FRIEND TO NAVIGATION Plainly expressing to the capacity of the simpler so the whole mystery or foundation of the same Art, for whose sake, the Author hath only penned this Treatise, being himself a faithful good willer thereto.   woodcut of ship 
Printed at London, by T.C. 1628.   

 drawing of astrological map   

TO ALL THE MOST NOBLE AND GENEROUS SEAMEN of England, happiness in this life, and eternally in the Heavens.  
VRania having brought me (Right Noble) to the sight of your most admirable Art, I could not choose but say, none could do so great a work, but the maker of all things. And seeing it framed of so many excellent pieces, to say; none but the Divine providence can uphold it: Whereupon I endeavoured to enable myself in some knowledge thereof; and (though poor) I Purchased some books, and in diverse years bestowed much money; employing my 〈◊〉 hours that I could spare in this study and practise. And in the mean time, much sickness, and death of my friends, bringing me vnder●●● intolerable bondage 〈…〉, made me come nearer this Honourable City; but with the loss of almost my whole estate: and in the way had both my Books and goods, all lost and spoilt in the Thames water. Nevertheless (although almost utterly unabled and ●nlearned) I have not ceased to do good if possible I might (knowing that no man is borne for himself.) Therefore I set forth in a bill at my door, to Teach the Art of Navigation, to the poorer sort gratis: And not so content I have here framed a good order there of (as I think) to sand abroad to those that cannot come to me. May it please you therefore under your favourable 〈…〉 direction, to admit it your presence once only: to ask you leave, and to give you accounted what I do among your servants as is most meet. And for your favourable entertainment of it, your 〈◊〉 respect to me for my good will, and your kind acceptance, I rest,  
Your Honour's servant to be commanded, john Skay.   

To the Ignorant and Honest Reader.  
HOnest Reader, this work intended for thy good, despise not, but rather accept it thankfully, seeing if it were better I ●ould have given it unto thee; and more, while I live I rest always ready to do thee what good I may. Thou hast here 〈◊〉 Chapters: The first is of the Cosmographical description of the world: The second is of the measure of the heavens: The third of the Elements: The fourth is of the ground of the Art, The Jnstrument, Astronomical propositions, with the Geographical 〈◊〉, and Hidrographicall description of the Earth and S●a: The fift Chapter is of certain considerations briefly set do●ne: The sixth is of Shipping and going out of the Harbour: The seventh of the journal observation and projection. The eight and ninth is of propositions of Navigation Arithmetically, Geometrically, and Jnstrumentally showed: The tenth is of the motion of the Moon, and of coming into the Harbour. If thou profit by it, give God the glory, to whom be praise for ever, Amen.  
Vale, From my house in Saint Thomas Hospital, May the first, 1628. john Skay.  
 drawing of geometric equation   

A FRIEND OF NAVIGATJON.  

