The lesser circles, of principal use, are the two tropics and two polar circles. Of these circles some are fixed, and always obtain the same position; others moveable, according to the position of the observer. The fixed circles are the equator and the ecliptic, with their parallels and secondaries; which are usually delineated upon the surfaces of the globes. The moveable circles are the horizon, with its parallels and secondaries. The horizon is that broad wooden circle surrounding the globe, and dividing it into two equal parts called the upper and lower hemispheres. It has two notches, to let the brazen meridian slip up and down, according to the different heights of the pole. On the flat side of this circle are described the twelve signs, the months of the year, the points of the compass, &c. The brazen meridian is an annulus or ring of brass, divided into degrees, viz. each quadrant in 90 degrees. It divides the globe into two equal parts, called the eastern and western hemispheres. The quadrant of altitude is a thin pliable plate of brass, answering exactly to a quadrant of the meridian. It is divided into 90°, and has a notch, nut, and screw, to fix to the brazen meridian in the zenith of any place; where it turns round a pivot, and supplies the room of vertical circles. The hour-circle is a flat ring of brass, divided into twenty-four equal parts, or hour-distances; and on the pole of the globe is fixed an index, that turns round with the globe, and points out the hours upon the hour-circle. Lastly, there is generally added a compass and needle upon the pediment of the frame. The surface of the celestial globe may be esteemed a just representation of the concave expanse of the heavens, notwithstanding its convexity; for it is easy to conceive the eye placed in the centre of the globe, and viewing the stars on its surface; supposing it made of glass, as some globes are also that if holes were made in the centre of each star, the eye in the centre of the globe, properly placed, would view through each of the holes the very stars in the heavens represented by them. As it would be impossible to have any distinct notion of the stars, in respect to their number, order, and distances, without arranging them in certain forms, called constellations, this the first observers of the heavens took care to do; and these, like kingdoms and countries, upon the terrestrial globe, serve to distinguish the longitude of any place." Bring the place 2. To find a place, having a given latitude and longitude." Find the degree of longitude on the equator, and bring it to the brass meridian; then find the degree of latitude on the meridian, either north or south of the equator, as the given latitude is north or south; and the point of the globe just under that degree of latitude is the place required. globe that have the same latitude and 3. To find all the places on the the same longitude, or hour, with a given given place, London, to the meridian, place, as suppose London." Bring the and observe what places are just under the edge of it, from north to south; and and hour with it. all those places have the same longitude Then turn the globe just under the given degree of latitude round; and all those places, which pass on the meridian, have the same latitude with the given place. 4. "To find the antæci, periæci, and antipodes, of any given place, suppose London." Bring the given place, London, to the meridian, then count 513 the same degree of latitude southward, or towards the other pole, and the point thus arrived at will be the antæci, or where the hour of the day or night is always the same at both places at the same time, and where the season and lengths of days and nights are also equal, but at half a year distance from each other, because their seasons are opposite or conridian, set the hour index to twelve at trary. London being still under the menoon, or pointing towards London; then turn the globe just half round, or till the index point to the opposite hour, or twelve at night; and the place that comes under the same degree of the meridian where London was shows where the periæci dwell, or those people that have the same seasons and at the same time as London, as also the same length of days and nights, &c. at that time, but only their time or hour is just opposite, or twelve hours distant, being day with one when night with the other, &c. Lastly, as the globe stands, count down by the meridian the same degree of latitude south, and that will give the place of the antipodes of London, being diametrically under or opposite to it; and so having all its times, both hours and seasons, opposite, being day with the one when night with the other, and summer with the one when winter with the other. 