AN INTRODUCTION TO ASTRONOMY AND GEOGRAPHY: BEING A plain and easy TREATISE OF THE GLOBES. In VII Parts. Containing I The Definitions of the Lines, Circles, etc. upon the Globe or Sphere; and of several Terms of Art II. The Problems in Astronomy Methodically digested, with variety of Examples. III. The several affections, of Triangles, and their Solution upon the Globe; with the variety of Problems which every case contains. iv The whole Art of dialing demonstrated and performed two several ways. V The Erection of an Astrological Figure of the Heavens, according to the several ways of the Ancient and Modern Astrologers. VI & VII. The Explanation and Uses of the Terrestrial Globe, with a brief Geographical and Hydrographical Description of the Earth and Water. By WILLIAM LEYBOURN. London: Printed by J. C. for Robert Morden and William Berry; at the Atlas near the Royal Exchange in Cornhill, and between York-house and the New Exchange in the Strand, London. 1675. THE PUBLISHERS TO THE READER. THat nothing is at once brought forth and perfected, is an observation we may make, as from other things, so in a more especial manner from Arts and Sciences: but not to speak of all, which yet have had, in Succession of time, their Accessions to perfection; we shall instance only in these of Astronomy and Geography. And certain it is, should we either seriously compare the Works of the Ancients, with the more accurate observations and additions of our modern Astronomers; what just cause would appear, for that correction and alteration which is now made? And should we compare the Geographical Tables or Charts of the Ancients, with the more perfect discoveries of our later times; what defects and errors shall we there discover? We hope therefore you will not think strange, because we have so much dissented in our Globes, Maps, etc. from those that now only are accounted excellent: for as we cannot consent to copy those errors and abuses which the Dutch would impose upon the whole world, so we cannot altogether pass by our English Map-makers without censuring them for reasons which we might, and the lesle curious may by their own Works themselves perceive. But as we well know no Art is at once perfected, so we well know there are abroad in the world many Books and Instruments conducing to the improvement of the Mathematics, whereof some are superstuous, others burdensome. The consideration whereof was prevalent enough to have clouded this under the opace darkness of latency, had it not been drawn upon the Horizon of public view, by two Magnets. The time and season of its undertaking was to us too disadvantageous to the rendering a Work of this Nature so complete, and in such good order as was requisite; being necessarily forced and confined to our daily Employment, so that we could never have Effected it, had not Mr. Leybourn, knowing our occasions, willingly assented to Calculate and Methodise all the Problems for us; which Favour we cannot but thankfully acknowledge and confess. As for the Subject of this Treatise, it being Astronomy and Geography, needs no Commendation where there are noble Men, or noble Minds. All therefore we shall say of Astronomy is, that, 'tis of such important consequence, that without the knowledge and science thereof, how great incertainty in times and seasons, what distracting confusion in human affairs must we necessarily be involved in? And for Geography, all that need to be spoken of it is, that 'tis so noble a Study, and of such grand importance, as Kings and Princes have made it their highest concern to understand. Wherhfore that the one might be the more easily apprehended, and the other most truly represented to our view and fancy, the Ancients, with the consent of succeeding Ages, have with much pains and industry, not only invented, but also commended to Posterity the Sphere or Globe, which also have had their Corrections and Admendments. And here we must say something of those New Globes lately set forth by ourselves. True it is, we might boast of the help and assistance of the ablest Mathematicians, not only in England and Holland, but in other parts also; yet sure we are, that after many years' experience, not only in the making and projecting of Globes, Maps, etc. but also in examining and comparing of the Descriptions, Discoveries, Drafts, Journals, Observations, and Writings, as well of Ancient as Modern Geographers, Astronomers, Seamen, and others, wherein we have spared no pains or cost, that we might deliver the delineations of the Earth from the Ataxy and confusion of former Globes and Maps. We say, although 'tis impossible we should emit and set them forth in that exactness and perfection they aught, and we could wish; yet doubtless we have and shall expose both Globes, Spheres, Maps, etc. more accurate and complete than some tristing Toys, as well Foreign as Native; which have been sold at great rates by others, and that with much Ostentation and Boasting; to the abuse of some Gentlemen, perhaps better stored with Money, than Knowledge or Judgement in such things. As for the excellency of the Globe or Sphere, all we shall say is, that they have the Priority in Nature of all other Instruments, as most fit and convenient to the Understanding and Fancy; not to speak any thing of them as they analogically represent the Heavens and the Earth in proper genuine Figures: for, that the Heavens are round, is not doubted by any; and for the form of the Earth, the Opinion of its rotundity is now generally known and received. Our only Request therefore is, that if (in so great a Circumference as we have run) some Lines be not truly drawn from the Centre, that they may not draw an Obliqne Censure from the Reader, but rather thy Animadversion and Pardon for ROB. MORDEN, W. BERRY. Advertisement of Globes, Books, Maps, etc. Made and sold by Robert Morden and William Berry. A New Size of Globes about 15 Inches Diameter, very much Rectified and Corrected; agreeing with the Description of this Book. Price 4 l. Another Size of Globes about 10 Inches Diameter. Price 50 s. A new sort of the Copernican Spheres 20 Inches Diameter, comprehending both general and particular. Price 4 l. Ptolomean Spheres of 10 Inches and 15 Inches Diameter, very much different from what were formerly made. Price 3 l. and 2 s. Concave Hemispheres about 3 Inches Diameter, which serve as a Case for the Terrestrial Globe, and may be carried in the Pocket. Price 15 s. A new and large Map of England, Scotland, and Ireland 6 Foot long, 4 Foot and ½ deep; wherein Ireland and Scotland are both new, and England much Corrected. Price pasted upon cloth, etc. 12 s. The Genuine Use and Effects of the Gun; showing, 1. From any shot of any Piece, to find the greatest Random; and to strike any place upon Ascents or Descents within the reach of the Piece. 2. The Complication and Relation of all Guns. 3. The Nature, Property and Use of Morter-pieces. 4. The shooting of Granado's out of Long Guns, and many other things of great consequence to this Art; as well Experimentally as Mathematically demonstrated, by Robert Anderson; with Tables of Projection, etc. Price 2 s. 6 d. The Description and Use of the Planetary System, with Tables for finding the places of the Planets, Eclipses, with all other Motions and Appearances of the Heavens for ever. Made more easy and ready than formerly, by Thomas Street. The Description and Use of Sutton's Large Quadrant, for finding the Hour and Azimuth universally. Also all sorts of Maps, Sea-plats, Mathematical Books, Projections, Instruments, etc. Astronomical Definitions. CONTAINING The Rudiments of Astronomy, AND Explanation of the Sphere & Globe. small insert between pages with illustration of globe 1 What a Sphere is. A Sphere (omitting the Geometrical denomination) in relation to my present purpose, I define to be, An Analogical representation of the Heavens and of the Earth, made of several Circles so fitted together, as thereby the better to express and represent to the Fancy the Systeme or Hypothesis of the Visible World. Of which there are chief three sorts invented; viz. the Ptolomean, Copernican, and Tychonian. A Description of the Ptolomean Sphere. This Systeme (which was so called from Ptolemy the Prince of Astronomers, being indeed the very Founder of that Art and Science) supposeth the Earth to be fixed in the Centre , and that all the Celestial Bodies do move round that in their Diurnal and Annual revolutions. This Sphere or Systeme of the World the Ancients divided into two Parts or Regions; Elemental, and Celestial. The first consisteth of four Parts: 1. The Earth. 2. The Water. Which together maketh one perfect Body. And above these, are the other two Elements, viz. 3. the Air; and above that, 4. the Fire. The second consisteth of many parts, viz. 1. The Moon. 2. Mercury. 3. Venus. 4. The Sun. 5. Mars. 6. Jupiter. 7. Saturn. 8. The Orb or Sphere of the Starry Heaven: all which Orbs or Circles are imagined to be carried about upon the Axis of the World, by the rapture of the Primum Mobile, as is represented by this Figure. Of the several Motions of the Planets. Besides the Diurnal motion by which the Planets are carried round about from East to West in 24 hours by the rapture of the Primum Mobile upon the Poles of the World, they have also a free and proper motion of their own, from West to East, according to the succession of the Signs upon the Poles of the Ecliptic, each of them in a several manner and space of time: viz. The Moon, which is the lowest of the Planets, makes her Revolution in 29 days and 8 hours. The next is Mercury, who is never far distant from the Sun, and finisheth his motion in 88 days. Than Venus, which in her preceding the Sun is called Lucifer, but in her following the Sun is called The Evening Star, finisheth her revolution in 225 days. Next the Sun, who performeth his revolution in the space of a year, viz. 365 days, and about 6 hours. The next is Mars, who finisheth his course in one year, and 321 days. Than Jupiter, who maketh his revolution in 11 years, 10 months, and 16 days. Lastly, Saturn the highest of all the Planets, and therefore goeth the greatest circuit, finisheth his periodical course in 29 years, 5 months, 15 days. Now all these Orbs in our large Ptolomean Spheres are placed upon the Poles of the Ecliptic, the better to correspond with their real places, and demonstrate their true motions in the Heavens. Fol. 2. small insert between pages with illustration of the Ptolemaic system Ptolomean Sphere A description of the Copernican Sphere. This Hypothesis is now known to the world by this name from Copernicus a Borussian, and Canon of the Cathedral Church of Worms, who seriously addressing himself to the illustration of Astronomy, revived the long-neglected Systeme of the World excogitated by Pythagoras, a Figure whereof is here inserted Of the Motion of the Planets. In this Systeme the rest of the Planets are mo●ed round the Sun, which is the Centre of the whole Universe, from whence he may the more equally communicate and distribute both light and heat to the rest of the opacous Planetary Bodies encircling him. The first next the Sun is Mercury, who finishes his Circuit about the Sun in 88 days. The next is Venus, who moveth round the Sun in 225 days. Next is the Earth, which is supposed to move with a double Motion: one about its own Centre, upon the Pole of the Equinoxial, from West to East, in 24 hours; whereby all its parts are alternately enlightened, and Day & Night successively enjoyed: the other Motion is made upon the Poles of the Zodiac or Ecliptic, in the space of a year; whereby all places by course enjoy Spring, Summer, Autumn, and Winter. The Earth is encompassed with a Circle which doth not include the Sun, representing the Orb of the Moon, who in 29 days and a half finisheth her period round the Earth. The next is Mars, who is moved above the Earth and Moon, and ends his course round the Sun in the space of one year and 320 days, or thereabouts; and appears far greater when he is in Opposition of the Sun, than he doth at other times; at which time he is five times nearer the Earth, than he is in his greatest remotion. The next is Jupiter, who with his four companions moveth above Mars, and is about 12 years encompassing his period round the Sun. The seventh is Saturn, who with his Ring and Moons is about 30 years in finishing his revolution round the Sun. And lastly, the Sphere of the Fixed Stars, which with their several distances encircle all the Planetary Orbs , and to us incommensurable. I shall not now need pled for the truth of this Systeme, nor speak much of its excellency: only this, That hereby is taken away that incredible daily motion of the Planets, and vast Orb of the fixed Stars; which are, if the Ancients Supputation should be true, many hundred times swifter than the Shot out of any Gun: for the motion of the Sun in one hour must be 785715 English miles, and in one minute 13095 miles more; and the fixed Stars must move in an hour 135 13686 English miles, and in one minute 225228 miles. And also hereby is annihilated all those Epicycles which the old Astronomers blindly tied to the motion of the Sun; and all those Eccentricks, Differents, and Equants, with the Circles of Inclination, Deviation, and Reflection. This Hypothesis, without any absurdity or perplexity of fictitious Orbs, or impertinent motions, exactly solves all the variety of appearances in the Heavens. Fol. 3. small insert between pages with illustration of the Copernican system copernican Sphere A description of the Tyohonian Sphere. This Hypothesis derives its name from that Noble Dane, Tycho Brahe who with an Heroical Bravery enterprised no lesle than the Instauration of the whole Science of Astronomy from its very Fundamentals; laying such solid Foundations, as that he came very near the height of his noble hopes of building the whole Theory of Astronomy anew: but being prevented by death, he could not accomplish his noble Design. He, I say, excogitated this Hypothesis; which all those that cannot allow of the Pholomaick, nor adhere to the Copernican, may accept and approve of: wherein he makes the Earth the Centre to the Sun, Moon and Stars, which also have their motion round the Earth; but the rest of the Planets to move round the Sun as their Centre: Saturn in opposition to the Sun to be nearer to the Earth than Venus in her Apogaeum, and Mars in opposition is nearer to the Earth than the Sun itself; as may be seen by the Figure. But for a further illustration, or satisfaction to the more Curious, I shall commend them to a sight of those New Spheres lately set forth by the Publishers of this Book, being more exactly and accurately made and contrived than those formerly done, and will be procured at a more reasonable rate; and in operation may be applied to Practical uses, as the Globe is. What a Globe is. A Globe is an Artificial representation of the Starry-Heaven, or the Earth and Water, under the form and figure of Roundness which they are supposed to have; showing, in a just proportion and distance, every particular Constellation in the Heavens, and each several Region or Country on the Earth. Of the Poles. A Globe hath two Poles: the one is called the Arctic, or North-Pole; the other the Antarctick, or South-Pole; representing the Poles of the World. Of the Axis. From the Centre of the Globe is imagined a line thorough both Poles, which is called the Axis of the world, and is represented by the two Wires in the Poles of the Globe. Of the Brazen Meridian. Every Globe is hung by the Axis at both the Poles in a Brazen Meridian, which is divided into 360 Degrees, or four times 90 equal parts. The reason why this Circle is thus divided into four Nineties, is because the Elevation of the Pole, or Latitude of the place, cannot be above 90 degrees. small insert between pages with illustration of a globe Of the Hour-circle, and Index. Upon every Meridian is fitted a small Brass-circle, whose centre is the Pole of the Globe, and is divided into 24 equal parts, representing the hours of the Day and Night; which in the revolution of the Globe is pointed to with an Index, which is fitted on the Axis of the the Globe. Of the Quadrant of Altitude. There is also another appendent relating to the Meridian, called The Quadrant of Altitude; which is a thin Brass-plate divided into 90 equal parts or degrees, and fitted with a Nut and a Screw, to move to any degree upon the Meridian. Of the Horizon. Besides the body of the Globe, there is also annexed a certain Frame of wood, which is called the Horizon; in the upper Plane whereof are several Circles delineated. The first or inner Circle is divided into equal Parts or Signs; every Sign having its name, nature and character placed to it; and every Sign subdivided into 30 equal parts called Degrees, and numbered with 10 20 30. Next to the Circle of Signs is a Calendar or Almanac according to the Old Style used by us here in England, called the Julian; which is divided into the 12 Months of the year, viz. January, February, etc. to December; every Month being subdivided into its number of days: whereunto is annexed the Festival days. Next is a Calendar of the New Style, now used in many foreign places, instituted by Pope Gregory the 13; in which the Months begin 10 days sooner than they do in the other. The last is a Circle of the Winds, divided into 32 equal parts, called Points of the Compass, according to the number of Winds which are observed by our modern Navigators; by which they design forth the Quarters of the Heavens, and the Coasts and Bearings of Countries. Of the Celestial Globe. THe Starry Heaven, that glorious Canopy embroidered with those sparkling Diamonds which hung upon the dusky cheeks of the Night, as a rich Jewel in an Aethiops ear, is represented unto us by the Celestial Globe, because upon its convexity are artificially placed all the Stars and other appearances, in that order and place as they are naturally situate in the concavity of that Orb or Heaven. Of the Lines, Circles, etc. upon the Celestial Globe. And first, Of the Aequinoxial. This is a great Line encircling the Globe, equally distant from both the Poles, and is divided into 360 equal parts or degrees: it is the same Line with the AEquator upon the Terrestrial Globe; only that remains fixed and unmoveable, this variable or at lest must be imagined to move in the Heavens. Fol. 5 small insert between pages with illustration of the Tychonic system Tyconian Sphere Of the Motion of the Aequinoxial. For one of these two must needs be granted; either the Motion of the Eighth Sphere, or Starry Orb, from West to East upon the Poles and Axis of the Ecliptic; or else the progress of the Equinoxial points into the precedent Signs. Now that the first is not to be admitted, appears manifestly, because that the Fixed Stars have not at all changed their situation in respect of Latitude: Therefore the other must needs be granted, viz. the Motion of the Equinoxial. Hence it comes to pass, that the Stars being fixed in their own Orb, move not, but are by the precession of the Equinox left behind the Equinoxial Colour; and so altar their Longitudes. For the first Star of Aries, which in the time of Meton the Athenian (who lived about 431 years before Christ's time) was in the very Vernal Intersection; in the year 1572, when Tycho observed it, was found to be in 27 deg. 37 min. of Aries; and will, according to his Opinion, finish its revolution in 25412 years. Of the Ecliptic. The Equinoxial is crossed or cut in two opposite points by an obliqne Circle, called the Ecliptic; which divides the Globe into two equal parts, called Hemispheres; the one the Northern, the other the Southern. This Circle is divided into 12 equal parts, which are called the 12 Signs; every part being marked with the Figure, Character, and Name of the Sign belonging to it: and each of these Signs is divided into 30 equal parts called Degrees. Under this Circle the Sun and the rest of the Planets finish their several Courses. It is called Via Solis, because the Sun always goes under it in its annual course: but the rest of the Planets have all of them their latitudes and deviations from this Line; by reason of which their digressions and extravagancies, the Ancients assigned the Ecliptic 12 degrees of Latitude: but modern Astronomers, by reason of the evagations of Mars and Venus, have added to each side two degrees more; so that the whole Latitude is confined to 16 degrees: which breadth is also called by some the Zodiac. Of the Poles of the Ecliptic. As all the Meridian's described upon the Globe meet in the Poles of the World; so all the Circles of Longitude being drawn thorough the 12 Signs, meet in the Poles of the Ecliptic; each Pole of the Ecliptic being distant from its correspondent Pole of the World 23 deg. 30 min. the one is called the North-pole, the other the South, according to their position next the North or South-poles of the world. Of the Colours. The Colours are two great Circles cutting one another at Right-angles in the Poles of the World, the one passing by the beginning of Aries and Libra, two Equinoxial-signes, and is therefore called the Equinoxial-Colure; the other passing thorough Cancer and Capricorn, two Solstitial-signes, and is therefore called the Solstitial Colour. This passeth thorough the Poles of the World, and also of the Ecliptic; but the other passeth thorough the Poles of the World only. The Colours divide the Ecliptic into four equal parts; viz. into Aries, Cancer, Libra, and Capricorn; which are called the Cardinal points: for according to the Sun's approach unto any of them, the Season of the year is altered into Spring, Summer, Autumn, and Winter. Of the Tropics. Parallel to the Equinoxial, there are two small Circles, which are called the Tropics, and are 23 deg. and a half distant from the Equinoxial, and are the bounds of the Ecliptic. That on the North-side of the Ecliptic is called The Tropic of Cancer, where the Sun hath the greatest North-declination, and makes our longest day and shortest night; which is about the 11 or 12 of June. The other, on the Southside of the Equinoxial, is called The Tropic of Capricorn; in which point the Sun hath its greatest Southern declination, making our shortest day; and longest night; which is about the 11 or 12 of December. Of the Arctic and Antarctick Circles. Twenty three degrees and a half from either Pole, are described two small Circles: that near the North-pole is called The Arctic Circle, the other in the South is called The Antarctick Circle. Of the Number of the Stars. Encircling the Terrestrial Orb at unmeasurable distance, sparkle the innumerable Lights or Stars in the immense Expansion of the Firmament. And although the Number of them for multitude seems innumerable, yet the greatest and more visible may be numbered and named. The Number that Astronomers have at present taken notice of, is about 1400; which we have inserted in our New Celestial Globe, according to their Right Ascensions and Declinations, and are sufficient for any use or purpose whatsoever. Of the Magnitudes of the Stars. For the better distinction of the bigness of the Stars, they are divided into 6 degrees of Magnitude. The biggest and brightest are called Stars of the first Magnitude; those next inferior in bigness and brightness are called Stars of the second Magnitude: and so the Stars gradually decrease to the sixth Magnitude, which is the smallest, except some few, which are called occult, or nebulose. And these several Magnitudes are expressed on the Globe in several shapes, as may be seen in a small Table placed for that purpose on our Celestial Globe. Of the Images or Constellations described upon the Celestial Globe. Astronomers desirous to bring the Stars into Order and Method, have reduced many Stars into one Constellation, the better to tell where to seek them; and being found, how to express them. The number of Constellations now drawn upon the Globe are 64: viz. In the Northern Hemisphere, Vrsa minor 20 Vrsa major 55 Draco 32 Cepheus 11 Boötes 28 Corona Boreal. 8 Hercules 28 Lyra 11 Cygnus 27 Cassiopoeia 45 Perseus 33 Auriga 27 Serpentarius 38 Serpens 13 Sagitta 8 Aquilea 12 Antinous 7 Delphinus 10 Equuleus 4 Pegasus 23 Andromeda 23 Triangulus 4 Coma Berenice 15 In all, 472 In the 12 Signs of the Zodiac, Aries 21 Taurus 49 Gemini 29 Cancer 15 Leo 40 Virgo 39 Libra 18 Scorpio 26 Sagittarius 28 Capricornus 29 Aquarius 41 Pisces 36 In all, 371 In the Southern Hemisphere, Cete 21 Orion 62 Eridanus 42 Lepus 13 Canis major 15 Canis minor 5 Argo navis 68 Hydra 24 Crater 8 Corvus 7 Centaurns 40 Crusero 5 Lupus 25 Ara 10 Corona Austral 17 Columbus 10 Piscis Austral. 13 Grus 13 Phoenix 15 Indus 12 Pavo 23 Avis Indica 11 Musca 4 Chameleon 10 Triangulum 5 Piscis volans 7 Dorado 7 Toucan 8 Hydrus 21 In all, 510 To these is lately added, by Sir Charles Scarborough, a Constellation called Cor Caroli Regis, figured with a Crowned Heart; being a very considerable and notable Star of the second Magnitude, and unformed, lying between Vrsa major and Coma Berenice's: and 'tis well worthy the observation and notice of the more Curious. Of the Via Lactea, or Milky way. This is a broad White Circle that is seen in the Heavens, and is described between two tracts of small pricks running through several Constellations round the Globe. It is caused by a great number of little Stars constipated in that part of Heaven so small and thick, that we can perceive nothing but a confused light. About the Southern Pole are also discovered two white spots, seeming to be only two white Clouds, being a pale assembly of very small Stars, as in the Galaxy or Milky way; which are more clearly discovered by the Telescope, otherwise inconspicuous to our eyes. An Explanation of several words of Art And first, Of Azimuths. AZimuths, or Vertical Circle●, are great Circles passing through the Zenith and Nadir: and as the Meridian's cut the Equinoxial at Right-angles, so the Azimuths cut the Horizon at Right-angles; and are numbered by degrees from the East to the West-point, towards the North and South in the Horizon. Almicanters. These are Circles parallel to the Horizon, whose Poles are in the Zenith and Nadir; and are called Circles of Altitude, because when the Sun, Moon, or any Star is in any number of degrees above the Horizon, it is said to have so many degrees of Altitude. And these Almicanters, or Circles of Altitude, are numbered upon the Vertical Circle, from the Horizon upwards towards the Zenith. Amplitude. The Amplitude is an Arch of the Horizon, or the number of degrees contained between the true East and West-points in the Horizon, and the rising or setting points of the Sun, Moon, or Stars. Ascension Is the rising point of any Star, or any part or point of the Equinoxial above the Horizon. Right Ascension. The Right Ascension of the Sun, Moon, or Star, is the number of degrees upon the Equinoxial comprehended between the first point of Aries and the Arch of another Meridian passing through the centre of the Sun, Moon, or Star which is upon the Meridian at the time proposed. Obliqne Ascension. Obliqne Ascension is the Arch or number of degrees of the Equinoxial, which riseth with the Sun, Moon, or any Star in an Obliqne Sphere. Obliqne Descension. Obliqne Descension is the Arch or number of the degrees of the Equinoxial which setteth with the Sun, Moon, or Star. Ascensional difference Is the number of degrees after the substraction of the Obliqne Ascension from the Right Ascension; or the difference between the Ascension of any point of the Ecliptic in a Right Sphere, and the Ascension of the same point in an Obliqne Sphere. Apogaeum Signifies that point of the Heavens where the Sun or any other Planet is farthest from the centre of the Earth. Perigaeum Is that point of the Heavens wherein the Sun or any Planet is nearest the centre of the Earth. A Circle Considered as it hath some ground in the nature of the Heavens or the Earth, at lest by Application, is by Astronomers and Geographers divided into 360 parts or degrees: not because this division hath any ground in Nature more than another, but because this number is most commodious for the distinction of Circles, and fittest for Calculation, because no number suffers more parts or divisions than this. Of the Compass of the Earth. Now because the whole Circumference of a Circle is 360 degrees, therefore the Compass of the Earth, according to this supposition, must be 21600 miles. But this opinion seems to be taken (or rather mistaken) from Ptolemy, who allows 500 Stadiums' in a deg. and to every Stadium 600 feet: so that it being proved that 5 Egyptian or Alexandrian feet are longer than 6 of the Italian, and the Roman or Italian is longer than our English; it is evident that there is no sufficient ground for this opinion, and therefore aught to be better considered of. A Constellation. A Constellation or Asterism is a certain number of Stars gathered together into one form, representing some living creature, or other thing, whereby they are particularly known. Compliment Is here usually taken for the remainder of the number of degrees and minutes that any part of a Circle wants of 90 degrees. As, suppose the Latitude of a Star be 50 degrees, that being substracted from 90 degrees, or the fourth part of a Circle, called a Quadrant, there remains 40 degrees for the Compliment thereof. Of a Degree. 'Tis a common Opinion, that 5 of our English Feet make a Geometrical Pace; 1000 of those Paces make an Italian or English Mile; and 60 of those Miles in any great Circle upon the Spherical surface of the Earth or Sea, make a Degree. So that a degree of the Heavens contains upon the surface of the Earth, according to the Opinion of the Ancients, 60 English Miles. 60 Italian Miles. 20 French Leagues. 20 Dutch Leagues. 15 German Miles. 