SECT. III.

The USE of the Globes.

Problem I. To find the Latitude and Longitude of any given Place upon the Globe; and on the contrary, the Latitude and Longitude being given, to find the Place.

1.

Turn the globe round its axis, ’till the given place lies exactly under the (Eastern side of the brass) meridian, then that degree upon the meridian, which is directly over it, is the Latitude; which is accordingly North or South, as it lies in the Northern or Southern hemisphere, the globe remaining in the same position.

That degree upon the equator which is cut by the brazen meridian, is the Longitude required from the first meridian upon the globe. If the longitude is counted both ways from the first meridian upon the globe, then we are to consider, whether the given place lies Easterly or Westerly from the first meridian, and the longitude must be expressed accordingly.

The Latitudes of the following places: and upon a globe where the longitude is reckoned both ways from the meridian of London, their longitudes will be found as follow:

Latitude.
Deg.		Longitude.
Deg.
Rome41¾North.13 East.
Paris48¾N.2½E.
Mexico20N.102 W.
Cape Horn58S.80 W.

2. The Latitude and Longitude being given to find the Place.

Seek for the given longitude in the equator, and bring that point to the meridian; then count from the equator on the meridian the degree of latitude given, towards the arctic and antarctic Pole, according as the latitude is Northerly or Southerly, and under that degree of latitude lies the Place required.

Prob. II. To find the Difference of Latitude betwixt any two given Places.

Bring each of the places proposed successively to the meridian, and observe where they intersect it, then the number of degrees upon the meridian, contained between the two intersections, will be the Difference of Latitude required. Or, if the places proposed are on the same side of the equator, having first found their latitudes, subtract the lesser from the greater; but if they are on contrary sides of the equator, add them both together, and the difference in the first case, and the sum in the latter, will be the difference of latitude required.

Thus the difference of latitude betwixt London and Rome will be found to be 9¾ degrees; betwixt Paris and Cape Bona Esperance 83 degrees.

Prob. III. To find the Difference of Longitude betwixt any two given Places.

Bring each of the given places successively to the meridian, and see where the meridian cuts the equator each time; the number of degrees contained betwixt those two points, if it be less than 180 degrees, otherwise the remainder to 360 degrees, will be the difference of longitude required. Or,

Having brought one of the given places to the meridian, bring the index of the hour circle to 12 o’clock; then having brought the other place to the meridian, the number of hours contained between the place the index was first set at, and the place where it now points, is the difference of longitude in time betwixt the two places.

Thus the difference of longitude betwixt Rome and Constantinople will be found to be 19 degrees, or 1 hour and a quarter; betwixt Mexico and Pekin in China, 240 degrees, or 9⅓ hours.

Prob. IV. Any Place being given to find all those Places that are in the same Latitude with the same Place.

The latitude of any given place being marked upon the meridian, turn the globe round its axis, and all those places that pass under the same mark are in the same latitude with the given place, and have their days and nights of equal lengths. And when any place is brought to the meridian, all the inhabitants that lie under the upper semicircle of it, have their Noon or mid-day at the same point of absolute time exactly.

Prob. V. The day of the Month being given; to find the Sun’s Place in the Ecliptic, and his Declination.

1. To find the Sun’s Place: Look for the day of the month given in the kalendar of months upon the horizon, and right against it you’ll find that sign and degree of the ecliptic which the Sun is in. The Sun’s place being thus found, look for the same in the ecliptic line which is drawn upon the globe, and bring that point to the meridian, then that degree of the meridian, which is directly over the Sun’s place, is the Declination required; which is accordingly either North or South, as the Sun is in the Northern or Southern signs. Thus,

Sun’s Place. Declination.
Deg.Min.Deg.Min.
April 23♉3001232 N.
July 31♌7511820 N.
October 26♏2491228 S.
January 20 ♒0492007 S.

Prob. VI. To rectify the Globe for the Latitude, Zenith, and the Sun’s Place.

1. For the Latitude: If the place be in the Northern hemisphere, raise the arctic Pole above the horizon; but for the South latitude you must raise the antarctic; then move the meridian up and down in the notches, until the degrees of the latitude counted upon the meridian below the Pole, cuts the horizon, and the globe is adjusted to the latitude.

2. To rectify the Globe for the Zenith: Having elevated the globe according to the latitude, count the degrees thereof upon the meridian from the equator, towards the elevated Pole, and that point will be the zenith or the vertex of the place; to this point of the meridian fasten the quadrant of altitude, so that the graduated edge thereof may be joined to the said point.

3. Bring the Sun’s place in the ecliptic to the meridian, and then set the hour index to XII at Noon, and the globe will be rectified to the Sun’s Place. If you have a little mariner’s compass, the meridian of the globe may be easily set to the meridian of the place.

Prob. VII. To find the Distance between any two given places upon the Globe, and to find all those places upon the globe that are at the same distance from a given place.

Lay the quadrant of altitude over both the places, and the number of degrees intercepted between them being reduced into miles, will be the distance required: Or, you may take the distance betwixt the two places with a pair of compasses, and applying that extent to the equator, you’ll have the degrees of distance as before.

Note, A geographical mile is the ¹/₆₀th part of a degree; whereof if you multiply the number of degrees by 60, the product will be the number of geographical miles of distance sought; but to reduce the same into English miles, you must multiply by 70, because about 70 English miles make a degree of a great circle upon the superficies of the Earth.

Thus, the distance betwixt London and Rome will be found to be about 13 degrees, which is 780 geographical miles.

If you rectify the globe for the latitude and zenith of any given place, and bring the said place to the meridian; then turning the quadrant of altitude about, all those places that are cut by the same point of it, are at the same distance from the given place.

Prob. VIII. To find the angle of position of Places, or the angle formed by the meridian of one Place, and a great circle passing through both the Places.

Having rectified the globe for the latitude and zenith of one of the given places, bring the said place to the meridian, then turn the quadrant of altitude about, until the fiducial edge thereof cuts the other place, and the number of degrees upon the horizon, contained between the said edge and the meridian, will be the angle of position sought.

Thus, the angle of position at the Lizard, between the meridian of the Lizard and the great circle, passing from thence to Barbadoes is 69 degrees South-Westerly; but the angle of position between the same places at Barbadoes, is but 38 degrees North-Easterly.

SCHOLIUM

The angle of position between two places is a different thing from what is meant by the bearings of places; the Bearings of two places is determined by a sort of spiral line, called a Rhumb Line, passing between them in such a manner, as to make the same or equal angles with all the meridians through which it passeth; but the angle or position is the very same thing with what we call the azimuth in astronomy, both being formed by the meridian and a great circle passing thro’ the zenith of a given place in the heavens, then called the azimuth, or upon the Earth, then called the angle of position.

From hence may be shewed the error of that geographical paradox, viz. If a place A bears from another B due West, B shall not bear from A due East. I find this paradox vindicated by an author, who at the same time gives a true definition of a rhumb line: But his arguments are ungeometrical; for if it be admitted that the East and West lines make the same angles with all the meridians through which they pass, it will follow that these lines are the parallels of latitude: For any parallel of latitude is the continuation of the surface of a Cone, whose sides are the radii of the sphere, and circumference of its base the said parallel; and it is evident, that all the meridians cut the said surface at right (and therefore at equal) angles; whence it follows, that the rhumbs of East and West are the parallels of latitude, though the case may seem different, when we draw inclining lines (like meridians) upon paper, without carrying our ideas any farther.

Prob. IX. To find the Antœci, Periœci, and Antipodes to any given place.

Bring the given place to the meridian; and having found its latitude, count the same number of degrees on the meridian from the equator towards the contrary Pole, and that will give the place of the Antœci. The globe being still in the same position, set the hour index to XII at noon, then turn the globe about ’till the index points to the lower XII; the place which then lies under the meridian, having the same latitude with the given place, is the Periœci required. As the globe now stands, the Antipodes of the given place are under the same point of the meridian, that its Antœci stood before: Or, if you reckon 180 degrees upon the meridian from the given place, that point will be the Antipodes. Let the given place be London, in the latitude of 51½ degrees North, that place which lies under the same meridian and the latitude 51½ degrees South, is the Antœci; that which lies in the same parallel with London, and 180 degrees of longitude from it, is the Periœci, and the Antipodes is the place whose longitude from London is 180 degrees, and latitude 51½ degrees South.

Prob. X. The Hour of the Day at one place being given; to find the correspondent Hour (or what o’Clock it is at that time) in any other place.

The difference of time betwixt two places is the same with their difference of longitude; wherefore having found their difference of longitude, reduced into time (by allowing one hour for every 15 degrees, &c.) and if the place where the hour is required lies (Easterly/Westerly) from the place where the hour is given, (add/subtract) the difference of longitude reduced into time (to/from) the hour given; and the sum or remainder will accordingly be the hour required. Or,

Having brought the place at which the hour is given to the meridian, set the hour index to the given hour; then turn the globe about until the place where the hour is required comes to the meridian, and the index will point out the hour at the said place.

Thus when it is Noon at London, it is

H. M.
At Rome052P. M.
Constantinople  207P. M.
Vera-Cruz530A. M.
Pekin in China750P. M.

Prob. XI. The Day of the Month being given, to find those places on the globe where the Sun will be Vertical, or in the Zenith, that day.

Having found the Sun’s place in the ecliptic, bring the same to the meridian, and note the degree over it; then turning the globe round, all places that pass under that degree will have the Sun vertical that day.

Prob. XII. A place being given in the Torrid Zone, to find those two Days in which the Sun shall be Vertical to the same.

Bring the given place to the meridian, and mark what degree of latitude is exactly over it; then turning the globe about its axis, those two points of the ecliptic, which pass exactly under the said mark, are the Sun’s place; against which, upon the wooden horizon, you’ll have the days required.

Prob. XIII. To find where the Sun is Vertical at any given time assigned; or the Day of the Month and the Hour at any Place (suppose London) being given, to find in what place the Sun is Vertical at that very time.

Having found the Sun’s declination, and brought the first place (London) to the meridian, set the index to the given hour, then turn the globe about until the index points to XII at noon; which being done, that place upon the globe which stands under the point of the Sun’s declination upon the meridian, has the Sun that moment in the Zenith.

Prob. XIV. The Day, and the Hour of the Day at one place, being given; to find all those places upon the Earth, where the Sun is then Rising, Setting, Culminating (or on the meridian) also where it is Day-light, Twilight, Dark Night, Midnight; where the Twilight then begins, and where it ends; the height of the Sun in any part of the illuminated hemisphere; also his depression in the obscure hemisphere.

Having found the place where the Sun is vertical at the given hour, rectify the globe for that latitude, and bring the said place to the meridian.

Then all those places that are in the Western semicircle of the horizon, have the Sun rising at that time.

Those in the Eastern semicircle have it setting.

To those who live under the upper semicircle of the meridian, it is 12 o’clock at noon. And,

Those who live under the lower semicircle of the meridian, have it at midnight.

All those places that are above the horizon, have the Sun above them, just so much as the places themselves are distant from the horizon; which height may be known by fixing the quadrant of altitude in the zenith, and laying it over any particular place.

In all those places that are 18 degrees below the Western side of the horizon, the twilight is just beginning in the morning, or the day breaks. And in all those places that are 18 degrees below the Eastern side of the horizon, the twilight is ending, and the total darkness beginning.

The twilight is in all those places whose depression below the horizon does not exceed 18 degrees. And,

All those places that are lower than 18 degrees, have dark night.

The depression of any place below the horizon is equal to the altitude of its Antipodes, which may be easily found by the quadrant of altitude.

Prob. XV. The Day of the Month being given; to show, at one view, the length of Days and Nights in all places upon the Earth at that time; and to explain how the vicissitudes of Day and Night are really made by the motion of the Earth round her axis in 24 hours, the Sun standing still.

The Sun always illuminates one half of the globe, or that hemisphere which is next towards him, while the other remains in darkness: And if (as by the [last problem]) we elevate the globe according to the Sun’s place in the ecliptic, it is evident, that the Sun (he being at an immense distance from the Earth) illuminates all that hemisphere, which is above the horizon; the wooden horizon itself, will be the circle terminating light and darkness; and all those places that are below it, are wholly deprived of the solar light.

The globe standing in this position, those arches of the parallels of latitude which stand above the horizon, are the Diurnal Arches, or the length of the day in all those latitudes at that time of the year; and the remaining parts of those parallels, which are below the horizon, are the Nocturnal Arches, or the length of the night in those places. The length of the diurnal arches may be found by counting how many hours are contained between the two meridians, cutting any parallel of latitude, in the Eastern and Western parts of the horizon.

