The New Southern Constellations.

Columba NoachiNoah’s Dove10
Robur CarolinumThe Royal Oak12
GrusThe Crane13
PhœnixThe Phenix13
IndusThe Indian12
PavoThe Peacock14
Apus, Avis IndicaThe Bird of Paradise11
Apis, MuscaThe Bee or Fly4
ChamæleonThe Chameleon10
Triangulum AustralisThe South Triangle5
Piscis volans, PasserThe Flying Fish8
Dorado, XiphiasThe Sword Fish6
ToucanThe American Goose9
HydrusThe Water Snake10

Hevelius’s Constellations made out of the unformed Stars.

Hevelius.Flamsteed.
LynxThe Lynx1944
Leo minorThe Little Lion 53
Asterion & CharaThe Greyhounds2325
CerberusCerberus4
Vulpecula & AnserThe Fox and Goose2735
Scutum SobieskiSobieski’s Shield7
LacertaThe Lizard1016
CamelopardalusThe Camelopard3258
MonocerosThe Unicorn1931
SextansThe Sextant1141

The Milky Way.

400. There is a remarkable track round the Heavens, called the Milky Way from its peculiar whiteness, which is owing to a great number of Stars scattered therein; none of which can be distinctly seen without telescopes. This track appears single in some parts, in others double.

Lucid Spots.

401. There are several little whitish spots in the Heavens, which appear magnified, and more luminous when seen through telescopes; yet without any Stars in them. One of these is in Andromeda’s girdle, first observed A. D. 1612, by Simon Marius; and which has some whitish rays near its middle: it is liable to several changes, and is sometimes invisible. Another is near the Ecliptic, between the head and bow of Sagittarius: it is small, but very luminous. A third is on the back of the Centaur, which is too far South to be seen in Britain. A fourth, of a smaller size, is before Antinous’s right foot; having a Star in it, which makes it appear more bright. A fifth is in the Constellation of Hercules, between the Stars ζ and η, which spot, though but small, is visible to the bare eye if the sky be clear and the Moon absent.

Cloudy Stars.
Magellanic Clouds.

402. Cloudy Stars are so called from their misty appearance. They look like dim Stars to the naked eye; but through a telescope they appear broad illuminated parts of the sky; in some of which is one Star, in others more. Five of these are mentioned by Ptolemy. 1. One at the extremity of the right hand of Perseus. 2. One in the middle of the Crab. 3. One unformed, near the Sting of the Scorpion. 4. The eye of Sagittarius. 5. One in the head of Orion. In the first of these appear more Stars through the telescope than in any of the rest, although 21 have been counted in the head of Orion, and above 40 in that of the Crab. Two are visible in the eye of Sagittarius without a telescope, and several more with it. Flamsteed observed a cloudy Star in the bow of Sagittarius, containing many small Stars: and the Star d above Sagittary’s right shoulder is encompassed with several more. Both Cassini and Flamsteed discovered one between the Great and Little Dog, which is very full of Stars visible only by the telescope. The two whitish spots near the South Pole, called the Magellanic Clouds by Sailors, which to the bare eye resemble part of the Milky-Way, appear through telescopes to be a mixture of small Clouds and Stars. But the most remarkable of all the cloudy Stars is that in the middle of Orion’s Sword, where seven Stars (of which three are very close together) seem to shine through a cloud, very lucid near the middle, but faint and ill defined about the edges. It looks like a gap in the sky, through which one may see (as it were) part of a much brighter region. Although most of these spaces are but a few minutes of a degree in breadth, yet, since they are among the fixed Stars, they must be spaces larger than what is occupied by our solar System; and in which there seems to be a perpetual uninterrupted day among numberless Worlds which no human art ever can discover.

Changes in the Heavens.

403. Several Stars are mentioned by antient Astronomers, which are not now to be found; and others are now visible to the bare eye which are not recorded in the antient catalogues. Hipparchus observed a new Star about 120 years before Christ; but he has not mentioned in what part of the Heavens it was seen, although it occasioned his making a catalogue of the Stars; which is the most antient that we have.

New Stars.

The first New Star that we have any good account of, was discovered by Cornelius Gemma on the 8th of November A. D. 1572, in the Chair of Cassiopea. It surpassed Sirius in brightness and magnitude; and was seen for 16 months successively. At first it appeared bigger than Jupiter to some eyes, by which it was seen even at mid-day: afterwards it decayed gradually both in magnitude and lustre, until March 1573, when it became invisible.

On the 13th of August 1596, David Fabricius observed the Stella Mira, or wonderful Star, in the Neck of the Whale; which has been since found to appear and disappear periodically, seven times in six years, continuing in its greatest lustre for 15 days together; and is never quite extinguished.

In the year 1600, William Jansenius discovered a changeable Star in the Neck of the Swan; which, in time became so small as to be thought to disappear entirely, till the years 1657, 1658, and 1659, when it recovered its former lustre and magnitude; but soon decayed, and is now of the smallest size.

In the year 1604 Kepler and several of his friends saw a new Star near the heel of the right foot of Serpentarius, so bright and sparkling that it exceeded any thing they had ever seen before; and took notice that it was every moment changing into some of the colours of the rainbow, except when it was near the horizon, at which time it was generally white. It surpassed Jupiter in magnitude, which was near it all the month of October, but easily distinguished from it by a steady light. It disappeared between October 1605 and the February following, and has not been seen since that time.

In the year 1670, July 15, Hevelius discovered a new Star, which in October was so decayed as to be scarce perceptible. In April following it regained its lustre, but wholly disappeared in August. In March 1672 it was seen again, but very small; and has not been visible since.

In the year 1686 a new Star was discovered by Kirch, which returns periodically in 404 days.

In the year 1672, Cassini saw a Star in the Neck of the Bull, which he thought was not visible in Tycho’s time; nor when Bayer made his Figures.

Cannot be Comets.

404. Many Stars, besides those above-mentioned, have been observed to change their magnitudes: and as none of them could ever be perceived to have tails, ’tis plain they could not be Comets; especially as they had no parallax, even when largest and brightest. It would seem that the periodical Stars have vast clusters of dark spots, and very slow rotations on their Axis; by which means, they must disappear when the side covered with spots is turned towards us. And as for those which break out all of a sudden with such lustre, ’tis by no means improbable that they are Suns whose Fuel is almost spent, and again supplied by some of their Comets falling upon them, and occasioning an uncommon blaze and splendor for some time: which indeed appears to be the greatest use of the cometary part of any system[[86]].

Some Stars change their Places.

Some of the Stars, particularly Arcturus, have been observed to change their places above a minute of a degree with respect to others. But whether this be owing to any real motion in the Stars themselves, must require the observations of many ages to determine. If our solar System changeth its Place, with regard to absolute space, this must in process of time occasion an apparent change in the distances of the Stars from each other: and in such a case, the places of the nearest Stars to us being more affected than of those which are very remote, their relative positions must seem to alter, though the Stars themselves were really immoveable. On the other hand, if our own system be at rest, and any of the Stars in real motion, this must vary their positions; and the more so, the nearer they are to us, or the swifter their motions are; or the more proper the direction of their motion is, for our perception.

The Ecliptic less oblique now to the Equator than formerly.

405. The obliquity of the Ecliptic to the Equinoctial is found at present to be above a third part of a degree less than Ptolemy found it. And most of the observers after him found it to decrease gradually down to Tycho’s time. If it be objected, that we cannot depend on the observations of the antients, because of the incorrectness of their Instruments; we have to answer, that both Tycho and Flamsteed are allowed to have been very good observers: and yet we find that Flamsteed makes this obliquely 21⁄2 minutes of a degree less than Tycho did, about 100 years before him: and as Ptolemy was 1324 years before Tycho, so the gradual decrease answers nearly to the difference of time between these three Astronomers. If we consider, that the Earth is not a perfect sphere, but an oblate spheroid, having its Axis shorter than its Equatoreal diameter; and that the Sun and Moon are constantly acting obliquely upon the greater quantity of matter about the Equator, pulling it, as it were, towards a nearer and nearer co-incidence with the Ecliptic; it will not appear improbable that these actions should gradually diminish the Angle between those Planes. Nor is it less probable that the mutual attractions of all the Planets should have a tendency to bring the planes of all their Orbits to a co-incidence: but this change is too small to become sensible in many ages.

CHAP. XXI.
Of the Division of Time. A perpetual Table of New Moons. The Times of the Birth and Death of Christ. A Table of remarkable Æras or Events.

406. The parts of time are Seconds, Minutes, Hours, Days, Years, Cycles, Ages, and Periods.

A Year.

407. The original standard, or integral measure of Time, is a year; which is determined by the Revolution of some Celestial Body in its Orbit, viz. the Sun or Moon.

Tropical Year.

408. The time measured by the Sun’s Revolution in the Ecliptic, from any Equinox or Solstice to the same again, is called the Solar or Tropical Year, which contains 365 days 5 hours 48 minutes 57 seconds; and is the only proper or natural year, because it always keeps the same seasons to the same months.

Sidereal year.

409. The quantity of time, measured by the Sun’s Revolution, as from any fixed Star to the same Star again, is called the Sidereal Year; which contains 365 days 6 hours 9 minutes 141⁄2 seconds; and is 20 minutes 171⁄2 seconds longer than the true Solar Year.

Lunar Year.

410. The time measured by twelve Revolutions of the Moon, from the Sun to the Sun again, is called the Lunar Year; it contains 354 days 8 hours 48 minutes 37 seconds; and is therefore 10 days 21 hours 0 minutes 20 seconds shorter than the Solar Year. This is the foundation of the Epact.

Civil Year.

411. The Civil Year is that which is in common use among the different nations of the world; of which, some reckon by the Lunar, but most by the Solar. The Civil Solar Year contains 365 days, for three years running, which are called Common Years; and then comes in what is called the Bissextile or Leap-Year, which contains 366 days. This is also called the Julian Year on account of Julius Cæsar, who appointed the Intercalary-day every fourth year, thinking thereby to make the Civil and Solar Year keep pace together. And this day, being added to the 23d of February, which in the Roman Calendar, was the sixth of the Calends of March, that sixth day was twice reckoned, or the 23d and 24th were reckoned as one day; and was called Bis sextus dies, and thence came the name Bissextile for that year. But in our common Almanacks this day is added at the end of February.

Lunar Year.

412. The Civil Lunar Year is also common or intercalary. The common Year consists of 12 Lunations, which contain 354 days; at the end of which, the year begins again. The Intercalary, or Embolimic Year is that wherein a month was added, to adjust the Lunar Year to the Solar. This method was used by the Jews, who kept their account by the Lunar Motions. But by intercalating no more than a month of 30 days, which they called Ve-Adar, every third year, they fell 33⁄4 days short of the Solar Year in that time.

Roman Year.

413. The Romans also used the Lunar Embolimic Year at first, as it was settled by Romulus their first King, who made it to consist only of ten months or Lunations; which fell 61 days short of the Solar Year, and so their year became quite vague and unfixed; for which reason, they were forced to have a Table published by the High Priest, to inform them when the spring and other seasons began. But Julius Cæsar, as already mentioned, § [411], taking this troublesome affair into consideration, reformed the Calendar, by making the year to consist of 365 days 6 hours.

The original of the Gregorian, or New Style.

414. The year thus settled, is what we still make use of in Britain: but as it is somewhat more than 11 minutes longer than the Solar Tropical Year, the times of the Equinoxes go backward, and fall earlier by one day in about 130 years. In the time of the Nicene Council (A. D. 325.) which was 1431 years ago, the vernal Equinox fell on the 21st of March: and, if we divide 1431 by 130, it will quote 11, which is the number of days the Equinox has fallen back since the Council of Nice. This causing great disturbances, by unfixing the times of the celebration of Easter, and consequently of all the other moveable Feasts, Pope Gregory the 13th, in the year 1582 ordered ten days to be at once struck out of that year; and the next day after the fourth of October was called the fifteenth. By this means the vernal Equinox was restored to the 21st of March; and it was endeavoured, by the omission of three intercalary days in 400 years, to make the civil or political year keep pace with the Solar for time to come. This new form of the year is called the Gregorian Account or New Style; which is received in all Countries where the Pope’s Authority is acknowledged, and ought to be in all places where truth is regarded.

Months.

415. The principal division of the year is into Months, which are of two sorts, namely Astronomical and Civil. The Astronomical month is the time in which the Moon runs through the Zodiac, and is either Periodical or Synodical. The Periodical Month is the time spent by the Moon in making one compleat Revolution from any point of the Zodiac to the same again; which is 27d 7h 43m. The Synodical Month, called a Lunation, is the time contained between the Moon’s parting with the Sun at a Conjunction, and returning to him again; which is in 29d 12h 44m. The Civil Months are those which are framed for the uses of Civil life; and are different as to their names, number of days, and times of beginning, in several different Countries. The first month of the Jewish Year fell according to the Moon in our August and September, Old Style; the second in September and October, and so on. The first month of the Egyptian Year began on the 29th of our August. The first month of the Arabic and Turkish Year began the 16th of July. The first month of the Grecian Year fell according to the Moon in June and July, the second in July and August, and so on, as in the following Table.