17½ Spanish Leagues. But according to several Experiments made, the quantity of a Degree is thus variously found to be; by Alhazard the Arabian 73 English Miles. Fernlius' 68 English Miles. Wilbrordus 70 English Miles. Gassendus 73 English Miles. Oughtred 66 English Miles. Norwood 69 English Miles. Declination Is the number of degrees that the Sun or any Star is distant from the Equinoxial towards either Pole; and hath a double denomination, viz. North or South-declination, according as the Sun or Star is on the North or Southside of the Equinoxial. Of Eclipses. An Eclipse is a privation or want of light in an opacous or dark Body which is beheld or aspected by a luminous Body: and these Eclipses are of two sorts. Of the Eclipse of the Sun. The Eclipse of the Sun is nothing else but the interposition of the Moon between our sight and the Sun. For the Moon being of a dark, solid, and opacous body, coming between the Sun and the Earth, doth thereby hid more or lesle of the Sun's bright-shining body from our sight, so that the Earth (improperly the Sun) is Eclipsed. This Eclipse cannot be universal (as the Moon's Eclipse is) but may appear in one Country a greater Eclipse, in another lesser, and in othersome no Eclipse at all. For seeing the Sun far exceedeth the Earth in bigness, and the Earth far exceedeth the Moon; therefore the Cone of the Earth's shadow cannot take away or hid the whole body of the Sun from all parts of the Earth, but one part only shall observe the same to be total, or of a like quantity. Of the Eclipse of the Moon. The Moon having no light but what she receiveth of the Sun, can never be Eclipsed but at the Full: yet not at every Full, but when she is diametrically opposite to the Sun, and the Earth in the midst between them both. For the Earth being a solid and opacous Body, casteth its shadow to that point which is opposite to the Sun. So that Conjunction and Opposition only makes an Eclipse; which so happens, when the centre of the Earth and the centres of both the Sun and Moon shall be in the same line: which can only be, where the Moon's Eccentrick cutteth the Sun's in that line which is called the Ecliptic. This intersection, which can be but in two places, is called the Nodes, or Dragon's head and tail. Nor does this intersection keep one certain place, but moving, make a Circle of 18 years. So that Eclipses being Periodical, an Eclipse of the Moon happening now, shall 18 years hence come to pass in the same Sign again. Horizon Is taken from the boundary or termination of the light; and is twofold: Natural, and Astronomical. The Natural Horizon is that apparent Circle which divides the visible part of Heaven from the invisible; extending itself in a strait line from the superficies of the Earth every way round about the place you stand upon, dividing the Heavens into two unequal parts; which is designed out by the sight, and is sometimes greater or lesser, according to the condition of the place. The Astronomical Horizon is that great Circle that divides that part of Heaven which is above is, from that part which is under us, exactly into two equal parts, always certain and the same, passing thorough the Centre of the Earth, whose Poles are the Zenith and Nadir. In this Circle, the Azimuths or Vertical Circles are numbered: and by this Circle our days and nights are measured. This Circle is represented by the upper Plane of the Wooden Horizon. Of the Sun's Longitude. The Longitude of the Sun is an Arch of the Ecliptic comprehended between the Circle of Longitude passing thorough the first point of Aries, and another Circle of Longitude passing thorough the Centre of the Sun. Of the Longitude of a Star or Planet. The Longitude of a Star or Planet is properly an Arch of the Ecliptic comprehended between the Semicircle of Longitude passing thorough the beginning of the Sign the Star or Planet is in, and the Semicircle of Longitude passing thorough the Centre of the Star or Planet. Of the Latitude of the Moon, Star or Planet. The Latitude of the Moon, Star or Planet, is their distance from the nearest point of the Ecliptic either North or South, or an Arch of a Semicircle of Longitude comprehended between the Ecliptic and the Star or point enquired after. Magnetical Meridian Is an Azimuth that passeth by the points of the Needle touched with the Loadstone, or that which passeth through the Poles of the Magnetical Variation. Magnetical Azimuth of the Sun Is an Arch of the Horizon contained between the Magnetical Meridian and the Azimuth of the Sun. Azimuth of the Sun Is an Arch of the Horizon comprehended between the Azimuth passing thorough the Centre of the Sun, and the true Meridian. Of the several Positions of the Globe or Sphere. WHereas in the resolving Problems (of what kind soever) upon the Globes, some there are which may be performed the Globe being in any position or situation, having no need of Rectification; as in Astronomy, the Longitude, Place, or Declination of the Sun; the Longitude, Latitude, Declination and Distances of Stars upon the Celestial Globe. In Geography, the Longitude, Latitude, and Distance of places by the Terrestrial Globe. Also several Problems may be resolved by the help of some one single Circle, either such as is described upon the Globe itself, or is otherwise appendent unto it. As in Geography, the difference of Longitude between Country and Country by the Equinoxial or Hour Circle; the Latitude and difference of Latitude of Places, by the Meridian. In Astronomy, the Declination of the Sun or Stars, by the Meridian; the distance of Stars in the Heavens, or of Places upon the Earth or Sea, by the Quadrant of Altitude. The day of the Month, Sun's place and declination upon the Horizon; with divers other the like Conclusions. But for the resolving of most Problems, it will be requisite to have the Globe placed or fitted according to some assigned Position or Elevation, such as the tenor of the Question shall require. And, besides the Globe itself, and the Circles described upon it, such also as are appendent unto it must come in use; as the General Meridian, Horizon, Quadrant of Altitude, Hour-circle, and Circle of Position, must be accordingly rectified. Now there are but Three Positions in which the Globe can be said to be rectified; of which, two are particular, and the third more general. For, Of a Direct Sphere. I. The Globe may be so placed in the Frame of the Horizon, that both the Poles thereof may rest upon (or he directly in) the North and South points of the Horizontal Circle, neither Pole having any Elevation, the Zenith-point being in the Equinoxial Circle, and the Axis of the world directly in the plain of the Horizon; and so the people living under that Circle have no Latitude, the Pole having no Elevation. The Globe or Sphere being in this position, is said to be in a Direct position, the Zenith being directly in the Equinoxial Circle, and the Poles of the world directly in the North and South points of the Horizon: and this direct position of the Sphere is particular to those who live under the Equinoxial, who have one benefit and privilege above all the inhabitants of the world besides, for that they can see both the Poles, and behold all the Stars (in both Hemispheres) to rise, culminate, and set. Of a Parallel Sphere. II. The Globe may be so placed in the Frame of the Horizon, that one of the Poles shall be in the Zenith, and the other in the Nadir-points; that is, either Pole shall be 90 deg. (or one quarter of a Circle) distant from the Horizon on either side thereof; and in this position will the Equinoxial Circle be in the Horizon, and the Axis of the world will cut both the Equinoxial and the Horizon at Right-angles: one Pole having 90 degrees of Elevation, and the other as much of Depression. The Globe or Sphere being in this position, is said to be Parallel; because the Equinoxial, and all the Circles of Declination, (which now are Circles of Altitude also) and the Axis of the world itself, do lie all of them Parallel to the Horizon. And those people (if any be) that inhabit these parts of the world, see only those Stars that are between the Equinoxial and the Elevated Pole; that is, if the South-Pole be elevated, they see the Southern; and if the North-Pole be elevated, they see the Northern Constellations only. And those people have but one day and one night in the whole year, and those most miserable cold, not to be imagined: for the Sun, at his highest, never extendeth to 24 degrees of Altitude; which is no more than it is with us in London at Noon upon the 14 or 15 of February. And this position of the Sphere is particular to these inhabitants only. Of an Obliqne Sphere. III. The third (and most usual) position of the Sphere or Globe is more general; for it hath relation to all people living between the Equinoxial and either of the Poles: and according as the Poles of the Globe are elevated or depressed, accordingly are the people said to be situate: Thus, if the Globe be placed in the Frame of the Horizon so that the Pole be elevated 10 degrees above the Horizon; than is the Globe elevated or fitted to resolve such Questions or Problems Astronomical as relate to those people who (have the Pole elevated, or) do live in the Latitude of 10 degrees. This position of the Globe or Sphere is called Obliqne, because the Axis of the World, the Equinoxial, and all the rest of the Parallels of Declination, are cut by the Horizon at Obliqne Angles; whereas in the two former positions they cut one the other at Right Angles. Now the people which live in these middle Latitudes, (I mean between the Arctic and Antarctick Circles) which (yet to our knowledge) is the most habitable part of the world, do see Stars both of the Northern and Southern Hemispheres; but yet they see not all the Stars in either Hemisphere: for in any Obliqne Latitude, the inhabitants see, and may observe all such Stars that are of the contrary Hemisphere to them, whose declinations are lesser than the compliment of their Latitude; and all those Stars of the contrary hemisphere, whose declination is greater than the compliment of their Latitude they never see, for they never ascend above their Horizon: and on the contrary, those Stars of their own Hemisphere whose declinations exceed the compliment of their Latitude, do never set, but are continually above their Horizon. Fol. 27. small insert between pages with illustration of a globe tilted on its axis Of Time Time is the measure of all our motions, and is divided by the two greater Lights of Heaven, into Hours, Days, Weeks, Months, and Years. Of a Year. A Year, though it might have been as truly said of any other Star or Planet, is now made proper to the Sun or Moon, whose Revolution in the Ecliptic is the general definition of this part of Time; so that every Month, in the stricter sense, should be taken for a Lunar Year. Of a Lunar Year. The Lunar Year is that space of time wherein the Moon measureth the Zodiac Twelve times, or maketh Twelve Conjunctions with the Sun; which she finisheth in the space of 354 days, 8 hours, and some odd minutes, 11 days or thereabouts before the Sun. Of the Sun's Year. The Sun's Year is the Revolution of his motion in the Ecliptic, which if it be accounted in the Zodiac, was called Annus Temporalis, because it distinguished the four Times or Seasons of the year. 'Tis otherwise called Annus Tropicus, because the Astronomers of old reckoned this Year from the Tropics. But if the Revolution of the Sun be accounted from any Fixed Star to the same again, the Year is than called Annus Sydereus. The precise quantity of this Year is determined of all to be 365 days: but the surplus of hours and minutes hath very much and vainly exercised the most Curious. Julius Caesar looking upon it as a matter no way below his great consideration, after consultation with several Mathematicians, allotted 365 days and 6 hours for this Revolution, reserving every year the 6 odd hours to make a day for the Bissextile or Leap-year, being every fourth in order; so that the three first years should consist of 365 days, and the fourth of 366; which by infallible experience is found too much. For the motion of the Sun through the Zodiac, from one fixed point to the same again, according to more exact Computation, is determined to consist of 365 days, 5 hours, 49 minutes, and 4 seconds; which falls short of 6 hours by 10 minutes and 56 seconds in every year; which must of necessity breed a difference of so many minutes and seconds every year betwixt the Year which the Sun describes in the Zodiac, and that which is reckoned upon in the Calendar. Hence it comes to pass, that the Vernal Equinox, which in the Emperor's time fell out to be upon the 24 of March, now falls about the 11 of March, twelve days backward, and more; so that if it be let alone, it will get back to the first of March, and so to February, till Easter falls to be upon Christmas day, which aught also to anticipate the 25 of December in our common Julian year, and be celebrated about thirteen days sooner. Therefore the Pontificians do, according to the Gregorian Emendation, precede us by ten days in their account, which is called The New Style. But why they went no higher than the Nicene Council in the Correction of the Year, and so fell short in the true time, the reason I must leave to them to give. Of Civil and Natural days. The Civil day is that space of time containing just 24 hours, reckoned from 12 of the clock on one day, to 12 of the clock the next day; in which space of time the Equinoxial makes a diurnal revolution upon the Poles of the world. The Natural day is that space of Time wherein the Sun moveth from the Meridian of any place to the same Meridian again. Of an Hour. An Hour is the twenty fourth part of a day and a night; or the space of time that 15 degrees of the Equator takes up in the passing through the Meridian. For the whole Equator, which contains 360 degrees, passeth through the Meridian in 24 hours, which are Mathematically divided into minutes, seconds, thirds, etc. Of the Zenith. The Zenith (or Senith in the Arabic) is the Pole of the Horizon, an imaginary point just over our heads, viz. the Vertical point, or highest top-point of the Heaven from our Horizon, and equidistant from it round about 90 degrees. Nadir. The Nadir, or Nathir, a foot-point perpendicularly imagined from our feet down through the Centre of the Earth, is that point in Heaven which is directly under our feet. Geographical Definitions. A Description of the Circles, Lines, etc. Drawn upon the Superficies of the Terrestrial GLOBE. Of the Equator. THe Equator, or the Line under the Equinoxial, is a great Circle encompassing the very middle of the Globe between the two Poles, dividing it into two equal parts from North to South; and is divided, as the Equinoxial, into 360 equal parts or degrees; and is called The Equator, either because it is equally distant from the Poles of the World, or rather because when the Sun comes to this Line, which is twice in a year, viz. in the beginning of Aries, which with us is about the 10 and 11 of March; and again in Libra, which is about the 13 and 14 of September, he makes equality of days and nights throughout the world. Of the Meridian's. The Meridian's are Circles passing through the Poles of the Globe, and cut the Equator at Right Angles: infinite in number, because all places from East to West have several Meridian's; but the number of Meridian's delineated upon our Globes are 24. The first or chief Meridian, which is as it were the Landmark of the whole Earth in our new Globes, gins at the Island of S. Michael, one of the Isles of the Azores; from whence is reckoned the beginning of Longitude, and is the only Circle passing through the Poles of the Globe, which is divided into twice 90 degrees, numbered from the Equator towards both the Poles. The lesser Meridian's are those black Circles which pass through the Poles of the Globe, succeeding the great Meridian at 10 and 10 deg. in most Globes, but in ours at 15 and 15 degrees difference, and are numbered in the Equator with 15 30 etc. to 360, round the Globe. Of the Parallels. The Parallels are Circles running East and West round about the Globe like as the Equator; only the Equator is a great Circle, and these are every one lesle than the other, diminishing gradually, until they terminate and end in the Pole. As the Meridian's are infinite, so are the Parallels, and are delineated through every tenth degree of the first Meridian, numbered from the Equator to either of the Poles with 10 20 to 90. Of the Lines under the Ecliptic, Tropics, and Polar Circles. Crossing the Equator obliquely in the middle, is the Ecliptic: the utmost extent of it towards the North, noteth ou● the Tropic of Cancer; towards the South, the Tropic of Capricorn; each of them distant from the Equator 23 degrees and a half. Parallel to the Tropics, and at the same distance from the Poles as they are from the Equator, are drawn the Arctic and Antarctick Circles, offering themselves to sight by their Names, and distinctions of Breadth and Colour; being, as the Tropics are, represented by more full Lines than the Parallels are. ADVETISEMENT. AS for the Land, Seas, Islands, Rivers, Rumbs, with other Geographical Definitions relating to the Terrestrial Globe; I shall refer you to my following Description of Universal Maps and Charts; wherein is fully and methodically explained the whole Rudiments of Geography; being very necessary for the better illustration and understanding of the Terrestrial Globe. Having thus given you a brief Account of the several Positions of the Globe or Sphere, and of the Explanations of the Lines, Circles, etc. it will be convenient now to show you the Uses thereof, in the Solution of Problems of divers kinds: in the performance whereof, I shall be both brief and plain. And whereas I formerly said, that divers Problems may be resolved upon either Globe, it being in any position whatsoever (which will plainly appear hereafter) yet it will be more convenient to have the Globe fitted and accommodated with all its necessary Appendants before you make use of it. And therefore my first business shall be, to show how to fix and set the parts of the Globe together, fit for use in any assigned Latitude or par● of the World: and this I call Rectifying of th● Globe. How to Rectify the GLOBES, fitting them for Use In any LATITUDE or place of the World. BEing provided of a pair of Globes, the Meridian, Horizon and Hour-circle truly turned and divided; also the Ball truly hung, and the Meridian and Horizon (in all positions) cutting each other at Right Angles, the Papers truly joined in their Pasting, etc. all which are to be performed by the Workman, (though the Buyer aught also to have inspection thereinto) you may proceed to Rectify them in this manner. 1. Put the Brass-Meridian into the two Notches that are in the North and South-parts of the Horizon; the Graduated or divided part thereof towards the East-point, and the blank or plain side of the Meridian towards the West-point of the Horizon; and let the Meridian rest in the Notch which is in the foot or bottom of the Horizon. 2. Place the Hour-wheel about the Pole, so that the Hour-lines of 12 and 12 do lie directly over the East or Graduated side of the Meridian, and that the point of the Axis do pass directly through the Centre of the Hour-circle; so shall the two Twelves, one of them represent 12 at Noon, and the other 12 at Midnight; and the two Six, the one 6 in the Morning, and the other 6 at Night. Than put the little Index or Pointer upon the Axis, so that it may move as you turn the Globe about, and so is your Hour-wheel Rectified. 4. Elevate the Pole of your Globe (whether the North or South-pole) according to the Latitude of that part of the world you are in: as for Example, for London, whose Latitude is 51 deg. 30 min. North; the Meridian being in the Notches of the Horizon, and also in that in the foot of the Frame, as is before directed, Move the Meridian up or down in the Notches, till you find 51 deg. 30 min. of the Meridian justly to touch the upper part of the North-part of the Horizon: for than is your Globe set exactly to the Latitude of 51 deg. 30 m. 4. For the Rectifying of the Quadrant of Altitude, this also must have respect to the Latitude: Wherhfore, the Latitude being 51 deg. 30 min. count 51 deg. 30 min. upon the South-part of the Meridian, from the Equinoxial-circle towards the North (or elevated) Pole; and put on the Nut which is at the end of the Quadrant, so that the edge of the divisions of the Quadrant may lie directly under the degrees of the Latitude, viz. under 51 deg. 30 min. and than screw the Nut fast; and so is the Quadrant of Altitude Rectified also. These are the four principal things that almost in all Cases must be Rectified: for the Circle of position, that seldom comes in use but in Dialling and Astrology; and so it is not fit to cumber the Globe with it at other times: but when there is occasion for it, you must rectify it as followeth. 5. The two ends of the diameter of this Circle are to be placed in the North and South-points of the Horizon; and it is to move up and down between the Meridian and the Horizon, and there to be fixed or held at any acquired place. And this is all the Rectifying that this Circle requires, and is (as I said before) but seldom to be used; and therefore they are not usually made or sold with the Globes. Astronomical Problems. PROB. I The day of the Month known, either according to the Julian or Gregorian Accounted, to find the Sun's place in the Ecliptic. SEek the day of the Month (in either Account) according as you find them placed in the Calendar, and right against it, in the innermost Circle next to it, you shall have the degree and minute in which the Sun shall be that day at noon. EXAMPLE. Let the day proposed be the 18 of October (in the Julian, or the 28 of October in the Gregorian Account) which is S. Luke's day: Find this Month and Day in the Calendar, and right against it (in the innermost Circle) you shall find 5 deg. 32 min. of Scorpio, in which Sign and degree the Sun will be upon that 18 or 28 of October. In like manner, upon the 24 of April (Julian) or the 4 of May (Gregorian) (which is all one day) the Sun will be found to be in 14 deg. 32 min. of Taurus. And so of any other day, as in the following Table, which showeth, that upon Jul. Greg d. m. January 7 17 The Sun's place at Noon will be in 28 9 Capricorn ♑ Februa●y 12 22 4 38 Pisces ♓ March 23 Apr 3 13 23 Aries ♈ May 10 26 5 39 Gemini ♊ July 27 Aug. 7 14 16 Leo ♌ Sep●emb●r 3 13 21 0 Virgo ♍ November 5 15 23 41 Scorpio ♏ PROB. II. By knowing the Place of the Sun in the Ecliptic, the day of the Month in either of the Accounts may be obtained: As followeth. SEek the sign, degree and minute in which the Sun is, in the innermost Circle of the Horizon; and right against it, you shall have the day of the month in both Accounts. EXAMPLE. Let the Sun be in 24 deg. of Gemini; look in the innermost Circle for the sign Gemini, and against it in the Calendar you shall find the 4 of June (Julian) or the 14 of June (Gregorian) which is the day of the month. In like manner, when the Sun is in 14 deg. 53 min. of Capricorn, the day of the month will be found the 25 of December (Julian) or the 4 of January (Gregorian.) PROB. III. The Latitude (51 deg. 30 min.) and the Sun's place in the Ecliptic (viz. in 29 of Taurus) being given, to find The Sun's Declination. Definition THe Sun Declination is an Arch of the Meridian, comprehended between the Equinoxial Circle, and that point of the Ecliptic in which the Sun is. Practice. Bring the 29 deg. of Taurus to the Brass-Meridian; than shall the degrees of the Meridian contained between the Equinoxial and this point, be 20. So that the declination of the Sun is 20 deg. North, because the Sun is in a Northern Sign. In like manner, deg. min. deg. min. When the Sun is in deg. 16 00 ♈ The Sun's declination will be found to be 15 46 North. 25 00 ♏ 19 05 South. 29 00 ♒ 11 52 South. 13 00 ♌ 5 9 North. PROB. IU. The Sun 's Amplitude. Definition THe Amplitude, is an Arch of the Horizon comprehended between the East or West-points thereof, and that point upon which the Sun doth Rise or Set. Practice. Bring 29 deg. of Taurus to the Horizon, and there you shall find 33 deg. 20 min. to be contained between that, and the East or West-point, towards the North, because the Sun is in a Northern sign, that is very near the North-East by East point of the Compass, as appears by the points upon the Horizon: and that is the Amplitude. An in like manner, d. m. d. m. When ☉ is in deg. 16 00 ♌ The Sun's Amplitude will be from the East or West sound 26 28 Northward 25 00 ♏ 31 41 Southward 29 00 ♒ 19 17 Southward 13 00 ♈ 7 34 Northward PROB. V The Sun's Right Ascension. Definition THe Right Ascension of the Sun or of a Star, is that Arch of the Equinoxial which is contained between the beginning of Aries, and that point of the Equinoxial which comes to the Meridian, with that point of the Ecliptic in which the Sun or Star is. Practice. Bring the 29 deg. of Taurus to the Meridian; so shall you find (upon the Equinoxial) 56 deg. 50 min. to be contained between the beginning of Aries and the Meridian: And such is the Sun's Right Ascension when he is 29 deg. of Taurus. In like manner, deg. min. deg. min. When the Sun is in deg. 16 00 ♌ The Sun's Right Ascension will be 138 26 25 00 ♏ 232 38 29 03 ♒ 330 51 13 00 ♈ 11 57 PROB. VI The Obliqne Ascension. Definition. THe Obliqne Ascension of the Sun, or of a Star, is that Arch of the Equinoxial which is comprehended between the beginning of Aries and that point of the Equinoxial which comes to the East-point of the Horizon, with that point of the Ecliptic in which the Sun or Star is. Practice. Bring 29 degr. of Taurus to the East-side or semicircle of the Horizon, than shall you find 20 deg. 30 min. of the Equinoxial to be contained between the beginning of Aries and the East-point of the Horizon; and that is the Obliqne Ascension of the Sun or Star, it being in the 29 degree of Taurus. In the same manner, deg. min. deg. min. The Sun being in 16 00 ♌ The Obliqne Ascension will be found to be 117 10 25 00 ♏ 258 27 29 00 ♒ 346 30 13 00 ♈ 005 26 PROB. VII. To find the Ascensional Difference. Definition THe difference of Ascension is no other than the difference of degrees between the Right and Obliqne Ascension. Wherhfore subtract the lesser from the greater, and the remainder will be the Ascensional Difference, which will be found to be 17 deg. 14 min. Or, the Ascensional difference is that space of time contained between 6 of the clock either in the morning or the evening, and the time of the Sun 's Rising or Setting. Wherhfore, PROB. VIII. To find the time of the Sun's Rising and Setting. Practice. BRing 29 deg. of Taurus to the Meridian, and set the Index of the Hour-wheel to 12 of the clock Southward: than turn the body of the Globe Eastward, till 29 deg. of Taurus touch the East-side of the Horizon; and than with the Index of the Hour-wheel point out 11 min. after 4 in the morning, at which time the Sun Riseth. And if you turn the body of the Globe about Westward, till the 29 deg. of Taurus doth touch the Westside of the Horizon, than shall the Index of the Hour-circle point at 49 min. after 7 at night, at which time the Sun setteth. PROB. IX. To find the Length of the Day and Night. Practice. TUrn the Globe about till 29 deg. of Taurus touch the East-side of the Horizon; and than set the Index of the Hour-circle to the North (or undermost) 12. Than turn the Globe Westward, till 29 deg. of Taurus touch the Horizon on the Westside, and than shall the Index of the Hour-wheel point at 3 hours 38 min. more than 12 hours. So that the day is than 15 hours and 38 min. long. And if you count the hours between the North 12 and the Hour-Index, you shall find them to be 8 hours and 22 min. which is the length of the night, the Sun being in 29 deg. of Taurus. And so, ☉ being in Ascens. diff. Sun's rise Sun's set length of day length of night d. m. d. m. h. m. h. m. h. m. h. m. 16. 00 ♌ you'll find 21. 12 4. 35 7. 25 14. 50 9 12 25. 00 ♏ 25. 52 7. 43 4. 17 8. 34 15. 28 29. 00 ♒ 15. 00 7. 00 5. 00 10. 00 14. 00 13. 00 ♈ 6. 30 5. 34 6. 26 12. 52 11. 8 PROB. X. The Sun's Meridian-Altitude, and his depression at Midnight. THis may be effected, by adding or substracting of the Sun's Declination to or from the Compliment of the Latitude: For, definite. It is an Arch of the Meridian, comprehended between the intersection of the Meridian with the Horizon, and that part of the Meridian upon which the Sun is at Noon or Midnight. Practice. Turn the Globe about, till the 29 degree of Taurus be just under the Meridian; than shall you find the number of degrees of the Meridian which are comprehended between that point and the Horizon to be 58 deg. 30 min. which is the Meridian-Altitude. And if you bring the 29 deg. of Scorpio, which is the opposite point of the Ecliptic in which the Sun is, to the Meridian, the number of the degrees of the Meridian between that point and the Horizon will be found to be 18 deg. 30 min. which is the Sun's depression at midnight. In like manner, d. m. d. m. When the Sun is in 16 00 ♌ You shall find the Meridian Altitude 54. 36 & the depres. 22. 24 25. 00 ♏ 19 25 57 25 29. 00 ♒ 26. 38 50. 22 13. 00 ♈ 43. 39 33. 21 PROB. XI. When the Twilight gins and ends. Definition THe Twilight beginneth when the Sun is 18 deg. below the Horizon before its Rising, and it endeth when the Sun comes to be 18 deg. below the Horizon after its Setting. Practice. The Globe Rectified, and the Sun in 29 deg. of Taurus, find the opposite point thereunto, which is the 29 deg. of Scorpio, and bring that point, as also the Quadrant of Altitude, both of them on the Westside of the Meridian; and than move both the body of the Globe, and the Quadrant of Altitude also, till the 29 deg. of Scorpio lie directly under 18 deg. of the Quadrant of Altitude: which done, keep them both together, and than see how many hours the Index is removed from 12, which you shall find to be 1 hour and 8 minutes. So that Twilight gins at 8 min. after 1 in the morning. And this being taken from 4 ho. 11 min. the time of the Sun's Rising that day, there will remain 3 ho. 3 min. which is the length or continuance of the Twilight. Also if you double the time of the beginning of Twilight 1 ho. 8 min. you shall have the length of dark night, which will be but 2 ho. 16 min. In like manner, if you would know when the Twilight endeth after Sunsetting, you must bring the 29 deg. of Scorpio (the point opposite to the Sun) on the East-side of the Meridian, making it and 18 deg. of the Quadrant of Altitude to meet, than the Index will show 10 hours 52 min. and till that time of night doth Twilight continued And so, The Sun in will be at Twilight last till deg. ho. mi. aft. midnight ho. min. at night. 29 ♋ 0. 10 11. 50 8 ♐ 5. 52 6. 8 2 ♎ 4. 6 7. 54 0 ♉ 2. 41 9 19 And if you go about to find the time of the beginning and end of Twilight; all the time that the Sun is passing from 2 deg. of Gemini to 30 deg. of Cancer, which is from about the 12 of May to the 12 of July, you shall find that there will be no Twilight at all, but all that time continual day for all that space of time the Sun never descendeth so much as 18 deg. under the Horizon, in the Latitude of 51 deg. 30 min. PROB. XII. What Altitude the Sun shall have at 6 of the Clock in the morning or evening. Definition THe Altitude of the Sun, or of a Star, etc. is an Arch of an Azimuth or Vertical Circle, passing through the Zenith and Nadir points and the body of the Sun or Star, counted from the Horizon to the Sun or Star: and all such Azimuths or Vertical Circles are represented by the Quadrant of Altitude. ¶ Note, that this Problem is only in use when the Sun is in the six Northern signs: for the Sun is never above the Horizon at six, when he is in Southern signs. Practice. Bring the 29 deg. of Taurus to the Meridian, and set the Index of the Hour-circle to 12; than turn the Globe about Eastward, till the Index of the Hour-circle come just to 6 a clock: than holding the Globe there, lay the Quadrant of Altitude just over the 29 deg. of Taurus, and there you shall find it to cut 15 deg. 30 min. of the Quadrant. And such Altitude shall the Sun have at 6 of the clock in the morning, and the same at 6 at night. And so, When the Sun is in The Sun's Altitude at 6 will be found to be deg. min. deg. min. 16. 00 ♌ 12. 32 13. 00 ♈ 4. 2 PROB. XIII. What Azimuth the Sun shall have at 6 of the Clock. Definition THe Azimuth is an Arch of the Horizon comprehended between the East, West, North or South points thereof, and the intersection of a Vertical Circle passing through the Sun or Star whose Azimuth you seek. ¶ Note, this Problem is of Use only when the Sun is in Northern signs. Practice. Bring the 29 of Taurus to the Meridian, and set the Index of the Hour-wheel to 12; than move the Globe till the Index lie upon 6; and holding the Globe there, lay the Quadrant of Altitude just over 29 deg. Taurus: than shall you find, that there are 77 deg. 14 min. of the Horizon contained between the intersection of the North-part of the Meridian, and the Quadrant of Altitude, which is the Azimuth from the North: or 12 deg. 46 min. from the East, which is the Azimuth from the East: or 102 deg. 46 min. from the South, which is its Azimuth therefrom. In like manner, Sun in The Sun's Azimuth at 6 will be found to be from the d. m. d. m. d. m. d. m. North East or West South 16. 0 ♌ 79.49 10.11 100.11 13. 0 ♈ 86. 47 3. 13 93. 