1′ 46° 18′ 36° 42′ 26° 54′ 17° 10′ 8° 4′
10 20 63° 2′ 30° 31′ 54° 1′ 45° 28′ 35° 5′ 26° 6′ 16° 20′ 7° 12′
♊︎ ♌︎ 61° 12′ 58° 49′ 52° 34′ 44° 7′ 34° 39′ 24° 50′ 15° 6′ 5° 50′
20 10 58° 48′ 56° 30′ 50° 29′ 42° 14′ 32° 53′ 23° 6′ 13° 20′ 3° 57′
10 20 55° 52′ 53° 42′ 47° 57′ 39° 55′ 30° 41′ 20° 57′ 11° 11′ 1° 40′
♉︎ ♍︎ 53° 30′ 50° 30′ 45° 1′ 37° 14′ 28° 10′ 18° 28′ 8° 40′
20 10 58° 51′ 46° 48′ 41° 44′ 34° 13′ 25° 19′ 15° 43′ 5° 44′
10 20 44° 58′ 43° 12′ 38° 15′ 31° 0′ 22° 18′ 12° 48′ 2° 59′
♈︎ ♎︎ 41° 0′ 39° 20′ 34° 37′ 27° 38′ 19° 9′ 9° 47′
20 10 37° 2′ 35° 26′ 30° 58′ 24° 15′ 15° 58′ 6° 42′
10 20 33° 9′ 31° 40′ 27° 24′ 20° 55′ 12° 51′ 3° 44′
♓︎ ♏︎ 29° 30′ 28° 4′ 23° 58′ 17° 42′ 9° 50′ 0° 54′
20 10 26° 8′ 24° 46′ 20° 51′ 14° 45′ 7° 6′
10 20 23° 12′ 21° 52′ 18° 5′ 12° 12′ 4° 43′
♒︎ ♐︎ 20° 48′ 19° 30′ 15° 48′ 10° 3′ 2° 42′
10 10 18° 48′ 17° 44′ 14° 6′ 8° 27′ 1° 13′
20 20 17° 52′ 16° 38′ 13° 3′ 7° 27′ 0° 19′
♑︎ 30 17° 30′ 15° 15′ 12° 42′ 7° 8′

## We now proceed to shew the Construction of the aforesaid Dial upon a Plane which afterwards may be made Cylindrical, or wrapped round a Cylinder; or this Dial may be made upon the Surface of a Cylinder itself, if the Lines be drawn thereon as upon a Plane.

Describe the Right-angled Parallelogram ABCD upon a brass Plate or Sheet of Paper, whose Breadth AB or CD let be nearly equal to the Circumference of the Cylinder it is to be wrapped round, and prolong the Line AB, upon which assume AE for the Length of the Style, which shall determine the Length of the Cylinder. Then about the Point E, as a Center, with the Radius EA, make a circular Arc equal to the Sun’s Meridian Altitude at his entrance into Cancer, and draw the occult Line ED, determining the Length or Height of the Cylinder AD; but if this length was given, and the Length of the Style required, you must describe an Arc about the Point D, equal to the Complement of the Sun’s greatest Meridian Altitude, which, if the greatest Altitude be 64 Deg. 30 Min. will be 25 Deg. 30 Min. and draw the occult Line DE, which shall determine the Length of the Style EA, proportioned to the Length of the Cylinder.

This being done, divide the Arc AF into Degrees and Minutes, and draw occult Lines thro’ each of the Points of Division, from the Center E to the Line AD, that so this Line may be made a Scale of Tangents. But this Line may be otherwise divided, by supposing the Radius AE 100 or 1000 equal Parts, according to the Length of the Cylinder, and taking the correspondent Tangents from printed Tables, and laying them off from A.

Things being thus ordered, divide the Sides AB, DC, into 6 equal Parts, and join the points of Division by five parallel right Lines, which will represent the beginnings of the twelve Signs; then trisect each of these parallel Spaces for the 10th and 20th Degree of each Sign. Now by this means the Beginnings of the Months may be set upon your Dial, because there will be no sensible Error in fixing the Sun’s Entrance into every Sign the 20th Day of every Month (N. S.). Then to prick down the Hour-Points upon all these Lines one after another, you must use the foregoing Table: for example, to prick down the Hour-Point of 10 in the Morning, or 2 in the Afternoon, upon the Line AD representing the Summer Tropick, you will find by the Table, that the Sun’s Altitude at the Time of the Summer Solstice at the Hours of 10 or 2, is 55 Deg. 19 Min. therefore you must take the Tangent of 55 Deg. 19 Min. from your Scale of Altitudes AD, and lay off from the Side AB upon the said Tropick, and then you will have a Point therein thro’ which the proposed Hour-Line must pass. Again, To prick down the Hour-Point of the said Hour of 2 upon another Parallel, suppose on that of the 1st Degree of Leo or Gemini, you will find by the Table that the Sun’s Altitude will then be 52 Deg. 34 Min. and the Tangent of these Degrees being taken from the Scale of Altitudes AD, and laid off upon the said Parallel from AB, will give a Point therein thro’ which the Hour-Line of 2 must pass. And if you proceed in this manner, and find Points in the other Parallels, and likewise on their Divisions of every 10th and 20th Degree; these Points joined will give the curved Hour-Line of 10 in the Morning, or 2 in the Afternoon.

And thus likewise may be found Points in the Parallels thro’ which the other Hour-Lines must pass; which being done, you must join all those belonging to the same Hour by an even Hand, and mark the Characters of the Signs, the first Letters of the Months, as likewise the Hour-Figures, each in their respective Places, as per Figure, and your Dial will be finished; which afterwards must be wrapped about the Cylinder, or bent Cylindrically, so that the Lines representing the two Tropicks be parallel between themselves.

The Style is fastened to a Chapiter serving as an Ornament, and must be moveable on the Line AB, that so it may be placed at right Angles on the Degree of the Sign or Day of the Month. This Dial being placed upright, or hung by a Ring, turn it to the Sun, so that the Shadow of the Style may fall down right upon the Parallel of the Day you desire to know the Hour in, and then the Extremity thereof will shew the Hour or Part.

