Mathematical Instruments
Book VIII. Ch. VI.

Of the Construction and Uses of a Moon-Dial, and a Nocturnal or Star-Dial.

Of the construction of an horizontal dial for shewing the Hour of the Night by the Moon.

Fig. 12

This is called a Moon-Dial, because by it you may tell in the Night by the Shadow of the Moon, what Hour-Circle the Sun is in. It consists of two Pieces or Plates of Brass, or other solid Matter, of a bigness at pleasure. The under-Plate H, is in figure of a Parallelogram, and the upper one A is circular, turns about the shadowed Space L, and the Center B, and has a horizontal Dial drawn upon it for the Latitude of the Place, according to the Rules before prescribed for drawing horizontal Dials. The under Plate hath a Circle thereon divided into 30 unequal Parts, for the Days of a Lunar Month. These Divisions are made thus; let DE be the equinoctial Line by which the horizontal Dial was drawn, and F the Center of the equinoctial Circle (or the Center by which the equinoctial Line is divided.) About this Center describe a dotted Circle, and divide it into 30 equal Parts, or half of it into 15, and having laid the Edge of a Ruler on the Center F, lay it over each Point of the Divisions of the said Circle one after another, and prick down Points upon the equinoctial Line; then lay the Ruler to the Center B, and on each Point of Division of the equinoctial Line, and divide the Circle H; and when you have divided half of it, transfer the same Divisions on the other Semi-Circle, and by this means the whole Circle will be divided into 30 unequal Parts for the 30 Days of a Lunar Month, about which Numbers must be graved, as they appear per Figure. This being done, place the Axis BC answering to the Elevation of the Pole, and dispose it so that when it is set up it may not hinder the Hour-Plate from turning about the Center B.

The Use of this Dial.

The Moon’s Age must be found by an Ephemeris, or by the Epact, that so the Point of the Hour-Line of 12 on the horizontal Dial may be applied to the Day of her Age in the Circle H of the under Plate.

But before we go any further, you must observe, that the Moon by her proper Motion recedes Eastwards from the Sun every Day about 48 Minutes of an Hour, that is, if the Moon is in Conjunction with the Sun on any Day upon the Meridian, the next Day she will cross the Meridian about three quarters of an Hour and some Minutes later than the Sun: and this is the Reason that the Lunar Days are longer than the Solar ones; a Lunar Day being that Space of Time elapsed between her Passage over the Meridian, and her next Passage over the same; and these Days are very unequal on account of the Irregularities of the Moon’s Motion.

Now when the Moon is come to be in Opposition to the Sun, she will again be found in the same Hour-Circle as the Sun is; so that if, for example, the Sun should be then in the Meridian of our Antipodes, the Moon would be in our Meridian, and consequently would shew the same Hour on our Sun-Dials as the Sun would, if it was above the Horizon. But this Conformity would be of small duration, because of the Moon’s retardation of about two Minutes every Hour. If moreover the Sun, at the Time of the Opposition, be just letting above our Horizon, the Moon being diametrically opposite to it will he just rising, &c. and therefore to remedy the said Retardation, we have divided the Circle H into 30 Parts.

Now the Point of the Hour-Line of 12 on the horizontal Dial being put to the Moon’s Age, as above directed, and the under-Plate set North and South by means of a Compass or meridian Line, the Shadow of the Style will shew the Hour of the Night; but to have the Hour more exact, you must know whether it is the first, second or third Quarter of the Moon’s Day that you seek the Hour in, that so the Point of the Hour-Line of 12 may be set against a proportionable part of one of the 30 Spaces or Lunar Days of the Circle H.

The Table on the under-Plate H, is used for finding the Hour of the Night by the Shadow of the Moon upon an ordinary Dial. To make this Table, draw 4 Parallel right Lines or Curves of any length, and divide the Space II into twelve equal Parts for 12 Hours, and the two other Spaces KK into 15, for the 30 Lunar Days.

The Use of this Table.

First observe what Hour the Shadow of the Moon shews upon a Sun-Dial; then find the Moon’s Age, and seek the Hour correspondent thereto in the Table, and add the Hour shewn by the Sun-Dial thereto; then their Sum, if it be less than 12, or else it’s excess above 12, will be the true Hour of the Night. For example; Suppose the Hour shewn upon the Sun-Dial by the Moon, be the 6th, and her Age be 5 or 20 Days, against either of these Numbers in the Table you will find 4, which added to 6 makes 10, and so the Hour of the Night will be 10. Again, Suppose the Moon shews the Hour of 9 upon the Sun-Dial, when she is 10 or 25 Days old, against 10 and 25 in the Table you will find 8, which added to 9, makes 17, from which 12 being taken, the Remainder 5 will be the true Hour sought. And so of others.