13. PROB. XIV. At what hour the Sun shall be upon the East or West Azimuths. ¶ This Problem is only of use when the Sun is in Northern signs. Practice. BRing the 39 deg. of Taurus to the Meridian, and the Index to 12 of the Clock. Also bring the beginning of the degrees of the Quadrant of Altitude to the East-point of the Horizon, and turn the Globe about till the 29 degree of Taurus do touch the degrees of the Quadrant of Altitude; than shall the Index point at 7 min. passed 7, at which time in the morning will the Sun be exactly upon the East Azimuth, or point of the Compass. And if you carry the Quadrant of Altitude to the West-point of the Horizon, and turn the Globe about till 29 deg. of Taurus touch the edge of degrees thereof, the Hour-IndeX will point at 4 of the clock and 53 min. at which time in the afternoon will the Sun be upon the West-Azimuth or point of the Compass. In the same manner, d. m. d m. d. m. When the Sun is in 16.00 ♌ It will be due East at 6.57 West at 5.03 13.00 ♈ 6.17 5.43 PROB. XV. What Altitude the Sun shall have when he is upon the East or West-Azimuths. ¶ This Problem is in use only when the Sun is in Northern signs. Practice. BRing 29 deg. of Taurus to the Meridian, and the Quadrant of Altitude to the East or West-points of the Horizon: Than turn the Globe about, till the 29 deg. of Taurus touch the Quadrant of Altitude, and you shall find it to touch at 25 deg. 55 min. of the Quadrant: and such Altitude hath the Sun, when he is upon the East or West-Azimuth. In like manner, d. m. d. m. When the Sun is in 16.00 ♌ It's Altitude when East or West will be found 20.19 13.00 ♈ 6.36 PROB. XVI. What Altitude the Sun shall have at any time of the day. Practice. BRing 29 deg. of Taurus to the Meridian, and set the Hour-Index to 12 a clock. Than turn about the Globe till the Hour-Index point to the given hour (suppose 9 in the morning, or 3 in the afternoon:) there keep the Globe; and laying the Quadrant of Altitude over the 29 deg. of Taurus, you shall find 43 deg. cut thereby; and such Altitude shall the Sun have at 9 in the morning, or 3 in the afternoon. And by this Problem the Sun's Altitudes in any sign or degree of the Ecliptic at all hours may be found; as in this following Synopsis or Table. So the Sun being in the beginning of Cancer Gemin. or Leo. Taurus or Virgo. Aries or Libra. Scorpio or Pisces. Aquar. or Sagitt. Capric. At the ho. d.m. d.m. d.m. d.m. d m. d.m. d.m. XII. His Altitude will be 62 00 58 42 50 00 38 30 27 1 18 18 15 00 XI. I. 59 43 56 34 48 12 36 58 25 40 17 6 13 52 X. II. 53 45 50 55 13 12 32 37 21 51 13 38 10 30 IX. III. 45 42 43 ● ●6 0 26 7 15 58 8 12 5 26 VIII. iv 36 41 34 13 27 31 18 8 8 33 1 15 VII V 27 17 24 56 18 18 9 17 0 6 VI 18 11 15 40 9 0 V VII. 9 32 6 50 IU. VIII. 1 32 This Table will be of good use to such as have occasion to make Cylinders or Quadrants, to find the hour of the day; or Rings and other Instrumental Dial's; and to insert the Tropics and other Parallels of the Sun's course in fixed Sun-dyals', etc. PROB. XVII. What Altitude the Sun shall have, he being upon any Azimuth. Practice. SEt the Quadrant of Altitude to the Azimuth you intent to find the Altitude upon; (suppose the 30 deg. of Azimuth from the South towards the East:) bring the Quadrant of Altitude thither; and keeping of it there, turn the Globe about till 29 deg. of Taurus touch the degrees of the Quadrant of Altitude, and you shall find them to concur at 55 deg. 34 min. of the Quadrant: and such is the Sun's Altitude, when he is 30 deg. from the South-part of the Meridian either Eastward or Westward. And by this means you may find the Azimuth at all times. An Example of the Sun's Altitude upon every Tenth Azimuth from the South, in the beginning of each Sign, here followeth. The Sun in the beginning of Cancer Leo or Gemin. Virgo or Taurus Libra or Aries Scorpio or Pisces. Sagitt or Aquar. Capric d. Az d.m. d.m. d.m. d.m. d.m. d.m. d. m. It's Altitude on hese degr. of Azimuth from the South, viz. 0 will be found to be 62 0 58 42 50 0 38 30 27 0 28 18 15 0 10 61 43 58 24 49 38 38 4 26 30 17 45 14 25 20 60 51 57 28 48 33 36 46 25 10 16 5 12 41 30 59 52 55 52 46 40 34 34 22 27 13 15 9 45 40 57 20 53 29 43 51 31 21 18 48 9 14 5 34 50 54 3 50 12 40 11 27 5 13 58 3 57 0 6 60 49 56 45 53 35 23 21 41 8 0 70 44 40 40 25 29 27 15 13 1 0 80 28 11 33 46 21 29 7 52 90 30 38 26 10 14 25 100 22 27 18 2 6 45 110 14 14 9 58 120 6 34 2 30 This Table hath the like use for the making of Cylinders, Quadrants, and other Instruments that give the Azimuth by the height of the Sun, as the preceding Table hath for the Hour, and may be so applied. The Latitude (51 deg. 30 min.) the Sun's place in the Ecliptic, (29 deg. 0 min. of Taurus) and his Altitude (12 deg.) being given: To find PROB. XVIII. The Sun 's Azimuth at any time. Practice. THe Globe being Rectified, etc. and the Quadrant of Altitude fixed, and brought to the Horizon; Turn 29 deg. of Taurus toward the East, if in the morning; or towards the West, if in the evening, till it come to lie just under deg. of the Quadrant of Altitude; and than note at what degree in the Horizon the Quadrant of Altitude resteth; which will be at 16 deg. 8 min. from the East if in the morning, or 16 deg. 8 min. from the West if in the afternoon Northward, which is the Azimuth from the East or West towards the North. And this Azimuth, if reckoned by the Points of the Compass upon the Horizon, will be E. by N. 4 deg. 53 min. Northward, if in the morning; or W. by N. 4 deg. 53 min. Northward, if in the evening, when the Sun is in 29 deg. of Taurus, and hath 12 deg. of Altitude. Now if you count the degrees of the Horizon between the Quadrant of Altitude and the North-part of the Meridian, you shall find them to be 73 deg. 52 min. which is the Azimuth from the North: And if you count them from the South-part of the Meridian, you shall find them to be 106 deg. 8 min. which is the Azimuth from the South. In like manner; deg. min. The Latitude being 51 30 The Sun's place 1 00 Aquarius. The Sun's Altitude 12 00 Than will the Azimuth be found to be 56 deg. from the East or West towards the South; which (by the Points of the Compass upon the Horizon) will appear to be S. E. by S. if in the morning, or S. W. by S. if in the evening. PROB. XIX. The Hour of the day. Practice. BRing 29 deg. of Taurus to the Meridian, and set the Index to 12● clock: Than, if it be in the forenoon, set the Quadrant of Altitude on the East-side of the Meridian▪ but on the Westside, if it be in the afternoon. And turn the Globe about, till the 29 deg. of Taur●s meet with 12 deg. of the Quadrant of Altitude and than shall the Index of the Hour-circle poin● at 5 a clock and 36 min. if it be in the morning or at 24 min. after 6 of the clock, if it be at night And that is the true hour of the day. In like manner, deg. min. The Latitude being 51 30 The Sun's place 1 00 Taurus. The Altitude 36 00 Than will the hour of the day be found to be either 9 in the morning, or 3 in the afternoon. And which of these hours it is, may best be known by a second observation of the Altitude: for if the Altitude do increase, it is the forenoon; but if it decrease, it is the afternoon. Again, deg. min. The Latitude being 52 30 The Sun's place the beginning of Taurus. The Sun's Altitude 25 56 The hour of the day would be found to be either 8 min. passed 4 in the afternoon. Or if it were in the forenoon, 52 min. after 7 in the morning. PROB. XX. To find the Difference of Latitude, or, to know how many degrees the Pole must be Elevated or Depressed, to make the Longest day in any Latitude an Hour longer, or the Shortest an Hour shorter than it is in your Latitude. LEt it be required to find in what Latitude the Longest day shall be an hour longer than it is at London. Practice. Rectify the Globe to the Meridian of London, the Index of the Hour-circle to 12, and the Solstitial Colour to the Meridian; so shall the number of hours contained between the Brass-Meridian and the Horizon, upon the Tropic of Cancer, be half the length of the Longest day at London, namely, 123 deg. which in time is 8 ho. and 13 minutes for the Semidecimal Arch of the Artificial day at London. At this intersection of the Tropic of Cancer with the Horizon, make a small mark upon the Tropic, and move the Globe Westward, till the Equinoxial Circle hath passed 7 deg. 30 min. or one half hour of time from the Meridian; and than observe where the Tropic of Cancer intersects with the Horizon again, and there upon the Tropic make another small mark. Than bring the Solstitial Colour back again to the Meridian, and there fix the Ball of the Globe; and move the Brass-Meridian in the Horizon upwards, till your second prick which you made upon the Tropic do just touch the Horizon; than will the Globe stand at that Latitude in which the Longest day will be an hour longer than the Longest day at London is; and that Latitude will be found be to be 56 deg. 27 min. But if you would know in what Latitude the Longest day shall be one hour shorter than at London, than having made a mark upon the Tropic at the termination of the Longest day there, move the Globe Eastward, till 7 deg. 30 min. have passed the Meridian; and where the Tropic and Horizon intersect, make there a mark upon the Tropic, and bring the Solstitial Colour to the Meridian, and fix the body of the Globe there: than depress the Meridian in the Horizon, till your second mark do touch the Horizon; than will the Globe rest at that Latitude wherein the Longest day will be one hour shorter than the Longest day at London is; which Latitude will be found to be 35 deg. 58 min. PROB. XXI. How much must the Sun's Declination increase or decrease, to make the Day Artificial one hour longer or shorter than it was at the time proposed. SUppose in Latitude 51 deg. 30 min. the Sun to have 10 degrees of North-Declination, and I would know how much the Declination must increase Northward, to make the day one hour longer than it is. Practice. Elevate the Globe to the Latitude of London, 51 deg. 30 min. and bring the Equinoxial Colour to the Meridian, and bring the Sun's place, or the parallel of 10 deg. his Declination, to the Horizon; and upon the Horizon make a small mark, to which bring the Equinoxial Colour, and at this intersection make a mark upon the Colour. Than (if the days lengthen, move the Globe Eastward, or Westward if they shorten) till 7 deg. 30 min. of the Equinoxial have passed the Meridian; and than, again, where the former Colour intersects with the Horizon, make another mark, upon the Colour. This done, bring the Colour to the Meridian again, and see what number of degrees of the Meridian (or degrees of the Colour itself) are contained between the two pricks; which you shall find to be 4 deg. and 40 min. And so much must the Declination increase Northward, to make the day lengthen one hour, at that time of the year. PROB. XXII. To find what number of days are contained between the days of Lengthening or shortening one hour, at any time of the year. LEt the time be as in the last Example, where the Sun hath 10 deg. of North-declination; which will be about the 5 of April. Practice. Having made two pricks in the Colour, as in the last Example, Bring the first prick to the Horizon, where you shall find it to stand against the 5 of April. Than move the Globe till the other prick touch the Horizon; which it will do about the 19 of April, at which time the day will be one hour longer than it was upon the 5 of April: between which two days, there are contained 13 complete days, and parts of both the two other days. So that you may conclude, that in 14 days time (at that season of the year) the days do lengthen one hour, and shorten as much when the Sun is in opposite signs. PROB. XXIII. To find the length of the Longest or Shortest day in any Latitude. Definition. THe Longest Artificial day in any Latitude, is made by the Sun 's passing thorough the Tropic of Cancer in all places that have North-Latitude, or thorough the Tropic of Capricorn to all that have South-Latitude. Practice. To find the length of the Longest day at London, in Latitude 51 deg. 30 min. elevate the Globe thereto, and bring the Solstitial Colour to the Meridian, and the Hour-Index to 12. Than, count the number of hours upon the Tropic of Cancer, that are contained thereupon between the intersection thereof with the Horizon, on the East-side, and its intersection on the Westside; which you will find to be 16 hours, and almost half an hour, viz. 26 min. for the length of the Longest day. And the number of hours contained between the East and West intersections of the Tropic of Capricorn and the Horizon, is the length of the Shortest day, which at London will be found to be 7 hours, and somewhat above half an hour, viz. 34 min. Half the length of the Day is the time of the Sun's Setting, and half the length of the Night is the time of the Sun's Rising. PROB. XXIV. Of the Reason of the Inequality of Days Natural, and Days Civil. Definition. A Natural day is that space of time in which the Sun moveth from the Meridian of some one place or Country, to the same Meridian again. These days are not always of an equal length, but are longer at some times of the year than at other times, but at all times they are longer than the Civil day is. A Civil day being that space of time containing just 24 hours, reckoned from 12 a clock one day, unto 12 of the clock the next day; in which time the Equinoxial maketh one entire Revolution about the Axis of the world. The difference between these two sorts of Days is but small; and there is a double cause for this small inequality. 1. Because the Sun 's apparent motion differs from his true motion, as being slower when he is in his Apogaeum, than in his Perigaeum; he moving scarce 58 min. in a day when he is in the one, and above 61 min. when he is in the other; and so increaseth in Right Ascension. 2. The difference of Right Ascensions in several eequal parts of the Ecliptic: for when the Sun is near either of the Tropics, the Ascensional Differences are greater than when the Sun is about the Equinoxial: for about Aries or Libra, the Right Ascension of 10 deg. is but 9 deg. 10 min. whereas the Right Ascension of 10 deg. of Cancer or Capricorn is 10 deg. and 53 min. Now this difference being so small, in one days time it is not perceptible by the Globe; wherefore if you would find this difference, it would be requisite to take some number of days, as 10, 20, or 30, and in them it will be apparent. And to effect it by the Globe, do thus. Practice. First, find the place of the Sun, both at the beginning and ending of those days you would compute the difference of 2. Find the Right Ascensions answerable to each place in the Ecliptic, as also the difference of Right Ascensions answerable to the motion of the Sun in each respective number of days. 3. Compare the difference of the Right Ascensions together, by substracting the lesser from the greater; and the difference converted into time, shall be the number of minutes that the one number of days exceedeth the other. EXAMPLE. Let it be required to find what difference there is in the length of the first 20 days of December, and the first 20 days of March. d.m. d. m. d. m. d.m. Sun's place Decemb. 1 ♐. 20.07 It's Right Ascens. is 259.01 22. 24 21 ♑ 10.33 281.25 4.02 Differ. of Ascens. March 1 ♓ 21.24 352.06 18. 22 21 ♈ 11.11 10.28 Here by this Synopsis you may perceive that the difference of these 20 days taken at several times of the year is 4 deg. 2 min. which converted into time is 16 minutes, that is, a quarter of an hour and one minute; and so much longer are the first 20 days in December, than the first 20 days of March. And by this means you shall find that the Month of January is longer than the Month of June (both Months consisting of 31 days) by 1 deg. 20 min. which is only 4 1/● min. in time. d.m. d.m. d.m. d.m. Sun's place Januar. 1 ♑ 21.47 It's Right Ascens. is 293.29 31. 24 31 ♒ 22.17 324.53 01.20 Differ. of Ascens. June 1 ♊ 20.40 79.43 30. 04 31 ♋ 18.16 109.47 Thus the Month of January is longer than the Month of June by 4 minutes and ●/● of a minute of time. ¶ All the Problems may be performed upon either Globe; the Horizon, Meridian, Quadrant of Altitude, Hour-circle, and most of the Circles upon the Globes themselves, being in both Globes the same. But it is most proper to use the Terrestrial Globe for Geographical and Nautical Problems; and the Celestial Globe for such as concern Astronomy: and these following are chief such, and therefore best to be wrought by the Celestial Globe. PROB. XXV. To find the Longitude and Latitude of any Star. Definition. THe Longitude of a Star is an Arch of the Ecliptic, contained between the beginning of Aries, and the intersection of an arch of a great Circle, which passeth through both the Poles of the Ecliptic, and also through the body of that Star. The Latitude of a Star is that part of an arch of a great Circle which passeth through both the Poles of the Ecliptic, and through the body of the Star, and is contained between the Ecliptick-line and that Star. Practice. For the Longitude, screw the Quadrant of Altitude over that Pole of the Ecliptic which is nearest to the Star whose Longitude you seek. Than laying the Quadrant just over the centre of the Star, Note, that the Poles of the Ecliptic are distant from the Poles of the world 23½ deg. on either side. look what degrees of the Ecliptic, (counting them from the beginning of Aries) and those degrees are the degrees of the Star's Longitude. So the Quadrant of Altitude skrewed over the North-pole of the Ecliptic, and laid upon the bright Star Capella, the Quadrant shall cut 77 deg. 16 min. of the Ecliptick-Circle, counted from the beginning of Aries; and that is that Star's Longitude. For the Latitude, the Quadrant fitted as before, and laid over the centre of Capella, the Star shall out 22 deg. 50 min. of the Quadrant of Altitude, and such is the Latitude of that Star, North, for that it lies on the North-side of the Ecliptick-Line. PROB. XXV. To find the Right Ascension and Declination of a Star. Definition THe Right Ascension of a Star is that Arch of the Equinoxial which is contained between the beginning of Aries and that point which comes to the Meridian with that Star. The Declination of a Star is an Arch of the Meridian contained between the Equinoxial and any Star. Practice. For the Right Ascension, (the Globe being rectified) bring Capella to the Meridian, and than shall you find 73 deg. 7 min. of the Equinoxial contained between the beginning of Aries and the Meridian; and that is the Right Ascension of Capella. For the Declination, bring Capella to the Meridian, so shall you find 45 deg. 37 min. of the Meridian contained between the Equinoxial and Capella; and that is the Declination of that Star▪ And in this manner you may find the Longitude, Latitude, Right Ascension, and Declination of any other Star upon the Celestial Globe: As in this following Table of the principal Fixed Stars of the first Magnitude, you shall find Stars names. Longit Latit. R. Asc. Decl. d. m. d. m. d. m. d. m. Arcturus 119. 39 ●1. 2 B 210. 13 20. 58 B Lucida Lyra 280. 43 61. 47 B 276. 27 38. 30 B Algol 51. 37 22. 22 B 41. 46 39 39 B Capella 77. 16 22 50 B 73. 7 45. 37 B Aldebaran 65. 12 5. 31 A 64. 17 15. 48 B Regulus 145. 17 0. 26 B 147. 43 13. 33 B Cauda Leonis 167. 3 12. 18 B 173. 4 16. 25 B Spica Virgin. 199. 16 1. 59 A 196. 56 9 31 A Antares 245. 13 4. 27 A 242. 23 25. 37 A Fomahant 329. 1● 21. 00 A 339. 46 31. 17 A Regel 72. 1● ●1. 11 A 74. 44 8. 37 A Syrius 99 3● 39 3● A 97. 42 16. 14 A protion 111. 18 15 57 A 110. 34 6. 3 B PROB. XXVI. To find the distance of two Stars. Practice. 1. IF the two Stars be both of them under the same Meridian, Bring them under the General (or Brass) Meridian, and see what degrees of the Meridian are contained between them, for that is their distance. 2. If they lie not under the same Meridian, but have the same declination, or lie in the same Parallel, Bring one of them to the Meridian, and see what degrees of the Equinoctial are cut thereby: than bring the other Star to the Meridian, and count what degrees of the Equinoctial are contained between the Meridian and the degrees before found, for that is the distance of those two Stars. 3. If the two Stars do neither lie under the same Meridian, nor in the same Parallel, Than lay the Quadrant of Altitude (it being lose) to both the Stars, and the degrees of the Quadrant contained between the two Stars is their distance. And if the Quadrant be too short, you may use the Circle of Position, or take their distance with a pair of Calope-Compasses, and measure their distance upon the Equinoctial, or any other great Circle. Thus The Right Shoulder of Auriga, and the Right Shoulder of Orion, being under the same Meridian, their distance will be found to be 37 deg. 38 min. Also Arcturus and the Lion's Neck, being near in the same Parallel, their distance will be found to be 57 degrees. Likewise Lyra the Harp, and Marchad in the Wing of Pegasus, will be found to be distant 63 degrees. PROB. XXVII. To know what Stars will be upon the Meridian at any hour of the Night. Practice. THe Sun being in 29 deg. of Taurus what Stars will be upon the Meridian at 10 a clock and 12 min. at Night, bring 29 deg. of Scorpio (which is the opposite Sign to Taurus) to the Meridian, and set the Index of the Hour-Circle to 12. than turn the Globe about Westward till the Index point at 12 min. after 10 a clock, and there hold the Globe; and all those Stars which lie under the Brass Meridian are than upon the Meridian, of which Arcturus is the Chief. PROB. XXVIII. To know what day in the year any Star shall be upon the Meridian at 12 a clock at Night. Practice. BRing the Star to the Meridian, and mark what degree of the Ecliptic is just under the Meridian at the same time: Than find that degree of the Ecliptic in the Horizon, and note what day of the year standeth against it, for that day of the year will that Star be upon the South-part of the Meridian at 12 at night: and when the Sun is in the opposite Point of the Ecliptic, the same Star will be upon the North-part of the Meridian at 12 at noon. PROB. XXIX. The Sun's Place, and the Altitude of a known Star given, To find the hour of the Night. THe Sun being in 21 deg. of Capricorn, the Altitude of the Great Dog 14 deg. I demand the hour of the Night. Practice. The Globe Rectified, etc. bring 21 deg. of Capricorn to the Meridian, and the Index to 12 a clock. Than move the Globe and Quadrant of Altitude so together, that the Great Dog meet with 14 deg. of the Quadrant; and than shall the Index point at 8 of the clock and 22 minutes; which is the true hour of the Night. And thus deg. d. d.m. When the Sun is in 20 ♐ and the Altitude of The Bull's eye 39 The ho will be 7.12 20 ♏ The Bulls eye 30 9 2 5 ♊ Arcturus 50 11.3 PROB. XXX. The Altitude of Aldebaran (or any other Star) being given in a known Latitude; To find the Star's Azimuth. Practice. THe Quadrant of Altitude being fixed in the Zenith, move it and the Globe, till the degrees of Altitude given do meet with the centre of the Star; than shall the end of the Quadrant of Altitude show you upon the Horizon the Azimuth in which the Star than is. And thus, if you bring the Quadrant of Altitude on the East-side of the Globe, moving it and the Globe both, till the centre of Aldebaran do meet just with 42 deg. of the Quadrant, you shall than find the Quadrant of Altl●ude to rest at 33 deg. of the Horizon, counted from the East; or at 57 deg. if you count them from the South: and that is the Azimuth of Aldebaran when he hath 42 deg. of Altitude; and that is near the S. E. by E. point of the Compass. The Latitude of the Place (51 deg. 30 min.) and the Declination of a Star (suppose the Bull's eye, Aldebaran) given: To find PROB. XXXI. It's Right Ascension. Practice. THe Globe Rectified to the Latitude, etc. bring Aldebaran to the Meridian: than count how many degrees of the Equinoxial are contained between the Meridian and the beginning of Aries; which will be 64 deg. 17 min. and that is the Right Ascension of that Star; which in time (by allowing 15 deg. for an hour, and 1 deg for 4 min. of time) is 4 hours 16 min. i● s Right Ascension in time. And in the same manner may you find d. m. h. m. The Right Ascension of Arcturus to be 210.13 in time 14. 1 Syrius 97.42 6.30 Algol 39.39 2.38 PROB. XXXII. It's Ascensional Difference. Practice. BRing the Star to the Meridian, and the Hour-Index to 12. Than bring the Star either to the East or Westside of the Horizon, and there you shall find 1 hour and 27 min. contained between the Index and 6 a clock: and such is the Ascensional Difference of Aldebaran. In like manner, you may find d. m. h. m. The Ascensional Difference of Arcturus to be 28 40 or in time 1.55 Syrius 21.28 1.26 Algol — — Algol his Declination being more than the Compliment of the Latitude, never rises nor sets, but is always above the Horizon. PROB. XXXIII. It's Amplitude. Practice. BRing Aldebaran to the Horizon on either side of the Globe, and you shall find it to touch the Horizon at 25 deg. 56 min. from the East or West Northward; which is the Amplitude of the Bull's eye's rising or setting. And according to the points of the Compass it riseth E.N.E. 2 deg. 26 min. Northerly; and sets W.N.W. 2 deg. 26 min. Northerly. And thus may you find d. m. That Arcturus Riseth from the East or West Northward 35. 6 Syrius Southward 26.41 Algol Never rises or sets PROB. XXXIV. The Semidiurnal Arch, and the time that Aldebaran (or any other Star) continues above the Horizon. Practice. BRing Aldebaran to the Meridian, and set the Hour-circle to 12. Than turn the Globe Westward, till Aldebaran touch the Horizon: than shall the Hour-Index point at 7 ho. 27 min. And so long time is Aldebaran above the Horizon, before he comes to the Meridian; and continues so many hours and minutes above the Horizon, after he hath passed the Meridian, and sets in the West. And those 7 ho. and 27 min. is the Semidiurnal Arch of that Star; which doubled, is 14 ho. 54 min. And so long doth that Star continued above the Horizon after the time of his rising. And in this manner you may find d. m. h. m. The Semidiurnal Arch of Arcturus to be 7.55 And his continuance above the Horizon 15.50 Syrius 4.34 9 8 Algol 12.00 24.00 PROB. XXXV. At what hour (any time of the year) Aldebaran comes to the Meridian. Practice. LEt the time be the first of January, at which time the Sun is in 22 deg. of Capricorn. Bring 22 deg. of Capricorn to the Meridian, and set the Hour-Index to 12. Than turn the Globe about till Aldebaran be under the Meridian, and than you shall find the Index to point at 42 min. after 8 of the clock, at which time Aldebaran will be upon the Meridian that night. In like manner you may find, that h m. Upon October 28 Arcturus will be upon the Meridian at 11.10 January 21 Syrius 9 33 January 1 Algol 7.12 PROB. XXXVI. At what hour (at any time of the year) Aldebaran (or any other Star) riseth or setteth. Practice. LEt the time be January 1. By the last before-going, you found that Aldebaran came to the Meridian at 8 ho. 42 min. And by the last but one, you found his Semidiurnal Arch to be 7 ho. 27 min. This being taken from 8 ho. 42 min. the time of his being South, leaveth 1 hour 15 minutes, the time of its Rising: so that upon the first of January Aldebaran did rise at 15 min. after 1 in the afternoon. Again, if you add his Semidiurnal Arch 7 hours 27 min. to the time of its being South 8 hours 42 min. the sum will be 16 hours 9 min. from which take 12 hours, and the remainder will be 4 hours 9 min. So that Aldebaran will set at 9 min. after 4 of the Clock the next morning. And in like manner you may find, that h. m. h. m. Upon October 28 Arcturus Rises at 3. 8 Sets at 7. 12 January 21 Syrius 5. 3 2. 3 January 1 Algol — — Algol never Rises nor Sets. PROB. XXXVII. At what Horary distance from the Meridian Aldebaran will be due East or West: And what Altitude he shall than have. Practice. BRing Aldebaran to the Meridian, and the Hour-Index to 12. and the Quadrant of Altitude to the West-point of the Horizon: than turn the Globe Eastward, till the Centre of the Star be just under the edge of the Quadrant; than shall the Index point at 5 hours and 40 min. So that when Aldebaran is due East or West, he will be 5 hours 40 min. of time short of, or gone beyond the Meridian. And when the Centre of Aldebaran is just under the Edge of the Quadrant of Altitude, you shall find it to touch 20 deg. 21 min. And such is the Altitude of Aldebaran when he is upon the East or West Azimuth. In like manner may you find, that h. m. d. m. Arcturus will be upon the East or West Azimuth, when he is distant from the Meridian 4. 49 and his Altitud will be 27. 13 Syrius 5. 04 20. 56 Algol 3. 15 54. 37 PROB. XXXVIII. What Altitude and Azimuth Aldebaran (or any other Star) shall have when six hours distant from the Meridian. Practice. BRing Aldebaran to the Meridian, and the Index to 12; Than turn the Globe about till the Index point at 6: than lay the Quadrant of Altitude over the Centre of the Star, and you shall find it to lie under 12 deg. 18 min. of the Quadrant: and such is the Altitude of Aldebaran. At the same time look what degrees of the Horizon are cut by the Quadrant of Altitude, and you shall find 8 degrees between it and the East or West Points Northwards. And such is the Azimuth of Aldebaran. And according to this Rule you shall find, that when Altit. Azim. d. m. d. m. Arcturus is 6 hours distant from the Merid. his 16. 15 76. 36 from the North. Algol 12. 38 79. 43 Syrius is never 6 hours distant from the Meridian, nor any other Star that hath South-declination. PROB. XXXIX. To find what Altitude and Azimuth any Star hath when he is at any horary distance from the Meridian. Practice. THis is no other than the last. For having brought the Star to the Meridian, and the Index to 12, move the Globe till it come to the designed hour. Than the Quadrant of Altitude being laid over the Star, shall at the same time show you both the Altitude and Azimuth thereof as before. This needeth no Example. PROB. XL. Having the Azimuth of a Star, to find at what horary distance that Star is from the Meridian, and what Altitude that Star than hath. Practice. BRing the Star to the Meridian, the Index to 12, and the Quadrant of Altitude to the Given Azimuth; than turn the Globe about till the Centre of the Star lie just under the Quadrant of Altitude; the Index at that time shall give the horary distance, and Quadrant the Altitude of the Star. Example: Aldebaran being seen upon 80 degrees of Azimuth from the Northwestward, that is, near upon the W. by N. Point of the Compass; the Star brought to the Meridian, and the Quadrant of Altitude to 80 degrees, and the Hour-Index to 12. If you bring the Star to the Quadrant of Altitude, you shall find the Index to point at 6 hours, which is the Stars horary distance from the Meridian. And the Quadrant of Altitude will show 12 deg. 18 min. the Altitude of Aldebaran at that time. PROB. XLI. Concerning the Poetical Rising and Setting of the fixed Stars viz. The Cosmical Acronical Heliacal Rising and Setting. And how to find either of them by the Globes. 1. Of the Cosmical Rising and Setting. Definition. A Star is said to Rise Cosmically, wh●● it Riseth with the Sun, or with that degree of the Ecliptic in which the Sun than is And the Cosmical Setting is, when a Star sets in the Morning, or goeth down under the West- Horizon in the Morning at such time as the Sun is Rising in the East. Practice. Upon the 27 of May, the Sun the● being in 17 degrees of Gemini, I would know what Stars do than Rise and Set Cosmically. Rectify the Globe to your Latitude, and bring the 11 degree of Aquarius to the East-part of the Horizon: than look what Stars are about the edge of the Eastern Semicircle of the Horizon, for a●● those Stars do that day Rise Cosmically. An● those Stars which touch or are near the Rim o● the West Semicircle of the Horizon do Set at that time Cosmically. So shall you find May 27 Aldebaran or the Bull's eye, with divers other smaller Stars, Rising, and The right leg of Serpentarius and several other smaller Stars Setting Cosmically. 2. Of the Acronical Rising and Setting of the Stars. Definition. A Star is said to Rise Acronically when it Riseth in the East- Horizon, at such time as the Sun goes down or Sets in the West- Horizon; and the Acronical Setting is when a Star goeth down under the Horizon with the Sun. Practice. Upon the 18 of October, in the Latitude of 51 deg. 30 min. the Sun than being in 5 deg. of Scorpio, I would know what Stars do on that day Rise and Set Acronically. Rectify the Globe to the Latitude, bringing the place of the Sun 5 deg. of Scorpio to the West-part of the Horizon, than shall all those Stars which you see on the Verge of the East-side of the Horizon, be Rising Acronically. And all those that are about the Verge of the Western part of the Horizon are than Setting Acronically. And so upon the forementioned day, you shall find A Star in the Whale's Tail, and several other smaller Stars Rising, and The Tail of the Lion, the South Ballance, and several other smaller Stars Setting Acronically. 3. Of the Heliacal Rising and Setting of the Stars. Definition. A Star is said to Rise Heliacally, when having been some time combust, (or hid under the Sunbeams) gins now to appear, it being at greater distance from the Sun. And a Star is said to Set Heliacally, which hath some small time before been seen, but now, by the near approach of the Sun, becomes cumbust and hidden under his Beams. Now to know when a Star gins to become cumbust, & when to be freed from his cumbustment of the Sunbeams, no certain Rule can be given: for the Magnitude of the Star, the difference of the Climate, the cloudiness or serenity of the Air may much altar. But the Opinion of the Ancient Astronomers was, that deg. A Star of the 1 Magnitude may be seen when the Sun is but 12 below the Horizon. 2 13 3 14 4 15 5 16 6 17 And those which are only Nebulous, cannot be seen till the Sun be 18 deg. under the Horizon. Practice. Rectify the Globe to the Latitude, and the Quadrant of Altitude in the Zenith; than bring the given Star (suppose Regulus, or the Lion's Heart) to the East-side of the Horizon, and the Quadrant of Altitude to the Westside; than Regulus being a Star of the First Magnitude (by the former Rule of the Ancients) may be seen when the Sun is but 12 deg. below the Horizon: wherefore see what degree of the Ecliptic do●h cut the Quadrant of Altitude in 12 degrees, which you shall find to be 9 deg. of Pisces, the opposite degree to which is 9 deg. of Virgo; to which Sign and degree when the Sun cometh, (which will be about the 23 of August,) than will Regulus, the Lion's heart, Rise Heliacally: Than for the Heliacal Setting, bring the Star to the Westside of the Horizon, and turn the Quadrant of Altitude to the East-side, and see what degree of the Ecliptic is elevated upon the Quadrant, as the Magnitude of the Star you deal with doth require. For when the Sun comes to the opposite degree of the Ecliptic, that Star shall Set Heliacally. So shall Rise Heliacally upon and Set Heliacally upon The Pleyades June 4 April 20 Aldebaran June 26 April 22 Arcturus Sept. 26 Novem. 