NoThe Jewish year.Days NoThe Egyptian year.Days
1TisriAug.-Sept.301ThothAugust 2930
2MarchesvanSept.-Oct.292PaophiSeptemb. 2830
3CasleuOct.-Nov.303AthirOctober 2830
4TebethNov.-Dec.294ChojacNovemb. 2730
5Shebat Dec.-Jan.305TybiDecemb. 2730
6Adar Jan.-Feb.296MechirJanuary 2630
7Nisan or AbibFeb.-Mar.307PhamenothFebruar. 2530
8JiarMar.-Apr.298ParmuthiMarch 2730
9SivanApril-May309PachonApril 2630
10TamuzMay-June2910PayniMay 2630
11AbJune-July3011EpiphiJune 2530
12ElulJuly-Aug.2912MesoriJuly 2530
Days in the year354>Epagomenæ or days added5
In the _Embolimic year after Adar they added a month called Ve-Adar of 30 days.Days in the year365
NoThe Arabic and Turkish year.Days NoThe ancient Grecian year.Days
1MuharramJuly 16301HecatombæonJune-July30
2SapharAugust 15292MetagitnionJuly-Aug.29
3Rabia I.Septemb. 13303BoedromionAug.-Sept.30
4Rabia II.October 13294PyanepsionSept.-Oct.29
5Jomada I.Novemb. 11305MæmacterionOct.-Nov.30
6Jomada II.Decemb. 11296PosideonNov.-Dec.29
7RajabJanuary 9307GamelionDec.-Jan.30
8ShasbanFebruary 8298AnthesterionJan.-Feb.29
9RamadanMarch 9309ElapheloblionFeb.-Mar.30
10ShawalApril 82910MunichionMar.-Apr.29
11DulhaadahMay 73011ThargelionApril-May30
12DulheggiaJune 52912SchirrophorionMay-June29
Days in the year354Days in the year354
The Arabians add 11 days at the end of every year, which keep the same months to the same seasons.

Weeks

416. A month is divided into four parts called Weeks, and a Week into seven parts called Days; so that in a Julian Year there are 13 such Months, or 52 Weeks, and one Day over. The Gentiles gave the names of the Sun, Moon, and Planets to the Days of the Week. To the first, the Name of the Sun; to the second, of the Moon; to the third, of Mars; to the fourth, of Mercury; to the fifth, of Jupiter; and to the sixth, of Saturn.

Days

417. A Day is either Natural or Artificial. The Natural Day contains 24 hours; the Artificial the time from Sun-rise to Sun-set. The Natural Day is either Astronomical or Civil. The Astronomical Day begins at Noon, because the increase and decrease of Days terminated by the Horizon are very unequal among themselves; which inequality is likewise augmented by the inconstancy of the horizontal Refractions § [183]: and therefore the Astronomer takes the Meridian for the limit of diurnal Revolutions; reckoning Noon, that is the instant when the Sun’s Center is on the Meridian, for the beginning of the Day. The British, French, Dutch, Germans, Spaniards, Portuguese, and Egyptians, begin the Civil Day at mid-night: the antient Greeks, Jews, Bohemians, Silesians, with the modern Italians, and Chinese, begin it at Sun-setting: And the antient Babylonians, Persians, Syrians, with the modern Greeks, at Sun-rising.

Hours

418. An Hour is a certain determinate part of the Day, and is either equal or unequal. An equal Hour is the 24th part of a mean natural Day, as shewn by well regulated Clocks and Watches; but those Hours are not quite equal as measured by the returns of the Sun to the Meridian, because of the obliquity of the Ecliptic and Sun’s unequal motion in it § [224-245]. Unequal Hours are those by which the Artificial Day is divided into twelve Parts, and the Night into as many.

Minutes, Seconds, Thirds, and Scruples.

419. An Hour is divided into 60 equal parts called Minutes, a minute into 60 equal parts called Seconds, and these again into 60 equal parts called Thirds. The Jews, Chaldeans, and Arabians, divide the Hour into 1080 equal parts called Scruples; which number contains 18 times 60, so that one minute contains 18 Scruples.

Cycles, of the Sun, Moon, and Indiction.

420. A Cycle is a perpetual round, or circulation of the same parts of time of any sort. The Cycle of the Sun is a revolution of 28 years, in which time, the days of the months return again to the same days of the week; the Sun’s Place to the same Signs and Degrees of the Ecliptic on the same months and days, so as not to differ one degree in 100 years; and the leap-years begin the same course over again with respect to the days of the week on which the days of the months fall. The Cycle of the Moon, commonly called the Golden Number, is a revolution of 19 years; in which time, the Conjunctions, Oppositions, and other Aspects of the Moon are within an hour and half of being the same as they were on the same days of the months 19 years before. The Indiction is a revolution of 15 years, used only by the Romans for indicating the times of certain payments made by the subjects to the republic: It was established by Constantine, A.D. 312.

To find the Years of these Cycles.

421. The year of our Saviour’s Birth, according to the vulgar Æra, was the 9th year of the Solar Cycle; the first year of the Lunar Cycle; and the 312th year after his birth was the first year of the Roman Indiction. Therefore, to find the year of the Solar Cycle, add 9 to any given year of Christ, and divide the sum by 28, the Quotient is the number of Cycles elapsed since his birth, and the remainder is the Cycle for the given year: if nothing remains, the Cycle is 28. To find the Lunar Cycle, add 1 to the given year of Christ, and divide the sum by 19; the Quotient is the number of Cycles elapsed in the interval, and the remainder is the Cycle for the given year: if nothing remains, the Cycle is 19. Lastly, subtract 312 from the given year of Christ, and divide the remainder by 15; and what remains after this division is the Indiction for the given year: if nothing remains, the Indiction is 15.

The deficiency of the Lunar Cycle, and consequence thereof.

422. Although the above deficiency in the Lunar Cycle of an hour and half every 19 years be but small, yet in time it becomes so sensible as to make a whole Natural Day in 310 years. So that, although this Cycle be of use, when rightly placed against the days of the month in the Calendar, as in our Common Prayer Books, for finding the days of the mean Conjunctions or Oppositions of the Sun and Moon, and consequently the time of Easter; it will only serve for 310 years Old Style. For as the New and Full Moons anticipate a day in that time, the Golden Numbers ought to be placed one day earlier in the Calendar for the next 310 years to come. These Numbers were rightly placed against the days of New Moon in the Calendar, by the Council of Nice, A. D. 325; but the anticipation which has been neglected ever since, is now grown almost into 5 days: and therefore, all the Golden Numbers ought now to be placed 5 days higher in the Calendar for the O.S. than they were at the time of the said Council; or six days lower for the New Style, because at present it differs 11 days from the Old.

Days Jan.Feb.MarchAprilMayJuneJulyAugustSept.Octob.Nov.Dec.
1 9 917176 11 19
2 17 61414311 19
3 176176 311 1988
4 6 614143 198 16
5 14 31111198 16
6 143143 19 1655
7 3 3111119 816 13
8 11 1988165513
9 11191119 13 2
10 19881616513 210
11 198 5132210
12 816816165 10 18
13 51313210 187
14 165165 21018187
15 5 513132 7 15
16 13 21010187 15
17 132132 187 1544
18 2 2101018 15 12
19 10 1877154412
20 10181018 15 1211
21 187715 412 9
22 187 154412119
23 715715 12 91717
24 154412 19 6
25 154 12 1917176
26 4 412 1 6 14
27 12 1199176 14
28 12112 9 176141433
29 1 19 17 3 11
30 1766143 11
31 9 9 143 11 19

How to find the day of the New Moon by the Golden Number.

423. In the annexed Table, the Golden Numbers under the months stand against the days of New Moon in the left hand column, for the New Style; adapted chiefly to the second year after leap-year as being the nearest mean for all the four; and will serve till the year 1900. Therefore, to find the day of New Moon in any month of a given year till that time, look for the Golden Number of that year under the desired month, and against it, you have the day of New Moon in the left hand column. Thus, suppose it were required to find the day of New Moon in September 1757; the Golden Number for that year is 10, which I look for under September and right against it in the left hand column I find 13, which is the day of New Moon in that month. N. B. If all the Golden Numbers, except 17 and 6, were set one day lower in the Table, it would serve from the beginning of the year 1900 till the end of the year 2199. The first Table after this chapter shews the Golden Number for 4000 years after the birth of Christ, by looking for the even hundreds of any given year at the left hand, and for the rest to make up that year at the head of the Table; and where the columns meet, you have the Golden Number (which is the same both in Old and New Style) for the given year. Thus, suppose the Golden Number was wanted for the year 1757; I look for 1700 at the left hand of the Table, and for 57 at the top of it; then guiding my eye downward from 57 to over against 1700, I find 10, which is the Golden Number for that year.

A perpetual Table of the time of New Moon to the nearest hour, for the Old Style.

424. But because the lunar Cycle of 19 years sometimes includes five leap-years, and at other times only four, this Table will sometimes vary a day from the truth in leap-years after February. And it is impossible to have one more correct, unless we extend it to four times 19 or 76 years; in which there are 19 leap years without a remainder. But even then to have it of perpetual use, it must be adapted to the Old Style, because in every centurial year not divisible by 4, the regular course of leap-years is interrupted in the New; as will be the case in the year 1800. Therefore, upon the regular Old Style plan, I have computed the following Table of the mean times of all the New Moons to the nearest hour for 76 years; beginning with the year of Christ 1724, and ending with the year 1800.

This Table may be made perpetual, by deducting 6 hours from the time of New Moon in any given year and month from 1724 to 1800, in order to have the mean time of New Moon in any year and month 76 years afterward; or deducting 12 hours for 152 years, 18 hours for 228 years; and 24 hours for 304 years, because in that time the changes of the Moon anticipate almost a complete natural day. And if the like number of hours be added for so many years past, we shall have the mean time of any New Moon already elapsed. Suppose, for example, the mean time of Change was required for January 1802; deduct 76 years and there remains 1726, against which in the following Table under January I find the time of New Moon was on the 21st day at 11 in the evening: from which take 6 hours and there remains the 21st day at 5 in the evening for the mean time of Change in January 1802. Or, if the time be required for May, A. D. 1701, add 76 years and it makes 1777, which I look for in the Table, and against it under May I find the New Moon in that year falls on the 25th day at 9 in the evening; to which add 6 hours, and it gives the 26th day at 3 in the Morning for the time of New Moon in May, A. D. 1701. By this addition for time past, or subtraction for time to come, the Table will not vary 24 hours from the truth in less than 14592 years. And if, instead of 6 hours for every 76 years, we add or subtract only 5 hours 52 minutes, it will not vary a day in 10 millions of years.

Although this Table is calculated for 76 years only, and according to the Old Style, yet by means of two easy Equations it may be made to answer as exactly to the New Style, for any time to come. Thus, because the year 1724 in this Table is the first year of the Cycle for which it is made; if from any year of Christ after 1800 you subtract 1723, and divide the overplus by 76, the Quotient will shew how many entire Cycles of 76 years are elapsed since the beginning of the Cycle here provided for; and the remainder will shew the year of the current Cycle answering to the given year of Christ. Hence, if the remainder be 0, you must instead thereof put 76, and lessen the Quotient by unity.

Then, look in the left hand column of the Table for the number in your remainder, and against it you will find the times of all the mean New Moons in that year of the present Cycle. And whereas in 76 Julian Years the Moon anticipates 5 hours 52 minutes, if therefore these 5 hours 52 minutes be multiplied by the above found Quotient, that is, by the number of entire Cycles past; the product subtracted from the times in the Table will leave the corrected times of the New Moons to the Old Style; which may be reduced to the New Style thus:

Divide the number of entire hundreds in the given year of Christ by 4, multiply this Quotient by 3, to the product add the remainder, and from their sum subtract 2: this last remainder denotes the number of days to be added to the times above corrected, in order to reduce them to the New Style. The reason of this is, that every 400 years of the New Style gains 3 days upon the Old Style: one of which it gains in each of the centurial years succeeding that which is exactly divisible by 4 without remainder; but then, when you have found the days so gained, 2 must be subtracted from their number on account of the rectifications made in the Calendar by the Council of Nice, and since by Pope Gregory. It must also be observed, that the additional days found as above directed do not take place in the centurial Years which are not multiples of 4 till February 29th, O. S. for on that day begins the difference between the Styles; till which day therefore, those that were added in the preceding years must be used. The following Example will make this accommodation plain.

Required the mean time of New Moon in June, A.D. 1909, N.S.

From 1909 take 1723 Years, and there rem. 186
Which divided by 76, gives the Quotient 2 and the remainder 34
Then, against 34 in the Table is June5d8h0mAfternoon.
And 5h 52m multiplied by 2 make to be subtr. 1144
Remains the mean time according to the Old Style, June5d 9h16mMorning.
Entire hundred in 1909 are 19, which divided by 4, quotes 4
And leaves a remainder of 3
Which Quotient multiplied by 3 makes 12, and the remainder added makes 15
From which subtract 2, and there remains 13
Which number of days added to the above time Old Style, gives June18d9h16mMorn.N.S.

So the mean time of New Moon in June 1909 New Style is the 18th day at 16 minutes past 9 in the Morning.

If 11 days be added to the time of any New Moon in this Table, it will give the time thereof according to the New Style till the year 1800. And if 14 days 18 hours 22 minutes be added to the mean time of New Moon in either Style, it will give the mean time of the next Full Moon according to that Style.

A Table shewing the times of all the mean Changes of the Moon, to the nearest Hour, through four Lunar Periods, or 76 years. M signifies morning, A afternoon.

