The Pathway to perfect SAILING. Showing briefly the six principal Points or Grounds of NAVIGATION. Written by Mr. RICHARD POLTER, one of the late principal Masters of the Navy Royal. And now published for the Common good of all Masters, Pilots, and other Seamen whatsoever. Whereunto is added, A NAUTICAL DISCOURSE necessary to be known of all SEAMEN to prove the way of a Ship (upon the Superficies of the Sea) outward and homeward to be both one, returning by the opposite Point of the Meridian Compass: And also to prove the East and West directed by the Meridian Compass to lead in a Magnetical parallel. First penned by JOHN BASSET deceased, a Teacher of Navigation at Chattam in Kent. And now put forth to public view with some Addition, By HENRY BOND. LONDON, Printed by T. FORCET for GEORGE HURLOCK, and are to be sold at his shop near St. Magnus' corner. 1644. compass rose THE PATHWAY to perfect SAILING. Showing the six principal Points or Grounds, concerning Navigation. 1 The Card, 2 Compass, 3 Tide, 4 Time, 5 Wind, 6 Way. TO deliver the opinions (concerning the Card and Compass) of a great sort of Sea-Matters and others, even in these times such is their knowledge, being altogether ignorant even vain and frivolous, that I will not spend the time once in nominating any such their knowledge, but will proceed to the matter. The Compass to be rectified two ways. FIrst, there is to be delivered two ways for a man's farther knowledge, that the Compass is to be rectified, as felloweth. The Compass is said to be rectified when the wire is set right under the Flower-deluce, for the Flower-deluce and his opposite point delivereth the true Meridian. Also, the Compass is said to be rectified, when the Wire is set right under the Flower-deluce, the Flower-deluce and his opposite point, delivereth the true Mutation or Variation of the Compass. Further, there are four opinions concerning the Compass: which four opinions and ways conceived of the Compass, being rectified from the true Meridian that it is presupposed to lead in, which I will give a taste of for this present, and so leave them to more convenient places. The first Opinion of the Compass. THe most Absurdest way of the Compass, is according to flat Cards-used in Navigation: which some sort of men (and the greatest number hold opinion, that the Compass leadeth not otherwise then according to these usual Plaits or Cards right lined, whose lines generally are parallel each to other, and that the East and West of the Compass leadeth in a parallel, which opinion is absurd: for how can the thing that is false deliver the way of truth, as hereafter followeth. The 2 Opinion of the Compass. This way of the Compass (in deed) which is more perfect than the former (according to the judgement of Astronomers) is called the Astronomical deducture of the Compass and by other sort of men called the paradoxal Compass, because the lineaments are spiral: which spiral lines by some are called Helicall lines. This Astronomical deducture (as above said) some sort of men hold to be principallest. In this judgement the Meridian's have their coartation, but the Compass (the East and West thereof) is said to lead in a parallel, therefore though nearer the truth then the former, yet absurd. The 3 Opinion of the Compass. Is the demonstrature of the Compass, which is more excellent than the two former, being known aright: which demonstrature of the Compass, some sort of men seeth (but with heads filled with infirmities, etc.) and wanting knowledge, knoweth not the meaning thereof, and but very few understand the goodness that is to be delivered thereby, yet the way of a Ship is not according to the present view thereof. The 4 Opinion of the Compass. Last of all (which fewest or none know) is the only true and excellent way, the way of the Compass according to the difference never yet delivered by any, nor known unto those that make themselves most artificiallest. I am now to deliver my reasons or examples concerning these four opinions (above said) which are to condemn the small knowledge, opinion or judgement of men, concerning them: and to justify and maintain the good knowledge that hereafter shall be delivered. The Card and Compass handled together. WHereas, before I have delivered a taste of the four opinions concerning the Compass, the first opinion relying only upon the truth of the Card, Therefore it is most necessary for me, next of all to deliver the absurdities of the Card, or to lay the Card open in his colours, that when the truth thereof is seen and known, then that the way of the Compass (according to the Card used in navigation) may be left, and a more true way taken how of etc. Now to proceed as followeth. The 1 Absurdity. A general Card strait or right lined, as it is, representeth, yea delivereth the whole universal world at the view thereof, as a plain flat or level and not otherwise, and maintaineth a scale correspondent thereunto, and how absurd this delivery is, hereafter is proved. The 2 Absurdity. This general Card maintaineth a Compass in the midst that shall extend itself, and the points thereof to the extremes of the Card, and so delivereth courses by one Compass to sail to the extremes of the world, delivered thereby if it were navigable, even to any part thereof, and to return the same way again: which is absurd, as by the demonstrature and way of the difference following is proved. The 3 Absurdity. This Card will direct by half a Compass made on the side of the Card, over the whole world, and the return to be the same way, which is more absurd. The 4 Absurdity. Yet this Card will direct farther, that is by a quarter of the Compass made in one of the corners of the Card over the whole world, which will be made most absurd and monstrous by the grounds before spoken of. The 5 Absurdity. This Card also maintaineth from the Poles themselves an East and West line, a Meridian and seven points of either side, whereby the North Pole delivereth from it a Southeast quarter and a South-west quarter, and the South Pole the like, which is absurd: my reason to prove it to be so, is this, because there passeth no lines imaginary from the Poles, but only Meridian's according to the sphere. The 6 Absurdity. This Card also maintaineth all the parallels of the East and West therein, to be of one length, which is absurd: for parallels are less towards the Poles: likewise y Card justifieth she parallels to make right angles with the Meridian's even to the Poles, which is absurd, for parallels are inferior Circles unto the Meridian's, which are great Circles, therefore cannot make right Angles with them. The 7 Absurdity. This Card also maintaineth all Meridian's to be parallels each to other, and thereby maketh a polar line of East and West in the Pole itself as long as the Equinoctial, which is an error of 360 degrees, or 21600 miles, which is most absurd: my reason is this, because the Pole itself is but an imaginary point. The 8 Absurdity. Again, there cannot be a right sign (for the way of a Ship) delivered from the plain of a Meridian in the Card, for the delivery of the longitude the Meridian's being parallels each to other (and wrested from their nature) as they be, neither can there be a second right 〈◊〉 (for the way of a Ship delivered from the East and West of a Card) for the delivery of the Latitude: because the parallels therein cannot deliver neither the plain of a great Circle nor small: therefore the Card absurd. The 9 Absurdity. Likewise this presupposed way of the Compass, according to the Card delivereth upon any point of the Compass, for any Segment in Navigation a less way in Longitude and Latitude than the Astronomical deducture of the Compass, or way upon the difference doth but if the Longitude of this first way be compared and reckoned after the coarting of the Meridian's for any segment in Latitude and Longitude: than it is more swifter in Longitude then any opinions else delivered of the Compass: therefore absurd. The 10 Absurdity. All the lines in a Card are supposed to be Segments or pieces of great Circles, which is absurd and false: for example, great Circles must cross themselves at opposite points of necessity, and therefore cannot be parallels each to other: likewise I am sure they count them not spiral lines, for spiral lines are not parallel either to other. The 11 Absurdity. The Equinoctial and Meridian, in a Card graduated as they be by even degrees, both in Longitude and Latitude, being in plano not lawfully projected, is absurd. The 12 Absurdity. Likewise, the Card delivereth this rule for his truth, upon what point of the Compass soever, the points being of one quality, in what Latitude or Longitude soever you are in like distantes for the differences of degrees in Longitude: which is absurd, and confuted by the Astronomical deducture, and by the way upon the difference. The 13 Absurdity. Also the Card delivereth this for his truth, upon what point of the Compass soever, the point being of one kind, in what Latitude or Longitude soever: like distances for the raising or laying of a degree in Latitude, which is absurd and false, as shall be delivered by examples upon the difference. The 14 Absurdity. This Card delivereth no truth in Longitude: nor so near the truth as is to be delivered, therefore in places that are distant from other, the difference of time in no sort is to be delivered thereby, therefore more absurd, then that which will deliver the time more certain. The 15 Absurdity. Whereas one Compass (as before) in the midst of the Card, the point thereof being extended to the extremes thereof, and widest there, it is as contrary to the truth as falsehood is; for the points of the Compass being extended in delatation but to the Quadrant, which is 90 deg. according to the demonstrature in plano, not lawfully projected, is absurd. But some of the points to be extended (without) or beyond the Quadrant 105 deg. too much, and continued in delatation, is absurd, but a half Compass in the East or West part of the Card, as the Card delivereth, some of which points, from his half Compass being extended from the Quadrant in delatation 276 deg. 45 min. too much, is more absurd. Nay a quarter of the Compass delivered from one corner of the Card, as the Card delivereth, some points thereof are extended in delatation beyond the Quadrant 300 deg. (of a great Circle) too much, which is most absurd, as the 15 part in the Astronomical deducture, and first part of the demonstrature justifieth. The 16 Absurdity. Whereas in this Card, the Compasses in the midst, sides or corners thereof, extending their delatation and coartation accordingly beyond the quadrant, as aforesaid, which is absurd: yet the delatation and coartation within the quadrant, is also absurd: as for example; Prove from the centre of any of the Compasses in the Card, at 30 deg, delatation, and see whether the degrees there, be half so big as they are in the quadrant (as they should be) or not, and you shall find 3 of the deg. there, at 30 deg. delatation, to make but a degree in the quadrant, which is ⅓ part false: but it delivereth at 45 deg. delatation 2 deg. there, to be 1 deg. on the quadrant, which delatation is 15 deg. false, or out of his due place. The 17 Absurdity. It is a great matter, how absurd the Card is: for let a Ship in her navigation in any Longitude or latitude whatsoever, sail in the line of North-east or the like points, as Northwest and South-West, before she can accomplish one Quadrant, she must use so much way in her navigation, as cometh to 127 deg. 15 min. of a 〈◊〉 Circle, which is 37 deg. 15 min. too much only in that distance: but a quadrant according to the Sphere is about 90 deg. of a great Circle, which is the truth: therefore the Card absurd. The 18 Absurdity. A Ship in her navigation differing in longitude according to the Card altereth the pole from the first place on the polar line to the meridian on the Card which the Ship is said to be in, which is absurd, as by the 18. part in the Sphere is justified. The 19 Absurdity. It is a servable that the Card (being so monstrous) flowing with nothing but absurd ties that one error thereof being 〈◊〉 myle● (as aforesaid) should be so greatly ●n request as it is being no life or help in the world ●n it to that which is 〈◊〉 in navigation: how it is possible for the ●smography, Geography o● 〈◊〉 thereof to be true: surely in this Card the one is as true as the ●ther and no tr●er therefore all false and ●bsurd. So that the Compass is 〈◊〉 from the good nature it hath 〈◊〉 t●sel●e or the delivery of a ●hips way, by the absurd opinion delivered to be or received from the Card. If I could alleadye more condemnations for the Card and the opinions of the compass (being accordingly) ●et enough suf●ceth. ●●lesse the matter in the Card and knowledge thereby received) were better: And thus I lea●● the Card and the judgement by it at the Compass. The Spherical Description of the Globe, with the Astronomical deducture of the Compass accordingly which by some sort of men, is called the paradoxal Compass. NO is will I proceed with the Astronomical deducture of the Compass: and because it dependeth upon the lineaments of the Sphere which is the ground thereof, as the Card is of the Compass used thereby in navigation therefore I will entreat somewhat concerning both the Sphere and deducture of 〈◊〉 compass upon it: and as there are 19 absurdities before delivered, for the confuting or delivering the Card in his colours: so many parts I will touch in this Astronomical deducture, that the difference (between them) may be seen. But this in brief by the way: The Astronomical deducture cleareth a way from it many absurdities, by cearting of the Meridian's (only) it maintaineth absurdly the East and West of the Compass, to lead in a parallel as hereafter shall appear. The 1. part. THis whole description of the world, in Globe or round form spherically lined with meridians (one being graduated) likewise with parallels and spiral lines, which by some men are called Rhombes, by others called Helicall lines, and by some called Parado●all lines, 〈◊〉 down with the Geography, the scale thereof being the graduated meridian, which is correspondent thereunto, is very profitable to the knowledge thereof, and confuteth the flat Card. The 2. 