19 Concerning the Cosmical and Acronical Rising of the Stars, see the following Table. A TABLE, Showing the time of the Year when 50 eminent Stars do Rise both Cosmically and Acronically. Calculated for the Year 1670, and may serve for many Years past, and to come, without any considerable Error. Names of the STARS. Their Cosmical Rising. Their Acronical Rising. Marchab. Pegasi Jan. 1 March 5 Right Shoulder of Aquarius 6 Feb. 1● Extreme Star in the Wing of Pegasus 28 15 Following Tail of the Goat Feb. 5 Jan. 2● Bright Star in the Ramshead March 2 April 20 The following Horn of the Ram 5 19 The former Horn of the Ram April 10 13 North-Tail of the Whale 12 March 15 The brightest of the Pleyades 22 May 10 The Star in the Knot in the Net of ♓ May 1 April 29 North-Horn of the Bull 15 June 7 North-Eye of the Bull 20 May 1● Belly of the Whale 23 March 16 Bulls Eye, Aldebaran 27 May 11 South-Horn of the Bull June 5 June 1● Lower Head of Gemini, Hercules 21 July 20 Bright Foot of Gemini 25 June 5 Middle Star in Orion's Girdle July 2 May 4 North- Asellus 12 July 25 Presepe 14 19 South- Asellus 17 1● Lesser Dog, protion July 18 June 6 Great Dog, Palilicium 30 May 3 Lions Heart August 8 August 9 Lions Back 10 Octob. 19 Hydra's Heart 20 June 16 Lions Tail 22 Octob. 16 Hares Thigh Sept. 9 April 14 Viademiatrix 13 Novem. 8 Arcturus 15 Decem. 13 Virgins Girdle 19 Octob. 18 Bright Star of the Crown 28 Jan. 7 Virgins Spike Octob. 3 Sept. 24 Right Shoulder of Hercules 6 Jan. 9 Left Shoulder of Hercules 10 21 Head of Hercules 21 8 North Ballance 21 Novem. 17 South Ballance 23 Octob. 24 Swans Bill 31 Feb. 16 Right Shoulder of Ophiucus Novem. 5 Jan. 4 Left Knee of Ophiucus 6 Decemb. 6 Lower Wing of the Swan 4 March 11 Vultures Tail 11 Jan. 27 Right Knee of Ophiucus 16 Decemb. 6 Scorpions Heart 22 Novemb. 4 The Eagle 25 Decem. 31 Pegasus Scheat Decem. 11 March 24 Adromeda's Girdle 22 April 29 Adromeda's Head 25 6 Upper Horn of the Goat 25 Jan. 14 TRIGONOMETRICAL PROBLEMS. INTRODUCTION. SPherical Triangles are best represented by the Circles upon, and those appendent to, the Globes; with whose nature, affections, and uses, the Practitioner aught throughly to be acquainted, before he enter upon the Solution of such Triangles by the Canons or Tables, either of Natural or Artificial Sins and Tangents, or by other Instruments, as Projections, Planispheres, or the like; the Globe being the Original from whence all these do proceed. For by the true understanding of the Uses of the Globes, it will not be difficult to Project the Sphere in Plano upon any Circle, as I have in some measure taught how to do upon the Plain of the Meridian, in my Geometrical Exercises, and shall here farther illustrate by the Circles upon the Globe itself; upon which the Sides and Angles of all Spherical Triangles, are most Naturally represented, and most Expeditiously measured. But before I come to the Practice hereof, give me leave to premise such general and necessary Definitions and Affections, as do in any case belong to Spherical Triangles. And (1) Of such Lines or (rather) Arches of Circles, of which Spherical Triangles are framed and measured. (2) Of the Kinds, Parts and Affections of such Triangles so constituted: how the things given or required in them are represented upon the Globe; with the Variety of Questions, that the solution of every Spherical Triangle will afford. With (3) The solution of several Problems for Practice. Of the Definitions and Affections of Triangles. 1. ALl Triangles do consist of six parts, viz. of three Sides, and as many Angles. 2. The three sides of a Spherical Triangle do consist of three Arches of Great Circles of the Sphere or Globe, each A b being lesle than a Semicircle, or 180 deg. 3. A Great Circle of the Globe, is that which divideth the Globe into two equal parts; such are the Meridian's, Equinoctial, Ecliptic, the Colours, etc. 4. The three Angles of a Spherical Triangle, are measured by three Arches of great Circles described upon the three angular Points of the Triangle; and are either Right, Acute, or Obtuse. 5. A Right-angled Spherical Triangle, is that which hath either One, or more Right Angles, which contains 90 deg. An Ac e-angled Spherical Triangle is that whose three Angles be all of them acute; that is, each of them lesle than 90 degrees. 6. An Obtuse Angle is that Spherical Triangle whose Angles are all obtuse, each exceeding 90 deg. or else mixed, some Obtuse and some Acute. 7. The Compliment of a Side or Angle to a Quadrant or 90 deg. is so much as that Side or Angle wants of 90 degrees. 8. The Compliment of a Side or Angle to a Semicircle or 180 deg. is so much as that Side or Angle wants of 180 degrees. 9 All Spherical Triangles which have not One or more Right Angles, are called Obliqne angled Spherical Triangles. Hence it followeth, That 1. If two Great Circles of the Sphere shall pass by each others Poles, those two Circles shall intersect each other at Right Angles. But 2. If two Great Circles of the Sphere do intersect each other, and do not pass through each others Poles, those two Circles shall intersect each at other Obliqne Angles. Thus The General Meridian and the Horizon, The Colours and the Eq inoctial The Azimuth or Vertical Circles & the Horizon, The Circle of Latitude and the Ecliptic, do intersect each other at Right Angles. But The Colours and the Horizon The Vertical Circles and Equinoctial The Circles of Longitude Horizon, and the Equinoctial do intersect each other at Obliqne Angles. And the Angles made by these obliqne intersections, are on the one side of the intersection Acute, and on the other side Obtuse. So 3. Two Arches of Great Circles intersecting each other, shall make on both sides the intersection, either two Right Angles or two Obliqne Angles, the one Acute, the other Obtuse; of which, both of them being taken together, shall be equal to two Right Angles, or 180 degrees. Thus, in the Latitude of 51 deg. 30 min. d. m. The Meridian intersecting with the Horizon doth make an Angle of 90.00 on the one side of the intersection, The Ecliptic 62.00 The Equinoctial 38.30 d. m. The Meridian and an Angle of 90.00 on the other side of the intersection. The Ecliptic 118.00 The Equinoctial 141.00 And both the Angles on either side together, are equal to two Right Angles, or 180 degrees. But here note, That these Angles do Vary in every Latitude, and are not always the same. 4. If a Spherical Triangle have three Right Angles, the three sides of that Triangle shall be all Quadrants, each containing 90 deg. So (in a Parallel Sphere) The Brass Meridian cutting the Horizon in the North and South-points of the Horizon at Right Angles, and the Equinoctial Colour in the East and West-points of the Horizon at Right Angles, and the Meridian and Colour intersecting each other in the Poles of the World, at Right Angles also: do constitute a Spherical Triangle with three Right Angles; and the three sides of this Triangle, (for this Reason) shall be all of them Quadrants: for upon the Meridian, between the Equinoctial (which is also the Horizon in a Parallel Sphere) and the Pole, is contained 90 degrees: also upon the Colour between the Equinoctial and the Pole is also 90 degrees, and between the East and North or South-points of the Horizon is also contained 90 degrees: So that the three Angles being Right Angles, the three Sides are also three Quadrants. 5. If a Spherical Triangle hath two Right Angles, the Sides opposite to those Angles shall be Quadrants; and the third side shall be the measure of the other Angle. Such Triangles as these do seldom come in Practice, for few or no Questions can arise out of them; but of Right angled Spherical Triangles, those that have One Right, and Two Acute Angles, most Questions are resolvable by. 6. An Acute angled Spherical Triangle, hath all its sides lesle than Quadrants, or 90 degrees. 7. An Obtuse angled Triangle, having all its Angles Obtuse, hath all its Sides more than Quadrants: If mixed, the Side or Sides opposite to the Obtuse Angle or Angles shall be Greater, and the Side or Sides opposite to the Acute Angle or Angles shall be Lesle than Quadrants. 8. The Sides of Spherical Triangles, are of the same affeciion or kind with their opposite Angles. 9 In Right-angled Spherical Triangles, the Side which is opposite to the Right Angle, I call the Hypotenuse; and the other two Sides which contain the Right Angle, I call the Sides or Legs. 10. In Obliqne angled Spherical Triangles, I call the Sides Sides, and the Angel's Angles, without any other distinction. 11. In the Solution of Right-angled Spherical Triangles, there are usually 16 Cases, which will all be reduced to Five; for that by the Globe three things are at once found. And in Obliqne Angled Spherical Triangles, there are usual 12 Cases, which by the Globe will be reduced to 6; the Globe answering two at one Position. Thus much of the Definitions and Affections of Spherical Triangles. Of the Solution of Spherical Triangles upon the Globe. IN the following Scheme or Figure of the Sphere or Globe, it being in an Obliqne Position, viz. Elevated to the Latitude of London, 51 deg. 30 min. you have upon the Superficies thereof divers Spherical Lines and Circles, by the intersections whereof, are constituted divers Spherical Triangles; some whereof are Right-angled, and others Oblique-angled: Of which I shall make use only of Two; one for the Five Cases of Right-angled Triangles, the other for the Six Cases of Oblique-angled Triangles. The Triangle which I shall make use of for the Five Cases of Right-angled Triangles, shall be the Triangle P. O. ☉, Right-angled at O. In which Right angled Spherical Triangle P. O. ☉. The Side P. ☉. being an Arch of the Brass Meridian, we suppose to be the Latitude of London, 51 deg. 30 min. ☉. O. being an Arch of the Horizon, we suppose to be the Amplitude of the Sun's Rising or Setting, from the North-part of the Meridian 56 deg. 40 min. P. ☉. being the Arch of another Meridian, or Hour-Circle, (which the Equinoctial Colour will best supply or represent) we suppose to be the Sun's distance from the Pole, or the Compliment of his Declination, 70 deg. 00 min. The Angle. ☉. P.O. is the hour from midnight, whose measure is to be reckoned upon the Equinoctial, between the Equinoctial Colour and the Brass Meridian, and will be 62 deg. 45 min. or in time 4 hours 11 min. P. ☉. O. is the Angle of the Sun's Position at the time of the Question, whose measure is the Arch of a Great Circle, comprehended between the Compliment of the side ☉. O. on the Horizon, and the Compliment of the Side ☉. P. on the Equinoctial Colour; both those Sides being continued to Quadrants, on the contrary Side of the Brass Meridian, and will contain 56 deg. 39 min. P.O. ☉. is the Right Angle, whose measure is the deg. of the Quadrant of Altitude contained between the East or West-points of the Horizon and the Zenith, which is 90 deg. The several parts of the Triangle being thus declared, let us now proceed to the several Cases which arise out of this, and every Right-angled Spherical Triangle. Of Right-angled Spherical Triangles. CASE I. The two Sides (or Legs) P. O. 51 deg. 30 min. and ☉. O. 56 deg. 40 min. being given to find the Hypotenuse P. ☉. and the Angles ☉. P. O. and. P. ☉. O. SEeing that the Side P. O. is an Acrh of the Brass Meridian, and contains 51 deg. 30 min. count thereupon from the Pole, 51 deg. 30 min. and bring those degrees to the Horizon. Than the Side ☉. O. being an Arch of the Horizon, and contains 56 deg. 40 min. count upon the Horizon, from the Meridian, 56 deg. 40 min. and turn the Globe about till the Equinoctial Colour do touch those degrees of the Horizon, so shall you have the true Triangle perfectly described upon the Globe itself: For from the Pole to the Horizon, upon the Meridian, is equal to P. O. From the Horizon to the Pole, upon the Colour is equal to P. ☉. And from the Colour to the Meridian, upon the Horizon is equal to ☉. O. Now (1. P. O. being 51 deg. 30 min. and O. ☉. 56 deg. 40 min. the degrees of the Colour counted from the Pole to the Horizon, will be found 70 degrees; for the Hypotenuse P. ☉. which was required. (2.) For the Angle ☉. P. O. count the degrees of the Equinoctial which are contained between the Colour and the Brass Meridian, and you shall find them to be 62 deg. 45 min. or 4 hours 11 min. for the quantity of the Angle ☉. P. O. (3.) For the Angle P. ☉. O. count the Compliment of ☉. P. 20 degrees upon the Equinoctial Colour, on the other Side of the Meridian (so shall that Point be distant from the Point ☉. 90 deg.) Also, count the Compliment of the Side ☉. O. viz. 33 deg. 20 m. upon the Horizon, on the other Side of the Meridian, (which Point will be 90 deg. distant from the Point ☉. upon the Horizon) So shall the distance between these two Points, (measured by the Quadrant of Altitude, or by Compasses) contain, 56 deg. 39 min. for the Angle P. ☉. O. Or, This Angle P. ☉. O. may be otherwise found in the same manner as you found the Angle ☉. P. O. (but than you must altar the Position of the Globe) For, if you Elevate the Globe to 56 deg. 40 min. the Side ☉. O. and bring the Equinoctial Colour to 51 deg. 30 min. of the Horizon, which is the Side P. O. (for so have you turned the Triangle upsidedown) than will the degrees of the Equinoctial intercepted between the Equinoctial Colour and the Brass Meridian, be 56 deg. 39 min. as before, and equal to the Angle P. ☉. O. Or this is the usual Way taught by others, whereby by the Globe must be often Rectified upon every small occasion. But I embrace the first way, as being both Exact and Natural: but having declared both ways, I leave every one at liberty to use that which he best liketh. So by this one Data you have resolved three Problems of the Sphere: for you have found 3 1. The Side P. ☉. the Compliment of the Sun's Declination 70 degrees. 2. The Angle ☉. P. O. the hour from Midnight 4 hours 11 minutes. 3. The Angle P. ☉. O. the Angle of the Sun's position 56 degrees 39 minutes. CASE. II. The Hypotenuse P. ☉. 70 degrees, and the Side (or Leg) P. O. 51 deg. 30 min. being given; to find the other Leg, O. ☉. and the Angles ☉. P. O. and O. ☉. P. THis Case differeth little from the former: for P. O. being an Arch of the Meridian, and containing 51 deg. 30 min. the Meridian being set thereto in the Horizon, turn the Globe about till 70 deg. of the Equinoctial Colour do touch the Horizon. So shall you find the degrees of the Horizon intercepted between the Colour and the Meridian to be 56 deg. 40 min. for the Side ☉. O. The Angles at P. and ☉. are to be found in all respects by their measures, as in the former Case. And by this one Data, you have resolved three other Problems of the Sphere, for by it you have found 6. 1. ☉. O. The Sun's Amplitude from the North, 56 deg. 40 min. 2. ☉. P. O. The hour from midnight 62 d g. 45 min. or 4 h. 11 min. 3. P. ☉. O. The Angle of the Sun's position, 56 deg. 39 min. And if instead of Leg P. O. there had been given the Leg ☉. O. you might have than found [by bringing 70 degrees of the Equinoctial Colour, to meet with 56 deg. 40 min. of the Horizon] 4. P. O. The Latitude 51 deg. 30 min. 5. ☉. P. O. The hour from midnight. 6. P. ☉. O. The Angle of the Sun's position 56 deg. 39 min. CASE III. The Hypotenuse P. ☉. 70 deg. and the Angle P. 62 deg. 45 min. at being given, to find the two Legs P. O. and ☉. O. and the Angle P. ☉. O. COunt the Angle P. 62 deg. 45 min. upon the Equinoctial from the Colour, and bring that point to the Brass Meridian, and there keep the Globe fast in the Meridian; than move the Brass Meridian upwards or downwards in the Horizon, till 70. deg. of the Equinoctial Colour do justly touch the Horizon, than shall the Brass Meridian rest in the Horizon; at 51 deg. 30 min. for the Side P. O. and the degrees of the Horizon intercepted between the Equinoctial Colour and the Meridian will be 56 deg. 40 min. for the Side ☉. O. and the distance between 33 deg. 20 min. of the Horizon, and 20 degrees of the Equinoctial Colour shall be 56 deg. 30 min. for the Angle P. ☉. O. So that by this single Data there are also three other Problems resolved, viz. there is found 6. 1. The P. O. the Latitude 51 deg. 30 min. 2. The Side ☉. O. the Amplitude from the North 56 deg. 40 min. 3. The Angle P. ☉. O. the Angle of the Sun's position. And if instead of the Angle at P. the Angle at ☉. had been given you might than have found [by counting 20 deg. upon the Equinoctial Colour, and to that point apply 56 deg. 39 min. of the Quadrant of Altitude, (or a pair of Compasses opened to that distance.) than turn the Meridian in the Horizon, and the Globe about his Axis, till 70 deg. of the Colour and 00 deg. of the Quadrant of Altitude do touch the Horizon on either side of the Meridian: for than will the Meridian rest in the Horizon, at 51 deg. 30 min.] 4. The Side ☉. O. the Amplitude from the North 56 deg. 40 min. 5. The Side P. O. the Latitude 51 d. 30 m. 6. The Angl● ☉. P. O. the hour from midnight 62 deg. 45 min. CASE IV. One Leg ☉. O. 56 deg. 40 min. and one Angle P. ☉. O. 56 deg. 39 min. being given; to find the Side P. O. the Hypotenuse P. ☉. and the Angle ☉. P. O. COunt upon the Horizon from the Brass Meridian 56 deg. 40 min. the given Side (or Leg) ☉. O. also from the Brass Meridian, on the contrary side thereof, count 33 deg. 20 min. the Compliment of ☉. O. to which degrees apply 00 deg. of the Quadrant of Altitude. Than move the Brass Meridian up and down in the Horizon, till the Equinoctial Colour cuts 56 deg. 40 min. and 56 deg. 39 min. of the Quadrant of Altitude do intersect the Colour. So will the Brass Meridian rest in the Horizon at 51 deg. 30 min. For the Side P. O. the Horizon will cut the Equinoctial Colour in 70 deg. For the Hypotenuse P. ☉. and the degrees of the Equinoctial intercepted between the Equinoctial Colour and the Brass Meridian, will be 62 deg. 45 min. for the quantity of the Angle ☉. P. O. which was required. And by this Data, there are three Problems resolved: for there is found 1● 1. The Side P. O. the Latitude 51 d. 30 m. 2. The Angle ☉. P, O. the hour from Midnight 62 deg. 45 min. 3. The Side P. ☉. the Sun's distance from the Pole 79 deg. But if the Given Leg had been ☉. O. and the Given Angle ☉. P. O. [by counting the Angle P. upon the Equinoctial, and bringing that point to the Meridian, and moving the Meridian up and down in the Horizon, till the Equinoctial Colour did touch 56 deg. 40 min. of the Horizon] you should than have found 4. The Hypotenuse P. ☉. the Sun's distance from the Pole 70 deg. 5. The Side P. O. the Latitude 51 d. 30 m. 6. The Angle P. ☉. O. the Angle of the Sun's position 56 deg. 39 min. Also, if the Hypotenuse P. ☉. and the Angle ☉. P. O. had been given [by counting the Angle ☉. P. O. upon the Equinoctial, and bringing if to the Meridian, and moving the Meridian in the Horizon, till 70 deg. of the Equinoctial Colour did touch the Horizon] you might than found 7. The Side ☉. O. the Sun's Amplitude from the North 56 deg. 40 min. 8. The Side ☉. P. the Sun's distance from the Pole 70 deg. 9 The Angle P. ☉. O. the Angle of the Sun's position 56 deg. 39 min. And again, if the given Side (or Leg) had been P. O. and the Angle O. ☉. P. [by setting the Brass Meridian to 51 d. 30 m. in the Horizon, and turning the Body of the Globe about till the Equinoctial Colour and the Horizon do make an Angle of 56 deg. 39 min.] you might than found 10. The Hypotenuse P. ☉. the Sun's distance from the Pole 70 deg. 11. The Side ☉. O. the Sun's Amplitude from the North 56 deg. 40 min. 12. The Angle ☉. P. O. the hour from midnight 62 deg. 45 min. CASE V The two Acute Angles ☉. P. O. 62 deg. 45 min, and P. ☉. O. 56 deg. 39 min. being given, to find the Side P. O. P. ☉. and ☉. O. COunt the quantity of the Angle ☉. P. O. 62 deg. 45 min. upon the Equinoctial, from the Equinoctial Colour, bringing those degrees to the Meridian, and there fix the Globe; than move the Meridian upward or downward in the Horizon, till the intersection of the Equinoctial with the Horizon do make an Angle of 56 deg. 39 min. So shall the Side P. O. be found to be 51 deg. 30 min. the Leg ☉. P. 56. deg. 40 min. and the Hypotenuse P. ☉. 70 deg. And from this Data, three Problems are resolved: for you have found 3. 1. The Side P. O. the Latitude 51 d. 30 m. 2. The Side ☉. P. the Amplitude from the North 56 deg. 40 min. 3. The Hypotenuse P. ☉. the Sun's distance from the Pole 70 deg. This last Case may best be resolved by changing of the Angles into Sides, as shall be hereafter taught. These are the Five Cases of Right-angled Spherical Triangles: And here you see that in this one Right-angled Spherical Triangle, by the several Parts given in these Five Cases, there are 30 Spherical Problems resolved: Namely, 3 by the First Case, 6 by the Second Case, 6 by the Third Case, 12 by the Fourth Case, and 3 by the Fifth Case. And so many are resolvable in every Right-angled Spherical Triangle. Of Oblique-angled Spherical Triangles. THe Triangle which I shall make use of for the resolving of the Six Cases of Oblique-angled Spherical Triangles, shall be the Obliqne Triangle Z. P. E. Obtuse angled at Z. In which Triangle The Side Z. P. being an Arch of the Brass Meridian, we will suppose to be the Compliment of the Latitude of London, containing 38 deg. 30 min. P. E. being an Arch of a Meridian (or Hour-Circle) we will suppose to be the Sun's distance from the Pole, or the Compliment of the Sun's Declination Northward 70 deg. Z. E. being an Arch of an Azimuth (or Vertical Circle) let be the Compliment of the Sun's, or a Star's Altitude, and to contain 47 deg. 12 min. The Angle Z. P. E. is the hour from Noon, whose measure is to be reckoned upon the Equinoctial, between the Equinoctial Colour and the Brass Meridian, and will be found to be 45 deg. or ●3 ho. in time. E. Z. P. is the Sun's Azimuth from the North-part of the Meridian, and is to be measured upon the Horizon between the North-part of the Brass Meridian, and the Quadrant of Altitude, and will be found to be 15 deg. 25 min. Z. E. P. is the Angle of the Sun's Position at the time of the Question, and may be measured either by turning of the Quadrant of Altitude to the contrary side of the Meridian, and counting thereupon the Compliment of the side E. Z. 42 deg. 48 min. and the Compliment of the side Z. P. 20 deg. upon the Colour; and the distance of these two Points shall be the measure of the Angle Z. E. P. which will be found to contain 36 deg. 52 min. Or you may turn the Triangle, and place Z. E. in the place where before Z. P. was placed, and than may that Angle be measured upon the Equinoctial between the Colour and the Brass Meridian, as the Angle Z. P. E. was. The several Sides and Angles of this Triangle being discovered, I will now come to show the several Cases which will arise out of this, and every Obliqne Spherical Triangle: But first let me show you How to express the Triangle upon the Globe. Elevate the Pole of the Globe to the Compliment of any of the Sides of the Triangle, as in this Example, to 51 deg. 30 min. (which i● the Compliment of the Side Z. P.) than count the Side Z. P. 38 deg. 30 min. from the Pole, and thereto screw the Quadrant of Altitude. This done, count the Side E. P. 70 deg. upon the Equinoctial Colour from the Pole, and the Side Z. E. upon the Quadrant of Altitude from the Zenith downward, and so move the Globe and Quadrant of Altitude together, till the numbers counted upon both of them concur in one Point; and so shall you have your Triangle exactly delineated upon the Globe; which being done, I proceed to the Six Cases of Oblique-angled Spherical Triangles, and to show the Variety of Problems that will naturally arise out of the solving of every such Obliqne Triangle. CASE I. The Side Z. P. 38 deg. 30 min. The Side E. P. 70 deg. and the Side Z. E. 47 deg. 12 min. being given, to find the Angles, ELevate the Globe to the 51 deg. 30 min. the Compliment of Z. P. and count the Side Z. P. the Compliment of the Latitude 38 deg. 30 min. from P. to Z. and there fasten the Quadrant of Altitude. Than count the Side E. P. the Compliment of the Sun's declination 70 deg. upon the Colour from the Pole downwards: Also count the Compliment of the Sun's Altitude 47 deg. 12 min. The Side Z. E. upon the Quadrant of Altitude downwards; and move the Globe and Quadrant together till 70 deg. of the Colour, and 47 deg. 12 min. of the Quadrant do meet: So is your Triangle represented upon the Globe. Now to find the several Angles. (1.) For the Angle at Z. count the number of degrees of the Horizon, which are contained between the North-part of the Meridian, and the Quadrant of Altitude; and you shall find them to be 115 deg. 35 min. and that is the quantity of the Angle E. Z. P. and is the Sun's Azimuth from the North-part of the Meridian. (2.) For the Angle Z.P.E. count the number of degrees of the Equinoctial, which are contained between the South-part of the Meridian and the Colour; which you will find to be 45 deg. and that is the quantity of the Angle Z. P. E. which is the hour from Noon, namely 9 in the Morning, or 3 in the Afternoon- (3.) For the Angle Z. E. P. (either change the Triangle by elevating the Globe anew, or) count 42 deg. 48 min. the Compliment of Z. E. upon the Quadrant of Altitude (it being brought to the other side of the Meridian) and also count the 20 deg. the Compliment of E. P. upon the Colour (on the other side of the Meridian) the distance between these two points, measured by Compasses or otherwise, will be found to contain 36 deg. 52 min. equal to the Angle Z. E. P. which is the Angle of the Sun's or Stars position at the time of the Question. And thus by this one Data, you have resolved three Problems: For you have found 3. 1. The Angle E. Z. P. the Sun's Azimuth 115 deg. 35 min. 2. The Angle Z. P. E. the hour 45 deg. or 9, or 3 a clock. 3. The Angle Z. E. P. the Sun's Angle of position 36 deg. 52. min. CASE II. The two Sides E.Z. 47 deg. 12 min. Z.P. 38 deg. 30 min. and the Angle comprehended between them being given, to find the other parts of the Triangle, COunt P. Z. 38 deg. 30 min. (the Globe being elevated to the Compliment thereof) upon the Meridian, from P. to Z. and there fix the Quadrant of Altitude: Than from the North-part of the Meridian upon the Horizon, count the quantity of the Angle E. Z. P. 115 deg. 35 min. and thereto bring the Quadrant of Altitude: Than count the other given Side E. Z. 47 deg. 12 min. upon the Quadrant of Altitude downwards; and turn the Globe about till the Equinoctial Colour cut the Quadrant of Altitude in 47 deg. 12 min. So is your Triangle delineated upon the Globe. Now to find the several Parts. (1.) The degrees of the Colour contained between the Pole and the Quadrant being 70, is the Side E. P. (2.) The degrees of the Horizon between the Meridian and Quadrant, being 115 deg. 35 min. is the Angle E. Z. P, And (3.) The distance between the Compliment of E. Z. upon the Quadrant of Altitude, and the Compliment of E P. upon the Colour (they being both continued to Quadrants, on the other Side of the Meridian) will be found 36 degrees 52 minutes, which is the Angle Z. E. P. Thus by this Data you have found 9 1. E. P. The Compliment of the Sun's Declination. 2. E. Z. P. The Sun's Azimuth from the North. 3. Z. E. P. The Angle of the Suns or Stars Position. But if the Sides Z. P. P.E. and Angle Z. P. E. had been given, there would than have been found 4. P. E. Z. The Angle of Position. 5. E. Z. The Com●●ment of the Sun's Altitude. 6. E. Z. P. The Azimuth from the North. Or if the Sides Z. E. P. E. and Angle Z. E. P. had been given, there would than be found 7. E. Z. P. The Sun's Azimuth from the North. 8. Z. P. The Compliment of the Latitude. 9 Z. P. E. The Hour from Noon. CASE III. The two Angles E. Z. P. 154 deg. 25 min. and Z.P.E. 45 deg. with the Side Z.P. 38 deg. 30 min. comprehended between them, given, to find the other parts of the Triangle, COunt 38 deg. 30 min. the Side Z. P. upon the Meridian from the Pole, and thereto screw the Quadrant of Altitude; than count 45 deg. the Angle Z. P. E. upon the Equinoctial, (beginning at the Colour) and bring those 45 deg. to the South-part of the Brass Meridian. Again, count 115 deg. 35 min. the Angle E. Z. P. upon the Horizon, from the North-part of the Meridian, and thereto bring the Quadrant of Altitude. And so have you represented your Triangle upon the Globe. Now to find the other Sides, and Angle. (1.) The degrees contained between the Pole and the Colour upon the Quadrant; namely, 47 deg. 12 min. will be the Side Z. E. (2.) And the degrees between the Pole and the Quadrant upon the Colour, viz. 70, will be the Side E. P. (3.) And the distance between the Compliments of the Sides E. Z. and E. P. being continued to Quadrants on the other Side of the Meridian; namely, 36 deg. 52 min. will be the Angle Z. E. P. And by this Data, you have found 9 1. Z. E. The Compliment of the Sun's Altitude. 2. E. P. The distance of the Sun from the Pole. 3. Z. E. P. The Angle of the Sun's Position. But if the Side E. P. and the Angle Z. E. P. and Z. P. O. had been given, there would have been found. 4. E. Z. The Compliment of the Sun or Stars Altitude. 5. E. Z. P. The Sun's or Stars Azimuth from the North. 6. Z. P. The Compliment of the Latitude. And if the Side Z. E. and the Angles P. Z. E. and P. E. Z. had been given, you might than find 7. Z. P. The Compliment of the Latitude. 8. Z. P. E. The Hour from Noon. 9 P. E. The Compliment of the Sun's or Stars Declination. CASE IU. The two Sides Z. P. 38 deg. 30 min. and E. P. 70 deg. with the Angle Z. E. P. 36 deg. 52 min. being given, to find the other Side and Angles. TO resolve this Case, you may (if you please) revert the Triangle, by elevating the Globe to 20 deg. the Compliment of the given Side P. E. than counting 70 deg. the Side E. P. upon the Meridian from the Pole, there screw the Quadrant of Altitude. Than upon the Horizon, count 36 deg. 52 min. the given Angle E. and to them bring the Quadrant of Altitude. So have you fixed the Angle Z. E. P. Than turn the Globe about, till 38 deg. 30 min. the other given Side P. Z. do touch the Quadrant, which it will do in 47 deg. 12 min. for the Side E. Z. And for the Angle P. the degrees of the Equinoctial between the Meridian and the Colour, viz. 45, is the Angle at P. and for the Angle at Z. it may be found by reverting of the Triangle again, or by the Compliments of the Sides extended on the other side of the Meridian, as hath been before taught. Lastly, for the Side E. Z. you have it upon the Quadrant 47 deg. 12 min. The Triangle being thus delineated, you have found (1.) the Side E. Z. 47 deg. 12 min. (2.) the Angle P. 45 deg. (3.) the Angle E. Z. P. 154 deg. 25 min. ¶ I thought good in this place to insert this manner of Change, not only for variety, but because in this and the next Case the Triangle may be more readily delineated upon the Globe. And in this one Data you have found 18 1. E. Z. The Complem. of the Sun's Altitude. 2. Z. P. E. The hour from Noon. 3. E. Z. P. The Azimuth of the Sun or a Star from the North. But if the Sides Z. P. and E. P. and the Angle E. Z. P. had been given, you would than have found 4. E. Z. The Compliment of the Suns or Stars Altitude. 5. Z. E. P. The Angle of the Suns or Stars Position. 6. Z P. E. The hour from Noon. And if there had been given E. P. E. Z. and E.Z.P. than would have been found 7. Z P. The Compliment of the Latitude. 8. Z. P. E. The hour from Noon. 9 Z. E. P. The Angle of Position. In like manner, if P.E. E.Z. and Z.P.E. had been given, you would have found 10. Z. P. The Compliment of the Latitude. 11. E. Z. P. The Azimuth from the North. 12. Z. E. P. The Angle of Position. Again, if there had been given E.Z. Z.P. and Z. P. E. you might than find 13. P. E. The Sun's distance from the Pole. 14. Z.E.P. The Angle of Position. 15. E. Z. P. The Azimuth from the North. Lastly, If E. Z. and Z. P. and the Angle Z. E. P. had been given, there would have been found 16. P. E. The Compliment of the Suns or Stars declination. 