Part 7

Thus when it is _Noon_ at _London_, it is

H. M. { _Rome_ 0 52 P. M. At { _Constantinople_ 2 07 P. M. { _Vera-Cruz_ 5 30 A. M. { _Pekin_ in _China_ 7 50 P. 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.