Description and Use of a New Celestial Planisphere
The
DESCRIPTION and USE
OF A NEW
CELESTIAL PLANISPHERE,
ADAPTED TO THE
Latitude & Parallel of London :
INSPECTED, AND APPROVED, BY THE ASTRONOMER ROYAL
AND BEING
PROJECTED upon the PLANE of the EQUINOCTIAL,
Proves every Way convenient for illustrating that most apparent and USEFUL PART of ASTRONOMY,
The DIURNAL MOTIONS of the SPHERES.
WITH A
CONCISE APHORISM OF THE ANNUAL MOTION OF THE EARTH AROUND THE SUN;
THE NATURE OF
The PRECESSION of the EQUINOXES;
AND THE
CONSEQUENT DIRECT MOTION OF THE STARS.
With all other PRECEPTS and EXAMPLES which are requisite in such a WORK.
CALCULATED AS WELL FOR THE INSTRUCTION OF PRIVATE STUDENTS, AS TO ACCOMPANY, AND SUCCEED THE USE OF THE GLOBES IN THE FINISHING SCHOOLS,
By SOLVING all the most useful PROBLEMS in ASTRONOMY,
With the utmost Facility and Ease.
By STEPHEN GODWIN,
Teacher of the Mathematics, in London.
SOUTHWARK; Printed by W. KEMMISH, KingStreet, BOROUGH
And Sold by the AUTHOR, at No. 20, Bride Lane; also by Smith, 172 Strand; Leathwait, at the Royal Exchange; Saffel, Compton Street, Soho; Syeds, Compass & Quadrant Maker, &c. 17, Rotherhithe Wall; Snart, Optician, 215, Tooley Street; and Messrs. Muggeridge, 92, Borough.—1802.
ADVERTISEMENT.
ANY Gentlemen who travel, or reside at times in Foreign Parts, may have correct Drawings at the shortest notice; or Temporary Horizons, for any Latitude, to apply to this Planisphere.
I also make other Circular Planispheres, in Pairs, to shut up together, for the Pocket, projected both on the South and North Poles, which contain all the Stars in both Hemispheres, of any note.—Also, Mercator’s, and other Charts, chiefly adapted to solve the Diurnal and Horary Motions of the Spheres, fitted to all Latitudes, and are equally useful to the young Academic, the private Student, and the Traveller, as they convey a general, clear, and pleasing Idea of the Positions and Motions of the Spheres.
N. B. Lessons of Instruction given on the most liberal Terms.
The PREFACE.
THERE have been many Planispheres invented and published, at different periods, within a century or two past, for different uses; some for Geography, some for Astronomy and Chronology, and some applicable to a few detached articles of all the three; but all that I have seen are very unsatisfactory, and some very erroneous: yet there are some Planispheres now extant, which are perfect in their kind, but not very useful, they being projected upon the Plane of the Meridian, are not capable of a Rotatory Motion, and, consequently rendered very limited: others have been projected on the Planes of the Horizon, or the Ecliptic, but none of them are adapted for, or capable of showing the Diurnal Motions; and, consequently, of very little utility, in comparison of what is wanted: my seeing these wants and defects, set me to consider, that, as the Earth's Diurnal Rotation was made upon the Poles of the Equator, so a Planisphere, projected upon the Plane of the Equinoctial, would answer my utmost wishes, and would be capable, not only to solve the daily apparent Motions of the Heavens, but also far more other Astronomical Problems than any of the abovementioned can do: this Planisphere is a true representation of the Celestial Sphere, in the parallel of London, extending to the Southern Horizon, or fifteen degrees below the Tropic of Capricorn; it is fitted with a circular and movable Index, consisting of an Horizon, an Hour Circle, the prime Vertical, the Meridian, and Equinoctial Colure, with a straight director above it, and moving on the center of the projection, or Pole of the Equinoctial; it generally solves each Problem at one remove, and in less time than can possibly be done by the Globe, or by any Planisphere hitherto extant, they being applicable to the use of Scales and Compasses, or of some troublesome and tedious Index: in short, a sight only, of the construction of this Planisphere, will sufficiently convince every discerning person of its superior utility and readiness in all the purposes of Astronomy to which it is intended; also, the World will readily be aware, that the Diurnal and Horary Motions of the Spheres is the most useful, the most general, and practical part of Astronomy; it is that which attracts the attention of every thinking being; all can discern the apparent Diurnal Motions of the Heavens, and would wish the phenomena solved to their capacities; indeed, most men are, more or less, deeply concerned in this part of Astronomy, either for profit or pleasure, or both. Who does not make some use or other of the rising, culminating, (or, passing the Meridian) and setting of the Sun, Moon, and Stars?—The merchant, the mariner, the traveller, and every public and private student, down to the peasant, who knows and regulates his times and seasons, his labour and rest, by the Sun, Moon and Stars, all have their interests in this part of Astronomy, the Diurnal and Horary Motion of the Spheres; and what further benefit and pleasure may we not expect to accrue among these respective classes, when this plain, easy, and familiar Treatise, (with the Planisphere) falls into their hands? when, by only turning the hour and minute given, and found on the Index, to the day of the month, the Planisphere stands rectified for that time, and represents the true situation of the Heavens, and the Problem is performed in a quarter of a minute of time, or less; and, by continuing to move the Index to a good regulated clock or watch, not only the relative positions and situations of the Celestial Bodies with respect to each other, and to the prime Circles of the Sphere, but their Horary Progressive Motion from East to West, is most beautifully seen by day and night perpetually; which, with the extensive variety of uses that this Planisphere is applicable to, cannot fail to enlighten and enlarge the understandings of every possessor of it, in the most rapid degree; and I humbly conceive and flatter myself, that if this little Work should be made an auxiliary Study in our Finishing Schools, great benefit will be derived therefrom in the education of the rising generation; as the precepts and examples are easy, pleasing and engaging, and are calculated to allure, rather than to deter the minds of Youth from the study thereof; which, if this should be the case, my utmost wishes will be answered, and my labour amply rewarded.