5. "To find the distance of two places on the globe." If the two places be ei ther both on the equator, or both on the same meridian, the number of degrees in the distance between them, reduced into miles, at the rate of seventy English miles to the degree, (or more exactly sixty-nine and one-fifth,) will give the distance nearly. But in any other situations of the two places, lay the quadrant of altitude over them, and the degrees counted upon it, from the one place to the other, and turned into miles as above, will give the distance in this case. 6. To find the difference in the time of the day at any two given places, and thence the difference of longitude." Bring one of the places to the meridian, and set the hour index to twelve at noon; then turn the globe till the other place comes to the meridian, and the index will point out the difference of time; then, by allowing fifteen degrees to every hour, or one degree to four miles of time, the difference of longitude will be known. Or the difference of longitude may be found without the time, thus: First bring the one place to the meridian, and note the degree of longitude on the equator cut by it; then do the same by the other place; which gives the longitudes of the two places; then subtracting the one number of degrees from the other gives the difference of longitude sought. 7." The time being known at any giv. en place, as suppose London, to find what hour it is in any other part of the world." Bring the given place, London, to the meridian, and set the index to the given hour; then turn the globe till the other place come to the meridian, and look at what hour the index points, which will be the time sought. 8. "To find the sun's place in the ecliptic, and also on the globe, at any giv en time." Look into the calendar on the wooden horizon for the month and day of the month proposed, and immediately opposite stands the sign and degree which the sun is in on that day. Then, in the ecliptic drawn upon the globe, look for the same sign and degree, and that will be the place of the sun required. 9. "To find at what place on the earth the sun is vertical, at a given moment of time at another place, as suppose London." Find the sun's place on the globe by the last problem, and turn the globe about till that place come to the meridian, and note the degree of the meridian just over it Then turn the globe till the given place, London, come to the meridian, and set the index to the given moment of time. Lastly, turn the globe till the index points to twelve at noon; then the place of the earth, or globe, which stands under the before noted degree, has the sun at that moment in the zenith. 10. "To find how long the sun shines, without setting, in any given place in the frigid zones." Subtract the degrees of latitude of the given place from ninety, which gives the complement of the latitude, and count the number of this complement upon the meridian fromthe equator towards the pole, marking that point of the meridian; then turn the globe round, and carefully observe what two degrees of the ecliptic pass exactly under the point marked on the meridian. Then look for the same degrees of the ecliptic on the wooden horizon, and just opposite to them stand the months and days of the months corresponding, and between which two days the sun never sets in that latitude. If the beginning and end of the longest night be required, or the period of time in which the sun never rises at that place; count the same complement of latitude towards the south or farthest pole, and then the rest of the work will be the same in all respects as above. of the horizontal refraction of the sun, Note, that this solution is independent which raises him rather more than half a degree higher, by that means making the day so much longer, and the night the shorter; therefore, in this case, set the mark on the meridian half a degree higher up towards the north pole than what the complement of latitude gives; then proceed with it as before, and the more exact time and length of the longest day and night will be found. 11. "A place being given in the torrid zone, to find on what two days of the year the sun is vertical at that place." Turn the globe about till the given place come to the meridian, and note the degree of the meridian it comes under. Next turn the globe round again, and note the two points of the ecliptic passing under that degree of the meridian. Lastly, by the wooden horizon, find on what days the sun is in those two points of the ecliptic; and on these days he will be vertical to the given place. 12. "To find those places in the torrid zone to which the sun is vertical on a given day." Having found the sun's place in the ecliptic, as in the eighth problem, turn the globe to bring the same point of the ecliptic on the globe to the meridian; then again turn the globe round, and note all the places which pass under that point of the meridian; which will be the places sought. After the same manner may be found what people are ascii for any given day. And also to what place of the earth, the moon, or any other planet, is vertical on a given day; finding the place of the planet on the globe by means of its right ascension and declination, like finding a place from its longitude and latitude giv en. 13. "To rectify the globe for the latitude of any place." By sliding the brass meridian in its groove, elevate the pole as far above the horizon as is equal to the latitude of the place; so for London, raise the north pole fifty-one and a half degrees above the wooden horizon: then turn the globe on its axis till the place, as London, come to the meridian, and there set the index to twelve at noon. Then is the place exactly on the vertez, or top point of the globe, at ninety degrees every way round from the wooden horizon, which represents the horizon of the place. And if the frame of the globe be turned about