The Sun’s Altitude may be shewn likewise by this Instrument thus: Put the Style upon the Scale of Altitudes, keeping the Cylinder suspended or horizontally placed, and turn it about so that the Style be towards the Sun; then the Shadow of the Extremity thereof shall shew the Sun’s Altitude above the Horizon.

The abovesaid Parallelogram may serve likewise as a Dial, without being wrapped round a Cylinder, or turned up cylindrically, if the Style be so adjusted as to slide along the Line AB, that so it may be set over the Day of the Month, or Parallel of the Sign the Sun is in. This is easily done, in making a little Slit along the Top of the Plate, and flatting the Foot of the Style, so that it may Hide in the said Slit without varying it’s Length. Now if this Parallelogram be placed upright, and the Line AB level (which may be easily done by means of a Plumb-Line fastened to one of the Sides), and you hold it thus in your hand, or suspend it by a Ring, so that it be directly exposed to the Sun, and the Shadow of the Style falls upon the Parallel of the Sign or Month; then the Extremity of the Shadow of the said Style will fall upon the Hour.

## The Construction and Use of a Dial drawn on a Quadrant.

This Figure represents a Portable Dial drawn on a Quadrant, whose Construction we have thought fit to lay down here, since it is made, as well as the Cylindrical Dial, by means of a Table of the Sun’s Altitude calculated for the Latitude of the Place the Dial is made for.

### The Use of this Quadrant.

Direct the Plane of the Instrument towards the Sun in such manner, that his Rays may pass thro’ the Holes of the Sights GG, and then the Plumb-Line freely playing, will shew the Time of Day by intersecting the Parallel that the Sun is in. But if you put a little Bead or Pin’s Head upon the Plumb-Line, then you may extend the Thread from the Center, and slide the Bead thereon, and fix it over the Degree of the Sign or Day of the Month, and holding up the Quadrant, as before, the Bead will fall upon the Hour of the Day.

## The Construction and Use of a Particular right-lined Dial.

### The Use of this Dial.

Slide the Cursor, and fix the Hole in which the Thread is fastened over the Degree of the Sign the Sun is in, or the Day of the Month; then slip the Bead or Pin’s-head along the Thread, until it be upon the Hour-Line of 12. This being done, hold up your Instrument, lifting it higher or lower ’till the Sun shines thro’ the Holes of the Sights R and S, and the Thread freely plays upon the Plane thereof; then the Bead will fall upon the Hour of the Day.

## The Construction and of an Universal right-lined Dial.

This right-lined Dial, which serves for all Latitudes, is made of a bigness at pleasure, upon a very even Plate of Brass or other solid Matter. The Construction of it is thus: Draw the Lines AB, CD, cutting each other at right Angles in the Point E, about which, as a Center, describe the Quadrant AF, which divide into 90 Deg. and with the Point E for the Vertex, make a Trigon of Signs according to the Method explained in Chap. 2. Divide each Sign into 3 Parts, each being 10 Deg. and set the first Letters of the Months to the Places corresponding to them, by supposing (as we have already) that the Sun’s Entrance into every Sign is the 20th Day of the Month (N. S.); for Example, his Entrance into ♈︎ the 20th of March, his Entrance into ♉︎ the 20th of April, &c This may be without any sensible Error in so small an Instrument. Now draw dotted Lines from the Center E thro’ the Divisions of the Quadrant AF, to the Line AG, which will divide it into Points, from which Parallels must be drawn to the Line AB, which shall be the different Latitudes or Elevations of the Pole, which must be only marked between the two Tropicks, as you see in the Figure, wherein they are drawn to every 5th Deg. On both sides the Point B lay off upon the Line BH, the Divisions that the Radii of the Signs of the Trigon make on the Line aa, representing the Latitude of 45 Deg. that so the Representation of another Zodiack may be made upon the Line BH.

Now the manner of drawing the Hour-Lines upon this Dial is thus: Draw Lines thro’ every 15th Deg. of the Quadrant AF, parallel to ED, which is the Hour-Line of 6; and these Parallels will be the Hour-Lines from 6 in the Evening to 6 in the Morning AL being the Hour-Line of Midnight. And if the parallel Spaces, be laid off on the other side of the Hour-Line of 6, you will have the Hour-Lines from 6 in the Morning to 6 in the Evening. And for drawing the half Hours, divide each 15th Deg. of the Quadrant AF into half, and draw Parallel Lines between the Hour-Lines.

The Hour-Lines may be yet otherwise drawn, by means of a Circle, whose Diameter is the Line AB, and whose Circumference is divided into 24 equal Parts for the 24 Hours of the Day, or into 48, for the Half-Hours. For then if right Lines be drawn thro’ the opposite Points of Division, parallel to ED, we shall have the Hour-Lines, and those of the Half-Hours, as we have said in the Construction of the former right-lined Dial.

About the Point I, as a Center, draw an occult Quadrant, which divide into 90 Deg. and laying a Ruler to the Center I, and on each Division mark the same Degrees upon the Sides GQ, and GS of the Instrument. Note, By means of these Divisions we may find the Sun’s Altitude above the Horizon, as we shall shew by and by. RR are two Sights fixed on the Side GH. And the Piece K is a small Arm or Index, made of 3 Blades of Brass, so joined to each other by headed Rivets, that they may have a Motion either to the right or left: at the sharp end of this Ann is made a very little Hole, thro’ which goes a Thread with a Plummet at the end thereof, and a little Bead or Pin’s Head thereon. This little Arm is fastened to the Instrument with a headed Rivet, that so it may have a Motion at the Place K.

### The Use of this Dial.

If the Hour of the Day be to be found by this Instrument, you must adjust the End a of the Index on the Intersection of the Line of the Latitude of the Place, and the Degree of the Sign the Sun is in, or the Day of the Month; then extend the Thread, and slide the Bead to the same Degree of the Sign in the little Zodiack, drawn on the Hour-Line of 12BI. This being done, hold the Instrument up until the Sun shines thro’ the Sights RR, and the Thread freely playing upon the Plane of the Instrument, the Bead will fall upon the Hour of the Day.