It will appear obvious to every one, that the Want of such a Planisphere, as an easy and sure Guide by which to demonstrate the Theory of Astronomy, hath always been the grand obstacle, the fatal check to the Study of this Science: it is not every Student that can, or that chuses to purchase a Pair of Globes, and to study Astronomy in bare and dry Authors, hath always been found an hopeless task, which hath deterred many a good genius from attempting this Study at all: but, I presume, and shall venture to affirm, that the encouragement every one will meet with at the first use of this Planisphere, will do away every objection, and will lead them on, by pleasing steps, through this short, pertinent, and useful Introduction, into the wide campaign of the most sublime Science of Astronomy.The
PRINCIPLE AND USES
of the new
CELESTIAL PLANISPHERE.
Take any Sector, or Gunter's Rule, (but the Sector is the best, because it can be set to any Radius) and, with a pair of compasses, take the parallel distance of 60°, and 60° in the Chords, or the parallel distance of 45° in the SemiTangents, opening the Sector to any Radius as will best suit your Plane; with this distance sweep a circle in the middle of your Plane, which is the Equinoctial, and must be divided into 360 equal parts, or degrees; and also into 24 hours, at the rate of 15° per hour; beginning where the degrees do; this circle is the Right Ascension in time, and degrees—in this case, the center of the Sector is the pole of the projection, and the parallel of 45° is the radius of the Equinoctial; and, as the Tropic of Cancer is 23° 28' from the Equinoctial towards the Pole, take the parallel distance of these degrees (counting them doubly) and sweep this circle concentric with the Equinoctial, and this is the parallel of Cancer, which will be 66° 32' from the Pole, or 33° 16' on the Sector. Then, to find the Tropic of Capricorn, you must make 45° on these lower Tangents a parallel distance of 45° on the upper Tangents, and take 23° 28' beyond that, for the parallel of Capricorn. Then, the Arctic Circle is 23° 30' from the Pole: and these are all the concentric Circles in the Plane, except the Circle of the Months, which is near the outward edge of the Planisphere. The next Circle is the Ecliptic, the apparent annual path of the Sun, and it is drawn thus:— set off the Secant of 23° 30' from Cancer, on the Solstitial Colure towards the Pole, and this shall be it’s center; and it’s Pole is at the intersection of the Arctic Circle, and the Meridian or Solstitial Colure; in this Circle, the signs of the Zodiac are placed; and, in tables of the Right Ascension of the Ecliptic, you will find the proper divisions of this Circle into signs and degrees; then lay a thread, or an Index, from the Pole, through every degree of Right Ascension of the Ecliptic, and it will give points in this Circle where all the Signs and Degrees do fall; the fixed Stars are laid down the same way by their right Ascension and Declination as found in the tables of the Stars, and those in my Planisphere are all of the first and second Magnitude, and their places computed up to the present year 1802, inclusive.— Then through the first points of Cancer and Capricorn, draw a straight line, and this will be the Meridian and Solstitial Colure; then again through the first points of Aries and Libra draw another line, which is the Equinoctial Colure and Hour Circle of six, these two Circles do intersect each other in the Pole of the world, and you will find the situation and names of all these Circles in the Planisphere; the next is the Circle of Months; and if you look on the horizon of any new Globe, or in any Almanac, you will there see the Sun's place to every day of the year; then lay an index from the center of the Plane through every point of the Ecliptic into the Circle of Months, and mark the days accordingly, and it is done: the index consists of an Hour Circle, next to the Circle of Months; and, as you hold the South point from you, the left hand hours are the morning and forenoon hours, and the right hand are the afternoon hours, to twelve at night, which is at the bottom next to you, due North; then the Meridian reaches from 12 to 12 through the Pole, and on the southern part is a scale of meridional Altitude from the Horizon up to the Zenith of London; and, on the northern part, is a scale of Declination, North and South; the Circle of the Horizon hath the points of the Compass, the Azimuth, and the Amplitude; and note, that the inward edge of the Circle is the sensible and apparent Horizon; next is the Prime Vertical, or Azimuth of East and West, and on it is a scale of Altitude when due East or West, up to the Zenith; the next is the Equinoctial Colure and Hour Circle of six; and above this is a small index, or director, to be used, as will be taught among the Problems. This may suffice for a description of the principle and construction of the Planisphere. I must now proceed to it's uses, as it relates to Astronomy: and here I must beg, that the more proficient in the science will not think I insult their judgment, by laying down the first rudiments of it. I wish first to inform, and bring forward young minds, and then I mean to advance, and find amusement for them, if not instruction.
The Circle of the Zodiac is divided into twelve parts, called Signs, and each sign into 30°; their names and characters are as follow:
♈  ♉  ♊  ♋  ♌  ♍ 
Aries.  Taurus.  Gemini.  Cancer.  Leo.  Virgo. 
♎  ♏  ♐  ♑  ♒  ♓ 
Libra.  Scorpio.  Sagittary.  Capricorn.  Aquarius.  Pisces. 