till the compass needle point to twentytwo and a half degrees, or two points west of the north point (because the variation of the magnetic needle is nearly twentytwo and a half degrees west), so shall the globe then stand in the exact position of the earth, with its axis pointing to the north pole. 14. "To find the length of the day or night, or the sun's rising or setting, in any latitude; having the day of the month given." Rectify the globe for the latitude of the place; then bring the sun's place on the globe to the meridian, and set the index to twelve at noon, or the upper twelve, and then the globe is in the proper position for noon-day. Next turn the globe about towards the east till the sun's place come just to the wooden horizon, and the index will then point to the hour of sun-rise; also turn the globe as far to the west side, or till the sun's place come just to the horizon on the west side, and then the index will point to the hour of sun-set. These being now known, double the hour of setting will be the length of the day, and double the rising will be the length of the night. And thus also may the length of the longest day, or the shortest day, be found for any latitude. 15. "To find the beginning and end of twilight on any day of the year, for any latitude.” It is twilight all the time from sun-set till the sun is eighteen degrees below the horizon, and the same in the morning from the time the sun is eighteen degrees below the horizon till the moment of his rise. Therefore, rectify the globe for the latitude of the place, and for noon, by setting the index to twelve, and screw on the quadrant of altitude. Then take the point of the ecliptic opposite the sun's place, and turn the globe on its axis westward, as also the quadrant of altitude, till that point cut this quadrant in the eighteenth degree below the horizon; then the index will shew the time of dawning in the morning; next turn the globe and quadrant of altitude towards the east, till the said point opposite the sun's place meet this quadrant in the same eighteenth degree, and then the index will shew the time when twilight ends in the evening. 16. "At any given day, and hour of the day, to find all those places on the globe where the sun then rises, or sets, as also where it is noon-day, where it is day-light, and where it is in darkness." Find what place the sun is vertical to, at that time and elevate the globe according to the latitude of that place, and bring the place also to the meridian; in which state it will also be in the zenith of the globe. Then is all the upper hemisphere, above the wooden horizon, enlightened, or in day light; while all the lower one, below the horizon, is in darkness, or night: those places by the edge of the meridian, in the upper hemisphere, have noon-day, or twelve o'clock; and those by the meridian below, have it midnight: lastly, all those places by the eastern side of the horizon have the sun just setting, and those by the western horizon have him just rising. Hence, as in the middle of a lunar eclipse, the moon is in that degree of the ecliptic opposite to the sun's place; by the present problem it may be shown what places of the earth; then see the middle of the eclipse, and what the beginning or ending, by using the moon's place instead of the sun's place in the problem. 17. "To find the bearing of one place from another, and their angle of position." Bring the one place to the zenith, by rectifying the globe for its latitude, and turning the globe till that place come to the meridian; then screw the quadrant of altitude upon the meridian at the zenith, and make it revolve till it come to the other place on the globe; then look on the wooden horizon for the point of the compass, or number of degrees from the south, where the quadrant of altitude cuts it, and that will be the bearing of the latter place from the former, or the angle of position sought. 18. "The day and hour of a solar or lunar eclipse being given, to find all those places in which the same will be visible." Find the place to which the sun is vertical at the given instant; and elevate the globe to the latitude of the place; then, in most of those places above the horizon will the sun be visible during his eclipse; and all those places below the horizon will see the moon pass through the shadow of the earth in her eclipse. 19. "The length of a degree being gi. gen, to find the number of miles in a great circle of the earth, and thence the diameter of the earth." Admit that one degree contains 69 English statute miles; then multiply 360(the number of degrees in a great circle) by 69 and the product will be 25,020, the miles which measure the circumference of the earth. If this number be divided by 3.1416, the quotient will be 7,963,86 miles, for the diameter of the earth. 100 20. The diameter of the earth being known, to find the surface in square miles, and its solidity in cubic miles." Admit the diameter be 7,964 miles; then multiply the square of the diameter by 3.1416, and the product will be 199,250,205 very near, which are the square miles in the surface of the earth. Again, multiply the cube of the diameter by 0,5236, and the product 264,465,789,170 will be the number of the cubic miles in the whole globe of the earth. 