Speculum Nauticum. A LOOKING glass FOR seamen: Wherein they may behold a small Instrument called the Plain SCALE▪ whereby all Questions nautical, and Propositions astronomical are very easily and demonstratively wrought. Published for the use and benefit of such as will make good use of it. By John Aspley, Student in physic, and Practitioner of the mathematics, in the city of London. The fourth Edition corrected. LONDON, Printed by Thomas Harper, and are to be sold by George Hurlock, at his shop at Saint Magnus corner. 1647. TO THE worshipful THE MASTER, WARDENS, And Assistants, of the Trinity House in Deptford Strand. ALbeit (Right worshipful) there be more essential natures contained in each part both of this macrocosm, and microcosm, existing in their absolute being, than the understanding of man can fully apprehend; yet I doubt not, but those which may with labour and diligence be known, & those also which of ingenious spirits and notable wits have been invented, and by them artificially & methodically taught, (tending not only to manifest profit in the commonwealth, but also to the great increase and setting forth of God's divine power, wisdom, goodness, providence, and increase of Virtue ought of all men to be embraced, (and especially of those which have any government, public charge, or authority in the Common wealth.) In regard that the nearer men approach to such excellent virtues, the nearer (without doubt) do they come unto goodness, to felicity, and to God himself. Hence saith the Prophet (unto men seated in eminent places) Dixi vos dii estis, so that those which either through arrogance, or ignorance deride, and contemn those Arts (which with great dexterity, care, and industry have been found out, and left unto us by the love of our Predecessors) do both offer contempt unto the goodness of God, and do much endamage and annoy all human society. So on the contrary part, they that do by all means further those so profitable Disciplines do both render true honour unto GOD, and do greatly advance the good of the commonwealth wherein they live. Now in regard there is no study (Divinity excepted) wherein the wit of man may be better employed then in the motion of the stars, and in the knowledge of their situation, place, and being, together with their wonderful effects: In regard whereof I was incited to employ some of my time in the study thereof, and at last considering that of the Orator, that Non solum nobis nati sumus, &c. We are not born for ourselves only, but our friends challenge a part in us, and our Countries come in for a share, especially those honours and graces of our country, those that traffic in the deep, and have their business in the great waters, those that are unto this Island as a wooden wall, the Sea-chariots, and the horses of England: these, I say, may claim justly to the fruits of our labours, whatsoever they be which have not altogether been abhorrent from the mathematical studies: considering this, I could do no less than bring in this my mite into their treasury, and Si quid Ars mea efficere possit, (if my skill can stand them in any stead) to further that so much deserving Science of Navigation. Accept therefore, I beseech you, this young son of my studies, this little handful of paper, wherein is contained not Anacreon's wanton Odes, or Ovid's lascivious Elegies, the incendiaries of lust, but a pure spark of chaste Vestalfire, a small part of the mathematics dedicated from a serviceable affection to your worships, that under the shield of your protections it may live secure, from the desperate stab of of critical persons, and envious spirits, who not only like snarling satyrs deride and contemn those so liberal Sciences, but swallow up with despite (if it were possible) the professors thereof. For rescue from such malignant spirits my Book flies to the shadow of your favour, which if you shall afford unto it, my labours shall be all sacrificed unto you, and I rest. Your Worships bound servant, John ASPLEY. To the Reader. HAving ever since I came unto understanding (courteous Reader) practised myself in the mathematical studies: and having attained unto my desires therein, I am willing to impart some of that knowledge which God hath bestowed upon me, unto the open view of the World: for the manifestation whereof, I have freely given unto thee this small Book, being the first fruit of my labours, containing such astronomical and nautical Questions as are wrought by the plain Scale: which if I should find to receive as free acceptance from thee, expect more of my labours in the same kind, and until then Irest. Thy friend in affection, John ASPLEY. Speculum Nauticum. OR THE seaman's glass. THE FIRST BOOK. CHAP. I. BEing intended in this Treatise of the plain Scale, to declare the manner of projection of the sphere, in plano, I have thought fitting first, to show unto you some terms of Geometry which are necessary for the unlearned to know (for whose sake chiefly I write this Treatise) before they enter into the definition of the sphere. First, therefore I intend to relate unto you, what a point or prick is, and afterward a Line both right and crooked, and such sorts thereof as are appertinent unto the operations and use of this Scale. Punctum, or a Point, is the beginning of things, or a prick supposed indivisible, void of length, breadth, and depth; as in the figure following is noted by the Point or Prick A. Linea, or a Line, is a supposed length, or a thing extending itself in length, not having breadth nor thickness, as is set forth unto you by the Line B AD. dem: 8. Parallela, or a parallel Line, is a Line drawn by the side of another Line, in such sort that they may be equidistant in all places. And of such Parallels two only belong unto this work of the plain Scale, that is to say, the right lined parallel, and the circular parallel. Right lined parallels, are two right lines equidistant one from another, which being drawn forth infinitely, would never touch or meet one another, as you may see in the figure where the line H. I. is parallel unto the line C. E. and the line G. F. is parallel unto them both. A circular parallel is a Circle drawn either within or without another Circle upon the same Centre, as you may plainly see by the two Circles B. C. D. E. and X. V. Y. W. These Circles are both drawn upon the Centre A. and therefore are parallel the one unto the other. There is another kind of parallel also, which is called a serpentine parallel, but because it is not belonging unto the use of this Scale, I will omit it and so proceed unto the rest. Perpendiculum, or Perpendicular is a Line raised from, or let fall upon another Line, making equal angles on both sides, as you may see declared in the figure, wherein the Line A. C. is perpendicular unto the line B. A. D. making equal angles in the point A. Diameter Circuli, or the Diameter of a Circle, is a right Line drawn thorough the Centre of any circle, in such sort that it may divide the Circle into two equal parts, as you may see the line B. A. D. is the Diameter of the Circle B. C. D. E. because it passeth thorough the Centre A. and the two ends thereof do divide the Circle into two equal parts, in the two extremes B. & D making the semicircle B. C. D. equal unto the semicircle D. E. B. Semidiameter circuli, or the Semidiameter of a Circle, is half of the Diameter, and is contained betwixt the Centre, and the one side of the Circle, as the Line A D, in the Semidiameter of the Circle B C D E. Semi-circulus, or a semicircle, is the one half of a Circle, drawn upon his Diameter, and is contained upon the Superficies, or Surface of the Diameter, as the Semicircle B C D, which is half of the Circle B C D E, and is contained above the Diameter B A D. quadrants circuli, is the fourth part of a Circle, and is contained betwixt the Semidiameter of the Circle, and a line drawn Perpendicular, unto the Diameter of the same Circle, from the Centre thereof, dividing the Semicircle into two equal parts, of the which parts, the one is the quadrant, or fourth part of the same Circle. As for example, the Diameter of the Circle B C D E, is the line B A D, dividing the Circle into two equal parts: then from the Centre A, raise the Perpendicular A C, dividing the Semicircle likewise, into two equal parts, so as A B C, or A C D, the quasired. CHAP. II. The manner how to raise a Perpendicular from the middle of a Line given. DRaw first a ground line whereupon you would have a Perpendicular raised, then open your Compasses unto any distance (so it exceed not the end of your line,) placing one foot of the said Compasses, in the point from whence the Perpendicular is to be raised, and with the other foot make a mark in the line on both sides, then removing your Compasses, unto any other distance that is greater, and set one foot thereof, in one of the marks, and with the other foot make a mark over the middle point, then with