In all those places that are in the Western semicircle of the horizon, the Sun appears rising: For the Sun, standing still in the vertex (or above the brass meridian) appears Easterly, and 90 degrees distant from all those places that are in the Western semicircle of the horizon; and therefore in those places he is then rising. Now, if we pitch upon any particular place upon the globe, and bring it to the meridian, and then bring the hour index to the lower 12, which in this case, we’ll suppose to be 12 at noon; (because otherwise the numbers upon the hour circle will not answer our purpose) and afterwards turn the globe about, until the aforesaid place be brought to the Western side of the horizon; the index will then shew the time of the Sun rising in that place. Then turn the globe gradually about from West to East, and minding the hour index, we shall see the progress made in the day every hour, in all latitudes upon the globe, by the real motion of the Earth round its axis; until, by their continual approach to the brass meridian (over which the Sun stands still all the while) they at last have noon day, and the Sun appears at the highest; and then by degrees, as they move Easterly the Sun seems to decline Westward, until, as the places successively arrive in the Eastern part of the horizon, the Sun appears to set in the Western: For the places that are in the horizon, are 90 degrees distant from the Sun. We may observe, that all places upon the Earth, that differ in latitude, have their days of different length (except when the Sun is in the equinoctial) being longer or shorter, in proportion to what part of the parallels stands above the horizon. Those that are in the same latitude, have their days of the same length; but have them commence sooner or later, according as the places differ in longitude.

Prob. XVI. To explain in general the alteration of Seasons, or length of the Days and Nights made in all places of the World, by the Sun’s (or the Earth’s) annual motion in the Ecliptic.

It has been shewed in the [last problem], how to place the globe in such a position as to exhibit the length of the diurnal and nocturnal arches in all places of the Earth, at a particular time: If the globe be continually rectified, according as the Sun alters his declination, (which may be known by bringing each degree of the ecliptic successively to the meridian) you’ll see the gradual increase or decrease made in the days, in all places of the World, according as a greater or lesser portion of the parallels of latitude, stands above the horizon. We shall illustrate this problem by examples taken at different times of the year.

1. Let the Sun be in the first point of ♋ (which happens on the 21st of June) that point being brought to the meridian, will shew the Sun’s declination to be 23½ degrees North; then the globe must be rectified to the latitude of 23½ degrees; and for the better illustration of the problem, let the first meridian upon the globe be brought under the brass meridian. The globe being in this position, you’ll see at one view the length of the days in all latitudes, by counting the number of hours contained between the two extreme meridians, cutting any particular parallel you pitch upon, in the Eastern and Western part of the horizon. And you may observe that the lower part of the arctic circle just touches the horizon, and consequently all the people who live in that latitude have the Sun above their horizon for the space of 24 hours, without setting; only when he is in the lower part of the meridian (which they would call 12 at night) he just touches the horizon.

To all those who live between the arctic circle and the Pole, the Sun does not set, and its height above the horizon, when he is in the lower part of the meridian, is equal to their distance from the arctic circle: For example, Those who live in the 83d parallel have the Sun when he is lowest at this time 13½ degrees high.

If we cast our eye Southward, towards the equator, we shall find, that the diurnal arches, or the length of days in the several latitudes, gradually lessen: The diurnal arch of the parallel of London at this time is 16½ hours; that of the Equator (is always) 12 hours; and so continually less, ’till we come to the Antarctic Circle, the upper part of which just touches the horizon; just those who live in this latitude have just one sight of the Sun, peeping as it were in the horizon: And all that space between the antarctic circle and the South Pole, lies in total darkness.

If from this position we gradually move the meridian of the globe according to the progressive alterations made in the Sun’s declination, by his motion in the ecliptic, we shall find the diurnal arches of all those parallels, that are on the Northern side of the equator, continually decrease; and those on the Southern side continually increase, in the same manner as the days in those places shorten and lengthen. Let us again observe the globe when the Sun has got within 10 degrees of the equinoctial; now the lower part of the 80th parallel of North latitude just touches the horizon, and all the space betwixt this and the pole, falls in the illuminated hemisphere: but all those parallels that lie betwixt this and the arctic circle, which before were wholly above the horizon, do now intersect it, and the Sun appears to them to rise and set. From hence to the equator, we shall find that the days have gradually shortened; and from the equator Southward, they have gradually lengthened, until we come to the 80th parallel of the South latitude; the upper part of which just touches the horizon; and all places betwixt this and the South Pole are in total darkness; but those parallels betwixt this and the antarctic circle, which before were wholly upon the horizon, are now partly above it; the length of their days being exactly equal to that of the nights in the same latitude in the contrary hemisphere. This also holds universally, that the length of one day in one latitude North, is exactly equal to the length of the night in the same latitude South; and vice versa.

Let us again follow the motion of the Sun, until he has got into the equinoctial, and take a view of the globe while it is in this position. Now all the parallels of latitude are cut into two equal parts by the horizon, and consequently the days and nights are of equal lengths, viz. 12 hours each, in all places of the world; the Sun rising and setting at six o’clock, excepting under the two Poles, which now lie exactly in the horizon: Here the Sun seems to stand still in the same point of the heavens for some time, until by degrees, by his motion in the ecliptic, he ascends higher to one and disappears to the other, there being properly no days and nights under the Poles; for there the motion of the Earth round its axis cannot be observed.

If we follow the motion of the Sun towards the Southern tropic, we shall see the diurnal arches of the Northern parallels continually decrease, and the Southern ones increase in the same proportion, according to their respective latitudes; the North Pole continually descending, and the South Pole ascending, above the horizon, until the Sun arrives into ♑, at which time all the space within the antarctic circle is above the horizon; while the space between the arctic circle, and its neighbouring Pole, is in total darkness. And we shall now find all other circumstances quite reverse to what they were when the Sun was in ♋; the nights now all over the world being of the same length that the days were of before.

We have now got to the extremity of the Sun’s declination; and if we follow him through the other half of the ecliptic, and rectify the globe accordingly, we shall find the seasons return in their order, until at length we bring the globe into its first position.

The two foregoing problems were not, as I know of, published in any book on this subject before; and I have dwelt the longer upon them, because they very well illustrate how the vicissitudes of days and nights are made all over the world, by the motion of the Earth round her axis; the horizon of the globe being made the circle, separating light and darkness, and so the Sun to stand still in the vertex. And if we really could move the meridian, according to the change of the Sun’s declination, we should see at one view, the continual change made in the length of days and nights, in all places on the Earth; but as globes are fitted up, this cannot be done; neither are they adapted for the common purposes, in places near the equator, or any where in the Southern hemisphere. But this inconvenience is now remedied (at a small additional expence) by the hour circle being made to shift to either Pole; and some globes are now made with an hour circle fixed to the globe at each Pole between the globe and meridian, so as to have none without side to interrupt the meridian from moving quite round the wooden horizon.

Prob. XVII. To shew by the globe, at one view, the longest of the Days and Nights in any particular places, at all times of the Year.

Because the Sun, by his motion in the ecliptic, alters his declination a small matter every day; if we suppose all the torrid zone to be filled up with a spiral line, having so many turnings; or a screw having so many threads, as the Sun is days in going from one tropic to the other: And these threads at the same distance from one another in all places, as the Sun alters his declination in one day in all those places respectively: This spiral line or screw will represent the apparent paths described by the Sun round the Earth every day; and by following the thread from one tropic to the other, and back again, we shall have the path the Sun seems to describe round the Earth in a year. But because the inclinations of these threads to one another are but small, we may suppose each diurnal path to be one of the parallels of latitude, drawn, or supposed to be drawn upon the globe. Thus much being premised, we shall explain this Problem, by placing the globe according to some of the most remarkable positions of it, as before we did for the most remarkable seasons of the year.

In the [preceding problem], the globe being rectified according to the Sun’s declination, the upper parts of the parallels of latitude, represented the Diurnal Arches, or the length of the days all over the world, at that particular time: Here we are to rectify the globe according to the latitude of the place, and then the upper parts of the parallels of declination are the diurnal arches; and the length of the days at all times of the year, may be here determined by finding the number of hours contained between the two extreme meridians, which cut any parallel of declination in the Eastern and Western points of the horizon; after the same manner, as before we found the length of the day in the several latitudes at a particular time of the year.

1. Let the place proposed be under the equinoctial, and let the globe be accordingly rectified for 00 degrees of latitude, which is called a direct position of the sphere. Here all the parallels of latitude, which in this case we will call the parallels of declination, are cut by the horizon into two equal parts; and consequently those who live under the equinoctial, have the days and nights of the same length at all times of the year; and also in this part of the Earth, all the Stars rise and set, and their continuance above the horizon, is equal to their stay below it, viz. 12 hours.

If from this position we gradually move the globe according to the several alterations of latitudes, which we will suppose to be Northerly; the lengths of the Diurnal Arches will continually increase, until we come to a parallel of declination, as far distant from the equinoctial, as the place itself is from the Pole. This parallel will just touch the horizon, and all the heavenly bodies that are betwixt it and the Pole never descend below the horizon. In the mean time, while we are moving the globe, the lengths of the diurnal arches of the Southern parallels of declination, continually diminish in the same proportion that the Northern ones increased; until we come to that parallel of declination which is so far distant from the equinoctial Southerly, as the place itself is from the North Pole. The upper part of this Parallel just touches the horizon, and all the Stars that are betwixt it and the South Pole never appear above the horizon. And all the nocturnal arches of the Southern parallels of declination, are exactly of the same length with the diurnal arches of the correspondent parallels of North declination.

2. Let us take a view of the globe when it is rectified for the latitude of London, or 51½ degrees North. When the Sun is in the tropic of ♋, the day is about 16½ hours; as he recedes from this tropic, the days proportionably shorten, until, he arrives into ♑, and then the days are at the shortest, being now of the same length with the night, when the Sun was in ♋, viz. 7½ hours. The lower part of that parallel of declination, which is 38½ degrees from the equinoctial Northerly, just touches the horizon; and the Stars that are betwixt this parallel and the North Pole, never set to us at London. In like manner the upper part of the Southern parallel of 38½ degrees just touches the horizon, and the Stars that lie betwixt this parallel and the Southern Pole, are never visible in this latitude.

Again, let us rectify the globe for the latitude of the Arctic Circle, we shall then find, that when the Sun is in ♋, he touches the horizon on that day without setting, being 24 hours compleat above the horizon; and when he is in Capricorn, he once appears in the horizon, but does not rise in the space of 24 hours: When he is in any other point of the ecliptic, the days are longer or shorter, according to his distance from the tropics. All the Stars that lie between the tropic of Cancer, and the North Pole, never set in this latitude; and those that are between the tropic of Capricorn, and the South Pole, are always hid below the horizon.

If we elevate the globe still higher, the circle of perpetual Apparition will be nearer the equator, as will that of perpetual Occultation on the other side. For example, Let us rectify the globe for the latitude of 80 degrees North: when the Sun’s declination is 10 degrees North; he begins to turn above the horizon without setting; and all the while he is making his progress from this point to the tropic of ♋, and back again, he never sets. After the same manner, when his declination is 10 degrees South, he is just seen at noon in the horizon; and all the while he is going Southward, and back again, he disappears, being hid just so long as before, at the opposite time of the year he appeared visible.

Let us now bring the North Pole into the Zenith, then will the equinoctial coincide with the horizon; and consequently all the Northern parallels are above the horizon, and all the Southern ones below it. Here is but one day and one night throughout the year, it being day all the while the Sun is to the Northward of the equinoctial, and night for the other half year. All the Stars that have North declination, always appear above the horizon, and at the same height; and all those that are on the other side, are never seen.

What has been here said of rectifying the globe to North latitude, holds for the same latitude South; only that before the longest days were, when the Sun was in ♋, the same happening now when the Sun is in ♑; and so of the rest of the parallels, the seasons being directly opposite to those who live in different hemispheres.

I shall again explain some things delivered above in general terms, by particular problems.

But from what has been already said, we may first make the following observations:

1. All places of the Earth do equally enjoy the benefit of the Sun, in respect of time, and are equally deprived of it, the Days at one time of the Year, being exactly equal to the Nights at the opposite season.

2. In all places of the Earth, save exactly under the Poles, the Days and Nights are of equal length (viz. 12 hours each) when the Sun is in the equinoctial.