17. Z. P. E. The hour from Noon. 18. E. Z. P. The Azimuth from the North. CASE V The two Angles E. Z. P. 115 deg. 35 min. Z. P. E. 45 deg. with the Side P. E. 70 deg. being given, to find the rest of the Triangle, ELevate the Globe to 20 deg. the Compliment of the Side E. P. Than count the given Side E. P. 70 deg. upon the Meridian, from P. to Z. and there six the Quadrant. Than the given Angle E. P. Z. being 45 deg. count 45 deg. upon the Equinoctial from the Colour, and bring that point under the Meridian: So have you constituted the Side E. P. and the Angle E. P Z. Now for the Angle E. Z. P. you must either revert the Triangle again, or find it by the Compliments of the Sides Z. E. and Z P. extended to Quadrants on the other side of the Meridian, which you shall find to be 115 deg. 35 min. The Triangle thus constituted, (1.) For the Side Z. P. the degrees of the Colour between the Pole and the Quadrant being 38 deg. 30 min. is that Side. (2.) The degrees of the Quadrant of Altitude, between the Zenith and the Colour, being 47 deg. 12 min. is the Side E. Z. (3.) The degrees of the Horizon between the Meridian and the Quadrant of Altitude being 36 deg. 52 min. is the quantity of the Angle Z. E. P. By which single Data, you have found 18 1. Z. P. The Compliment of the Latitude. 2. E. Z. The Compliment of the Altitude. 3. Z. E. P. The Angle of Position. But if the Angles E. Z. P. Z. P. E. and the Side Z. E. had been given, than might you find 4. Z. P. The Compliment of the Latitude. 5. P. E. The Sun's distance from the Pole. 6. Z. E. P. The Angle of Position. Or had there been given Z. P. E. Z. E. P. and the Side E. P. you might from thence find 7. Z. P. The Compliment of the Latitude. 8. E. P. The Compliment of the Sun or Stars declination. 9 E. Z. P. The Sun's Azimuth from the North. And if the Ang●●s Z. P. E. Z. E. P. with the Side Z. P. had been given, than would have been found 10. Z. E. The Compliment of the Sun's Altitude. 11. P. E. The Sun's distance from the Pole. 12. E. Z. P. The Sun's Azimuth from the North. But if the Angles Z. E. P. E. Z. P. and the Side Z. P. had been given, than you might find 13. Z. E. The Sun's distance from the Zenith. 14. P. E. The Sun's distance from the Pole. 15. Z. P. E. The hour from Noon. And lastly, had there been given the Angles Z. E. P. E. Z. P. and the Side P. F. than might be found 16. Z. E. The Compliment of the Suns or Stars Altitude. 17. Z. P. The Compliment of the Latitude. 18. Z. P. E. The hour from Noon. CASE VI The three Angles E. Z. P. 115 deg. 35 min. the Angle E. P. Z. 45 deg. and the Angle Z. E. P. 36 deg. 52 min. being given, to find the three Sides. THis Case may best be resolved by turning the Angles of the Triangle into Sides, as shall be taught by and by: and so by this Data, you may find 1. Z. E. The Compliment of the Sun's Altitude. 2. Z. P. The Compliment of the Latitude. 3. E. P. The Compliment of the Sun's declination. And thus have you out of this one Obliqne Spherical Triangle, by the variety that these Six Cases afford, not lesle than Threescore Problems resolved, viz. 3 in the First Case, 9 in the Second Case, 9 in the Third Case, 18 in the Fourth Case, 18 in the Fifth Case, and 3 in the Sixth Case; in all 60. And so many Varieties or Changes are there in every Oblique-angled Spherical Triangle.— And besides these Varieties, this Triangle Z E P. is not peculiar only to the appellations that I have here given them, but to other purposes also. For, This Oblique-angled Triangle is not capable only of resolving the forementioned Astronomical Questions, but may be applied to Geographical or Nautical Questions also. For, 1. The Side Z P, may represent the Compliment of the Latitude of that Town or City whose Zenith-point is Z. 2. The Side E Z, may represent the distance between those two Cities or Towns. 3. The Side P E will be the Compliment of the Latitude of that City or Town at E. 4. The Angle E P Z, is the difference of Longitude between the two places E and Z. 5. The Angle P Z E, the Point of the Compass leading from Z to E. 6. The Angle P E Z, the Point of the Compass leading from E to Z. And in this Triangle the same things being given, they may be varied as before, and afford as many Questions in Geography or Navigation as in Astronomy: Namely 60. Moreover, In the same Triangle, if you imagine the Side P Z to be 23 deg. 30 min. the distance of the Pole of the World, from the Pole of the Ecliptic: Than will 1. The Side Z P, be the distance of the Pole of the World from the Pole of the Ecliptic. 2. The Side P E will be the Compliment of the declination of a Star at E. 3. The Side Z E, will be the Northern Latitude of the Star at E. 4. The Angle E P Z. will be the Compliment of the Stars Right Ascension. 5. The Angle P Z E, is the quantity of the Stars Longitude. 6. The Angle P E Z, is the Angle of the Stars Position. And this way it will afford 60 Varieties more, as is before intimated. Theorem. The Sides of any Spherical Triangle, may be turned into Angles; & contra, the Compliment of the Greatest Side, or Greatest Angle (to a Semicircle) being taken for the Greatest Side, or Greatest Angle. Demonstration. LEt A B C, be a Spherical Triangle Obtuse-angled at B. And let D E be the Measure of the Angle A H I be the Measure of the Angle C G F be the Measure of the Angle B viz. of its Compliment, F B G, it being the Angle of the Triangle. Now, K L is equal to D E L M is equal to F G K M is equal to H I Because K D and L E are Quadrants, and their Common Compliment is L D L G and F M be Quadrants, and their Common Compliment is L F K I and M H are Quadrants, and their Common Compliment is K H. Therefore, The Sides of the Triangle K L M, are equal to the Angles of the Triangle A B C, taking for the Greatest Angle A B C, the Compliment thereof F B G. It may also be demonstrated, That the Sides of the Triangle A B C, are equal to the Angles of the Triangle K L M, by the converse of the former. For The side A B is equal to O P the measure of the Angle M L K. The side B C is equal to F H the measure of the Angle L M K. The side A C is equal to D I the measure of the Angle D K 1 of the Compliment of the Obtuse Angle D K 1 For, A D and C I are Quadrants, & their Common Compliment is C D A P and O B be Quadrants, & their Common Compliment is A O B F and C H are Quadrants, & their Common Compliment is C F. Therefore, The Sides may be turned into Angles, and the contrary; which was to be demonstrated. And by this Conversion, may the 5th Case of Right, and the 6th Case of Obliqne Spherical Triangles be resolved. How a Perpendicular is to be let fall in any Obliqne Spherical Triangle, thereby dividing it into two Right-angled Triangles. THis Problem is not of any Use in the Solution of Triangles by the Globe, as is evident by what hath been already delivered concerning the Solving of them: But for that in Trigonometrical Calculations, there is a necessity for so doing, and the doing of it not lesle difficult to conceive or imagine; and seeing how naturally, and lively it is represented upon the Globe, I will therefore here insert it. And it is grounded upon the First Theorem at the beginning of this Tract of Trigonometry. Viz. If two Great Circles of the Sphere shall pass through each others Poles, those two Circles shall cut each the other at Right Angles. In the Oblique-angled Triangle Z E P, before made use of, let there be given, (1.) the Side P Z. (2.) The Side E Z, and (3.) The Angle E P Z. and let it be required to find the Side P E. This Problem is done by the Globes, by the 4th Case, as you may there see, without the help of any Perpendicular; but in Calculation it is wholly necessary, and therefore may well deserve the place of a Problem here. The Triangle being described upon the Globe, let it be required to let fall a Perpendicular from the Angle Z, upon the Side E P: The Measure of the Given Angle P, being upon the Equinoctial, see upon the Globe where the Side P E (being extended) cuts the Equinoctial, and from that Point count 90 degrees upon the Equinoctial, and that Point shall be the Pole of the Circle (P E.) (For the Poles of all Great Circles are a Quadrant, or 90 degrees distant from their Peripheries:) wherefore the Quadrant of Altitude being fixed in Z, bring it to this Point found in the Equinoctial, and than will it cut the Side E P in the Point where the Perpendicular must fall, which will be at K, 29 degrees 21 minutes distant from P. CONCLUSION. IN the foregoing Precepts, I have made use only of two Triangles for the Solution of the Five Cases of Right, and the Six Cases of Oblique-angled Spherical Triangles; namely, of the Triangle P ☉ ☉ for Right-angled, and Z E P for Oblique-angled. Yet in the Figure of the Globe by the intersection of the several Circles thereof, there are divers other Triangles (both Right and Obliqne) constituted, all which the foregoing Rules and Cases will sufficiently Solve. A few of the Principal I will mark in the Globe by numerical Figures, for distinction, and give you an account of what Circles they are composed, and what Questions of the Globe are Resolvable thereby. The Triangle that I have made use of for Right-angled Triangles, viz. P ☉ O, I have marked with the Figure 1. And the Obliqne Triangle Z E P, with the Figure 2; and shall say no more of them, having sufficiently dealt with them already; but come to give you an account of such other as I have marked in the Figure. And, The first that I shall take notice of is, the Triangle A K M, marked with the Figure 3. Right-angled at M, which is constituted of A K an Arch of the Equinoctial. A M an Arch of the Ecliptic, and K M an Arch of a Circle of Longitude passing through the Poles of the Ecliptic. In this Triangle, The Side A M, is the Suns or a Star's Longitude, or distance from the Equinoctial Point A. The side K M is the South Latitude of a Star at M, or the Sun's South declination. A K is the Right Ascension. The Angle K A M is the Sun's greatest declination. A K M, the Angle the Circle of Longitude makes with the Equinoctial. A M K is a Right Angle. The second Triangle that I shall take notice of is A ☉ B, Right-angled at B, and marked with the Figure 4; and which is constituted of A ☉, an Arch of the Horizon. A B, an Arch of the Equinoctial, and B ☉, an Arch of a Meridian or Hour-circle. In this Triangle, The Side A ☉ is the Amplitude of the Suns Rising or Setting from A, the East or West-Points of the Horizon. B ☉ is the Sun's declination North. A B is the Ascensional difference. The Angle ☉ B A is a Right Angle. ☉ A B is the Compliment of the Latitude. A ☉ B is the Angle of the Sun's Position at his Rising. The third Triangle that I here take notice of, is the Triangle A L F, Right-angled at F, and marked with the Figure 5. The which is constituted of A L, an Arch of the Equinoctial Colour. A F, an Arch of the Horizon, and L F, an Arch of an Azimuth or Vertical-Circle. In this Triangle The side A F is the Sun's Azimuth being East or West. A L is the Sun's Declination North. L F is the Sun's Altitude at Six a clock. The Angle L F A is a Right Angle. L A F is the Latitude. A L F is the Angle of Position. The fourth and last Triangle that I shall mention, is A D C, Right-angled at C, and marked with the Figure 6, and is composed of A D, an Arch of the Prime Vertical-Circle, or Azimuth of East or West. A C, an Arch of the Equinoctial. D C, an Arch of a Meridian or Hour-circle. In which Triangle, The Side A D is the Sun's Altitude when he is due East or West. A C is the Ascensional difference. D C is the Sun's Declination North. The Angle A D C is a Right Angle. C D A an Angle of Position. D A C is the Latitude. Thus have you an account of Four other Right-angled Spherical Triangles, with the Affections or Natures of their Respective Sides and Angles; out of each of which, by the Varieties that will arise from the Five Cases, may be deduced 30 Problems, and in all the Four Triangles 120 Problems, all which may be performed according to the Directions of the Five Cases of Right-angled Spherical Triangles; the Exercising whereof I leave to the Practitioner.— There are divers other Triangles may be found both Right and Oblique-angled, but these as the principal I commend to the Practice of the young Tyro. HOROLOGIOGRAPHICAL Problems. INTRODUCTION. Of the distinction of Plains, upon which Dial's are to be made. ALl Plains upon which Dial's are made, in any Latitude or part of the World, do either lie 1. Parallel to the Horizon, or 2. are Perpendicular to the Horizon, or 3. do cut the Horizon at Obliqne Angles: and of these sorts of Plains there are several Varieties, excepting the first. 1. A Plain that lieth Parallel to the Horizon, is said to be an Horizontal Plain; and of this kind there is no Variety. 2. Of Plains that are Perpendicular to the Horizon there are Two Varieties: For, insert with six illustrations Fig. 1. depiction of the geometric construction of an horizontal dial as described on page 127 Fig. 2 depiction of the geometric construction of a vertical south dial as described on page 130 Fig. 3. depiction of the geometrica construction of a vertical east or west dial as described on page 132 Fig. 4. depiction of the geometric construction of a south vertical declining dial as described on page 136 Fig. 5. depiction of the geometric construction of a direct west reclining dial as described on page 151 Fig. 6. depiction of the geometric construction of a north or south reclining plane as described on page 155 Secondly, If any such Perpendicular Plain do stand in any other Azimuth or Vertical-Circle between the South and the East, or the North and the West, so that one Face beholdeth the Southeast, and its opposite the North-West; than these Plains are called Vertical Plains declining from the North or South, towards either the East or West. 3. Of Plains that cut the Horizon at Obliqne Angles, and yet do lie directly in the prime Vertical Circle, or Azimuth of East or West, there are six Varieties. For First, A plain beholding the South, may fall back, (or Recline) from the Zenith, just into the Pole, and than will the Plain lie Parallel to the Axis of the World, and such a Plain is called a Polar Plain. Secondly, It may so fall back (or Recline) from the Zenith, that it may fall between the Zenith and the Pole, or between the Pole and the Horizon; and these three sorts are called South Reclining Plains. Reclining equal to lesle than more than the Pole or Co-Latitude of the Place. Thirdly, A Plain beholding the North, may fall back (or Recline) from the Zenith equal to the Equinoctial, and so the Plain will lie parallel to the Equinoctial Circle, and is therefore called an Equinoctial Plain. Fourthly, It may so fall back (or Recline) that it shall rest between the Zenith and the Equinoctial, or between the Equinoctial and the Horizon; and these three sorts are called North-reclining Plains. Reclining equal to lesle than more than the Equinoctial, or Latitude of the Place. Fifthly, Of Plains that cut the Horizon at Obliqne Angles, and do lie directly in the Meridian or Azimuth of North and South, there is only one Variety: for all such Plains will fall between the Zenith, and the East or West-points of the Horizon; and so are called East or West Recliners. Sixthly, Of Plains that cut the Horizon at Obliqne Angles, and do not lie in the prime Vertical-Circle or Azimuth of East and West, but in some other intermediate Azimuth or Vertical-Circle between the South and the East or West; there are also of those six Varieties:— And of these, those that behold the South, are called South-Recliners, declining East or West:— And those that behold the North, are called North-Recliners, declining either East or West. Now of these six sorts, all Plains besides the forementioned must necessarily be one of these following: For, 1. The South Reclining Plain, by reason of its Declination, may so chance as to fall Just into the Pole-point, and so is called a Polar declining Plain. Between the Zenith and the Pole. Between the Pole and the Horizon. And these are called South Declining Plains. Reclining equal to lesle than more than the Pole. 2. The North Reclining Plain, by reason of its Declination, may so chance as to fall Just at the Intersection of the Meridian with the Equinoctial. Between the Zenith and Equinoctial. Between the Horizon and the Equinoctial. And these are called North Declining Plains. Reclining equal to lesle than more than the Equinoctial. Seventhly, All Plains that do Decline and Recline also, have their opposite Faces, and those are called North or South Declining Inclining Plains; and the same Dial that serves for a South Declining Reclining Plain, will serve also for a North Declining Inclining Plain. And Dial's upon any of these Variety of Plains may be made by the Globes, as I shall come now to show. PROB. I. To make an Horizontal Dial in any Latitude. I. The Operation by the Globe. ELevate the Globe to the Latitude of the place for which you would make your Dial, (suppose for London, in the Latitude of 51 deg. 30 min.) Than bring the Vernal Equinoctial Colour (which is the first point of Aries also) to the Meridian, and (if you will) the Index of the Hour-Circle to 12. This done, 1. Turn the Globe about Westward, till the Hour-Index points at 1 a clock, or rather [till 15 degrees of the Equinoctial come to be just under the Meridian,] and there keeping the Globe, look upon the Horizon how many degrees thereof are cut by the Equinoctial Colour; which you shall d. m. Latitude 51. 30 d. m. 12 00. 00 11 1 11. 50 10 2 24. 20 9 3 38. 3 8 4 53. 35 7 5 71. 6 6 90. 0 find to be 11 deg. 50 min. which set down in a little Table, as you see here is done; for this 11 deg. 50 min. is the distance that the hour-lines of 11 and 1 a clock are distant from the Meridian upon the Dial Plain. 2. Turn the Globe more Westward, till 30 degrees of the Equinoctial comes to the Meridian, and than see what degrees of the Horizo n are cut by the Equinoctial Colour; which you will find to be 24 deg. 20 min. which note down in a Table as before, for that is the hour-distance of 10 and 2 a clock from the Meridian. 3. Turn the Globe still more Westward, till 45 degrees of the Equinoctial come to the Meridian, and than shall the Equinoctial Colour cut 38 deg. 3 min. of the Horizon counted from the Meridian, which is the distance of 9 and 3 a clock. Do thus with the oath r hours of 8 and 4, of 7 and 5, and so shall the Colour cut 90 degrees at 6 a clock, or when 90 degrees of the Equinoctial comes to the Meridian. And this being done, your Dial is so far made as the Globe can assist you. Now for II. The Geometrical Construction of this Dial. FIG. I. 1. Upon the Plain on which you design to draw your Dial, draw a Right line A B, representing the Meridian of your Globe, and the Hour-line of 12 of the clock. 2. Assign towards one end of this Line a point as C, representing the Centre of your Dial, and through that Point draw another Line at Right angles to A B, which shall be the Hour-line of 6 a clock, as the Line H K; and upon the Point C, describe a Semicircle D E F, according to the Radius of some Line of Chords. Than If you know not what a Line of Chords is, or know not how to use it, read my Geometrical Exercises lately printed. 1. Laying your little Table (before made) before you, you found Latit. 51 deg. 30 min. Take 51 deg. 30 min. from your Line of Chords, and set that distance upon the Semicircle from E to G, and draw the Line C G for the Style or Cock of your Dial. 2. Than seeing that 11 and 1 a clock are distant from the Meridian 11 deg. 50 min. take 11 deg. 50 min. from your Chord, and set it upon the Semicircle from E to 11, and from E to 1, and draw the Lines C 11, and C 1, for the Hour-lines of 11 and 1. 3. The distance of 10 and 2 a clock being 24 deg. 20 min. take 24 deg. 20 min. out of your Line of Chords, and set it upon the Semicircle from E to 10, and from E to 2, and draw the Lines C 10, and C 2, for the Hour-lines of 10 and 2. 4. Do thus with the rest of the Hour-lines of 9 and 3, 8 and 4,— 7 and 5. So have you all the Hour-lines between 6 in the morning and 6 at night; and for the Hour-lines of 4 and 5 in the morning, and of 7 and 8 at night, draw the same Hour-lines before 6 through the Centre, as in the Figure, and they shall be the true Hour-lines: And so is your Dial finished. The Style must stand upright upon 12 of the clock, not inclining on either side. PROB. II. To make a Vertical South Dial. I. The Operation by the Globe. THe Globe being set to the Latitude, and the Quadrant of Altitude in the Zenith, 1. Bring the Equinoctial Colour to the Meridian, and (if you will) the Index of the Hour-wheel to 12. 2. Turn the Globe about Westward, Comp. of the Latit. 38. d. 30 m. d. m. 12 00.00 11 1 9.28 10 2 19.45 9 3 31.54 8 4 47. 9 7 5 66.42 6 90.00 till 15 degrees of the Equinoctial be under the Meridian, than shall the Equinoctial Colour cut 9 deg. 28 min. of the Horizon counted from the Meridian; and that is the Hour-distance of 11 and 1 a clock from 12. 3. Turn the Globe more Westerly, till 30 degrees of the Colour be under the Meridian, than shall the Colour cut upon the Horizon 19 deg. 45 min. which is the distance of 10 and 2 a clock from 12. 4. Turn the Globe still Westerly, till 45 deg. 60 deg. and 75 deg. come under the Meridian; and so shall you find that the Colour will cut the Horizon in 31 deg. 54 min. for 9 and 3 a clock, and the rest, as in this Table. And so is the Globular work of your Dial finished. II. The Geometrical Construction of this Dial. FIG. II. 1. Draw a Right Line L M upon your Plain for the Meridian, and Hour-line of 12; and another Perpendicular thereto, as N P for the Hour-line of 6 and 6; and where these Lines cross (as at O,) is the Centre of your Dial. 2. Upon O, as a Centre, with the Radius of a Line of Chords, describe a Semicircle N Q P, and than taking 38 deg. 30 min. (which is the Compliment of the Latitude of the place) out of the Line of Chords, set it from Q to R, and draw the Line O R for the Cock or Style of your Dial. 3. Laying your Table before you against 11 and 1 of the clock, you find 9 deg. 28 min. Take therefore 9 deg. 28 min. out of your Chord, and set that distance upon the Semicircle from Q to 11 and from Q to 1, and draw the Lines O 11 and O 1, for the Hour-lines of 11 and 1. 4. Also take 19 deg. 45 min. and set them from Q to 10, and from Q to 2, and draw O 10, and O 2, for the hours of 10 and 2. 5. Do the like for the hours of 9 and 3, 8 and 4, 7 and 5, and your Dial is finished. The Style must stand over 12, and must point downwards towards the South-Pole. PROB. III. To make a Vertical direct North- Dial. THe North Vertical Dial, is the same with the South, only the Style must point upwards towards the North-Pole, and the hours about Midnight, as 9, 10, 11, 12: 1, 2, and 3 must be left out, and 4 and 5 in the Morning; and 7 and 8 at Night must be drawn through the Centre: So is your North-Dial also finished. PROB. iv To make a direct Vertical East or West Dial. I. The Operation by the Globe. THe Globe rectified to the Latitude, the Index to 12, the Quadrant of Altitude in the Zenith: If you turn the Quadrant of Altitude so about till the graduated edge thereof do behold the direct East or West-points of the Horizon, you shall find that it will lie in the very Plain of the Meridian-Circle, and so the Pole will have no elevation over it; for turning the Globe about, the Equinoctial Colour will not cut the Quadrant of Altitude in any particular degree, but it will cut all the degrees thereof at the same time; wherefore the Hour-lines of these Plains will make no Angles at the Pole, and therefore must be parallel one to the other, which the Globe evidently demonstrates, but will not conveniently give the parallel distance of each from other, they being nearer or farther off of each other according as the Style is proportioned to the Plain, which I shall now come to show in II. The Geometrical Construction of these Dial's. FIG. III. Let the Plain upon which you would make an East or West Dial, be A B C D. 1. Upon D (or any where towards the lower part of the Line B D, for an East Dial, or of A C for a West) with 60 degrees of your Chord, describe an Arch F G, upon which set the Compliment of the Latitude of the place, viz. 38 deg. 30 min. from F to G, and draw the Line D G E for the Equinoctial. 2. Towards the upper part of this Line, as at P, assume any point, and through it draw the Line 6 P 6 perpendicular to the Equinoctial, for the Hour-line of Six.— Also, towards the lower part of the same Line, assume another point, as L, and through it draw the Line 11 L 11 for the Hour-line of Eleven. 3. With 60 degrees of your Chords, upon the point L, describe a small Arch of a Circle, as H K, and upon it (always) set 15 degrees (or one hours' distance) from H to K, and draw the Line L K M, cutting the Hour-line of Six in M. 4. Upon M as a Centre, with 60 degrees of your Chord, describe an Arch of a Circle N O, which divide into five equal parts in the points ☉ ☉ ☉ ☉. 5. Lay a Ruler upon M, and each of these points ☉ ☉ ☉ ☉, and the Ruler will cut the Equinoctial-line E D in the points ****, through which points, if you draw Right Lines parallel to the Hour-line of 6, they shall be the Hour-lines of 7, 8, 9, and 10 of the clock, the Hour-lines of 6 and 11 being drawn before. 6. For the Hour-lines of 4 and 5 in the Morning, before 6, they retain the same distance from 6, as do the hours of 7 and 8; and thus is your Dial finished. The Style must stand upon the Hour-line of 6, and be elevated so high as is the length of the Line M P, and may either be a pin of Wyre, or a plate of Brass or Iron. The West-Dyal is the same with the East, only changing the names of the hours. For 4, 5, 6, 7, 8, 9, 10, 11 in the morning, in the East-Dyal; Must be changed to 8, 7, 6, 5, 4, 3, 2, 1 in the afternoon, in the West-Dyal: Which is all the difference. PROB. V To make a Vertical Dyal declining from the South, Eastward, or Westward; 30 degrees in the Latitude of 51 deg. 30 min. I. The Operation by the Globe. THe Globe being Rectified to the Latitude of the place, the Quadrant of Altitude in the Zenith, the Index of the Hour-Circle at 12, and the Equinoctial Colour brought under the Meridian; 1. Count the Declination of the Plain upon the Horizon, from the East or West-points thereof (according as the Plain declines) towards the South: namely, 30 degrees; and to that point of the Horizon bring the Quadrant of Altitude, and there keep it. 2. Turn the Globe about till the Index of the Hour-wheel cuts 11 of the clock, or rather (as I said before) till 15 degrees of the Equinoctial have passed the Meridian, and than shall you find the Equinoctial Colour to cut the Quadrant of Altitude at 9 deg. 50 min. if you count the degrees from the Zenith point downwards. 3. Turn the Globe farther about, till 30 degrees of the Equinoctial be passed the Meridian, and than shall you find the Colour to cut the Quadrant of Altitude at 18 deg. 14 min. counted from the Zenith downwards as before. 4. Do the like with all the Hours from Noon. Hour-distances upon the Plain d. m. 12 00 00 11 1 09 50 10 2 13 14 9 3 26 19 8 4 34 56 7 5 44 56 6 6 57 49 5 7 75 37 rest of the hours, and you shall find that at the several 15 degrees of the Equinoctial, the Equinoctial Colour will cut such degrees of the Quadrant of Altitude as are expressed in this Table, if you count them from the Zenith downwards, as is before directed. This done; 5. Bring the Quadrant of Altitude to the other side of the Meridian, and set it to 30 degrees, the Plains declination, counted from the East or West-points Northward, as you did before towards the South, which will be in the just opposite point of the Horizon to which it was before; and also, bring the Equinoctial Colour under the Meridian. Than, 6. Turn the Globe about (the contrary way to what you did before) till 15 degrees of the Equinoctial be passed the Meridian, and than shall you find the Equinoctial Colour to cut at 12 deg. 23 min. of the Quadrant of Altitude counted from the Zenith. And so continuing turning Hours from Noon. Hou● distances on the Plain. a. m. 12 00 00 1 11 12 23 2 10 29 19 3 9 52 42 4 8 80 07 the Globe about till 30, 45, and 60 degrees of the Equinoctial have passed the Meridian, you shall find the Equinoctial Colour to cut the Quadrant of Altitude at such degrees as are expressed in this Table. The Hour-distances upon the Plain being thus attained, there are two other requisites in all upright declining Dial's also to be found by the Globe, before the Dyal can be finished. Namely, 1. The distance of the Substile from the Meridian. 2. The height of the Pole above the Plain, or the height of the Style above the Substile. To find both which, Bring the Equinoctial Colour to the Plains declination 30 degrees counted upon the Horizon from the South-Eastward; and the Quadrant of Altitude to 30 degrees counted in the Horizon from the East-Northward: So shall the Quadrant cut the Colour at Right Angles. And The number of degrees of the Quadrant contained between this Intersection and the Zenith (which here is 21 deg. 41 min.) is the distance of the Substile from the Meridian. And the degrees of the Colour contained between this Intersection and the Pole (which here is 32 deg. 37 min.) is the height of the Pole above the Plain. II. The Geometrical Construction. FIG. iv 1. Draw a Line C D for the Meridian and Hour-line of 12, and at Right Angles thereunto draw another Line, as A B, for the Horizontal-line, crossing the former in the point C, the Centre of the Dyal. 2. Upon the point C, describe the Semicircle A G B. 3. Take 21 deg. 41 min. from your Chord, and (because the Plain declines Eastward) set it from G to E; also, take from your Chord 32 deg. 37 min. (which is the height of the Pole above the Plain) and set it from E to F, and draw the Line C E for the Substile, and C F for the Style of the Dyal. 4. Having recourse to the Tables of Hour-distances, take 9 deg. 50 min. and set it from G to 11; also take 18 deg. 14 min. and set it from G to 10, and so the rest in that Table.— Also out of the other Table take 12 deg. 23 min. and set them from G to 1, etc. 5. From the Centre C draw Lines through the several points 5, 6, 7, 8, 9, 10, and 11; and also through 1, 2, 3, and 4, they shall be the true Hour-lines; and so have you finished this Dyal. And in the making of this Dyal, you have made four Dial's; as I have intimated at large in my Art of dialing, Part 1. Chap. 7. For, If you hold the Paper upon which the Southeast declining Dyal is drawn, against the light, than shall you discover the Style to stand on the Right hand of the Plain, whereas it now stands on the Left hand; so the same Hour-lines, Substile, Stile and all, being drawn on the backside of the Paper, and those that are the Forenoon-hours in the East-decliner numbered as the Afternoon-hours in the West-decliner, that is, call 11, 1, and 10, 2, and 9, 3, etc. as in the Tables; so shall the South-Dyal declining East 30 degrees, become a South-Dyal declining Westward 30 degrees. And if you turn the South-East-Dyal upsidedown, so that the Style may point upwards towards the North-Pole, (and leave out the hours about 12, as 9, 10, 11, and 1, 2, and 3, which in North-dyals' represent 9, 10, and 11 at night, and 1, 2, and 3 in the morning; all which time (in those middle Latitudes) the Sun is under the Horizon) it will become a North-Dyal declining Eastward 30 degrees. Also if you turn the South declining West Dyal upsidedown, and leave out the hours about Midnight, as 9, 10, 11, 12, 1, 2, and 3, it will than become a North-Dyal declining Westward 30 degrees. Now for such South or North Dial's as do dedecline far towards the East or West, as 60, 70, 80, or 85 degrees, there you shall find that the hour-distances will fall so near together, that they will be of no competent distance one from another, except they be extended very far from the Centre; and therefore the old way hath been (in such Cases) to draw the Dyal upon the Floor of a Room, extending the Substile, Stile, and Hour-lines till they appear of a competent distance from each other, and than according to the bigness of your Dyal-plain, to cut of the Hour-lines, Style, and Substile, and so transfer them from the Floor to the Plain upon which the Dyal is to be made: but this way being too Mechanical for an Artist to exercise, I have in my foresaid Art of dialing showed a more artificial way of performing this work Geometrically, by which (although the Dyal should decline 80 or 88 degrees) upon a quarter of a sheet of paper you may draw your Dyal, and have the Style of a competent height, and all the Hour-lines at a convenient distance one from another. And so let this suffice to be said in this place concerning Upright declining Dial's; for I intent not here to teach the Art of dialing, but show the Use of the Globes. PROB. VI Of direct South Reclining Dial's. TO find the distances of the Hour-lines upon these Plains by the Globes, this is the natural way. The Operation by the Globe. Having set the Globe to the Latitude, the Index of the Hour-circle to 12, the Quadrant of Altitude to the Reclination, with the end thereof in the East o● West points of the Horizon, and brought the Equinoctial Colour to the Meridian,— Turn the Globe ●bout, till the 1 hour, or 15 degrees of the Equinoctial hath passed the Meridian, and than see what degrees of the Quadrant of Altitude are cut by the Equinoctial Colour; for those degrees counted from the upper part of the Quadrant downwards, are the degrees of the first hours distance (as of XI or I of the clock) from the Meridian, and so for all the rest of the Hour-distances. This I say is the natural way; but the more artificial and better way, will be to refer such Reclining Plains to a New Latitude where they may become Horizontal Plains; and that may be easily effected, as I shall presently show. I formerly said, that a South-Plain may so recline, that the Reclination thereof may be either Lesle than Equal to Moore than the Compliment of the known Latitude. Now to refer any of these Plains to a new Latitude where they may become Horizontal Plains, observe, 1. If the Reclination of the Plain be lesle than the Compliment of the known Latitude, Subtract the Plains Reclination from the Compliment of the Latitude, and the Remainder shall be a New Latitude, where the Reclining Plain shall be an Horizontal Plain; and, in this Case, the South-Pole (in North-Latitudes) is always elevated. 