the same distance of your compass, set one foot in the other mark upon the Line, and with the other foot make another arch of a Circle over the middle point, so that it may cross the first arch, and from the meeting of these two arches, draw a right Line unto the middle point, from which the Perpendicular was to be raised, which Line shall be the Perpendicular desired. Example, suppose your Base or ground Line whereupon a Perpendicular is to be raised by the Line F L K, and from L, the Perpendicular is to be raised, set one foot of your Compasses in the point L, and with the other, make the marks G, and M, on both sides of the point L, then opening your Compasses wider, set one foot in the point M, and with the other draw the arch S, over the point L, then with the same distance of your Compasses, set one foot in G, and with the other make the arch R. crossing the arch S. in the point T. then from T. draw the Line T. L. which Line is Perpendicular unto the Line F. L. K. from the point L, which is the Perpendicular desired. CHAP. III. To let a Perpendicular fall from any Point assigned, unto the middle of a Line. LEt the line whereupon you would have a Perpendicular let fall, be the Line L F K, and the Point assigned, to be the point T, from whence you would have a Perpendicular let fall upon the Line F L K, first set one foot of your Compasses in T, and open your Compasses unto any distance, so that it be more than the distance T L, which here we suppose to be the distance T M, then make in the Line F L K, the marks G, and M, then with your Compasses, take the one half of G M, which is in the point L, then from L, draw a Line unto the point T, so the Line T L, shall be the Perpendicular, which was desired to be let fall from the assigned point T, unto the middle of the Line F L K. CHAP. IV. To raise a Perpendicular upon the end of a Line. SUppose the Line whereupon you would have a Perpendicular raised, be the Line F L K, and from the point F, a Perpendicular is to be raised: first open your Compasses unto any distance, which here we put to be the distance F G, & set one foot of your compasses in the point F, and with the other draw the arch D E G, then set one foot of your Compasses in the point G, and with the other draw the arch E, then placing one point of your Compasses in F, with the other draw the arch D B, then place your Compasses in D, and with the same distance, draw the arch A, cutting the arch D B, in C, then draw a Line from C, unto the end of the Line F L K, unto the assigned point F, so shall the Line G F, be a Perpendicular, raised from the end of the Line F L K, and from the assigned point F. CHAP. V. To let a Perpendicular fall from any point assigned unto the end of a Line. LEt the Line F L K, be the Base or ground Line, and from the point I, a Perpendicular is to be let fall upon the end of the Line K, first from the assigned point I, draw a Line unto any part of the Base, which shall be the Line IHM, then find the middle of the Line IM, which is at H, place therefore one foot of your Compasses in the point H, and extend the other unto I, with which distance draw the arch INK, upon the Centre H, cutting the Base or ground-line in the point K, then draw the Line KI, which Line shall be the Perpendicular desired. NOW I doubt not but you understand the way to let fall, or to raise, any manner of Perpendicular Line, either from, or upon any part of a Line: therefore now I intend to proceed unto the main point here aimed at, which is, to declare, and make known unto you the several operations performed by the plain Scale, which though it be in use with very few, yet it is most necessary with seamen, because all questions in Navigation are thereby easily and plainly wrought. And also all questions in Astronomy (belonging unto the expert and industrious seamen) may both speedily and easily be wrought by the same Scale: in regard whereof I have declared in this little Book, that knowledge (which God hath been pleased to bestow upon me) concerning the necessary use and practice thereof; hoping that you will as kindly accept it, as it is freely offered unto your courteous considerations. The figure of the plain Scale. CHAP. VI. Of the description of the Scale. THis Scale usually is divided into three parts, the first whereof is a Scale of equal Leagues, divided into Degrees, or Leagues from 1 unto 100 and upwards, at your pleasure, and numbered with 10 20 30 40, and so forth unto the end. All these divisions are equal one unto another, and is in use for to measure the leagues that any ship hath run upon any course, or the leagues that she hath raised or depressed the Pole, or departed the Meridian, as in the work hereafter shall be more fully declared. The second part of the Scale, is the single cord of a Circle, or the Cord of 90, and in dividing into 90, unequal divisions, representing the 90, deg. of the Quadrant: and are numbered with 10, 20, 30, 40, &c. unto 90. This Cord is in use to measure any part, or arch of a Circle not surmounting 90 degrees: The number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance, all Circles are to be drawn, whereupon 60 of these degrees are the Semidiameter of any Circle, that is drawn upon that Radius. The third part of the Scale is divided into 8 parts, representing the 8 points of the mariner's compass, contained in one quarter of a Circle, if the Circle be drawn upon the same Radius, and every one of the aforesaid points, is (for exactness sake) subdivided into 8 smaller parts. I have likewise caused two other lines to be placed upon the back side of the Scale, which I do call the first and second Lines of Longitudes: the first is divided into 20 unequal parts, or leagues, which 20 Leagues are equal unto the Cord of 90. The use of this first Line of Longitude, is to show how many Leagues and Miles in any parallel, do answer unto one degree of the equinoctial. The second Line of Longitude is divided into 100 proportionable parts, or into 100 unequal Leagues; and every league is subdivided into miles, and half miles. The use of this Line is thus, when you have found by the first Line of Longitudes, how many Leagues and Miles do answer unto a degree of the Equinoctial in any latitude you desire: this second line will show you how many degrees any number of leagues in that parallel, will answer unto a degree in the equinoctial Circle. Thus having showed you the parts of the Scale, and unto what use they do generally serve, I will proceed to declare the particular use thereof, in the Art of Navigation, as followeth. CHAP. VII. To find how much any Ship hath raised or depressed the Pole, knowing the course she hath made, and the leagues she hath sailed. THe Course is Southwest and by South, the Leagues sailed are 100, the difference of Latitude is demanded. In the first Demonstratiou, draw first the Line A B, and upon the Centre A, raise a Perpendicular A F. Then opening your Compasses unto the Radius of your Scale, and set one Foot thereof in the centre A. and with the other draw the Arch K C B, then in regard your course is Southwest & by South, that is three points from the South, take three of the eight points of the compass with your Compasses, and place them from K, unto C, then draw the Line A C D, and place the distance of the Leagues you have sailed (which) are 100 upon the Line A C D, from A, unto D. Then from D, draw the Line D F. parallel unto A B, cutting the Meridian A K F, in the point F, then take the distance of F A, and apply it unto the Scale of equal Leagues, and you shall find it just 83 Leagues, or 4 Deg. 9, Min. which are the degrees you have altered the Latitude, which degrees and minutes (if the Latitude from whence you departed, be South) must be added unto the Latitude from whence you departed, and you shall have the Latitude that you are in: contrariwise, subtract them (if the Latitude from whence you departed, be North) and you have likewise the Latitude that you are in. CHAP. VIII. The distance of Latitude and Leagues sailed being given, to find the distance meridional, and consequently the difference of Longitude. Sailing from the North parallel of 56. deg. and 5. min. 100 Leagues betwixt South and West, until the Pole be depressed 4 deg. 9 min. the difference of longitude is demanded. dem. prim. IN the first demonstration draw the Quadrant A K C B, as is taught in the last Chapter. Then reduce your degrees of Latitude into Leagues, which is done by multiplying of them by 20 the product will be 83 leagues, which leagues being applied unto the Meridian Line A K F, they will end in the point F, Then from F, draw the Line D F, parallel unto A B, Then open your Compasses unto the distance of 100 leagues of your Se●lt of equal parts, and set one foot of your Compasses in the point A, and with the other draw the arch G H, cutting the Line FD, in D, so shall the distance D F, be the distance of the Meridian, from the