Yrs of the Cyc.A.D.JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemb.OctoberNovemb.Decemb.
D.H.D.H.D.H.D.H.D.H.D.H.D.H.D.H.D.H.D.H.D.H.D.H.
11724145A135M136A127M118A108M99A810M610A611M412A41A
2172532M12A33M14A14M296M287A278M258A259M2310A2311M
305A
317262111A2011M2112A201A201M182A183M164A155M145A136M127A
41727118M99A119M910A911M712A70A61M41A42M23A24M
315A
51728306M287A297M278A278M259A2510M2311A2211M2112A201A202M
61729182A173M184A174M165A156M147A127M118A119M90A911M
71730711A60A81M61A62M43A43M24A25M307M288A289M
306A
81731269A2510M2610A2511M2411A230A231M212A202M193A184M175A
91732165M146A157M138A138M119A1110M911A811M712A61A62M
10173342A33M44A34M25A16M308M288A279M2610A2511M2411A
307A
111734230A221M231A222M212A203M194A185M165A166M147A148M
121735129A119M1210A1111M1011A90A91M72A62M53A44M35A
13173625M----17M299M289A2710M2611A250A2312A231A222M213A
316A308A
141737203M184A204M185A185M166A167M148A138M129A1110M1011A
151738911M712A91A81M72A63M54A45M25A26M308M298A
317A
161739289M2610A2811M2612A260A251M242A233M213A214M195A196M
171740176A167M168A159M149A1310M1211A110A912A91A82M73A
18174163M44A64M45A45M26A27M308M289A2810M2611A2611M
317A
1917422412A231A252M233A233M214A215M196A186M177A168M159A
201743149M1210A1411M1212A120A111M102A93M73A74M55A56M
21174436A27M28A19M3010M2811A280A2612A251A252M233A233M
309A
221745214A205M215A206M196A187M178A168M149A1410M1211A120A
2317461012A91A112M93A93M74A75M56A46M37A28M19A
3110M
2417472910A2811M2911A280A2712A261A262M243A233M224A215M206A
251748196M177A188M169A169M1410A1411M1212A110A111M92A93M
26174973A64M75A66M56A47M38A29M3010M2911A280A2712A
319A
271750261A252M263A254M244A235M226A217M197A198M179A1710M
2817511510A1411M1511A140A1312A121A122M103A93M84A75M66A
29175256M37A48M29A29M3011M2912A280A271M262A253M243A
3110A
301753234M215A236M217A217M198A199M1710A1610M1511A140A141M
311754121A112M123A114M104A95M86A77M57A58M39A310M
321755110A----111A2912A291A282M273A253M244A245M226A226M
3111M310A
331756207A198M199A189M1710A1611M1512A141A131M122A113M104A
34175794M75A96M77A77M58A59M310A210M114A301M291A
310A
351758282M263A283M264A264M245A246M227A217M208A199M1810A
3617591710M1511A170A161M151A142M133A122M104A105M86A87M
37176067A58M59A410M310A211M112A301M282A283M264A264M
311A
381761245A236M247A238M229A2110M2010A1911M1711A170A161M152A
391762142M123A143M124A124M105A106M87A77M68A59M410A
401763311M112A30A21M11A293A294M274M265M256A247M237A
312M
411764228M209A2110M1911A1911M1712A171A162M142A143M124A125M
421765105A96M106A97M87A78M69A510M310A311M112A11A
311M
431766292A283M294A285M275A266M257A248M228A229M2010A2011M
4417671811A170A191M172A172M153A154M135A126M116A107M98A
45176888M69A710M511A511M312A31A22M303M294A285M275A
312A
461769266M247A267M248A248M229A2210M2011A1911M1812A171A172M
471770152A143M154A145M135A124M117A108M88A89M610A611M
481771411M30A51M32A32M13A14M295M276A277M258A259M
305A
491772239A2210M2210A2111M2011A190A191M172A162M153A144M135A
501773125M106A127M108A108M89A89M610A511M412A31A32M
51177412A----14A295A296M277A278M258A249M2310A2211M2111A
313M315M
521775200A191M202A193M183A174M165A156M136A137M118A119M
53177699A810M810A711M612A50A51M32A22M13A295A295M
314M
541777276A267M278A269M259A2410M2311A220A2012A201A192M183A
551778173M154A175M156A156M137A138M119A109M910A811M712A
56177960A51M62A53M43A34M25A16M297M288A279M269A
306A
5717802510M2311A2411M2212A220A211M202A193M173A174M155A156M
581781136A127M138A129M119A1010M911A80A612A61A52M43A
59178233M14A35M16A16M298M289A279M2510A2511M2312A230A
307A
601783221M202A222M203A203M184A185M166A156M147A138M129A
611784119M910A1011M812A80A71M62A53M33A34M15A16M
306A
621785297M278A299M2710A2710M2511A250A241M221A222M203A203M
631786184A175M185A176M166A157M148A139M119A1110M911A90A
641787712A61A82M63A63M44A45M26A16M308M289A289M
307A
6517882610A2511M2512A241A241M222A223M204M194M185A176M16 7A
661789157M138A159M1310A1310M1111A110A101M81A82M63A64M
67179044A35M45A36M26A17M309M289A2710M2611A250A2412A
308A
681791231A222M233A224M214A205M196A187M167A168M149A1410M
6917921210A1111M1112A101A101M82A83M64A54A45A36M27A
70179317M----19M2910M2811A270A271M251A242M233A224M214A
308A3010A
711794205M186A206M187A187M168A169M1410A1310M1211A110A111M
72179591A82M93A84M74A65M56A47M27A28M3010M2910A
319A
7317962811M2612A270A261M251A242M233A224M204A205M186A187M
741797167A158M169A1510M1410A1311M1212A111A101M92A83M74A
75179864M45A66M47A47M28A29M3010M2811A280A271M261A
3110A
761799252M233A254M235A235M216A216M198A188M179A1610M1511A
118001411A1212A130A121M111A102M93A84M64A65M46A47M

The year 1800 begins a new Cycle.

Easter Cycle, deficient.

425. The Cycle of Easter, also called the Dionysian Period, is a revolution of 532 years, found by multiplying the Solar Cycle 28 by the Lunar Cycle 19. If the New Moons did not anticipate upon this Cycle, Easter-Day would always be the Sunday next after the first Full Moon which succeeds the 21st of March. But, on account of the above anticipation § [422], to which no proper regard was had before the late alteration of the Style, the Ecclesiastic Easter has several times been a week different from the true Easter within this last Century: which inconvenience is now remedied by making the Table which used to find Easter for ever, in the Common Prayer Book, of no longer use than the Lunar difference from the New Style will admit of.

Number of Direction.
To find the true Easter.

426. The earliest Easter possible is the 22d of March, the latest the 25th of April. Within these limits are 35 days, and the number belonging to each of them is called the Number of Direction; because thereby the time of Easter is found for any given year. To find the Number of Direction, according to the New Style, enter [Table V] following this Chapter, with the compleat hundreds of any given year at the top, and the years thereof (if any) below an hundred at the left hand; and where the columns meet is the Dominical Letter for the given year. Then, enter [Table I], with the compleat hundreds of the same year at the left hand, and the years below an hundred at the top; and where the columns meet is the Golden Number for the same year. Lastly, enter [Table II] with the Dominical Letter at the left hand and Golden Number at the top; and where the columns meet is the Number of Direction for that year; which number, added to the 21st day of March shews on what day either of March or April Easter Sunday falls in that year. Thus, the Dominical Letter New Style for the year 1757 is B ([Table V]) and the Golden Number is 10, ([Table I]) by which in [Table II], the Number of Direction is found to be 20; which, reckoned from the 21st of March, ends on the 10th of April, and that is Easter Sunday in the year 1757. N. B. There are always two Dominical Letters to the leap-year, the first of which takes place to the 24th of February, the last for the following part of the year.

Dominical Letter.

427. The first seven Letters of the Alphabet are commonly placed in the annual Almanacks to shew on what days of the week the days of the months fall throughout the year. And because one of those seven Letters must necessarily stand against Sunday it is printed in a capital form, and called the Dominical Letter: the other six being inserted in small characters to denote the other six days of the week. Now, since a common Julian Year contains 365 Days, if this number be divided by 7 (the number of days in a week) there will remain one day. If there had been no remainder, ’tis plain the year would constantly begin on the same day of the week. But since one remains, ’tis as plain that the year must begin and end on the same day of the week; and therefore the next year will begin on the day following. Hence, when January begins on Sunday, A is the Dominical or Sunday Letter for that year: then, because the next year begins on Monday, the Sunday will fall on the seventh day, to which is annexed the seventh Letter G, which therefore will be the Dominical Letter for all that year: and as the third year will begin on Tuesday, the Sunday will fall on the sixth day; therefore F will be the Sunday Letter for that year. Whence ’tis evident that the Sunday Letters will go annually in a retrograde order thus, G, F, E, D, C, B, A. And in the course of seven years, if they were all common ones, the same days of the week and Dominical Letters would return to the same days of the months. But because there are 366 days in a leap-year, if this number be divided by 7, there will remain two days over and above the 52 weeks of which the year consists. And therefore, if the leap-year begins on Sunday, it will end on Monday; and the next year will begin on Tuesday, the first Sunday whereof must fall on the sixth of January, to which is annexed the Letter F, and not G as in common years. By this means, the leap-year returning every fourth year, the order of the Dominical Letters is interrupted; and the Series does not return to its first state till after four times seven, or 28 years: and then the same days of the month return in order to the same days of the week.

To find the Dominical Letter.

428. To find the Dominical Letter for any year either before or after the Christian Æra[[87]]: In [Table III] or IV for Old Style, or V for New Style, look for the hundreds of years at the head of the Table, and for the years below an hundred (to make up the given year) at the left hand: and where the columns meet you have the Dominical Letter for the year desired. Thus, suppose the Dominical Letter be required for the year of Christ 1758, New Style, I look for 1700 at the head of [Table V], and for 58 at the left hand of the same Table; and in the angle of meeting, I find A, which is the Dominical Letter for that year. If it was wanted for the same year Old Style, it would be found by [Table IV] to be D. But to find the Dominical Letter for any given year before Christ, subtract one from that year and then proceed in all respects as just now taught, to find it by [Table III] Thus, suppose the Dominical Letter be required for the 585th year before the first year of Christ, look for 500 at the head of [Table III], and for 84 at the left hand; in the meeting of these columns is FE, which were the Dominical Letters for that year, and shews that it was a leap-year; because, leap-year has always two Dominical Letters.

To find the Days of the Months.

429. To find the day of the month answering to any day of the week, or the day of the week answering to any day of the month; for any year past or to come: Having found the Dominical Letter for the given year, enter [Table VI], with the Dominical Letter at the head; and under it, all the days in that column to the right hand are Sundays, in the divisions of the months; the next column to the right are Mondays; the next, Tuesdays; and so on to the last column under G, from which go back to the column under A, and thence proceed towards the right hand as before. Thus, in the year 1757, the Dominical Letter New Style is B, in [Table V], then in [Table VI] all the days under B are Sundays in that year, viz. the 2d, 9th, 16th, 23d, and 30th of January and October; the 6th, 13th, 20th, and 27th of February, March and November; the 3d, 10th, and 17th, of April and July, together with the 31st of July: and so on to the foot of the column. Then, of course, all the days under C on Mondays, namely the 3d, 10th, &c. of January and October; and so of all the rest in that column. If the day of the week answering to any day of the month be required, it is easily had from the same Table by the Letter that stands at the top of the column in which the given day of the month is found. Thus, the Letter that stands over the 28th of May is A; and in the year 585 before Christ the Dominical Letter was found to be FE § [428]; which being a leap-year, and E taking place from the 24th of February to the end of that year, shews by the Table that the 25th of May was on a Sunday; and therefore the 28th must have been on a Wednesday: for when E stands for Sunday, F must stand for Monday, G for Tuesday, A for Wednesday, B for Thursday, C for Friday, and D for Saturday. Hence, as it appears that the famous Eclipse of the Sun foretold by Thales, by which a peace was brought about between the Medes and Lydians, happened on the 28th of May, in the 585th year before Christ, it certainly fell on a Wednesday.

Julian Period.

430. From the multiplication of the Solar Cycle of 28 years into the Lunar Cycle of 19 years, arises the great Julian Period consisting of 7980 years; which had its beginning 764 years before the supposed year of the creation (when all the three Cycles began together) and is not yet compleated, and therefore it comprehends all other Cycles, Periods and Æras. There is but one year in the whole Period which has the same numbers for the three Cycles of which it is made up: and therefore, if historians had remarked in their writings the Cycles of each year, there had been no dispute about the time of any action recorded by them.

To find the year of this Period.
And the Cycles of that year.

431. The Dionysian or vulgar Æra of Christ’s birth was about the end of the year of the Julian Period 4713; and consequently the first year of his age, according to that account, was the 4714th year of the said Period. Therefore, if to the current year of Christ we add 4713, the Sum will be the year of the Julian Period. So the year 1757 will be found to be the 6470th year of that Period. Or, to find the year of the Julian Period answering to any given year before the first year of Christ, subtract the number of that given year from 4714, and the remainder will be the year of the Julian Period. Thus, the year 585 before the first year of Christ (which was the 584th before his birth) was the 4129th year of the said Period. Lastly, to find the Cycles of the Sun, Moon, and Indiction for any given year of this Period, divide the given year by 28, 19, and 15; the three remainders will be the Cycles sought, and the Quotients the numbers of Cycles run since the beginning of the Period. So in the above 4714th year of the Julian Period the Cycle of the Sun was 10, the Cycle of the Moon 2, and the Cycle of Indiction 4; the Solar Cycle having run through 168 courses, the Lunar 248, and the Indiction 314.

The true Æra of Christ’s birth.