2. "To find the sun's place in the ecliptic." Find the day of the month in the calendar on the horizon, and right against it is the degree of the ecliptic, which the sun is in for that day. 3. To find the sun's declination." in the ecliptic to the meridian, and that Rectify the globe, bring the sun's place degree which it cuts in the meridian is the declination required. 4. "To find the sun's right ascension." Bring the sun's place to the meridian, and the degree of the equinoctial cut by the meridian is the right ascension required. 5. "To find the sun's amplitude." Bring the sun's place to the horizon, and the arch of the horizon intercepted between it and the east or west point is the amplitude, north or south. 6. To find the sun's altitude for any given day and hour." Bring the sun's place, to the meredian; set the hour-index to the upper XII; then turn the globe till the index points to the given hour, where let it stand; then screwing the quadrant of altitude in the zenith, lay it over the sun's place, and the D 66 8. To find the time when the sun rises or sets." Find the sun's place for the given day; bring it to the meridian, and set the hour-hand to XII; then turn the globe till the sun's place touches the east part of the horizon, the index will shew the hour of its rising; after that turn the globe to the west part of the horizon, and the index will shew the time of its setting for the given day. 9. "To find the length of any given day or night." This is easily known by taking the number of hours between the rising and setting of the sun for the length of the day; and the residue, to twenty-four, for the length of the night. 10. "To find the hour of the day, having the sun's altitude given." Bring the sun's place to the meridian, and set the hour-hand to XII; then turn the globe in such a manner, that the sun's place may move along by the quadrant of altitude (fixed in the zenith) till it touches the degree of the given altitude, where stop it, and the index will shew on the horary circle the hour required. 11. "To find the place of the moon, or any planet, for any given day." Take White's ephemeris, and against the given day of the month you will find the degree and minute of the sign which the moon or planet possesses at noon, under the title of geocentric motions. The degree thus found being marked, in the ecliptic on the globe by a small mark, or other wise, you may then proceed to find the declination, right ascension, latitude, longitude, altitude, azimuth, rising, south. ing, setting, &c. in the same manner as has been shewn for the sun. 12. "To explain the phenomena of the harvest-moon.' In order to this we need only consider, that when the sun is in the beginning of Aries, the full moon on that day must be in the beginning of Libra; and since when the sun sets, or moon rises, on that day, those equinoxal points will be in the horizon, and the ecliptic will then be least of ali inclined thereto, the part or arch which the moon describes in one day, viz. 13°, will take up about an hour and a quarter ascending above the horizon; and therefore, so long will be the time after sunset, the next night, before the moon will rise. But at the opposite time of the year, when the sun is in the autumnal, and the full moon in the vernal equinox, the ecliptic will, when the sun is setting, have the greatest inclination to the horizon; and therefore, 13° will in this case soon ascend, viz. in about a quarter of an hour and so long after sun-set will the moon rise the next day after the full: whence, at this time of the year, there is much more moon-light than in the spring; and hence this autumnal full moon came to be called the harvest-moon, the hunter's or shepherd's moon all which may be clearly shewn on the globe 13. "To represent the face of the starry firmament for any given hour of the night." Rectify the globe and turn it about, till the index points to the given hour; then will all the upper hemisphere of the globe represent the visible half of the heavens, and all the stars on the globe will be in such situations as exactly correspond to those in the heavens; which may therefore be easily found, as will be shewn in the sixteenth problem. 14. "To find the hour when any known star will rise, or come upon the meredian." Rectify the globe, and set the index to x11; then turn the globe till the star comes to the horizon or meri. dian, and the index will show the hour required. 15. "To find at what time of the year any given star will be on the meridian at XII at night." Bring the star to the meridian, and observe what degree of the ecliptic is on the north meridian under the horizon; then find in the calendar on the horizon the day of the year against that degree, and it will be the day required. 16. "To find any particular star.”— First find its altitude in the heavens by a quadrant, and the point of the compass it bears on; then, the globe being rectified, and the index turned to the given hour, if the quadrant of altitude be fixed on the zenith, and laid towards the point of the compass on which the star was ob served, the star required will be found at the same degree of altitude on the said quadrant, as it was by observation in the heavens. The invention of globes is of great an tiquity. Some allusions to the celestial globe may be found as early as Hipparchus's time, in the writings of Pliny and |