3. and 4. part. This Globe maintaineth a Compass, being in the Zenith of the earth and water, to ertend itself according to the domon●●rature in delatation to the Horizon and no farther: deviding it into 32 parts or points correspondent thereunto, but maketh the way of a Ship out and home to be all one, and spirally according to this Astronomical opinion, that the East and West of the Compass leadeth in a parallel is absurd, and causeth great difference, which in this place and time cannot be delivered. The 5. part. This Globe maintaineth the Poles of the world, to be but imaginary pricks or points, delivering or from which points pro●eedeth only meridians, which is true. The 6. part. This Globe maintaineth all the parallels of Latitude therein, to be in their d●e form, proportion, or Magnitude biggest nearest the equator, and smallest nearest the Poles disallow parallels, to make right angles with meridians, which is true. The 7, part. In which Globe, likewise the meridians hath their delatation and Coartation, admitting no polar line, which is true. The 8. part. This Globe, according to the East and west, leading in a Parallel, delivereth one right sign (for the way of a Ship) from the plain of the meridian (as it should) for the delivery of the longitude the difference respected, and the other right sign, from the plain of the parallel for the delivery of the latitude, which is absurd: for if the plains of a great Circle and a small to deliver the place of a Ships being, agreeth not, neither is true, as the difference justifieth. The 9 part. The way of the compass, according to this astronomical delivery, for any sagments in navigation upon what Point soever, is more swifter in latitude, and more flower in longitude, than the delivery used in navigation by the Card, delivering the point of a Ships being, by the signs as aforesaid, which is absurd. The 10. part. This opinion of East and West, leading in a parallel, delivereth no sagments of any spiral lines, to be sagments of great circles, which is absurd. The 11. part. The Equinoctial and meridian, in the Globe graduated as they be, by even degrees both in longituds and Latituds, being in round Globe form, is very true. The 12. part. This opinion delivereth upon any spiral line (of one quality) which are the points of the Compass, according to this judgement, not like distances for the difference of degrees in longitude, but is different in every place between the equator and the poles yet not the truth, as the difference justifieth. The 13. part. And for this Astronomical delivery by any the spiral points thereof, to raise or lay degrees in latituds: it difforsth little from thecard, which is absurd. The 14. part. This Globe, the meridians thereof delivereth or distinguisheth itself into degrees of longitude, by which degrees is delivered the difference of time between any two places or distances, yet not very true, as in the delivery of the time shall appear. The 15. part. As by this Globe (before) the compass according to the demonstrature from any one place, delivereth delatation to the Horizon where it is greatest, so from the Fonitor or horizon to the Nadir, all the Arches of depression suffer coartation so that the intersections or cross of those points of demonstrature, are in the Zenith and his Nadir, which is true: therefore the 15 absurdity in the Card proved by this to be most absurd. The 16. part. By this Globe a compass in the Zenith thereof delivering the Azimuthes to the horizon which is at 90 degrees from the zenith you shall find these Azimuths at 30 degrees delatation, There a degrees maketh 1 degree in the horizon, and all the rest of the Azimuthes or degrees of them, is delivered with truth, either in delatation or coartation, therefore the 15 absurdity in the Card is acknowledged. The 17. part. This Globe delivereth by any point of demonstrature, 90 degrees and no more to the quadrant, therefore confuteth the Card according to the 17 absurdity. The 18. part. This Globe maintaineth the poles of the world to continue in their places which is true. The 19 part. The Cosmography, Geography and Geometry, of the Globe, is fare more perfect than that in the card, yet if the Cosmography and Geography were set down according to the difference, it would be more perfect: but then there must be more descriptions than one: and thus I leave the Astronomical deducture of the Compass. This always to be noted, look what Lineaments is said to be in the Sphere of the Heavens, as meridians for the delivery of the time, and a help for the delivery of the Longitude the equinoctial and all the parallels, for the delivery of the latituds, the horizon, and all the Almicanters of Altitude, for the delivery of heights, the vertical circles or Azimuthes, for horizontal distances, spiral lines according to the difference for the delivery of a Ships way and such like: is imagined to be in the earth and waters correspondent to these in the Heavens, for the travellers help and comfort: the conceit of which aright, is mo● excellent. Hear followeth the demonstrature of the Compass, which is a notable knowledge and light in Navigation, being rightly conceived and used as it ought. The 1. part. IN any latitude and longitude whatsoever, the demonstrature of the Compass representing as it doth, the Zenith of the earth and water, and divideth the horizon, not only into 32 parts, as in the 2. 3. and 4. absurdities of the Card is set down, but also into 360. parts, nay rather into infinite parts called vertical Circles or Azimuths going with delatation from the Zenith or Vertical point to the horizon where the delatation is greatest, and from this horizon or Fenitor to the Nadir (as is said in the 15 part of the Astronomical deducture) these Azimuthes or Vertical circles (in the Hemisphere, not seen, being arches of depression) suffer coartation, so that the intersections or cross of these Azimuthes one another, is only in the Zenith and his Nadir. The 2. part. This demonstrature is moneable or alterable, according to a Ships differing in longitude and latitude, because a Ship caristh her Zenith with her, so that the division of the horizon according to this demonstrature is variable, according to the distance (from any one place) that the Ship is remeved: etc. A●confuting of the 2. absurdity of the Card. The 3. part. The devisson that the Zenith, horizon and Nadir hath, or is divided withal, is only by almicanter circles, which otherwise are called circles of Altitude or height (being above the horizon) all which are parallels to the horizon, but those from the Horizon to the Nadir, notwithstanding parallel to the Horizon are circles of depression, any one of which Almicanters, may be delivered upon any degree or minute of the height above the Horizon, likewise taken (in some working spherically) in profundity and depth under the Horizon. The 4. part. This demonstrature delivereth upon any Azimuth whatsoever, the distance from the Zenith to the Horizon to be but 90 degrees. The 5. part. The East and west of this demonstrature (from the zenith) 90 degrees in longitude, toucheth the Equinoctial, in the horizon at opposite points. The 6. part. Now the quality of the Globe is this, the pole thereof being elevated or set to his purposed latitude in his horizon, according to this demonstrature, delivereth the obliquity of the Sphere, which in truth is a confuting of the East and west to lead in a parallel: as for example. The 1 Example. The Globe set to the parallel of 50 degrees north latitude, than a perpendienler line imagined to descend, from this Zenith of 50 degrees North latitude, delivereth his Nadir to be in 50 degr. south latitude, so that it delivereth the opposite part of the parallel to be from the Nadir or the arch of the meridian, contained between the opposite part of the parallel and the nadir, to be a 100 degrees obliqne or different, and from the place of being in the zenith upon the meridian, to the oposite part of the parallel directly, is but 80 deg. of a great circle, so that this opposite part of the parallels is 10 deg. above the Horizon, which is obliqne to the zenith and Nadir, also 90 degr. North from the east and west of the demonstrature, in the equinoctial. So that a Ship being in (or having) this zenith before spoken of, the perpendicularity of the ship, being in this Zenith and Nadir line, and the opposite part of this Parallel so obliqne unto her, as before how is it possible, for the East and west of the compass to lead in a parallel. The 2. Example. Likewise when the ship, the right line of her caping in the vertical circle of east and west extended both the ways, crosseth the equator at opposite points, 90 degr. different in longitude, and the Ship being a segment of that great circle, and no sagment of a small circle as parallels are how is it possible, then for the east and west to lead in a parallel; it is not possible as the way of the difference justifieth. etc. The third Example. Being in the parallel of 80 degrees North latitude, * If the 〈◊〉 sail upo● vertical 〈◊〉 of East o● that the 〈◊〉 pass doth 〈◊〉 whé the sh● 'gins its co● but as the 〈◊〉 is guided b● magnetical Wires or 〈◊〉 dle, so tha● Line or its 〈◊〉 makes eq● Angel's wi● very mag● call Merid● The way 〈◊〉 ship leading the East o● doth descry magnetical ralell. a perpendicular line delivereth the Nadir to be in 80 deg. S. latitude, so that the arch of the meridian contained between the nadir and the opposite part of the parallel is 160 degrees. But the arch of the meridian, contained between the point of being, and the opposite part of the parallel is but 20 degrees. So that the opposite part of the parallel is 70 degrees above the horizon: and likewise 90 deg. North from the east and west, according to the demonstrature in the equinoctial: ☞ Here you see the obliquity of the opposite part of the parallel to the zenith, and his nadir; showing how unnatural it is to deliver the way of a Ship, leading in the east and west to describe a parallel. And whereas before I have delivered the East and west of the demonstrature of the Compass in the horizon, and that the north and south is likewise very plain to be delivered; therefore I will omit them: and will give examples of the demonstrature in the latitude 51 degrees 30 min. and 80 degrees North latitude, only of the north-east, southeast, south-west, and northwest to the Horizon, as bereafter followeth. North-east in the Latitude 51 Degrees 30 Minutes. FRom our point of being in our Meridian, which is the beginning of our longtitude, our Zenith being in the parallel of 51 degrees 30 min. north latitude: I am to demonstrate 3 sagments of a great circle containing 30 degrees a piece by the demonstrature north-east: the end of which first sagment of 30 degrees, shall touch the Meridian in longitude Eastward, 54 degrees from the first, and there likewise shall touch the parallel of north latitude 64. degrees. And the end of the second sagment of 60 degrees in longitude shall touch the 105 degrees 30 min. or Meridian in longitude eastward from the first, and there likewise toucheth the 50. deg. 30. min. parallel north. Likewise the end of the third sagment, at 90 degrees in longitude, shall touch the 128 degree or Meridian in longitude Eastward from the first, and there likewise toucheth the 26 degree ward 13 min. parallel North: also it there toucheth the horizon 45 deg. either from the cardines East or North. Northwest. Likewise the demonstrature of the Northwest cutteth the Meridian's and parallels in longitude Westwards as the demonstrature of North-east cutteth them East-wards, only that it cutteth the Horizon 45 deg. either from the cardines West or North. Southeast in the same Latitude 51 deg. 30 min. IN our place of being (as aforesaid) I am to demonstrate three sagments of a great Circle containing 20 deg. a piece by the demonstrature Southeast, the end of which first sagment of 30 deg. shall touch the 23 deg. 30 min. or Meridian in longitude Eastward from the first and there likewise toucheth the 27 degrees 20 min. parallel North. And the end of the second sagment of 60 deg. in Longitude. wall touch the 3 deg. 40 min. or meridian in longitude Eastward from the first and there likewise toucheth in 〈◊〉 deg. 25 min. N. latitude. Likewise the end of the third sagment at 90 deg in Longitude, shall touch the 51 degree 40 min. or meridian, in Longitude eastward from the first and there likewise toucheth the 26 deg. 15 m. parallel south: Also there toucheth the Horizon 45 degrees either from the cardines East or south. South-west. Likewise the demonstrature of the South-west toucheth the meridians and parallel in longitude westward as the demonstrature of southeast toucheth them eastward, only that it cutteth the horizon 45 deg. either from the cardines south or west. Examples in the Parallel of 80 deg. North latitude beginning with the first example of North-east in the north latitude 8 deg. FRom our point of being in our meridian which is the beginning of our longitude our zenith being in the parallel of 80. deg. north latitude I am to demonstrate; sagments of a great circle, containing ●0 deg. apéece by the demonstrature north-east the end of which first sagment of 30 degrees, shall touch the meridian in longitude eastward 119 degrees 30 minutes from the first and there likewise shall 〈◊〉 the parallel of North latitude 60 de●r. And the end of the second sagment. 160. deg. in longitude, wall touch the 130. deg. ●. min. or meridian in longitude eastward from the first, & there likewise toucheth the 36. deg. 45. mi. parallel north. Likewise t●t end of the 3. sagment at 90. deg. in Longitude shall touch the 134 deg. 30 min. or meridian in longitude eastward from the first & there likewise toucheth the 7 parallel north Also there toucheth the horizen 45 deg. either from the cardines east or north. Northwest. Likewise, the demonstrature of northwest cutteth the meridians and paraellls in longitude westward, as the demon●rature of north east, cutteth them eastward only, that it cutteth the horizon 45 deg. either from the cardines west or north. Example of the Southeast in the same Latitude 8. deg. IN our place of of being, (as aforesaid) I am to demonstrate three sagments of a great circle, containing 30 deg. apéece, by the demonstrature southeast, the end of which first sagment of 30 deg. shall touch the 35 deg. 40 mi. or meridian in longitude eastward from the first and there likewise toucheth the 52 degree 10 min. parallel north. And the end of the second sagment of 60 deg. in longitude wall touch the 41 degrees 40 min. or meridian in longitude Eastward from the first, and there likewise toucheth the 22 degrees ● min. parallel north. Likewise, the end of the third sagment at 90 deg. in longitude wall touch the 45 deg. 30 min. or meridian in longitude eastward from the first, and there likewise toucheth the 7 parallel South. Also there tuecheth the horizon 45 deg. either from the cardines East or south. South-west. Likewise the demonstrature of south-west, cutteth the meridians and parallels in longitude westward, as the demonstrature of the southeast cutteth them eastward, only that it cutteth the horizon 45 deg. either from the cardines south or west. And as for the arches of depression, from the horizon or senitor to the nadir of these points before spoken of, the delivery of them is even as easy as the former, and so is the arch of altitude, or arch of depression upon any Azimuth whatsoever. Only this following to be respected as a preparative: before I deliver any nice examples according to the difference, it is necessary for me to explain in a more familiar sort (at large) some part of the difference only according to the demonstrature, for the more confuting of the Card, and the opinion holden thereof for the way out and home to be all one. It is delivered in the 15 absurdity of the Card, that upon some points of the