CHAP. I Of the Cosmographical description of the World.  
THat which the only wise God made f●● man's behoof, even all the world of nothing, Genesis the first, is said to be round by all modern Writers: it is proved by reason, and the holy Scripture saith it, Psal. 98. It was divided in two parts by the same God, the part seen, and the part unseen: But by faith search the Scriptures, there is the part and use in this our Christian Navigation discovered.  
The part seen is well defined to be a Book, In which we may see and learns to praise God in his works. And is said to be of two parts, Celestial, and Elemental: In the Celestial part may be considered the Arts, as Music, the harmony of the Spheres: of Soul and Body, for whose health Physic is next, in which we consider that Miraculous Medicine of preserving life, be it Philosopher stone, Salt, Virgin-earth, or other denomination, or Mathematical, as the knowledge of Points, Lines, Circles, Signs, Constelations, Planets, and their Influence and power over bodies Elemental.  
It may be said and that by good reason, I should begin at the Center of the Earth, & so consider of the things contained therein, with the Sea, The face of the Earth and Sea, and the things thereon; The Air covering the other two, and the Fire enclosing them all, making one round body of these Elemental parts, being as a Center to the rest, and so p●●●e upwards unto the first m●●●er, showing the relation and agreement they have together, I living now nearer the Center of the Earth (as it is observed) by many thousand miles. But if I end my work so high may be proud (as some are) or dazelled in the Celestial brightness, being mortal, be cast down with Pl●●●●.  
In the name of God therefore, and in his fear, I will begin with the first mover, which going with great violence, turneth the whole frame of the heaven within it, round in 24. hours, from East to West: Which heaven is said to have no impression in it, being almost invisible, yet carrying the light and darkness, making difference of time in days and nights, according to the difference of Longitude and Latitude of places on Earth.  
In it are all the circles and points in the whole fabric of the world said to be described, their plains defending and meeting in the Center of the Earth, or on their other Centors.  
The two principal points, are the two poles of the world, the North pole elevated with us here at London 51. degrees, 30. or there about; and the South pole is right opposite to it. Therefore depressed just as much (under the horizon of which I will speak anon.) These 〈◊〉 points are fixed fast and unmoveable, between which and through the Center of the Earth, there passeth▪ right line, which is called the Axletree of the world on which it turneth.  
Now we must consider both this right line, and all other right lines, cords, signs, great circles and parallels, to be divided into 360. degrees, each degree subdivided into 60. minutes, each minute into 60. Seconds, each Second in 60. thirds, &c. as the case may require.  
The Herman is a great circle, whose Center is in the midst of the Earth, and his plain reacheth not only to the face of the Earth, dividing the upper part from the lower, the day from the nights ●ut through the Or●e● of the planits and fixed stars, even to the first mover. This circle hath two poles, one is called Zon●th, and is right over our head, from the which if a right line pass through the Centre of the Earth, touching the concave superficies of the first Mover, it pointeth out the other Pole, called Mador. In this Circle we count the Amplitude of the Sun or Stars; and the Point of the Compass or Wind. All the Circles called Azimuthes, cross the Horizon at Rectangles and pass through the Poles thereof: His Parallels are Circle● of Altitude, in which are observed the position of the Planets and fixed Stars above the Earth, at all hours of the day or night: Or Circles of Depresion dividing the length of twilight.  
The Equator is a great Circle, deuiding the World into two equal parts, the North part and the South: His parallels sets out the Latitude of places on Earth, and declination in the Heavens. All the Meridian's do cross the Equator at Rectangles and go through the Poles of the World. On this is counted the Longitude of places on the Earth or Sea: For every 4 of equal time made by the equal motion of the Equator doth make 1 degree. And 7 deg. 30 min. of Longitude, either East or West, doth cause ● hour difference in time of day or night, and they that devil 15, or 30 deg. East have it 12 or 11 a clock, when we have it 10 or 11, and those that devil so much West of us, have it 10 or 11 when it is midnight with us. These that devil at the East part of our Horizon have it noon when it is but 6 in the morning with us, and when it is noon with us, it is 6 at night with them, but with them in the West part of our Horizon it is 6 in the morning. Our Antipodes have it ●oo●e when it is midnight with us, and when it is Summer with us, it is Winter with them.  
All other Circles not yet named are likewise supposed to be in the first Heaven, as in the rest of the inferior Orbs of which we suppose 11 after Maginus. And to good purpose: For the Lord of Heaven and Earth hath so laid the Foundations of the Earth that they cannot be moved, Ps. 14.2. Though Co●●●●● to bring some good purpose about hath imagined so.  
Let the tenth Heaven be (if you please) the Wa●ers above the Firmament, for there are so, Gen. 1.6, 7.   