2. If the Reclination be equal to the Compliment of the known Latitude, than the New Latitude is no Latitude; for the Plain lies in the very Axis of the World, and hath neither Pole elevated above it. 3. If the Reclination of the Plain be Moore (or Greater) than the Compliment of the known Latitude, Subtract the Compliment of the known Latitude from the Plains Reclination, and the Remainder shall be the New Latitude; and in this Case, the North-Pole (in North-Latitudes) is always elevated. Examples of these three Varieties. I. Of a South Plain, Reclining lesle than the Compliment of the Latitude. d. m. Latitude 51 30 South reclin. 20 00 New Latitude 18 30 Hours Hour-distances. d. m. 12 00 00 11 1 04 52 10 2 10 23 9 3 17 36 8 4 28 48 7 5 49 50 6 90 00 LEt a South-Plain in the Latitude of 51 deg. 30 min. North, recline 20 degrees: Now because 20 degrees is lesle than 38 deg. 30 min. the Compliment of the known Latitude, subtract 20 degrees from 38 deg. 30 min. and there will remain 18 deg. 30 min. which will be the New Latitude. Wherhfore i● you Rectify the Globe to 18 deg. 30 min. of Latitude, and make an Horizontal Dyal as is before taught in all respects, that Dyal shall serve for a South Dyal reclining 20 degrees in the Latitude o● 51 deg. 30 min. the Hour-distances being such as are expressed in this Table, and the height of the Style above the 12 a clock-line (or Substile) to be equal to the New Latitude, namely, 18 deg. 30 min. II. Of a South Plain Reclining equal to the Compliment of the Latitude. Thus if a South Plain in the Latitude of 51 deg. 30 min. shall recline 38 deg. 30 min. equal to the Compliment of the Latitude, the New Latitude (as is said before) shall be no Latitude, and so neither Pole have any Elevation; wherefore the Dyal is to be made in all respects as an East or West-Dyal is made, only that Hour-line which in the East or West-Dyal is the six a clock hour-line, must in these Dial's be the Hour-line of 12, etc. III. Of a South Plain Reclining more than the Compliment of the Latitude. Suppose a South Plain in the Latitude of 51 deg. 30 min. should recline from the Zenith 60 degrees; forasmuch as 60 degrees is more than 38 deg. 30 min. the Compliment of the known Latitude, subtract 38 deg. 30 min. from 60 degrees, and the remainder 21 deg. 30 min. will be the New Latitude. d. m. Latitude 51 30 South reclin. 60 00 New Latitude 21 30 Hours Hour-distances. d. m. 12 00 00 11 1 05 37 10 2 11 57 9 3 20 08 8 4 32 25 7 5 53 50 6 90 00 Wherhfore, if you make an Horizontal Dyal for the Latitude of 21 deg. 30 min. you shall find the Hour-distances to be such as are expressed in this Table, and shall serve for a South-Dyal Reclining from the Zenith 60 degrees in the Latitude of 51 deg. 30 min. And the Style of this Dyal must be elevated above the Substile (or Hour-line of 12) 21 deg. 30 min. equal to the New Latitude, and must point upwards towards the North-Pole, as must the Styles of all South-Plains which recline more than the Compliment of the Latitude. PROB. VII. Of North Reclining Dial's. THe Natural way of finding of the Hour-distances for North Reclining Dial's is thus. The Operation by the Globe. Rectify the Globe to the Latitude, set the Quadrant of Altitude to the Reclination, and the end of it to the East or West-points of the Horizon, the Index of the Hour-circle to 12, and the Equinoctial Colour to the Meridian— Than turn the Globe about till 15 degrees of the Equinoctial have passed the Meridian, and than see what degrees of the Quadrant of Altitude are cut by the Equinoctial Colour; for those degrees (counted from the upper end of the Quadrant of Altitude) are the first hours distance from 12, as of 11 and 1 of the clock: and so of all the rest of the hours. This is the natural way; but the better way will be to refer these North Reclining Plains to a New Latitude, as before you did the South-Recliners. Of these North Reclining Plains, there are Three Varieties, as there were of the South-Recliners: for the Reclination may be either Lesle than Equal to Moore than the Latitude of the Place. And to refer these to a New Latitude where they will be Horizontal Plains, you must observe, 1. If the Reclination of the Plain be lesle than the Latitude, add the Compliment of the Latitude to the Plains Reclination, and the sum shall be the New Latitude, and the North-Pole (in North-Latitudes) shall be always elevated. 2. If the Reclination be equal to the Latitude, add the Compliment of the Latitude and the Reclination together, and the sum shall be the New Latitude, which in this Case will always be 90 degrees. 3. If the Reclination be more than the Latitude, add the Compliment of the Reclination of the Plain, and the Latitude together, the sum of them shall be the New Latitude, and the North-Pole shall always be elevated. Examples of these three Varieties. I. Of a North-Plain Reclining lesle than the Latitude. LEt a North-Plain in the Latitude of 51 deg. 30 min. recline from the Zenith 20 degrees; add 20 degrees (the Reclination) to 51 deg. 30 min. (the Latitude) the sum will be 71 deg. 30 min. which is the New Latitude. Wherhfore, if you rectify the d. m. Latitude 51 30 North reclin. 20 00 New Latitude 71 30 Hours Hour-distances. d. m. 12 00 00 11 1 14 17 10 2 28 42 9 3 43 29 8 4 58 40 7 5 74 13 6 90 00 Globe to 71 deg. 30 min. of Latitude, and make an Horizontal Dyal as is before taught, you shall find the Hour-distances to be such as are expressed in this Table. And the Style must be elevated above the Substile (or Hour-line of 12) equal to the New Latitude; and so shall this Horizontal Dyal, made for the Latitude of 71 deg. 30 min. serve for a North-Dyal Reclining 20 degrees in the Latitude of 51 deg. 30 min. II. Of a North-Plain Reclining equal to the Latitude. Let a North Plain in the Latitude of London 51 deg. 30 min. recline from the Zenith 51 deg. 30 min. this added to the Compliment of the Latitude, viz. 38 deg. 30 min. the sum will be 90. so that 90 degrees is the New Latitude. Wherhfore, rectify the Globe to 90 degrees of Latitude, so shall the Pole be in the Zenith, and the Equinoctial in the Horizon: than turn the Globe about till 15 degrees of the Equinoctial Colour have passed the Meridian, and it will rest at 15 degrees of the Horizon, which is the first Hours distance from 12. for in these Plains the degrees of the Equinoctial and the degrees of the Horizon being the same, there is no more in the making of this Dyal, than to divide a Circle into 24 equal parts, and (being the New Latitude is 90 degrees) erect a Wire perpendicularly in the Centre, and the Dyal is finished. III. Of a North-Plain Reclining more than the Latitude. Suppose that in the Latitude of 51 deg. 30 min. a North-Plain should recline from the Zenith 60 degrees, add 51 deg. 30 min. (the Latitude) to 30 degrees (the Compliment of the Reclination) the sum will be 81 deg. 30 min. for the New Latitude.— d. m. Latitude 51 30 North reclin. 60 00 New Latitude 81 30 Hours Hour-distances. d. m. 12 00 00 11 1 14 51 10 2 29 44 9 3 44 41 8 4 59 43 7 5 74 51 6 90 00 Wherhfore, rectify the Globe to the Latitude of 81 deg. 30 min. and (according to former directions) make an Horizontal Dyal for that Latitude, and you shall find the Hour-distances to be such as are expressed in this Table; and such an Horizontal Dyal for the Latitude of 81 deg. 30 min. shall serve for a North-Dyal reclining 60 deg. in the Latitude of 51 deg. 30 min. The Style must be elevated above the 12 a clock hour-line equal to the New Latitude, namely, 81 deg. 30 min. PROB. VIII. Of East or West direct Recliners. SUch Plains as lie in the Meridian or Azimuth of North and South, and do recline from the Zenith towards the East or West-points of the Horizon, are called East or West-Recliners; and the Natural way of finding of the Hour-distances upon these Plains by the Globes, is thus. I. The Operation by the Globe. The Globe rectified to the Latitude of the place, the Quadrant of Altitude in the Zenith, the Equinoctial Colour brought to the Meridian, the Index of the Hour-wheel to 12, and the Semicircle of Position fixed to the Meridian in the North and South-points of the Horizon, elevate the Semicircle of Position to the Compliment of the Plains Reclination, counted upon the Quadrant of Altitude from the Horizon; than move the Globe about, till the Equinoctial Colour hath passed 15 degrees of the Equinoctial, and than the degrees of the Semicircle of Position, cut by the Equinoctial Colour (counted from the Meridian) shall be the degrees that the first Hour-line of 11 or 1 shall be distant from the Meridian, or Hour-line of 12.— Also turn the Globe about till the Equinoctial Colour hath passed 30 degrees of the Equinoctial Circle, and the degrees cut by the Colour upon the Semicircle of Position shall be the degrees that the second Hour-line shall be distant from 12, etc. This is the natural way of making of these Dial's by the Globe; but (to retain our former method) we will refer these East or West reclining Dial's to a New Latitude, where they shall be upright declining Plains; which to perform, is easy: For, In all East or West-Recliners, the New Latitude is always the Compliment of the known Latitude, and the New Declination is always the Compliment of the Reclination; so that an East or West-Plain in the Latitude of 51 deg. 30 min. reclining 40 degrees will be an upright North-Plain declining East or West 50 degrees in the Latitude of 38 deg. 30 min. for 38 deg. 30 min. is the New Latitude, it being the Compliment of the known Latitude, and 50 degrees is the New Declination, it being the Compliment of the Plains Reclination in the known Latitude; wherefore, if (according to the directions given you in the Fifth Problem) you make an upright Dyal, declining 50 degrees for the Latitude of 38 deg. 30 min. you shall find all the Hour-distances to be such as are expressed in this Table. d. m. East Reclining 40.00 Latitude 51.30 New Latitude 38.30 Declination 50.00 Distance of Substile from Meridian 43.55 Styles height 30.12 Hours from Noon. Hour-distances on the Plain. d. m. 12 00 00 1 11 15 14 2 10 26 12 3 9 34 57 4 8 42 42 5 7 50 21 6 6 58 39 7 5 68 43 8 4 82 18 11 1 22 09 10 2 48 43 9 3 75 52 The distance of the Substile from the Meridian to be 43 deg. 55 min. and the height of the Pole or Style above the Plain or Substile, to be 30 deg. 12 min. which declining Dyal in this New Latitude, shall be an East of West reclining Dyal in the known Latitude. But in the placing of the Dyal upon the Reclining Plain, there is this difference: For whereas in all Upright declining Plains, the Meridian or Hour-line of 12 is perpendicular to the Horizon, in all East or West-Recliners the Meridian (or Hour-line of 12) must lie parallel to the Horizon. And here note also, that all East-Recliners in the known Latitude, are North-East-decliners in the New Latitude, and all West-Recliners are North-West-decliners. All which may be seen in FIG. V And note farther, that upon all East and West-Recliners (how far soever) in North-Latitudes, the North-Pole is always elevated; and upon the East and West-Incliners opposite to them, the South Pole. Thus have you the manner of making of all sorts of direct Reclining or Inclining Dial's by the Globes two several ways; namely, by the Natural way, as they lie in respect of the Horizon, where they are to be placed Obliquely: And also a more artificial way, by referring them to New Latitudes and New Declinations, where they may become Horizontal or Upright declining Plains. And now let us proceed to PROB. IX. Of Declining, Reclining, or Inclining Plains. SUch Plains as do not directly behold the East, West, North, or South-points of the Horizon, nor do stand perpendicular thereunto, but fall back from the Zenith, these Plains are called North or South-Recliners. For the making of these kinds of Dial's by the Globes, the natural way followeth. I. The Operation by the Globe. For our Example, Let us suppose a Plain in the Latitude of 51 deg. 30 min. to decline from the North-part of the Meridian towards the West 72 degrees, and also to Recline from the Zenith 26 deg. 34 min. 1. Elevate your Globe to the Latitude of the place 51 deg, 30 min. the Quadrant of Altitude in the Zenith, the Hour-circle to 12, and the Equinoctial Colour to the Meridian. 2. The Declination being 72 degrees Westward, count upon the Horizon 20 degrees (the Compliment thereof, from the South-part of the Meridian Westward, and from the North-part of the Meridian Eastward; and to these two Points in the Horizon, bring the two extreme ends of your Circle of Position, (or which is far better, a narrow Plate of thin Brass, containing a complete Semicircle at lest, divided into degrees as the Quadrant of Altitude is, beginning the divisions at the middle, and numbering them both ways towards the ends) and there fix it. Than bring the Quadrant of Altitude to 20 degrees in the Horizon, counted from the East Southward.— Now (because the Plain Reclines 26 deg. 34 min.) count those degrees upon the Quadrant downwards, from the Pole, and to that Point bring down the thin Plate of Brass, representing your Plain, and there fix it; for now is your Globe prepared for the making of your Dyal. And, I. For the Hour-distances: If you turn the Globe about Eastward, till 15 degrees of the Equinoctial have passed the Meridian, you shall find upon the Plain intercepted between the Colour and the Meridian 26 deg. 3 min. which is the first Hours distance from 12 upon your Plain. Again, turn the Globe about still Eastward, and you shall find 44 deg. 30 min. intercepted, which is the second Hours distance upon the Plain. Also, if you turn about the Globe Westward till 15 degrees of the Equinoctial have passed the Meridian, there will be cut by the Colour upon the Plain 28 deg. 4 min. which is the first Hours distance on the other side of the Meridian. And so do with all the rest, and you shall find them as in this following Table. d. m. North declining 72.00 Reclining 26.34 Distance between the Merid. & Hor. 36.00 Distance between the Plain & Zen. 58.17 Height o'th' Pole above the Plain. 31.28 Distance of the Subst. & Merid. 82.04 d. m. 12 00 00 1 11 26 03 2 10 44 30 3 9 57 47 4 8 67 53 5 7 76 12 Substile. Substile. 6 6 84 15 7 5 87 38 8 4 78 25 9 3 66 48 10 2 50 46 11 1 28 04 Now (besides these Hour-distances) there must be Four other things found before we come to the Geometrical Construction of this Dyal; and those are 1. The Arch of the Plain (or the distance) between the Meridian and Horizon; and that may be found thus.— Take with your Compasses (or with a thin Plate of Brass or Horn divided) the distance upon the Plain, from the Intersection of the Plain with the Horizon, to the Intersection of the Plain with the Meridian; and those degrees, namely 36, are the distance required. 2. The Arch of the Meridian between the Plain and the Zenith. The which may be thus found. For the degrees of the Meridian intercepted between the Plain and the Zenith, viz. 58 deg. 17 min. is this Arch. 3. The height of the Pole or Style above the Plain.— And to find this, You must continued your Quadrant of Altitude from the Zenith below the Horizon, so much as is the Reclination of the Plain 26 deg. 34 min. And mark that Point, for it is the Pole of the Reclining Plain. Than a thin Plate of Brass, (for the Quadrant of Altitude will be (for the most part) too short) divided, being extended from the North-Pole towards the South-Pole) and passing through the Pole of the Plain, (which is the Point before found) mark where the Plate cutteth the Plain; and the number of degrees of the Plate that are contained between the Pole and the Plain, which will be here found 31 deg. 28 min. is the height of the Pole or Style above the Plain or Substile. 4. The distance of the Substile from the Meridian.— And that is, the number of degrees of the Plate representing the Plain, which are contained between the Plate which came from the Pole of the World to the Pole of the Plain, and the Intersection of the Plain with the Meridian, which in this Example will be found to be 82 deg. 4 min. II. The Geometrical Construction of this Dyal. FIG. VI 1. Upon the Plain upon which you intent to describe your Dyal, draw an Horizontal-line thereupon, as A B; and another perpendicular thereunto D G, for the Vertical line of your Plain, cutting the former A B in the point G, which point C make the Centre of your Dyal. 2. Upon the Centre C, with 60 degrees of a Line of Chords describe a Circle, as A F B G. than take 36 degrees (the distance of the Meridian from the Horizon,) and set it from A to 12 downwards, and from B to 12 upwards, and draw the Line 12 C 12 for the Hour-line of XII. 3. Take 82 deg. 4 min. (the distance of the Substile from the Meridian) and set them from 12 upwards to G on the left hand, and downwards from 12 to G on the right hand, drawing the Line G C G through the Centre, for the Substilar-line of both the Dial's:— And the height of the Pole or Style above the Plain being 31 deg. 28 min. set those degrees from G to H, both above and below, and draw the Line H C H quite through the Centre, for the Axis or Stiles of both Dial's. 4. Laying your Table of Hour-distances before you, and there finding the first Hours distance from 12 is 26 deg. 3 min. set that distance upon the Circle from 12 upwards to 1, and from 12 (on the other side) downwards to 11, and draw the Lines C 1, and C 11, for the Hour-lines of 11 and 1 a clock, which will both be but one straight line. Do thus with all the hours in the Table till you come to the Substile, and than, (beginning at the bottom of the Table) 5. Take 28 deg. 4 min. and set them from 12 on the Right hand downwards to 11, and from 12 on the Left hand upwards to 1, and draw the Line C 11 in the North Reclining Plain, for the Sun will never shine upon the South Inclining Plain at 1, otherwise you should have drawn it through the Centre as you did before. Do the like with all the rest of the hours, drawing such through the Centre as you find occasion for; which the sight of the Figure will inform you how to do, better than many Words. Lastly, Erect your Style perpendicular to your Substile, making an Angle therewith equal to the elevation thereof; namely, 31 deg. 28 min. and your Dial's are finished.— And in the making of these, you have made two others also, viz. A North Declining 72 degrees Eastward, and Inclining 26 deg. 34 min. and a South Declining Westward 72 degrees, and Inclining as the other; all which is done (and may easily be apprehended by any person) as is discovered in the 4th Problem of this Book for Upright declining Plains. Thus have you the manner of making all manner of plain Sun-dyals' by the Globes, not only by the Natural way, as they are Naturally represented upon the Globe; but by an Artificial way also, by referring them to New Latitudes, in which they shall become Horizontal Dial's, or at lest Upright Decliners. It resteth now that I should say something concerning the inserting of Tropics, Parallels, Italian, Babylonish, and Jewish Hours, as also Almicanters, Azimuths, and such other Spherical Lines and Arches as are oftentimes inserted upon Dyal-Plains for the delight and curiosity of the Ingenious: but these, with some other things concerning the Gnomonicks, I shall refer to another Treatise by itself. ASTROLOGICAL Problems. INTRODUCTION. Astrology consisteth principally of two parts, viz. the one Mathematical, as is the Astronomical part; the other Judiciary, as is the Astrological part. The Mathematical part teacheth how in a Scheme or Figure (as they call it) to represent the Face of the Heavens in Plano, for any hour of the day or night, at all times of the year, and in all parts of the world. The Astrological part teacheth how (from the sight of the said position of the Scheme or Figure of the Heavens at the time of its erection) to give a determinate Judgement of what was demanded upon that Erection of the Scheme or Figure: as of Annual Revolutions, Elections, the Nativity of a Person. The principal Authors that have given their Opinions concerning the dividing of the Heavens into 12 Mansions or Houses, are, 1 Ptolemy, 2 Alcabitius, 3 Campanus, and 4 Regiomontanus: Which last way, is now generally received and practised among the Astrologers of these times, and by them termed the Rational way of Regiomontanus. Now, because (as I said before) that the Erection of a Figure of the Heavens is the Mathematical or Astronomical part of Astrology, I shall therefore show how by the Globes to Erect a Figure of the Heavens according to the various ways of the Four forementioned Authors, and that for the same Latitude and Time. PROB. I How to Erect a Figure of the Heavens in the Latitude of London 51 deg. 30 min. N. for the 10th day of March, at 49 min. after 9 in the Forenoon; at which time the Sun enters into the first scruple of Aries this Year 1675. I. According to the (esteemed) Rational way of Regiomontanus. Definition 1 THe Heavens are divided into XII Houses or Mansions, by 12 Semicircle of Position; for which purpose, to some Globes, there is made one of Brass, which is fixed in the Intersections of the Meridian and Horizon, by the Elevation or Depression whereof the Heavens may be divided into parts or houses through each degree of the Ecliptic. Definition 2 Of these XII Houses or Mansions of Heaven, Four are called Cardinal, as (1.) the Horoscope, or Ascendent, or Cuspis of the First House. (2.) The Medium Coeli, or Angle of the South, or Cuspis of the Tenth House. (3.) The Descendent, or Angle of the West, or the Cuspis of the Seventh House. (4.) The Imum Coeli, or the Angle of the North, or the Cuspis of the Fourth house. Definition 3 Regiomontanus divides the Heavens into XII Houses according to his way, by the Circle of Positions passing through every 30th degree of the Equinoctial, and cutting the Ecliptic at several points, which are the Cuspises of the several Houses:— So that when the Globe is set to the Latitude, and the Hour-wheel Rectified and brought to the Given hour, you have the Cuspises of the Four Cardinal houses given: For, The degree of the Ecliptic cut by the East-side of the Horizon, South-part of the Meridian, Westside of the Horizon, North-part of the Meridian, Gives the Cuspis of the First Tenth Seventh Fourth House. The Cuspises of the other 8 Houses are found by the motion of the Circle of Position, as shall be showed by and by. Definition 4 The Houses are denominated by 1, 2, 3, 4, etc. to 12, from the Ascendent downward to the Imum Coeli, up again to the Descendant, and again by Medium Coeli down to the Ascendant. As in the following Scheme. A Figure of the XII HOUSES. an illustration of the 12 astrological houses Let this suffice for Definition, and now we will come to the Practice by the Globes. Practice. FIrst, to the day proposed, the 10th of March, find (by the first Astronomical Problem) the Sun's place in the Ecliptic at noon, which you shall find to be in 0 deg. 5 min. of Aries. Secondly, Set the Globe to the Latitude 51 deg. 30 min. Thirdly, Bring the Sun's place at Noon (0 deg. 5 min. of Aries) to the Meridian. Fourthly, Turn the Globe about till the Hour-Index point to the hour given, viz. to 49 min. after 9 in the Morning. Lastly, The Globe being in this Position, fix it. The Globe being fixed in this Position, you shall find that the East-semicircle of the Horizon doth cut the Ecliptic in 0 deg. 29 min. of Cancer, which is the Sign than Ascending, and must be placed upon the Cuspis of the first House. Than cast your eye upon the Intersection of the South-part of the Meridian and the Ecliptic, and there you shall find the Ecliptic cut by the Meridian in 25 deg. of Aquarius, and that point of the Ecliptic is than in the Medium Coeli, and must be set upon the Cuspis of the Tenth House. Also you shall find that the West-semicircle of the Horizon cuts the Ecliptic in 00 deg. 29 min. of Capricorn; which point is than Descending, and must be placed upon the Cuspis of the Seventh House. Lastly, You shall find that the North-part of the Meridian doth cut the Ecliptic in the 25th deg. of Leo, which point is than upon the Imum Coeli, and must be placed upon the Cuspis of the Fourth House. Thus have you found the Points of the Ecliptic which do occupy the Cuspises of the Four Cardinal Houses: Now for the other Eight Houses. Let the Globe still rest in its former Position, and than, First, Bring the Circle of Position to its place on the East-side of the Horizon; and being there fixed, raise it upwards towards the Meridian, till 30 deg. of the Equinoctial be intercepted between the Horizon and the Circle of Position; and than you shall find that the Circle of Position will intersect the Ecliptic in 20 deg. of Taurus; which degrees must be set upon the Cuspis of the Twelfth House. Secondly, Move the Circle of Position yet ●igher towards the Meridian, till 30 deg. more of the Equinoctial be intercepted between it and the Horizon, (in all 60 deg.) and when it so doth, you shall find the Circle of Position will cut the Ecliptic in 36 deg. of Pisces; which point must be set upon the Cuspis of the Eleventh House. The Meridian gives the Cuspis of the Tenth House in 25 deg. of Aquarius, as before. Thirdly, Move the Semicircle of Position from the East-side of the Horizon to the Westside, and move it downwards from the Meridian, till 30 deg. of the Equinoctial be intercepted between the Meridian and Circle of Position, and than you shall find that the Circle of Position will intersect the Ecliptic in 7 deg. of Aquarius; which point must be set upon the Cuspis of the Ninth House. Fourthly, Move the Circle of Position yet lower by 30 deg. i e. 60 deg. from the Meridian downwards, and than you shall find the Position-Circle to cut the Ecliptic in 21 deg. of Capricorn; which point must be set upon the Cuspis of the Eighth House. The Descendant or Cuspis of the Seventh House is the Intersection of the Westside of the Horizon and Ecliptic, which is in 00 deg 29 min. of Capricorn, as before. And thus have you found the Cuspises of the Four Houses above the Horizon, beside the Ascendant and the Medium Coeli; viz. of the 12, 11 9 and 8 houses. Now the Cuspises of the Four other Houses under the Earth have the same degrees of the opposite Signs upon them: For, 20 deg. of Taurus being upon the Cuspis of the 12 house 26 deg. of Pisces 11 7 deg. of Aquarius 9 21 deg. of Capricorn 8 20 deg. of Scorpio will be on the Cusp of the 6 house. 26 deg. of Virgo 5 7 deg. of Leo 3 21 deg. of Leo 2 For the Six Signs Aries, Taurus, Gemini, Cancer, Leo, Virgo, are opposite to Libra, Scorpio, Sagitt. Capric. Aquar. Pisces. And this is the manner how (by the Globe) to erect a Figure according to the (reputed) Rational way of Regiomontanus. Now if you would insert the places of the Planets into your Figure, (for it is them that the ginger principally giveth Judgement by) your best way will be to have recourse to some good Ephemeris (if you cannot Calculate them from Astronomical Tables) and so may you find the places of the several Planets at the time for which this Figure was Erected (viz. March 10. 1675. 49 min. after 9 in the forenoon) to be as followeth. Viz. deg. ♄ Saturn is in 28 Aries. ♃ Jupiter 19 Sagitarius. ♂ Mars 17 Gemini. ☉ Sol 00 Aries. ♀ Venus 18 Aquarius. ☿ Mercury 15 Aries. ☽ Luna 24 Capricorn. ☋ Dragons Tail 17 Capricorn. ☊ Dragons Head 17 Cancer. Having thus obtained the places of the Planets either by Calculation, by Ephemerideses, or Instrument, you may place them in their Correspondent places in your Figure, as is here done, and so is your Figure ready to give your Judgement upon. Lat. N. 51 d. 30 m. A Figure of Heaven at the Sun's entrance into ♈, March 10. 49 min. after 9 in the Forenoon 1675, according to REGIOMONTANUS. an illustration of the heavens at the sun's entrance into Aries according to Regiomontanus II. To Erect a Figure of the Heavens for the forementioned time, Anno 1675, March 10 day at 49 min. passed 9 in the Morning. According to Campanus. Definition. THe Cuspises of the Four Cardinal Houses according to Campanus, viz. the Ascendent, Mid-heaven, Descendent, and Imum Coeli, are the same as they were according to Regiomontanus: But, as Regiomontanus divided the Houses by the Circle of Positions passing through each 30th degree of the Equinoctial, and intersecting the Ecliptic in the Cuspises of the several Houses; Campanus divides the 12 Houses of Heaven by the Position-Circles passing through each 30th degree of the prime Vertical Circle, (or Azimuth of East and West) and so intersecting the Ecliptic in the Cuspises of the several Houses. So that to Erect a Figure according to Campanus way, you must do as followeth. Viz. Practice. You must first set the Globe to the Latitude, bring the Sun's place at noon for the day given to the Meridian, and the Hour-index to 12, and the Circle of Position to the East-side of the Horizon; than turn the Globe about till the Index point at the given hour, and than fix the Globe. Thus far as in the former way of Regiomontanus; and the Cuspises of the First, Fourth, Seventh, and Tenth Houses will be the same as in his way. But now to proceed according to Campanus. The Globe seated and fixed in this Position, bring the Quadrant of Altitude to the Zenith, and there fix it; and bring the lower part of the Quadrant of Altitude to the East-point of the Horizon, and there keeping of it fixed: 1. Elevate the Circle of Position, till it cut 30 deg. of the Quadrant of Altitude, and than see what degrees of the Ecliptic are cut by the Position-Circle, which you shall find to be 22 deg. of Aries; which point must be set upon the Cuspis of the Twelfth House. 2. Move the Circle of Position yet 30 deg. higher upon the Quadrant of Altitude, namely, to 60 deg. from the Horizon, and than see what degree of the Ecliptic is cut thereby, and you shall find that the Position-circle cuts the Ecliptic in 12 deg. of Pisces, which must be set upon the Cuspis of the Eleventh House. The Meridian cuts the Ecliptic in 25 deg. of Aquarius, which is the Cuspis of the Tenth House, or Mid-heaven, as before. 3. Remove the Circle of Position, and also the Quadrant of Altitude, from the East to the Westside of the Horizon, and there move him from the Meridian downwards, till 30 deg. of the Quadrant of Altitude be intercepted between the Zenith and the Circle of Position, and than see where the Position-circle cuts the Ecliptic, which will be in 12 deg. of Aquarius; and those degrees must be set upon the Cuspis of the Ninth House. 