Meridian, from whence you departed, which (being applied unto the Scale) is 56 leagues. Then in regard you sailed from the North parallel of 56 deg. and 5 min. until you had depressed the Pole 4 deg. 5. min. subtract therefore 4 deg. 9 min. from 56 deg. 5 min. and there remaineth 51 deg. 56 min. which is the latitude of the place you are in, and in that parallel have you departed the first Meridian 56 leagues. Then opening your Compasses unto 51 deg. 56 min. of your Cord, and apply it unto the first Line of Longitudes, and you shall find that 12 leagues and one mile (in that parallel) do alter one degree of Longitude. Then set one foot of your Compasses in the second Line of Longitude, at 12 Leagues, one Mile, and extend the other unto one degree of that Line; then with that distance set one foot of your Compasses in 56 leagues of the aforesaid Line, and the other will extend unto 4 degrees 33 min. which is the distance meridional desired. CHAP. ix.. distance of Latitude and distance meridional, given to find the rhomb. Sailing from the North parallel 69 degrees 20 min. until the Pole be depressed four degrees and 9 min. and the distance meridional, or difference of Longitude, six degrees to find the rhomb is required. By the first Demonstration, draw the Quadrant A K C B, then turn your four degrees nine minutes into Leagues, it maketh eighty three Leagues; which place upon the line A K F, from A, unto E, then subtract the difference of the two Latitudes, from the number of the first Latitude, and it leaves the second Latitude 62 deg. 11 min. Then opening the Compasses unto the middle Latitude, which is sixty four degrees, and fifteen minutes of the Cord, applying it unto the first Line of Longitudes, and you shall find eight Leagues, two miles, and four seconds to answer unto one degree: then set your Compasses in one degree, in your second Line of Longitudes, and extend the other foot unto eight Leagues, and two miles, and four seconds: then with that distance of the Compasses, place the one foot at six degrees of that line, and turn the other upward, and it will extend unto fifty six leagues, therefore open your Compasses to the distance of fifty six Leagues, in the line of equal leagues, and apply them from the point P, upon the line F D, parallel unto A B, from F, unto D, then from the point D, draw the line D A. cutting the Quadrant K C B, in C, so shall K C, be the distance of the rhomb from the South Westward, which is just thirty three degrees, forty five minutes from the South, which is Southwest and by South, the rhomb desired. CHAP. X. By the Latitude of two places, and distance upon the rhomb to find the leagues sailed. The Pole depressed three degrees thirty minutes, the rhomb the fourth from the Meridian. IN the second Demonstration draw the line A E, then from A, raise the Perpendicular A C, then opening the Compasses to the distance of the Radius, placing one foot thereof in the Centre A, and with the other draw the Quadrant B D E, Then reduce your three degrees, thirty minutes into leagues, counting for every degree twenty Leagues, and for thirty minutes ten Leagues, so they make seventy Leagues; then open your Compasses unto seventy degrees in the line of equal parts, and place them upon the line A B C, from A, unto C, then from C, draw the line C F, parallel unto A E. Then in regard your rhomb was the fourth from the South, take four of the eight points of the compass, and place them upon the Quadrant from B, unto D, then from A, by the point D, draw the line A D F, cutting the line C F, in F. So shall the distance betwixt A, and F, be the number of leagues (upon the fourth rhomb) before you can either raise or depress the Pole three degrees thirty minutes, which is here found to be ninety nine Leagues. CHAP. XI. To find the distance of any Island from you, that you may discern by two Stations, knowing the point of the compass, the Island beareth unto each of the Stations. Suppose (being at Sea) you discover an Island bearing South-west of you, which place let it be your first Station, and seventy Leagues sailing South observing the Island to bear West of you, which let be the second Station, the demand is to find the Island from both the Stations. IN the second Demonstration let A, be the first Station, and upon the Centre A, draw the quadrant A B D E, then in regard you found the Island to bear South-west from you, therefore take four of your eight points of compass, and place them upon your Quadrant from E, unto D, then from the Centre A, by the point D, draw the Line A D F, representing the visual Line passing betwixt your sight and the Island, being in the first Station. Then sailing seventy leagues South, which is from A, your first Station, unto C, the second Station: then observing the Island to bear West of you, therefore from the point C, the second Station, draw the Line dem: 2 C F, parallel unto A E, cutting the Line A D E, in point F, so shall the point F, be the place of the Island desired, and the distance A F, is the distance of the Island from the first Station, which is just ninety nine Leagues off the Line of equal parts. And likewise the distance from C, unto F, is the distance of the Island from the second Station, which is here found to be just seventy Leagues: and by this manner of work you may find the distance of any Island from you, which you may discern either by Sea or Land. CHAP. XII. Sailing from the South Longitude of 60 degrees, 51 minutes, and from Latitude 25 degrees, 24 min. 99 Leagues, upon a South-west course: the Latitude and Longitude of the second place is demanded. IN the second Demonstration, draw the Quadrant A B C D E, as is formerly taught: then in regard you sail South-west, take four points of the compass from your Scale, & place them from B, unto D, then by the point D, draw the line A D F, then place your ninety nine Leagues upon the Line A D F, from A, unto F, so shall E, be the place of your Ship. Then from F, draw the Line F C, parallel unto A E, cutting the line A B C, in C, so let the distance C A, be Leagues that you have run South, which is 70 Leagues. or 3 deg. 30 min. which being added to the latitude from whence you departed, makes sixty four degrees and twenty one minutes for the Latitude of the second place: then take the distance C F, and apply it unto the Line of equal parts, and you shall find it likewise seventy Leagues: then opening your Compasses unto the middle Latitude 62 degrees, 36 minutes in the Line of Cordes, and apply it unto the first Line of Longitudes, you shall find that nine leagues and 0 miles, and 38 seconds, do alter a degree of longitude, then placing one foot of your Compasses in the second line of longitudes, at 9 leagues and thirty eight seconds, and extend the other to one, then keeping the distance of the Compasses, set one foot in the seventy leagues of the same line, and the other foot will extend unto 7 degrees 37 min. which being subtracted from the longitude from whence you departed, leaves seventeen degrees, and forty seven minutes for the Longitude of the second place. CHAP. XIII. A Ship sailing from the North parallel of fifty degrees, having an hundred Leagues to sail South-west, and by West, by the way is enforced by contrary winds, to sail upon several points of the compass, first sailing thirty leagues upon a direct course, than West northwest twenty Leagues, than South sixty Leagues, the question is to find the Latitude of the second place, how far it is to the place whereunto you are bound, the distance of the rhomb that is betwixt them, the distance that you are from your first Meridian, and thereby the difference of Longitude. IN the third Demonstration, draw the Line A D, and from the point A, raise the perpendicular A B, then open your Compasses unto the Radius of your Scale, and place one foot thereof in the centre A, & with the other draw the Quadrant B C D, then take three points of the Compasses, and and place them upon the Quadrant from D, unto C, then from the Centre A, by the point C, draw the line ACL, 100 Leagues in length, which is the true course you are to sail. Then in regard you sailed 30 leagues direct, take thirty degrees from your Scale of equal parts, and place them upon the line AEC, extending from A, unto E, then in regard you turned your Course, West, Northwest, from the Centre E, draw the Line EG, parallel unto AD, and again from the Centre E, draw the Line EH, Perpendicular to EG, and parallel to AB, then with the distance of the Radius, set one foot of your compasses in the Centre E, and with the other draw the quadrant G MH, and in regard you sailed West, Northwest, which is 2 points from the West, dem: 3 Northward, take from your Scale two points of the compass, and place them upon the Quadrant GM, H, from G, unto M, then from the Centre E, unto the point M, draw the line EFM, then take twenty leagues with your Compasses from the Scale of equal parts, and