3. Those who live under the equinoctial, have the days and nights of equal lengths at all times of the year.

4. In all places between the equinoctial and the Poles, the days and nights are never equal, but when the Sun is in the equinoctial points ♈ and ♎.

5. The nearer any place is to the equator, the less is the difference between the length of the artificial days and nights in the said place; and the more remote the greater.

6. To all the inhabitants lying under the same parallel of latitudes the days and nights are of equal lengths, and that at all times of the year.

7. The Sun is vertical twice a year to all places between the tropics; to those under the tropics, once a year; but never any where else.

8. In all places between the Polar Circles, and the Poles, the Sun appears some number of days without setting; and at the opposite time of the year he is for the same length of time without rising; and the nearer unto, or further remote from the Pole, those places are, the longer or shorter is the Sun’s continued presence or absence from the Pole.

9. In all places lying exactly under the Polar Circles, the Sun, when he is in the nearest tropic, appears 24 hours without setting; and when he is in the contrary tropic, he is for the same length of time, without rising; but at all other times of the year, he rises and sets there, as in other places.

10. In all places lying in the (Northern/Southern) hemisphere, the longest day and shortest night, is when the Sun is in the (Northern/Southern) tropic, and on the contrary.

Prob. XVIII. The Latitude of any place, not exceeding 66½ degrees, and the day of the Month being given; to find the time of Sun-rising and setting, and the length of the Day and Night.

Having rectified the globe according to the latitude, bring the Sun’s place to the meridian, and put the hour index to 12 at noon; then bring the Sun’s place the Eastern part of the horizon, and the index will shew the time when the Sun rises. Again, turn the globe until the Sun’s place be brought to the Western side of the horizon, and the index will shew the time of Sun-setting.

The hour of Sun-setting doubled, gives the length of the day; and the hour of Sun-rising doubled, gives the length of the night.

Let it be required to find when the Sun rises and sets at London on the 20th of April. Rectify the globe for the latitude of London, and having found the Sun’s place corresponding to May the 1st, viz. ♉ 10¾ degrees, bring ♉ to 10¾ degrees to the meridian, and set the index to 12 at noon; then turn the globe about ’till ♉ 10¾ degrees be brought to the Eastern part of the horizon, and you’ll find the index point 4¾ hours, this being doubled, gives the length of the night 9½ hours. Again, bring the Sun’s place to the Western part of the horizon, and the index will point 7¼ hours, which is the time of Sun-setting; this being doubled, gives the length of the day 14½ hours.

Prob. XIX. To find the length of the longest and shortest Day and Night in any given place, not exceeding 66½ degrees of Latitude.

Note, The longest day at all places on the (North/South) side of the equator, is when the Sun is in the first point of (Cancer/Capricorn) Wherefore having rectified the globe for the latitude, find the time of Sun-rising and setting, and thence the length of the day and night, as in the [last problem], according to the place of the Sun: Or, having rectified the globe for the latitude, bring the solstitial point of that hemisphere, to the East part of the horizon, and set the index to 12 at noon; then turning the globe about ’till the said solstitial point touches the Western side of the horizon, the number of hours from noon to the place where the index points (being counted according to the motion of the index) is the length of the longest day; the complement whereof to 24 hours, is the length of the shortest night, and the reverse gives the shortest day and the longest night.

Longest Day. Shor. N.
Deg.Hours.Hours.
Thus in Lat. 4515½8½
51½16½7½
6018½5½

If from the length of the longest day, you subtract 12 hours, the number of half hours remaining, will be the Climate: Thus that place where the longest day is 16½ hours, lies in the 9th Climate. And by the reverse, having the Climate, you have thereby the length of the longest day.

Prob. XX. To find in what Latitude the longest Day is, of any given length, less than 24 hours.

Bring the solstitial point to the meridian, and set the index to 12 at noon; then turn the globe Westward, ’till the index points at half the number of hours given; which being done, keep the globe from turning round its axis, and slide the meridian up or down in the notches, ’till the solstitial point comes to the horizon, then that elevation of the Pole will be the latitude.

If the hours given be 16, the latitude is 49 degrees; if 20 hours, the latitude is 63¼ degrees.

Prob. XXI. A place being given in one of the Frigid Zones (suppose the Northern) to find what number of Days (of 24 hours each) the Sun doth constantly shine upon the same, how long he is absent, and also the first and last Day of his appearance.

Having rectified the globe according to the latitude, turn it about until some point in the first quadrant of the ecliptic (because the latitude is North) intersects the meridian in the North point of the horizon; and right against that point of the ecliptic on the horizon, stands the day of the month when the longest day begins.

And if the globe be turned about ’till some point in the second quadrant of the ecliptic cuts the meridian in the same point of the horizon, it will shew the Sun’s place when the longest day ends, whence the day of the month may be found, as before: Then the number of natural days contained between the times the longest day begins and ends is the length of the longest day required.

Again, turn the globe about, until some point in the third quadrant of the ecliptic cuts the meridian in the South part of the horizon; that point of the ecliptic will give the time when the longest night begins. Lastly, turn the globe about, until some point in the fourth quadrant of the ecliptic cuts the meridian in the South point of the horizon; and that point of the ecliptic will be the place of the Sun when the longest night ends.

Or, the time when the longest day or night begins, being known, their end may be found by counting the number of days from that time to the succeeding solstice; then counting the same number of days from the solstitial day, will give the time when it ends.

Prob. XXII. To find in what Latitude the longest Day is, of any given length less than 182 Natural Days.

Find a point in the ecliptic half so many degrees distant from the solstitial point, as there are days given, and bring that point to the meridian; then keep the globe from turning round its axis, and move the meridian up or down until the aforesaid point of the ecliptic comes to the horizon; that elevation of the Pole will be the latitude required.

If the days given were 78, the latitude is 71½ degrees.

This method is not accurate, because the degrees in the ecliptic do not correspond to natural days; and also because the Sun does not always move in the ecliptic at the same rate; however, such problems as these may serve for amusements.

Prob. XXIII. The day of the Month being given, to find when the Morning and Evening Twilight begins and ends, in any place upon the Globe.

In the [foregoing problem], by the length of the day, we mean the time from Sun-rising to Sun-set; and the night we reckoned from Sun-set, ’till he rose next morning. But it is found by experience, that Total Darkness does not commence in the evening, ’till the Sun has got 18 degrees below the horizon; and when he comes within the same distance of the horizon next morning, we have the first Dawn of Day. This faint light which we have in the morning and evening, before and after the Sun’s rising and setting, is what we call the Twilight. [4] Having rectified the globe for the latitude, the zenith, and the Sun’s place, turn the globe and the quadrant of altitude until the Sun’s place cuts 18 degrees below the horizon (if the quadrant reaches so far) then the index upon the hour circle will shew the beginning or ending of twilight after the same manner as before we found the time of the Sun-rising and setting, in [Prob. 18]. But by reason of the thickness of the wooden horizon, we can’t conveniently see, or compute when the Sun’s place is brought to the point aforesaid. Wherefore the globe being rectified as above directed, turn the globe, and also the quadrant of altitude, Westward, until that point in the ecliptic, which is opposite to the Sun’s place, cuts the quadrant in the 18th degree above the horizon; then the hour index will shew the time when day breaks in the morning. And if you turn the globe and the quadrant of altitude, until the point opposite to the Sun’s place cuts the quadrant in the Eastern hemisphere, the hour hand will shew when twilight ends in the evening. Or, having found the time from midnight when the morning twilight begins, if you reckon so many hours before midnight, it will give the time when the evening twilight ends. Having found the time when twilight begins in the morning, find the time of Sun-rising, by [Prob. 18], and the difference will be the duration of twilight.

Thus at London on the 12th of May twilight begins at three quarters past one o’clock: The Sun rises at about half an hour past four: Whence the duration of twilight now is 2¾ hours, both in the morning and evening. On the 12th of November, the twilight begins at half an hour past six, being somewhat above an hour before Sun-rising.

Prob. XXIV. To find the time when total Darkness ceases, or when the Twilight continues from Sun-setting to Sun-setting, in any given place.

Let the place be in the Northern hemisphere; then if the complement of the latitude be greater than (the depression) 18 degrees, subtract 18 degrees from it, and the remainder will be the Sun’s declination North, when total darkness ceases. But if the complement of the latitude is less than 18 degrees, their difference will be the Sun’s declination South, when the twilight begins to continue all night. If the latitude is South, the only difference will be, that the Sun’s declination will be on the contrary side.

Thus at London, when the Sun’s declination North is greater than 20½ degrees, there is no total darkness, but constant twilight, which happens from the 26th of May to the 18th of July, being near two months. Under the North Pole the twilight ceases, when the Sun’s declination is greater than 18 degrees South, which is from the 13th of November, ’till the 29th of January: So that notwithstanding the Sun is absent in this part of the world for half a year together, yet total darkness does not continue above 11 weeks; and besides, the Moon is above the horizon for a whole fortnight of every month throughout the year.

Prob. XXV. The day of the Month be given; to find those places of the Frigid Zones, where the Sun begins to shine continually without setting; and also those places where he begins to be totally absent.

Bring the Sun’s place to the meridian, and mark the number of degrees contained betwixt that point and the equator; then count the same number of degrees from the nearest Pole (viz. the North Pole, if the Sun’s declination is Northerly, otherwise the South Pole) towards the equator, and note that point upon the meridian; then turn the globe about, and all the places which pass under the said point, are those where the Sun begins to shine constantly, without setting on the given day. If you lay the same distance from the opposite Pole towards the equator, and turn the globe about, all the places which pass under that point, will be those where the longest night begins.

The Latitude of the place being given, to find the hour of the day when the Sun shines.

If it be in the summer, elevate the Pole according to the latitude, and set the meridian due North and South; then the shadow of the axis will cut the hour on the Dial plate: For the globe being rectified in this manner, the hour circle is a true Equinoctial Dial; the axis of the globe being the Gnomon. This holds true in Theory, but it might not be very accurate in practice, because of the difficulty in placing the horizon of the globe truly horizontal, and its meridian due North and South.

If it be in the winter half year, elevate the South Pole according to the latitude North, and let the North part of the horizon be in the South part of the meridian; then the shade of the axis will show the hour of the day as before: But this cannot be so conveniently performed, tho’ the reason is the same as in the former case.

To find the Sun’s altitude, when it shines, by the Globe.

Having set the frame of the globe truly horizontal or level, turn the North Pole towards the Sun, and move the meridian up or down in the notches, until the axis casts no shadow; then the arch of the meridian, contained betwixt the Pole and the horizon, is the Sun’s altitude.

Note, The best way to find the Sun’s altitude, is by a little quadrant graduated into degrees, and having sights and a plummet to it: Thus, hold the quadrant in your hand, so as the rays of the Sun may pass through both the sights, the plummet then hanging freely by the side of the instrument, will cut in the limb the altitude required. These quadrants are to be had at the instrument-makers, with lines drawn upon them, for finding the hour of the day, and the azimuth; with several other pretty conclusions, very entertaining for beginners.

The Latitude and the Day of the Month being given, to find the hour of the day when the Sun shines.

Having placed the wooden frame upon a level, and the meridian due North and South, rectify the globe for the latitude, and fix a needle perpendicularly over the Sun’s place: The Sun’s place being brought to the meridian, set the hour index at 12 at noon, then turn the globe about until the needle points exactly to the Sun, and casts no shadow, and then the index will shew the hour of the day.

Prob. XXVI. The Latitude, the Sun’s Place, and his Altitude, being given; to find the hour of the Day, and the Sun’s Azimuth from the Meridian.

Having rectified the globe for the latitude, the zenith, and the Sun’s place, turn the globe and the quadrant of altitude, so that the Sun’s place may cut the given degree of altitude: then the index will show the hour, and the quadrant will cut the azimuth in the horizon. Thus, if at London, on the 21st of August, the Suns altitude, be 36 degrees in the forenoon, the hour of the day will be IX, and the Sun’s azimuth about 58 degrees from the South part of the meridian.

The Sun’s Azimuth being given, to place the Meridian of the Globe due North and South, or to find a Meridian Line when the Sun shines.

Let the Sun’s azimuth be 30 degrees South-Easterly, set the horizon of the globe upon a level, and bring the North Pole into the zenith; then turn the horizon about until the shade of the axis cuts as many hours as is equivalent to the azimuth (allowing 15 degrees to an hour) in the North-West part of the hour circle, viz. X at night, which being done, the meridian of the globe stands in the true meridian of the place. The globe standing in this position, if you hang two plummets at the North and South points of the wooden horizon, and draw a line betwixt them, you will have a meridian line; which if it be on a fixed plane (as a floor or window) it will be a guide for placing the globe due North and South, at any other time.