If the Time of the Sun’s rising and setting in all the Signs of the Zodiack for the Latitudes denoted upon the Instrument be required, fix the End a of the Index on the Intersection of the Line of the Latitude of the Place, and the Degree of the Sign the Sun is in; then the Thread freely falling parallel to the Hour-Lines, will shew the Hour of the rising and setting of the Sun. For Example, The End of the Index being fixed on the Intersection of the Sign of ♋︎, and the Line of the Latitude of 49 Deg. the Thread will fall along the Hour-Line of 4 in the Morning, or 8 in the Evening: and this shews, that about the 20th of June, (N. S.) the Sun rises at Paris, at 4 in the Morning, and sets at 8 in the Evening, and so of others.

The Elevation of the Pole is found thus: Place the End of the Index on the Point I, and raise or lower the Instrument until the Sun’s Rays pass thro’ the Holes of the Sights; then the Thread freely playing, will shew the Sun’s Altitude upon the Degrees on the side QS or QG.

All these kinds of Dials, that shew the Hour of the Day by the Sun’s Altitude, are convenient in this, that they shew the Time of Day without a Compass; but their common Imperfection is, that about Noon the Hour cannot be exactly determined by them, unless by several Observations to know whether the Sun increases or decreases in Altitude, and consequently whether it is before or after Noon.

## The Construction of an horizontal Dial for Several Latitudes.

This Dial, which is made upon a very even and smooth Plate of Brass, or other solid Plate 24, Matter, hath a little Piece of Brass in form of a Bird, the lower part of which is adjusted in two little knuckles, that so it may be rendered moveable, and lie down upon the Plane of the Dial. This Bird is kept upright by means of a Spring that is underneath the Dial-Plate, which going thro’ a little square Hole in the Plate, keeps the Bird firm upon it’s Foot. There is a Style or Axis going into the Bird, which is double, the lower End. of which goes into a little Knuckle at the Center of the Dial, that so the said Style may be raised or lowered, according to the Latitude. There is on the Style a circular Arc, whereon the Degrees are set down from 35 or 40 to 60. There is a Slit made along this divided Arc, passing by the Eye of the Bird, that so it’s Bill may be set to the Degree of the Pole’s Elevation, and fixed there. The Dial-Plate is hollowed in circular, that so a Compass may be added thereto, fastened underneath by two Screws. The Needle and the Glass covering it, are placed in the same manner as in other Compasses, of which we Have already spoken.

The Surface of this Dial is divided into 4 or 5 Circumferences for the like Number of different Latitudes, according to some one of the Methods before laid down for drawing of horizontal Dials, whereof that by the Calculation of Angles is most in use for such small Dials as these. They may be drawn also by means of a Plat-form, upon which are several Dials divided by the Rules before given. But this is well known to the Instrument-Maker.

The outmost Circumference, which is divided for 55 Deg. of Latitude, may well enough serve for those Places contained between the 58th and 53d Deg. of Latitude. The second, which is divided for 50 Deg. of Latitude, may serve for Places contained between the 53d and the 47th Deg. of Latitude. The third, which is divided for 45 Deg. may serve for Places between the 47th and 42d Deg. And the fourth, which is divided for 40 Deg. serves for Places contained between the 42d and 38th Deg. of Latitude.

When a 5th Dial is drawn upon the Plate for the Latitude of 35 Deg. this serves for all Places contained between the 37th and the 32d Deg. of Latitude. Now by means of a good Map of the World, or Globe, you may see at what Places these Dials will be in use; for that which is made for one Latitude, will serve for all Places round about the Earth, having the same North and South Latitude. We commonly grave underneath the Dial a Table of the principal Cities of the World with their Latitudes and Longitudes; that so the convenient Circumference on the Plate may be chose, and the Axis of the Dial raised to the proper Elevation of the Pole.

### The Use of this Dial.

To find the Hour of the Day, raise or lower the Style, so that the End of the Bill of the little Bird may answer to the Degree of the Elevation of the Pole marked on the Style, as at Paris against the 49th Degree. The Style being thus raised, place the Dial parallel to the Horizon, that is, level, and turn it so to the Sun ’till the North Point of the Needle usually marked with a little Ring, fixes itself over the Line of Declination, whereon is a Flower-de-luce, and North is writ. Then the Shadow of the Style will shew the Hour of the Day upon the Circumference divided for the Latitude of the Place. You must take care not to see the Dial near Iron, for this changes the Direction of the Needle.

## The Construction and of a Ring Dial.

Take a very round Ring of Brass, or other solid Matter, about two Inches in Diameter, Four or five Lines in breadth, and of a convenient Thickness, and assume the Point A at pleasure thereon (whereat there is a little Hole), about which, as a Center, describe a Quadrant ADC, which divide into 90 Degrees. Then find the Sun’s Altitudes in the foregoing Table at every Hour when he is in the Equinoctial for the Latitude of Paris, and laying a Ruler from the Center A thro’ those Altitudes assumed on the Quadrant, you may draw Lines which will divide the concave Surface of the Ring into the Hour-Points. Now this Dial will be very good for the Times of the Equinox, it being suspended by the Ring B, so that the Line AD is upright.

But one of these Dials may be made for shewing the Hour of the Day at any other time of the Year, if the Hole A be made moveable. For doing which, make the Arcs AE, AI, 23 Deg. for the Signs ♉︎, ♍︎, ♏︎, and ♓︎, AF, AK, 40 Deg. 26 Min. for the Signs ♊︎, ♌︎, ♒︎, and ♐︎, and the Arcs AG, AL, 47 Deg. for the Signs ♋︎ and ♑︎. (The reason why we assume these Arcs double, is, because Angles at the Circumference are but half those at the Center.) Now by this means we shall have a Kind of Zodiack upon the convex Surface of the Ring, whereon must be marked the Signs in their proper Places, or else the first Letters of the Months, that so the Hole A may be put to the Degree of the Sign, or the Day of the Month.

You must describe likewise 7 Circles in the concave Surface of the Ring, whereof that in the Middle will be for the Equinoctial, and the others for the other Parallels. This being done, about the Points A, E, F, G, I, K, L, as so many Centers, describe Quadrants of 90 Deg. upon which Quadrants assume the Altitudes of the Sun every Hour when he is in every of the Signs, and produce the Radii drawn from the Centers to the Points of Assumption, until they cut the Circumferences in the concave Part of the Ring, and you will have Points thereon for the Hour-Lines, which must be joined.