place them upon the line EFM, from E, unto F, then is your Ship in the point F. Lastly, in regard you run South sixty Leagues, from F, draw a Line parallel unto the Meridian AB, which is the line FI, then take from your Scale of equal parts 60 Leagues, and place them from F, unto I, then is your Ship in the point I, than last of all is to be found how far it is to the place whereunto you were bound, the distance of the rhomb that is betwixt you, the degrees and minutes you have raised the Pole, the distance of departure from the first Meridian, and thereby the difference of Longitude, and that you may so do, first draw the Line OIK. Perpendicular unto the line IF, in the point I, and with your Compasses opened unto the distance of the Radius, set one foot of your Compasses in the Centre I, and with the other draw the Quadrant KNF, then in regard your ship is in the point I, and the place whereunto you are bound in the point L, therefore from I, thorough the point L, draw the Line ILN, cutting the Arch KNF, in the point N, therefore let IL, be the Leagues you have unto the place whereunto you are bound, which is forty one Leagues and a half, and the rhomb the distance KN, which is West, and by North, and three degrees unto the Northward, so likewise is the line AO, the number of Leagues you have run due South, which is 68 Leagues and one Mile, or 3 degrees and 25 minutes, which being taken from 50 degrees, the parallel from which you departed, leaves 46 degrees and 35 minutes for the parallel you ate in. Last of all, shall the line IO, be the Leagues that you have departed your first Meridian, which are 42 leagues and one mile: then open your Compasses unto 48 degrees 17 minutes and 30 seconds of the line of Cords, which is the middle Latitude, and apply it unto the first line of Longitudes, you shall find that 13 Leagues 0, miles, fifty six seconds do answer unto a degree of Longitude in that parallel, then setting one foot of your Compasses in 13 Leagues, and 56 seconds in your second line of Longitudes, extending the other unto one degree, then with the same distance of your Compasses, set one foot in 42 Leagues and one mile of the same line, and the other will show you 3 degrees and 13 minutes, which is the difference of the Longitude desired. CHAP. IV. Two Ships departing from one parallel, and Port, the one in saying eight Leagues betwixt the North, and the West, hath raised the Pole two degrees, the other in sailing a hundred Leagues betwixt the North, and West, hath raised the Pole four degrees, I demand by what rhombs the said Ships have sailed, and the rhomb, and distance that is betwixt them. IN the fourth Demonstration, draw the Quadrant ABC DE, then in regard the first ship hath raised the Pole two degrees, which is forty leagues, take forty leagues of your Scale, and apply them unto the Meridian line AG L, from A, unto G, then from the point G, draw the line GF, parallel unto AB, then opening your compasses unto 80 Leagues set one foot in the Centre A, with the other, make a mark in the line GF, which will be at F, so shall F, be the place of the first ship: the second ship hath raised the Pole four degrees, which is eighty Leagues, therefore place eighty Leagues upon the Meridian line AGL, from A, unto L, and from the point L, draw the line LM, parallel unto GHF, then open your Compasses unto the distance of a hundred Leagues, which are the Leagues the second ship did run, and set the foot of your Compasses in the Centre A, and with the other make a mark in the line LM, which will be at M, then draw the line MA, which is the course of the second Ship, and the line FA, is the course of the first ship, then from F, let a Perpendicular fall, being Perpendicular to the line GF, which is the line FK, then opening your Compasses unto the Radius of your Scale, set one foot in the Centre F, and with the other draw the Quadrant HIK, likewise from F, the place of the first ship, draw a line by the point M, the place of the second ship: cutting the quadrant KHI, in I, so let IK, be the course that is betwixt them, that is, if you will sail from the first ship unto the second, you must sail North, and by East, and 41 minutes to the Eastward, likewise let FM, be the distance that is betwixt them, which in this Demonstration is 40 Leagues, two miles, so shall BC, be the course of the first ship from the West Northward, which here is found to be 30 degrees and one minute from the West Northward, or Northwest, by West & 3 deg. and 44 min. to the Westward. Lastly, the Arch E D, is in the distance of the course that the second ship made from the North Westward, which is found by this Demonstration to be north-west, and by North, and three degrees five minutes to the Westward. CHAP. XV. Two Ships departing from one parallel and Port, in the parallel of 47 degrees 56 minutes, the first in sailing 80 Leagues betwixt the North and West, hath raised the Pole two degrees, I demand by what course the second ship must run, and how much she shall alter in her first Meridian or Longitude, to bring herself 40 Leagues and two Miles North and by East, and forty one Minutes to the Eastward of the first ship. IN the fourth Demonstration draw the Quadrant ABCD E, then multiply your two degrees, you have altered your latitude by twenty, and it maketh forty Leagues; which forty Leagues set upon the line AEL, from A, unto G, then from the point G, draw the line GF, parallel unto AB, then open your Compasses unto the distance of 80 Leagues, which are the leagues your first ship did run, and place one foot of your Compasses in the Centre A, and with the other make a mark in the line GF, which will be at the point F, then from the Centre A, unto the point F, draw the line of, representing the distance of the course of the first ship 80 Leagues: Then from F, let fall a perpendicular FK, and upon the Centre F, upon the Radices of the Scale draw the Arch HIK. Then in regard you must bring the second ship dem: 4 North and by East, and 41 minutes Eastward of the first ship, take 11 degrees, 56 minutes from your Scale of cords, and place them from K, unto I upon the Quadrant KIH, Then from F, draw the line IF, and upon the line FI, place the distance that you must bring the second ship from the first (which is 40 leagues and two miles) from F, unto M, So is M, the place of your second ship. Then from M, draw the line ML, parallel unto FG, cutting the line AGL, in L, then draw the line MA, cutting the Quadrant BDE, in D, So shall the arch DE, be the course that the second ship must run, to bring herself 40 leagues, and two miles North and by East, and 41 minutes East of the first ship. Then to know what you have altered the Latitude, first take the distance LA, and apply it unto the Scale of equal parts, and you shall find it to be 80 leagues, which is just 4 degrees, which you have altered your latitude, or Poles elevation: which 4 degrees added unto the Latitude you departed from it, makes 51 degrees, 56 minutes for the Latitude that your second Ship is in, then take the distance LM, and apply it to the Scale, it gives 60 leagues; then open your Compasses unto the distance of the middle Latitude, which is 39 degrees, 51 minutes of the cord, and apply it unto the first line of Longitudes, and it gives 12 Leagues, and two miles and a half, to alter one degree of Longitude in that parallel: Then set one foot of your Compasses in 12 Leagues, two miles and an half in the second line of Longitudes, extending the other foot unto one degree, and with the same distance upon the same Line, set one foot of the Compasses in 60 leagues (the leagues that you are from the Meridian) and the other foot will extend unto four degrees forty minutes, which is the difference of the Longitude. CHAP. XVI. To find by Demonstration how many miles or minutes of any parallel, doth answer unto one degree of the equinoctial. LEt the Latitude given be 58 degrees 54 minutes, therefore having drawn the Quadrant ABC, from B, upon the cord BEC, set the Latitude of the place 58 degrees 54 minutes unto the point D. Then from the point D, draw the line DF, parallel unto BA, So shall the length of the line DF, be the number of miles which answer unto one degree of Longitude in the parallel of 58 degrees 5 minutes, which being dem: 5 applied unto your Scale of equal parts, gives thirty one miles. So likewise the ark BE, being the Latitude of 52 degrees, ending in the point E, give the lines EG. For the miles that answer unto one degree of Longitude in the parallel of 52 degrees were found by the Scale to be 36 miles, and 56 seconds, or 56/60 of a mile. THE SECOND BOOK OF THE PLAIN SCALE, wherein is declared the definition of the sphere, a description of the six great Circles, and also of the four lesser Circles, and last of all, certain Questions astronomical, performed by the said Scale. CHAP. I. De Sphaera. The figure of the plain Scale. ASpheare according to the