4. Let down the Circle of Position 30 deg. more upon the Quadrant of Altitude, that is, 60 deg. distant from the Zenith, and than look what degrees of the Ecliptic are cut by the Position-Circle, which you will find to be 28 deg. of Capricorn; which degrees must be set upon the Cuspis of the Eighth House. The Cuspis of the Seventh House is 00 deg. 29 min. of Capricorn, being the Intersection of the Westside of the Horizon with the Ecliptic, as before. And by this Artifice you have obtained the degrees belonging to the 1, 12, 11, 10, 9, and 8 houses above the Horizon, according to the way prescribed by Campanus: Now the other Six Houses under the Horizon, viz. the 2, 3, 4, 5, 6 and 7, are easily found, they being the opposite Points of the Ecliptic to the Six above the Horizon. So, d. m The Cuspis of the 1 house being Cancer 0 29 12 Aries 22 0 11 Pisces 12 0 10 Aquarius 25 0 9 Aquarius 12 0 8 Capricorn 28 0 The Cuspis of the 7 house will be Capricorn 0 29 6 Libra 20 0 5 Virgo 12 0 4 Leo 25 0 3 Leo 12 0 2 Cancer 28 0 And thus have you the Cuspises of the several Houses according to the way prescribed by Campanus; which Figure having the Planets placed therein, will stand as in the following Scheme. Lat. N. 51 d. 30 min. A Figure of Heaven at the Sun's entrance into Aries, March the 10th 1675. at 49 min. after 9 in the Forenoon, according to CAMPANUS. an illustration of the heavens at the sun's entrance into Aries according to Campanus III. To Erect a Figure of Heaven for the forementioned time, viz. March 10. 1675, at 49 min. after 9 in the Forenoon. According to Alcabitius. Definition.] ALcabitius would have the XII Houses of Heaven to be divided by Domifying Circles, or Circles of Position drawn from the Poles of the World through every 30th deg. of the Equator, beginning at the point of the Ecliptic Ascending; and so counting 30 deg. upon the Equinoctial from thence, shall be the Cuspises of the several Houses.— Wherhfore, to Erect a Figure according to this mode, do as followeth. Practice. You must Rectify the Globe to the Latitude, bring the Sun's place at noon to the Meridian, the Index of the Hour-wheel to 12 at noon, and turn the Globe about to the hour given, and than fix it; so shall the Ascendent be the same as in the two former ways, viz. 00 deg. 29 min. of Cancer. Than look what Meridian passeth through the Ascendent, and count 30 deg. more upon the Equinoctial; and that Meridian where it passeth through the Ecliptic shall be the Cuspis of the Twelfth House, and 30 deg. forwarder that Meridian shall cut the Ecliptic in the Cuspis of the Eleventh House, and so onward till you come to the Cuspis o● the Descendent. And thus, If from that Meridian which passeth through 00 deg. 29 min. of the Ecliptic you count 30 deg. more upon the Equinoctial, you shall find that d. m. The First Meridian of 30 deg. distant from the Ascendent, will cut the Ecliptic in 0 29 Cancer, Second 2 29 Gemini, Third 2 29 Taurus, Fourth 0 29 Aries, Fifth 28 29 Aquarius, Sixth 28 29 Capricorn, Which are the Cusps of the 1 houses. 12 11 10 9 8 And the opposite Signs and degrees of the Ecliptic shall give the Cuspises of the Six other under the Earth; namely, of the 7, 6, 5, 4, 3, and 2. And thus your Figure being erected, and the Planets placed therein, you will find it to be as in the following Scheme. Lat. N. 51 d. 30 m. A Figure of Heaven at the Sun's entrance into Aries, March the 10th 1675. at 49 min. after 9 in the Forenoon, according to ALCABITIUS. an illustration of the heavens at the sun's entrance into Aries according to Alcabitius iv To Erect a Figure for the forementioned time, viz. March 10. 1675. at 49 min. after 9 in the Forenoon, the Sun than entering Aries in the Meridian of London. According to the way prescribed by Ptolemy. Definition.] PTolomy adviseth that the Heavens should be divided into 12 Houses or Mansions, by Domifying Circles drawn through the Poles of the Ecliptic, and through every 30th deg. thereof from the Ascendant downwards, round about.— Wherhfore to Erect a Figure of the Heavens according to this way of Ptolemy, do thus. Practice. Rectify the Globe and Hour-circle as before, and you shall have the same Ascendant, viz. 00 deg. 29 min. of Cancer to Ascend, which is the Cuspis of the First House; than 30 deg. forwarder, downwards, will be 00 deg. 29 min. of Leo, for the Cuspis of the Second House, etc. d. m. So the Cuspis of the 1 house will be 0 29 Cancer, 2 0 29 Leo, 3 0 29 Virgo, 4 0 29 Libra, 5 0 29 Scorpio, 6 0 29 Sagittary. And the Cuspis of the 7 house 0 29 Capricorn. 8 0 29 Aquarius. 9 0 29 Pisces. 10 0 29 Aries. 11 0 29 Taurus. 12 0 29 Gemini. And so a Figure Erected for the forementioned time, according to this prescription of Ptolemy, and the Planets placed therein, will appear as in the following Scheme or Figure. Lat. N. 51 d. 30 m. A Figure of Heaven at the Sun's entrance into Aries, March the 10th 1675. at 49 min. after 9 in the Forenoon, according to PTOLEMY. an illustration of the heavens at the sun's entrance into Aries according to Ptolemy And thus have I shown you the manner of Erecting of a Figure of the Heavens according to the Prescriptions of the four forementioned Authors, and have placed the Planets, Dragons-head and Tail, and the place of Fortune in each of them; by which you may see, that according to these four Varieties of Erections, the Planets keep not in the same Houses, as by the following Synopsis appears. For, is in accor. to Regiomon. accor. to Campan. accor. to Alcabit. accor. to Ptolemy Saturn XI House, XII House, X House, X House. Jupiter VI VI VI VI Mars XII XII XII XII Sol XI XI IX IX Venus IX IX VIII VIII Mercury XI XI X X Luna VIII VII VII VII Dragons-tail VII VII VII VII Dragons-head I I I I Part of Fort. XI XII X X PROB. II. Having the Longitude and Latitude, or the Right Ascension and Declination of a Star, Planet, or Comet, how to find the Place thereof upon the Globe, and to insert it therein if need require. WHat the Longitude, Latitude, Right Ascension and Declination of a Star or Planet is, you have heretofore defined; and in the last Problem you are taught how to Erect a Figure of the Heavens Four several ways. Now if you would see (upon the Globe itself) in what House any of the Planets are, without Erecting a Figure, you were best to make Marks, or set Characters of those Planets, Comets, or the like, upon the Globe, before you delineate your Figure upon Paper; which to effect, do thus. I. By the Longitude and Latitude given. If the Latitude of the Star, Planet, or Comet be Northward Southward Elevate the North-pole South-pole of the Globe to 66 d. 30 m. than will the Pole of the Ecliptic be in the Zenith, and the Ecliptic circle will lie in the very Plain of the Horizon; in which Position of the Globe, screw the Quadrant of Altitude in the Zenith, over the Pole of the Ecliptic.— This done, bring the point of Longitude of the Planet in the Ecliptic to the Quadrant of Altitude, and count the Latitude of the Planet upon the Quadrant, and under the degree of Latitude shall be the point upon the Globe in which the Planet or Star in the Heavens is. And so the Longitude of the Seven Planets being as it is expressed in the following Table, such Longitude and Latitude they had at the time of the Erection of the former Figure, March 10. 1675, at 49 min. after 9 in the Forenoon. Now, if you find their respective Points upon the Globe (as I have now shown you how to do) you may than, not only see in what House each of them shall be, but also what Fixed Stars are there, and what Fixed Stars are in the other Houses also. The Longitude of ♄ Saturn 27.49 ♈, And the Latitude 2.17 M.A. ♃ Jupiter 18.55 ♐, 0.38 S. D. ♂ Mars 17.24 ♊, 1.48 S. A. ☉ Sol 00.00 ♈, 0.00. ♀ Venus 17.56 ♒, 3.44 S. D. ☿ Mercury 15.15 ♈, 1.07 S. A. ☽ Luna 24 47 ♑, 0.35 S. D. II. By the Right Ascension and Declination given. Suppose a Star, Planet, or Comet to have 147 deg. 43 min. of Right Ascension, and 33 deg. 33 min. of North Declination, as the Star Regulus, or the Lion's Heart hath, and you would find its place upon the Globe; Count 147 deg. 43 min. the Stars Right Ascension upon the Equinoctial from the beginning of Aries, and bring that point of the Equinoctial to the General Meridian; and keeping the Globe there, count 33 deg. 33 min. the Stars Declination upon the Meridian upwards, (because the Stars Declination is Northward) and that point shall be the place of Regulus upon the Globe. And thus may any part, or point, in the Heavens be found upon the Globe, if either the Longitude and Latitude, or the Right Ascension and Declination of that point be first known. PROB. III. To know in what House, or under what Circle of Position, any Star, Planet, or point of the Ecliptic is. Rectify the Globe to the Latitude, the Quadrant of Altitude to the Zenith, the place of the Sun in the Ecliptic to the Meridian, and the Hour-circle to 12; than turn the Globe about to the hour given, and bring the Quadrant of Altitude to the East or West points of the Horizon, and there fix the Globe: Than move the Circle of Position upwards, till it touch the Star, Planet, or other Points of the Ecliptic, which you desire to know the Circle of Position of, than shall the Position-circle cross the Quadrant of Altitude in the number of that Circle of Position in which that Star, Planet, or other Point of the Ecliptic is. Thus, If at the time of the Erection of the former Figure it were required to know in what Circle of Position (or house) the Pleyades or Seven Stars was; Rectify the Globe as before, and bring the Position-circle to the Pleyades, than shall the Circle of Position cut the Quadrant of Altitude in 42 deg. and under that Circle of Position is the Pleyades or Seven Stars at that time, and so consequently in the XIIth House. I might here show how to find the place of the Thing which Astrologers call the Part of Fortune. Also how (as they call it) to Direct a Figure, and to find out Revolutions, etc. But forasmuch as these things are not so Mathematical as to require a Globe, or other Mathematical Instruments, either to Demonstrate them by, or to ease the Operation; I shall refer them that have occasion for such things, to the Arithmetical working of them by the Pen, as most convenient, and reserve the Globes for other purposes. PROB. IU. Of the Planetary hours, how to find the Length thereof, and what Planet at is that Reigneth any common hour of the day or night. Definition 1 A Planetary hour for the day, is the 12th part of the Artificial day counted from the time of the Suns Rising to its Setting; and a Planetary hour for the night, is the 12th part of the Artificial night counted from the time of the Sun Setting to the time of its next Rising: So that the Planetary hours are not of the same length all the year long, as the Common hours of 60 min. are, but are some times of the year longer, and sometimes shorter.— For, when the Artificial day is above 12 hours long, (as it is all the time that the Sun is in the Six Northern Signs) than doth a Planetary hour contain more than 60 min.— And when the Artificial day is lesle than 12 hours long, (as it is all the time that the Sun is in the Six Southern Signs) than doth the Planetary hour contain lesle than 60 min.— But when the Sun is in the Equinoctial, and the Artificial day and night are equal (each containing just 12 hours) than the Common hour and the Planetary hour are the same, either of them containing just 60 min. definite. 2 The Planetary hours take their denominations from the Planetary names of the days of the week, as Sunday hath for its Planetary name Sol. Monday Luna. Tuesday Mars. Wednesday Mercury. Thursday Jupiter. Friday Venus. Saturday Saturn. So that upon whatsoever day of the week you would know what Planetary hour it is, the first Planetary hour of that day or night is called by the Planetary name of that day of the week; as, if it be Monday, than Luna governs the first Planetary hour that day and night; if Sunday, Sol; if Wednesday, Mercury, etc. I. To find the length of a Planetary hour at any time. Practice. For the day proposed, (after you have rectified the Globe to the Latitude, etc.) bring the Sun's place in the Ecliptic for that day to the East-side of the Horizon, and see what degree of the Equinoctial is cut by the Horizon; than bring the Sun's place to the Meridian, and than again see what degree of the Equinoctial is than cut by the Horizon: the difference of those degrees being divided by Six, because there are 6 Planetary hours between Sunrising and our 12 at noon (which is always the Sixth Planetary hour) and that Quotient shall be the number of minutes contained in a Planetary hour all that day. Thus for Example, on Tuesday the 27 of July 1675, the Sun's place will be found to be in 14 deg. of Leo; bring 14 deg. of Leo to the East-part of the Horizon, and you shall find that 115 deg. of the Equinoctial are (at that time) cut by the Horizon; which degrees note down, or mark upon the Globe.— Than bring 14 deg. of Leo to the Meridian, and than you shall find 226 deg. of the Equinoctial cut by the Horizon; so that if you take with your Compasses, or count the number of degrees of the Equinoctial contained between 115 deg. and 226 deg. you shall find them to be 111 deg. [or if you subtract 115 from 226, which is easiest, the difference will be 111 also] and so many degrees of the Equinoctial do pass the Meridian in Six Planetary hours: Wherhfore divide 111 by 6, and in the Quotient you shall find 18 deg. 30 min. to pass th● Equinoctial in One Planetary hour; and so counting 15 deg. for 1 hour, and 1 deg. for 4 min. of time, you shall find the length of the Planetary hour to contain One Common hour and 14 min. more, which is one hour and a quarter wanting one minute.— And the length of a Planetary hour for the night will contain only 46 min. which is lesle than One Common hour by a quarter wanting one minute. II. To find what Planetary hour of the day or night it is. On the forementioned day, Tuesday the 27 of July 1675, Let it be required to know what Planet ruleth that day at 5 of the clock in the afternoon. The length of the Planetary hour for that day is 1 hour 14 min. Wherhfore, the Globe being Rectified, bring the Index of the Hour-circle to 5 a clock, and than count the number of degrees which were cut by the Horizon in the last operation, and the degrees of the Equinoctial now at the Horizon, and you shall find them to be 187 deg. which reduced into minutes of time (by multiplying them by 4) giveth 748, which 748 min. being divided by 74, the number of minutes contained in one Planetary hour, the Quotient will be 10 hours and 8 min. showing that there are 10 Planetary hours passed, since the Sunrising, and that there are 8 min. of the Eleventh hour passed also. III. To find what Planet governeth that hour of the day. For the effecting hereof, the Globe standeth you in no stead at all; wherefore observe the Table following and its Use. The TABLE. Governors of the day. Sunday. Monday. Tuesday. Wednesday. Thursday. Friday. Saturday. Governors of the night. Sol. 1 12 9 0 10 0 11 Jupiter. Venus. 2 0 10 0 11 1 12 Mars. Mercury. 3 0 11 1 12 2 0 Sol. Luna. 4 1 12 2 0 3 0 Venus. Saturn. 5 2 0 3 0 4 1 Mercury. Jupiter. 6 3 0 4 1 5 2 Luna. Mars. 7 4 1 5 2 6 3 Saturn. Sol. 8 5 2 6 3 7 4 Jupiter. Venus. 9 6 3 7 4 8 5 Mars. Mercury. 10 7 4 8 5 9 6 Sol. Luna. 11 8 5 9 6 10 7 Venus. Saturn. 12 9 6 10 7 11 8 Mercury. Jupiter. 0 10 7 11 8 12 9 Luna. Mars. 0 11 8 12 9 0 10 Saturn. The use of the aforesaid Table. Having by former Rule found what Planetary hour it is, if you would know what Planet it is that Reigneth that hour, in the head of this Table seek the day of the week (in this Example Tuesday) and the Planetary hour of the day (in this Example 10 hours in the Forenoon:) in the same Column, and right against it in the first Column on the left hand, you shall find Venus, which shows that at 5 of the clock in the afternoon that day Venus governeth, and hath governed 8 min. of her hour. Also in the same Table you shall find against 10 under Tuesday in the last Column towards the right hand, the word Mars, showing that Mars governeth the 10th Planetary hour for the night, and hath governed 8 min. of his hour. POSTSCRIPT. FOrasmuch as the two first Sections of the foregoing Problem are for the more part performed by Arithmetical computation, and not Globular Operation; I have (for that it may be satisfactory to some persons) added a Table, whereby, if you know but at what hour the Sun riseth at any time of the year, you may know readily the length of the Planetary hour, and also at any Common hour of the day or night, what Planetary hour it is, and also what Planet ruleth that hour. A TABLE, Showing the length of The PLANETARY HOUR For any hour of the Day or Night At any time of the Year, etc. In the Forenoon. ☉ Ris. I TWO III IV V VI H. M. h. m. h. m. h. m. h. m. h. m. H. 6 0 7 0 8 0 9 0 10 0 11 0 12 5 47 6 50 7 52 8 54 9 56 10 58 12 5 35 6 39 7 43 8 48 9 52 10 56 12 5 22 6 28 7 35 8 41 9 47 10 54 12 5 10 6 18 7 27 8 35 9 43 10 52 12 4 57 6 8 7 18 8 29 9 39 10 50 12 4 45 5 58 7 10 8 23 9 35 10 48 12 4 34 5 48 7 3 8 17 9 31 10 46 12 4 23 5 39 6 55 8 12 9 28 10 44 12 4 12 5 30 6 48 9 6 9 24 10 43 12 4 2 5 23 6 42 8 2 9 21 10 40 12 3 55 5 16 6 37 7 58 9 18 10 39 12 3 48 5 10 6 32 7 54 9 16 10 38 12 3 43 5 6 6 29 7 52 9 14 10 37 12 3 40 5 3 6 27 7 50 9 13 10 36 12 6 0 7 0 8 0 9 0 10 0 11 0 12 6 13 7 11 8 9 9 7 10 4 11 2 12 6 25 7 21 8 17 9 13 10 8 11 4 12 6 38 7 30 8 24 9 19 10 13 11 6 12 6 50 7 42 8 23 9 25 10 17 11 8 12 7 3 7 53 8 42 9 32 10 20 11 10 12 7 15 8 3 8 50 9 38 10 25 11 12 12 7 26 8 12 8 57 9 43 10 29 11 14 12 7 37 8 21 9 5 9 49 10 34 11 16 12 7 48 8 30 9 12 9 54 10 36 11 18 12 7 57 8 38 9 18 9 59 10 39 11 20 12 8 5 8 44 9 23 10 4 10 41 11 21 12 8 12 8 50 9 28 10 6 10 45 11 22 12 8 17 8 54 9 31 10 9 10 46 11 23 12 8 20 8 58 9 34 10 10 10 46 11 24 12 In the Afternoon. ☉ Ris. VII VIII IX X XI XII. H. M. h. m. h. m. h. m. h. m. h. m. h. m. 6 0 1 0 2 0 3 0 4 0 5 0 6 0 5 47 1 2 2 4 3 6 4 8 5 10 6 13 5 35 1 4 2 8 3 13 4 17 5 21 6 25 5 22 1 6 2 13 3 19 4 25 5 32 6 38 5 10 1 8 2 17 3 25 4 33 5 42 6 50 4 57 1 11 2 21 3 32 4 42 5 43 7 3 4 45 1 13 2 25 3 38 4 50 6 6 7 15 4 34 1 14 2 29 3 43 4 57 6 12 7 26 4 23 1 16 2 32 3 49 5 5 6 21 7 37 4 12 1 18 2 36 3 44 5 12 6 31 7 48 4 2 1 20 2 39 3 59 5 18 6 38 7 57 3 55 1 21 2 42 4 3 5 23 6 44 8 5 3 48 1 22 2 44 4 6 5 28 6 50 8 12 3 43 1 23 2 46 4 9 5 31 6 54 8 17 3 40 1 24 2 47 4 10 5 33 6 57 8 20 6 0 1 0 2 0 3 0 4 0 5 0 6 0 6 13 1 58 2 56 2 54 3 51 4 49 5 47 6 25 1 56 2 50 2 48 3 43 4 39 5 35 6 38 1 54 2 47 2 41 3 35 4 28 5 21 6 50 1 52 2 43 2 34 3 37 4 18 5 10 7 3 0 50 1 39 2 29 3 18 4 8 4 57 7 15 0 48 1 35 2 23 3 10 3 58 4 45 7 26 0 46 1 31 2 17 3 3 3 48 4 38 7 37 0 44 1 28 2 12 2 55 3 39 4 21 7 48 0 43 1 24 2 6 2 48 3 30 4 13 7 57 0 41 1 21 2 2 2 42 3 23 4 3 8 5 0 39 1 18 1 57 3 6 3 15 3 55 8 12 0 38 1 16 1 54 3 2 3 10 3 48 8 17 0 37 1 14 1 51 2 8 3 5 3 43 8 20 0 36 1 13 1 49 2 27 3 3 3 40 Before Midnight. ☉ Ris. I TWO III IV V VI H. M. h. m. h. m. h. m. h. m. h. m. H. 6 0 7 0 8 0 9 0 10 0 11 0 12 5 47 7 11 8 9 9 7 10 4 11 2 12 5 35 7 21 8 17 9 13 10 8 11 4 12 5 22 7 30 8 25 9 19 10 13 11 6 12 5 10 7 42 8 33 9 25 10 17 11 8 12 4 57 7 53 8 42 9 32 10 21 11 10 12 4 45 8 3 8 50 9 38 10 25 11 12 12 4 34 8 12 8 57 9 43 10 29 11 14 12 4 23 8 21 9 5 9 49 10 32 11 16 12 4 12 8 30 9 12 9 54 10 36 11 18 12 4 2 8 38 9 18 9 59 10 39 11 20 12 3 55 8 44 9 23 10 2 10 41 11 21 12 3 48 8 50 9 28 10 6 10 44 11 22 12 3 43 8 54 9 31 10 9 10 45 11 23 12 3 40 8 57 9 33 10 10 10 47 11 23 12 6 0 7 0 8 0 9 0 10 0 11 0 12 6 13 6 47 7 51 8 54 9 56 10 58 12 6 25 6 39 7 43 8 48 9 52 10 56 12 6 38 6 28 7 35 8 41 9 47 10 54 12 6 50 6 18 7 27 8 35 9 43 10 52 12 7 3 6 8 7 18 8 29 9 39 10 50 12 7 15 5 58 7 10 8 23 9 35 10 48 12 7 26 5 48 7 3 8 17 9 31 10 46 12 7 37 5 39 6 55 8 12 9 28 10 44 12 7 48 5 30 6 48 8 6 9 24 10 42 12 7 57 5 23 6 42 8 2 9 21 10 41 12 8 5 5 16 6 37 7 58 9 18 10 39 12 8 12 5 10 6 32 7 54 9 16 10 38 12 8 17 5 6 6 29 7 52 9 14 10 37 12 8 20 5 3 6 27 7 50 9 13 10 36 12 After Midnight. ☉ Ris. VII VIII IX X XI XII. H. M. h. m. h. m. h. m. h. m. h. m. h. m. 6 0 1 0 2 0 3 0 4 0 5 0 6 0 5 47 1 58 1 56 2 54 3 51 4 41 5 47 5 35 1 56 1 52 2 48 3 43 4 39 5 35 5 22 1 54 1 47 2 41 3 35 4 28 5 22 5 10 1 52 1 43 2 35 3 27 4 18 5 10 4 57 0 50 1 39 2 29 3 18 4 8 4 57 4 45 0 48 1 35 2 23 3 10 3 58 4 45 4 34 0 46 1 31 2 17 3 3 3 48 4 34 4 23 0 44 1 28 2 12 2 55 3 39 4 23 4 12 0 42 1 24 2 6 2 48 3 30 4 13 4 2 0 41 1 21 2 2 2 41 3 23 4 3 3 55 0 39 1 18 1 57 2 36 3 15 3 55 3 48 0 38 1 16 1 54 2 32 3 10 3 48 3 43 0 37 1 15 1 51 2 28 3 5 3 43 3 40 0 37 1 14 1 50 2 27 3 3 3 40 6 0 1 0 2 0 3 0 4 0 5 0 6 0 6 13 1 2 2 4 3 7 4 9 5 11 6 13 6 25 1 4 2 8 3 13 4 17 5 21 6 25 6 38 1 6 2 13 3 19 4 25 5 32 6 38 6 50 1 8 2 17 3 25 4 33 5 42 6 50 7 3 1 11 2 21 3 32 4 43 5 53 7 3 7 15 1 13 2 25 3 38 4 50 6 3 7 15 7 26 1 14 2 29 3 44 4 57 6 12 7 26 7 37 1 16 2 32 3 49 5 5 6 21 7 37 7 48 1 18 2 36 3 54 5 12 6 30 7 48 7 57 1 20 2 39 3 59 5 18 6 38 7 57 8 5 1 21 2 42 4 2 5 23 6 44 8 5 8 12 1 22 2 44 4 6 5 28 6 50 8 12 8 17 1 23 2 46 4 9 5 31 6 54 8 17 8 20 1 24 2 47 4 10 5 33 6 57 8 20 The Table described. EAch Page of the Table consisteth of 7 Columns: in the first Column of each Page, towards the left hand, is placed the time of the Suns Rising, and in the Six subsequent Columns are placed the beginnings and continuances of of the Planetary hours.— The first of the four Pages contain the Six Planetary hours for the Day, viz. I, TWO, III, IV, V, and VI, from Sunrising to Noon.— The second Page contains the other Six Planetary hours for the Day, viz. VII, VIII, IX, X, XI, and XII, from Noon to Sun setting.— The third Page contains the first Six Planetary hours for the Night, viz. the I, TWO, III, IV, V, and VI, from Sunsetting till Midnight.— And the fourth Page contains the other Six Planetary hours for the Night, viz. the VII, VIII, IX, X, XI, and XII, from Midnight till Sunrising the next Morning. Thus much for the Description. Now followeth The Use of the Table. LEt it be required to find what Planet Ruleth upon Friday the 27 of March, at 10 of the clock in the Forenoon. First, You must find the time of the Suns Rising for the day proposed, by the Astronomical Problem for that purpose; which you shall find to be at 22 min. after 5 in the Morning. Secondly, Find this 5 hours and 22 min. in the first Column towards the left hand, and look along that line towards the right hand, and you shall find under the first Planetary hour 6 hou. 28 min. which is the time of the first Planetary hour; under the second, 7 hou. 35 min. for the third 8 hou. 41 min. etc. Now, because 10 of the clock is the time that I require the Planetary hour, I continued looking along that line, till I find 10 of the clock, and I find 10 hou. 54 min. to stand under the fifth Planetary hour; which shows that it is the fifth Planetary hour, and that that hour did begin at 47 min. after 9 of the clock, and will continued till 54 min. after 10 of the clock. Now to know what Planet it is that Ruleth at that time, repair to the foregoing Table, and find Friday in the head thereof, and the Common hour of the day given (in our Example 10.) under Friday; and right against 10 towards the left hand, you shall find Luna, which showeth that Luna Rules, and will continued Ruling till 54 min. after 10 of the clock. The like is to be understood of all the rest: and if it be the Night-hour, in the little Table you must find the name of the Ruling Planet on the right hand of the Table. Note, That if you cannot find the very exact time of Sunrising in the Table, you must make use of that which is nearest to it, which will be sufficient for this purpose. GEOGRAPHY AND NAVIGATION Made easy: OR, A plain Description and Use of the Terrestrial GLOBE. INTRODUCTION. THe Terrestrial or Earthly Globe is an artificial Representation of the Earth and Water under that form and figure of roundness which they are known to have describing the Situations, and measuring the distances of all their parts. The Land drawn out upon a Globe, is bounded and distinguished from the Water with an irregular line which runs turning and winding into Creeks and Angles like as the shore which it represents: that side which is left uncoloured, is the limits of the water; the other side of the Line which encircles the Colours, is the bounds of the Land, which is either Continent or Island. A Continent is a great quantity of Land not environed or separated by the Sea, in which many Kingdoms and Countries are contained; as Europe, Asia, etc. An Island is a part of the Earth clasped in the embraces of the Sea, and hooped as it were with a watery Girdle, as Great Britain and Ireland. These again are subdivided into Peninsula's, Isthmus', Promontories, Capes, etc. A Peninsula or Pene-Insula is a part of Land which being almost environed and encompassed round with water, is joined to the firm Land by some little Isthmus, as Africa is joined to Asia, or Morea to Greece. An Isthmus is a little narrow neck of Land betwixt two Seas, joining a Peninsula to the Continent; as that of Darien in America, or Corinth in Greece. A Promontory is Mons in Mari prominens, a high Hill or Mountain lying out as an Elbow of land into the Sea, the utmost end of which is called a Cape; as the Cape of Good Hope, and Cape Verde. The Land drawn upon the Superficies of the Terrestrial Globe, is divided in four principal Parts or Quarters, viz. Europe, Asia, Africa, and America. Of EUROPE. Europe as it is now divided, contains these Kingdoms or States. Cities. Rivers. England. London, York, Oxford, Cambridge, Canterbury, Bristol. Thames, Savern. Humber, Trent. Scotland. Edenbrugh, St. Andrews. Tweed, Froth, Tay. Ireland. Dublin, Waterford, Galloway, Limerick. Shannon Sure. Blackwater, Barrow France. Paris, Lions, Orleans, Bordeaux, Tholouse, Aix. Seine, Loire. Garone, Rosne. Spain. Madrid, Siville, Toledo, Saragosse. Ebre, Gaudalquivir. Gaudiana, Douro. Portugal. Lisbon, Braganza. Tagus. Belgia, or the 17 Provinces. Amsterdam, the Hague, Antwerp, Brussels. , Isel, Lis, Eschant. Italy. Rome, Florence, Venice, Naples, Genua, Milan. Po. Arhe. Tiber. Savoy. Chambery, Turin. Doire, Lisire. The Swisseses. Geneve, Basel, Zurich. Rus, Aar. Denmark. Copenhagne, Sleswick. Bergen, Christiane. Sley, Eyder. Sweden. Stockholm, Gottenburg, Upsal, Calmer. Wenar, Veter. Poland. Cracovia, Danzick, Vilna, Warzovia. Duna, Neimen, Vistula, Neiper. Neister, Bog. The Empire of Germany. Vienna, N●r●mberg, Hambrough, Prague, Collen, Heidelburg. Danube, Elbe. Rhine, Veser. Oder. Russia. Moscow, Novogrodt, Archangel, Smolensko. Wolga, Dwina. Tanais, Boristenes. Turkey in Europe. Constantinople, Belgrade, Adrianople, Saloniche. Danube, Dravus. Savus, Tebiscus. Petite Tertaria. Cers and Cassa. Islands. Great Britain, Ireland, Wight, Man, Zealand, Candia, Sicily, Sardinia, the Hebrides, Orcades, etc. Of ASIA. Asia as it is now divided, contains these Empires or Countries. Cities. Rivers. The Turkish Empire. Aleppo, Smyrna. Damascus, Jerusalem, Bagdat, Mosul. Tigris. Euphrates. Georgia. Fazo, Testis. Fazis, Kur. The Arabia's. Mecca, Medina, Anna, Mocha. Chaiber, Nageran. The Persian Empire. Ispahan, Casbin, Tauris, Suras, Ferrabat, Herat. Teus, Pulimalon, Ilment, Brandemer. The Moguls Empire. Agra, Lahor, Delli, Amadabat, Cambaja, Surrat. Indus, Ganges, Jemini, Guenga. India within Ganges. Goa, Calicut, Cochin, Negapatan, Narsinga, Golconda, Mazulpatan. Ganges. India without Ganges. Pegue, Sian, Mallaca, Cambodia. Ganges, Caor, Ava, Martaban. Tartary. Astracan, Sarmachand, Cascar, Balch, Tanchut. Oxas, Chesel, Obey. China. Pekin, Canton, Nanking, Hancheu. Croceus, Ta, Kiang. Japan. Meaco, Jedo, Firando, Nangasacque. Islands. The Maldives, Ceylon, Sumatra, Java, Borneo, Celebes, the Philippines, and the Molucces. Of AFRICA. Africa as it is now divided, contains these Parts or Countries. Cities or principal places. Rivers. Barbary. Fez, Morocco, Salle, Tanger, Algiers, Tunis. Suba, Tenlis, Sus, Omirabib. Egypt. Cairo, Alexandre, Rossette, Damiete. Nilus. Biledulgerid. Segelmess, Taradant, Biledulgerid, Dara. Ghir. The Deserts of Sarra. Zanbaga, Zuenziga, Terga, Lempta, Berdoa. Ghir. The Land of the Blacks. Tombolu, Agades, Borno, Zanfara. Niger, Senega, Gambea, Grande. Kingdom of Nubie. Nubie, Darga, Dancala, Gorham. Nilus, Ghir. Guinee. St. George de la Mine, Cape Cors, Settera, Benin. Sweiro, Mancha, Calabar, Benin. Aethiopia, Amara, Caxumo, Cafales, Arquico. Nilus. Congee. St. Salvador. Zaire, Coanza, Coango. Caffares. Cefala, Mono, Motapa, Cape of Good Hope. Spirito Sancto, Rio d'Infanta. Zanguebar. Mosambique, Quiloa, Mombaza, Melinde, Adel. Islands. The Maderas, Canaries, Cape Verde, St. Helena, Madagascar, Babelmandel, Zocotora, Malta, etc. Of AMERICA. America as it is now divided, contains these Parts or Countries. Cities or principal places. Rivers. The Coast of the N.W. passage. Carlton Island, Fort Charles. Prince Ruperts Riu. Canada, or Nova Fraacia. Quebeck, Breast, Hurones, Port-royal. Canada. New Engl. and New York. Boston, Plymouth, New York, Milford. Hudsons, Conecte. Marimack, paccat. Maryland and Virginia. St. Maries, James Town. ●ames, York, Patomeck, Rapahanok. Carolina and Florida. Charles Town, St. Matthew. Albemarle, Ashly, Clarendon, Cooper. New Mexico. New Mexico. B. del Norto. New Spain. Mexico, Valadolid, St. Jago, Gaudalajara. Panuco, Baran●a, Qacatulca, Sal. Castilladel, or Panama, Porto Bello, Cartagena, St. Martha. St. Martha, St. Magdalen. Guiana. Manoa, St Thome. Oronoque Surinam, Wiapoca. Perve. Lima, Casco, Potosi, La plata. Maragnon, Patinie, Chili. St. Jago, Imperiai, Baldivia. Copayapo. The Country over Amazons. Homagus, Yorimen, Topinambes, Coropa. Topacalma, Amazons. Paraguay. Buenos Airs, Cuidat, Real, St. Jago, Estero. Rio de plata, Perana. Brasil. Pairaba, Permambuco, Baha, Spirito Sancto. Grandee S. Francisc. Ilheos', Janiero. Islands. Hispaniola, Cuba, Jamaica, Newfoundland, the Caribee- Islands, Long Island, Magellan Island, etc. And thus much shall serve for the Description of the Land upon the Terrestrial Globe. Of Water and its Parts. THe Earth (as was said defore) is encompassed about with the Water, which is either Ocean, Seas, Straitss, Creeks, Lakes, or Rivers. The Ocean is a general Collection or Rendezvouz of all waters. The Sea is a part of the Ocean, and is either Exterior, lying open to the shore, as the British or Arabian Seas; or Interior, lying within the Land, to which you must pass through some Straight, as the Mediterranean or Baltic Seas. A Strait is a narrow Part or Arm of the Ocean, lying betwixt two shores, and opening a way into the Sea, as the Straitss of Gibraltar, the Hellespont, etc. A Creek is a small narrow part of the Sea that goeth up but a little way into the Land, otherwise called a Bay, a Station or Road for Ships. A Lake is that which continually retains and keeps water in it, as the Lake Nicurgua in America, and Zaire in Africa. A River is a small Branch of the Sea flowing into the Land, courting the Banks while their Arms display to embrace her silver Waves. Of the Names of the Ocean. FIrst of all, according to the four Quarters it had four Names; from the East it was called the Eastern-Ocean, from the West the Western, from the North the Northern, and from the South the Southern. But besides these more general Names, it hath other particular appellations, according to the Regions or Countries it boundeth upon, and the Nature of the Sea: viz. as it lies extended toward the East, it is called the Chinean Sea, from the adjacent Country of China; so the Archipelago of St. Lazarus, from the multitude of Islands. Toward the South, 'tis called Oceanus Indicus, or the Indian Sea, because upon it lies the Indians. The Golph of Bengala, from Bengala a City in the Indieses. Where it touches the Coast of Persia, it is called Mare Perficum: so also Mare Arabicum, from Arabia; so towards the West, is the Aethiopian Sea. Than the Atlantic Ocean, from Atlas a Mountain or Promontory in Africa; by the Spaniards called Mar deal zur, as also Mare Pacificum; and on the other side of America is called by them Mar deal Nort. Where it touches upon Spain, it was called Oceanus Cantabricus, now the Bay of Biscay. The Sea between England and France is called the Channel; between England and Ireland, the Irish Sea, by some St. George's Channel: between England and Holland, it is called by some the Germane Ocean, by others the British Seas; Vulgò, the Narrow Seas: beyond Scotland it is called Marc Caledonium; higher towards the North, it is called the Hyperborean or Frozen Sea; more Eastward upon the Coast of Tartary, the Tartarian Sea, etc. And this shall suffice concerning the Ocean, or Exterior Seas. The Names of the Inland Seas. THe Baltic Sea, of old Sinus Codanus, by some called the East-Sea, by the Inhabitants the Belt, lying between Denmark and Sweden, the entrance whereof is called the Sound. Secondy, The Euxine Sea, or the Black Sea, by some Mare Caucasium, Scythicum, Sarmaticum, Colchicum; by the Turks, Caradinizi: to which joins Meotis Palus, now Mar de Zabacken. The third, is the Caspian or Hyrcanian Sea; by the Turks, Mar de Sala; by the Persians', the Sea of Backu; by the Moscovites, Chwalenskei Mare. The length is from North to South, and the breadth from East to West, contrary to all the Ancient Geographers; which is certainly discovered, not only by the exact observation of Olearius, but also by his curious inquiries of the true Situation of its Maritime places according to the Longitude and Latitude of the Persians', as also by the Astronomical Calculations of Mr. Graves, etc. The fourth, is the Arabian Golph, Mare Erythaeum, or Rubrum, Vulgò the Red-sea; by others Mare Rasso, and Mar de Mecca. The fifth, is the Persian Golph, or the Golph the Elcatife. The sixth, is Mare Mediterraneum; by the English, the Straitss; by the Spaniards, Mar de Levant: the beginning or entrance of it is called the Straitss of Gibraltar, rather Gibraltarec, olim Gaditanus. This Sea hath many names, as it toucheth upon several Countries; the particular account whereof I shall refer to Geographical description of Maps, etc. The length of it is by our New Globes not 37 deg. of the Equinoctial from Tanger to Scanderone: by other Globes and Maps it is more than 42 deg. of the Equinoctial. And this shall suffice for a Description of the Water, and its Parts. Now that all Places, Cities, Towns, Seas, Rivers, Lakes, etc. may be readily found out upon the Globe, all Geographers do or should place them according to their Longitude and Latitude: the use of which in the absolute sense, is to make out the position of any place in respect of the whole Globe, or to show the Situation and distance of one place from & in respect of any other. An Advertisement concerning Longitude. To say the truth, by reason of the variety of Meridian's, the Longitudes are grown to such an uncertainty and confused pass, that 'tis not every man's work to set them down. This indeed I have observed, that many Geographers, or rather discribers of particular places, tell us that such a place is so many degrees of Longitude; but from what Meridian, others must guests. Some particularly profess to follow Mercator: but what are most men the wiser for this? for Mercator 's Meridian was not always the same; sometimes through the Canary Islands, sometimes through the Azores. Others again will tell you their Meridian shall pass through the Azores; but whether from that of St. Michael, or that of Corvo, is not set down; and yet 6 deg. of difference. I shall therefore take this course: First, set down the several Meridian's observed. Secondly, the difference of Longitude betwixt these Meridian's. Lastly, which of these I have fixed upon. 1. The Great Meridian by Ptolemy and most of the ancient and Greek Geographers was made to pass through Junonca one of the Fortunate, now thought to be the Canary Islands. 