Prob. XXVII. The Latitude, Hour of the Day, and the Sun’s place being given, to find the Sun’s Altitude and Azimuth.

Rectify the globe for the latitude, the zenith, and the Sun’s place, then the number of degrees contained betwixt the Sun’s place and the vertex, is the Sun’s meridional zenith distance; the complement of which to 90 degrees, is the Sun’s meridian altitude. If you turn the globe about until the index points to any other given hour, then bringing the quadrant of altitude to cut the Sun’s place, you will have the Sun’s altitude at that hour; and where the quadrant cuts the horizon, is the Sun’s azimuth at the same time. Thus May the 1st at London, the Sun’s meridian altitude will be 61½ degrees; and at 10 o’clock in the morning, the Sun’s altitude will be 52 degrees, and his azimuth about 50 degrees from the South part of the meridian.

Prob. XXVIII. The Latitude of the place, and the day of the Month being given; to find the depression of the Sun below the Horizon, and the Azimuth at any Hour of the Night.

Having rectified the globe for the latitude, the zenith, and the Sun’s place, take a point in the ecliptic exactly opposite to the Sun’s place, and find the Sun’s altitude and azimuth, as by the [last problem], and these will be the depression and the altitude required. Thus, if the time given be the 1st of December, at 10 o’clock at night, the depression and azimuth will be the same as was found in the [last problem].

Prob. XXIX. The Latitude, the Sun’s Place, and his Azimuth being given, to find his Altitude, and the Hour.

Rectify the globe for the latitude, the zenith, and the Sun’s place, then put the quadrant of altitude to the Sun’s azimuth in the horizon, and turn the globe ’till the Sun’s place meet the edge of the quadrant, then the said edge will shew the altitude, and the index point to the hour. Thus, May the 21st at London when the Sun is due East, his altitude will be about 24 degrees, and the hour about VII in the morning; and when his azimuth is 60 degrees South-Westerly, the altitude will be about 44½ degrees, and the hour about 2¾ in the afternoon.

Thus, the latitude and the day being known, and having besides either the altitude, the azimuth, or the hour; the other two may be easily found.

Prob. XXX. The Latitude, the Sun’s Altitude, and his Azimuth being given; to find his Place in the Ecliptic and the Hour.

Rectify the globe for the latitude and zenith, and set the edge of the quadrant to the given azimuth; then turning the globe about, that point of the ecliptic which cuts the altitude, will be the Sun’s place. Keep the quadrant of the altitude in the same position, and having brought the Sun’s place to the meridian, and the hour index to 12 at noon, turn the globe about ’till the Sun’s place cuts the quadrant of altitude, and then the index will point the hour of the day.

Prob. XXXI. The Declination and Meridian Altitude of the Sun, or of any Star being given; to find the Latitude of the Place.

Mark the point of declination upon the meridian, according as it is either North or South from the equator; then slide the meridian up or down in the notches, ’till the point of declination be so far distant from the horizon, as is the given meridian altitude; that elevation of the Pole will be the latitude.

Thus, if the Sun’s, or any Star’s meridian altitude be 50 degrees, and its declination 11½ degrees North, the latitude will be 51½ degrees North.

Prob. XXXII. The Day and Hour of a Lunar Eclipse being known; to find all those Places upon the Globe where the same will be visible.

[5] Find where the Sun is vertical at the given hour, and bring that point to the zenith; then the Eclipse will be visible in all those places that are under the horizon; Or, if you bring the Antipodes to the place where the Sun is vertical, into the zenith, you will have the places where the Eclipse will be visible above the horizon.

Note, Because Lunar eclipses continue sometimes for a long while together, they may be seen in more places than one hemisphere of the Earth; for by the Earth’s motion round its axis, during the time of the eclipse, the Moon will rise in several places after the eclipse began.

Note, When an eclipse of the Sun is central, if you bring the place where the Sun is vertical at that time, into the zenith, some part of the eclipse will be visible in most places within the upper hemisphere; but by reason of the short duration of Solar eclipses, and the latitude which the Moon commonly has at that time (tho’ but small) there is no certainty in determining the places where those eclipses will be visible by the globe; but recourse must be had to calculations.

Prob. XXXIII. The Day of the Month, and Hour of the Day, according to our way of reckoning in England, being given; to find thereby the Babylonic, Italic, and the Jewish, or Judaical Hour.

1. To find the Babylonic Hour (which is the number of hours from Sun-rising.) Having found the time of Sun-rising in the given place, the difference betwixt this and the hour given, is the Babylonic Hour.

2. To find the Italic Hour (which is the number of hours from Sun-setting.) Subtract the hour of Sun-setting from the given hour, and the remainder will be the Italic Hour required.

3. To find the Jewish Hour (which is ¹/₁₂ part of an Artificial Day.) Find how many hours the day consists of; then say, as the number of hours the day consists of is to 12 hours, so is the hour since Sun-rising to the Judaical hour required.

Thus, if the Sun rises at 4 o’clock (consequently sets at 8) and the hour given be 5 in the evening, the Babylonish hour will be the 13th, the Italic the 21st and the Jewish hour will be nine and three quarters.

The converse being given, the hour of the day, according to our way of reckoning in England, may be easily found.

The following Problems are peculiar to the Celestial Globe.

Prob. XXXIV. To find the Right Ascension and Declination of the Sun, or any Fixed Star.

Bring the Sun’s place in the ecliptic to the meridian; then that degree of the equator, which is cut by the meridian, will be the Sun’s Right Ascension; and that degree of the meridian, which is exactly over the Sun’s place, is the Sun’s Declination.

After the same manner, bring the place of any Fixed Star to the meridian, and you will find its Right Ascension in the equinoctial, and Declination of the meridian.

Thus, the right ascension and declination is found, after the same manner as the longitude and latitude of a place upon the Terrestrial Globe.

Note, The right ascension and declination of the Sun vary every day; but the right ascension, &c. of the Fixed Stars is the same throughout the year[6].

The Sun’s Right Ascension. Declin.
Deg. Deg.
Thus on January 3131417⅓ S.
April 514¼6 N.
July 21120¼20½ N.
November 26 242¼21 S.
R.Asc. Dcl.
Deg. Deg.
Aldebaran6516 N.
Spica Virginis197¾9¾ S.
Capella7445⅔
Syrius, or the Dog-Star98¼16⅓

Note, The declination of the Sun may be found after the same manner by the Terrestrial Globe, and also his right ascension, when the equinoctial is numbered into 360 degrees, commencing at the equinoctial point ♈: But as the equinoctial is not always numbered so, and this being properly a Problem in Astronomy, we choose rather to place it here.

By the converse of this problem, having the right ascension and declination of any point given, that point itself may be easily found upon the globe.

Prob. XXXV. To find the Longitude and Latitude of a given Star.

Having brought the solstitial colure to the meridian, fix the quadrant of altitude over the proper Pole of the ecliptic, whether it be North or South; then turn the quadrant over the given Star, and the arch contained betwixt the Star and the ecliptic, will be the latitude, and the degree cut on the ecliptic will be the Star’s longitude.

Thus the latitude of Arcturus will be found to be 31 degrees North, and the longitude 200 degrees from ♈, or 20 degrees from ♎: The latitude of Fomalhaut in the Southern Fish, 21 degrees South, and longitude 299½ degrees, or ♑ 29½ degrees. By the converse of this method, having the latitude and longitude of a Star given, it will be easy to find the Star upon the globe.

The distance betwixt two Stars, or the number of degrees contained betwixt them, may be found by laying the quadrant of altitude over each of them, and counting the number of degrees intercepted; after the same manner as we found the distance betwixt two places on the Terrestrial Globe, in [Prob. VII].

Prob. XXXVI. The Latitude of the Place, the Day of the Month, and the Hour being given; to find what Stars are then rising or setting, what Stars are culminating, or on the meridian, and the Altitude and Azimuth of any Star above the Horizon; and also how to distinguish the Stars in the Heavens one from the other, and to know them by their proper Names.

Having rectified the globe for the latitude, the zenith, and the Sun’s place, turn the globe about until the index points to the given hour, the globe being kept in this position.

All those Stars that are in (Eastern/Western) side of the horizon, are then (Rising/Setting).

All those Stars that are under the meridian, are then culminating. And if the quadrant of altitude be laid over the center of any particular Star, it will show that Star’s altitude at that time; and where it cuts the horizon, will be the Star’s azimuth from the North or South part of the meridian.

The globe being kept in the same elevation, and from turning round its axis, move the wooden frame about until the North and South points of the horizon lie exactly in the meridian; then right lines imagined to pass from the center thro’ each Star upon the surface of the globe, will point out the real Star in the heavens, which those on the globe are made to represent. And if you are by the side of some wall whose bearing you know, lay the quadrant of altitude to that bearing in the horizon, and it will cut all those Stars which at that very time are to be seen in the same direction, or close by the side of the said wall. Thus knowing some of the remarkable Stars in any part of the heavens, the neighbouring Stars may be distinguished by observing their situations with respect to those that are already known, and comparing them with the Stars drawn upon the globe.

Thus, if you turn your face towards the North, you will find the North Pole of the globe points to the Pole Star; then you may observe two Stars somewhat less bright than the Pole Star, almost in a right line with it, and four more which form a sort of quadrangle; these seven Stars make the constellation called the Little Bear; the Pole-Star being in the tip of the tail. In this neighbourhood you will observe seven bright Stars, which are commonly called Charles’s Wane; these are the bright Stars in the Great Bear, and form much such another figure with those before-mentioned in the little Bear: The two foremost of the square lie almost in a right line with the Pole Star, and are called the Pointers, so that knowing the Pointers, you may easily find the Pole Star. Thus the rest of the Stars in this constellation, and all the Stars in the neighbouring constellations may be easily found, by observing how the unknown Stars lie either in quadrangles, triangles, or strait lines from those that are already known upon the globe.

After the same manner the globe being rectified, you may distinguish those Stars that are to the Southward of you, and be soon acquainted with all the Stars that are visible in our hemisphere.

SCHOLIUM.

The globe being rectified to the latitude of any place, if you turn it round its axis, all those Stars that do not go below the horizon during a whole revolution of the globe, never set in that place; and those that do not come above the horizon never rise.

Prob. XXXVII. The latitude of the place being given; to find the Amplitude, Oblique Ascension and Descension, Ascensional Difference, Semi-diurnal Arch, and the time of continuance above the horizon, of any given point in the heavens.

Having rectified the globe for the latitude, and brought the given point to the meridian, set the index to the hour of 12; then turn the globe until the given point be brought to the Eastern side of the horizon, and that degree of the equinoctial which is cut by the horizon at that time, will be the Oblique Ascension; and where the given point cuts the horizon, is the Amplitude Ortive: If the globe be turned about until the given point be brought to the Western side of the horizon, it will there show the Amplitude Occasive; and where the horizon cuts the equinoctial at that time, is the Oblique Descension.

The time between the index at either of these two positions, and the hour of 6; or half the difference between the oblique ascension and descension is the Ascensional Difference.

If the place be in North latitude and the declination of the given point be (North/South) the ascensional difference reduced into time, and (added to/subtracted from) 6 o’Clock, gives the Semi-diurnal Arch; the complement whereof to a semicircle, is the Semi-nocturnal Arch. If the place be in South latitude, then the contrary is to be observed with respect to the declination.

The semi-(diurnal/nocturnal) arch being doubled, gives the time of continuance (above/below) the horizon. Or the time of continuance above the horizon, may be found by counting the number of hours contained in the upper part of the horary circle, betwixt the place where the index pointed when the given point was in the Eastern or Western parts of the horizon. If the given point was the Sun’s place, the index pointed the time of his rising and setting, when the said place was in the Eastern and Western parts of the horizon, as in [Prob. 18]. Or the time of Sun-rising may be found by adding or subtracting his ascensional difference, to or from the hour of six, according as the latitude and declination are either contrary or the same way.

Thus, at London, on the 31st of May, the Sun’s

These things for the Sun vary every day; but for a Fixed Star the day of the month need not be given, for they are the same all the year round.

Prob. XXXVIII. The Latitude and the Day of the Month being given; to find the Hour when any known Star will be upon the meridian, and also the time of its rising and setting.

Having rectified the globe for the latitude of the Sun’s place, bring the given Star to the meridian, and also to the East or West side of the horizon, and the index will shew accordingly when the Star culminates, or the time of the rising or setting.