description of Theodosius, is a certain solid Superficies, in whose middle is a Point, from which all lines drawn unto the Circumference are equal; which Point is called the Centre of the sphere, by which Centre a right Line being drawn, and extending himself on either side unto that part of the Circumference whereupon the sphere is turned, is called Axis Sphaera, or the Axletree of the World. A sphere accidentally is divided into two parts, that is to say, in Sphaeram rectam, & Spharam obliquam. Sphera recta, or a right sphere, is only unto those that dwell under the equinoctial, Quibus neuter Polorum magis altero elevatur: that is, to whom neither of the Poles of the World are seen, that lie hid in the Horizon. Sphaera obliqua, or an oblique sphere, is unto those that inhabit on either side of the equinoctial, unto whom one of the Poles is ever seen, and the other hid under the Horizon. The Circles whereupon the sphere is composed are divided into two sorts: that is to say, in Circulos majores & minores. Circuli majores, or the greater Circles, are those that divide the sphere into two equal parts and they are in number six: viz. the equinoctial, the middle of the zodiac, or the ecliptic Line, the two colours, the Meridian and the Horizon. Minores vero Circuli, or the lesser Circles, are such as divide the sphere into two parts, unequally: and they are four in number; as the tropic of Cancer, the tropic of Capricorn: the Circle Arctic, and the Circle Antarctic. CHAP. II. Of the four greater Circles. THe equinoctial is a Circle that crosseth the Poles of the World at right angles, and divideth the sphere into two equal parts, and is called the equinoctial, because when the sun cometh unto it, which is twice in the year, viz. In principio Arietus & Libra, that is, March and September) the days and nights are equal throughout the whole World, whereupon it is called, Equator dici & noctis, the equal proportioner of the day and night artificial: and in the figure is described by the Line caesar. The Meridian is a great Circle, passing thorough the Poles of the World, and the Poles of the Horizon, or Zenith point over our heads, and is so called, because that in any time of the year, or in any place of the World, when the sun (by the motion of the Heavens) cometh unto that Circle, it is noon or 12 of the Clock. And it is to be understood, that all towns and places that lie East and West one of another, have every one a several Meridian: but all places that lie North and South one of another, have one and the same Meridian. This Circle is declared in the figure following by the Circle BCDE. The Horizon is a Circle, dividing the superior Hemisphere from the inferior, whereupon it is called Horizon. that is to say, the bonds of sight, or the farthest distance that the eye can see, and therefore it is also called Circulus Hemispheri. The orisons are divided into two sorts, viz. Rectus & obliquus, a right and an oblique, or a declining Horizon: whereof those have a right Horizon which have the equinoctial for their Zenith, and the Poles of the World in their Horizon: Because the Horizon (hiding both the Poles of the World) is a Circle supposed to be drawn by the Poles of the World, dividing the equinoctial at right angles, as in the figure following you may plainly see. First, imagining the Circle XVYW, to be the earth, and those that inhabit at the Point V, have the line BD, for their Horizon, cutting the equinoctial CAE, at right angles in A, and therefore is called Horizon rectus & Sphaerarecta, a right Horizon, and a right sphere. Those have an oblique sphere, or an oblique Horizon to whom one of the Poles are visual, or elevated above the Horizon, and have the other hid under the Horizon, and in regard such a Horizon doth cross the equinoctial at oblique angles, it is called Horizon obliquns, or a declining Horizon, as for example: Those that inhabit at the point S, have T, for their Zenith, and KAL, for their Horizon, dividing the equinoctial CAE, at oblique angles, making the angle contained betwixt the Horizon AK, and the equinoctial A C, an angle of thirty eight degrees, and 28 minutes, and the angle contained betwixt the Horizon all, and the Pole AD, an angle of 51 deg. 32 minutes, which is the elevation of the Pole for those that inhabit at S, those and all other have an oblique sphere, except they inhabit just under the equinoctial Circle, unto whom only doth a right sphere belong. Colurus Solstitiorum, or the Summer colour is a Circle passing by the Poles of the World, and by the Poles of the ecliptic, and by the head of Cancer and Capricorn, whereupon the first scruple of Cancer where the colour crosseth the ecliptic Line, is called Punctus Solstitiae aestivalis, or the point of the Summer Solstice: to which place when the Sun cometh: he can approach no nearer unto our Zenith, but returneth unto the Equator again. Arcus vero Coluri, The ark of the colour contained betwixt the Summer Solstice and the Equator, is called the greatest declination of the Sun, which Ptolemie found to be 23 degrees, 31 minutes: but by the observation of Copernicus it was found to vary, for he found the declination sometimes to be 23 degrees 52 minutes, and in the process of time, to be but 23 degrees, 28 minutes. And in these our days (by the observation of Tycho de Brake, and that late famous Mathematician, Master Edward Right) it is found distant from the equinoctial 23 degrees 31 minutes 30 seconds. The other colour passeth by the Poles of the World, and by the first point of Aries and Libra, whereupon it is called, Colurus distinguens Equinoxia. These two colours do cross each other at right angles in the Poles of the world, whereupon these Verses were made. Haec duo Solstitia faciunt Cancer, Capricornus, Sed noctes aequant Aries & Libra diebus. dem: 6 CHAP. III. Of the four lesser Circles. THe Sun having ascended unto his highest solstitial Point doth describe a Circle, which is the nearest that he can approach unto the North Pole, whereupon it is called, Circulus Solstitii aestivalis, the Circle of the Summer Solstice, or the tropic of Cancer, and is noted in the figure before, by the line H. Y. I. The Sun also approaching unto the first scruple of Capricornus, or the Winter Solstice, describeth another Circle, which is the utmost bounds that the Sun can depart from the equinoctial Line towards the Antarctic Pole, whereupon it is called, Circulus Solstitii hyemalis, sive Tropicus hyemalis, vel Caepricorni: the Circle of the Winter Solstice, the Winter Tropic, or the tropic of Caepricorn, and is described in the figure by the Line G. X. F. So much as the ecliptic declineth from the equinoctial, so much doth the Poles of the ecliptic decline from the Poles of the World, whereupon the Pole of the ecliptic, which is by the North Pole of the World, describeth a certain circle as it passeth about the Pole of the World, being just so far from the Pole as the tropic of Cancer, is from the Equator, and it is the third of the lesser Circles, and is called, Circulus Arcticus, or the Circle of the North Pole, and is described in the Diagram, in the second Chapter by the Line P. O. The fourth and last of the lesser Circles, is described in like manner, by the other Pole of the ecliptic, about the South Pole of the World, and therefore called Circulus Antarcticus, the Antarctic Circle, or the Circle of the Antarctic or South Pole, and is demonstrated in the former figure, by the line N. M. CHAP. IV. Certain questions astronomical performed by the plain SCALE. The true place of the Sun, given to find his declination. THe Sun being in the head of Taurus, his declination is desired: by the seventh Demonstration, draw the line A. D. then upon the Centre A. raise the Perpendicular A. B. then opening your Compasses to the Radius of your Scale, and place one foot in the Centre A. and with the other draw the Quadrant B. C. D. then opening your Compasses unto the greatest declination of the Sun, and place it upon the Quadrant from D. unto K. then from the point K. draw the Line K. H. cutting the line A. B. in H. then with the distance A. H. draw the small Quadrant G. E. H. and in regard the Sun is in the head of Taurus, which is thirty degrees from the beginning of Aries, let A. D. be the Equator, dem: sep: and D the beginning of Aries, D. C. 30 degrees, or longitude of the sun, then from the point C. draw the line C. A. cutting the Quadrant G. E. H. in E. then from E. draw the line E. L. parallel to A. D. cutting the Quadrant B. C. D. in L. so shall the arch L. D. be the declination of the Sun desired, which in this demonstration is found to be 11 deg. and 31 minutes. CHAP. V. The declination of the Sun, and quarter of the Eclipse that he possesseth being given, it is desired to find his true place. The declination is 10 degrees 31 minutes, the first quarter that he possesseth, is betwixt the head of Aries and Cancer. FIrst by the