Compass, the Card delivereth a way outward, to be 390 degrees of a great circle, and to return homeward (to the first place) again, the like distance upon the same line, which shall be proved most false by the examples following: which examples shall be from the latitude of 50 degr. and from the first example, as before in the latitude 51 degrees 30 min. yet these deliver not the truth neither, as by examples upon the difference, hereafter follow. The 1 Example in North Latitude 50 deg. IN which latitude, our zenith or place of being in one meridian, where we make our beginning of longitude and place of departure; I am now to deliver from this our zenith or place of being, the demonstrature of northwest for one sagment of 30 degrees in longitude: which sagment end of 30 degr. Northwest, there cutteth the 11 degr. 30 minutes or meridian westward in longitude, and the 63 degree parallel north. Our Zenith now being altered and the latitude 63 degrees as aforesaid, we will make proof to return back again to our first place by the line southeast the like sagment of 30 degrees in longitude, where this sagment end cutteth the 26 degree 20 minutes or meridian in longitude eastward, and there likewise cutteth the 37 degree 40 minutes parallel north. So that this line of northwest extended to 30 degrees as aforesaid, and from thence returning southeast to 30 degrees likewise there is difference in longitude westward, from the first meridian or place of departure 25 degrees 10 minutes or meridians, and in latitude southward from the first 12 deg. 20 minutes. Here you see the difference of the way out and home, only for 30 degrees according to the demonstrature, and only for the lines of northwest and southeast, the rest of the points, hath their like difference in their kind. The 2. example in north Latitude 51 deg. 30 min. I Am now to deliver from this our zenith or place of being, the demonstrature of the north-east to 90 degr. which line of north-east there cutteth the 128 degree or meridian in longitude eastward from the first, and there likewise cutteth the 26 deg. 15 minutes parallel north, as in the demonstrature is delivered. Our Zenith now being altered, and in the latitude 26 degr. 15 minutes as before said, we will make a proof to return back again to the first place, by the demonstrature south-west to 90 degr. where this line of south-west cutteth but the 65 degr. 30 minutes meridian in longitude westward and there likewise cutteth the 39 degree 30 minutes parallel south. So that this line of north-east extended to 90 degrees as aforesaid, and from thence returning south-west to 90 degr. likewise, there is difference in longitude eastward from the first, 62 degrees 30 minutes, and in latitude southward from the first 91 deg. Here you see also the difference of the way out and home, only for 90 degr. according to the demonstrature, and only for the lines of North-east and south-west, and all the rest of the points is the like in their kind. And so I leave the demonstrature. Now will I deliver Example worthy the noting, of the way of the Compass according to the difference. THe East or west of that Compass, rectified to the true meridian upon the difference, maketh a spiral line in delatation to the equinoctial from what parallel soever, by sagments (of great circles) correspondent for this purpose: which sagments maketh right angles with the meridians, and which sagments delivereth a right sign from the plain of the vertical circle of east & west for the way of a Ship in her navigation, and the meridian according to this difference delivereth the other. This part of the difference only confuteth the general Card and the lines of inclination, with the Geography of the Globe. This difference, the east and west thereof delating as aforesaid, ruleth notwithstanding the variation of the Compass, generally spoken of, as for example: the Compass here with us at London, is set at half a point variation Eastward, where it should be 10 degrees 38 minutes 45 seconds by my own observations, (which was made in the year 1586.) which maketh the west to be half a point to the northward of the west: therefore in going from Silly (which is in latitude 50 degrees 15 minutes or there abouts) west by the Compass (which is in truth west half north) with Cape Race, in Newfoundland the places distant some 600 leagues from other causeth a falling more southerly into the latitude 46 degr. 30 minutes or thereabouts, which showeth the way of the difference to rule in this distance. And by a Compass rectified to the truemeridian that is a Compass that the north and south thereof, delivereth or pointeth according to that true meridian of Silly on which meridian is delivered the arch of altitude or almicanter of the Sun's height at n●ne, by which or such Compass, Cape Race heareth from Silly due west and by north and there toucheth the parallel of north latitude 46 degrees 30 minutes, likewise toucheth the meridian in longitude westward from Silly the 45 degrees 40 minutes according to the coarting of the meridians in this distance. And to speak the truth, in this distance there is but little or novariation of the Compass to be respected for the compass at Silly set at the whole variation as it ought to be, yet in sailing by that compass, you run some 400 leagues before the north and south points thereof, delivereth the true meridian: and in sailing the other 200 leagues, the compass is varied westward a point and a half or there abouts, which is no more than will answer the variation eastward, as before, so that in this distance, the nutation of the Compass, eastward, & westward, considered aright, is as nothing to be respected, (but the one set against the other,) as by the examples following is proved. The first Example. Let a compass be rectified to the nutation and sail from Silly therewithal to Cape Race, west: & when you come on the coast of Newfoundland you shall be delated from the parallel of Silly, only according to the difference, and not otherwise. The second Example. Let the compass be set at half a point nutation, as most commonly it is and sail by the Compass from Silly west with Cape Race, and you wall perforce (keeping that course directly) fall into more southerly latitude, which is into 46 degrees 30 min. or thereabouts, according to the difference, and not otherwise. The third Example. Let the Compass be rectified to the true meridian, and sail by that Compass from Silly west with; Cape Race and you shall likewise fall into the latitude 44 degr. or thereabouts, according to the difference. And as for this way (above said) is delivered a delating from the parallel in going west: so likewise in returning east from thene again, you shall likewise delate from that parallel again, according to the difference: which delivery overthroweth the whole Card. Again, look how the difference of the east and west ruleth in the distance and differedce in longitude (as aforesaid) so in any other distance and difference in longitude, it is likewise to be respected. Also this difference of the east and west, is the ground from whence the difference is delivered, for all the rest of the points of the Compass. So that you see, this way upon the difference, delivereth way outward to any place, different unto the way homeward, being not both alike, as by the examples of the north-east, and returning south-west: likewise of the northwest, and returning southeast, as hereafter appeareth. The 1 part of the 1 example from the Equator. Being in the Equinoctial in any one meridian, I will there make my beginning of longitude: from which equator and meridian of longitude, I will deliver the line of inclination, north-east continuing it to the latitude 75 degr. 6 minutes 14 seconds according to the difference. The 2 part of the 1 Example. The line of the north-east continued to the latitude 75 degrees 6 minutes 14 seconds, according to the difference, endeth in the longitude 131 degrees from the first, reckoned according to the meridians in this distance coarted. The 3 part of the 1. Example. Being in the latitude 75 degrees 6 minutes 14 seconds, and in the longitude 131 degrees as aforesaid, I am now to return back again by the line of inclination, south west to the Equinoctial. Conclusion of the 1 Example. Being returned to the Equinoctial as aforesaid, I do ●nd the length of the line of inclination south-west homeward, to be shorter than the line of north-east outward, by 130 leagues, and I am in longitude eastwerd from the first place on the Equinoctial 70 leagues, therefore the way out and home not all one. Another Example North latitude 50 degrees. I sail northwest 50 leagues, at the 50 leagues end I have altered my longitude from the first, 2 degrees 38 minutes, and my latitude 1 degree 40 minutes. I return back again southeast 50 leagues, and being returned, I find myself in less longitude or westward from the first, 3 minutes, and in latitude more than the first 6 minutes. Another Example in the North latitude 60 degrees. I sail northwest 50 leagues, at this sagment end, my longitude from the first is 3 degrees 30 minutes, and latitude from the first 1 degree 50 minutes, I return back again southeast 50 leagues, and being returned, I find myself in less longitude or westward from the first, 5 minutes, and in latitude more than the first 10 minutes. And for the more confuting of the way out and home to be all one, and the more justifying the delating from the parallel to be true, I will deliver some more examples upon the east, for a sagment of 20 leagues, in the latitude 51 degrees 30 minutes, and 75 degrees, delivering the horizontal distance between the east (according as is said) to lead in a parallel, and the way of the east according to the difference, as followeth. In Latitude 51 deg. 30 minutes, my first place. From any one meridian of longitude, I take a sagment of the parallel in this latitude of 20 leagues, presupposed by some to be East from the first place: and again, I do departed from my first place of being 20 leagues eastward according to the difference: now at this 20 leagues end, I am delated from the parallel according to the way upon the difference, which maketh my place (now of being) to hear from the first place, two Azimuthes, and 〈◊〉 more southerly. As for Example. To explain it more briefly, I imagine one line from the first place east, according to the parallel 20 leagues: imagine another line from the first place East by the Compass, according to the way upon the difference: now the horizontal distance between these two lines at 20 leagues end, from the first place, shallbe (as before) two Azimuths 〈◊〉. In the latitude 75 my first place and meridian of longitude, I take a sagment of the parallel in this latitude, of 20 leagues, presupposed likewise by many to be East from the first place: and again, I do departed from my first place of being 〈◊〉 leagues east, according to the way upon the difference: now at this twenty leagues end, my horizontal distance between my place of being, and place (as aforesaid) on the parallel, from the first place, in 3 azimuths' 〈◊〉. Likewise this way of the difference delivereth upon any Aziemuth, or point of Compass, sagments of great Circles, different from other, correspondent as they ought to be for the difference in longitude as hereafter by three examples of the North northwest following appeareth, which confuteth the 12 absurdity of the Card. Three Example of the North North west, for the difference of the length of the Sagments in longitude from the Equinoctial, to 80 degrees in latitude, as followeth. The first Example from the Equator. BEing in the Equinoctial in one meridian, the sagment of N. northwest to reach to the next Meridian which is one degree in longitude, is in length 52 leagues ⅓. The 2 Example from 40 deg. of latitude. Being in 40 degr. of latitude, and in one Meridian, the sagment of north northwest, to touch the next meridian, is in length 17 leagues ⅓. The 3 Example from 80 degrees of Latitude. Being in 80 degrees of Latitude, and in one meridian, the sagment of north northwest, to touch the next meridian, is in length 〈◊〉 leagues ⅔. Also, this way of the difference delivereth upon any Azimuth or point of the compass, sagments of great circles, different from other, for the laying or raising of a degree in latitude whatsoever, as by 3 examples of the west, north west following appeareth, which confuteth the 13 absurdity of the card. The 1 example from the Equator. I Being in the equinoctial, am required to deliver a sagment of the west north west, only to raise a degree in latitude, which according to the difference is 52 leagues and ⅔. The 2 example from 40 deg. of Latitude. I being in 40 degrees of latitude the sagment of west northwest to raise a deg. in latitude upon the difference is 56 leagues. The 3 example from 80 deg. of Latitude. I being in 80 degrees of latitude the Segment of a west northwest, to raise a deg. in latitude upon the differ once is 60 leagues. And that will I leave the way of the Compass upon the difference for this time. And because the variation, or nutation concerneth the compass and is a thing delivered in print, meant belike to some purpose: therefore I will touch this variation, or nutation, in some few words as hereafter followeth. If a man for his delivery of the variation in print, would observe by a needle touched by some, who maketh the common or ordinary Compasses, I take it, precise fellows, would soon be delivering forth, that the variation so set in print, were wide from the truth. Because with some men in the touching of a Needle, or Compass, though the stone they touch withal, be not, the best, neither shall an inch in breadth of the north part of that stone, break any square with them, by whose compasses many time shipping at the Seas be endangered. It were better for a man for the security of his charge, or that purposed to set down the variation in print, to take a needle touched by a better stone and a more perfect man to handle the touching thereof: truly when Robert Norman died (who had a good stone) Sea man had a great loss, yet Master Mullinux of Lambeth, who having a better stone was as careful & as precise in hi● 〈◊〉 concerning the touching of Needles, & Compasses, as over Master Norman was. Notwithstanding the variation (by such a good Needle) set down in print and delivered as a general thing, * This 〈◊〉 to pass 〈◊〉 mistakin g● true place 〈◊〉 the Poles c● Stone, f● Master G● brand, an● thers with 〈◊〉 observing 〈◊〉 variation 〈◊〉 two severa● Needles to 〈◊〉ed with tw● verall Sto● found the 〈◊〉 riation to 〈◊〉 the same, 〈◊〉 the like I 〈◊〉 seen befor● since. H. B● would been but ●ested at: and made a thing indeed that a man might spend much time to no purpose and less edifying to the Seamen as some have done: my reason is this, because this Stone (though a notable one) and I have not seen a better, and good to make abservations withal to be kept to a man's self, or out of print or for the amplifying of some note in writing: ☞ It cannot deliver the variation of another Stone; for in truth the