432. The vulgar Æra of Christ’s birth was never settled till the year 527; when Dionysius Exiguus, a Roman Abbot, fixed it to the end of the 4713th year of the Julian Period; which was certainly four years too late. For, our Saviour was undoubtedly born before the Death of Herod the Great, who sought to kill him as soon as he heard of his birth. And, according to the testimony of Josephus (B. xvii. c. 8.) there was an eclipse of the Moon in the time of Herod’s last illness: which very eclipse our Astronomical Tables shew to have been in the year of the Julian Period 4710, March 13th, 3 hours 21 minutes after mid-night, at Jerusalem. Now, as our Saviour must have been born some months before Herod’s death, since in the interval he was carried into Ægypt; the latest time in which we can possibly fix the true Æra of his birth is about the end of the 4709th year of the Julian Period. And this is four years before the vulgar Æra thereof.

The time of his crucifixion.

In the former edition of this book, I endeavoured to ascertain the time of Christ’s death; by shewing in what year, about the reputed time of the Passion, there was a Passover Full Moon on a Friday: on which day of the week, and at the time of the Passover, it is evident from Mark xv. 42. that our Saviour was crucified. And in computing the times of all the Passover Full Moons from the 20th to the 40th year of Christ, after the Jewish manner, which was to add 14 days to the time when the New Moon next before the Passover was first visible at Jerusalem, in order to have their day of the Passover Full Moon, I found that the only Passover Full Moon which fell on a Friday, in all that time, was in the year of the Julian Period 4746, on the third day of April: which year was the 33d year of Christ’s age, reckoning from the vulgar Æra of his birth, but the 37th counting from the true Æra thereof: and was also the last year of the 402d Olympiad[[88]], in which very year Phlegon an Heathen writer tells us, there was the most extraordinary Eclipse of the Sun that ever was known, and that it was night at the sixth hour of the day. Which agrees exactly with the time that the darkness at the crucifixion began, according to the three Evangelists who mention it[[89]]: and therefore must have been the very same darkness, but mistaken by Phlegon for a natural Eclipse of the Sun; which was impossible on two accounts, 1. because it was at the time of Full Moon; and 2. because whoever takes the pains to calculate, will find that there could be no regular and total Eclipse of the Sun that year in any part of Judea, nor any where between Jerusalem and Egypt: so that this darkness must have been quite out of the common course of nature.

From the co-incidence of these characters, I made no doubt of having ascertained the true year and day of our Saviour’s death. But having very lately read what some eminent authors have wrote on the same subject, of which I was really ignorant before; and heard the opinions of other candid and ingenious enquirers after truth (which every honest man will follow wherever it leads him) and who think they have strong reasons for believing that the time of Christ’s death was not in the year of the Julian Period 4746, but in the year 4743; I find difficulties on both sides, not easily got over: and shall therefore state the case both ways as fairly as I can; leaving the reader to take which side of the Question he pleases.

Both Dr. Prideaux and Sir Isaac Newton are of opinion that Daniel’s seventy weeks, consisting of 490 years (Dan. chap. ix. v. 23-26) began with the time when Ezra received his commission from Artaxerxes to go to Jerusalem, which was in the seventh year of that King’s reign (Ezra ch. vii. v. 11-26) and ended with the death of Christ. For, by joining the accomplishment of that prophecy with the expiation of Sin, those weeks cannot well be supposed to end at any other time. And both these authors agree that this was Artaxerxes Longimanus, not Artaxerxes Mnemon. The Doctor thinks that the last of those annual weeks was equally divided between John’s ministry and Christ’s. And, as to the half week, mentioned by Daniel chap. ix. v. 27. Sir Isaac thinks it made no part of the above seventy; but only meant the three years and an half in which the Romans made war upon the Jews from spring in A.D. 67 to autumn in A.D. 70, when a final Period was put to their sacrifices and oblations by destroying their city and sanctuary, on which they were utterly dispersed. Now, both by the undoubted Canon of Ptolemy, and the famous Æra of Nabonassar, which is so well verified by Eclipses that it cannot deceive us, the beginning of these seventy weeks, or the seventh year of the reign of Artaxerxes Longimanus, is pinned down to the year of the Julian Period 4256: from which count 490 years to the death of Christ, and the same will fall in the above year of the Julian Period 4746: which would seem to ascertain the true year beyond dispute.

But as Josephus’s Eclipse of the Moon in a great measure fixes our Saviour’s birth to the end of the 4713th year of the Julian Period, and a Friday Passover Full Moon fixes the time of his death to the third of April in the 4746th year of that Period, the same as above by Daniel’s weeks, this supposes our Saviour to have been crucified in the 37th year of his age. And as St. Luke chap. iii. ver. 23. fixes the time of Christ’s baptism to the beginning of his 30th year, it would hence seem that his publick ministry, to which his baptism was the initiation, lasted seven years. But, as it would be very difficult to find account in all the Evangelists of more than four Passovers which he kept at Jerusalem during the time of his ministry, others think that he suffered in the vulgar 30th year of his age, which was really the 33d; namely in the year of the Julian Period 4743. And this opinion is farther strengthened by considering that our Saviour eat his last Paschal Supper on a Thursday evening, the day immediately before his crucifixion: and that as he subjected himself to the law, he would not break the law by keeping the Passover on the day before the law prescribed; neither would the Priests have suffered the Lamb to be killed for him before the fourteenth day of Nisan when it was killed for all the people, Exod. xii. ver. 6. And hence they infer that he kept this Passover at the same time with the rest of the Jews, in the vulgar 30th year of his age: at which time it is evident by calculation that there was a Passover Full Moon on Thursday April the 6th. But this is pressed with two difficulties. 1. It drops the last half of Daniel’s seventieth week, as of no moment in the prophecy; and 2. it sets aside the testimony of Phlegon, as if he had mistaken almost a whole Olympiad.

Others again endeavour to reconcile the whole difference, by supposing, that as Christ expressed himself only in round numbers concerning the time he was to lie in the grave, Matt. xii. 40. so might St. Luke possibly have done with regard to the year of his baptism: which would really seem to be the case when we consider, that the Jews told our Saviour, sometime before his death, Thou art not yet fifty years old, John vii. 57. which indeed was more likely to be said to a person near forty than to one but just turned of thirty. And as to his eating the above Passover on Thursday, which must have been on the Jewish Full Moon day, they think it may be easily accommodated to the 37th year of his age; since, as the Jews always began their day in the evening, their Friday of course began on the evening of our Thursday. And it is evident, as above-mentioned, that the only Jewish Friday Full Moon, at the time of their Passover, was in the vulgar 33d, but the real 37th year of Christ’s age; which was the 4746th year of the Julian Period, and the last year of the 202d Olympiad.

Æras or Epochas.

433. As there are certain fixed points in the Heavens from which Astronomers begin their computations, so there are certain points of time from which historians begin to reckon; and these points or roots of time are called Æras or Epochas. The most remarkable Æras are those of the Creation, the Greek Olympiads, the building of Rome, the Æra of Nabonassar, the death of Alexander, the birth of Christ, the Arabian Hegira, and the Persian Jesdegird: All which, together with several others of less note, have their beginnings in the following Table fixed to the years of the Julian Period, to the age of the world at those times, and to the years before and after the birth of Christ.

Julian Period.Y. of the World.Before Christ.
1. The creation of the world, according to Strauchius76413949
2. The Deluge, or Noah’s Flood242016562293
3. The Assyrian Monarchy by Nimrod253717732176
4. The Birth of Abraham 2712 19482001
5. The beginning of the Kingdom of the Argives285620921857
6. The begin. of the Kingdom of Athens by Cecrops315723931556
7. The departure of the Israelites from Egypt3216 24521497
8. Their entrance into Canaan, or the Jubilee325624921457
9. The destruction of Troy352928651184
10. The beginning of King David’s reign365328891060
11. The foundation of Solomon’s Temple369629321017
12. The Argonautic expedition37763012937
13. Arbaces, the first King of the Medes38383074175
14. Mandaucus the second38653101848
15. Sosarmus the third39153151798
16. Artica the fourth39453181768
17. Cardica the fifth39963232718
18. Phraortes the sixth40573293656
19. Cyaxares the seventh40803316633
20. The beginning of the Olympiads39383174775
21. The Catonian Epocha of the building of Rome39613197752
22. The Æra of Nabonassar39673202746
23. The destruction of Samaria39903226723
24. The Babylonish captivity41333349600
25. The destruction of Solomon’s Temple41243360589
26. The Persian monarchy founded by Cyrus41543390559
27. The battle of Marathon42243460489
28. The begin. of the reign of Art. Longimanus42493485464
29. The beginning of Daniel’s 70 weeks42563492457
30. The beginning of the Peloponnesian war42823518431
31. The death of Alexander43903626323
32. The restoration of the Jews45483784129
33. The corr. of the Calendar by Julius Cæsar4669390544
34. The beginning of the reign of Herod4673390940
35. The Spanish Æra4675391138
36. The battle at Actium4683391930
37. The taking of Alexandria4683391930
38. The Epoch of the title of Augustus4686392227
39. The true Æra of Christ’s birth470939454
40. The death of Herod471039463
41. The Diony. or vulg. Æra of the birth of Christ47133949AD0
42. The true year of Christ’s death4746398233
43. The destruction of Jerusalem4783401970
44. The Dioclesian persecution50154251302
45. The Epoch of Constantine the Great50194255306
46. The Council of Nice50384274325
47. The Epocha of the Hegira53354571622
48. The Epoch of Yesdejerd53444580631
49. The Jellalæan Epocha579150271078
50. The Epocha of the reformation623054661517

Tab. I. Shewing the Golden Number (which is the same both in the Old and New Style) from the Christian Æra to A.D. 4000.

Years less than an Hundred.
Hundreds of Years. 0123456789101112131415161718
19202122232425262728293031323334353637
38394041424344454647484950515253545556
57585960616263646566676869707172737475
76777879808182838485868788899091929394
9596979899
019003800 12345678910111213141516171819
10020003900 67891011121314151617181912345
20021004000 11121314151617181912345678910
3002200&c. 16171819123456789101112131415
4002300-- 23456789101112131415161718191
5002400-- 78910111213141516171819123456
6002500-- 12131415161718191234567891011
7002600-- 17181912345678910111213141516
8002700-- 34567891011121314151617181912
9002800-- 89101112131415161718191234567
10002900-- 13141516171819123456789101112
11003000-- 18191234567891011121314151617
12003100-- 45678910111213141516171819123
13003200-- 91011121314151617181912345678
14003300-- 14151617181912345678910111213
15003400-- 19123456789101112131415161718
16003500-- 56789101112131415161718191234
17003600-- 10111213141516171819123456789
18003700-- 15161718191234567891011121314

Tab. II. Shewing the Number of Direction, for finding Easter Sunday by the Golden Number and Dominical Letter.

G. N.12345678910111213141516171819
A26195261233191226195261252612331912
B2713627133420132720627136201334206
C281472114352172821728147211428217
D291582215292282915829151221529228
E30162231630239301692316223930239
F241732410312410311710241732410311710
G25184251132181132184251842511321811

This Table is adapted to the New Style.

Tab. III. Shewing the Dominical Letters, Old Style, for 4200 Years before the Christian Æra.

Before ChristHundreds of Years.
Years less than an Hundred.0100200300400500600
7008009001000110012001300
1400150016001700180019002000
2100220023002400250026002700
2800290030003100320033003400
3500360037003800390040004100
0285684D CC BB AA GG FF EE D
1295785EDCBAGF
2305886FEDCBAG
3315987GFEDCBA
4326088B AA GG FF EE DD CC B
5336189CBAGFED
6346290DCBAGFE
7356391EDCBAGF
8366492G FF EE DD CC BB AA G
9376593AGFEDCB
10386694BAGFEDC
11396795CBAGFED
12406896E DD CC BB AA GG FF E
13416997FEDCBAG
14427098GFEDCBA
15437199AGFEDCB
164472 C BB AA GG FF EE DD C
174573 DCBAGFE
184674 EDCBAGF
194775 FEDCBAG
204876 A GG FF EE DD CC BB A
214977 BAGFEDC
225078 CBAGFED
235179 DCBAGFE
245280 F EE DD CC BB AA GG F
255381 GFEDCBA
265482 AGFEDCB
275583 BAGFEDC

Tab. IV. Shewing the Dominical Letters, Old Style, for 4200 Years after the Christian Æra.

After ChristHundreds of Years.
Years less than an Hundred.0100200300400500600
7008009001000110012001300
1400150016001700180019002000
2100220023002400250026002700
2800290030003100320033003400
3500360037003800390040004100
0285684D CE DF EG FA GB AC B
1295785BCDEFGA
2305886ABCDEFG
3315987GABCDEF
4326088F EG FA GB AC BD CE D
5336189DEFGABC
6346290CDEFGAB
7356391BCDEFGA
8366492A GB AC BD CE DF EG F
9376593FGABCDE
10386694EFGABCD
11396795DEFGABC
12406896C BD CE DF EG FA GB A
13416997ABCDEFG
14427098GABCDEF
15437199FGABCDE
164472 E DF EG FA GB AC BD C
174573 CDEFGAB
184674 BCDEFGA
194775 ABCDEFG
204876 G FA GB AC BD CE DF E
214977 EFGABCD
225078 DEFGABC
235179 CDEFGAB
245280 B AC BD CE DF EG FA G
255381 GABCDEF
265482 FGABCDE
275583 EFGABCD

Tab. V. The Dominical Letter, New Style, for 4000 Years after the Christian Æra.