seventh Demonstration draw the Quadrant A. B. C. D. as is taught in the former Chapter, then set the greatest declination of the Sun upon the Cord from D. unto K. which is 23 deg. and 31 minutes, then from K. draw the line K. H. parallel unto the Equator D. A. cutting the line B. A. in the point H. So shall H. A. be the sign of the Suns greatest declination, then with the distance A. H. draw the Circle, G. E. H. then from D. upon the Cord D. B. C. set the declination of the Sun, which is 11 degrees, 31 min. from D. unto L. then draw the line L. E. parallel unto A. D. cutting the Quadrant G. E. H. in E. Then from the Centre A. by the point E. draw the line A. E. C. cutting the Quadrant B. C. D. in C. So shall the ark C. D. be the distances of the sun from the head of Aries, which is here found to be just 30 deg. which is in the beginning of Taurus. CHAP. VI. By the elevation of the Pole, and declination of the Sun, to find the amplitude of the Sun, or his true rising, or settling from the East or West point. BY the eight Demonstration, first draw the line B. D. then upon the centre A. draw the Circle B. C. D. E. then from A. raise the perpendicular C. A. E. then is your Circle divided into four equal parts: then suppose the elevation of the Pole to be 51 deg. 32 min. which must be placed upon the Circle, from D. unto F. then from the point F. by the centre A. draw the line F. A. G. representing the Pole of the World F. being the North Pole, and G. the South Pole, then subtract 51 deg. 32 min. from 90. deg. and the remainder is the height of the equinoctial, which, is 38 deg. 28 min. which must be placed upon the Circle from the Horizon B. unto the point I: ● then from I. by the Centre A. draw the line I. A. H. representing the equinoctial Circle. Then from I. unto M. set the declination of the Sun, being here supposed 14 deg. 52. min. North, then from the point M. draw the line, or parallel of declination M. I. N. parallel unto the Equator I. A. H. cutting the Horizon B. D. in T. then from T. raise the perpendicular T. V. cutting the Circle B. C. D. E. in V, so shall the distance C. V. be the true amplitude of the sun desired, which here is found to be 24 deg. 21 min. North. CHAP. VII. By the Amplitude of the Sun to find the Variation of the compass. HAving found the amplitude of the Sun by the last Chapter, first observe with a compass, or rather with a Semicircle upon what degree and minute the Sun riseth or setteth, beginning to reckon from the East or West, & ending at the North or South at 90 degrees: and when you have diligently observed the magnetical rising or setting, by the Semicircle, or by some other like fitting Instrument: and also the true amplitude found, as is declared in the last Chapter, the difference of these two amplitudes, is the variation of the compass: But when the sun riseth upon the same Degree of the compass, as is found by the Scale, the variation is nothing, but the Needle pointeth directly unto the Poles of the World, which by Master Mulinux was affirmed to be at the Westernmost part of Saint Michael's, one of the Islands of the Vs, from whence he will have the Longitude reckoned. Secondly when the sun is in the equinoctial Circle, where he hath no amplitude, look what distance the compass maketh the Sun to rise from the East or West of the compass, the same distance is the Compasses variation, from the North or South. Thirdly, if the sun rise more to the South of the compass, or setteth more to the North by the compass, than is showed by the Scale, the difference betwixt the amplitude given by the Scale, and the amplitude given by the Needle, is the Variation of the compass from the North Westward. Fourthly, if the compass showeth the Sun to rise more Northward, of set more Southward, than is showed by the Scale, the difference is the variation of the compass, from the North Eastward. Fiftly, if the Scale show the amplitude of the sun rising Southerly, and the compass show it to be Northerly, add both the Amplitudes together, and they show you the variation Westerly. CHAP. VIII. The place of the Sun being given, to find his declination by a whole Circle. ACcording unto the right Demonstration, first draw the Circle B. C. D. E. then draw the Horizon B. A. D. and then the equinoctial I. A. H. as is before taught: and then the tropic of Cancer K. L. 23 degrees and a half from the equinoctial: then draw the tropic of Capricorn P. O. of like distance from the equinoctial, and after from K. to O. draw the ecliptic Line K. A O. And when you have thus laid down the sphere, suppose the Sun to be in the tenth degree of Taurus, at which time his declination is desired. And in regard the sun is more near unto the tropical point Cancer, then unto Capricorn; first find how many degrees he is from the tropic of Cancer, and you shall find him to be 50 degrees; therefore take with your Compasses 50 degrees from the cord, and apply it from the tropical point Cancer at K. unto V. upon one side, and upon P. on the other side: then draw the Line V. P. cutting the ecliptic K. O. in the point R. then from R. draw the Line M. R. N. parallel unto the equinoctial I. A. H. and cutting the Quadrant B. C. in the point M. So shall the ark M. I. be the declination of the Sun desired, which being applied unto your Scale, gives you 14 degrees and 52 minutes. CHAP. ix.. The elevation of the Pole, and declination of the Sun, given to find his height in the vertical Circle. The Pole is elevated 51 deg. 32 minutes, the declination of the Sun is 14 degrees 52 minutes North, his height in the vertical Circle is found as followeth. FIrst according unto the former Chapter draw the Circle B. C. D. E. then the Horizon B. A. D. and after the vertical Line C. A. E. then the Pole of the World F. G. and likewise the Equator I. A. H. this being done, place the declination of the Sun 14 degrees, 52 minutes, upon the Circle from I. unto M. and also from H. unto N. then draw the Line M. N. cutting the line C. A. E. in S. then from S. draw the Line S. W. parallel unto the Horizon B. A. D. cutting the Meridian Circle B. C. D. E. in W. So shall the distance D. W. be the height of the Sun in the vertical Circle, for the time demanded which by this proposition is found to be 19 degrees and 8 minutes. CHAP. X. The elevation of the Pole, and the Amplitude of the Sun, being given, to find the declination. The elevation of the Pole is 51 deg. 32 min. the sun's Amplitude is 24 deg. 21 minutes, the declination is found as followeth by the eight Demonstration. dem: 8 FIrst upon the centre A. draw the Circle B. C. D. E. then draw the Line B. A. D. representing the Horizon, dividing the Circle into two equal parts: then draw the Line C A E, perpendicular to B A D. representing the East & West point of the compass, then placing the elevation of the Pole 51 degrees, and 32 minutes, from D unto F, from F, by the Centre A, draw the Line F A G, which let be the Pole or axletree of the World; then from B unto I, and from D unto H, set the compliment of the Poles elevation: which shall represent the equinoctial, in regard it maketh right angles with the Pole of the World, in the Centre A. Then from C unto V, place the amplitude of the Sun, which is 24 degrees, and 21 minutes: then from V. let fall the perpendicular V T, cutting the Horizon B A D, in the point T: then from the point T, draw the Line M T N, parallel unto the equinoctial I A H: and cutting the Circle B C D E, in the point M and N, so let the distance of I and M, or H and N, be the declination of the Sun, which was desired: which being applied unto your Scale, gives you 14 degrees, and 52 minutes. CHAP. XI. The elevation of the Pole, the declination of the Suum, and hour of the day being given to findethe Almicanter is desired. The elevation of the Pole is 30 degrees, the declination of the Sun is 20 degrees North, the hour is nine in the morning, at which time the Almicanter is found, as followeth. BY the ninth Demonstration, first upon the centre A, draw the circle BCDE, then draw the Line B. D. for the Horizon, then place your Poles elevation which is 30 degrees upon the circle from D, unto R, then from R by the Centre A, draw the Line RAS, representing the Pole of the World, then from B unto F, place the compliment of the Poles elevation, which is 60. degrees, and from the point F, by the Centre A, draw the line FAH, representing the equinoctial Line, and then set the declination of the Sun from F, unto L, and from L, draw the Line LPO, parallel unto the Equator FAH, cutting the Pole of the world in the point P, then set one foot of your Compasses in the point P, and extend the other either unto L, or unto O, and with the same distance of your Compasses, upon the centre P, draw the circle LNOQ, which is called the hour circle, so shall L, be the point of 12 a clock at noon, N, the place of six a clock afternoon, O the place of 12 a clock at midnight, and Q, the place of six