Variations delivered by many stoves are different: you shall not have two Stones alike qualitied, or that will deliver one or a like Variation, but the variation of every stone differeth from other: there cannot generally be set down a certain variation for any one place, which let suffice for this time. Therefore, that man that was conceited to set the Variation in print, as a general thing: though it were myself, all things to nothing, I would there in my delivery, likewise he condentning all men's knowledge saving my own, to justify my do. But to the matter, the variation or nutation of the compass as it shall at any time or place be found, is a thing to be noted, yet my delivery is of it, that it is not surpassing all other knowledge, neither the overthrow of good knewledge, neither will I accept of it as a thing notable above all the rest, my reason is this, because the way upon the difference, being more excellent, over ruleth it, which indeed they impute to variation, which is untrue: and thus will I leave the nutation of the Compass. And whereas before in my delivery of the nutation, I had forgotten to give a taste of the error, which is likewise set in print, and conceiveth the nice delivery of the said nutation: I thought it now therefore good (though late) not to overpasse it but to give knowledge thereof: it is said that the middle point between any two Azimuthes observed upon equal elevations in forenoon and afternoon, is the true Meridian. For the confuting hereof I will deliver you an example in the north Latitude 51 degrees 32 minutes as followeth. The Sun being in her swift declination, in or near the Equanor, I purpose to make two observations: the former observation to be 2 min, before 8 of the clock in the forenoon, the Almicanter delivered then by the centre of the Sun, being 18 deg. in elevation, & the horizontal distance eastward from the true meridian delivered by the Azimuth of the Sun to be 66 degr. 38 min. The second observation in the afternoon, the Sun having the same Almicanter 18 degrees: the declination increasting respected, for 8 hours, which is 8 min. North declination, maketh the time to be 3 min. after 42 clock in the afternoon, wherein there is a min. of time different from the South and the horizontal distance westward from the south, to be 66 deg. 53 min. So that I find this last observation to be farther from the true Meridian, than the former by 15 min. but if the declination were decreasing, than the last observation will be nearer the true meridian than the former by 15 minutes, which maketh a difference of variation, sometime too much, or sometime too little, by 15 minute. good, which is an error and therefore not the truth. Concerning the Tides. EVen as men's minds for the most part are settled in the absurdities of the Card and compass accordingly, so are they settled likewise in an absurd reckoning of their Tides, as hereafter by the grace of God shall appear, divided into two parts, first by the delivery of the Cpact, next according to the time observed, by the point of the Compass, as follweth. The first part. These men doth not only know the prime by the date of our Lord, and the Epact by the prime, and the day of conjunction by the epact which is tolerable, according to the order of the Epact, allowing 30 days to every Moon: or on the 30 day end, to be Conjunction, which is notwithstanding partly at random, being without respect of the precise time of the conjunction. But they justify also the departure of Sun and Moon in 24 hours to be continually 48 min. as though the motions of the Sun and Moon continued all one, or that 48 minute. were their mean departure. For they bring the 32 Points, every point being a 11 degr. and 〈◊〉 a sunder, the whole being 260 degr. to 30 whole days, allowing to every 24 hours the departure to be (as aforesaid) 48 min. for 30 times 48 minutes is 24 hours: and 4 minutes of time is answerable to one degree of the equator: therefore the departure is 12 degrees in 24 hours, allowing the just revolution of the Equator to be in 24 hours, neither more nor less, for 30 times 12 degrees is 360 degrees, in which their reckoning in my conceit, the course of the Sun in those 30 days, which is about 29 degrees 30 min. is forgotten or not respected, but either they must of necessity, grant that the revolution of the Equator, and the departure of the Sun and Moon, to be as hereafter shall be delivered: or else they must allow the Moon's error in her yearly course, to be 5 days 15 hours 2 minute. too much, which is an absurdity, and maketh the synodical error more than it should be by 11 hours, 15 minute. 56 sec. 49 thirds, but if they do acknowledge, (which I take it, is in few or none of their heads) that the whole revolution of the Equator is turned about in 24 hours, (as hereafter wall be delivered) and so do reckon the odd time, to be for the mean course of the Sun in the 30 days, yet they must acknowledge by that reckoning, that the conjunction of the Sun and Moon, must happen almost at like times of every year, or at leastwise, make the difference sooner of the yearly period of the conjunction, to be but 5 days 6 hours 9 minutes, which indeed should be 10 days, 21 hours, 11 minutes, as hereafter shall be delivered. The 2 part. 1 Also these men I justify in reckoning their Tides, in what place and time soever: that where it floweth a southeast moon in conjunction or opposition (and possible by a false Compass) than it floweth till 9 a clock: or where it floweth an East Moon in conjunction or opposition than it floweth 6 a clock, & so it's their judgements generally of all the rest of the points of the Compass. 2. And to justify my say to be true, they have set forth, printed tide tables thereof, and in those tables also, have set their judgements, that in conjunction or opposition it floweth 48 min. after the time upon any point as aforesaid, which is absurd. Therefore these their judgements, for the security of their charge, had as much need of refining as the East and west of the Compass, as is said before, to lead in parallel. Now to the matter, for the refining of the first part as aforesaid, let this first part following suffice. To know the times of the Conjunction otherwise then by the Epact, is delivered by Ephemerideses, or Almanacs, but rather known by that man that hath the knowledge himself in the Theorickes of the Sun and Moon: which knowledge ought to be in a man that would be artificial to refine these absurdities. In Mesl●nes Astronomy the metions of the Sun and Moon are distinguished into three parts; Slow, swift, and mean: that is to say; in Apogaeum slow, when they are farthest from the earth: in Perigeun, swift, when they are nearest the earth: and in their means between the Apogaeum and Perigenm, notwithstanding by these three motions there is knowledge understood, that the motions of the Sun and Moon are always different, that is to say, the Sun from this place in Apogeum, to his Perigeum: likewise the Moon from her Apogeum and Perigeum of her epicicle, & from her Apogeum in her almost an oval form of the centre of her Epicicle, to her Perigeum thereof being the like qualitied. To justify the Motions to be the truer: it is also delivered in this Astrovomy, that the Sun hath three Orbs or particular Spheres, as followeth. First an Excentricke which is called the difference of the body of the Sun. second is called the difference of the Apogeum of the eccentric the which 2 doth contain the eccentric between them: and 3. maketh the whole sphéere of the Sun to be concentric. I could delate more of this theoric than I purpose, but it would be to small effect concerning the tides, only respect this that follows. Three reasons to be delivered for this Hypothesis as aforesaid to be true. 1 First reason which causeth the Sun to be in an excentricke, is the difference of the Sun's motion: which is proved sometimes swift, sometimes slow, as aforesaid. 2 Second reason is, because the Diameter of the Sun, is proved sometime bigger (as in Perigeum 33 min. 44 sec.) then at some other time by 2 minutes 6 seconds, therefore nearer at sometimes then at other times. 3 The third reason is the inequality of the Epicicles because it is proved that the Sun farthest from the earth causeth the Eclipses to be longer and nearer the earth to be sooner. Mesline delivereth also in his Astronomy, that the greatest excentricity of the Sun is 48 semidiamiters of the earth. And the least excentricity which is near in this our age about the 9 degree of Cancer, is 37 semidiamiters of the earth. So the difference is a 11 semidiamiters of the earth, and the semediamiter of the earth is 3436 miles and 〈◊〉. So that the sun is nearer unto us at some time, then at some other times by 74 semediamiters of the earth. Mesline delivereth likewise, one yearly Period of the sun to have 365 days 6 hour. 9 min. 39 sec. and this year he calleth Sideriall or starry, the diurnal mean motion of the Sun accordingly to be 59 min. 8 sec. 11 thirds 22 fourth's 16 fifths. The motions of the Sun as followeth. Her swift motion in 24 hours, is— 1 deg. 1 min. 16 second. Her mean motion in 24 hours, is— 0— 59— 8 Her slow motion in 24 hours, is— 0— 57— 0 Mesline delivereth likewise in his Astronomy, that the Moon hath five Orbs or particular Spheres as followeth: 1 First, an Excentricity. 2 Second, the difference of the Epicicles centre. 3 Third, the difference of the excentricity in Apogeum and Perigeum. 4 Fourth is the Epicicle which carrieth the body of Luna. 5 Fifth, which is Concentric called aequans Luna. Four reasons following to prove this Hypothesis to be true. 1 First reason which causeth the Moon to be in an eccentric, is the difference of the Moon's motion, which is proved sometime swift, sometime slow. 2 Second reason, is because the Diameter of the Moon is proved sometime bigger as in Perigeum 35 miu. 38 seconds then at some other time by 5 min. 38 sec. therefore nearer at some times then at other times. 3 Thirdly, the inequality of the Eclipses, because the moon farthest from the earth, causeth the Eclipses to be longer, and nearer the earth to be sooner. 4 Fourthly, the inequality of the Paralax, because the Moon sarthest from the earth, causeth the Horizontali paralax, to be the less, and nearer to be more. And the greatest excentricity of the Moon is 10 semidiamiters of the earth, and 〈◊〉 a little less. And the semidiamiter of the Epicicle is 5 semidiamiters of the earth and 〈◊〉. So that the Moon is nearer unto us at sometime, then at some other times by 30 semidiamiters of the earth, and 〈◊〉, and likewise by the whole diameter of the Circulus parvus: which is 20 semidiamiters and 〈◊〉. For the Moon maketh a Circulus parvus, (contrary to her own motion) about the centre of the world, according to the semidiamiter of the eccentric, which is 10 semidiamiters, and 8 min. 30 seconds. Likewise the Moon hath latitude of the Ecliptic on either side 5 degrees, which for the tides is greatly to be respected. Mesline delivereth one mean periodical period of the Moon to have 27 days, 7 hours 24 minutes. Also delivereth the mean time to make up the periodical period to a mean synodical period to have days, 2 days, 5 hours 20 minutes. So that the mean synodical month of the Moon to be in conjunction again with the Sun hath days, 29 days 12 hours, 44 minutes 3 sec. 11 thirds. The motions of the Moon as followeth. The swift motion of the Moon, sometimes in 24 hours is 15 degrees 0 min. The mean motion in 24 hours, is 13 degrees 30 minutes. And her slow motion in 24 hours, is 12 degrees 0 minutes. Now having delivered so much as needeth of the Theoricks of the Sun and Moon concerning the Tides, we are come to the point to know their departure, which is as followeth. The departure of the Sun and Moon. The Moon presently after the conjunction departeth Eastward from the Sun in Apogeum in 24 hours 11 deg. 3, min. which is answerable in min. to 44 min. 12 seconds. The Moon presently after the conjunction, departeth from the Sun in Perigeum in 24 hours 13, degrees 58 minutes 44 seconds answerable in minutes, to 55 min. 55 seconds. The Moon presently after the conjunction departeth from the Sun in mean motion, in 24 hours 12 degrees, 30 min. 52 seconds, answerable in min. to 50 min. 3, seconds 〈◊〉. Sometimes the departure of the Sun and Moon, when the Sun is in her slow motion, and the Moon in her swift motion, in 24 hours, is 14 degrees, 3 minute. which maketh the departure in minutes answerable to be 56 min. 12 seconds. And sometime the departure of the Sun and Moon, when the Sun is in his swift motion, and the Moon in her slow motion, in 24 hours, is but 10 degrees, 58 min. 44 seconds, which maketh the departure in min. answerable to be 43 min. 55 sec. And the mean motion, between both these motions, next above in 24 hours, is 50 min, 2 seconds, 30 thirds. Which motion is justified twice, as above: therefore I conclude the mean departure of the Sun and moon in 24 hours to be 50, min, 3, seconds 30 thirds. And the hourly departure accordingly to be 2 min. 5 secon. 8 thirds, 45 fourth's. Stadius affirmeth in his Ephemerideses, that in 24 hours the whole equator and 50 minutes 8 seconds is turned about, which is most certain. Likewise in a mean be affirmeth that the conjunction of the Sun and Moon in one period, at any one time in the year, shall differ from the same time the next year, and be sooner 10 days, 21 hours 11 min. as for example: In the 12 mean synodical months, the time that wanteth of 12 times, 30 days, is 5 days 15 hours 2 minutes, which maketh 360 days, the remainder of the days of the whole year, is 5 days, 6 hours, 9 minutes, which both together is 10 days 21 hours 11 min. Now for the refining of the first 2 parts, let these last 2 parts following suffice, concerning their error in reckoning their tides delivering the time by the points of the Compass: for the confuting hereof. I will deliver six examples, three examples in the north latitude 30 degr. and the other 3 examples in the north latitude 51 degrees 32 minutes. North latitude 30 degrees, the Sun in Capricorn. 1 The Sun at southeast, it is then ¼. before 9 a clock in the morning, but if the Moon hath 5 degrees South latitude, than it is 24 minutes before 9 a clock: if the Moon have 5 deg. North latitude, than it is 16 min. past 9 a clock. Sun in the Equator. The Sun at Southeast, it is then 14 min. past 10 aclock in the morning: but if the Moon have 5 degrees south latitude, than it is 7 min. before 10: if the Moon have 4 degrees north latitude, than it is 30 minutes past 10 a clock Sun in Cancer. The Sun at southeast, it is then 32 min. past a 11 a clock in the morning, but if the Moon have 5 degr. south latitude, than it is 15 minute. past a 11: if the Moon hath 5 degr. north latitude, than it is 54 minutes past a 11 of the clock. And at an east Sun, it is then 18 min. past 9 a clock in the morning, but if the Moon have 5 degrees South latitude than it is 24 min. past 8 a clock: if the Moon have 5 degr. north latitide, than it is 40 min past 10 a clock. So that in this north latitude of 30 degrees, there is difference of time in the Moons being sentheast &, 3 hours. 