After Christ.Hundreds of Years.
Years less than an Hundred.100200300400
500600700800
900100011001200
1300140015001600
1700180019002000
2100220023002400
2500260027002800
2900300031003200
3300340035003600
3700380039004000
CEGB A
1295785BDFG
2305886ACEF
3315987GBDE
4326088F EA GC BD C
5336189DFAB
6346290CEGA
7356391BDFG
8366492A GC BC DF E
9376593FACD
10386694EGBC
11396795DFAB
12406896C BE DG FA G
13416997ACEF
14427098GBDE
15437199FACD
164472 E DG FB AC B
174573 CEGA
184674 BDFG
194775 ACEF
204876 G FB AD CE D
214977 EGBC
225078 DFAB
235179 CEGA
245280 B AD CF EG F
255381 GBDE
265482 FACD
275583 EGBC
285684 D CF EA GB A

Tab. VI. Shewing the Days of the Months for both Styles by the Dominical Letters.

Week Day.ABCDEFG
January 31
October 31		1234567
891011121314
15161718192021
22232425262728
293031
1234
Feb. 28-29
March 31
Nov. 30
567891011
12131415161718
19202122232425
262728293031
1
April 30
July 31
2345678
9101112131415
16171819202122
23242526272829
3031
12345
August 31
6789101112
13141516171819
20212223242526
2728293031
12
Septemb. 30
Decemb. 31
3456789
10111213141516
17181920212223
24252627282930
31
123456
May 31
78910111213
14151617181920
21222324252627
28293031
123
June 30
45678910
11121314151617
18192021222324
252627282930

CHAP. XXII.
A Description of the Astronomical Machinery serving to explain and illustrate the foregoing part of this Treatise.

Fronting the Title Page.
The Orrery.

434. The Orrery. This Machine shews the Motions of the Sun, Mercury, Venus, Earth, and Moon; and occasionally, the superior Planets, Mars, Jupiter, and Saturn may be put on; Jupiter’s four Satellites are moved round him in their proper times by a small Winch; and Saturn has his five Satellites, and his Ring which keeps its parallelism round the Sun; and by a Lamp put in the Sun’s place, the Ring shews all the Phases described in the 204th Article.

The Sun.
The Ecliptic.

In the Center, No 1. represents the Sun, supported by it’s Axis inclining almost 8 Degrees from the Axis of the Ecliptic; and turning round in 251⁄4 days on its Axis, of which the North Pole inclines toward the 8th Degree of Pisces in the great Ecliptic (No. 11.) whereon the Months and Days are engraven over the Signs and Degrees in which the Sun appears, as seen from the Earth, on the different days of the year.

Mercury.

The nearest Planet (No. 2) to the Sun is Mercury, which goes round him in 87 days 23 hours, or 8723⁄24 diurnal rotations of the Earth; but has no Motion round its Axis in the Machine, because the time of its diurnal Motion in the Heavens is not known to us.

Venus.

The next Planet in order is Venus (No. 3) which performs her annual Course in 224 days 17 hours; and turns round her Axis in 24 days 8 hours, or in 241⁄3 diurnal rotations of the Earth. Her Axis inclines 75 Degrees from the Axis of the Ecliptic, and her North Pole inclines towards the 20th Degree of Aquarius, according to the observations of Bianchini. She shews all the Phenomena described from the 30th to the 44th Article in Chap. I.

The Earth.

Next without the Orbit of Venus is the Earth (No. 4) which turns round its Axis, to any fixed point at a great distance, in 23 hours 56 minutes 4 seconds of mean solar time ([221] & seq.) but from the Sun to the Sun again in 24 hours of the same time. No. 6 is a sidereal Dial-Plate under the Earth; and No. 7 a solar Dial-Plate on the cover of the Machine. The Index of the former shews sidereal, and of the latter, solar time; and hence, the former Index gains one entire revolution on the latter every year, as 365 solar or natural days contain 366 sidereal days, or apparent revolutions of the Stars. In the time that the Earth makes 3651⁄4 diurnal rotations on its Axis, it goes once round the Sun in the Plane of the Ecliptic; and always keeps opposite to a moving Index (No. 10) which shews the Sun’s daily change of place, and also the days of the months.

The Earth is half covered with a black cap for dividing the apparently enlightened half next the Sun, from the other half, which when turned away from him is in the dark. The edge of the cap represents the Circle bounding Light and Darkness, and shews at what time the Sun rises and sets to all places throughout the year. The Earth’s Axis inclines 231⁄2 Degrees from the Axis of the Ecliptic, the North Pole inclines toward the beginning of Cancer; and keeps its parallelism throughout its annual Course § [48], [202]; so that in Summer the northern parts of the Earth incline towards the Sun, and in the Winter from him: by which means, the different lengths of days and nights, and the cause of the various seasons, are demonstrated to sight.

There is a broad Horizon, to the upper side of which is fixed a Meridian Semi-circle in the North and South Points, graduated on both sides from the Horizon to 90° in the Zenith, or vertical Point. The edge of the Horizon is graduated from the East and West to the South and North Points, and within these Divisions are the Points of the Compass. On the lower side of this thin Horizon Plate stand out four small Wires, to which is fixed a Twilight Circle 18 Degrees from the graduated side of the Horizon all round. This Horizon may be put upon the Earth (when the cap is taken away) and rectified to the Latitude of any place: and then, by a small Wire called the Solar Ray, which may be put on so as to proceed directly from the Sun’s Center towards the Earth’s, but to come no farther than almost to touch the Horizon, the beginning of Twilight, time of Sun-rising, with his Amplitude, Meridian Altitude, time of Setting, Amplitude, and end of Twilight, are shewn for every day of the year, at that place to which the Horizon is rectified.

The Moon.

The Moon (No. 5) goes round the Earth, from between it and any fixed point at a great distance, in 27 days 7 hours 43 minutes, or through all the Signs and Degrees of her Orbit; which is called her Periodical Revolution; but she goes round from the Sun to the Sun again, or from Change to Change, in 29 days 12 hours 45 minutes, which is her Synodical Revolution; and in that time she exhibits all the Phases already described § [255].

When the above-mentioned Horizon is rectified to the Latitude of any given place, the times of the Moon’s rising and setting, together with her Amplitude, are shewn to that place as well as the Sun’s; and all the various Phenomena of the Harvest Moon § [273] & seq. made obvious to sight.

The Nodes.

The Moon’s Orbit (No. 9.) is inclined to the Ecliptic, (No. 11.) one half being above, and the other below it. The Nodes, or Points at 0 and 0 lie in the Plane of the Ecliptic, as described § [317], [318], and shift backward through all it’s Signs and Degrees in 182⁄3 years. The Degrees of the Moon’s Latitude, to the highest at NL (North Latitude) and lowest at SL (South Latitude) are engraven both ways from her Nodes at 0 and 0; and, as the Moon rises and falls in her Orbit according to its inclination, her Latitude and Distance from her Nodes are shewn for every day; having first rectified her Orbit so as to set the Nodes to their proper places in the Ecliptic: and then, as they come about at different, and almost opposite times of the year § [319], and then point towards the Sun, all the Eclipses may be shewn for hundreds of years (without any new rectification) by turning the Machinery backward for time past, or forward for time to come. At 17 Degrees distance from each Node, on both Sides, is engraved a small Sun; and at 12 Degrees distance, a small Moon; which shew the limits of solar and lunar Eclipses § [317]: and when, at any change, the Moon falls between either of these Suns and the Node, the Sun will be eclipsed on the day pointed to by the annual Index (No. 10,) and as the Moon has then North or South Latitude, one may easily judge whether that Eclipse will be visible in the Northern or Southern Hemisphere; especially as the Earth’s Axis inclines towards the Sun or from him at that time. And when, at any Full, the Moon falls between either of the little Moon’s and Node, she will be eclipsed, and the annual Index shews the day of that Eclipse. There is a Circle of 291⁄2 equal parts (No. 8.) on the cover of the Machine, on which an Index shews the days of the Moon’s age.

[PLATE IX]. Fig. X.

There are two Semi-circles fixed to an elliptical Ring, which being put like a cap upon the Earth, and the forked part F upon the Moon, shews the Tides as the Earth turns round within them, and they are led round it by the Moon. When the different Places come to the Semi-circle AaEbB, they have Tides of Flood; and when they come to the Semicircle CED they have Tides of Ebb § [304], [305]; the Index on the hour Circle (No. 7.) shewing the times of these Phenomena.

There is a jointed Wire, of which one end being put into a hole in the upright stem that holds the Earth’s cap, and the Wire laid into a small forked piece which may be occasionally put upon Venus or Mercury, shews the direct and retrograde Motions of these two Planets, with their stationary Times and Places as seen from the Earth.

The whole Machinery is turned by a winch or handle (No. 12,) and is so easily moved that a clock might turn it without any danger of stopping.

To give a Plate of the wheel-work of this Machine, would answer no purpose, because many of the wheels lie so behind others as to hide them from sight in any view whatsoever.

Another Orrery.
[PLATE VI]. Fig. I.

435. Another Orrery. In this Machine, which is the simplest I ever saw, for shewing the diurnal and annual motions of the Earth, together with the motion of the Moon and her Nodes; A and B are two oblong square Plates held together by four upright pillars; of which three appear at f, g, and g2. Under the Plate A is an endless screw on the Axis of the handle b, which works in a wheel fixed on the same Axis with the double grooved wheel E; and on the top of this Axis is fixed the toothed wheel i, which turns the pinion k, on the top of whose Axis is the pinion k2 which turns another pinion b2, and that other turns a third, on the Axis a2 of which is the Earth U turning round; this last Axis inclining 231⁄2 Degrees. The supporter X2, in which the Axis of the Earth turns, is fixed to the moveable Plate C.

In the fixed Plate B, beyond H, is fixed the strong wire d, on which hangs the Sun T so as it may turn round the wire. To this Sun is fixed the wire or solar ray Z, which (as the Earth U turns round its Axis) points to all the places that the Sun passes vertically over, every day of the year. The Earth is half covered with a black cap a, as in the former Orrery, for dividing the day from the night; and, as the different places come out from below the edge of the cap, or go in below it, they shew the times of Sun-rising and setting every day of the year. This cap is fixed on the wire b, which has a forked piece C turning round the wire d: and, as the Earth goes round the Sun, it carries the Cap, Wire, and solar Ray round him; so that the solar Ray constantly points towards the Earth’s Center.

On the Axis of the pinion k is the pinion m, which turns a wheel on the cock or supporter n, and on the Axis of this wheel nearest n is a pinion (hid from view) under the Plate C, which pinion turns a wheel that carries the Moon V round the Earth U; the Moon’s Axis rising and falling in the socket W, which is fixed to the triangular piece above Z; and this piece is fixed to the top of the Axis of the last mentioned wheel. The socket W is slit on the outermost side; and in this slit the two pins near Y, fixed in the Moon’s Axis, move up and down; one of them being above the inclined Plane YX, and the other below it. By this mechanism, the Moon V moves round the Earth T in the inclined Orbit q, parallel to the Plane of the Ring YX; of which the Descending Node is at X, and the Ascending Node opposite to it, but hid by the supporter X2.

The small wheel E turns the large wheels D and F, of equal diameters, by cat-gut strings crossing between them: and the Axis of these two wheels are cranked at G and H, above the Plate B. The upright stems of these cranks going through the Plate C, carry it over and over the fixed Plate B, with a motion which carries the Earth U round the Sun T, keeping the Earth’s Axis always parallel to itself; or still inclining towards the left-hand of the Plate; and shewing the vicissitudes of seasons, as described in the [tenth chapter]. As the Earth goes round the Sun the pinion k goes round the wheel i, for the Axis of k never touches the fixed Plate B; but turns on a wire fixed into the Plate C.

On the top of the crank G is an Index L, which goes round the Circle m2 in the time that the Earth goes round the Sun; and points to the days of the months; which, together with the names of the seasons, are marked in this Circle.

This Index has a small grooved wheel L fixed upon it, round which, and the Plate Z, goes a cat-gut string crossing between them; and by this means the Moon’s inclined Plane YX with its Nodes is turned backward, for shewing the times and returns of Eclipses § [319], [320].

The following parts of this machine must be considered as distinct from those already described.

Towards the right hand, let S be the Earth hung on the wire e, which is fixed into the Plate B; and let O be the Moon fixed on the Axis M, and turning round within the cap P, in which, and in the Plate C the crooked wire Q is fixed. On the Axis M is also fixed the Index K, which goes round a Circle h2, divided into 291⁄2 equal parts, which are the days of the Moon’s age: but to avoid confusion in the scheme, it is only marked with the numeral figures 1 2 3 4, for the Quarters. As the crank H carries this Moon round the Earth S in the Orbit t, she shews all her Phases by means of the cap P for the different days of her age, which are shewn by the Index K; this Index, turning just as the Moon O does, demonstrates her turning round her Axis as she still keeps the same side towards the Earth S § [262].

[PL. VIII.]

At the other end of the Plate C, a Moon N goes round an Earth R in the Orbit p; but this Moon’s Axis is stuck fast into the Plate C at S2; so that neither Moon nor Axis can turn round; and as this Moon goes round her Earth she shews herself all round to it; which proves, that if the Moon was seen all round from the Earth in a Lunation, she could not turn round her Axis.

N. B. If there were only the two wheels D and F, with a cat-gut string over them, but not crossing between them, the Axis of the Earth U would keep its parallelism round the Sun T, and shew all the seasons; as I sometimes make these Machines: and the Moon O would go round the Earth S, shewing her Phases as above; as likewise would the Moon N round the Earth R; but then, neither could the diurnal motion of the Earth U on its Axis be shewn, nor the motion of the Moon V round that Earth.