Note, These Divisions may be separately drawn, and afterwards transferred on the Ring.

### The Use of this Dial.

Place the moveable Hole at the Degree of the Sign wherein the Sun is; then holding the Ring suspended, turn it towards the Sun, so that his Rays passing thro’ the Hole A, may fall upon the convenient Circumference of the Sign in the concave Part of the Ring, and then you will have the Hour of the Day shewn.

## To describe the Hour-Lines upon another sort of Ring.

The fourth Figure represents this Ring compleat, and the Parallelogram ABCD, represents it laid open or stretched upon a Plane, that so the Hour-Lines may be pricked down thereon before it be turned up circularly.

This Ring is made of a blade of Brass, or other solid Matter, being in length proportionable to the Bigness you would have the Ring, and at least 4 or 5 Lines broad, with a proportionable Thickness, and whose Extremes AC, BD, are cut at right Angles. About the Points C and D describe two Quadrants AL, MB, and divide each of them into 9 equal Parts; and from each opposite Division draw the Parallels of the Signs, whereof the Line CFD shall be for ♈︎ and ♎︎, AEB for the two Tropicks, and the others for the other Signs placed according to their order. Then bisect the Parallelogram ABCD by the Line EF, and draw the Line GH separately equal to EB, that so a Scale may be made thereof, which must be divided into 9 equal Parts, each of which must be subdivided into 10 equal Parts more by little dots, and so the said Scale will be divided into 90 equal Parts, answering to the 90 Deg. of a Quadrant. This being done, take the Degrees of the Sun’s Altitude from the above posited Table of Altitudes, at every Hour when the Sun is in the Equinox, and the Solstices, for the Horizon of Paris. For example, When the Sun is in the 1st Deg. of ♋︎, his Meridian Altitude is 64 Deg. 29 Min. take 64$$\frac{1}{2}$$ equal Parts from the Scale GH between your Compasses, and lay them off upon the Brass Blade both ways from E to the Points I and K, as likewise from the Point F to the Points L and M, and join the Points IL and KM, by right Lines: then take from the Table the Sun’s Altitude at the Hours of 1 and 11, when he is in the Summer Solstice, viz. 61 Deg. 54 Min. which here may be taken for 62 Deg. and opening your Compasses to the Extent of 62 equal Parts of the Scale, lay them off upon AB from K towards E, and you will have a Point of the Hour-Lines of 11 and 1; likewise take 41 equal Parts or Degrees, for the Sun’s Meridian Altitude when he is in the Equinoctial, and lay them off from M to O, and from L to N, and the Points N and O are those thro’ which the two Hour-Lines of 12 must be drawn. Moreover, take 39 Deg. 20 Min. the Sun’s Altitude when he is in the Equinox, at the Hours of 11 and 1, from the Scale, and lay them off from the said Points M and L upon the said Line CD, and you will have two Points in the Line CD, thro’ which the Hour-Lines of 11 and 1 must be drawn. And in this manner may Points be found in this Line, thro’ which the other Hour-Lines must pass.

But now to find Points in the Line AB, or Tropick of Capricorn, on this side the Point E, thro’ which the Hour-Lines must be drawn, (for the Points of the same Line, on the other side of E, for the Tropick of Cancer may be found in the same manner as the Points for the Hour-Line of 11 and 1 was) you must take 17$$\frac{1}{2}$$ Degrees, or equal Parts from the Scale, viz. the Sun’s Meridian Altitude, when he is in the Tropick of Capricorn, and lay them off from I to P, and P will be the Point thro’ which the Hour-Line of 12 must pass; and so may the Points be found thro’ which the other Hour-Lines must be drawn. Now if the Points found in the Lines AB, and CD, thro’ which the Hour-Lines pass, be joined by right Lines; these right Lines will be the Hour-Lines.

But if you have a mind to be exacter, you may take the Degrees of the Sun’s Altitudes at every Hour when he enters, and is in each 10th and 20th Degree of every Sign, and then find Points on the respective Parallels on the Dial thro’ which the Hour-Lines must be drawn, which will not be right Lines but Curves; and in this case the Dial will be exacter.

Having drawn the Hour-Lines, you must Number them on both sides the Lines AB, CD, and also set down the Characters of the Signs, and the first Letters of the Months, each in their proper Place. When this is done, you must drill two little Holes in the Points R and S (viz. the middles of the Lines IL, KM) in a conical Figure, the greater Bases being outmost, that so the Sun’s Rays may better come thro’ them; afterwards round or turn up the said Blade circularly, solder the Extremities AC, BD together, and place a Button, with a Ring in the middle of the Junction of the said Extremities, so that the whole Instrument be in equilibrio; which that it may, you must turn the outside thereof.

### The Use of this Instrument.

Hold the Ring suspended, and turn the Hole proper for the Time of Year towards the Sun, so that his Rays may fall upon the Parallel of the Sign he is in, the Day wherein you use the Instrument; and then the Hour of the Day will be shewn thereon by a bright Spot or Point of Light.

Note, The Hole S is in use from the 20th of March, (N. S.) to the 22d of September, and the Hole R for the other six Months. We likewise write upon the convex Superficies of the Ring near the little Holes, the 20th of March, and the 22d of September, as appears in Figure 3, and, lastly, observe that these two last Dials are proper but for one Latitude.

## The Construction and Use of the universal Astronomical Ring-Dial.

This Instrument, whose Use is to find the Hour of the Day in any part of the Earth, by a bright Spot of the Sun’s Light, is made of Brass or other Metal, and consists of two Rings, or flat Circles turned both within side and without. The Diameter of these Rings, which ought to be broad and thick proportionable to their bignesses, are from two to six Inches. The outward Ring A represents the Meridian of any Place wherein one is, and there are two Divisions of 90 Degrees thereon, which are diametrically opposite to each other, one whereof serves from our North Pole to the Equator, and the other from the Equator to the South Pole.

The innermost Ring represents the Equator, and ought to turn very exactly within the outward one, by means of two Pivots or Pins put into Holes made diametrically opposite in the two Rings at the Points of the Hour of 12.