Thus at London, on the 21st of January, Syrius will be upon the meridian, at a quarter past ten in the evening; rises at 5¼ hours, and sets at three quarters past two in the morning.

By the converse of this problem, knowing the time when any Star is upon the meridian, you may easily find the Sun’s place. Thus, bring the given Star to the meridian, and set the index to the given hour; then turn the globe ’till the index points to 12 at noon, and the meridian will cut the Sun’s place in the ecliptic. Thus when Syrius comes to the meridian at 10½ hours after noon, the Sun’s place will be ≈ ¼ deg.

Prob. XXXIX. To find at what time of the Year a given Star will be upon the Meridian, at a given Hour of the Night.

Bring the Star to the meridian, and set the index to the given hour, then turn he globe ’till the index points to 12 at noon, and the meridian will cut the ecliptic in the Sun’s place; whence the day of the month may be easily found in the kalendar upon the horizon.

Prob. XL. The Day of the Month, and the Azimuth of any known Star being given; to find the Hour of the Night.

Having rectified the globe for the latitude and the Sun’s place, if the given Star be due North or South, bring it to the meridian, and the index will show the hour of the night. If the Star be in any other direction, fix the quadrant of altitude in the zenith, and set it to the Star’s azimuth in the horizon; then turn the globe about until the quadrant cuts the center of the Star, and the index will shew the hour of the night.

The bearing of any point in the heavens may be found by the following methods.

Having a meridian line drawn in two windows, that are opposite to one another, you may cross it at right angles with another line representing the East and West; from the point of the intersection describe a circle, and divide each quadrant into 90 degrees; then get a smooth board, of about 2 feet long, and ¾ foot broad (more or less, as you judge convenient) and on the back part of it fix another small board crossways, so that it may serve as a foot to support the biggest board upright, when it is set upon a level, or an horizontal plane. The board being thus prepared, set the lower edge of the smooth, or fore side of it, close to the center of the circle, then turn it about to the meridian, or to any azimuth point required (keeping the edge of it always close to the center) and casting your eye along the flat side of it, you will easily perceive what Stars are upon the meridian, or any other bearing that the board is set to.

Prob. XLI. Two known Stars having the same Azimuth, or the same Height, being given; to find the Hour of the Night.

Rectify the globe for the latitude, the zenith, and the Sun’s place.

1. When the two Stars are in the same azimuth, turn the globe, and also the quadrant about, until both Stars coincide with the edge thereof; then will the index shew the hour of the night; and where the quadrant cuts the horizon, is the common azimuth of both Stars.

2. If the two Stars are of the same altitude, move the globe so that the same degree on the quadrant will cut both Stars, then the index will shew the hour.

This problem is useful when the quantity of the azimuth of the two Stars in the first case, or of their altitude in the latter case, is not known.

If two Stars were given, one on the meridian, and the other in the East or West part of the horizon; to find the Latitude.

Bring that Star which was observed on the meridian, to the meridian of the globe, and keep the globe from turning round its axis; then slide the meridian up or down in the notches, ’till the other Star is brought to the East or West part of the horizon, and that elevation of the Pole will be the Latitude sought.

Prob. XLII. The Latitude, Day of the Month, and the Altitude of any known Star being given; to find the Hour of the Night.

Rectify the globe for the latitude, zenith, and Sun’s place: Turn the globe, and the quadrant of altitude, backward or forward, ’till the center of that Star meets the quadrant in the degree of altitude given; then the index will point the true hour of the night; and also where the quadrant cuts the horizon, will be the azimuth of the Star at that time.

If the Latitude, the Sun’s Altitude, and his Declination (instead of his Place in the Ecliptic) are given; to find the Hour of the Day and Azimuth.

Rectify the globe for the latitude and zenith, and having brought the equinoctial colure to the meridian, set the index to 12 at noon; which being done, turn the globe and the quadrant, until the given declination in the equinoctial colure, cuts the altitude on the quadrant; then the index will shew the Hour of the day, and the quadrant cut the Azimuth in the horizon.

If the Altitude of two Stars on the same Azimuth were given; to find the Latitude of the Place.

Set the quadrant over both Stars at the observed degrees of altitude, and keep it fast upon the globe with your fingers; then slide the meridian up or down in the notches, ’till the quadrant cuts the given azimuth in the horizon; that elevation of the Pole will be the latitude required.

Prob. XLIII. Having the Latitude of the place, to find the degree of the Ecliptic, which rises or sets with a given Star; and from thence to determine the time of its Cosmical and Achronical rising and setting.

Having rectified the globe for the latitude, bring the given Star to the Eastern side of the horizon, and mark what degree of the ecliptic rises with it: Look for that degree in the wooden horizon, and right against it, in the kalendar, you will find the month and day when the Star rises Cosmically. If you bring the Star to the Western side of the horizon, that degree of the ecliptic which rises at that time, will give the day of the month when the said Star sets Cosmically. So likewise against the degree which sets with the Star, you will find the day of the month of the Achronical setting; and if you bring it to the Eastern part of the horizon, that degree which sets at that time will be the Sun’s place when the Star rises Achronically.

Thus, in the latitude of London, Syrius, or the Dog-Star, rises Cosmically the 10th of August, and sets Cosmically the 10th of October. Aldebaran, or the Bull’s Eye, rises Achronically on the 22d of May, and sets Achronically on the 19th of December.

Prob. XLIV. Having the Latitude of the place, to find the time when a Star rises and sets Heliacally.

Having rectified the globe for the latitude, bring the Star to the Eastern side of the horizon, and turn the quadrant round to the Western side, ’till it cuts the ecliptic in 12 degrees of altitude above the horizon, if the Star be of the first magnitude; then that point of the ecliptic which is cut by the quadrant, is 12 degrees high above the Western part of the horizon, when the Star rises; but at the same time the opposite point in the ecliptic is 12 degrees below the Eastern part of the horizon, which is the depression of a Star of the first magnitude, when she rises Heliacally; or has got so far from the Sun’s beams, that she may be seen in the morning before Sun-rising. Wherefore look for the said point of the ecliptic on the horizon, and right against it will be the day of the month when the Star rises Heliacally. To find the Heliacal setting, bring the Star to the West side of the horizon, and turn the quadrant about to the Eastern side, ’till the 12th degree of it above the horizon, cuts the ecliptic; then that degree of the ecliptic which is opposite to this point, is the Sun’s place when the Star sets Heliacally.

Thus you will find that Arcturus rises Heliacally the 28th of September, and sets Heliacally December the 2d.

Prob. XLV. To find the place of any Planet upon the globe; and so by that means, to find its place in the Heavens: Also to find at what Hour any Planet will rise or set, or be on the meridian at any one Day in the Year.

You must first seek in an Ephemeris (White’s Ephemeris will do well enough) for the place of the Planet proposed on that day; then mark that point of the ecliptic, either with chalk, or by sticking on a little black patch; and then for that night you may perform any problem, as before, by a Fixed Star.

Let it be required to find the situation of Jupiter among the Fixed Stars in the heavens, and also what time he rises and sets, and comes to the meridian on the 19th of May, 1757, N. S. at London.

Looking for the 19th of May, 1757, in White’s Ephemeris, I find that Jupiter’s place at that time is in about 12 degrees of ♏; latitude about 1¼ degree North. Then looking for that point upon the Celestial globe, I find that ♃ is then nearly in conjunction with the bright Star in the Southern Balance, and about 1 degree North of it.

To find when he rises and sets, and comes to the meridian: Having put a little black patch on the place of Jupiter, elevate the globe according to the latitude, and having brought the Sun’s place to the meridian, set the hour index to 12 at noon; then turn the mark which was made for Jupiter, to the Eastern part of the horizon, I find ♃ will rise somewhat more than half an hour after three in the afternoon; and turning the globe about, I find it comes to the meridian a little before eleven at night; and sets almost a quarter past six next morning.

This example being understood, it will be easy to find when either of the other two superior Planets, viz. Mars and Saturn, rise, set, and come to the meridian.

I shall conclude this subject about the Globes with the following problems.

Prob. XLVI. To find all that space upon the Earth, where an Eclipse of one of the Satellites of Jupiter will be visible.

Having found that place upon the Earth, in which the Sun is vertical at the time of the eclipse, by [Prob. 13], elevate the globe according to the latitude of the said place; then bring the place to the meridian, and set the hour index to 12 at noon. If Jupiter be in consequence of the Sun, draw a line with black lead, or the like, along the Eastern side of the horizon, which line, will pass over all those places where the Sun is setting at that time; then count the difference betwixt the right ascension of the Sun, and that of Jupiter, and turn the globe Westward, ’till the hour index points to this difference; then keep the globe from turning round its axis, and elevate the meridian, according to the declination of Jupiter. The globe being in this position, draw a line along the Eastern side of the horizon; then the space between this line, and the line before drawn, will comprehend all those places of the Earth where Jupiter will be visible, from the setting of the Sun, to the setting of Jupiter.

But if Jupiter be in antecedence of the Sun (i. e. rises before him) having brought the place where the Sun is vertical, to the zenith, and put the hour index to 12 at noon, draw a line on the Western side of the horizon; then elevate the globe according to the declination of Jupiter, and turn it about Eastward, until the index points to so many hours distant from noon, as is the difference of right ascension of the Sun and Jupiter. The globe being in this position, draw a line along the Western side of the horizon; then the space contained between this line, and the other last drawn, will comprehend all those places upon the Earth where the Eclipse is visible, between the rising of the Sun, and that of Jupiter.

The DESCRIPTION of the Great Orrery, lately made by Mr. Thomas Wright, Mathematical Instrument-Maker to his late Majesty, and now by Benjamin Cole, his Successor.

The Orrery is an Astronomical Machine, made to represent the motions of the Planets. These machines are made of various sizes, some having more Planets than others; but I shall here confine myself to the description of that above-mentioned.

In the Introduction we gave a short account of the Order, Periods, Distances, and Magnitudes of the Primary Planets; and of the Distances and Periodical Resolutions of the Secondary Planets round their respective Primaries. We shall here explain their Stations, Regradations, Eclipses, Phases, &c. but first let us take a general view of the Orrery.

The Description of the Orrery.

Vide [Frontispiece].

The frame which contains the wheel-work, &c. that regulates the whole Machine, is made of fine ebony, and is near four feet in diameter; the outside thereof is adorned with twelve pilasters, curiously wrought and gilt: Between these pilasters the twelve Signs of the Zodiac are neatly painted, with gilded frames. Above the frame is a broad ring, supported with twelve pillars: This ring represents the Plane of the Ecliptic, upon which there are two scales of degrees, and between those the names and characters of the twelve Signs. Near the outside is a scale of months and days, exactly corresponding to the Sun’s place at noon, each day throughout the year.

Above the ecliptic stands some of the principal circles of the sphere, according to their respective situations in the heavens, viz. N° 10, are the two Colures, divided into degrees, and half degrees; N° 11, is one half of the Equinoctial Circle, making an angle with the ecliptic of 23½ degrees. The Tropic of Cancer, and the Arctic Circle, are each fixed parallel, and at their proper distance from the equinoctial. On the Northern half of the ecliptic is a brass semicircle, moveable upon two points fixed in ♈ and ♎: This semicircle serves as a moveable horizon, to be put to any degree of latitude upon the North part of the meridian. The whole machine is also so contrived, as to be set to any latitude, without in the least affecting any of the inside motions: For this purpose there are two strong hinges (N° 13,) fixed to the bottom frame, upon which the instrument moves, and a strong brass arch, having holes at every degree, thro’ which a strong pin is to be put, according to the elevation. This arch and the two hinges, support the whole machine, when it is lifted up according to any latitude; and the arch at other times lies conveniently under the bottom frame.

When the machine is set to any latitude (which is easily done by two men, each taking hold of two handles, conveniently fixed for that purpose) set the moveable horizon to the same degree upon the meridian, and you may form an idea of the respective altitude, or depressions of the Planets, above or below the horizon, according to their respective positions, with regard to the meridian.