2. By the Arabian and Nubian Geographers through the utmost point of the Western shore near Hercules Pillars. 3. Ortelius in his sheet- Europe makes London to lie in 28 deg. of Longitude; but in the Sheetmaps of France and Belgia it lies in but 21 deg. so that his first Meridian to me is yet unknown. The Spaniards since the Conquest of the West-indieses, contrary to all other, accounted their Longitude from East to West, beginning at Toledo. 4. Our Modern Geographers, as Mercator, Cambden, Speed and others, removed it into the Azores; some placing it at St. Michael's, others at Corvo. 5. Blaew the Dutch Geographer gins his Longitudes from Tenerif one of the Canary Islands; but upon his Great Map, the Great Meridian passeth through Tercera-Isle, one of the Azores; which the rest of their Common Map-makers, De Wit, Visher, etc. as well as many of our English, are bound to follow through ignorance, transcribing as well his Errors as his Copies for the best. 6. Sanson the French Geographer, for some Reasons (best known to himself) gins his Longitudes at Ferro one of the Canary-Islands; and therefore Blome his Translator is bound to follow it, though possibly he cannot tell so much, and yet the Kingdoms great pretender to Geography. 7. The English Hydrographer tells us, that with a great deal of Reason and Consideration he placed his first Meridian at Graciosa one of the Islands of the Azores; but it is delineated upon his Globes and Maps through Tercera, almost 2 deg. more Eastward: a small mistake, that another must come after him to tell himself what Meridian he went by. Secondly, The differences of these several Meridian's I find are thus stated. From Ptolemy's Meridian to the Arabian Meridian was by Abalfeda in his Introduction to his Geography accounted to be 10 deg. of the Equator; Briet saith but 8 deg. d. m. From the Pico of Tenerife to Toledo is 15. 55 the Spanish. Meridian. Graciosa 10.25 the mistaken Engl. Tercera 9 0 the supposed D●tch Palma or Ferro 2. 5 the French Corvo 13 25 Mercator or others St. Michael 8. 5 And this last is the Meridian from which the Longitudes are reckoned in the new Terrestrial Globe, and in several Maps that are lately set forth by the Publishers of this Book. Geographical Problems. PROB. I. To find the Longitude. Definition.] LOngitude is the distance of a place from the first Meridian reckoned in the degrees of the Equator, beginning, as was said, in this New Terrestrial Globe, at St. Michael's Island in the Azores. Practice. Bring the place (that is, the mark of the place) suppose London, to the Brazen Meridian; than count how many degrees of the Equator are contained between the first Meridian and that of London cut by the Brazen Meridian, which you will find to be 28 deg. which is the Longitude required. And in this manner you shall find d. m. London to be distant from the first Meridian by this New Globe 28 0 Jerusalem 66 30 Jedo in Japan 167 0 Rio de la plata 32 0 Mexico 75 0 Charlton Isle 51 30 d. m. By other Globes and Maps 26 0 reckoned from the same Meridian. 73 30 178 0 21 0 86 0 65 50 PROB. II. To find the Latitude of a place. Definition. THe Latitude of a place, is the distance of the Equator from the parallel of of that place, reckoned in the degrees of the Great Meridian; and is either North or South, according as it lies between the North or South-poles of the Equator. Practice. To find the Latitude, bring the mark of the place, for example, suppose London, to the Brazen Meridian; than count the number of degrees upon the Meridian, contained between the Equator and the place given. Thus you shall find the Latitude of d. m. d. m. London to be 51.30 By other Globes and Maps 51.30 Labour in the Moguls Country to be 31.30 23.30 The South-part of the Caspian Sea to be 37.0 41.0 Astracan, on the Nor. part of the Caspian Sea to be 46.0 49.0 The North-part of China to be 42.0 52.0 Delli in India to be 28.0 21.0 PROB. III. The Longitude and Latitude of any place being known, to find the true Situation of it, though not expressed upon the Globe. Practice. BRing the degree of the Equator that answereth to the Longitude of the place to the Meridian, and than reckon the Latitude of the place upon the degrees of the Meridian towards either Pole according as it is either North or South Latitude; and right under that degree and minute upon the Meridian, is the true Situation of the place enquired after. PROB. iv To find what time or hour of the Day or Night it is in any part of the Earth. Definition. BY reason of the earth's diurnal motion round the Sun in 24 hours, the Sun enlightening but one half of it at the same time, it comes to pass that when it is Morning in one place, it is Noon in a second, Night in a third, and Midnight in a fourth, according to their several Situations in respect of East and West one from the other. This difference of time is known by the number of degrees contained in the Equator between any two places proposed converted into hours and minutes, reckoning 15 deg. to an hour, etc. but more readily by the Globe thus: Practice. Suppose at London, at 12 of the clock at Noon, you would know what a clock it is at Mexico in the West-indieses; bring London to the Meridian, and set the Index of the Hour-circle to 12. than turn the Globe Eastward, because London is East of Mexico, till you bring Mexico to the Meridian; than see what hour the Index points at, for that is the hour than at Mexico. Thus you will find, h. m. when it is 12 a clock at London, it is at Mexico 5 10 a clock before Noon. Charlton Island 6 45 Rio de la plata 8 10 Jerusalem 2 35 a clock after Noon. Surrat 5 15 Jedo 9 18 And thus by knowing what difference of time there is between place and place at 12 a clock, the like difference is to be understood of all other hours. PROB. V To find the distance of any two places upon the Globe one from another. Practice. LAy the Quadrant of Altitude upon both the places required; than count the number of degrees of the Quadrant of Altitude contained between the two places: which being found, multiply them by 60, gives the distance in English Miles. Thus you will find d. m. d. m. The distance of London from Mexico 81.30 by our new size of Globes. 91.30 by the great Dutch Globes and English. Jerusalem 33.30 38.0 Surrat 65.50 71.40 Jedo 85.10 92.30 Rio de la plata 100.20 106.0 Charlton Island 46.20 54.0 If you find (as you needs must) that the proportion of Miles upon these new Globes do very much differ from those distances set down by other Authors, you are desired not to think much; for the Longitudes are not yet exactly agreed on: the perfection is not one man's, nor one ages work, and must be waited for. Where you find the places upon this Globe to agreed with others, you have cause to suspect they have lain upon the leeses of time, not as yet inquired into: where you find them to disagree, you may conclude that they have been brought to a truer correction and amendment. PROB. VI To find the Position, or what Point of the Compass any two places are Scituate one from another. Definition. THe Position is an Angle which is made by the meeting of the Meridian of one place, with the Vertical Circle of another. Practice. To find this out, you are to elevate the Pole to the Latitude of one of the places, suppose London; than bring it to the Meridian, and it will fall out to be directly in the Zenith, for the Elevation is always equal to the Latitude; than fasten the Quadrant of Altitude to the Zenith, and turn it about till it fall upon the other place, suppose the Isle of Tenerif, and the end of the Quadrant where it toucheth the Horizon will show that the Isle Tenerif beareth from London S S W: so also the bearing of Barbadoes from the Lizard to be S W, half a point Westerly; and the opposite point N E, half a point Easterly, the bearing of the Lizard from the Barbadoss. PROB. VII. To know at any time in what place of the Earth the Sun is in their Zenith. This must be to such Inhabitants of the Earth only that inhabit in the Torrid Zone between the Tropics. Practice. BRing the place you are in, suppose London, to the Meridian, and the Index to the hour 12; than consider the time of the day, which suppose to be half an hour after 5 in the afternoon, the Sun having than 10 deg. of North-Declination: than because it is afternoon, turn the body of the Globe Eastward, till the Index hath passed 5 hours and ½, from 12, that will be to 7 a clock and ½, and there stay the Globe; than see what place or Country is under the Meridian that cuts 10 deg. of North Declination, and you will find Nombre di dios upon the Isthmus of Panama in the West-indieses. But if it were required the same day at half an hour after 6 in the Morning, than you should have turned the Globe Westward till the Index had passed 6 hours and ½: and than under the Meridian, and upon the parallel of 10 deg. North-Declination, you will find it near 4 or 5 little Islands close by the Westside of Mallacca in the East-indieses, where the Sun will be in Zenith at that time. Having found in what place of the Earth the Sun is in the Zenith, elevate the Globe to the Latitude of the place either North or South; than bring that place to the Meridian, so shall all places cut by the Horizon have the Sun in their Horizon; those to the Westward shall have the Sunrising in their Horizon; those to the Eastward shall have the Sunsetting. In those Countries that are above the Horizon it is daylight, and in those but 18 deg. below the Horizon, it is twilight; but in those Country's further below the Horizon, it is at that time midnight. PROB. VIII. The difference of Longitude being known, to find what degree of the Ecliptic culminates at any other place at any time proposed. March the 10th at 10 of the clock before noon here at London, I would know what degree of the Ecliptic culminates than at Jerusalem. Practice. ELevate the Globe to the Latitude of your place, viz. 51 deg. 30 min. than bring the Sun's place for that day, viz. ♈ 0 deg. to the Meridian, and the Index to 12. than turn the Globe Eastward till the Index point at the given hour, viz. 10 of the clock, and you will find the 28 deg. of ♒ than culminating here at London. Next turn about the Globe Westward until 33 deg. 30 min. of the Equator be passed through the Meridian, or till the Index of the Hour-circle be moved 2 hours 35 min. which is the difference of Longitude given, and than you will find the Meridian cut the Ecliptic at almost 4 deg. of ♈: so that I say, ♈ 4 deg. is the point of the Ecliptic that is than culminating at Jerusalem. PROB. IX. To find out the several Positions of the Inhabitants of the Earth, the distinction of Shadows, the different Habitations, etc. THe Longitude and Latitude of a place once resolved on, the Position of the Sphere you cannot miss of; for if the place you try for have no Latitude, it must of necessity lie under the Equator; and therefore in a Right Position: if it have lesle or more Latitude, the Position is Obliqne; if the place have 90 deg. of Latitude, the Position is Parallel: the Reasons were told before, and may evidently be discerned upon the Globe. For the Climes and Parallels, and consequently the length of the longest day, the foreknowledge of the Latitude leadeth you directly; for they who are under the Equator, have their day always 12 hours, and their night 12 hours long. Now as each Country declines from the Equator towards either of the Poles, so the days vary their length in Summer, and the nights theirs in Winter: according therefore to the different lengthening of their days, the Ancients did distinguish the Earth into several portions or parts, which they called Climates and Parallels: every Clime contains two Parallels; so that where the longest days are half an hour longer than at the Equator, the first Climate gins; and where they are increased an hour longer than at the Equator, the second Climate gins, which by the 20th Problem you will find to be at 8 deg. 34 min. for the first, and at 16 deg. 43 min. where the second gins; and so for any of the rest. The Tropics and Polar circles divide the Surface of the Globe into 5 parts o● spaces, which are called Zones, whereof one is contained within the Arctick-circle, another compassed by the Antartick-circle, and are called the Frigid Zones; The other two lying between the Arctick-circle and Tropic of Cancer, and between the Antarctick-circle, and the Tropic of Capricorn, are called the Temperate Zones; and the other lying between the two Tropics, is called the Torrid or Mid-zone. Knowing these, you may easily conclude upon the distinction of Shadows: for those of the Frigid Zones are termed Periscii, because there their Shadows have a Circular motion. Those of the Temperate Zone are called Heteroscii, because their Meridian-shadows bend toward either Pole; towards the North to those that devil within the Tropic of Cancer, and the Arctick-circle; towards the South to those that devil within the Tropic of Capricorn and the Antarctick-circle. The Inhabitants of the Torrid Zone were called Amphriscii, because the Noon-shadows according to the time of the year do sometimes fall towards the North when the Sun is in the Southern signs, and sometimes towards the South when the Sun is in the Northern signs. To find out the other distinction of Habitation, viz. Antaeci, Periaeci, Antipodes. LEt London be the place; bring it to the Meridian, where you find it to be 51 deg. 30 min. elevated above the Equator, accounted so many degrees of Southern Latitude below the Equator, and you meet with the Antaeci (if any be.) Remove London from the Meridian 180 deg. and you shall find your Periaeci under the Meridian where London was before, and your Antipodes is in the place where their Antaeci stood before. NAUTICAL Problems. INTRODUCTION. INTRODUCTION. THere be four things upon which the practice of Navigation is principally grounded, viz. 1. Longitude, 2. Latitude, 3. Course, 4. Distance. As for the Longitude, though it may be found by the other, yet hitherto there hath not been published any general Rule true and practicable, whereby the Longitudes of places may be immediately and ordinarily found out of themselves. The Latitudes of places may be immediately found out by observation of Sun or Stars, as shall be showed hereafter. The third thing to be considered in the Art of Navigation, is the Course or Line by which the Ship must go, which dependeth upon the Winds; the designation of these, upon the certain knowledge of one principal, which considering the Situations of the Earth, aught to be North or South, which now is found by the Needle touched with the Loadstone, being thereby endued with such a Magnetical Virtue, that if left to its liberty 'twill seat itself in a situation North or South. The North and Southwind thus assured by the motion of the Needle, the Mariner supposeth his Ship to be upon some Horizon or other, the Centre whereof is the Ship; so that crossing this North and South-line at Right Angles, showeth the East and West; so you have the 4 Cardinal Winds cross each of these, and you have the 8 Whole Winds, another division makes 16, which again divided, makes 32 in all: And these Lines which a Ship following the direction of the Magnetical Needle describes upon the Surface of the Water, were by the Portugals called Rumbs, and is still continued. These Rumbs are represented upon the Globe by those Helispherical or Spiral Lines that you see divided into 32 parts, with a Flower-de-luce always pointing to the North. The finding of the Rumb and Distance of a Ship in any place from whence she hath departed, is the last of the four things propounded as necessary in this Art of Navigation; which how to perform, shall be also showed in the following Problems: But first of the Latitude. PROB. I. To find the Latitude. I. By the Sun's Declination, and Merididian-Altitude. WHen the Sun is in the Equinoctial, having no Declination, and the Meridian-Altitude is observed on the Southside of the Meridian, The Meridian-Altitude taken from 90 degr. leaves the Elevation of the North-pole. North-side of the Meridian, The Meridian-Altitude taken from 90 degr. leaves the Elevation of the South-pole. Place these 2 Schemes upon folio. 222. depiction of the sun's declination depiction of a meridian altitude When the Sun's Declination is North, If the Meridian Altitude be lesle than 90 deg. and the Sun upon the Southside of the Meridian, the Sun's Declination being taken from the Meridian Altitude, leaves the height of the Equinoctial; which taken from 90 deg. gives the Latitude North. South, If the Meridian Altitude be lesle than 90 deg. and the Sun upon the Southside of the Meridian, add the Meridian Altitude and the Declination together, their sum is the height of the Equinoctial; which taken from 90 deg. leaves the Latitude North. But if the sum of the Declination and Altitude exceed 90 deg. take 90 therefrom, the remainder is the Latitude South. When the Sun's Declination is North. If the Meridian Altitude be lesle than 90 deg. and the Sun upon the Southside of the Meridian, add the Altitude and Declination together, their sum is the height of the Equinoctial; which taken from 90 deg. leaves the Latitude South.— But if the sum be above 90 deg. take 90 deg. therefrom, the remainder is the Latit. North. South, If the Meridian Altitude be lesle than 90 deg. and the Sun upon the North-side of the Meridian, subtract the Declination from the Meridian Altitude, the remainder is the height of the Equinoctial; which taken from 90 deg. leaves the Latitude South. When the Sun's Declination is North, South, if the Meridian Altitude be just 90 d. the Sun's Declination is the Latitude North. South. If the Meridian Altitude be observed under the Pole, within the bounds of the Polar Circles, in such Case the Sun's Declination must be taken from 90 deg. and what remains is his distance from the Pole; which being added to the Meridian Altitude, the sum is the Latitude of the place. ¶ Note here, that whatsoever is said concerning finding of the Latitude by the Sun's Declination and Meridian Altitude, the same is to be performed by the Meridian Altitude of any known Star: and the manner how to effect it, will best be seen by the Globe. II. By the Meridian Latitude of a known Star. Suppose that being at Sea I should observe Algol to be upon the South side of the Meridian, and to have Altitude 62 deg. and I would know in what Latitude I than was. Arithmetically. The Declination of Algol is 39 deg. 39 min. North, the Compliment whereof is 50 deg. 21 min. his distance from the Pole; add this distance and his Altitude 62 deg. together, the sum is 112 deg. 21 min. which taken from 180 deg. leaves 67 deg. 39 min. for the Latitude of the place North. By the Globe. Bring Algol to the Meridian, and from the Centre of the Star, downwards, count his Altitude 62 deg. and mark that point upon the Meridian: than bring that point to the South-part of the Horizon, and you shall find the North-pole to be elevated 67 deg. 39 min. which is the Latitude you are than in. In like manner, if you should observe d. m. The Bull's Eye upon the South-part of the Meridian, having Altitude 51 0 Spica Virgins 32 0 The Great Dog 72 0 d. m. you would be in Latitude 54 48 N. 67 31 S. 34 14 N. III. By observing of two Stars, one being upon the Meridian, and the other Rising or Setting. Bring the Star which you see upon the Meridian unto the Meridian, and there holding the Globe fast, move the Meridian in the Horizon, till you see the other Star on the East or West, Rising or Setting, as you observed it, and than shall the Globe stand at the Latitude you are in. So if you should see Regulus upon the Meridian, and Lyra rising towards the East, the Latitude would be found 37 deg. 50 min. iv By the Altitude of two known Stars, being both of them upon the same Azimuth or point of the Compass. Lay the Quadrant of Altitude, or rather your thin Brass Semicircle to both the Stars, at the proper degrees of Altitude, as you observed them to be in the Heavens; (for the difference of their Altitudes is equal to their distance.) Than turn the Globe about in the Horizon, till the Quadrant or thin Plate of Brass do touch the Horizon in that Azimuth (or point of the Compass) on which you observed the Stars to be, so shall the Globe rest at the Latitude you are in. So Capella and Scheder, Capella being 20 deg. high, and Scheder 66 deg. h gh, and both of them upon the North-East-point of the Compass, you will find yourself to be by this observation in the Latitude of 40 deg. 0 min. North-Latitude. V By the Altitude, Azimuth, and Declination of the Sun, or of a known Star. Suppose a fixed Star, as that in the Right Knee of Hercules, having 47 deg. 9 min. of Declination, should be observed at Sea to have 122 deg. of Azimuth from the North-part of the Meridian, and to be 60 deg. high; and from hence the Latitude were required: Elevate the Globe to the Stars Altitude 60 deg. and from the Pole count the Compliment thereof 30 deg. to which screw the Quadrant of Altitude: also count 122 deg. upon the Horizon from the North-part of the Meridian, and to those degrees bring the Quadrant of Altitude, and there keep it; than turn the Globe about, till 47 deg. 9 min. the Stars Declination (counted upon the Equinoctial Colour from the Equinoctial) do cut the Quadrant of Altitude, and those degrees will cut the Quadrant of Altitude in 71 deg. 13 min. and that is the Latitude in which you than are. VI By the Suns, or a Stars Declination and Amplitude. Let the Sun or a Star have 10 deg. of North Declination, and let the Amplitude thereof at its Rising or Setting be observed to be 57 deg. from the North. Elevate the Globe to 33 deg. (the Compliment of the Amplitude) and count the Amplitude itself 57 deg. upon the Meridian from the Pole forward, and thereto screw the Quadrant of Altitude, and bring the other end thereof to the East or West-points of the Horizon; than count 80 deg. (the Compliment of the Stars or Sun's Declination) from the Pole upon the Equinoctial Colour, and bring those degrees to the Quadrant of Altitude; so shall 80 deg. of the Colour cut 71 deg. 24 min. of the Quadrant, and that is the Latitude you are than in. VII. By the Sun's Ascensional difference, and Amplitude. If the Ascensional difference be 27 deg. 7 min. and the Amplitude 33 deg. 20 min. and the Latitude were required; Elevate the Globe to 27 deg. 7 min. the Ascensional difference, and from the Pole count the Compliment thereof 62 deg. 53 min. and thereto screw the Quadrant of Altitude; than bring the Equinoctial Colour to the Meridian, and count upon the Quadrant of Altitude upwards, the Compliment of the Amplitude 57 deg. 40 min. which degrees bring to the Equinoctial, and than shall the Quadrant of Altitude cut upon the Horizon 51 deg. 30 min. counted from the East or West, which is the Latitude of the place. And if you count the degrees of the Equinoctial comprehended between the Meridian and the Quadrant of Altitude, you shall find them to be 20 deg. 5 min. which is the Sun's Northerly Declination at that time. VIII. By the Suns or a Stars Declination, and the time that he is upon the East or West-points of the Compass. Let the Sun or a Stars Declination be 15 deg. North, and let the time that the Sun is upon the East or West-points of the Compass, be 59 deg. 45 min. which in time is 3 hours 56 min. and hence let the Latitude be required. Elevate the Globe to 15 deg. the Declination, and screw the Quadrant of Altitude to 75 deg. the Zenith-point; than count 59 deg. 45 min. or 3 hours 56 min. (the hour) upon the Horizon from the South, Eastward or Westward; and thereto bring the Quadrant of Altitude: Than look what degrees or the Quadrant are cut by the Equinoctial, and you shall find 28 deg. counted from the Zenith, and that is the Latitude sought. And thus have you several ways both by the Sun and Stars to find the Latitude at any time. I will now proceed to some other Problems for finding the Rumb and distance, which as was said is the last of the four things necessary in the Art of Navigation. PROB. II. Any two places given, to find their Rumb. Definition. THose Lines which a Ship following the direction of the Magnetical Needle describeth on the surface of the Sea, are called Rumbs, and are (as was said) described upon the Terrestrial Globe by certain Spiral Lines; for the better understanding whereof, I shall premise these few Propositions: First, The Needle touched with the Loadstone pointeth out the common Intersection of the Horizon and Meridian; the one end respecting the North, the other the South, as aforesaid. Secondly, A Circle drawn through the Vertex of any place that is distant from the Equator, cannot cut divers Meridian's at equal Angles. Thirdly, A great Circle drawn through the Vertical-point of any place, and inclining to the Meridian, maketh greater Angles with all other Meridian's, than with that from whence it was drawn. Fourthly, If we sail upon any point of the Compass except North or South, we often change our Horizon and Meridian. Fifthly, The same Rumb cutteth all Meridian's of all places at equal Angles, and respecteth the same Quarters of the World in every Horizon. Sixthly, The potions of the same Rumb intercepted between any two Parallels whose difference of Latitude is the same, are also equal to each other: therefore an equal Segment of the same Rumb equally changeth the difference of Latitude in all places; so that in an equal space passed in one and the same Rumb, one of the Poles is equally elevated, and the other depressed. Seventhly, Rumbs though never so far continued, do not pass through the Poles, but wind about the Poles until they lose themselves. Hence you may understand if your Ship be directed under the North or South-Rumb, your course will be always under the same Meridian; if under the East or West-Rumb, you will either describe the Equator, or a Circle parallel to it: If your Vertical point be under the Equator, your Ship will describe an Arch or Segment of the Equator; but if your Zenith or Vertical point be distant from the Equator either North or South, your course will than describe a Parallel as far distant from the Equator, as the Latitude of the place is whence you first set forwards: But if your Voyage be to be made under the Rumb which inclineth to the Meridian, your course will than be neither in a greater or lesser Circle, but your Ship will describe a kind of a crooked spiral Line. Practice. Find the two places upon the Globe, and when you have found them, see what one Rumb-line passeth through both of them, and that is the Rumb or point of bearing of those two places one from the other. So C. deal Gade on the Coast of Zanguebar and C. Cormorin are both of them found upon the W S W and E N E Rumb, and that is their Point of bearing, or Rumb required. If you can find no one Rumb that passeth through both your places, than you must look what Rumb-line upon the Globe runneth most parallel to both the given places, and conclude that to be the point or Rumb of those two places bearing one from another.— So if the two places were Seirra Liona in Africa, and the Island S. Helena, if you look upon the Globe you shall find no one Rumb-line to pass through both the places, but that Rumb to which the places lie most parallel, is the N N W and S S E Rumb; and so Sierra Liona beareth from S. Helena N N W, and on the contrary S. Helena is situate from Sierra Liona S S E, which is the Rumb required. PROB. III. Having the distance sailed, and the Rumb you have sailed upon given, to find the difference of the two places both in Longitude and Latitude. HAving found the Rumb upon which you made your course, make a small mark thereupon, for the place you departed from: than from the Equinoctial take the number of miles or leagues you have sailed upon that Rumb (allowing 20 leagues for a deg.) and set that distance upon the Rumb from the former point made; and at the termination of your number of miles or leagues, make a second mark upon the Rumb: than bringing the place, or point, you departed from to the Meridian, you shall there find the Latitude of that place or point, and the Meridian cutting the Equinoctial will show you the Longitude of that place or point.— Do so by bringing the second point or place to the Meridian, and there shall you find the Latitude, and upon the Equinoctial the Longitude of that place or point. Now if you subtract the lesser Latitude from the greater, you have the difference of Latitude; and the lesser Longitude substracted from the greater, gives the difference of Longitude. PROB. IU. The Latitude of two places, and the Rumb that the two places bear each from other given, to find the difference of Longitude of those two places, and also their distance upon the Rumb. Practice. FIrst, find the Rumb upon the Globe, and turn the Globe about till that Rumb doth cut the Meridian in the Latitude of the first place from whence you departed, and there make a mark or point upon the Rumb, and at the same time see also what degrees of the Equinoctial are cut by the Meridian; for that is the Longitude of the first point. Secondly, Turn the Globe about, till the same Rumb does cut the Meridian in the Latitude of the second place, and there make another mark upon the Rumb; and also see what degrees of the Equinoctial are cut by the Meridian, which degrees are the Longitude of the second point or place: and the lesser Longitude being substracted from the greater, gives the difference of Longitude of the two places or points. Than for the distance upon the Rumb, the distance between the two points before made, being measured upon the Equinoctial, and reduced to miles or leagues, shall give the distance upon the Rumb. ¶ Here note, that the distance upon the Rumb being entirely taken and applied to the Equinoctial, will give the distance in the Arch of a great Circle, and not really in the Rumb, for the distance upon the Rumb will be always greater than the great Circular distance: wherefore the better way will be, to take in a pair of Compasses one, two, three (or some small number of) degrees of the Equinoctial, and run that distance over upon the Rumb-line from point to point; and the number of all those returns of the Compasses (reduced to miles or leagues) shall be the near distance of the two places upon the Rumb. FINIS.