There is a thin Riglet (called a Bridge) with a Cursor marked C, composed of two little Pieces that slide in an Aperture made along the Middle of the said Bridge, and which are kept together by two small Screws. Thro’ the Middle of this Cursor is a very little Hole drilled, that so the Sun may shine thro’ it. Now the Middle of the said Bridge may be considered as the Axis of the World, and the Extremities as the Poles of the World; and there are drawn on one side thereof the Signs of the Zodiack with their Characters, and on the other side the Days and Names of the Months, or only their first Letters, being placed according to the respect they have to the Signs. The Signs are divided into every 10th or 5th Degree, according to their Declination, by means of a Trigon already divided, the Vertex of which, or Extremity of the Radius of the Equinoctial, being within side the Equinoctial Circle, as at the Point F. The two Pieces DD which are screwed to the outermost Ring, serve to support the Bridge or Axis which is moveable round, and are so ordered as that the innermost Ring may lie exactly within the outermost, and they both make as it were but one. The two Pieces E are also screwed on the outermost Ring, and serve as Props to keep the Equinoctial Circle or inward Ring at right Angles to the Meridian or outermost Ring.

We shall not here repeat the manner of dividing the two Quadrants into Degrees, and the Equinoctial Circle into Hours, Halves, and Quarters, having sufficiently spoken of this elsewhere. We shall only add, that all the Divisions of the Equinoctial Circle must be drawn upon the concave Side thereof, which may be done by means of a Piece of Steel turned up square, according to the Curvature of the Circle.

Near the outward Edges, on each of the two flat Sides of the Meridian, is made a Groove for the Piece G to slide therein, the Middle of which is bent inwards, that so it may go into the said Grooves. The two Sides of this Piece, which must be well hammered that they may have a good Spring, are made flat, in order to press against the convex Surface of the Meridian, that thereby the Piece G may be held fall on any Degree of Division of the Meridian. The Button thro’ which the Ring of Suspension H goes, is riveted to the Middle of the Piece G, so that it may turn round very freely, and by this means the Instrument be very perpendicularly suspended by the Ring H: for this is one of the principal things in which the Exactness of the Instrument consists.

### The Use of the Astronomical Ring-Dial.

Place the short Line a on the Middle of the hanging Piece G over the Degree of the Latitude of the Place you are in upon the Meridian Circle, for example, over the 49th Deg. at Paris; and then put the Line crossing the little Hole of the Cursor on the Bridge to the Degree of the Sign, or the Day of the Month you desire to know the Hour of the Day in. This being done, open the Instrument so that the two Rings or Circles be at right Angles to each other, and suspend it by the Ring H, so that the Axis of the Dial represented by the Middle of the Bridge be parallel to the Axis of the World.

Turn the flat Side of the Bridge towards the Sun, so that his Rays coming thro’ the little Hole in the Middle of the Cursor, fall exactly on a Line drawn round the Middle of the concave Surface of the Equinoctial Circle, or innermost Ring; and then the bright Spot or luminous Point shews the Hour of the Day in the said concave Surface of the Ring.

Note, The Hour of 12 cannot be shewn by this Dial, because the outermost Circle or Ring being then in the Plane of the Meridian, it hinders the Sun’s Rays from falling upon the innermost or Equinoctial Circle. You must observe likewise, that when the Sun is in the Equinoctial, you cannot then tell the Hour of the Day by this Dial, because his Rays fall parallel to the Plane of the said Equinoctial Circle. But this is but about one Hour every Day, and four Days in the Year.

## The Construction and Use of a Ring-Dial with three Rings.

This Instrument differs from the precedent one in nothing but only a third Ring or Circle, carrying the Sun’s Declination. The Ring A represents the Meridian of the Place you would use the Dial in; the Ring B represents the equinoctial Circle; and the Ring D, which turns exactly within the said Equinoctial Circle, produces the same Effect, as the Bridge representing the Axis of the World in the precedent Instrument. The two Extremities of the Diameter of this last Ring, or the two Points of the Circumference thereof, whereat it is fastened to the Meridian, answer to the two Poles of the World, On the opposite Parts DD of the Circumference of this Circle, is denoted a double Trigon of Signs, whose Center is the Vertex wherein all the Radius’s reunite, the Arcs of each of which are subdivided into every 10th or 5th Degree, to which may be likewise subjoined the Days of the correspondent Months.

The Index E is fastened to the Center of the innermost Ring; having two Sights riveted to the Extremities thereof, each having a small Hole drilled therein, for the Sun’s Rays to pass thro’. Note, Dials composed in this manner shew the Hour of 12, because the Index is without the Plane of the Meridian Circle: and when we make them large, as 9 or 10 Inches in Diameter, we divide the Equinoctial Circle into every 5th or every 2d Minute.

This Dial hath a Piece F like as the former Dial has, going into a Groove made on each side the Meridian, to be did to the Latitude of the Place. We sometimes set these Dials upon Pedestals, nearly like those of Spheres, which are slid to the Latitude; and in this Case they are placed upon an horizontal Plain; we likewise add Compasses to them, by which means the Variation of the Needle may be exactly known.

### The Use of this Dial.

Place the little Line in the middle of the hanging Piece F to the Latitude of the Place wherein you have a mind to know the Hour of the Day, and the fiducial Line of the Index on the Day of the Month, or Degree of the Sign the Sun is in. Then open the Equinoctial Circle at tight Angles to the Meridian, and holding the Instrument suspended, raise or lower the innermost Circle, so that the Sun’s Rays may go thro’ the Holes of the two Sights; then the Line which is drawn along the middle of the Convexity of the said Circle, will shew the Hour or Part drawn in the middle of the Concavity of the Equinoctial Circle, even at all times of the Day.

This may likewise be done something more convenient, when the Instrument is placed horizontally upon it’s Pedestal.

## The Construction of an universal inclined Horizontal, and Equinoctial Dial.

This Instrument consists of two Plates of Brass, or other solid Matter, whereof the under-one A is hollowed in about the Middle, to receive a Compass fastened underneath with Screws. The Plate B is moveable by means of a strong Joint at the Plate C. Upon this Plate is drawn a horizontal Dial for some Latitude greater than any one of those the Dial is to be used in, and having a Style thereon proportionable to that Latitude; for when the said Plane B is raised by means of the Quadrant, the horizontal Plane must always have a less Latitude than that the Dial is made for, or otherwise the Axis of the Style will have an Elevation too little.