The first six of these are called Northern Signs, and possess that half of the Ecliptic that is on the north side of the Equinoctial, beginning at the first point of Aries, and ending at the last point of Virgo; the latter six are called Southern Signs, and possess the Southern half of the Ecliptic, beginning at the first point of Libra, and end at the last degree of Pisces, where Aries begins again: and note, that these signs are different from the Constellations which bear their names; for, though they were formerly situated together, yet now, by a slow retrograde motion of the Equinoctial points, at the rate of about 50 seconds per year, or one degree in 72 years, the Constellation Aries is got into the Sign Taurus, and so of the rest, till the Stars of Pisces are now in Aries; this motion is made upon the Poles of the Ecliptic, contrary to the succession of the Signs, and is called the Precession of the Equinoxes. The beginning of the four Quarters of the Ecliptic are called Cardinal Signs, viz. Aries, Cancer, Libra, and Capricorn, because the Sun enters them at the beginning of the four Quarters of the year; the four Points of the Compass, called East, West, North, and South, are called Cardinal Points also; and if you bring Cancer or Capricorn to the Meridian, the Cardinal Signs will coincide with, and lay over the Cardinal Points of the Horizon. Aries and Libra are also called Equinoctial Points and Signs, because the days and nights are then equal, as you will see by the Planisphere; if you take the Sun's diurnal Arc on those two days that the Sun enters these Points, the two Meridians passing through these Points, are called Colures, and that which passeth through the Equinoctial Points is called the Equinoctial Colure; and that which is at right Angles to it, passing through the Solstitial Points, is called the Solstitial Colure, as you will see them named in the Planisphere. The use of the Prime Vertical in the Planisphere is very obvious; for when, by the diurnal rotation of the Earth, any celestial object comes to this circle in the eastern Hemisphere, it appears due East: and if it be in the western Hemisphere, it is due West, and its Altitude, in both Cases, is seen by inspection; also, when any celestial object comes to the southern part of the Meridian, its meridian Altitude is seen on the scale of Altitude; and when any object is rising or setting, its Azimuth and Amplitude are seen on the Horizon, with the Point of the Compass on which it rises or sets.
When the reader has studied and learnt the Principle and Construction of the Planisphere, which is easy to accomplish, he will find it is adapted, in the most easy and ready manner, for solving all the Problems to which it is applied, and I may venture to say, far more so than the Celestial Globe, it being inconvenient on the Globe to apply, and work the Hour Circle and the days of the months together, and to concentrate the Hour Circle and the Ecliptic is impossible; but, on this Planisphere, the Days of the Months and the Degrees of the Ecliptic are adjusted together, by which means the Day of the Month being given, is the same as the Sun's place in the Ecliptic, or the contrary, so that these may become synonymous terms in the Planisphere, and as the Hour Circle on the Index turns concentrical with the Circle of Months, and just within it, so most of the Problems are solved at one remove of the Index, and the answer appears in an instant, by inspection.
Having now described, and prepared the Planisphere for use, I shall next explain the motions of the Terraqueous Globe on which we live: and first, according to the true, or Copernican System of the Universe, this Globe is a Planet, revolving round the Sun in an orbit somewhat elliptical, and in the space of 365 days, 5 hours, and 49 minutes, being accompanied with the Moon, and hath the Sun at rest in the center of its orb.–Now, conceive at a vast distance, without this system, the sphere of the fixed Stars; and then imagine that on this, or some one day, the Sun appears to a beholder on this Earth, to be in a straight line, or in conjunction with some one of the Stars; but as the Earth revolves in its orbit, the Sun will apparently leave this Star, and seem to pass from the right to the left, or from West to East with respect to the Stars, 'till it hath passed over all the Stars in it's path quite round the Heavens in one year; and here it is evident to the understanding of every one, that this apparent motion of the Sun, arises from the real motion of the Earth in its orbit round the Sun, and the path, or circle that the Sun appears to describe in the Heavens, is called the Ecliptic; and note, that this same path would be traced out by the Earth in its annual motion, to a beholder in the Sun, with only this difference–that when the Sun appears to us to be in Aries, the Earth would appear to a solar spectator to be in Libra, and so always with six signs, or 180° difference;—while the Earth is performing its annual course round the Sun, it turneth round upon an imaginary axis in every 23 hours, 56 minutes, from West to East, with its inhabitants, to whose sight, all the celestial bodies must of course appear to move the contrary way, viz. from East to West; for, the sensible Horizon being an opaque, solid, and apparently level expanse, doth always conceal the lower Hemisphere from our sight, but as the whole Earth continues to revolve upon the Poles of the Equinoctial, from West to East, the eastern part of the Horizon will be depressed below those Stars, or celestial bodies, that were before below it, and invisible, and they will become visible to our sight, and are then said to rise: the same way the Western Horizon doth elevate itself above those bodies that are the most Western, and they are said to set; and so those bodies, that are on the eastern side of the Meridian, will, by this motion of the Earth, culminate, or transit the Meridian, and in turn will set; as also the imaginary Circles of the Sphere, such as the Equinoctial, the Ecliptic with the Sun, and Planets therein, will, by the rotation of the Earth, all appear to rise, culminate and set, causing day and night to all; this is called the Diurnal, and Horary Motions of the Spheres; this is the part of Astronomy that is peculiarly noticed by, and useful to every one; it is these motions, that my Planisphere is chiefly intended to illustrate, and render familiar to the capacity and practice of every Lover of Astronomy; and this leads me to the following Precepts, on the Properties and Uses of the Circles of the Planisphere.
Precept 1. Declination of the Celestial Bodies, is an Arc of the Meridian, beginning at the Equinoctial, and counting it North and South to the Poles, so any celestial object, while it is situated on the North side of the Equinoctial, is said to have North Declination; but, while they are on the South side of the Equinoctial, are said to have South Declination; and note! on my Planisphere, a Scale of Declination, North and South, is laid down upon the Meridian of the Index with 0° degrees, revolving in the Equinoctial.
Precept 2. Right Ascension in Time, and Degrees, is an Arc of the Equinoctial, beginning at the first point of Aries, and counting Eastward round the Planisphere, and, with the celestial bodies, it is that degree of the Equinoctial that comes to the Meridian with any of them respectively; and the Right Ascension of every sign, degree, and point of the Ecliptic, is to be found the same way, by bringing those points respectively to the Meridian, and the degree of the Equinoctial cut thereby, is the Right Ascension of that sign, or point required; and note! that 15° of this circle revolves through the Meridian in every hour, it being the 24th part of 360°, which the whole Circle contains.
Precept 3. Oblique Ascension, and Descension, is the degree of the Equinoctial cut by the Horizon, at the time of the rising, and setting of any of the celestial bodies, or of any part of the Ecliptic, as may be required.
Precept 4. Ascensional Difference, is the difference between the Right, and Oblique Ascension; and with the Sun, it shows how much it rises and sets, before, and after six o'clock.