a clock in the morning: Every one of the four quarters, must be divided into six equal parts, or hours, making the whole circle to contain 24 parts, representing the 24 hours of the day and night, then in regard the hour of the day was 9 of the clock, which is 3 hours' forenoon, take three of those 24 hours, and place them upon the circle LNOQ, from the Meridian point L, unto K, the nine a clock point in the morning, and unto M, the point of three a clock after noon, then draw the line MK, cutting the parallel of the Sun LO, in the point I, then from I, draw the line IG, parallel unto the Horizon BAD, which shall cut the Meridian circle BCDE, in the point G, so let the distance of G and B, be the Almicanter of the Sun which was desired, which in this Demonstration is found to be 48 degrees and 18 minutes. dem: 9 CHAP. XII. The elevation of the Pole, the Almicanter, and declination of the Sun, being given to find the hour of the day. The elevation of the Pole is 30 degrees, the declination of the Sun is 20 degrees, the Almicanter of the Sun is 48. degrees, and 18 minutes, the hour of the day is found as followeth, by the ninth Demonstration. FIrst, upon the centre A. draw the circle B. C. D. E. then draw the Diameter B. D. representing the Horizon, then from D. unto R. set 30 degrees, the elevation of the Pole, then from R. by the Point A. draw the line R. A. S. representing the Pole of the World, then draw the line F. A. H. crossing the Pole in A. at right Angles, cutting the meridian Line in F. then from F. set 20 degrees, the declination of the Sun unto L. and then from the point L, draw the Line LPO, representing the parallel of the sun, and cutting the Pole of the World in P, then placing one foot of your Compasses in P, extend the other unto L, with which distance of your Compasses, draw the hour circle I, NOQ, then from the Horizon at B, place the sun's Almicanter: (which is 48 degrees, and 18 minutes) upon the Quadrant BGL, from B, unto G, then from the point G, draw the line GI, parallel unto the Horizon BAD, cutting the line LO, in I, then from the point I, draw the line KIM, parallel to the Pole of the World QAN, cutting the circle LNO, in M, then let IN, be divided into 6 hours, whereof LM, are three: whereupon I conclude, that it is three hours from noon, that is, at nine a clock in the morning, or three in the afternoon. CHAP. XIII. The Almicanter, or height of the Sun being given, to find the length of the right shadow. ACcording unto the tenth Diagram, draw the Line of, and upon the centre A, raise the perpendicular AC; then upon the centre A, draw the Quadrant CDF. then suppose the height of your Gnomon, or substance yielding shadow be the Line AB, which is to be divided into 12 equal parts, which Gnomon I have here made just 12 degrees of the equal League of the Scale: then from B to the top of the Gnomon, draw the Line BE, parallel unto of, then set the Almicanter which is 45 degrees from F unto D, and from the point D, draw the Line DA, cutting the Line BE, in the point G, so shall BG be the length of the right shadow desired, which is here found to be 14 degrees and 18 minutes, which is but just the length of your Gnomon, and 2/12 and ⅓ of a twelfth over: Note that the right shadow, is the shadow of any Post, staff, or Steeple, that standeth at right angles with the Horizon, the one end thereof respecting the Zenith of the place, and the other the Nadir. CHAP. XIV. The Almicanter, or height of the Sun being given to find the length of the contrary shadow. BY the verse or contrary shadow, is understood the length of any shadow, that is made by a staff or Gnomon standing against any perpendicular wall, in such a manner that it may lie parallel unto the Horizon, the length of the contrary shadow, doth increase as the sun riseth in height: whereas contrariwise dem: 10: the right shadow doth decrease in length, as the Sun doth increase in height: the way to find the verse shadow is as followeth. First, draw your Quadrant as is taught in the last Chapter wherein let A, B, be the length of the Gnomon, likewise from B, draw the line B E, parallel unto A F, as before, then set your Almicanter from C, upon the Quadrant which is given to be 70 degrees, and it will extend from C, unto H, then from the point H, draw the line H A, cutting the line B E, in the point K, so shall K B, be the length of the contrary shadow, which here is found to be 34 degrees and 8 min. or twice so long as your Gnomon, and 10/12 and about ½ part of a twelfth more. CHAP. XV. The latitude of the place, the Almicanter, and declination of the Sun being given, to find the Azimuth. The latitude of the place is 51 degrees, 30 minutes, the declination of the Sun 20 degreès North, the Almicanter 38 degrees 30 minutes, the true Azimuth of the Sun is desired. FIrst upon the Centre Adraw the Circle B C D E, then draw the Diameter B A D, and from D, unto F, set the elevation of the Pole which is 51 degrees, and 30 minutes whose compliment is 38 degrees and 30 minutes, which must be placed from B, unto H, then from H, draw the line H A I, representing the equinoctial Line, and from F, draw the line F A G, representing the Pole of the World, then from H, unto P, and from I, unto Q, set the declination of the Sun, which is 20 degrees, and by those two points draw the line P Q, for the parallel of the sun's declination; then upon the Circle from B, unto H, set the sun's Almicanter, 38 degrees, and 30 minutes, then from H, draw the line H R, parallel unto the Horizon, cutting the sun's parallel P O, Q, in O, then draw the line T V A E, Perpendicular unto the line B A D, in the Centre A, and cutting the line H V R, in V, then set one foot of your Compasses in the point V, extend the other unto R, and with the same distance draw the Semicircle H, dem: 11: L R, then draw the concentric Circle upon the Radius of the Scale M T N, and where the line P O Q, and the line M O N, do meet in the point V, raise the Perpendicular O L, cutting the Semicircle H L R, in L, then lay the Scale from the Centre N, to the point L, and draw the Line L K, cutting the Semicircle M T N, in K, so shall K T, be the true distance of the Sun, from the East, or West point Southward, or the sun's true Azimuth, which is here found to be 72 degrees, and 10 minutes from the South part of the Meridian. CHAP. XVI. The place of the Sun being given to find the right ascension is desired. Suppose the Sun be in the 20 degree of Taurus, his right ascension is found as followeth. FIrst draw the line B A F, for the Pole of the world, then upon the Centre A, draw the Circle B C D E, then from the Centre A, raise the Perpendicular C A E, for the Equator: then place your greatest declination from C, unto Q, and from E, unto P, then draw the line Q A P, which doth represent the ecliptic line, then in regard the sun is in the 20 degree of Taurus, which is 40 degrees, from the head of Cancer, which 40 degrees place from Q, unto L, and unto K, then draw the line K L, cutting the ecliptic in I, then from the point I, draw the line H I, parallel unto C A E, cutting the Pole of the World in O, then set one foot of your Compasses in O, and extend the other unto G, with which distance draw the Semicircle H D G, then opening your Compasses unto the Radius of the Scale, and upon the Centre O, likewise draw the Circle H N F G, then draw the line I M, parallel unto A O D, cutting the Semicircle H M D G, in M, then lay your Scale from the dem: 12: Centre O, unto the point M, and draw the Line N M, cutting the concentric Circle in N, so shall the distance N F, be the right ascension, which is here found to be 42 degrees, 27 minutes. CHAP. XVIII The elevation of the Pole, and declination of the Sun, given to find the difference of the ascensions. The Poles elevation is 51 degrees 32 minutes, the declination of the Sun is 21 degrees. dem: 13: FIrst draw the Line B A K, representing the Horizon, then upon the Centre A, draw the Circle B C D E F, Then from K, unto D, set the elevation of the Pole which is 51 degrees, and 32 minutes: then from the point D, by the Centre A, draw the line D A F, representing the Pole of the world: then from R, unto C, set the compliment of the Poles elevation which is 38 degrees, and 26 minutes, then from C, by the Centre A, draw the line C A E, representing the equinoctial Line; then from C, unto G, and likewise from E, unto H, for the declination of the Sun, which is 21 degrees, then from G, unto H, draw the parallel of the sun's declination, cutting the Pole of the World in L, and the Horizon in I, then set one foot of your compasses in the point L, and extend the other unto G, then with that distance of your Compasses draw the Semicircle G M N H, then opening your Compasses unto the radius of your Scale, and upon the same centre draw a concentric circle, G X O H, then from I, where the declination of the Sun doth cut the Horizon, draw the line I N, parallel