Within the ecliptic, and nearly in the same place thereof, stands the Sun, and all the Planets, both Primary and Secondary. The Sun (Nº 1.) stands in the middle of the whole system, upon a wire, making an angle with the plane of the ecliptic, of about 82 degrees; which is the inclination of the Sun’s axis, to the axis of the ecliptic. Next to the Sun is a Small ball (Nº 2.) representing Mercury: Next to Mercury is Venus (Nº 3.) represented by a larger ball (and both these stand upon wires,) so that the balls themselves may be more visibly perceived by the eye. The Earth is represented (Nº 4.) by an ivory ball, having some of the principal meridians and parallels, and a little sketch of a map described upon it. The wire which supports the Earth, makes an angle with the plane of the ecliptic 66½ degrees, which is the inclination of the Earth’s axis to that of the ecliptic. Near the bottom of the Earth’s axis is a Dial Plate (Nº 9.) having an index pointing to the hours of the day, as the Earth turns round its axis.

Round the Earth is a ring, supported by two small pillars, which ring represents the Orbit of the Moon, and the division upon it answers to the Moon’s latitude; the motion of this ring represents the motion of the Moon’s Orbit, according to that of the Nodes. Within this ring is the Moon (Nº 5.) having a black cap or case, which by its motion, represents the Phases of the Moon according to her age. Without the Orbits of the Earth and Moon is Mars (Nº 6.) The next in order to Mars is Jupiter, and his four Moons (Nº 7); each of these moons is supported by a crooked wire fixed in a socket, which turns about the pillar that supports Jupiter. These satellites may be turned by the hand to any position; and yet when the machine is put in motion, they will all move in their proper times. The outermost of all is Saturn, and his five Moons (Nº 8.) These moons are supported and contrived after the same manner with those of Jupiter. The whole machine is put into motion by turning a small winch (like the key of a clock, Nº 14.) and all the inside work is so truly wrought, that it requires but very small strength to put the whole motion.

Above the handle there is a cylindrical pin, which may be drawn a little out, or pushed in, at pleasure: when it is pushed in, all the Planets, both primary and Secondary, will move according to their respective periods, by turning the handle: When it is drawn out, the motions of the Satellites of Jupiter and Saturn will be stopped, while all the rest move without interruption. This is a very good contrivance to preserve the instrument from being clogged by the swift motions of the wheels belonging to the Satellites of Jupiter and Saturn, when the motions of the rest of the Planets are only considered.

There is also a brass lamp having two convex glasses, to be put in the room of the Sun; and also a smaller Earth and Moon, made somewhat in proportion to their distance from each other, which may be put on at pleasure.

The lamp turns round in the same time with the Earth, and by means of the glasses cast a strong light upon her; and when the smaller Earth and Moon are placed on, it will be easy to shew when either of them may be eclipsed.

Having thus given a brief description of the outward part of this machine, I shall next give an account of the phænomena explained by it, when it is put into motion.

Of the Motions of the Planets in general.

Having put on the handle, push in the pin which is just above it, and place a small black patch (or bit of wafer) upon the middle of the Sun (for instance) right against the first degree of ♈; you may also place patches upon Venus, Mars, and Jupiter, right against some noted point in the ecliptic. If you lay a thread from the Sun to the first degree of ♈, you may set a mark where it intersects the orbit of each Planet, and that will be a help to note the time of their revolutions.

One entire turn of the handle answers to the diurnal motion of the Earth round her axis, as may be seen by the motion of the hour index, which is placed at the foot of the wire on which the terella is fixed. When the index has moved the space of ten hours, you may observe that Jupiter has made one revolution compleat round its axis; the handle being turned until the hour index has passed over 24 days, 8 hours, will bring the patch upon Venus to its former situation with respect to the ecliptic, which shews that ♀ has made one entire revolution round her axis. Mars makes one compleat revolution round its axis in 24 hours and about 40 minutes. When the handle is turned 25½ times round, the spot upon the Sun will point to the same degree of the ecliptic, as it did when the instrument was first put into motion. By observing the motions of the spots upon the surface of the Sun, and of the Planets in the heavens, their diurnal motion was discovered; after the same manner as we do here observe the motions of their representatives, by that of the marks placed upon them.

If while you turn the handle you observe the Planets, you will see them perform their motions in the same relative times as they really do in the heavens, each making its period in the times mentioned in the Tables, Page, 28, 27¼ turns of the handle will bring the Moon round the Earth, which is called a Periodic Month; and all the while she keeps the same face towards the Earth; for the Moon’s annual and diurnal motion are performed both in the same time nearly, so that we always see the same face or side of the Moon.

If before the instrument is put into motion, the satellites of Jupiter and Saturn be brought into the same right line from their respective primaries, you will see them, as you turn the handle, immediately dispersed from one another, according to their different celerities. Thus one turn of the handle will bring the first of Jupiter’s Moons about ⁴/₇ part round Jupiter, while the second has described but ²/₇ part, the third but above ¹/₇, and the fourth not quite ¹/₁₆ part, each of its respective orbits. If you turn the handle until the hour index has moved 18½ hours more, the first satellite will then be brought into its former position, and so has made one entire revolution; the second at the same time will be almost diametrically opposite to the first, and so has made a little more than half of one revolution; the others will be in different aspects, according to the length of their periods, as will be plainly exhibited by the instrument. The same observations may be made with respect to the satellites of Saturn.

The machine is so contrived, that the handle may be turned either way; and, if before you put it into motion, you observe the aspect (or situation with respect to each other) of the Planets, and then turn the handle round any number of times; the same number of revolutions being made backwards, will bring all the Planets to their former situations. I shall next proceed to particulars.

Of the Stations and Retrogradations of the Planets.

Retrograde Motion of the Planets.

The primary Planets, as they all turn round the Sun, at different distances, and in different times, appear to us from the Earth to have different motions; as sometimes they appear to move from West to East, according to the order of the signs, which is called their Direct Motion; then by degrees they slacken their pace, until at last they lose all their motion, and become Stationary, or not to move at all; that is, they appear in the same place with respect to the fixed Stars for some time together; after which they again begin to move, but with a contrary direction, as from East to West, which is called their Retrograde Motion; then again they become stationary, and afterwards reassume their direct motion. The reason of all these appearances is very evidently shewn by the Orrery.

Of the Stations, &c. of the Inferior Planets.

We shall instance in the Planet Mercury, because his motion round the Sun differs more from the Earth’s than that of Venus does.

When Mercury is in his superior conjunction (or when he is in a direct line from the Earth beyond the Sun) fasten a string about the axis of the Earth, and extend it over Mercury to the ecliptic; then turning the handle, keep the thread all the while extended over ☿, and you will find it move with a direct motion in the ecliptic, but continually slower, until Mercury has the greatest elongation from the Earth. Near this position, the thread for some time will lay over Mercury without being moved in the ecliptic, tho’ the Earth and Mercury both continue their progressive motion in their respective orbits. When Mercury has got a little past this place, you will find the thread must be moved backward in the ecliptic, beginning first with a slow motion, and then faster by degrees, until Mercury is in his inferior conjunction, or directly between the Earth and the Sun. Next this position of ☿, his retrograde motion will be the swiftest; but he still moves the same way, tho’ continually slower, ’till he has again come to his greatest elongation, where he will appear the second time to be stationary; after which he begins to move forward, and that faster by degrees, until he is come to the same position with respect to the Earth, that he was in at first. The same observations may be made relating to the motions of Venus. In like manner the different motions observed in the superior Planets may be also explained by the Orrery. If you extend the thread over Jupiter, and proceed after the same manner as before we did in regard to Mercury, you will find that from the time Jupiter is in conjunction with the Sun, his motion is direct, but continually slower, until the Earth is nearly in a quadrate aspect with Jupiter, near which position Jupiter seems to be stationary: After which he begins to move, and continually mends his pace, until he comes in opposition to the Sun, at which time his retrograde motion is swiftest. He still seems to go backward, but with a slower pace, ’till the Earth and he are again in a quadrate aspect, where Jupiter seems to have lost all his motion; after which he again resumes his direct motion, and so proceeds faster by degrees, ’till the Earth and he are again in opposition to each another.

[Plate 3. Fig. 1].

These different motions observed in the Planets, are easily illustrated, as followeth: The lesser circle round the Sun is the orbit of Mercury, in which he performs his revolution round the Sun, in about three months, or while the Earth is going thro’ ¼ part of her orbit, or from A to N. The numbers 1, 2, 3, &c. in the orbit of Mercury, show the spaces he describes in a week nearly, and the distance AB, BC, DC, &c. in the Earth’s orbit, do likewise show her motion in the same time. The letters A, B, C, &c. in the great orb, are the motions of Mercury in the Heavens, as they appear from the Earth. Now if the Earth be supposed in A, and Mercury in 12, near his superior conjunction with the Sun; a spectator on the Earth will see ☿, as if he were in the point of the Heavens A, and while ☿ is moving from 12 to 1, and from 1 to 2, &c. the Earth in the same time also moves from A to B, and from B to C, &c. All which time ☿ appears in the Heavens to move in a direct motion from A to B, and from B to C, &c. but gradually slower, until he arrives near the point G; near this place he appears stationary, or to stand still; and afterwards (tho’ he still continues to move uniformly in his own orbit, with a progressive motion) yet in the sphere of the fixed Stars he will appear to be retrograde, or to go backwards, as from G to H, from H to I, &c. until he has arrived near the point L, where again he will appear to be stationary; and afterwards to move in a direct motion from L to M, and from M to N, &c.

What has been here shewed concerning the motions of Mercury, is also to be understood of the motions of Venus; but the conjunctions of Venus with the Sun do not happen so often as in Mercury; for Venus moving in a larger orbit, and much slower than Mercury, does not so often overtake the Earth. But the retrogradations are much greater in Venus than they are in Mercury, for the same reasons.

[Fig. 2.]

The innermost circle represents the Earth’s orbit, divided into 12 parts, answering to her monthly motion; the greatest circle is in the orbit of Jupiter, which he describes in about 12 years; and therefore the ¹/₁₂ thereof, from A to N, defines his motion, in one of our years nearly; and the intermediate divisions, A, B, C, &c. his monthly motion. Let us suppose the Earth to be in the point of her orbit 12, and Jupiter in A, in his conjunction with the Sun; it is evident that from the Earth Jupiter will be seen in the great orb, or in the point of the Heavens A, and while the Earth is moving from 12 to 1, 2, &c. ♃ also moves from A to B, &c. all which time he appears in the Heavens to move with a direct motion from A to B, C, &c., until he comes in opposition to the Earth near the point of the Heavens E, where he appears to be stationary; after which ♃ again begins to move ’ (tho’ at first with a slow pace) from E through F, H, I to K, where again he appears to stand still, but afterwards he reassumes his direct motion from I thro’ K, to M, &c.

From the construction of the preceding figure it appears, that when the superior Planets are in conjunction with the Sun, their direct motion is much quicker than at other times; and that because they really move from West to East, while the Earth in the opposite part of the Heavens is carried the same way, and round the same center. This motion afterwards continually slackens until the Planet comes almost in opposition to the Sun, when the line joining the Earth and Planet, will continue for some time nearly parallel to itself, and so the Planet seems from the Earth to stand still; after which, it begins to move with a slow motion backward, until it comes into a quartile aspect with the Sun, when again it will appear to be stationary, for the above reasons; after that it will resume its direct motion, until it comes into a conjunction with the Sun, then it will proceed as above explained. Hence it also appears, that the retrogradations of the superior Planets are much slower than their direct motions, and their continuance much shorter; for the Planet, from its last quarter, until it comes in opposition to the Sun, appears to move the same way with the Earth, by whom it is then overtaken: After which it begins to go backwards, but with a slow motion, because the Earth being in the same part of the Heavens, and moving the same way that the Planet really does, the apparent motion of the Planet backwards, must thereby be lessened.

What has been here said concerning the motions of Jupiter, is also to be understood of Mars and Saturn. But the retrogradations of Saturn do oftener happen than those of Jupiter, because the Earth oftener overtakes Saturn; and for the same reason, the regressions of Jupiter do oftener happen than those of Mars. But the retrogradations of Mars are much greater than those of Jupiter, whose are also much greater than those of Saturn.

In either of the satellites of Jupiter or Saturn, these different appearances in the neighbouring Worlds are much oftener seen than they are by us in the primary Planets.

We never observe these different motions in the Moon, because she turns round the Earth as her center; neither do we observe them in the Sun, because he is the center of the Earth’s motion; whence the apparent motion of the Sun always appears the same way round the Earth.

Of the Annual and Diurnal Motion of the Earth, and of the increase and decrease of Days and Nights.

The Earth in her annual motion round the Sun, has her axis always in the same direction, or parallel to itself; that is, if a line be drawn parallel to the axis, while the Earth is in any point of her orbit, the axis in all other positions of the Earth will be parallel to the said line. This parallelism of the axis, and the simple motion of the Earth in the ecliptic, solves all the phænomena of different seasons. These things are very well illustrated by the Orrery.