Precept 5. Celestial Longitude is an Arc of the Ecliptic, beginning at the first point of Aries; and if the Sun, or any celestial object, be in any particular sign, degree, and minute, as if in 24 degrees, 35 minutes, and 49 seconds of Leo—we write it's Longitude thus—4^{s} 24° 35' 49". Observe the same rule in all other cases, as required.
Precept 6. Latitude of a Star, or Planet, is an Arc of a great Circle, intersecting the Ecliptic at right angles, and passing through the Poles thereof; that on the North side of the Ecliptic is called North Latitude, and the other side South.
Precept 7. Azimuths are Vertical Circles, passing through the Zenith and Nadir, intersecting the Horizon at right Angles, and counted from the South, and North, towards the East and West, or it is an Arc of the Horizon, intercepted between the North, or South Points, and the place where any celestial body rises or sets.
Precept 8. Amplitude of the Sun, or a Star, is an Arc of the Horizon, intercepted between the East or West Points, and that Point where the object rises or sets; and note! you will find the degrees of the Azimuths, and Amplitudes on the Horizon of the Index to the Planisphere; the degree of Amplitude at rising, is called Amplitude Ortive, and Occasive at setting, which must be denominated, whether it be North, or South.
Precept 9. Cosmical Rising, or Setting, is when a Star or Planet rises or sets when the Sun doth.
Precept 10. Achronical Rising, and Setting, is when a Star, or other object rises, when the Sun sets, or sets when the Sun rises.
Precept 11. Circles of perpetual Apparition, and Perpetual Occultation, are different parts, or portions of the celestial sphere in all the different Latitudes upon this Globe; and in this parallel, the Circle of Perpetual Apparition is all that part of the celestial sphere round the North Pole, that never sets below the Horizon, being from the parallel of 38° 28' declination Northward, and is always equal to the compliment of the Latitude; all the Stars within this Parallel revolve round and round the Pole, but never disappear, nor descend lower than the Northern part of the Horizon, and these are sometimes called CircumPolar Stars, and you will see them, with their names in the Planisphere, by turning the Index round about;—by the same cause, the Circles of Perpetual Occultation, is, all those Stars, near, and round the South Pole, which never rise above the Southern part of our Horizon, from 38° 28' Declination Southward, to the South Pole.
But, this subject will be better illustrated, and rendered apparent, by the solution of the following Problems with the Planisphere.
Problem 1. The day of the month, and hour of the day given, to rectify the Planisphere for use, suppose July 5th, at 4 hours, 36 minutes, P. M.?
Find the day of the month in the outermost circle, and bring the hour and minute on the index thereto, and it is done. This Problem is very short, and solved in an instant; but its effects are very extensive and great; for it presents to our view the true and perfect situation of the celestial sphere, together with all the Stars, and other celestial bodies, in their true positions with respect to each other, and to the Horizon, and Meridian, and all the other Circles of the Spheres: this effect cannot fail to delight and gratify every beholder, who will make their various uses of it accordingly, and by continuing to turn the Index by a good regulated clock, or watch, as often as we chuse to take an observation, we shall most beautifully discern and comprehend the diurnal rotation of the Earth upon its axis, from West to East, causing the apparent diurnal motion of the Heavens from East to West.
Problem 2. To find the Sun's place in the Ecliptic, suppose on August 14th, and November 29th, at noon. Turn the director to the days proposed, and you will find it points to 22° of Leo on August 14th, and 7° of Sagittary on November 29th, for answer.
Problem 3. To find the Sun's Declination on June 4th, and December 1st. Find the Sun's place by the second Problem, for each day, and bring these points of the Ecliptic to the Meridian of the Index North, and it gives 22° 30' North, on June 4th, and 21° 40' South, on December 1st.
Problem 4. To find the Sun's right Ascension in time, and degrees, on April 9th, and on September 2d. Lay the director to each day, and it intersects the Sun's place in the Ecliptic, and his right Ascension in the Equinoctial 18°, or, 1 hour, 12 minutes in time, on April 9th—and 161° 30', or 10 hours, 46 minutes, in time, on September 2d.
Problem 5. To find the Declination, and right Ascension of the fixed Stars, suppose CorScorpio, and the Virgin Spike, and CorLeo—Bring each Star to the Meridian of the Index, North, and it gives the right Ascension of CorScorpio 244° 20', or 16 hours, 17 minutes, 20 seconds, in time, with 26° South Declination.—Observe the same rule with all the other Stars.
Problem 6. To find the Right Ascension and Declination of the Moon, or a Planet, depends on two cases, thus—Seek the Planet's Longitude and Latitude in the Nautical Ephemeris, for the time given.—So, August 7th, at noon, the Moon's Longitude is 22° 39' of Scorpio, or 7 signs, 22° 39', with 4° 57' South Latitude; then find this Point in the Zodiac on the Planisphere, and take the Right Ascension and Declination of it, as in the last Problem by the Stars, and you will find the Moon's Right Ascension 261° 40', or 17 hours, 27 minutes in time, and 27° 45' South Declination, as was required.
Problem 7. To find the Sun's Oblique Ascension and Ascensional Difference on April 14th. Find his place in the Ecliptic 25° of Aries, bring it to the eastern Horizon, and you will see 11° of the Equinoctial rise with it, which is the Oblique Ascension required; then find the Sun's Right Ascension as before, 23° 40' and subtract the Oblique Ascension from it, and it leaves 12° 40', which is the Ascensional Difference required.—Observe the same rule with the Moon, Planet, or any Star.
Problem 8. To find the time of Sunrising, and length of the day and night, on February 4th and August 9th. Bring the Sun's place to the Horizon, Eastward, and lay the director through it into the Hour Circle, and it points to 25 minutes past 7. For rising on February 4th, subtract this from 12 hours, and the remainder is the Sun's Semidiurnal Arc, 4 hours, 35 minutes, which doubled, is 9 hours, 10 minutes, for the length of the day; but subtract this 9 hours, 10 minutes, from 24 hours, and the remainder is the Nocturnal Arc, or the length of the night, 14 hours, 50 minutes, as required; but, on August 9th, the Sun rises 25 minutes before 5; therefore, the Semidiurnal Arc is 7 hours, 25 minutes, which doubled, makes 14 hours, 50 minutes, for the length of the day, which, subtracted from 24 hours, leaves 9 hours, 10. minutes, for the length of the night required.