The Calculator.

436. In the year 1746 I contrived a very simple Machine, and described it’s performance in a small treatise upon the Phenomena of the Harvest Moon, published in the year 1747. I improved it soon after, by adding another wheel, and called it the Calculator. It may be easily made by any Gentleman who has a mechanical Genius.

Fig. I.

The great flat Ring supported by twelve pillars, and on which the twelve Signs with their respective Degrees are laid down, is the Ecliptic; nearly in the center of it is the Sun S supported by the strong crooked Wire I; and from the Sun proceeds a Wire W, called the Solar Ray, pointing towards the center of the Earth E, which is furnished with a moveable Horizon H, together with a brazen Meridian, and Quadrant of Altitude. R is a small Ecliptic, whose Plane co-incides with that of the great one, and has the like Signs and Degrees marked upon it; and is supported by two Wires D and D, which enter into the Plate PP, but may be taken off at pleasure. As the Earth goes round the Sun, the Signs of this small Circle keep parallel to themselves, and to those of the great Ecliptic. When it is taken off, and the solar Ray W drawn farther out, so as almost to touch the Horizon H, or the Quadrant of Altitude, the Horizon being rectified to any given Latitude, and the Earth turned round its Axis by hand, the point of the Wire W shews the Sun’s Declination in passing over the graduated brass Meridian, and his height at any given time upon the Quadrant of Altitude, together with his Azimuth, or point of Bearing upon the Horizon at that time; and likewise his Amplitude, and time of Rising and Setting by the hour Index, for any day of the year that the annual Index U points to in the Circle of Months below the Sun. M is a solar Index or Pointer supported by the Wire L which is fixed into the knob K: the use of this Index is to shew the Sun’s place in the Ecliptic every day in the year; for it goes over the Signs and Degrees as the Index U goes over the months and days; or rather as they pass under the Index U, in moving the cover plate with the Earth and its Furniture round the Sun; for the Index U is fixed tight on the immoveable Axis in the Center of the Machine. K is a knob or handle for moving the Earth round the Sun, and the Moon round the Earth.

As the Earth is carried round the Sun, its Axis constantly keeps the same oblique direction, or parallel to itself § [48], [202], shewing thereby the different lengths of days and nights at different times of the year, with all the various seasons. And, in one annual revolution of the Earth, the Moon M goes 121⁄3 times round it from Change to Change, having an occasional provision for shewing her different Phases. The lower end of the Moon’s Axis bears by a small friction wheel upon the inclined Plane T, which causes the Moon to rise above and sink below the Ecliptic R in every Lunation; crossing it in her Nodes, which shift backward through all the Signs and Degrees of the said Ecliptic, by the retrograde Motion of the inclined Plane T, in 18 years and 225 days. On this Plane the Degrees and Parts of the Moon’s North and South Latitude are laid down from both the Nodes, one of which, viz. the Descending Node appears at 0, by DN above B; the other Node being hid from Sight on this Plane by the plate PP; and from both Nodes, at proper distances, as in the other Orrery, the limits of Eclipses are marked, and all the solar and lunar Eclipses are shewn in the same manner, for any given year, within the limits of 6000, either before or after the Christian Æra. On the plate that covers the wheel-work, under the Sun S, and round the knob K are Astronomical Tables, by which the Machine may be rectified to the beginning of any given year within these limits, in three or four minutes of time; and when once set right, may be turned backward for 300 years past, or forward for as many to come, without requiring any new rectification. There is a method for its adding up the 29th of February every fourth year, and allowing only 28 days to that month for every other three: but all this being performed by a particular manner of cutting the teeth of the wheels, and dividing the month circle, too long and intricate to be described here, I shall only shew how these motions may be performed near enough for common use, by wheels with grooves and cat-gut strings round them, only here I must put the Operator in mind that the grooves are to be made sharp (not round) bottomed to keep the strings from slipping.

The Moon’s Axis moves up and down in the socket N fixed into the bar O (which carries her round the Earth) as she rises above or sinks below the Ecliptic; and immediately below the inclined Plane T is a flat circular plate (between Y and T) on which the different Excentricities of the Moon’s Orbit are laid down; and likewise her mean Anomaly and elliptic Equation by which her true Place may be very nearly found at any time. Below this Apogee-plate, which shews the Anomaly, &c. is a Circle Y divided into 291⁄2 equal parts which are the days of the Moon’s age: and the forked end A of the Index AB (Fig II) may be put into the Apogee-part of this plate; there being just such another Index to put into the inclined Plane T at the Ascending Node; and then the curved points B of these Indexes shew the direct motion of the Apogee, and retrograde motion of the Nodes through the Ecliptic R, with their Places in it at any given time. As the Moon M goes round the Earth E, she shews her Place every day in the Ecliptic R, and the lower end of her Axis shews her Latitude and distance from her Node on the inclined Plane T, also her distance from her Apogee and Perigee, together with her mean Anomaly, the then Excentricity of her Orbit, and her elliptic Equation, all on the Apogee Plate, and the day of her age in the Circle Y of 291⁄2 equal parts; for every day of the year pointed out by the annual Index U in the Circle of months.

Having rectified the Machine by the Tables for the beginning of any year, move the Earth and Moon forward by the knob K, until the annual Index comes to any given day of the month; then stop, and not only all the above Phenomena may be shewn for that day, but also, by turning the Earth round its Axis, the Declination, Azimuth, Amplitude, Altitude of the Moon at any hour, and the times of her Rising and Setting, are shewn by the Horizon, Quadrant of Altitude, and hour Index. And in moving the Earth round the Sun, the days of all the New and Full Moons and Eclipses in any given year are shewn. The Phenomena of the Harvest Moon, and those of the Tides, by such a cap as that in [Plate 9] Fig. 10. put upon the Earth and Moon, together with the solution of many problems not here related, are made conspicuous.

[PL. VIII.]

The easiest, though not the best way, that I can instruct any mechanical person to make the wheel-work of such a machine, is as follows; which is the way that I made it, before I thought of numbers exact enough to make it worth the trouble of cutting teeth in the wheels.

Fig. III.

Fig. 3d of [Plate 8] is a section of this Machine; in which ABCD is a frame of wood held together by four pillars at the corners, whereof two appear at AC and BD. In the lower Plate CD of this Frame are three small friction-wheels, at equal distances from each other; two of them appearing at e and e. As the frame is moved round, these wheels run upon the fixed bottom Plate EE which supports the whole work.

In the Center of this last mentioned Plate is fixed the upright Axis f FFG, and on the same Axis is fixed the wheel HHH in which are four grooves I, X, k, L of different Diameters. In these grooves are cat-gut strings going also round the separate wheels M, N, O and P.

The wheel M is fixed on a solid Spindle or Axis, the lower pivot of which turns at R in the under Plate of the moveable frame ABCD; and on the upper end of this Axis is fixed the Plate o o (which is PP, under the Earth, in Fig. I.) and to this Plate is fixed, at an Angle of 231⁄2 Degrees inclination, the Dial-plate below the Earth T; on the Axis of which, the Index q is turned round by the Earth. This Axis, together with the Wheel M, and Plate o o, keep their parallelism in going round the Sun S.

On the Axis of the wheel M is a moveable socket on which the small wheel N is fixed, and on the upper end of this socket is put on tight (but so as it may be occasionally turned by hand) the bar ZZ (viz. the bar O in Fig. I.) which carries the Moon m round the Earth T, by the Socket n, fixed into the bar. As the Moon goes round the Earth her Axis rises and falls in the Socket n; because, on the lower end of her Axis, which is turned inward, there is a small friction Wheel s running on the inclined Plane X (which is T in Fig. I.) and so causes the Moon alternately to rise above and sink below the little Ecliptic VV (R in Fig. I.) in every Lunation.

On the Socket or hollow Axis of the Wheel N, there is another Socket on which the Wheel O is fixed; and the Moon’s inclined Plane X is put tightly on the upper end of this Socket, not on a square, but on a round, that it may be occasionally set by hand without wrenching the Wheel or Axle.

Lastly, on the hollow Axis of the Wheel O is another Socket on which is fixed the Wheel P, and on the upper end of this Socket is put on tightly the Apogee-plate Y, (that immediately below T in Fig. I.) all these Axles turn in the upper Plate of the moveable frame at Q which Plate is covered with the thin Plate cc (screwed to it) whereon are the fore-mentioned Tables and month Circle in Fig. I.

The middle part of the thick fixed Wheel HHH is much broader than the rest of it, and comes out between the Wheels M and O almost to the Wheel N. To adjust the diameters of the grooves of this fixed wheel to the grooves of the separate Wheels M, N, O and P, so as they may perform their motions in the proper times, the following method must be observed.

The Groove of the Wheel M, which keeps the parallelism of the Earth’s Axis, must be precisely of the same Diameter as the lower Groove I of the fixed Wheel HHH; but, when this Groove is so well adjusted as to shew, that in ever so many annual revolutions of the Earth, its Axis keeps its parallelism, as may be observed by the solar Ray W (Fig. I.) always coming precisely to the same Degree of the small Ecliptic R at the end of every annual revolution, when the Index M points to the like Degree in the great Ecliptic; then, with the edge of a thin File give the Groove of the Wheel M a small rub all round; and by that means, lessening the Diameter of the Groove, perhaps about the 20th part of a hair’s breadth, it will cause the Earth to shew the precession of the Equinoxes; which, in many annual revolutions will begin to be sensible as the Earth’s Axis slowly deviates from its parallelism § [246], towards the antecedent Signs of the Ecliptic.

The Diameter of the Groove of the Wheel N, which carries the Moon round the Earth, must be to the Diameter of the Groove X as a Lunation is to a year; that is, as 291⁄2 to 3651⁄4.

The Diameter of the Groove of the Wheel O, which turns the inclined Plane X with the Moon’s Nodes backward, must be to the Diameter of the Groove k as 20 to 18225⁄365. And,

Lastly, the Diameter of the Groove of the Wheel P, which carries the Moon’s Apogee forward, must be to the Diameter of the Groove L as 70 to 62.

[PLATE IV].

But, after all this nice adjustment of the Grooves to the proportional times of their respective Wheels turning round, and which seems to promise very well in Theory, there will still be found a necessity of a farther adjustment by hand; because proper allowance must be made for the Diameters of the cat-gut strings: and the Grooves must be so adjusted by hand, as, that in the time the Earth is moved once round the Sun, the Moon must perform 12 synodical revolutions round the Earth, and be almost 11 days old in her 13th revolution. The inclined Plane with its Nodes must go once round backward through all the Signs and Degrees of the small Ecliptic in 18 annual revolutions of the Earth and 225 days over. And the Apogee-plate must go once round forward, so as its Index may go over all the Signs and Degrees of the small Ecliptic in eight years (or so many annual revolutions of the Earth) and 312 days over.

N. B. The string which goes round the Grooves X and N for the Moon’s Motion must cross between these Wheels; but all the rest of the strings go in their respective Grooves IM, kO, and LP without crossing.

The Cometarium.

437. The Cometarium. This curious Machine shews the Motion of a Comet or excentric Body moving round the Sun, describing equal Areas in equal times § [152], and may be so contrived as to shew such a Motion for any Degree of Excentricity. It was invented by the late Dr. Desaguliers.

Fig. IV.

The dark elliptical Groove round the letters abcdefghiklm is the Orbit of the Comet Y: this Comet is carried round in the Groove according to the order of letters, by the Wire W, fixed in the Sun S, and slides on the Wire as it approaches nearer to or recedes farther from the Sun, being nearest of all in the Perihelion a, and farthest in the Aphelion g. The Areas aSb, bSc, cSd &c. or contents of these several Triangles are all equal; and in every turn of the Winch N the Comet Y is carried over one of these Areas; consequently in as much time as it moves, from f to g, or from g to h, it moves from m to a, or from a to b; and so of the rest, being quickest of all at a, and slowest at g. Thus, the Comet’s velocity in its Orbit continually decreases from the Perihelion a to the Aphelion g; and increases in the same proportion from g to a.

[PLATE IV].

The elliptic Orbit is divided into 12 equal Parts or Signs with their respective Degrees, and so is the Circle n o p q r s t n which represents a great Circle in the Heavens, and to which all the fixed Stars in the Comet’s way are referred. Whilst the Comet moves from f to g in its Orbit it appears to move only about 5 Degrees in this Circle, as is shewn by the small knob on the end of the Wire W; but in as short time as the Comet moves from m to a, or from a to b, and it appears to describe the large space tn or no in the Heavens, either of which spaces contains 120 Degrees or four Signs. Were the Excentricity of its Orbit greater, the greater still would be the difference of its Motion, and vice versâ.

ABCDEFGHIKLMA is a circular Orbit for shewing the equable Motion of a Body round the Sun S, describing equal Areas ASB, BSC, &c. in equal times with those of the Body Y in its elliptical Orbit above mentioned; but with this difference, that the circular Motion describes the equal Arcs AB, BC, &c. in the same equal times that the elliptical Motion describes the unequal Arcs ab, bc, &c.

Now, suppose the two Bodies Y and I to start from the Points a and A at the same moment of time, and each having gone round its respective Orbit, to arrive at these Points again at the same instant, the Body Y will be forwarder in its Orbit than the Body I all the way from a to g, and from A to G; but I will be forwarder than Y through all the other half of the Orbit; and the difference is equal to the Equation of the Body Y in its Orbit. At the Points a, A, and g, G, that is, in the Perihelion and Aphelion, they will be equal; and then the Equation vanishes. This shews why the Equation of a Body moving in an elliptic Orbit, is added to the mean or supposed circular Motion from the Perihelion to the Aphelion, and subtracted from the Aphelion to the Perihelion, in Bodies moving round the Sun, or from the Perigee to the Apogee, and from the Apogee to the Perigee in the Moon’s Motion round the Earth, according to the Precepts in the 355th Article; only we are to consider, that when Motion is turned into Time, it reverses the titles in the Table of The Moon’s elliptic Equation.