Plate 3.

If you put on the lamp in the place of the Sun, you will see how one half of our globe is always illuminated by the Sun, while the other hemisphere remains in darkness; how Day and Night are formed by the revolution of the Earth round her axis; for as she turns from West to East, the Sun appears to move from East to West. And while the Earth turns in her orbit, you may observe that her axis always points the same way, and the several seasons of the year continually change.

To make these things plainer, we will take a view of the Earth in different parts of her orbit.

When the Earth is in the first point of Libra (which is found by extending a thread from the Sun, and over the Earth, to the ecliptic) we have the Vernal Equinox, and the Sun at that time appears in the first point of ♈. In this position of the Earth, two Poles of the world are in the line separating light and darkness; and as the Earth turns round her axis, just one half of the equator, and all its parallels, will be in the light, and the other half in the dark; and therefore the days and nights must be every where equal.

As the Earth moves along in her orbit, you will perceive the North Pole advances by degrees into the illuminated hemisphere, and at the same time the South Pole recedes into darkness; and in all places to the Northward of the equator, the days continually lengthen, while the contrary happens in the Southern parts, until at length the Earth is arrived in Capricorn. In this position of the Earth all the space included within the arctic circle falls wholly within the light, and all the opposite part lying within the antarctic circle, is quite involved in darkness. In all places between the equator and the arctic circle, the days are now at the longest, and are gradually longer, as the place are more remote from the equator. In the Southern hemisphere there is a contrary effect. All the while the Earth is travelling from Capricorn towards Aries, the North Pole gradually recedes from the light, and the South Pole approaches nearer to it; the days in the Northern hemisphere gradually decrease, and in the Southern hemisphere they increase in the same proportion, until the Earth be arrived in ♈; then the two Poles of the world lie exactly in the line separating light and darkness, and the days are equal to the nights in all places of the world. As the Earth advances towards Cancer, the North Pole gradually recedes from the light, while the Southern one advances into it, at the same rate. In the Northern hemisphere the days decrease, and in the Southern one they gradually lengthen, until the Earth being arrived in Cancer, the North frigid Zone is all involved in darkness, and the South frigid Zone falls intirely within the light; the days every where in the Northern hemisphere are now at the shortest, and to the Southward they are at the longest. As the Earth moves from hence towards Libra, the North Pole gradually approaches the light, and the other recedes from it; and in all places to the Northward of the equator, the days now lengthen, while in the opposite hemisphere they gradually shorten, until the Earth has gotten into ♎; in which position the days and nights will again be of equal length in all parts of the world.

You might have observed that in all positions of the Earth, one half of the equator was in the light, and the other half in darkness; whence under the equator, the days and nights are always of the same length: And all the while the Earth was going from ♎ towards ♈, the North Pole was constantly illuminated, and the South Pole all the while in darkness; and for the other half year, the contrary. Sometimes there is a semicircle exactly facing the Sun, fixed over the middle of the Earth, which may be called the horizon of the disk: This will do instead of the lamp, if that half of the Earth which is next the Sun be considered, as being the illuminated hemisphere, and the other half, to be that which lies in darkness.

[Plate 4.]

The great circle ♈, ♉, ♊ &c. represent the Earth’s annual orbit; and the four lesser circles ESQC, the ecliptic, upon the surface of the Earth, coinciding with the great ecliptic in the Heavens. These four lesser figures represent the Earth in the four cardinal points of the ecliptic, P being the North Pole of the equator, and p the North Pole of the ecliptic; SPC, the solstitial colure which is always parallel to the great solstitial colure ♋ ☉ ♑ in the Heavens; EPQ the equinoctial colure. The other circles passing thro’ P, are meridians at two hours distance from one another; the semicircle EÆQ is the Northern half of the equator; the parallel circle touching the ecliptic in S, is the tropic of Cancer; the dotted circle, the parallel of London, and the small circle, touching the Pole of the ecliptic, is the Arctic Circle. The shaded part, which is always opposite to the Sun, is the obscure hemisphere, or that which lies in darkness; and that which is next the Sun, is the illuminated hemisphere.

If we suppose the Earth in ♎, she will then see the Sun in ♈ (which makes our vernal equinox) and in this position the circle bounding light and darkness, which here is SC, passes thro’ the Poles of the World, and bisects all the parallels of the equator; and therefore the diurnal and nocturnal arches, or the length of the days and nights, are equal in all places of the world.

But while the Earth in her annual course, moves through ♏, ♐, to ♑, the line SC, keeping still parallel to itself, or to the place where it was at first, the Pole P will, by this motion, gradually advance into the illuminated hemisphere; and also the diurnal arches of the parallels gradually increase, and consequently the nocturnal ones decrease in the same proportion, until the Earth has arrived into ♑; in which position the Pole P, and all the space within the arctic circle, fall wholly within the illuminated hemisphere, and the diurnal arches of all the parallels that are without this circle, will exceed the nocturnal arches more or less, as the places are nearer to, or farther off from it, until the distance from the Pole is as far as the equator, where both these arches are always equal.

Again, while the Earth is moving from ♑ through ♒, ♓, to ♈, the Pole P begins to incline to the line, distinguishing light and darkness, in the same proportion that before it receded from it; and consequently the diurnal arches gradually lessen, until the Earth has arrived into ♈ where the Pole P will again fall on the horizon, and so cause the days and nights to be every where equal. But when the Earth has passed ♈, while she is going thro’ ♉, and ♊, &c. the Pole P will begin to fall in the obscure hemisphere, and so recede gradually from the light, until the Earth is arrived in ♋; in which position not only the Pole, but all the space within the arctic circle, are involved in darkness, and the diurnal arches of all the parallels, without the arctic circle, are equal to the nocturnal arches of the same parallels, when the Earth was in the opposite point ♑; and it is evident that the days are now at the shortest, and the nights the longest. But when the Earth has past this point, while she is going through ♌ and ♏, the Pole P will again gradually approach the light, and so the diurnal arches of the parallels gradually lengthen, until the Earth is arrived in ♎; at which time the days and nights will again be equal in all places of the World, and the Pole itself just see the Sun.

Plate 4.

Here we only considered the phænomena belonging to the Northern parallels; but if the Pole P be made the South Pole, then all the parallels of latitude will be parallels of South latitude, and the days, every where, in any position of the Earth, will be equal to the nights of those who lived in the opposite hemisphere, under the same parallels.

Of the Phases of the Moon, and of her Motion in her Orbit.

Nodes.

Dragon’s Head.

Dragon’s Tail.

Retrograde Motion of the Nodes.

The orbit of the Moon makes an angle with the plane of the ecliptic, of above 5¼ degrees, and cuts it into two points, diametrically opposite (after the same manner as the equator and the ecliptic cut each other upon the globe, in ♈ and ♎) which points are called the Nodes; and a right line joining these points, and passing through the center of the Earth, is called the Line of the Nodes. That node where the Moon begins to ascend Northward above the plane of the ecliptic, is called the Ascending Node, and the Head of the Dragon, and is thus commonly marked [Symbol]. The other node from whence the Moon, descends to the Southward of the ecliptic, is called the Descending Node, and the Dragon’s Tail, and is thus marked [Symbol]. The line of nodes continually shifts itself from East to West, contrary to the order of the signs; and with this retrograde motion, makes one revolution round the Earth, in the space of about 19 years.

Periodical Month.

Synodical Month.

The Moon describes its orbit round the Earth in the Space of 27 days and 7 hours, which space of time is called a Periodical Month; yet from one conjunction to the next, the Moon spends 29 days and a half, which is called a Synodical Month; because while the Moon in her proper Orbit finishes her course, the Earth advances near a whole sign in the ecliptic; which space the Moon has still to describe, before she will be seen in conjunction with the Sun.

When the Moon is in conjunction with the Sun, note her place in the ecliptic; then turning the handle, you will find that 27 days and 7 hours will bring the Moon to the same place; and after you have made 2¼ revolutions more, the Moon will be exactly betwixt the Sun and the Earth.

Phases of the Moon.

The Moon all the while keeps in her orbit, and so the wire that Supports her continually rises or falls in a socket, as she changes her latitude; the black cap shifts itself, and so shews the phases of the Moon, according to her age, or how much of her enlightened part is seen from the Earth. In one synodical month, the line of the nodes moves about 1½ degree from West to East, and so makes one entire revolution in 19 years.

Let AB be an arch of the Earth’s orbit, and when the Earth is in T, let the Moon be in N, in conjunction with the Sun in S, while the Moon is describing her orbit NAFD, the Earth will describe the arch of her orbit T t; and when the Earth has got into the point t, the Moon will be in the point of her orbit n, having made one compleat revolution round the Earth. But the Moon, before she comes in conjunction with the Sun, must again describe the arch n o; which arch is similar to T t, because the lines FN, f n, are parallel; and because, while the Moon describes the arch n o, the Earth advances forward in the ecliptic; the arch described by the Moon, after she has finished her periodical month, before she makes a synodical month, must be somewhat greater than n o. To determine the mean length of a synodical month, find the diurnal motion of the Moon (or the space she describes round the Earth in one day) and likewise the diurnal motion of the Earth; then the difference betwixt the two motions, is the apparent motion of the Moon round the Earth in one day; then it will be, as this differential arch is to a whole circle; so is one day to that space of time wherein the Moon appears to describe a compleat circle round the Earth, which is about 29½ days. But this is not always a true Lunation, for the motion of the Moon is sometimes faster, and sometimes slower, according to the position of the Earth in her orbit.

In one synodical month the Moon has all manner of aspects with the Sun and Earth, and because she is opaque, that face of hers will only appear bright which is towards the Sun, while the opposite remains in darkness. But the inhabitants of the Earth can only see that face of the Moon which is turned towards the Earth; and therefore, according to the various positions of the Moon, in respect of the Sun and Earth, we observe different portions of her illuminated face, and so a continual change in her[7] Phases.

Let S be the Sun, RTV an arch of the Earth’s orbit, T the Earth, and the circle ABCD, &c. the Moon’s orbit, in which she turns round the Earth in the space of a month; and let A, B, C, &c. be the centers of the Moon in different parts of her orbit.

Now if with the lines S A, S B, &c. we join the centers of the Sun and Moon, and at right angles to these draw the lines H O; the said lines H O will be the circles that separate the illuminated part of the Moon from the dark and obscure. Again, if we conceive another line I L to be drawn at right angles to the lines TA, TB, &c. passing from the center of the Earth to the Moon, the said line I L will divide the visible hemisphere of the Moon, or that which is turned towards us, from the invisible, or that which is turned from us; and this circle may be called the Circle of Vision.

Full Moon.

Half Moon.

New Moon.

Now it is manifest, that whenever the Moon is in the position A, or in that point of her orbit which is opposite to the Sun, the circle of vision, and the circle bounding light and darkness, do coincide, and all the illuminated face of the Moon is turned towards the Earth, and is visible to us; and in this position the Moon is said to be full. But when the Moon arrives to B, all her illuminated face is then not towards the Earth, there being a part of it, HBI, not to be seen by us; and then her visible face is deficient from a circle, and appears of a gibbous form, as in B. [Fig. 3]. Again when she arrives to C, the two forementioned circles cut each other at right angles, and then we observe a half Moon, as in C, [Fig. 3]. And again the illuminated face of the Moon is more and more turned from the Earth, until she comes to the Point E, where the circle of vision, and that bounding light and darkness, do again coincide. Here the Moon disappears, the illuminated part being wholly turned from the Earth; and she is now said to be in Conjunction with the Sun, because she is in the same direction from the Earth that the Sun is in, which position we call a New Moon. When the Moon is arrived to F, she again assumes a horned figure, but her horns (which before the change were turned Westward) have now changed their position, and look Eastward. When she has arrived to a quadrate aspect at G, she will appear bissected, like a half Moon, afterwards she will still grow bigger, until at last she comes to A, where again she will appear in her full splendor.

The same appearances which we observe in the Moon are likewise observed by the Lunarians in the Earth, our Earth seeing a Moon to them, as their Moon is to us; and we are observed by them to be carried round in the space of time that they are really carried round the Earth. But the same phases of the Earth and Moon happen when they are in contrary position; for when the Moon is in conjunction to us, the Earth is then in opposition to the Moon, and the Lunarians have then a full Earth, as we in a similar position have a full Moon. When the Moon comes in opposition to the Sun, the Earth, seen from the Moon, will appear in conjunction with her, and in that position the Earth will disappear; afterwards she will assume a horned figure, and so shew the same phases to the inhabitants of the Moon as she does to us.