Problem 9. To find the length of the longest and shortest days at London. Bring the first point of Cancer to the eastern Horizon, lay the director, through it, and it points to 3 hours, 47 minutes, A. M. or 13 minutes before 4 in the morning, subtract this from 12 hours, and it leaves 8 hours, 13 minutes for the Semidiurnal Arc, which doubled, gives 16 hours, 26 minutes for the length of the day; but, subtract this from 24 hours, and it leaves 7 hours, 34 minutes for the shortest night under Cancer; by the same rule, you will find this 7 hours, 34 minutes, to be the length of the day under Capricorn, and the 16 hours, 26 minutes, the length of the night as was sought.
Problem 10. To find the Meridional Altitude of the Sun on the 3d of October. Bring the Sun's place for that day, found by Problem 2d. to the Meridian, viz. 10° of Libra, and you will find 34° 40' Altitude, by the scale of Meridional Altitude, as required.
Problem 11. To find the Meridional Altitude of the fixed Stars, Fomahaut and Lyra. Bring them respectively to the Meridian, and you will find Fomahaut hath 7° 39' Altitude, and Lyra 77° 4', and Canis Minor hath 44° 37' Meridional Altitude.
Problem 12. What is the Zenith distance of the Sun on May 13th, at noon? Find the Sun's place 22° of Taurus, bring it to the Meridian, and it hath 56° 30' Meridional Altitude; subtract this from 90°, and the remainder 33° 30', is the Zenith distance required.
Problem 13. To find the Sun's Amplitude at rising, on the 15th of February. Bring the Sun's place 10^{s} 27° to the Eastern part of the Horizon, and you will see it intersect the Amplitude of 21° towards the South, the same with any Star or Planet.
Problem 14. To find the Right Ascension and Declination of any Point in the Ecliptic; suppose 20° of Leo. Bring that Point to the Meridian, and it hath 142° Right Ascension, and 14° North Declination.
Problem 15. To find what Point of the Ecliptic doth rise, culminate, and set, at any day, hour, and minute proposed; as September 4th, 3 hours, 42 minutes, P. M.—Rule.—Bring the day of the month to the hour and minute given, and you will find 5° of Capricorn rising, and 11° of Scorpio transiting the Meridian, Southward; and, of course, 5° of Cancer setting, and 11° of Taurus on the Meridian, North.
Problem 16. May 24th, in the morning, the Sun was observed due East; What is his Altitude, and the Hour of the Day?—Find the Sun's place, 2^{s} 3°. Bring it to the Prime Vertical, Eastward; lay the director through it into the HourCircle, and it shews the Time of the Morning to be 16^{m} past 7, with 27° of Altitude, as was required.
Problem 17. To find the Diurnal Arc of any Star, or Planet, suppose the Virgin’s Spike, (being the Star to the eastern Horizon). Lay the director through it into the HourCircle, and it points to 5 minutes before 7, which is 5 hours, 5 minutes, A. M. this is its Semi diurnal Arc, which, doubled, makes 10 hours, 10 minutes, for the Diurnal Arc, or time of this Star’s duration above our Horizon; which, subtracted from 24 hours, leaves 13 hours, 50 minutes, for the Nocturnal Arc, or time of its stay below our Horizon. Mind the same rule with all the fixed Stars, (but the Moon), and the swiftest of the Planets will require a reduction in their places in this Problem.
Problem 18. To find at what hour a Star rises on any day, suppose Oct. 14th. What time doth the Bull's Eye rise?——Bring the Star to the Horizon, and see what hour and minute points to the day proposed, and it is found to be 12 minutes before Eight at night, and sets 12 minutes before Ten in the morning.
Problem. 19. To learn to know the Stars by sight, at any hour, by the help of the Planisphere. Suppose July 20th. at 10^{h} &c. P. M. you are out of doors, and rectify the Planisphere to that hour; then suppose yourself to stand under that part of the heavens where is marked the Zenith of London in the Planisphere, and you will see the Dragon's Head in the Vertex, Lyra South by East, the Stars of Pegasus nearly East, with from ten to twenty degrees of Altitude, the Great Bear West North West; Arcturus West South West; the Virgin's Spike nearly setting WestSouthWest; CorScorpio SouthSouthWest; Atair in the Eagle SouthSouth East; Perseus and Capella near the northern Horizon; and you may see the names of the rest in the Planisphere, that are then above the Horizon; and by varying the time of the year in your observation, you will easily see, and learn the names of them all.—This is a useful Problem, and should often be repeated. Problem 20. To find the time of night. Being out from home, on January 20, I saw the great Dog Star, Sirius, on the southern Meridian, by a quadrant, or compass, then bring the Star to the meridian of the Planisphere, and lay the director to the day of the month, and it points to half past 10 at night, as was sought.
Problem 21. To find the time of night on the 15th of May. Being out from home, I observe the Star Atair, due East, with 11° Altitude.—What is the hour?—Bring the Star to the Prime Vertical, East, and turn the director to the day of the month, and it points to the hour, 20 minutes before 11 o'clock at night, which is the time required.
Problem 22. August 1st. being out, I observed the Sun was due West by the compass—What is the hour?—Find the Sun's Place, and bring it to the Prime Vertical, West; then lay the director through it, and it shows the time to be 4 minutes before 5, in the afternoon, as was sought.
Problem 23. What time of day or night, will Cor Leo appear due East on October 9th? Bring the Star to the Prime Vertical, Eastward; lay the director to the day of the month, and it shows the time to be 20 minutes before 4, in the morning, for answer.