Fig. V.

This curious Motion is performed in the following manner. ABC is a wooden bar (in the box containing the wheel-work) above which are the wheels D and E; and below it the elliptic Plates FF and GG; each Plate being fixed on an Axis in one of its Focuses, at E and K; and the Wheel E is fixed on the same Axis with the Plate FF. These Plates have Grooves round their edges precisely of equal Diameters to one another, and in these Grooves is the cat-gut string gg, gg crossing between the Plates at h. On H, the Axis of the handle or winch N in Fig. 4th, is an endless screw in Fig. 5, working in the Wheels D and E, whose numbers of teeth being equal, and should be equal to the number of lines aS, bS, cS, &c. in Fig. 4, they turn round their Axes in equal times to one another, and to the Motion of the elliptic Plates. For, the Wheels D and E having equal numbers of teeth, the Plate FF being fixed on the same Axis with the Wheel E, and the Plate FF turning the equally big Plate GG by a cat-gut string round them both, they must all go round their Axes in as many turns of the handle N as either of the Wheels has teeth.

’Tis easy to see, that the end h of the elliptical Plate FF being farther from its Axis E than the opposite end i is, must describe a Circle so much the larger in proportion; and therefore move through so much more space in the same time; and for that reason the end h moves so much faster than the end i, although it goes no sooner round the Center E. But then, the quick-moving end h of the Plate FF leads about the short end hK of the Plate GG with the same velocity; and the slow moving end i of the Plate FF coming half round as to B, must then lead the long end k of the Plate GG as slowly about: So that the elliptical Plate FF and it’s Axis E move uniformly and equally quick in every part of its revolution; but the elliptical Plate GG, together with its Axis K must move very unequally in different parts of its revolution; the difference being always inversely as the distance of any point of the Circumference of GG from its Axis at K: or in other words, to instance in two points, if the distance Kk be four, five, or six times as great as the distance Kh, the Point h will move in that position four, five, or six times as fast as the Point k does, when the Plate GG has gone half round: and so on for any other Excentricity or difference of the Distances Kk and Kh. The tooth i on the Plate FF falls in between the two teeth at k on the Plate GG, by which means the revolution of the latter is so adjusted to that of the former, that they can never vary from one another.

On the top of the Axis of the equally moving Wheel D, in Fig. 5th, is the Sun S in Fig. 4th; which Sun, by the Wire Z fixed to it, carries the Ball I round the Circle ABCD, &c. with an equable Motion according to the order of the letters: and on the top of the Axis K of the unequally moving Ellipsis GG, in Fig. 5th, is the Sun S in Fig. 4th, carrying the Ball Y unequably round in the elliptical Groove a b c d, &c. N.B. This elliptical Groove must be precisely equal and similar to the verge of the Plate GG, which is also equal to that of FF.

In this manner, Machines may be made to shew the true Motion of the Moon about the Earth, or of any Planet about the Sun; by making the elliptical Plates of the same Excentricities, in proportion to the Radius, as the Orbits of the Planets are whose Motions they represent: and so, their different Equations in different parts of their Orbits may be made plain to sight; and clearer Ideas of these Motions and Equations acquired in half an hour, than could be gained from reading half a day about such Motions and Equations.

The improved Celestial Globe.
[PLATE III]. Fig. III.

438. The Improved Celestial Globe. On the North Pole of the Axis, above the Hour Circle, is fixed an Arch MKH of 231⁄2 Degrees; and at the end H is fixed an upright pin HG, which stands directly over the North Pole of the Ecliptic, and perpendicular to that part of the surface of the Globe. On this pin are two moveable Collets at D and H, to which are fixed the quadrantal Wires N and O, having two little Balls on their ends for the Sun and Moon, as in the Figure. The Collet D is fixed to the circular Plate F whereon the 291⁄2 days of the Moon’s age are engraven, beginning just under the Sun’s Wire N; and as this Wire is moved round the Globe, the Plate F turns round with it. These Wires are easily turned if the Screw G be slackened; and when they are set to their proper places, the Screw serves to fix them there so, as in turning the Ball of the Globe, the Wires with the Sun and Moon go round with it; and these two little Balls rise and set at the same times, and on the same points of the Horizon, for the day to which they are rectified, as the Sun and Moon do in the Heavens.

Because the Moon keeps not her course in the Ecliptic (as the Sun appears to do) but has a Declination of 51⁄3 Degrees on each side from it in every Lunation § [317], her Ball may be screwed as many Degrees to either side of the Ecliptic as her Latitude or Declination from the Ecliptic amounts to at any given time; and for this purpose S is a small piece of pasteboard, of which the curved edge S is to be set upon the Globe at right Angles to the Ecliptic, and the dark line over S to stand upright upon it. From this line, on the convex edge, are drawn the 51⁄3 Degrees of the Moon’s Latitude on both sides of the Ecliptic; and when this piece is set upright on the Globe, it’s graduated edge reaches to the Moon on the Wire O, by which means she is easily adjusted to her Latitude found by an Ephemeris.

The Horizon is supported by two semicircular Arches, because Pillars would stop the progress of the Balls when they go below the Horizon in an oblique sphere.

To rectify it.

To rectify the Globe. Elevate the Pole to the Latitude of the Place; then bring the Sun’s place in the Ecliptic for the given day to the brasen Meridian, and set the Hour Index to XII at noon, that is, to the upper XII on the Hour Circle; keeping the Globe in that situation, slacken the Screw G, and set the Sun directly over his place on the Meridian; which done, set the Moon’s Wire under the number that expresses her age for that day on the Plate F, and she will then stand over her place in the Ecliptic, and shew what Constellation she is in. Lastly, fasten the Screw G, and laying the curved edge of the pasteboard S over the Ecliptic below the Moon, adjust the Moon to her Latitude over the graduated edge of the pasteboard; and the Globe will be rectified.

It’s use.

Having thus rectified the Globe, turn it round, and observe on what points of the Horizon the Sun and Moon Balls rise and set, for these agree with the points of the Compass on which the Sun and Moon rise and set in the Heavens on the given day; and the Hour Index shews the times of their rising and setting; and likewise the time of the Moon’s passing over the Meridian.

This simple Apparatus shews all the varieties that can happen in the rising and setting of the Sun and Moon; and makes the forementioned Phenomena of the Harvest Moon ([Chap. xvi.]) plain to the Eye. It is also very useful in reading Lectures on the Globes, because a large company can see this Sun and Moon going round, rising above and setting below the Horizon at different times, according to the seasons of the year; and making their appulses to different fixed Stars. But, in the usual way, where there is only the places of the Sun and Moon in the Ecliptic to keep the Eye upon, they are easily lost sight of, unless covered with Patches.

The Planetary Globe.
[PL. VIII.] Fig. IV.

439. The Planetary Globe. In this Machine, T is a terrestrial Globe fixed on its Axis standing upright on the Pedestal CDE, on which is an Hour Circle, having its Index fixed on the Axis, which turns somewhat tightly in the Pedestal, so that the Globe may not be liable to shake; to prevent which, the Pedestal is about two Inches thick, and the Axis goes quite through it, bearing on a shoulder. The Globe is hung in a graduated brasen Meridian, much in the usual way; and the thin Plate N, NE, E, is a moveable Horizon, graduated round the outer edge, for shewing the Bearings and Amplitudes of the Sun, Moon, and Planets. The brasen Meridian is grooved round the outer edge; and in this Groove is a slender Semi-circle of brass, the ends of which are fixed to the Horizon in its North and South Points: this Semi-circle slides in the Groove as the Horizon is moved in rectifying it for different Latitudes. To the middle of the Semi-circle is fixed a Pin which always keeps in the Zenith of the Horizon, and on this Pin the Quadrant of Altitude q turns; the lower end of which, in all Positions, touches the Horizon as it is moved round the same. This Quadrant is divided into 90 Degrees from the Horizon to the zenithal Pin on which it is turned, at 90. The great flat Circle or Plate AB is the Ecliptic, on the outer edge of which, the Signs and Degrees are laid down; and every fifth Degree is drawn through the rest of the surface of this Plate towards its Center. On this Plate are seven Grooves, to which seven little Balls are adjusted by sliding Wires, so that they are easily moved in the Grooves, without danger of starting out of them. The Ball next the terrestrial Globe is the Moon, the next without it is Mercury, the next Venus, the next the Sun, then Mars, then Jupiter, and lastly Saturn; and in order to know them, they are separately stampt with the following Characters; ☽, ☿, ♀,

To rectify it.

To rectify this Machine. Set all the planetary Balls to their geocentric places in the Ecliptic for any given time by an Ephemeris: then, set the North Point of the Horizon to the Latitude of your place on the brasen Meridian, and the Quadrant of Altitude to the South Point of the Horizon; which done, turn the Globe with its Furniture till the Quadrant of Altitude comes right against the Sun, viz. to his place in the Ecliptic; and keeping it there, set the Hour Index to the XII next the letter C; and the Machine will be rectified, not only for the following Problems, but for several others, which the Artist may easily find out.

PROBLEM I.
To find the Amplitudes, Meridian Altitudes, and times of Rising, Culminating, and Setting, of the Sun, Moon, and Planets.

It’s use.

Turn the Globe round eastward, or according to the order of Signs; and as the eastern edge of the Horizon comes right against the Sun, Moon, or any Planet, the Hour Index will shew the time of it’s rising; and the inner edge of the Ecliptic will cut it’s rising Amplitude in the Horizon. Turn on, and as the Quadrant of Altitude comes right against the Sun, Moon, or Planets, the Ecliptic cuts their meridian Altitudes in the Quadrant, and the Hour Index shews the times of their coming to the Meridian. Continue turning, and as the western edge of the Horizon comes right against the Sun, Moon, or Planets, their setting Amplitudes are cut in the Horizon by the Ecliptic; and the times of their setting are shewn by the Index on the Hour Circle.

PROBLEM II.
To find the Altitude and Azimuth of the Sun, Moon, and Planets, at any time of their being above the Horizon.

Turn the Globe till the Index comes to the given time in the Hour Circle; then keep the Globe steady, and moving the Quadrant of Altitude to each Planet respectively, the edge of the Ecliptic will cut the Planet’s mean Altitude on the Quadrant, and the Quadrant will cut the Planet’s Azimuth, or Point of Bearing on the Horizon.

PROBLEM III.
The Sun’s Altitude being given at any time either before or after Noon, to find the Hour of the Day, and the Variation of the Compass, in any known Latitude.

With one hand hold the edge of the Quadrant right against the Sun; and, with the other hand, turn the Globe westward, if it be in the forenoon, or eastward if it be in the afternoon, until the Sun’s place at the inner edge of the Ecliptic cuts the Quadrant in the Sun’s observed Altitude; and then the Hour Index will point out the time of the day, and the Quadrant will cut the true Azimuth, or Bearing of the Sun for that time: the difference between which, and the Bearing shewn by the Azimuth Compass, shews the variation of the Compass in that place of the Earth.

The Trajectorium Lunare.
[PL. VII.] Fig. V.

440. The Trajectorium Lunare. This Machine is for delineating the paths of the Earth and Moon, shewing what sort of Curves they make in the etherial regions; and was just mentioned in the 266th Article. S is the Sun, and E the Earth, whose Centers are 81 Inches distant from each other; every Inch answering to a Million of Miles § [47]. M is the Moon, whose Center is 24⁄100 parts of an Inch from the Earth’s in this Machine, this being in just proportion to the Moon’s distance from the Earth § [52]. AA is a Bar of Wood, to be moved by hand round the Axis g which is fixed in the Wheel Y. The Circumference of this Wheel is to the Circumference of the small Wheel L (below the other end of the Bar) as 3651⁄4 days is to 291⁄2; or as a Year is to a Lunation. The Wheels are grooved round their edges, and in the Grooves is the cat-gut string GG crossing between the Wheels at X. On the Axis of the Wheel L is the Index F, in which is fixed the Moon’s Axis M for carrying her round the Earth E (fixed on the Axis of the Wheel L) in the time that the Index goes round a Circle of 291⁄2 equal parts, which are the days of the Moon’s age. The Wheel Y has the Months and Days of the year all round it’s Limb; and in the Bar AA is fixed the Index I, which points out the Days of the Months answering to the Days of the Moon’s age, shewn by the Index F, in the Circle of 291⁄2 equal parts at the other end of the Bar. On the Axis of the Wheel L is put the piece D, below the Cock C, in which this Axis turns round; and in D are put the Pencils e and m, directly under the Earth E and Moon M; so that m is carried round e as M is round E.

It’s use.