18 min. and in the same latitude, difference from 6 a clock in the east 4 hours 40 min. The other 3 Examples in the North Latitude 51 deg. 32 min. the Sun in Capricornus. 1 The Sun at the Southeast &, it is then 20 minute. before 9 a clock in the morning: but if the Moon hath 5 degrees south latitude, than it is 18 minutes past 8: if the Moon have 5 deg. north latitude, than it is 51 min. past 8 a clock. Sun in the Equator. The Sun at the southeast, it is then 28 min. past 9 a clock in the morning: but if the Moon have 5 degrees south latitude, than it is 18 min. past 9: if the moon have 5, degr. north latitude than it is 38 min. past 9 a clock. Sun in Cancer. The Sun at the southeast, it is then 18 min past 10 a clock in the morning: but if the Moon hath 5 degrees south latitude, than it is 6 minute. past 10. If the Moon have 5 degrees north latitude, than it is 30 min. past 10 a clock. But at an East sun it is then 22 min. past 7 a clock in the morning, but if the Moon have 5 degrees South latitude, than it is 2 minutes past 7. If the moon have 5 degrees north latitude: than it is 42 minutes past seven a clock: likewise in this north latitude 51 degrees 32 minutes, there is difference of time in the Moon being southeast 2 hours, 2 minutes. And in this same Latitude, difference in the east from 6 a clock 1 hour 42 minutes: the mistaking of so much time, from the high water, may overthrow their charge in going into a harbour, where it is to be respected. 2 And whereas they deliver in their Tide Tables, a hoying of the tide in conjunction, aspect or opposition, to 48 m. after the order in that Tide Table, it is not so, but in a river or in-draft, where the Tides in their prime runneth quick, there Tides of floods will run 48 minutes after high water; and thus I will leave the delating of the Tides. Concerning the Time. SUch is the conceit or knowledge of the Seamen in these times, they take a common Compass, little respecting the error thereof, being touched with a Stone of no strength, and made wise by a man of no knowledge: which Compass they divide into 24 hours, to deliver unto them, (they observing the Sun therewith) the time they look for, which is as lame a delivery of the times as may be, and is somewhat touched before in the delivery of the Tides, and likewise as hereafter followeth. Also, they observe the time by an hour, or half hour glass, made by as careful a man, as the Compass before spoken off, who indeed careth but little what error more or less, is delivered in those glasses in 24 hours, nay in half an hour, with whom (as the proverb is) an inch breaketh no square. Which Compasses and glasses had need to be carefully respected, nay rather to be reform, for the security (as aforesaid) of the Navy of England, which is greatly endangered by them: and although these appendances for Navigation be mere false, yet the Marketfolks being not experienced by them, in consideration of the cheapness of them, holding themselves thereby, profitable servants to the owner. And although the time generally from one Moon to the other, cannot be perfectly delivered by any man, as hereafter shall be showed: yet by precise Instruments carefully made the time may be delivered more truly which is the greatest help we have in Longitude, and for a man most careful in making of his Compases, and running glasses, I commend Master Emerie Mollineux of Lambeth (while he lived) as for my conceit, which I have learned concerning the time, it is as hereafter followeth. First, for the confuting of the time delivered by the compass, as aforesaid, which Compass is delivered by 360 degrees, which indeed are Azimuths, and they justify as many Azimuths, which is 15 for one hour, as for another which is contrary, therefore I will deliver three examples to that effect in the Latitude 51 degrees 32 minutes as followeth. Sun in Capricornus. From the Sun rising to one hour of time, there is 12 Azimuths and a ¼, and from a 11 a clock to 12 at Noon, there is 14 Azimuths. Sun in the Equator. From the Sun rise to one hour of time, there is 12 Azimuthes, and from a 11 a clock, to 12 at noon, there is 18 Azimuths and ¾. Sun in Cancer. From the Sun rise to one hour of time, there is a 11 Azimuths and ⅓. and from a 11 a clock to 12 at noon, there is 28 Azimuths. Also followeth two examples, the one in the north latitude 51 degrees 32 minutes: the other in the north latitude, 66 degrees, concerning the difference of time, as followeth. The 1 Example in Latitude 51 degr. 32 min. Aries whole sign, hath declination a 11 degrees 20 min. which reacheth in longitude Eastward, according to the degrees on the Equator to the 28 meridian likewise this whole sign of Aries doth lengthen the days 1 hour and 58 min. Taurus, his whole sign, hath declination 8 degrees 50 min. which reacheth in longitude Eastward according to the degrees on the Equator, to the 29 and a half Meridian: also this whole sign of Taurus doth length the days one hour and 44 minutes. Gemini, his whole sign hath declination 3 degr. 18 min. which reacheth in Longitude eastward, according to the degrees on the equator to the 32 and a half Meridian, also this whole sign of Gemini doth length the days 1 hour 40 minutes. The 2 Example in North Latitude 66 degr. Aries whole sign, lengthen the days,— 3 hour. and 46 min. Taurus' whole sign, lengthen the days,— 3 hour. and 38 min. Gemini his whole sign lengthen the days,— 3 hour. and 0 min. But seeing that general time from one Noon to an other, cannot be delivered certain, but there shall be error, so that one 24 hours shall be longer, or shorter than an other by 1 hour and 28 min. as hereafter shall be delivered: what certain time then can be delivered by any Instruments for that purpose, which delivereth but a second error. Now will I deliver some causes, why the time from one noon to another cannot be of like time. Mesline in his Astronomy, delivereth that there is three Ecliptickes as followeth. First the ecliptic of the 10 Sphere which is fixed. Second, the ecliptic of the 9 Sphere which is mooveable. Thirdly the ecliptic of the 8 Sphere which is the true ecliptic of the starry firmament. Likewise he delivereth that the Sun in Aries of the 9 Sphere, is gone to the Eastward of the Sun in Aries of the 10 sphere, 27 degrees and better, but directly under it, in the same plain. Also that the Sun in Aries of the 8 spheres never is nearer to the Sun in Aries of the 9 Sphere, then by the semidiamiter of the small circle, which is 9 degrees. Now Mesline delivereth the difference of the judgements of the Astronomers, and that Ptolemy maketh the 8 and 9 Spheres all one, delivering the opinion of Prolomy and others, as followeth. The opinion of Prolomie concerning the motions of the fixestarres, comparing the observations made 400 years before him, with his own: he finding thereby that the fixed Stars had moved in consequence 4 deg. therefore Prolomie all owed to the whole revolution of the fixed stars, to accomplish one period, according to the rate of 36000 years, that was for every 100 years, one deg. Mesline delivereth also that the opinion of the Alfonfines, concerning our revolution, or accomplishing of one period of the 9 Spheere is in 49000 years, but according to Copernicus his observations, one periodical revolution is in 25816 years, delivering that an hour of the longest day is 63 minute. 40 seconds, such as an hour of the shortest day is only 60 min. Copernicus maketh (likewise) the year 365 days, 5 hours, 49 min. 15 seconds, 46 thirds, which he calleth the year Tropical, equally supputated, or counted to the middle Equinoctial. The Aifonsines in their account, wanteth in their yearly period 13 thirds. Mesline maketh the year 365 days, 6 hours, 9 minutes, 39 seconds, and this year he calleth syderial or starry, numbered under the Spheere of the fixed Stars, which is more than the year delivered by Copernicus by 20 min. 23 seconds 14 thirds. Delivering the causes of this difference to be this, making these beginning of the years to be in one point of the 10 and 9 sphéeres and the 9 Sphéere (before the year end) being remooved Easterly from the 10 spheere, the year end coming to that point of the 12 Sphere again, is to proceed from it to that point of the 9 sphere also before the year endeth. Mesline delivereth (the Sun being in the Apogeum part) that from middle motion to middle motion, the sun passeth 176 deg. 20 min. of the ecliptic: And the sun being in the Perigeum part from middle motion to middle motion, passeth 183 degr. 40 minutes of the ecliptic, which maketh a difference of 7 degr. 20 minutes. But from the points of true motions, the halves are equal, each 180 deg. of the Ecliptic: and the arch of time between the middle and middle motion of the Sun in Apogeum part, is 182 days and 6 hours. Also the Arch of time between the true motion and true motion of the Sun in the Perigeum part, is 179 days, which maketh the arch of time between the middle & the middle motion of the sun in Apogeum; more than in Perigeum by three days and 6 hours. Also maketh the arch of time, between the middle and true motion of the sun in both the mean motions, to be 4 days, and the arch of time different from the true motion, to the true motion of the sun in Apogeum, from the same points in his Perigeum to be 7 days and 6 hours. And as for the rest of the Theoricks of the Sun and Moon I have delivered sufficient in my delivery before, concerning the tides. But to conclude, Stadius in his Ephemerideses, folio 57, of the Equation of the natural days, delivereth as followeth, concerning the time. We have computed all the places of all the Planets, and also of the Sun and Poone in this our Ephemerideses, for the natural days (to wit,) for the space, in the which the whole equator is turned about, and moreover 59 minutes and 8 seconds: but because the apparent, and true day now and then doth exceed this time, and now and then is less: to wit, Partly when the Sun goeth swifter, or flower, by reason of his Prosthopherisis. Partly for that equal times of the Equator, in the diurnal coartation, or revolution, do not answer to equal sagments, or parts of the Zodiac, it cometh to pass, that this inequality of time also needeth a prosthopherists, the which notwithstanding cannot be always equal nor certain. For the conversion of the Circle of the Equator, though unto the middle and equal Equinoctium, which is the year divided into four equal parts, is evermore equally constant, yet nevertheless, unto the apparent Equinoctium which are the true equinoctiis which the eccentric yields, is found not equal. Neither doth the Sun's Apogeum occupy a stable seat, because the Apogeum flecteth in the ecliptic every 13 years 1 degree 27 minutes: moreover the Sun in his excentricity doth some hurt by his shifting for the excentricity is more and less by a eleven semidiamiters of the Earth, as aforesaid, and goeth about with the ecliptic. Wherefore also no limit can be appointed perpetually to this insquallity, but in this our Age all these causes conferred one with another, and added together, do show that the greatest day and less day do differ one from an other, one hour and 28 minutes, that is to say, 22 times (or Meridian's) and 20 minutes of the Equator. But seeing that this inequality of days cannot missever the rest of the motions, therefore it is necessary that we should admit them to be equal, unto the which equal days, the Tables of the motions s●ould be both made and set together, the which course we have taken in this our Ephemerideses, and for that cause we have borrowed the Canon of equation of days out of Erasmus Rheinholdus which agreeth nearest with our age, until the year 1600. Therefore according as you find the number in that cannon to add or subtract: that is the time that the day is either longer or shorter, than the equal day, or the day of the Ephemerideses, that is the first equated day, or day equated upon the difference of 88 minutes, whereas the second equation is of days once equated, and now doth equate them upon the difference of 44 minutes, whose half is 22 minutes, the greatest number within the cannon. Seeing there is a first error concerning the time delvered, even in the motions of the Heavens, as aforesaid, therefore these running Glasses delivering but a second error: the reason is this: because they cannot be made without their imperfections, had need to be most carefully made, and by the precisest workman: that the time delivered by them, may be but according to the second error, for the delivery of the Longitude, by which running glasses, next hereafter nominated, the longitude is better delivered then by any other Instruments. A glass whose sand is mettle and the metal said by some will not rust, notwithstanding in my opinion, it will rust somewhat, and be sometimes moister, then at other times, likewise the hole that the sand runneth thorough, will grow wider with the force of the sand, the rather being violated by the surges of the sea: which impersections considered, the glass must needs deliver the time, sometime shorter, and sometime longer, according to the weather, therefore a second error: yet this glass is more tolerable than the rest for this delivery, and is to be used before all other, of which glasses there may be divers sorts, for the delivery of more and less at pleasure. And because the running glasses with sand is more grosser, and that clocks and watches hath their more imperfections, than the former glass, I will omit them, and leave the delivery of the time for this present. Concerning the Winds according to Experience. DIfferences or diversities of winds, I have seen at sea, in some calm day (in Summer time) among divers shipping, being of one fleet, so that some six of them or more, have all had contrary winds till the wind was settled. Likewise in Summer time I have known in places of small distance, as at Hambrough, the wind at west south west, so much winds, that we have rid with our topmast down, and at Lee the very same time, the Ships then bound for Hambrough, had as much wind (by report) at east north east, but both those winds continued not long, and between which winds no doubt it was calm for the time. Also in Winter time, I have seen it often in being at sea, the wind being at South southeast, much wind and upon a sudden the wind hath altered to the northwest, or north northwest, very much wind, and so alterations of winds I have likewise seen upon other points of the Compass. And in sailing toward Headlands, or in sailing about Headlands I have found the wind sometime too large unto us, and other times again too scant unto us: Therefore the wind very difficult to delate upon, my farther reason is this, because it is delivered in the Scriptures that the wind bloweth, but from whence it cometh or whether it goeth no man knoweth. Yet notwithstanding, many times in Summer time, but especially in winter time, when grown stormy winds are settled, we may conjecture that they extend fare, and in my opinion (which opinion I hold