Note! That if, on the contrary of this Problem, the hour be given, then the Planisphere will discover what point the Star is upon: Example, observing the Stars in the Great Dog's hind part, above the Horizon, and having no compass, I refer to my watch, and find it 3 in the morning of the 15th of October, then rectify the Planisphere to the day and hour, and we find the Star in the Dog's flank, SouthEast by South, and that in the tail a quarter East; Aldebaran is on the Meridian, and the large Star in the Lion's tail, just rising on the EastNorthEast; thus, the points are gained by the Planisphere, without any other instrument, not only by land, but on the sea, between London and Holland, or in the Irish sea, or the Western Ocean to Newfoundland, or any other part, in or near this parallel, with useful and pleasing effects both to the landsman and mariner.
Problem 24. To find when the Sun rises or sets cosmically with any Star, suppose the Great Dog, Sirius. Bring the center of the Star to the eastern Horizon; lay the director through the intersection of the Horizon and the Ecliptic, and it points out the 11th of August for the day required; but to know when they set cosmically, or together, bring the Star to the western Horizon, and the director through the intersection of the Ecliptic and the western Horizon, and it points out the 15th of May to be the day sought.
Problem 25. To find when the Sun rises or sets achronically, suppose with the right shoulder of Orion. Bring the Star to the western Horizon, and lay the director through the intersection of the Ecliptic and the eastern Horizon, and it points to the 28th of November, for the day of Achronical rising; but, if you bring the Star to rise, and turn the director through the intersection of the Ecliptic and the western Horizon, it points out the 8th of January to be the day on which the Sun sets achronically with that Star.
Problem 26. To find what days of the year the Sun enters into each sign of the Zodiac? Rule—Begin at Aries, and lay the director through it into the circle of months, and it cuts the 20th of March, when the Sun enters that sign; then lay it through 0° of Taurus, and it points out the 20th of April; and so, whatever day is pointed to, when the director is laid through any point of the Ecliptic, that is the day in which the Sun will be in that sign and degree,
Problem 27. To find what time of the year the Little Dog Star will be on the Meridian at eleven o'clock at night?—Turn the Index, till the Star is on the Meridian; then look for eleven at night, and it points to the 25th of January, for the day sought; at which time, of the year, and hour of the night, we may see Procyon, Castor and Pollux, transit the southern Meridian.
Problem 28. To find a true Meridian line by the precise time of the Transit of any Star found by the Planisphere, thus: Drop a perpendicular line between the eye and the Star down to any fixed object before you on the Horizon, and a line drawn from the eye to that object is a true Meridian line.
Problem 29. To find what two days of the year, the days will be of any given length less than sixteen hours, suppose 10 hours, 20 minutes long?—Rule: Find the Semidiurnal Arc on the index, viz., 5 hours, 10 minutes, both East and West; lay the director to these hours and minutes, on both sides of the Meridian alternately; then turn the director and the Horizon together, till some point in the Ecliptic intersects them both, and the director will then lay over the day of the month required, viz. the 22d of February, and October 19, for answer. Again, what two days will be 13 hours, 30 minutes long each? Turn the director to 45 minutes after six Westward, and move it and the the Horizon together, till some point in the Ecliptic exactly intersects them both, and it points out September 1st. for one of the days; and, when brought to 15 minutes past 5, in the morning hours, or Eastern side of the Meridian, the director points out April 10th. for the other.
Problem 30. To find what two days in the year the Sun will have 50° Meridian Altitude at London? Turn the Meridian of the Index on each side of the Tropic of Cancer, till some point of the Ecliptic intersects it in the given Altitude, and in one case it points out August 23d. and in the other, April 21st. for answer.
Problem 31. To find the Longitude and Latitude of the Moon, Planets, or fixed Stars, refer to the 5th and 6th Precepts for a Rule. Example: Required to find the Moon's place in the Zodiac, on August 9th at noon? Look in the Nautical, or any good Ephemeris, for the Moon's Longitude, and we find it 8^{s} 19° 32', seek this place in the Ecliptic of the Planisphere, viz. 19° of Sagittary, but this is not her true place, for if you look in the next column in the Ephemeris, to the right hand, you find 5° 12' South Latitude; then bring the Longitude to the Scale of Declination, and with the compasses take 5°. 12'. Southward from this point, and set it off at right angles with the Ecliptic, with one foot in the Moon's Longitude, and the other will stand in the Moon's true place in the Zodiac; but if she had had North Latitude, then you must have taken the Latitude Northward on the Scale, and projected that distance on the North side of the Ecliptic, and that would be the Moon's true place in the Zodiac, where, if you make a point with a pencil, you may take the right Ascension, or Declination of the Moon, or a Planet at that time—the reason why the Scale of Declination will solve this Problem of celestial Latitude, is, the Radius of the Equinoctial and the Ecliptic being similar, and both on a tangent scale, the error, if any, is wholly imperceptible; and note! if the place of the Moon, or a Planet be required for any certain number of hours after noon, a reduction must be made in their Longitude and Latitude for that hour, and that Arc added to the place at noon, (if it be the Moon, or a Planet direct in motion), and that point sought in the Zodiac as before directed.
Problem 32. There are many secret ways by which Astronomers may date their letters to each other, some of them as follow. ☉ ♍ 5° 51'—if you receive a letter from your friend in the country thus dated, you look in the Nautical Almanac, and correct the Sun's motion the day in which it is in that sign and degree, and find it was on the 29th of August, at midnight, in which he wrote it.—Again, ♄ 13° 10' ♍—This answers to September 22, at six in the afternoon——Again, the Dragon's Head in ♓︎ 8° 8', this will direct you to the 7th November, at noon—or, ☽︎ ♋︎ 2° 17'—☿ 8^{s} 2° 12' this can be no other time than Decem. 10th, at noon; and many other ways you may invent from these examples, according to your fancy.