Instead of the Quadrant D we generally place but only an Arc from the Equator to 60 Degrees, which are numbered downwards, 60 being at the Bottom, and for this Latitude of 60 Deg. we commonly draw the aforesaid horizontal Dial. That Arc of 60 Deg. is fastened by two small Tenons, and may be laid down upon the Plate A, as likewise may the Style upon the Plate B, and both of these are kept upright by means of little Springs underneath the Plates. What remains of the Construction of this Dial, may be supplied from the Figure thereof.

### The Use of the inclined horizontal Dial.

Raise the upper Plate B to the Degree of Latitude or Elevation of the Pole of the Place wherein you are, by means of the Graduations on the Quadrant D. Then if the Plane A be set horizontal, so that the Needle of the Compass settles itself over it’s Line of Declination, the Shadow of the Axis will shew the Hour of the Day. Note, We grave the Names of several principal Cities, as likewise their Latitudes and Longitudes, underneath the two Plates, in order to avoid the Trouble of seeking them in Maps.

After the abovesaid manner, equinoctial Dials are made Universal throughout the whole Earth; but here we must have a whole Quadrant. The upper Plate is commonly in form of a hollowed Circle, which we divide into 24 equal Parts, for the Hours, each of which we subdivide into 4 equal Parts, for the Quarters; all these being drawn in the Concavity of the Circle.

There is a Piece that goes thro’ the Circle, carrying the right Style, which is kept fast in the middle of the Circle by means of a little Spring fastened underneath the Circle; and by this means the right Style may be raised above the said Circle, and lowered underneath it. And when the equinoctial Dial is drawn, we use the little Piece F for a Style, placed in the Center of the Circle. Note, The upper part of the Dial shews the Hour of the Day from the 22d of March, (N. S.) to the 22d of September, and the under part thereof the Hour of the Day, the other 6 Months of the Year.

### The Use of the equinoctial Dial.

You must place the Edge of the equinoctial Circle to the Degree of the Elevation of the Pole, by means of the Quadrant; then if the Dial be set North and South by means of the Compass, the Shadow of the Style will shew the Hour of the Day at all times of the Year, even when the Sun is in the Equinoctial, because the Circle is hollowed in.

## The Construction of an Azimuth Dial.

This Dial, which is commonly made in the Bottom of a Compass, is called an Azimuth Dial, because it is made by means of the Azimuth’s or Sun’s Vertical Circles, upon a Plate of Brass, or other solid Matter, parallel to the Horizon. First, draw the Line AB, representing the Meridian, upon which describe a Circle at pleasure, half of which we shall only use here for drawing the Morning Hour-Lines, because those of the Afternoon are drawn after the same way. Divide this Circle into Degrees, beginning from the Point A, representing the North Pole. Then trisect the Semi-Diameter AC, and take AD equal to two thirds thereof, which must be divided into 6 Parts, thro’ each Point of Division; about the Center C must be drawn concentrick Arcs, representing the Parallels of the Signs, the Arc H being the Summer Tropick, that nearest to the Center C the Winter Tropick, and each of the others for two Signs equally distant from the Tropicks, as appears per Figure.

The Parallels of the Signs may moreover be drawn, in describing a Semi-Circle upon the Line HD, which Semi-Circle being divided into 6 equal Parts, you must let fall dotted Parallels upon the Line HD; these Parallels will divide the said Line into unequal Parts, and if thro’ the Points of Divisions Arcs be described about the Center C, these Arcs will be the Parallels of the Signs at unequal Distances from each other.

Now for drawing the Hour-Lines, the following Table of the Sun’s Azimuths must be used; for example, to prick down a Point in the Tropick of Cancer, thro’ which the Hour-Line of 11 in the Morning must be drawn, you will find the Sun’s Azimuth will then be 30 Deg. 17 Min. and when he is in the first Degree of ♊︎, or last of ♌︎, his Azimuth at the same Hour is 27 Deg. 58 Min. and so of others. Therefore if a Ruler be laid on the Center C, and on the 30th Deg. and 27 Min. of the outward divided Limb, the Edge of the Ruler will cut the Parallel of ♋︎, in a Point thro’ which the Hour-Line of 11 must pass: then keeping the Ruler to the Center, move it, and lay it over the 27th Deg. and 58th Min. of the outmost Limb, and you will have a Point in the Parallel of ♊︎ and ♌︎ thro’ which the Hour-Line of 11 must pass; and in this manner may Points be found in the other Parallels thro’ which the Hour-Line of 11 must pass; and also Points in all the Parallels thro’ which the other Morning Flour-Lines must pass: each of which Points belonging to the same Hours being joined, you will have the curved Hour-Lines on one side of the Meridian. And to find the Points thro’ which the Afternoon Hour-Lines must pass, take the Distances of each Point in the Parallels from the Meridian, and transfer them on the same Parallels continued out on the other side of the Meridian, because the Sun’s Azimuth at any two Hours equally distant on each side the Meridian, is the same.

### The Use of the Azimuth Dial.

Turn the Side B towards the Sun, so that the Shadow of the right Style planted in a Point without the Compass, and parallel to the Line of Noon, may fall along the Meridian Line: then the Needle pointing exactly North and South, will shew the Hour of the Day in the Intersection thereof with the Parallel of the Sign the Sun is in, upon condition that the Needle has no Variation. But since the Needle varies now above 12 Degrees at Paris, you must place the Style in the Point E over the Line of Declination or Variation KI, and adjust the Shadow of the Style along the said Line of Variation, and by this means the Error arising from the Needle’s Variation will be avoided.