To find the Moon’s Age, you must first find the Golden Number; and this is done by adding 1 to the given Year, and dividing the Sum by 19, and the Remainder will be the Golden Number. Then you must find the Epact, by means of the Golden Number; and this is done thus: Divide the Golden Number by 3, and each Unit remaining being called 10, will be the Epact, if the Sum be less than 30; but if above 30, 30 being taken from it, and the Remainder added to the Golden Number will be the Epact. The Epact being found, the Moon’s Age may be had after this manner: If the Moon’s Age be sought in January, add 0 to the Epact; in February, 2; in March, 1; in April, 2; in May, 3; in June, 4; in July, 5; in August, 6; in September, 8; in October, 8; in November, 10; and in December, 10: and the Sum, if it be less than 30, or the Excess above 30, added to the Day of the given Month (rejecting 30 if need be) will be the Moon’s Age that Day. For example, To find the Moon’s Age the 14th Day of March, in the Year 1716, (O. S.) the Golden Number is 7, and the Epact 17; therefore adding 1 for March to 17, and the Sum will be 18; and if to this 18 be added 14 for the Day of the Month, the Sum will be 32, from which 30 being taken, and the Remainder 2 will be the Moon’s Age. Note, This way of finding the Moon’s Age is not so exact as we have it by the Ephemeris. Likewise observe, that vertical Moon-Dials may be made in the manner as the horizontal ones are, but the Divisions of 30 Parts upon equinoctial Dials must be equal, and the moveable Circle divided into 24 equal Parts, &c.

The Construction of a Nocturnal or Star-Dial.

Fig. 13

The 13th Figure shews the Disposition of the chief Stars composing the Constellation of Ursa Major, and Ursa Minor, about the Pole and the Pole-Star.

The Nocturnal we are going to mention, is made by the Consideration of the diurnal Motion, that the two Stars of Ursa Major, called his Guards, or the bright Star of Ursa Minor, make about the Pole, or the Pole Star, which at present is but about 2 Deg. distant from the North Pole.

Now to construct this Instrument, you must first know the right Ascension of the said Stars, or in what Days of the Year they are found in the same Hour Circle as the Sun is. This may be found, by Calculation, on a Globe, or a Celestial Planisphere, by placing the Star in question under the Meridian, and examining what Degree of the Ecliptick will be found at the same time under the Meridian. By this Method you will find that the bright Star or Guard of the Little Bear, was found twice in one Year with the Sun under the Meridian, viz. in the Year 1715, once the 8th of May, (N. S.) above the Pole, and again the 8th of November below the Pole. Therefore in the said two Days of the Year, the abovementioned Star will be in all the Hour-Circles at the same Time as the Sun is; and consequently will shew the same Hour. You will find also, that the two Guards of Ursa Major were found two other Days of the Year under the same Meridian or Hour-Circle as the Sun, viz. the first Day of September below the Pole, and the first Day of March above it. And in these two Days the said Stars will shew the same Hours as the Sun does; but because the fixed Stars return to the Meridian every Day about 1 Deg. sooner than the Sun, or four Minutes of an Hour, which is two Hours per Month, it is this, which is to be observed for having the Hour of the Sun, which is the Measure of our Days.

Fig. 14

These things being premised, it will not be difficult to make a Nocturnal or Star-Dial, in the following manner:

The Instrument is composed of two circular Plates applied on each other; the greater of which, having a Handle for holding up the Instrument when using, is about two Inches and a half in Diameter, and is divided into twelve Parts for the twelve Months of the Year, and each Month divided into every 5th Day; so that the middle of the Handle exactly answers to the Day of the Year wherein that Star which is used has the same right Ascension as the Sun has. If, for example, This Instrument be made for the two Guards of Ursa Major, the first Day of September must, be against the middle of the Handle; and if it be made for the bright Star of Ursa Minor, the 8th Day of November must be against the middle of the Handle. Therefore if you will have the Instrument serve for both these Stars, the Handle must be made moveable about the said circular Plate, that so it may be fixed according to necessity; and this is easy to do by means of two little Screws.

This being done, the upper lesser Circle must be divided into 24 equal Parts, or twice 12 Hours, for the 24 Hours of the Day, and each Hour into Quarters, according to the Order appearing in the Figure. These 24 Hours are distinguished by a like Number of Teeth, whereof thole whereat the Hours of 12 are marked are longer than the others, that so the Hours may be counted in the Night without a Light.

In the Center of the two circular Plates is adjusted a long Index A, moveable about the same upon the upper Plate. These three Pieces, viz. the two Circles and the Index, are joined together by means of a headed Rivet, and pierced so, that there is a round Hole thro’ the Center about two Inches diameter, for easy seeing the Pole-Star thro’ it. Note, The Motions of the upper-Plate and Index ought to be pretty stiff, that so they may remain where they are placed when the Instrument is using.

The Use of this Instrument.

Turn the upper circular Plate ’till the longest Tooth whereat is marked 12 be against the Day of the Month on the under Plate; then bringing the Instrument near your Eyes, hold it up by the Handle, so that it leans neither to the Right or Left, with it’s Plane as near parallel to the Equinoctial as you can; and looking at the Pole-Star thro’ the Hole in the Center of the Instrument, turn the Index about, ’till, by the Edge coming from the Center, you can see the bright Star, or Guard of the Little Bear, if the Instrument be adapted for that Star, and that Tooth of the upper Circle that is under the Edge of the Index, is at the Hour of the Night upon the Edge of the Hour-Circle; which may be known without a Light, by accounting the Teeth from the longest, which is for the Hour of 12.

You must proceed in this manner for finding the Hour of the Night, when the Instrument is made for the Guards of Ursa Major, which Stars are nearly in a right Line with the Pole-Star, are of the same Magnitude, and are very useful for finding the Pole-Star.

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