Lay the Machine on an even Floor, pressing gently on the Wheel Y to cause its spiked Feet (of which two appear at P and P, the third being supposed to be hid from sight by the Wheel) enter a little into the Floor to secure the Wheel from turning. Then lay a paper about four foot long under the Pencils e and m, cross-wise to the Bar: which done, move the Bar slowly round the Axis g of the Wheel Y; and, as the Earth E goes round the Sun S, the Moon M will go round the Earth with a duly proportioned velocity; and the friction Wheel W running on the Floor, will keep the Bar from bearing too heavily on the Pencils e and m, which will delineate the paths of the Earth and Moon, as in Fig. 2d, already described at large, § [266], [267]. As the Index I points out the Days of the Months, the Index F shews the Moon’s age on these Days, in the Circle of 291⁄2 equal parts. And as this last Index points to the different Days in it’s Circle, the like numeral Figures may be set to those parts of the Curves of the Earth’s Path and Moon’s, where the Pencils e and m are at those times respectively, to shew the places of the Earth and Moon. If the Pencil e be pushed a very little off, as if from the Pencil m, to about 1⁄40 part of their distance, and the Pencil m pushed as much towards e, to bring them to the same distances again, though not to the same points of space; then as m goes round e, e will go as it were round the Center of Gravity between the Earth e and Moon m § [298]: but this Motion will not sensibly alter the Figure of the Earth’s Path or the Moon’s.

If a Pin as p be put through the Pencil m, with its head towards that of the Pin q in the Pencil e, its head will always keep thereto as m goes round e, or as the same side of the Moon is still obverted to the Earth. But the Pin p, which may be considered as an equatoreal Diameter of the Moon, will turn quite round the Point m, making all possible Angles with the Line of its progress or line of the Moon’s Path. This is an ocular proof of the Moon’s turning round her Axis.

The Tide Dial.
[PLATE IX]. Fig. VII.
It’s use.

441. The Tide-Dial. The outside parts of this Machine consist of, 1. An eight-sided Box, on the top of which at the corner is shewn the Phases of the Moon at the Octants, Quarters, and Full. Within these is a Circle of 291⁄2 equal parts, which are the days of the Moon’s age accounted from the Sun at New Moon round to the same again. Within this Circle is one of 24 hours divided into their respective Halves and Quarters. 2. A moving elliptical Plate painted blue to represent the rising of the Tides under and opposite to the Moon; and has the words, High Water, Tide falling, Low Water, Tide rising, marked upon it. To one end of this Plate is fixed the Moon M by the Wire W, and goes along with it. 3. Above this elliptical Plate is a round one, with the Points of the Compass upon it, and also the names of above 200 places in the large Machine (but only 32 in the Figure to avoid confusion) set over those Points on which the Moon bears when she raises the Tides to the greatest heights at these Places twice in every lunar day: and to the North and South Points of this Plate are fixed two Indexes I and K, which shew the times of High Water in the Hour Circle at all these places. 4. Below the elliptical Plate are four small Plates, two of which project out from below its ends at New and Full Moon; and so, by lengthening the Ellipse shew the Spring Tides, which are then raised to the greatest heights by the united attractions of the Sun and Moon § [302]. The other two of these small Plates appear at low water when the Moon is in her Quadratures, or at the sides of the elliptic Plate, to shew the Nepe Tides; the Sun and Moon then acting cross-wise to each other. When any two of these small Plates appear, the other two are hid; and when the Moon is in her Octants they all disappear, there being neither Spring nor Nepe Tides at those times. Within the Box are a few Wheels for performing these Motions by the Handle or Winch H.

Plate XIII.

J. Ferguson inv. et del.

J. Mynde Sculp.

Turn the Handle until the Moon M comes to any given day of her age in the Circle of 291⁄2 equal parts, and the Moon’s Wire W will cut the time of her coming to the Meridian on that day, in the Hour Circle; the XII under the Sun being Mid-day, and the opposite XII Mid-night: then looking for the name of any given place on the round Plate (which makes 291⁄2 rotations whilst the Moon M makes only one revolution from the Sun to the Sun again) turn the Handle till that place comes to the word High Water under the Moon, and the Index which falls among the Afternoon Hours will shew the time of high water at that place in the Afternoon of the given day: then turn the Plate half round, till the same place comes to the opposite High Water Mark, and the Index will shew the time of High Water in the Forenoon at that place. And thus, as all the different places come successively under and opposite to the Moon, the Indexes shew the times of High Water at them in both parts of the day: and when the same places come to the Low Water Marks the Indexes shew the times of Low Water. For about two days before and after the times of New and Full Moon, the two small Plates come out a little way from below the High Water Marks on the elliptical Plate, to shew that the Tides rise still higher about these times: and about the Quarters, the other two Plates come out a little from under the Low Water Marks towards the Sun and on the opposite side, shewing that the Tides of Flood rise not then so high, nor do the Tides of Ebb fall so low, as at other times.

By pulling the Handle a little way outward, it is disengaged from the Wheel-work, and then the upper Plate may be turned round quickly by hand so, as the Moon may be brought to any given day of her age in about a quarter of a minute.

The inside work described.
Fig. VIII.

On AB, the Axis of the Handle H, is an endless Screw C which turns the Wheel FED of 24 teeth round in 24 revolutions of the Handle: this Wheel turns another ONG of 48 teeth, and on its Axis is the Pinion PQ of four leaves which turns the Wheel LKI of 59 teeth round in 291⁄2 turnings or rotations of the Wheel FED, or in 708 revolutions of the Handle, which is the number of Hours in a synodical revolution of the Moon. The round Plate with the names of Places upon it is fixed on the Axis of the Wheel FED; and the Elliptical or Tide-Plate with the Moon fixed to it is upon the Axis of the Wheel LKI; consequently, the former makes 291⁄2 revolutions in the time that the latter makes one. The whole Wheel FED with the endless Screw C, and dotted part of the Axis of the Handle AB, together with the dotted part of the Wheel ONG, lie hid below the large Wheel LKI.

Fig. 9th represents the under side of the Elliptical or Tide-Plate abcd, with the four small Plates ABCD, EFGH, IKLM, NOPQ upon it: each of which has two slits as TT, SS, RR, UU sliding on two Pins as nn, fixed in the elliptical Plate. In the four small Plates are fixed four Pins at W, X, Y, and Z; all of which work in an elliptic Groove oooo on the cover of the Box below the elliptical Plate; the longest Axis of this Groove being in a right line with the Sun and Full Moon. Consequently, when the Moon is in Conjunction or Opposition, the Pins W and X thrust out the Plates ABCD and IKLM a little beyond the ends of the elliptic Plate at d and b, to f and e; whilst the Pins Y and Z draw in the Plates EFGH and NOPQ quite under the elliptic Plate to g and h. But, when the Moon comes to her first or third Quarter, the elliptic Plate lies across the fixed elliptic Groove in which the Pins work; and therefore the end Plates ABCD and IKLM are drawn in below the great Plate, and the other two Plates EFGH and NOPQ are thrust out beyond it to a and c. When the Moon is in her Octants the Pins V, X, Y, Z are in the parts o, o, o, o of the elliptic Groove, which parts are at a mean between the greatest and least distances from the Center q, and then all the four small Plates disappear below the great one.

The Eclipsareon.
[Pl. XIII.]

442. The Eclipsareon. This Piece of Mechanism exhibits the Time, Quantity, Duration, and Progress of solar Eclipses, at all Parts of the Earth.

The principal parts of this Machine are, 1. A terrestrial Globe A turned round its Axis B by the Handle or Winch M; the Axis B inclines 231⁄2 Degrees, and has an Index which goes round the Hour Circle D in each rotation of the Globe. 2. A circular Plate E on the Limb of which the Months and Days of the year are inserted. This Plate supports the Globe, and gives its Axis the same position to the Sun, or to a candle properly placed, that the Earth’s Axis has to the Sun upon any day of the year § [338], by turning the Plate till the given Day of the Month comes to the fixed Pointer or annual Index G. 3. A crooked Wire F which points towards the middle of the Earth’s enlightened Disc at all times, and shews to what place of the Earth the Sun is vertical at any given time. 4. A Penumbra, or thin circular Plate of brass I divided into 12 Digits by 12 concentric Circles, which represent a Section of the Moon’s Penumbra, and is proportioned to the size of the Globe; so that the shadow of this Plate, formed by the Sun, or a candle placed at a convenient distance, with it’s Rays transmitted through a convex Lens to make them fall parallel on the Globe, covers exactly all those places upon it that the Moon’s Shadow and Penumbra do on the Earth: so that the Phenomena of any solar Eclipse may be shewn by this Machine with candle-light, almost as well as by the light of the Sun. 5. An upright frame HHHH, on the sides of which are Scales of the Moon’s Latitude or Declination from the Ecliptic. To these Scales are fitted two Sliders K and K, with Indexes for adjusting the Penumbra’s Center to the Moon’s Latitude, as it is North or South Ascending or Descending. 6. A solar Horizon C, dividing the enlightened Hemisphere of the Globe from that which is in the dark at any given time, and shewing at what places the general Eclipse begins and ends with the rising or setting Sun. 7. A Handle M, which turns the Globe round it’s Axis by wheel-work, and at the same time moves the Penumbra across the frame by threads over the Pullies L, L, L, with the velocity duly proportioned to that of the Moon’s shadow over the Earth, as the Earth turns on its Axis. And as the Moon’s Motion is quicker or slower, according to her different distances from the Earth, the penumbral Motion is easily regulated in the Machine by changing one of the Pullies.

To rectify it.

To rectify the Machine for use. The true time of New Moon and her Latitude being known by the foregoing Precepts § [355], [363], if her Latitude exceeds the number of minutes or divisions on the Scales (which are on the side of the frame hid from view in the Figure of the Machine) there can be no Eclipse of the Sun at that Conjunction; but if it does not, the Sun will be eclipsed to some places of the Earth; and, to shew the times and various appearances of the Eclipse at those places, proceed in order as follows.

To rectify the Machine for performing by the Light of the Sun. 1. Move the Sliders KK till their Indexes point to the Moon’s Latitude on the Scales, as it is North and South Ascending or Descending, at that time. 2. Turn the Month Plate E till the day of the given New Moon comes to the annual Index G. 3. Unscrew the Collar N a little on the Axis of the Handle, to loosen the contiguous Socket on which the threads that move the Penumbra are wound; and set the Penumbra by Hand till its Center comes to the perpendicular thread in the middle of the frame; which thread represents the Axis of the Ecliptic § [371]. 4. Turn the Handle till the Meridian of London on the Globe comes just under the point of the crooked Wire F; then stop, and turn the Hour Circle D by Hand till XII at Noon comes to its Index. 5. Turn the Handle till the Hour Index points to the time of New Moon in the Circle D; and holding it there, screw fast the Collar N. Lastly, elevate the Machine till the Sun shines through the Sight-Holes in the small upright Plates O, O, on the Pedestal; and the whole Machine will be rectified.

To rectify the Machine for shewing the Candle-Light, proceed in every respect as above, except in that part of the last paragraph where the Sun is mentioned; instead of which place a Candle before the Machine, about four yards from it, so as the shadow of Intersection of the cross threads in the middle of the frame may fall precisely on that part of the Globe to which the crooked Wire F points: then, with a pair of Compasses take the distance between the Penumbra’s Center and Intersection of the threads; and equal to that distance set the Candle higher or lower as the Penumbra’s Center is above or below the said Intersection. Lastly, place a large convex Lens between the Machine and Candle, so as the Candle may be in the Focus of the Lens, and then the Rays will fall parallel, and cast a strong light on the Globe.

It’s use.

These things done, which may be sooner than expressed, turn the Handle backward until the Penumbra almost touches the side HF of the frame; then turning it gradually forward, observe the following Phenomena. 1. Where the eastern edge of the Shadow of the penumbral Plate I first touches the Globe at the solar Horizon, those who inhabit the corresponding part of the Earth see the Eclipse begin on the uppermost edge of the Sun, just at the time of its rising. 2. In that place where the Penumbra’s Center first touches the Globe, the inhabitants have the Sun rising upon them centrally eclipsed. 3. When the whole Penumbra just falls upon the Globe, its western edge, at the solar Horizon, touches and leaves the place where the Eclipse ends at Sun-rise on his lowermost edge. Continue turning, and, 4. the cross lines in the Center of the Penumbra will go over all those places on the Globe where the Sun is centrally eclipsed. 5. When the eastern edge of the Shadow touches any place of the Globe, the Eclipse begins there: when the vertical line in the Penumbra comes to any place, then is the greatest obscuration at that place; and when the western edge of the Penumbra leaves the place, the Eclipse ends there; the times of all which are shewn on the Hour Circle: and from the beginning to the end, the Shadows of the concentric penumbral Circles shew the number of Digits eclipsed at all the intermediate times. 6. When the eastern edge of the Penumbra leaves the Globe at the solar Horizon C, the inhabitants see the Sun beginning to be eclipsed on his lowermost edge at its setting. 7. Where the Penumbra’s Center leaves the Globe, the inhabitants see the Sun set centrally eclipsed. And lastly, where the Penumbra is wholly departing from the Globe, the inhabitants see the Eclipse ending on the uppermost part of the Sun’s edge, at the time of its disappearing in the Horizon § [343].

N.B. If any given day of the year on the Plate E be set to the annual Index G, and the Handle turned till the Meridian of any place comes under the point of the crooked Wire, and then the Hour Circle D set by the hand till XII comes to its Index; in turning the Globe round by the Handle, when the said place touches the eastern edge of the Hoop or solar Horizon C, the Index shews the time of Sun-setting at that place; and when the place is just coming out from below the other edge of the Hoop C, the Index shews the time that the evening Twilight ends to it. When the place has gone through the dark part A, and comes about so to touch under the back of the Hoop C on the other side, the Index shews the time that the Morning Twilight begins; and when the same place is just coming out from below the edge of the Hoop next the frame, the Index points out the time of Sun-rising. And thus, the times of Sun-rising and setting are shewn at all places in one rotation of the Globe, for any given day of the year: and the point of the crooked Wire F shews all the places that the Sun passes vertically over on that day.

FINIS.