A Table of the Sun’s Azimuth or Distance from the Meridian every Hour of the Day, for the Latitude of 49 Degrees.
Hours
Signs XI
I
X
II
IX
III
VIII
IV
VII
V
VI
VI
V
VII
IV
VIII
♋︎ 30° 17′ 53° 40′ 70° 30′ 83° 57′ 95° 20′ 105° 56′ 116° 28′ 127° 56′
♌︎ ♊︎ 27° 58′ 50° 33′ 67° 34′ 81° 6′ 92° 45′ 103° 35′ 114° 56′
♍︎ ♉︎ 23° 30′ 43° 52′ 0° 29′ 74° 17′ 86° 21′ 97° 36′
♎︎ ♈︎ 19° 33′ 37° 25′ 52° 58′ 66° 57′ 78° 34′
♏︎ ♓︎ 16° 42′ 32° 25′ 46° 30′ 59° 28′ 71° 12′
♐︎ ♒︎ 14° 56′ 29° 11′ 42° 23′ 54° 26′
♑︎ 14° 19′ 28° 2′ 40° 48′

## The Construction and Use of the Analemmatic or Ecliptick Horizontal Dial.

This is called an Analemmatick Dial, because it is made by means of the Analemma, which is the Projection or Representation of the principal Circles of the Sphere upon a Plane. The 9th Figure is the Analemma; and the 10th Figure represents the Dial compleat, which shews the Hour of the Day without a Compass.

Now to project the Analemma; upon a very even smooth Plate of Brass, draw the Lines AB and CD, cutting each other at right Angles in the Point E, about which, as a Center, describe the Circle ACBD, representing the Meridian, it's Diameter CD, the Horizon, and AB the prime Vertical. Then assume the Arc DF equal to the Elevation of the Pole, which here is 49 Deg. and draw the Line EF representing the Axis of the World; likewise assume the Arc CG equal to the Height of the Equinoctial 41 Degrees, and draw the Line GE for the Equinoctial. Assume the Arcs GH, GI, each of 23 Deg. 30 Min. for the Sun’s greatest Declination, and draw the Line HI cutting the Equinoctial in the Point Y, about which, as a Center, describe the Circle HLIK, or only half of it, which divide into 6 equal Parts, and thro’ each Point of Division draw Parallels to the Equinoctial, which continue out to the Horizon; then from the Sections made by the said Parallels on the Meridian, let fall the Parallels M, N, O, and P to the Horizon, and from the Sections made by the said Parallels on the Axis, let fall the indefinite Perpendiculars Sc , Rb, Qa to the Horizon. This being done, take the Distance EM between your Compasses, with which setting one Foot in N, with the other make a small Arc upon the Line Qa, and with one Foot in O cut the Line Rb with the other; then, continually keeping the Compasses opened to the Extent EM, set one Foot in P, and cut the Line Sc in the Point C.

Now to construct the little Zodiack, take the Distance ♐︎C, and lay off from E towards A and B for the Tropicks of ♋︎ and ♑︎, again, lay off the Distance 46, from the Point E on one side, for the Parallel of ♊︎, and on the other side for the Parallel of ♒︎; and finally, take the Distance Xa, for marking the Parallel of ♉︎ on one side, and that of ♓︎ on the other, and then the little Zodiack may be formed, as per Figure. Now to prick down the Hour-Points, you must describe the Circle MTZV about the Center E, with the Distance EM, and divide the Circumference thereof into 24 equal Parts, as likewise the Circumference of the Meridian ACBD, and from each opposite Point of Division in the Meridian draw strait Lines parallel to AB, and in the Circle MTZV, strait Lines parallel to CD, and thro’ the Intersections of these Lines that are nearest to the Meridian, draw lightly an Ellipsis from Point to Point, as you see in the Figure. These Points of Section will be the Hour-Points, those for the Morning being on the left, and those for the Afternoon on the right; and to have the half and quarter Hour-Points, the two Circles ACBD, MTZV, must be divided into 96 equal Parts.

Things being thus prepared, transfer all the Hour-Points on another Brass Plate, and form the Ellipsis B thereon, by lightly drawing Lines from Point to Point, and grave the proper Numbers upon it, as they are marked in the 10th Figure. Likewise transfer the Trigon of Signs upon the said Plate, taking each of the Distances between your Compasses, the one after the other, so that the Signs ♈︎ and ♎︎ be in the Line of the Hour of 6, and place the Characters of the Signs thereon, as also the first Letters of the Months, each one in their order. When this is done, you must adjust a Cursor C so as to slide along the middle of the Trigon. This Cursor carries the right Style D, which rises and falls by means of two small Knuckles.

On the other part of this Plate, is drawn an horizontal Dial according to the common Rules, for the same Latitude the Analemma is made for, and we place the Style or Axis E thereon upon the Hour-Line of 12, which rises, falls, and is kept upright by means of a small Spring underneath the Plate.

### The Use of this Dial.

Set the Dial parallel to the Horizon, and put the Cursor with it’s right Style upon the Day of the Month, or Sign the Sun is in; then turn the Instrument until the same Hour be shewn upon the two Dials, which will be the Hour of the Day. If, for example, the Shadow of the Extremity of the right Style falls upon the nth Hour on the Analemmatick Dial, and at the same time the Shadow of the Style of the horizontal Dial falls likewise upon the 11th Hour, on the horizontal Dial; then the true Hour of the Day will be that of 11. The Conveniency of this Dial consists in this, that the Hour of the Day may be found thereby without a meridian Line, or Compass; but then it must be pretty large, to shew the Hour exactly.

## The Construction of an universal Polar, East, and West Dial.

This Instrument consists of a very strait and smooth circular Piece of Brass, or other Metal, pretty thick, that so it may preserve it’s perpendicular Weight, as likewise that a Groove may be made round the Limb thereof, for a hanging-Piece to slide about the same, like that on the astronomical Ring.

About the Center cf the said circular Piece describe the Circumference of a Circle, which divide into twice 90 Degrees. Then draw a right Line from the 90th Degree thro’ the Center, representing the Equinoctial, near the Top of which assume a Point at pleasure, thro’ which draw a right Line perpendicular to the Equinoctial-Line, which shall be the Hour-Line of 6. Then to have the other Hour-Lines, you must lay off the answerable Tangents upon the Equinoctial-Line both ways from the Point therein of the Hour-Line of 6; as the Tangent of 15 Deg. for the Hour-Points of 5 and 7; the Tangent of 30 Deg. for 4 and 8; the Tangent of 45 Deg. for 3 and 9, &c. and if Lines be drawn thro’ these Points parallel. to the Hour-Line of 6, these will be the Hour-Lines; and the Length of the right Style’ must be equal to the Radius or Tangent of 45 Deg. and must be placed upright upon the Hour-Line of 6, at the Point wherein it cuts the Equinoctial-Line.