for a truth) the wind than bloweth in a great circle, therefore cannot be parallel to any great circle of that nature, my reason is this, because great Circles doth cross themselves at opposite points, as in the demonstrature of the compass is delivered. Therefore this delivery confuteth the flat Cards, my reason is this, because the flat cards delivereth the winds to blow in parallels, according to the lineament thereof. Likewise by the Card the wind at East, and a Ship going west, either in the latitude 33 degrees, 35 minutes, and latitude 60 degrees, or in the north latitude 40 degr. 15 minute, or in what latitude else soever, it is said to lead in parallel, notwithstanding, the Card delivereth the going West to make right angles with the Meridian's which is very absurd, and that the wind continuing, they shall go west still asore the wind. Concerning the Wind by Sphericail working. The 1 Example. PResuppose a first place (which I will call our place of departure) in any one degree of Longitude whatsoever, and in the North latitude of 33 degrees, 35 minutes, in which place the east and west of the Compass, according to the demonstrature, is a Tangent unto this parallel, and crosseth the equator at opposite points 90 degrees in longitude Eastward and westward, which east and west points in the Equinoctial keep in mind. I purpose from this first place (as aforesaid) to sail to a second place according to the way of the west upon the difference, which shall be in Longitude from the first place 90 degrees, which 90 degrees, in these parallels is 75 degrees of a great circle or little more: now in this second place which is at this present my place of being, I find myself delated from the parallel according to the difference, where I purpose to deliver the East and west (likewise) according to the demonstrature, being also a Tangent unto this parallel, and crosseth the Equator at opposite points according to the former, but 90 degrees different in longitude, which two demonstrated semicircles do cross each other in 24 degrees 15 minutes of Latitude, 46 degrees in longitude from the first place, and 44 degrees in longitude from the second place. Now from my first place of departure, the second place doth bear according to the demonstrature of the Compass, west northwest, and 5 degrees Northerly, upon which point the two places are distant 73 degrees 15 minutes of a great circle, which is a nearer way than the way upon the West according to the difference as aforesaid, by 1 degree 45 minutes. And from my second place of being, the first place of departure doth bear according to the demonstrature of the Compass East north-east, and 7 degrees northerly, upon which point likewise the two places are distant 73 degrees 15 minutes. So by this delivery having continued the Ships caping or course west from the first place of departure to the second place, I do conclude that the wind being constant and stable in blowing from the first place to the second place, that at the second place the wind is altered of you two points and 7 degrees to the Northward, as aforesaid, though it were at East in the first place, which delivery in my judgement is the truth. But if the wind being at the first place, be at East, and in sailing to the second place, be variable and alterable, as the East and west by the demonstrature is variable by crossing the equator according to the difference in Longitude, than you shall go from the first place to the second place west as aforesaid (the wind being at east) right afore the wind, but this way in my judgement is impossible and not the truth. A second Example. Being in 60 degrees North latitude and in one Meridian of longitude, which is my first place, which hath his demonstrature of East and west accordingly, being a tangent unto this parallel, and in crossing the equator at opposite points, after the manner in the first Example, from whence I sail West accordingly to the difference to 90 degrees in longitude, which 90 degrees in longitude in these parallels is 45 degrees of a great circle, little more, which is my second place: in which second place I find myself there to be delated from the parallel according to the difference, and in which second place likewise I purpose to deliver the east and west according to the former, but 90 degrees different in longitude; and these two demonstrated semicircles, do cross each other in 50 degrees of latitude 46 degrees 20 minutes in longitude from the first place, and 43 degrees, 40 minutes in longitude from the second place. So that from my first place of departure, the second place doth bear (according to the demonstrature of the Compass) Northwest and by west 5 degrees 20 minutes northerly, upon which point the two places are distant 42 degrees 35 minutes of a great Circle, which is likewise a nearer way than the way of the west according to the difference, as aforesaid, 2 degr. 25 minutes. And from the second place of being, the first place of departure, doth bear according to the demonstrature of the Compass, north-east and by east and 8 deg. 20 minute. northerly, upon which point the two places are also distant 42 deg, 35 minute. of a great circle. So that by this delivery being at this second place, the wind is altered of me three points, and 8 degrees 20 minutes Northward, though it were at the East in the first place. The 3 Example. Being in 80 degrees 15 minutes North latitude and in one meridian of longitude, which is my first place, which hath his demonstrature of East and west accordingly, being a tangent unto this parallel, and in crossing the equator at opposite points, after the manner in the first example, from whence I sail west according to the difference to 90 degr. in longitude, which 90 degrees in longitude in these parallels, is 15 degrees of a great circle, little more which is my second place: in which second place I find myself there to be delated from the parallel, according to the difference. And in which second place, I purpose to deliver also the East and west, according to the demonstrature, which likewise crosseth the equator at opposite points according to the former, but 90 degrees difference in longitude, and these two demonstrated semicircles do cross each other in 75 degrees, 45 minu. in latitude, 48 degrees in longitude from the first place, and 42 degrees in longitude from the second place. So that from my first place of departure, the second place doth bear according to the demonstrature of the Compass northwest and by west, and 8 degrees northerly: upon which point the two places are distant 14 degrees of a great circle, which also is an nearer way than by the west as aforesaid by one degree. And from my second place of being, the first place of departure doth bear according to the demonstrature of the Compass north-East, one degree 20 minutes northerly, upon which point the two places are distant 14 degrees of a great circle. So that by this delivery being at the second place, the wind is altered of me 4 points and one degree 20 minutes to the northward, though it were at the East in the first place. A thing worth the Noting. In all the three examples, as aforesaid this thing worthy the noting might be delivered, yet I will deliver but one of them according to the second example in latitude 60 deg. in which second example or delivery, the intersection or crossing of these two semicircles of east & west according to the demonstrature from the first place of departure and second place, is in latitude 50 deg, which intersection is from the first place west, & from the second place east. Yet being in this intersection, the first place beareth from it north-east by east, and 5 degrees, 38 minutes northerly, and distant 27 degrees 32 minutes of this great circle. And the second place beareth from it northwest & by west, 1 degree 40 minutes northerly, and distant 26 degrees, 20 min. of this great circle according to the demonstrature of the compass: and thus much concerning the wind. Concerning the way of a Ship or Shipping at Sea. FIrst, there is to be understood and likewise to be respected by him that hath the charge (whatsoever in navigation) that there is two lawful and good navigable courses to be used at the Seas, and no more, either of which courses is very artificial (the keeping of them aright) & resteth in the good discretion of the said Master. being likewise artificial, which of them he will use. But if the said Master be not artificial, but a man of great conceit, the manner or way of these courses be too deep for his understanding, and therefore he not worthy to take charge at all. My reason is this, because all other courses whatsoever more than these two, are absurd, frivolous and false: the names of which two courses followeth. First the course upon the great Circle being the nearest way from place to place. The first course is to sail upon a great Circle which is after this manner: being in any one latitude and longitude whatsoever, which is your first place, and do purpose to sail from thence upon any Azimuths of demonstrature whatsoever, which are great circles, until you come to 90 degrees in longitude, which is unto the Horizon of your first place, in which way you are to respect every day at noon your Latitude (if you may) and likewise your time as before, or hereafter shall be delivered. For the delivery of your longitude, you are likewise to respect your Sagments, being of what length soever upon the Azimuths you sail, how you are, (upon any distance) to cross the meridians and parallels: for if you find yourself in such a latitude, as you should be in, and not in the right longitude, which the Azimuths you should sail upon giveth, then are you also wide and to seek reformation: also if you find yourself in such a longitude and not in the right latitude which your Azimuthes giveth, then are you also wide & to seek reformation, therefore being a harder way or course to sail upon the great circle (notwithstanding this way the nearest way, or course from place to place of all other ways) requiring always reformation with great Judgement, I will only deliver 6 examples from one meridian of longitude and latitude 51 deg. 32 min. upon the demonstrature, northwest and by north, continuing it from the first place to the Horizon by Sagments of 15 deg. of a great Circle a piece, only for a taste and your farther knowledge of this way and so will leave it, and hereafter will deliver the remainder of the second course or way, which is more easy, according to the difference, being the only and natural way indeed that the Compass leadeth in. The 1 Example. Being in the Latitude and longitude, as aforesaid, 15 degrees of a great Circle from thence, according to the demonstrature of N. west and by north, crosseth the 18 deg. 30 minutes, or meridian in longitude westward from the first, and the parallel of north latitude 62 degrees 50 minutes. The 2 Example. Other 15 degrees which maketh 30 deg. from the first according to the demonstrature of northwest and by north, toucheth the meridian of 50 degr. in longitude westward from the first, and the parallel of north latitude 69 deg 28 min. The 3 Example. Other 15 degrees which maketh 45 degr. from the first according to the demonstrature, of Northwest and by north, toucheth the Meridian of 92 deg. 50 minutes in longitude westward from the first, and the parallel of North latitude 66 degrees, 45 minutes. The 4 Example. Other 15 degrees which maketh 60 degrees from the first, according to the demonstrature of Northwest and by north, toucheth the Meridian of 118 degrees in Longitude westward from the first, and the parallel of north latitude 57 degrees, The 5 Example. Other 15 degrees which maketh 75 degrees from the first, according to the demonstrature of N. west and by north, toucheth the Meridian of 130 deg. 32 min. in longitude westward from the first, and the parallel of north latitude 44 degrees 35 minutes. The 6 Example. The other or last 15 degrees, which maketh 90 degrees from the first, and is the Horizon thereof according to the demonstrature of the northwest and by north, toucheth the Meridian of 139 degrees, 15 minutes in longitude westward from the first, and the parallel of north latitude 31 degrees 10 minutes. And as for the second course or way of the Compass upon the difference which we will rely upon, I have delivered sufficiently thereof, as a foresaid, and for that which is also needful besides concerning this way upon the difference to be delivered, hereafter in good sort and order followeth There is to be respected by a Master of a Ship, or the man that taketh charge in navigation, the goodness of the Compass which he saileth by, with the nutation thereof in any place if it be needful, otherwise it will cause the way of a Ship to be in an Azimuth contrary to his expectation, which will cause an error. And for the understanding of this nutation what it is, in any place may be delivered by a man of good conceit, as followeth, having a topographical instrument or otherwise called a Theodolite of brass, perfectly made, and the Needle good, being well touched with an excellent Stone, and handled for the purpose, delivereth the nutation very well: but if the sea-gate be so great that this Instrument cannot deliver it certain, then will no Instrument (set forth for this purpose) deliver it neither, then must you work thus, the height of the Sun in the Meridian at Noon observed by the Compass, the Compass itself, then delivereth the nutation from Noon, but rather thus: the latitude at Noon known, then upon any observation of the Sun by the Compass on any elevation about 3. a clock in the afternoon, the declination respected, the Compass delivereth the nutation more truer, my reason is this: because the descension of the Sun in his diurnal Arch at this time in the afternoon is more swifter towards the Horizon, than it is, the Sun being near the Meridian. Also that the said Master have with him of the best running Glasses that can be made, as aforesaid, to deliver the time unto him, as certain as may be, which must be regarded as a principal thing, the coarting of the Meridian's respected, for the delivery of the Longitude in any place, by which way the longitude is delivered after the best manner, the latitude being known, otherwise, the time not respected or regarded, the longitude will be delivered with the more error. Likewise the Azimuth of the way must be delivered or set down according to the difference, and not otherwise, and not above 20 leagues the Sagment: for the snorter the Segments be the truer the reckoning, which Azimuths of way upon the differance, delivereth the longitude also in reasonable sort the latitude being known, yet in this way is more absurdity, then in the former way for the delivery of the longitude. Moreover there is to be respected by him that hath the charge in navigation, the violence of winds, Sea-gates, Tide-gates, surrents, Edies or whatsoever else, that may either be helping of the ship in her way, or a hindrance of her way, which causeth an Azimuth of way, contrary to expectation, which not