Problem 33. To find the Jewish hours by the Planisphere, viz. they divide the day, and night, be it long or short, into twelve equal parts, from Sunrising to Sun setting, for the day, and from Sunsetting to its rising for the night; thus, in the Equinoctial, the Jewish hours, and the common hours are alike in length, viz. 60 minutes; but, it is evident, when the Sun is in Cancer, the twelfth part of the day will be more than 60 minutes, and the night hours less, and they continually vary between the Tropics: thus, suppose at London the longest day is 16 hours, 26 minutes—multiply this by 60, and divide the product 986^{m} by 12, and the quotient is 82^{m}^{2}⁄_{12} for the length of a Jewish hour by day under Cancer, and this is the length of the night hours under Capricorn; by this rule, we find a night hour under Cancer is 37 minutes, which is the length of a day hour under Capricorn; but, to find the length of these hours at all times of the year by night and day, by the Planisphere, do thus: suppose, for Example, on the first of August, at 10 hours, 12 minutes in the forenoon; I wish to know the exact time of day, according to the Jews?—Take the Semidiurnal Arc, by Problem 8, which is 7 hours, 37 minutes, multiply this by 60, and divide the product, 457 minutes by 6, and the quotient is 76^{m}^{1}⁄_{6}, for the length of a Jewish hour that day; then rectify the Planisphere to the hour and minute given, 10 hours, 12 minutes, A. M. and steady it there; then bring the director to the time of Sunrising before found, 23 minutes past 4, and carry it warily over 76 minutes, or 1 hour, 16 minutes, and it will then lay over 21 minutes before 6, and call this the first Jewish hour; then move it on 1 hour, 16 minutes more, to 5 minutes before 7, and call it the second hour; then to 11 minutes after 8, and it ends the 3d hour; then proceed to 27 minutes past 9, for the 4th hour; so now, as the director lays, you will plainly see there is not another whole hour, before you come to our 12 minutes past 10; so count the odd minutes, till the director comes in a straight line with the 1st of August, the hour and minute given, and the Sun's place, and you find 45 minutes towards the 5th Jewish hour, with 31 minutes to come to finish that hour: by this rule, you will come at the true Chronological time mostly made use of in the Scriptures, and may reduce it, by this last Problem, to our time of the day; so we find, at the Crucifixion of our Saviour, there was darkness from the 6th hour until the 9th hour; now, by this Problem, we learn, that the 6th hour ended at the Semidiurnal Arc, or when the Sun was on the Meridian at noon; and the 9th hour was a few minutes after 3, by an equal Hour Clock; inasmuch as it was near the Vernal Equinox. But, the Arithmetical way to find the Jewish hour and minute, is thus—find how many of our hours and minutes the day consists of; then say, as these hours and minutes are to 12 hours, so is the hours and minutes since Sunrising, to a fourth proportional, the Judaical hour required; but observe, the time gained as above, is only that shewed by a clock or watch that goes equal hours at the place where the occurrence happened; but to find the real time of day at another place, (suppose London), we must find the difference of Longitude of the two places, and turn it into time, at the rate of 15° per hour; so, Jerusalem, being 35° 20' East of London, or 2^{h} 21^{m} 20^{s} in time earlier, this must be subtracted from the time given by their equal hour clock, and the remainder will be the time at London, as though their ninth hour, at that time of the year, ended at 8^{m} after 3 in the afternoon, subtract 2^{h} 21^{m} 20^{s} from 3^{h} 8^{m}, and the remainder is 46^{m} 40^{s} after 12 at noon at London.
Problem 34. To find the Horary Angle, or the apparent time from noon by a single Altitude of the Sun; for which purpose, Mr. Syed's Patent Quadrant is the best instrument by land, as well as by sea, it having an artificial Horizon, which the common Hadley’s Quadrants have not; but if one should have a Collins's or a Sutton's Quadrant in hand, it will do in this case; but, if neither are in your possession, then the Planisphere will be sufficient for this work. Thus, hang a small plummet by a fine hair to the centerpin, then stick a fine pin in the morning hour line of 6, and another in the same line near the center, both perpendicular to the Plane, then hold the Planisphere up with the Southern 12 to the Vertex, and the East side next the Sun, and elevate the index, till the shadow of the outermost pin falls on the other, and then the hair will show the Altitude on the lower limb of the index, counting each division for a degree from the Northern 12 to the right hand.—Thus, for Example, on August the 13th, 1802, I find the Sun's Altitude in the forenoon 43°. Required the Horary Angle, or apparent time from noon?
Take the Secant of the Latitude rejecting the index,   51° 31'  0,20601 
And the Secant of the Declination,   14° 53'  0,01482 
Subtract the Declination from the Latitude, and the difference is,    36° 38' 
To which add the Complement of the Altitude,   47° 
And it makes   83° 38' 
And their difference is,   10° 22' 
Then take the half of   83° 38' 
Which is,    41° 49'=sine, 9,82396. 
And take half the difference,  5° 11'=sine, 8,95590 
Now add the Secants of the Latitude and the Declination to these two sines all into one sum, which makes    19,00069 
The half of which will be,   9,50034 
Which is the sine of —  18° 27' 
Which double, or multiply it by 2, and the product is,  36° 54' 
Which, convert into time, at 15° per hour, and it gives  2^{h} 27^{m} 36^{s} 
as the Log. of the length of the shadow, 5, ___ _  0,69897 
Is to the Log. of the Perpendicular, 4, _  0,60206 
So is Radius, ___ _  10,00000 
To the Tangent of the Sun's Alt. 38° 39'  9,90309 
and, note! if the shadow falls between the divisions, then take the proportional part of the logarithms, and work as before, as if the shadow falls on 5^{1}⁄_{4} parts, then take a quarter of the difference of the next less, and greater Arcs, and add it to the logarithm of the lesser Arc, and work with it as in the last Example; by which Rule you can never err a minute, either in the altitude or the time; and you may safely set the watch thereby. This Problem is universal, and is of the utmost utility to the practical Navigator, as well as for the exercise and amusement of every private student, by land.