At the Points CC, on the Hour-Line of 9 in the Morning, and 3 in the Afternoon, are adjusted two small Knuckles, in which is placed the Piece V, which may lie down upon the circular Piece, and likewise stand at right Angles to it. Upon this Piece are pricked down the Hour-Lines of a Polar Dial, from 9 in the Morning to 12, and from 12 to 3 in the Afternoon. We shall not here repeat the manner of drawing these Hour-Lines, for we have sufficiently spoken of this already, as likewise how to draw the Arcs of the Signs; only observe, that the Parallels of the Signs are divided into every 10th Deg. and the first Letters of the Names of the Months are set down in their proper Place.

The Style B is adjusted to the circular Piece with a Joint, that so it may be railed or lie flat upon the said Piece; but it must be raised so that the Extremity thereof may be exactly over the Point in the Equinoctial-Line cut by the Hour-Line of 6, and the Distance of the said Extremity from this Point equal to the Distance from the Hour-Line of 9 to the Hour-Line of 6.

### The Use of the said Dial.

If you have a mind to find the Hour of the Day before Noon, place the little Line on the middle of the hanging Piece L upon the Latitude of the Place; on that Quadrant on the Right-hand of the Style B, raise the Style so that the Extremity thereof be directly over the Intersection of the Equinoctial and the Hour-Line of 6, and it’s Distance from that Point of Intersection equal to the Distance from the Hour-Line of 9 to the Hour-Line of 6. Then holding the Dial suspended by it’s Ring, expose it to the Sun, so that the Shadow of the Extremity of the Style falls upon the Day of the Month; and you will have the Hour of the Day upon the East or Polar Dial. But if the Hour of the Day be required in the Afternoon, you must put the hanging Piece on the Latitude of the Place upon the Quadrant on the left side of the Style, and turn the Dial to the Sun so that the Shadow of the Extremity of the Style falls on the Degree of the Sign or Day of the Month. Then you will have the Hour of the Day as before.

Thus have I said down the Construction and Uses of Portable Dials, chiefly in use, which may be set North and South, without a Compass or Meridian Line. But before I close this Chapter, I shall briefly describe some other Portable Dials, which are curious enough, but are something difficult to make.

The first of these is a horizontal Dial of 2 or 3 Inches square, which we make of Brass or any other solid Metal, for a given Latitude, and whose Axis shewing the Hour, is a Thread fastened at one end to the Center of the said Dial, and the other end of which is hung to the top of a pretty thick Brass Blade, placed at the Extremity of the Dial near the Hour-Line of 12. This Blade may lie down upon the Plane of the Dial, and is kept upright by means of a Spring underneath the Dial; and the Height of the Notch wherein the Thread lies above the Plane of the Dial, is equal to the Tangent of the Latitude.

About a quarter of the Height of the said Blade is adjusted thereon a Circle or Ring, proportioned to the bigness of the Dial-Plate. This Ring is moveable by means of a Joint, and so may lie down upon the Blade, and the Blade upon the horizontal Dial-Plane; and when the Instrument is using, there is a Prop to keep this Ring at the Height of the Equinoctial, viz. 41 Deg. but when the Thread serving for an Axis is extended, it must exactly pass thro’ the Center of this Ring.

The Concavity of the Ring is divided into Hours, Halves, and Quarters, as the equinoctial Ring of the universal Ring-Dial is; and there is a Bead or Pin’s Head put upon the Thread, that so it may be moved to the Sign the Sun is in, and serve as a Cursor to shew the Hour of the Day in the middle of the Concavity of the Ring or Equinoctial.

Now to place the Bead to the Sign or proper Month, you must have a separate Brass Riglet, having the Signs of the Zodiack, as also the Days of the Months drawn thereon in the manner they were drawn upon the Bridge of the universal Ring-Dial; and having placed the said Riglet from the Center of the horizontal Dial along the Thread or Axis, slide the Bead to the Degree of the Sign the Sun is in, and then take away the Riglet, and so will the Bead be placed for shewing the Hour of the Day.

On the backside of the Blade is drawn an upright Line for a Plumb-Line to play on, that so the Dial may be set level. Note, This Dial may be rendered universal, if an Arc of a Circle divided into Degrees be adjusted behind the Blade by means of a Joint, so as it may lie upon the Blade, and the Point whereon the Plumb-Line is hung by the Center of the said Arc; for then the Dial may be set to the Latitude, by making the Plumb-Line fall upon the proper Degree on the circular Arc. It is proper also to observe, that the Hours from eight in the Evening to four in the Morning may be taken away from the Equinoctial Ring, that so this Dial may be of use at the Time of the Equinox.

### The Use of the aforesaid Dial.

Having placed the Bead to the Degree of the Sign the Sun is in, or Day of the Month, as before directed, expose the Dial to the Sun, and turn it to the right or left until the Shadow of the Bead falls upon the same Hour or Part, on the middle of the Concavity of the equinoctial Ring, as the Shadow of the Thread or Axis does on the horizontal Dial; and then that will be the true Time of the Day.

We make several other portable Dials, as horizontal Astrolabes, being Projections of the Sphere upon the Plane of the Horizon; other Astrolabes vertically used by means of a Plumb-Line; horizontal Dials made by means of the Sun’s Altitudes, which are likewise set North and South without a Compass, and whereon the Signs are drawn by right Lines issuing from the same Center, and the Hour-Lines, curve Lines; as likewise other portable Dials, which are curious enough, whose Construction and Figures we reserve for another time.

Horizontal Dials whereon are drawn the Signs, as that of Fig. 7. may likewise be set North and South without a Compass, if the Dial be so placed in the Sun, that the Shadow of the Extremity of the right Style falls upon the Degree of the Sign the Sun is in, or Day of the Month. But here there is this Inconveniency, that the Distance of the Parallel of Cancer from the adjacent Parallels is so small, that the Space of 10 Days there cannot be distinguished. So that when we have done all we can, it is scare possible to make a portable Dial that can set North and South without a Compass or Meridian Line, without falling into one of these Inconveniencies, either of having the Hour-Lines near Noon too nigh each other, or not exactly shewing the Hour of the Day at the Time of the Solstices, because of the small Difference that there is in the Sun’s Elevations and Declination at those times.