regarded, bringeth also an error. And for the observing of Latitudes at the Seas, it had need to be very precisely done, having a large Quadrant of brass with a mooveable perpendicular accordingly, delivereth it in the best sort: for in observing with Cross, staves there is error, my reason is this, the staff not lawfully projected to the great circle, delivereth error: my further reason also is this in observing with the staff, the eye is feign to behold and note two things at once, which is the centre of the Sun, and the Horizon, which is not possible to be done truly. And in a Sea-gate it can neither be precisely delivered by an Astrolabe, therefore the Quadrant best for this purpose: for in observing therewith, there is but only the centre of the Sun, to be respected. Also there must be great care had by him that hath the charge in navigation, unto the Steeridge of the ship that it be greatly respected of him, and of these principal men which goeth in the ship with him. which he doth appoint for this purpose, because the man at the Steridge may be negligent, and by some men that may be at the Steridge, as I have seen myself er● now, a 3 or 4 points of either side of the course, commanded to be kept, breaketh no square with him, which had or contrary Steridge to expectation, not counted of or noted, causeth a foul error. And whereas there are infinite sorts of shipping, for bigness, length, and draft, so they are as different in qualities, wherefore I will deliver as many qualities concerning all sorts of Shipping, as at this present I can remember, leaving the consideration of the rest, (not now remembered of me, unto him that hath the charge for the present) with those nominated according as he shall find the Ships quality to be which he goeth in, the qualities as followeth. Of Shipping there are divers sorts which have divers qualities the reason is this, because they are different in bigness and mould, and may be mistrived. To proceed, some of them are long, and some short, some of them flottie, and other some of great charge, or draft, some have a desire or cast to Portward, and some other to starbord, some of them good to sail (try) weather about without sails or hull, & other some qualitied to the contrary, some are desirous to have there stays and shrowds slacked, and some to have them set taught, some Ships desire to be trimmed a head, some a stern, and some others of an even keel, some stéereth hard every way, and other some easy of stéeridge any way: some Ships being in a sea-gate will stay, or wend to windward, and other some cannot, but must wend afore the wind, which is a great loss of way, long and short shipping being flotty, are laboursome in a sea-gate, but long or short being deep Shipping, is easy at Sea, some are fast Ships upon a wind, and some others are léeward, some are good Ships of asayle quarter winds, and some not so good: some are good a fore the wind, and some others to the contrary, some Ships are stiff sided which is a principal quality, and some other are tender sided, which is a bad quality, some Ships are of advantage most ways, and other some are to the contrary. Therefore the quality of a Ship is greatly to be regarded of him that hath the charge, for a Ship of advantage or disadvantage, causeth an Azimuth of way contrary to expectation, which not being respected causeth an error. Now for your diurnal by some called a Travers book, or book wherein you keep reckoning of the Ships way at the Seas, in my conceit and as I do use, is in manner as followeth. In the head or beginning of which Book, set down the Title thereof, with the Month, day, and date of our Lord, at the time of your beginning to enter into your navigation: after which being set down, divide the leaf and so the rest of the leaves up & down into 10 spaces or columes, and set down in every column as followeth. IN the first column of which Book, set down the Months and Days, according to the sequel of them as the time spendeth. In the second column, set down the Hours of time spent in sailing upon the Azimuths used. In the third column, set down the Azimuth of the wind, or that wind for the time it was in. In the 4 column, set down the Course which you have kept, that is the course on the true Azimuth, according to the difference. In the 5 column, set down the quantity of Leagues run upon the Azimuth as aforesaid. In the 6 column, set down the degrees of the Pole elevated. In the 7 col. also the min. if there be any, of the Pole elevated. In the 8 column, set down the degrees of Longitude delivered by the time and Latitude as aforesaid. In the 9 column, also the minutes (if there be any) of the Longitude so delivered. In the 10 column, set down your Discourse of things, according as you find occasion. And to Conclude. Every of which inconveniences, or rather good knowledge and understanding as aforesaid, being thus noted, known and regarded by him that hath the charge, he shall deliver in his Navigations the Ships way he goeth in with all advantages, who is worthy to be trusted with any charge, and to receive great commendations. But to the contrary, he that hath the charge, not respecting these things, as aforesaid, he delivereth the Ships way with all nusurdities and error, and to speak the truth, he is not worthy to take charge at all, but to have an inferior place, which he is more fit for: and thus will I leave delating of the way of Shipping at Sea. FINIS. AN APPENDIX, OR A necessary Discourse (concerning the Helisphericall line that a Ship describeth upon the Superficies of the Sea, being directed by the Magnetical Needle or Compass.) Between Mr. Geograph, and Mr. Nautae Enterloquitors. Nautae. WEll met, Mr. Geograph? Geograph. Mr. Nautae and so are you also well met, I have much desired a long while to have some discourse with you. Nau. With me Sir, about what I pray you? Geo. About that which I suppose you are well seen in, I mean the way of a Ship upon the Superficies of the Sea, being guided by the Magnetical Needle or Compass? Nautae. Sir, I am very willing to entertain your motion, and at this present to discourse with you, there are diversities of opinions, concerning the way, motion or moving of a Ship upon the Superficies of the Sea, which way, motion, or moving of a Ship is twofold, principally in Sailing. That is first spiral, alias Helisphericall, or else secondly if is under a great Circle, the spiral or Helisphericall way is produced by the Ships continual keeping of one course, the four Cardinal points excepted. But in the way of sailing under a great Circle, the course is always alterable. And whereas the most part of Seamen will thinks that the demonstration which is delivered by the common plain Sea-chart (because it is eastest done) is of all other the most exquisite for the truth thereof; Yet is the same indeed, ●ld with many gross errors, so that such men must and are often times of necessity greatly deceived of their expectation. Therefore for the better understanding of those that are Practitioners in the Art of Navigation, we will in this discourse make relation both of the spiral way of a Ship, in keeping always one course, with the Solution of such doubts, and answering such allegations and objections as may arise through want of judgement therein; as also of sailing under a great Circle with the various and changeable position thereof; let me therefore hear your opinion concerning the Rudiments and grounds thereof. Geograph. First, it is necessary for all Practitioners of Navigation to know, that the Earth and Water together do make a round or Spherical body being enclosed with one converity or superficies, and that the earth and the waters are not plain or flat form, but like unto a round Sphere or Globe, it may thereby easily be understood, that the courses extended from place to place, cannot be truly described with right lines, especially in a farre distance, because they are Segments of great Circles and are circularly extended, from one place to another. Hence it follows, that the superficies of the Sea being Spherical not plain or flat form, that a Ship departing from any assigned place between the Equator and either of the Poles, by any one Rhombe or point of the Mariner & Compass, the Meridian only excepted, I say that in keeping one and the same course she maketh her way neither in straight lyns, great Circle or lesser Circle, but in a spiral line composed of Segments of great Circles, and returning by the opposite of that Rhombe or point towards the place of her departure, she shall make her way in another spiral line, by reason whereof she cannot according to that course fall with the place of her departure. Nautae. That the course or way of a Ship is spiral yet concentric to the centre of the Earth, and therefore in Segments of great Circles, when it is continued by any one Rhombe or point of the Compass (the East and West, North and South, only excepted, and that when the Compass hath no variation) is undoubtedly true. But whereas you say, that in returning to the place of her departure by the opposite point or Rhombe she sailed out, she shall not fall with the place of her departure, (no current nor any such thing to hinder her) you err very much, yet I would willingly hear by what reason you can prove your opinion. Geograph. If you will grant that a Ship maketh her way in Segments of great Circles, then know hereby that all great circles do in every several Latitude make several Angles with the Meridian, also those circles that do in several Latitudes make like Angles with the Meridian, have their several declinations or greatest distance from the Equator. And by reason of these varieties both in Angles and declinations, or greatest distances from the Equinoctial, there must also be a variety in the courses made by opposite Angles, and yet both spiral. For being under the Equinoctial, an Azimuth of North-east toucheth the Hortzon in Latitude 45 deg. 0 min. North Latitude, 90 deg. distant from the place of being under the Eqinoctiall; Then coming to the second place in Latitude 45 deg. 0 min. North Latitude, and intending to return to the first place under the Equinoctial by the opposite point of North-east, which is Southwest 45 deg. 0 min. from the Meridian, that great circle or Azimuth intersecteth or cutteth the Equator at near 54 deg. 45 min, which distance is less (to return to the Equator by the opposite point) by 35 deg. 15 min. and is short of the place of departure; also being under the Equinoctial the North-east, Southeast, and Southwest Azimuths do require 1 deg. 24 min. 51 seconds 24 thirds to raise one degree of Latitude. But being in a parallel 60 deg. 0 min. North, the North-east and Northwest require 1 deg. 26 min. 13 seconds 3 thirds of distance to raise one degree of Latitude. And in the same parallel of 60 deg. 0 min. the Southeast and South-west, require no more but 1 deg. 23 min. 32 seconds 53 thirds, for one degree of the Poles depression. Hereby it appeareth that the Segments of North-east and Northwest, are greater to raise one degree, than the Segments of Southeast and South-west to depress one degree, by 0 deg. 2 min. 35 second 10 thirds. Moreover the greater Segments which do raise the Pole one degree in that Latitude, exceed these under the Equinoctial by 0 deg. 1 min. 21 seconds 39 thirds, and the lesser Segments which depress the Pole, are less than those of the Equinoctial by 0 deg. 1 min, 13 seconds 3 thirds, which might serve for sufficient satisfaction that the way outward and homeward are not alike, again for your better understanding, you may note, that being at the Equinoctial, a Segment of a great circle of 20 leagues, which maketh with the Meridian an Angle of 45 deg. 0 min. doth raise the Pole and differ the Longitude near 0 deg. 42 min. 25 seconds 3 thirds. And in parallel 60 deg. 0 min. North Latitude, a Segment of 20 leagues Southeast or South-west depresseth the Pole 0 deg. 42 min. 58 seconds 8 thirds, and differs the Longitude near 1 deg. 23 min. 4 seconds 6 thirds, and in the same parallel of 60 deg. 0 min. the like segment of 20 leagues distance Northeast or Northwest elevateth the Pole 0 deg. 41 min. 57 seconds 40 thirds, whereby it plainly appeare●, that if the way of a Ship be composed of Segments of great Circles, the way outward and homeward being made by opposite Angles are not alike, yet both spiral. Therefore a Ship making her way by any one Rhombe or point the Meridian only excepted, and returning by the opposite point thereof, cannot by course fall with the place of her departure. And further it must be considered that the greater Latitude is, and the greater the Angle of the course is in respect of the Meridian, the greater is the variety, and the East and West are most variable, moreover in North Latitude, if the course be between the South and the East or West, than the way homeward returning by the opposite to the Meridian of the place of departure shall be shorter than the way outwards, and falleth into a lesser Latitude according to the course, distance, and declination from the Equinoctial, but if the course be between the North and the East, and the North and the West, then in returning by the opposite, to the Meridian of the place departure, the way homewards shall be longer than the way outwards, falling likewise into a lesser Latitude, according to the course, distance, and the declination from the Equinoctial. Nautae. If the spiral or Helisphericall way of a Ship upon the Superficies of the Sea, being composed of Segments of great Circles, had also those Segments limited or honded to contain 20 or 30 leagues a piece, then should all your former allegations be true, but those Segments in regard of their smallness cannot be sensibly distinguished, neither can it be certainly said, that a Ship in keeping always one course, continueth under one great circle 1 league or 1 mile; for when the course is always continued according to any one point of the Compass, it maketh an obliqne Angle with the Meridian, and then so often as the Ship changeth her Zenith, so often she changeth likewise the great Circle she maketh her way in, that is to say, so many Zeniths as she passeth under, so many great Circles she maketh her way in, and each of those Circles make several Angles with the Equinoctial, and the greater the Latitudes are the greater are the Angles, for in the Latitude of 59 deg. 30 min. the vertical circle of South-west and Northeast maketh an Angle of 68 deg. 58 min. with the Equator, and in Latitude 60 deg 0 min. the Azimuth of South-west and Northeast maketh an Angle of 69 deg. 18 min. with the Equinoctial. Also in parallel 60 deg. 30 min. the South-west and Northeast Azimuthes make an Angle of 69 deg. 37 min. with the Equator, and in the Latitude of 68 deg. 58 min. it makes an Angle of 69 deg. 18 min. and in the Latitude of 69 deg 37 min. the foresaid great circles make right Angles with the Meridian, and are circles of West and East, yet notwithstanding the variable Angles that these great circles make with the Equinoctial, and the contrary Angles, that every great circle maketh with