Of the Eclipses of the Sun and Moon.

Eclipse.

An Eclipse is that deprivation of light in a Planet, when another is interposed betwixt it and the Sun. Thus, an eclipse of the Sun is made by the interposition of the Moon at her conjunction, and an eclipse of the Moon is occasioned by the shadow of the Earth falling upon the Moon, when she is in opposition to the Sun.

[Fig. 4.].

Lunar Eclipse.

Let S be the Sun, T the Earth, and ABC its shadow; now if the Moon, when she is in opposition to the Sun, should come into the conical space ABC, she will then be deprived of the solar light, and so undergo an eclipse.

Solar Eclipse.

[Fig. 5.]

In the same manner, when the shadow of the Moon falls upon the Earth (which can never happen but when the Moon is in conjunction with the Sun) that part upon which the shadow falls will be involved in darkness, and the Sun eclipsed. But because the Moon is much less than the Earth, the shadow of the ☽ cannot cover the whole Earth, but only a part of it. Let S be the Sun, T the Earth, ABC the Moon’s orbit, and L the Moon in conjunction with the Sun: Here the shadow of the Moon falls only upon the part DE of the Earth’s surface, and there only the Sun is intirely hid: but there are other parts EF, DG, on each side of the shadow, where the inhabitants are deprived of part of the Solar rays, and that more or less, according to their distance from the shadow. Those who live at H and I will see half of the Sun eclipsed, but in the spaces FM, GN, all the Sun’s body will be visible, without any eclipse. From the preceding figure it appears, that an eclipse of the Sun does not reach a great way upon the superficies of the Earth; but the whole body of the Moon may sometimes be involved in the Earth’s shadow.

[Fig. 6.]

Although the Moon seen from the Earth, and the Earth seen from the Moon, are each alternately, once a month, in conjunction with the Sun; yet, by reason of the inclination of the Moon’s orbit to the ecliptic, the Sun is not eclipsed every new Moon, nor the Moon at every full. Let T be the Earth, DTE an arch of the ecliptic, ALBF, the Moon’s orbit, having the Earth T, in its center; and let AGBG be another circle coinciding with the ecliptic, and A, B, the nodes, or the two points where the Moon’s orbit and the ecliptic cut each other. A the ascending node, and B the descending node. The angle GAL equal to GBL is the inclination of the Moon’s orbit to the ecliptic, being about 5¼ degrees. Now a spectator from the Earth at T, will observe the Sun to move in the circle AGBC, and the Moon in her orbit ALBF; whence it is evident, that the Sun and Moon can never be seen in a direct line, from the center of the Earth, but when the Moon is in one of the nodes A or B; and then only will the Sun appear centrally eclipsed. But if the conjunction of the Moon happens when she is any where within the distance A c of the nodes, either North or South, the Sun will then be eclipsed, more or less, according to the distance from the node A, or B. If the conjunction happens when the Moon is in b, the Sun will be then one half eclipsed; and if it happens when she is in c, the Moon’s limb will just touch the Sun’s disk, without hiding any part of it.

The shadow of the Earth at the place where the Moon’s orbit intersects it, is three times as large as the Moon’s diameter, as in [Fig. 4.] and therefore it often happens that eclipses of the Moon are total, when they are not central: And for the same reason the Moon may sometimes be totally eclipsed for three hours together; whereas total eclipses of the Sun can scarcely ever exceed four minutes.

The eclipses of the Sun and Moon are very well explained by the Orrery: Thus having put the lamp in the place of the Sun, and the little Earth and the little Moon in their proper places, instead of the larger ones, let the room wherein the instrument stands be darkened; then turning the handle about, you will see when the conjunction of the Moon happens. When she is in or near one of the nodes, her shadow will fall upon the Earth, and so deprive that part upon which it falls of the light of the Sun: If the conjunction happens when the Moon is not near one of the nodes, the light of the lamp will fall upon the Earth, either above or below the Moon, according to her latitude at that time. In like manner, when the full Moon happens near one of the nodes, the shadow of the Earth will fall upon the Moon; and if the Moon’s latitude be but small, her whole face will be involved in darkness. At other times, when the full Moon happens when she is not near one of her nodes, the shadow of the Earth will pass either above or below the Moon, and so by that means the Moon will escape being eclipsed.

Of the Eclipses of the Satellites of Jupiter.

The apparent diameters of the inferior Planets are so small, that when they pass betwixt us and the Sun, they only appear like small spots upon the Sun’s surface, without depriving us of any sensible quantity of his light. The shadow of the Earth likewise terminates before it reaches any of the superior Planets, so that they are never eclipsed by us; and the Earth when she is in conjunction with the Sun, only appears like a black spot upon his surface.

But Jupiter and his Moons mutually eclipse each other, as our Earth and Moon do; as also doth Saturn and his Moons. The satellites of Jupiter become twice hid from us, in one circulation round ♃; viz. once behind the body of Jupiter, i. e. when they are in the right line joining the centers of the Earth and ♃; and again they become invisible when they enter the shadow of Jupiter, which happens when they are at their Full, as seen from ♃, at which times they also suffer eclipses; which eclipses happen to them after the same manner as they do to our Moon, by the interposition of the Earth betwixt her and the Sun.

[Fig. 7.]

Let S be the Sun, ABT the Earth’s orbit; and C ♃ D, an arch of Jupiter’s orbit, in which let Jupiter be in the point ♃; and let CFDH be the orbit of one of Jupiter’s satellites, which we will here suppose to be the farthest from him. These satellites, while they move thro’ the inferior parts of their orbs, viz. from D thro’ H, I, to C, seem from the Earth and the Sun to have a retrograde motion; but when they are in the superior part of their orbit, they are then seen to move from West to East, according to their true motion. Now while they describe the superior part of their orbits, they will be twice hid from the Earth, once in the shadow of ♃, and once behind his body. If Jupiter be more Westerly than the Sun, that is, when the Earth is in A, they will be first hid in the shadow F, and afterwards behind the body of ♃ in G: But when the Earth is in B, then they are first hid behind ♃’s body in E, and afterwards fall into the shadow F. While the satellites describe the inferior parts of their orbit, they only once disappear, which may be either in I or H, according to the position of the Earth, in which places they cannot be distinguished from the body of Jupiter.

Plate 5.

When the satellites seen from ♃ are in conjunction with the Sun, their shadows will then fall upon ♃, and some part of his body be involved in darkness, to which part the Sun will be totally eclipsed.

By observing the eclipses of Jupiter’s satellites, it was first discovered that light is not propogated instantaneously, though it moves with an incredible swiftness: For if light came to us in an instant, an observer in T will see an eclipse of one of the satellites, at the same time that another in K would. But it has been found by observations, that when the Earth is in K, at her nearest distance from Jupiter, these eclipses happen much sooner than when she is in T. Now having the difference of time betwixt these appearances in K and T, we may find the length of time the light takes in passing from K to T, which space is equal to the diameter of the Earth’s annual orb. By these kinds of observations it has been found, that light reaches from the Sun to us in the space of eleven minutes of time, which is at least at the rate of 100,000 miles in a second.

FINIS.

AN INDEX OF THE
Astronomical Terms
Made Use of in this BOOK.

Acronical Rising and Setting of the Stars  Page [96]
Almacanthers[63]
Altitudes[ib.]
——— Meridian Altitude[63]
Amplitude[62]
Amphiscians[91]
Annual Motion[7]
Antœci[92]
Antarctic Circle[53]
——— Pole[ib.]
Antipodes[93]
Arctic Circle[52]
Arctic Pole[53]
Ascension[68]
——— Right[ib.]
——— Oblique[69]
Ascensional Difference[ib.]
Ascians[91]
——— Heteroscians[ib.]
Asterisms[36]
Atmosphere[81]
Axis[43]
——— of the World[49]
Azimuth[61]
Babylonish Hours[71]
Bissextile[78]
Circle[42]
——— Great Circles[ib.]
——— Parallel, or lesser Circles[43]
——— Secondary Circles[ib.]
Circles of the Sphere[47]
Climates[93]
Colures[53]
——— Equinoctial Colure[ib.]
——— Solstitial Colure[54]
Comets[29]
Conjunction[11], [207]
Constellations[36]
Cosmical rising and setting of the Stars[96]
Crepusculum[83]
Day, Natural and Artificial[69]
Declination[52]
Diurnal Motion7
Diurnal Arch[68]
Eclipses[208]
——— Solar[ib.]
——— Lunar[ib.]
Eclipses of Jupiter’s Satellites[212]
Ecliptic[53]
Egyptian Year[75]
Elongation[18]
Equator, or Equinoctial[48]
Equinoctial Points[53]
——— Precession of[55]
——— Vernal and Autumnal[70]
Excentricity[4]
Galaxy, or Milky Way[38]
Geocentric Place[19]
Globe[42]
——— Terrestrial[43]
——— Celestial[44]
Gregorian Account[80]
Heliacal rising and setting of the Stars[96]
Heliocentric Place[19]
Hemisphere[42]
——— Northern and Southern[49]
Heteroscians[91]
Horizon[58]
——— Sensible[ib.]
——— Rational[59]
Hour Circles[50]
Italian Hours[72]
Jewish Hours[ib.]
Julian Account[79]
Latitude, in Astronomy[56]
——— in Geography[84]
Longitude in Astronomy[56]
——— in Geography[87]
Meridian[50], [61]
Nadir[61]
Nodes[3], [202]
Nocturnal Arch[68]
Orbit[3]
Parallel of the Earth’s Semidiameter[23]
——— of the Earth’s Annual Orb[20]
Periœci[92]
Periscians[91]
Periodical Month[74], [202]
Phases of the Moon[201]
Planets[1]
——— Inferior and Superior[14]
Planetary Hours[72]
Poles[42]
——— of the World[49]
——— of the Ecliptic[56]
Polar Circles[52]
Points of the Compass[60]
——— Cardinal Points[59]
Primary Planets[5]
Retrograde Motion of the Planets[187]
——— of the Nodes[202]
Secondary Planets[5]
Sidereal Year[74]
Signs of the Zodiac[54]
——— Northern and Southern[ib.]
Solstices[71]
——— Summer and Winter Solstices[ib.]
Solstitial Points[53]
Sphere[42]
——— Parallel and Right[67]
——— Oblique[68]
Stationary[186]
Style Old[79]
——— New Style[80]
Synodical Month[74], [202]
Tropics (of Cancer and Capricorn)[52]
Twilights[83]
Vertical Circles[61]
——— Prime Vertical[62]
Zenith[61]
Zenith Distance[63]
Zones, Torrid, Temperate, and Frigid[90]

THE END

Directions to the Binder.

The great Orrery to face the Title. 
Plate I.Page 2
Plate II.28
The Globes35
Plate III.194
Plate IV.200
Plate V.214

A CATALOGUE

Of Mathematical, Philosophical,
and Optical Instruments,

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BENJAMIN COLE,

At his Shop, the Sign of the Orrery, No. 136,
in Fleet street, London.

l.s.d.

Variety of pocket cases of Drawing Instruments, in Silver,

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FOOTNOTES:

[1] By the Orbit of a Planet is commonly understood the Tract or Ring, described by its Center round the Sun, but by the Plane of the Orbit is meant a flat Surface extended every way thro’ the Orbit infinitely.

[2] N. B. According to Biachini’s Observations, Venus’s axis inclines 75 degrees from the perpendicular to the plane of the Ecliptic (which is 51½ deg. more than the axis of our Earth) her Tropics are only 15 deg. from her Poles, and her Polar Circles at the same distance from her Equator; so that the Sun’s greatest Declination on each side of her Equator is 75 deg. by which she must undergo a much greater variety of seasons than we do on our Earth.

[3] ☌ Is a mark commonly used for conjunction; thus ☌ with the ☉, is to be read conjunction with the Sun.

[4] [Prob. VI.]

[5] [Prob. XIII].

[6] The insensible change in the Longitude, Right Ascension, and Declination of the Fixed Stars, made by their slow motion, parallel to the ecliptic (being but 1 degree in 72 years) is not worth notice in this place.

[7] Phases of the Moon are those different appearances we observe in her, according to her position in respect to the Sun and Earth.

Transcriber’s Notes:

The cover image was created by the transcriber, and is in the public domain.

Illustrations were moved so as not to break up paragraphs.

Antiquated spellings were not corrected.

Typographical errors have been silently corrected.