unto the Pole of the World A M, cutting the circle G M H, in N, then lay your Ruler from the point I, unto the point N, and so draw the line N O, cutting the concentric circle G X O H, in O, so shall the distance of O, and X, be the difference of the ascensions, which is here found to be 28 degrees, and 54 minutes. CHAP. XVIII. The right ascension of the Sun or star being given, together with the difference of their ascensions, to find the oblique ascension. THe right ascension of any point of the Heavens being known, the difference of the ascension is either to be added thereunto or else to be subtracted from it, according as the star is situate in the Northern or Southern signs: As for example, if the Sun be in any of these six signs, Aries, Taurus', Gemini, Cancer, Leo, or Virgo, than the difference of the ascensions is to be subtracted from the right ascension, and the remainder is the oblique ascension. Suppose therefore the sun to be in the fourth degree of Gemini, where the right ascension is found to be four hours, and 8 minutes, or 62 degrees, and the difference of ascension where the Pole is elevated 51 degrees, is found to be one hour 53 minutes otherwise 28 degrees 53 minutes, which being taken from the right ascension, leaves two hours, and 16 minutes, or 33 degrees, and 42 minutes, which is the oblique ascension of the Sun in the fourth degree of Gemini. But if the Sun be upon the South side of the equinoctial, either in Libra, Scorpio, Sagitarius, Capricornus, Aquarius, or Pisces, than the difference of the ascensions is to be added unto the right ascension, and the Product will be the oblique ascension. Suppose the fourth degree of Sagitarius, is given, for which sign and degree the oblique ascension of the Sun is desired, his right ascension being then found to be 242 degrees, or 16 hours, 8, minutes, the difference of the ascension is one hour, 53 minutes, or 28 degrees, 18 minutes, which being added unto the right ascension, makes 18 hours, and one minute; or in degrees 270 degrees, and 18 minutes: which is the oblique ascension of the Sun, when he is in the fourth degree of Sagitarius. And if you would find the oblique descension, you must add the difference of the ascensions unto the right ascension, when the Sun is in these six signs, Aries, Taurus', Gemini, Cancer, Leo, Virgo: and contrariwise, when the sun is in the other six signs, you must subtract the difference from the right ascension, and you shall have the oblique descension of the Sun, or any star, whose right ascension and difference of ascensions is known. But it is to be understood, that this manner of operations doth serve no longer than you are upon the North side of the equinoctial. For if the South Pole be elevated, the work is contrary: for so long as the Sun is in any of the Northern signs, the difference of the ascensions is to be added unto the right ascension, to find the oblique ascension. And contrariwise, subtracted to find the oblique descension. Likewise if the Sun or star be in the Southern signs, then is the difference of ascensions, subtracted from the right ascension, to find the oblique ascension, and added, to find the oblique descension. A Description of some peculiar things, fit to be considered, by such as intend to practise the Art of Navigation, or astronomy. THE Zenith is an imaginary point in the Heavens over our heads, making right Angles with the Horizon, as the equinoctial maketh with the Pole. The Nader is a prick in the Heavens under our feet, making right Angles with the Horizon under the earth, as the Zenith doth above; and therefore is opposite unto the Zenith. The declination of the Sun, is the ark of a Circle contained betwixt the ecliptic and the Equinoctial, making right Angles with the equinoctial. But the declination of a star, is the ark of a Circle let fall from the Centre of a star, perpendicularly unto the equinoctial. The Latitude is the ark of a Circle contained betwixt the Centre of any Star, and the ecliptic Line making right Angles with the ecliptic, and counted either Northward, or Southward, according to the situation of the Sea, whether it be nearer unto the North or South Pole of the ecliptic. The Latitude of a Town or Country is the height of the Pole above the Horizon, or the distance betwixt the Zenith and the equinoctial. The Longitude of a Star, is that part of the ecliptic, which is contained betwixt the stars place in the ecliptic, and the beginning of Aries counting them from Aries according to the succession or order of the signs. The Longitude of a town or country are the number of degrees, which are contained in the equinoctial, betwixt the Meridian that passeth over the Isles of Azores, (from whence the beginning of Longitude is accounted) Eastwards, and the Meridian that passeth over the Town or Country desired. The Altitude of the Sun or Star, is the Arch of a circle contained betwixt the Centre of the Sun, or any star, and the Horizon. The Amplitude is that part of the Horizon which is betwixt the two East or west points, and the point of the compass that the Sun or any Star doth rise or set upon. Azimuths are Circles, which meet together in the Zenith, and cross the Horizon at right Angles, and serve to find the point of the compass, which the Sun is upon at any hour of the day, or the Azimuth of the Sun or Star, is a part of the Horizon contained betwixt the true East or West point, and that Azimuth which passeth by the Centre of the same star to the Horizon. The right asention of a star is that part of the equinoctial that riseth or setteth with the Star, in a right sphere, or in an oblique sphere, it is that portion of the equinoctial, contained betwixt the beginning of Aries, and that place of the equinoctial, which passeth by the Meridian with the centre of the star. The oblique ascension is a part of the equinoctial, contained betwixt the beginning of Aries, and that part of the equinoctial that riseth with the centre of a star, in an oblique sphere. The difference ascentional, is the difference betwixt the right and oblique ascension: or it is the number of degrees contained betwixt that place of the equinoctial that riseth with a Centre of a star, and that place of the equinoctial that cometh unto the Meridian, with the Centre of the same star. almicanterah, are Circles drawn parallel unto the Horizon, one over another until you come unto the Zenith: these are Circles that do measure the elevation of the Pole, or height of the Sun, Moon, or Stars above the Horizon, which is called the Almicanter of the Sun, Moon, or Star: the ark of the Sun or Stars Almicanter, is a portion of an Azimuth contained betwixt that Almicanter which passeth thorough the centre of the star, and the Horizon. A general Table for the Tides in all places. The moon's age. Hours and Minutes to be added. The moon's age. Hours and Minutes to be added. Days. Degrees. Minute's Days Degrees. Minutes. 1 0 48 16 0 48 2 1 36 17 1 36 3 2 24 18 2 24 4 3 21 19 3 12 5 4 0 20 4 0 6 4 48 21 4 48 7 5 36 22 5 36 8 6 24 23 6 24 9 7 12 24 7 12 10 8 0 25 8 0 11 8 48 26 8 48 12 9 36 27 9 36 13 10 24 28 10 24 14 11 12 29 11 12 15 0 0 30 0 0 The use of the Table of the Tides. FIrst it is to be understood, that by the swift motion of the first mover, the Moon and all the rest of the Stars and Planets, are turned about the World in 24 hours, upon which swift motion of the Moon, the daily motions of the Sea do depend, which motion of the Sea falleth not out always at one hour, the reason thereof is, because of the swift motion of the Moon in regard she goeth almost 13 degrees in 24 hours, and the Sun moveth scarce one degree: which gives every day 12 degrees, that the Moon cometh slower to any point in the Heaven than the sun: which 12 degrees makes 48 minutes of time for the difference of every full Sea, according unto the middle motion of the Moon, which difference is here set down in this Table for every day of the moon's age. Therefore if you would know the full Sea at any place in the World, first you must know at what hour it is full Sea at the new or full Moon; which hours and minutes keep in mind, then seek the age of the Moon as is before taught, and with the number of her age enter this Table, under the title of the moon's age, and having found her age in the Table, against it you shall find the hours and minutes which are to be added unto the time that the Moon maketh full Sea in any place, and the whole number of hours and minutes is the time that the Moon maketh full Sea in any place upon the day desired. As for ensample, I desire to know the full Sea at London bridge upon the 13 of July 1624. the age of the Moon being found as before, is eight days, then in the Table I find eight days, and against it 6 hours, and 24 minutes, which being added unto 3 hours, the full Sea upon the change day gives 9 a clock, 24 minutes for the time at the full Sea upon the 13 day of July 1624. FINIS.