Problem 35. To regulate and adjust the motions of the Planets from noon, or midnight, as found in the Nautical Almanac, to any other intermediate hour and minute of the day or night required, and consequently to find their true places in the Zodiac, in the Planisphere, and their situations, at all times, with respect to, and their progressive motion among the fixed Stars, &c. their names and characters are as follow:—☉, the Sun, is in the center, and ☿, Mercury, is nearest to the Sun of all the Planets, and performs its course round the Sun in about three months, or 87 days, 23 hours. ♀, Venus, whose Orb is next to Mercury, revolves round the Sun in 224 days, and 17 hours, or about eight months. The Planet, which is the third in order from the Sun, is our Earth, performing its annual orbit in 365 days, 5 hours, and 49 minutes: the next is ♂ Mars, which moves round his Orb in 686 days and 23 hours. Then ♃ Jupiter, performs his circuit in 4332 days and 12 hours, or about 12 years. Then ♄ Saturn, completes his revolution in 10759 days, and 7 hours, or a little less than 30 years. Then the Georgian Planet, whose Orb is exceeding remote, and motion very slow, as may be seen in the Nautical Almanac. Those Planets, whose Orbs are beyond the Orb of the Earth, are called Superior Planets; but Venus and Mercury, whose Orbs are between the Earth's Orb and the Sun, are called Inferior Planets. The Georgian, Saturn, Jupiter, and the Earth, have their secondary Planets, or Satellites accompanying them round the Sun; and moving round them also, in certain periods. Jupiter hath four, and the Earth one; which is the Moon, revolving round it in 27 days, 7 hours, and 43 minutes, which is also the exact time of her rotation round her axis, so that she always turneth the same side, or part of her Surface towards us; her Orbit is an Ellipsis, whose mean Radius is about 244258 miles, and whose Plane is inclined to the Plane of the Ecliptic, at an Angle of 5° 18' at the most; the Moon never remains in the Ecliptic, but intersects it twice in a month, in the points called her Nodes, or the Head and Tail of the Dragon; or, these are the points in which the Moon's Orbit bisects the Plane of the Ecliptic, and while she is in that half of her Orbit, which is on the South side of the Ecliptic, she is said to have South Latitude; and the point of the Ecliptic where the Moon passeth out of South, into North Latitude, is called her North Node, or Head of the Dragon, and then she hath North Latitude, till she arrives at her South Node, or, to the Dragon's Tail. The Nodes of the other Planets, according to Geocentric Vision, are, in those points of the Ecliptic where they have no Latitude, which are seen in the Nautical Almanac. I need not premise farther, in order to this Problem.
Required the Moon's true place on Sept. 15th, 1802, at 7^{h} 24^{m}, P. M. Find in the Nautical, or in any good Ephemeris, her Longitude at noon on the 15th, 1^{s} 10° 49', and subtract it from her Longitude on the 16th, at noon, 1^{s} 24° 41', and the difference is, 0^{s} 13° 52'; then say, by the rule of proportion, if 24^{h} give this Arc 0^{s} 13° 52', what will the time from noon, 7^{h} 21^{m} give?—Answer, 4° 16', which, added to her Longitude the 15th, at noon, makes it 1^{s} 15° 5', her true and corrected Longitude at the hour and minute required. But note! this is not the moon's true place in the Zodiac; for, against the 15th at noon, she hath 4° 33" North Latitude, which, subtract from her Latitude the 16th at noon, viz. 5° 4', and the difference is 31'; then say, if 24 hours, give this Arc 31', what will 7^{h} 24^{m} give?—Answer, 9': which, as her Latitude is increasing, must be added to the Latitude at noon, the 15th, 4° 33', and it makes 4° 42', the true Latitude sought; then seek the Longitude found, 1^{s} 15° 5', in the Ecliptic, in the Planisphere, and point off the Latitude 4° 42', at right Angles thereto Northward, as per Problem 31, for noon, and this point will be the true place of the Moon at the hour and minute required, situated in the Heavens, in the Angle between the Whale's Jaw, the Pleiades, and the Ram's North Horn; and you may now take her right Ascension and Declination with the utmost ease in an instant, by Problems 3 and 4; and, if you rectify the Planisphere for the day and hour given, you will find the Moon rising between those Stars on the EastNorthEast. The same rule must be used with all the Planets: and note! that when any Planet is direct in motion, or increasing in Latitude, then such increased Arcs, in any number of hours and minutes given, must be added to their place the preceeding noon, and the sum is the true place at the time required; but if the Planet be retrograde, or decreasing in Latitude, then such Arcs of motion must be subtracted from their place the preceeding noon, and the remainder is the true place of the Planet required. Example, in the Planet Mercury, November 2d, 1802, at 6^{h} 30^{m}, P. M. his Longitude at noon, 7^{s} 23° 49', and the third day at noon, 7^{s} 23° 7', which is less than the second day, and consequently he is retrograde, subtract, and take the difference, which is 42'; then say, if 24^{h} give 42', what will 6^{h} 30^{m} give?—Answer, 11' 20": subtract this from 7^{s} 23° 49', the Longitude at noon on the second day, and the remainder is, 7^{s} 23° 37' 40", the true Longitude of Mercury at the time required; then for the Latitude of the Planet, observe, on the first day it is 2° 16' South; and on the seventh day, 0° 38' South, and of course decreasing: subtract, and the difference is 1° 38': then say, if six days give 1° 38', what will 1 day, 6^{h} 30^{m} give?—Answer, 20' 21', to be subtracted from the Latitude on the first day, 2° 16', and it leaves 1° 55' 39" for the true South Latitude of the Planet at the day and hour proposed; seek these points in Longitude and Latitude in the Zodiac of the Planisphere, and that is the true and corrected place of Mercury for the time required, viz. in 7^{s} 23° 37' 40", and between the Stars in the Scales, and the Claws of the Scorpion, and you may take the right Ascension and Declination of the Planet at the time, as before taught; by this rule, we have the situation of all the Planets among the Stars at any time required, and by which, any person may learn to distinguish, and know them by sight from the fixed Stars, and to trace their progressive motions continually.
FINIS.

This work was published before January 1, 1924, and is in the public domain worldwide because the author died at least 100 years ago.