every new Meridian. I say that in regard those Segments that a Ship maketh her way in are so small and insensible, she shall in keeping one course outwards, produce a spiral or Helisphericall line, and returning by the opposite point thereof thee shall again pass under all those Zenithes, that she did in her may outward and in like Segments, and shall by the same line of inclination, fall again with the place of her departure. But when a Ship maketh an East or West way the line of her Caping maketh always right Angles with the Meridian, then shall those great Circles of whose Segments the Ships way is composed make like Angles with the Equator, that is to say equal to the Latitude, and the Ship shall according to that course run a parallel to the Equinoctial. Geograph. Me thinks that is strange that you will allow the East and West way of a Ship being made in Segments of great Circles to entersect the Equator at East and West, by reason whereof they are Touch-lines to the parallel of Latitude, and yet you will not allow or grant the East and West to make a spiral way as well as the rest. For how is it possible that the line of Inclination or way of a Ship being composed of Segments of great Circles, and those Touch-lines to the parallel of Latitude, so that the Ships Caping is quite contrary to the parallel, and maketh obliqne Angles therewith, and that especially in great Latitudes, how then is it possible that the East and West should lead in a parallel or produce a lesser Circle or any part thereof. Nautae. Take a small Compass sly, and fasten it to a thread that may pass thorough the North and South points thereof, and make a noose in the end of the thread and put it upon the Axis of the Globe at the Pole, then carrying the fly with the thread about the body of the Globe, and you shall see the centre of the fly describeth a parallel to the Equinoctial, and yet the East and West of the fly always respecteth the Equinoctial at 90 degrees 0 minute. of distance. And so would a Ship if she had a haser or some thing else fast about the Pole to attract her thereunto. Geograph. But by the Globe thus do fasten the quadrant of Altitude to the Equinoctial in the brass Meridian, and bring the beginning of the degrees of the quadrant to parallel 60 deg. 0 min. and then from that point where the beginning of the degrees of the quadrant do touch in parallel 60 deg. 0 min. along by the edge of the quadrant to the Equinoctial is the line of East and West, now with the point of an needle or some such thing prick of by the edge of the quadrant 1 deg 0 min. and make a mark there, then move ●●lobe until the beginning of the degrees of the quadrant do fit with that mark, and then as before prick of again 1 deg. 0 min, by the edge of the quadrant, and so proceed by 1 deg. 0 min. until you have gone round about the Globe, and that the point of the Needle fall in the first Meridian where you began, and you shall find the line of the Inclination to be dilated from Latitude 60 deg. 0 min. about 2 deg. 20 min. and for further proof hereof suppose yourself to be under the Equinoctial, and the Compass to have no variation, and the Ship to cape East or West, also the Mainmast to stand upright in the step, the head thereof pointing to the Zenith, and the heel to the Nadir or rather to the centre of the earth, and the mid-ship beam making right Angles with the mast to be parallel to the Axis of the world, I say, that this Ship proceeding East or West in this manner maketh her way in a great Circle to wit in the Equinoctial, and returning by the opposite point thereof shall again fall with the place of her departure. Now I say by the same reason, that if the said Ship being in any Latitude between the Equator and either of the Poles, in Caping East or West, her mid-ship beam shall then be a parallel to the plane of the Horizon, and also to the Axis of that great Circle or Circles which in her proceeding line of Inclination she maketh her way in; the head of the mainmast pointing to the Zenith, and the heel to the Nadir, and the line of her Caping maketh contrary Angles with every new parallel. Now if a Ship in sailing under one Circle must have her mid-ship beam always parallel to the Axis of that great Circle she maketh her way in, then in keeping directly under one parallel, her mid-ship beam must be always parallel to the Axis of the world, for that is the Axis of every parallel, and so likewise the maine-must being rectified perpendicularly in manner as aforesaid, must be also a parallel to the Equinoctials diameter, and make an Angle with the Horizon equal to the Latitude, the head thereof not respecting the Zenith, nor the heel the Nadir nor the Centre of the earth, but the Centre of the parallel of her Latitude, and in this manner a Ship may run in a parallel to the Equinoctial. But now this may stand with humane reason, I leave to your further construction. Nautae. Here in you are notably deceived as it shall presently at large be made plain and evident, for whereas you say that the way of a Ship, cannot describe a parallel to the Equinoctial except her mid-ship beam be parallel to the Axis of the world, I say that so long as the mid-ship-beame remaineth due North and South, that is to say parallel to the Meridian's Diameter in the plane of the Horizon, although the head of the mainmast (it being perpendicularly erected) point to the Zenith, and the heel to the Nadir, so long I say, her way shall describe a parallel to the Equinoctial, but as I said before you seem by all your former allegations to prove, that the way of a Ship; being composed of Segments of great Circles should have those Segments limited or terminated to contain 15, 20, or 30 leagues a piece, which if it were so, then should it be altogether according to your saying. But now far as much as there is some difficulty in the premises and few Mariners know how to censure thereof, I will therefore briefly prove by Arithmetical calculation, the East and West in any Latitude to lead in a parallel as well as the Equinoctial. Example. The parallel of 60 deg. 0 min. is equal to the length of half the Equinoctial or 180 deg. 0 min. of a great Circle; we will therefore in the same make our beginning, and from the first place being situate therein, produce 18 Segments which contain 10 deg. 0 min. a piece, which by Arithemeticall calculation may be thus found out. The Theorem. viz. AS the Radius is to the sine of the Latitude 60 deg 0 min. so is the sine of the Compliment of the distance the sine of 80 deg. 0 min. to the sine of the Latitude of that place where the first Segment of 10 deg. 0 min. endeth, and so again in like manner for the second Segment. viz. As the Radius is to the sine of the Latitude where the first Segment endeth, so is the sine of the Compliment of 10 deg. 0 min. to the sine of the Latitude where the second Segment endeth, and this is to be continued 18 times, so shall you find the last work to bring forth the sine of 41 deg. 06 min. but if you work by Logarithme sins, multiply the Legarithme sine of 80 deg. 0 min. (the Compliment of 10 deg. 0 min.) by 18, because there are 18 Segments, and the product add to the Logarithme sine of 60 deg. 0 min. the Latitude given, the sum will be the Logarithme sine of 41 deg. 06 min. the Latitude of the 18 Segment, which dilateth from Latitude 60 deg. 0 min. the sum of 19 deg. 0 min. wanting but 0 deg. 06 min. From whence we may see, that if great Segments have such great alterations, than lesser Segments must have their correspondent varieties proportional unto them, but mark what follows, and I make no doubt but that anon you will be of another opinion then formerly you have been concerning this matter, as from the aforesaid parallel 60 deg. 0 min. let there be produced 36 Segments according as was afore showed, each Segment containing 5 deg. 0 min. or 100 leagues a piece, and you shall find the end of the last Segment to fall in Latitude 49 deg. 41 min. which is dilated from parallel 60 deg. 0 min. but 10 deg. 59 min. where note that this dilatation is less than the former by 7 deg. 55 min. In like manner, in the same parallel 60 deg. 0 min. let there be produced 180 Segments of 1 deg. 0 min. or 20 leagues apèece, and you shall find the end of the last Segment to fall in Latitude 57 deg. 25 min. which is dilated but 2 deg 35 min. from parallel 60 deg. 0 min. Again, let 10800 Segments be produced in the same parallel of 60 deg 0 min. of 0 deg. 1 min. one minute apiece due East or West and working according to the former manner, the last Segment will end in 59 deg. 57 min. ½. which dilateth from parallel 60 deg. 0 min. but 2½. minutes, wherefore the consideration hereof may serve for a sufficient satisfaction plainly to prove that the East and West directed by the magnetical Needle or Compass doth lead in a Magnetical parallel, for as great Segments have their great varieties. and lesser Segments have their lesser alterations correspondent unto them so by the same reason insensible Segments must have insensible differences and the like reason holdeth for any other point of the Compass as well as for the East or West as I have formerly shown you, and at our next meeting I will set you down or show you the Theorems for operating of it. But you will say, here is in 10800 minutes a difference of 2½. minutes, and Segments of minutes in a man's judgement are so small that a Ship cannot make her way in lesser Segments and yet these Segments are not void of a sensible difference, I answer as before, that neither in sailing East or West, nor in the spiral or Melisphericall way by any other course or point of the Compass, a Ships continuance under a great Circle or Circles cannot be terminated, and whereas 10800 minutes do in the East or West from Latitude 60 deg. 0 min. produce a difference of 2½ minutes, I say in respect of ●0 great a distance, the difference is insousible. But if you please to take so much pains for the former parallel of 60 deg. 0 min. to make a trial from second to second, that is 〈◊〉 648000 Segments be produced East or West each Segment to contain one second and the end of the last Segment shall not be from the first place, so much as one second, and thus having proved sufficiently that the East and West being directed by the magnetical Needle or Compass, doth lead in a magnetical parallel, and also that in keeping one course the Ships way is spiral or Helisphericall, and returning by the opposite point thereof, the Ship shall again fall with the place of her departure, we will finish this discourse, and speak of some principal rules which of all seamen and Mariners ought to be known. Geograph. What is the first and most useful Proposition in the Pariners practise to be taken notice of. Nau. By the course and both Latitudes to find the difference of Longitude and the distance. Geo. For what reason is that proposition is sailing the primary and most useful. Nau. Because the course is commonly given, and the Latitudes may be known by observation, but the distance and the difference of Longitude by sailing may be supposed but not certainly known without the help of the former, and so likewise the distance in sailing East or West may be supposed, but not certainly known. Geo. I pray you Sir, let us then proceed, 〈◊〉 the practice without any further circumstance, that having both Latitudes and the course we may ●nde the difference of Longitude and the distance Nau. We will Sir. Suppose a ship to be in Latitude 50 deg. 0 min. North Latitude sails South South-west ½ point West, until she be in the Latitude of 47 deg. 0 min. I demand the difference of Longitude and the distance the Ship hath run. The Theorem. AS a mean proportional between the fines of the Compliments of both Latitudes is to the Tangent of the course, so is the difference of Latitude to the difference of Longitude, which by the Logarithmes is thus. Add the Logarithme tangent of the course 28 deg. 7 min. to 〈◊〉 Logarithme of the difference of Latitude 60 leagues, and from that sum subtract half the sine Compliment of 50 deg. 0 min. which is half the sine of 40 deg. 0 min. and half the sine Compliment of 47 deg. 0 min. which is half the sine of 43 deg. 0 min. added together, (I mean Logarithme sins) and the remainder shall be the Logarithme of the difference of Longitude. Geo. What is the second most useful proposition that a Mariner in his practice is to take notice of. Nau. By both Latitudes and the departure from the Meridian to find the difference of Longitude the course and the distance. Geo. Wherefore do you account this to be the second most useful proposition in the Mariners practice. Nau. Because all Mariners that keep their account by difference of Latitude and difference of Longitude (which only is the true way) after that they have ca●● up their Traverse by difference of Latitude and departure from the Meridian, do find their difference of Longitude as well as their course and their distance from their first place, where they began their Traverse this way. Geo. I pray you Sir, set me down the Theorems for the operating of this. Nau. I will Sir, which are these following. 1 As the sum of half the Logarithme sins of the Compliments of both Latitudes is to the departure from the Meridian, so is the Radius to the difference of Longitude. 2 As the difference of Latitude is to the departure from the Meridian, so is the Radius to the Tangent of the course. 3 As the sine of the Compliment of the course is to the Radius, so is the difference of Latitude to the distance that the ship hath run from the first place where she began her Traverse. Geo. What is the third, and as I remember you said the last useful peoposition to be taken notice of in the Mariner's practice. Nau. By having given the Latitudes of two places and their difference of Longitude to find the magnetical course or Rhomb, and the distance. Geo. How can this be useful for a Mariner in his practice. Nau. Because many times it chanceth that a Mariner is to sail from one port whose Latitude and Longitude he hath in Geographical tables (as in Mr. Hugh's his use of the Globes or in the Tables of the Seaman's Calendar) and is to sail so another port, whose Latitude and Longitude he hath also in the said Tables, and by this proposition be may examine the truth of his Sea-chart he sails by. Geo, Set me down the Theorems for this proposition, and I will trouble you no further at this time. Nau. Sir, I am in some haste because the time is farther spent than I supposed since we met, but I will perform your request, and then I will take my leave of you for this time. First, as the difference of Latitude is to the difference of Longitude, so is half the sins Compliments of both Latitudes (I mean of the Logarithme sins) to the Tangent of the course. Secondly as before, as the sine Compliment of the course is to Radius, so is the difference of Latitude to the distance run. Geo. Master Nautae, I thank you very kindly for your company, and your conference, you have informed my judgement very much in the matter of Navigation, Nau. Sir, I am very joyful of it, far you well. FINIS.