CHAP. II Of the measure of the heavens  
THe Zodiaque is a great broad circle crosing the heavens, like a b●n●●●re or girdle, of 12ᵈ. broad at the ●east; in the midst ●hereof is a little circle called the ecliptic line, which crosseth the Equator at two opposite points, swerveth from it 23ᵈ. 30. at the lest, as I ha●e ●ound by observation this present year 1627. So the poles of the Ecliptic are distant from the poles of the world 23ᵈ. 30. by reason of his obliquity: Between which poles there passeth a right live or Axletree on which the second ●ouer or tenth heaven is carried in his own motion (contrary to the first) from West to East, and is most slow, making his revolution in 3434 years, and 10. days.  
The third mover or ninth heaven, hath his two poles in the two points of the equator and ecliptic, and his motion is from North to South: making his revolution in 1717. years and 5 days. But it will be sufficient for our necessary use here to observe the motion of the 8. Sphere, wherein are all the Celestial bodies of the fixed Stars placed: whose motion is from West to East, (some say but one degree in 100 year. But the Planets moon in their Or●es 〈…〉, as 〈…〉 in 30. year, ♃ jupiter in 12. years, ♂ Mars in 2. year, ● Sol in one year, ♀ Venus and ☿ Mercury like ☉, Luna ☽ in a month from West to East.  
Diverse learned Mathematitions have set down the magnitude of the Stars to be fare bigger than the earth: making 6. differences in bigness be sides du●ke and obscure. The Planets also are said to be of 〈◊〉 bignesses and distances. The Orb of ♄ is 〈…〉 mile's, and his distance from ♃ 78721 〈◊〉 The Orb of ♃ 189●●54● m. His distance to ♂ 78721. m. That of ● 2630●800. m. his distance from ☉ 15725. m. ☉ 343996 4/●. m. His distance from ♀ 23437 ½ m. The Orb of ♀ 3274494 6/12. Her distance from ☿ 12812. m. ☿ Orb 253372 ●/●. m. His difference from ☽ 12812, The ☽ to be in bigness 105222 ●/11. her distance from the earth 15750. m. ♄ his bigness compared to the Earth i● as 95. to 1. and his distance to the Firmament 2●0445. m. 〈◊〉 his magnitude to the Earth is a● 〈…〉 91. to ●. And ♂ as ●. to 1. And the distant from the Firmament to the Earth 35 846 3½. m.  
It hath appeared manifestly that both by Sea and Land, who so travaileth 60. miles on a great Circle, altereth a degree in his Travail: which taketh 360. times though number of the whole circle, maketh 21600. miles the conpasse of the whole Earthland Sea. The Diameter than will be 6872 ●/11. the semyd●amiter 3436 4/1● m. which maketh the firmament to be from the Centre of the Earth 361899●●/●. Therefore it seemeth there is something to work upon to measure these things by.  
Likewise I may say, considering that from the earth to the Moon is 15750. miles, and the ☽ is the lowest Planet. Therefore the two uppermost elements of Air and Fire are both together 15750 mi. in thickness: Again, because the hill Atlas is said to reach to the middle region of the air, that hill (which I take to be Teneriffe) may easily be measured, and so the measure of all the rest is had.  
The motion of the Ecliptic causeth other four circles to be described, two are described of the motion of the Poles thereof, being distant from the ●●●es of the world 23. degrees 30. min. making two round circles about them: that next the North pole is called the circle Arctic, the other is called the circle Antarctic. The other two are described by the motion of the tow. Tropical points of the ecliptic, that on the North side of the equanox is called the tropic of Cancer, the other is called the tropic of Capricorn. The Zodiac is divided in 12. equal parts, beginning at the one intersection of the same with the equanoctiall circle, and so going round: And these are called the 12. signs of the Zodiac, for as much as they extend them to the full breadth of the Zodiac, and are the cause thereof.  
There are two other circles also of special use in this Art, one is the Equinoctial colour, the other is the Solititiall colour: These two are great circles, the first parteth the Equator and Zodiac even, so as the North signs are from the South signs parted by the first point of the first sign, and the first point of the sca●outh sign: and these signs are called Aries ♈, and Libra ♎, beginning the Spring and Autumn, the other passeth by the two tropical points of Cancer ♋, and Capricorn ♑, making the beginning of Summer and winter; both passing through the poles of the world, parting the Zodiac in four equal parts, and the equator also with all his parallels.  
The first point of ♈, beginneth the Spring: that is, when the Sun in his motion cometh first to that point, and going through three signs ♈, Taurus ♉, Gemini ♊: then coming to the first point of ♋, beginneth the Summer: and passing through other three signs ♋ Leo ♌, Virgo ♍, so cometh to Libra ♎, Scorpio ♏, Sagitarius ♐, to the first point of ♑: beginneth the winter, from thence through ♑, Aquarius ♒, Pisses ♓ making the whole year's revolution. And these 12. signs are called Constellations, and there are other Constellations, some North, some South, in which is had a number of Stars bearing names, according to their nature: and these constellations bear rule over diverse Regions, Countries and Cities.  
Now we may pass the Orbs of the other superior Planets, & only note their friendly aspect as we pass by them: And consider the ☽ motion with the ☉ for the Eclipses of use, for finding of longitude on earth or Sea. Her monthly motion, full change and quarters, for the ebbing and flowing of the waters, spring and nepe tides: her aspect with the planets, good or evil, for Physic: Her place in the Zodiac, for blood letting either for Man or Beast.   

C●AP. III Of the Elements.  
FIre is the most noble and superior of all the Elements, pure, subtle, most spiritual, putting heat into all substances, lightsome, having motion Lionous, choleric. Air is divided into three Regions, that next the Fire is most hot, that in the middle most cold, unto the which some hill tops do climb, as hath been seen, the lowest is that in which we live, the Clouds and the Fowls do fly, which God of his mercy make wholesome unto us. God said let the waters under heaven be gathered into one place, and let the dry land appear, Gen. 1.9. And God called the dry land Earth, and the gathering together of the waters, he (even God) called Sea, ver. 10. Sea, which together with the earth make one huge massive body, round, as may be proved by the artificial Globe,: but most truly by a perfect Artsman, who using his skill in his Travel, both by Sea and Land, is able to salve such small irrigularities, as high tides, low ebbs, high Mountains low Valleys: For experience is the mother of Acts, and th●●● 〈◊〉 partly conceive in this (I having a brother in Saint Christopher's Isle of the West Indies, who hath sent me Letters from Meri●rs Hope there) I went to Gresham College in London of purpose to look on the biggest Terrestrial Globe they had, and found such an I'll to lie in some 16. deg. of North Latitude, and in 321 deg. of longitude, or there about: But there I could find neither Cape, nor Bay, Port nor Haven; therefore no universal Map will serve that turn: And Master ♄, being there with me at that time, made a doubt, saying: How know I that Saint Christopher's Isle lieth there except I had seen it, or know some body that had been there? meaning no doubt some Artsman, who (I confess) may not only observe the longitude and latitude; but may also set out the Capes and Bays, Ports, Bounds, Rivers, and give the dimention of the superficis.  
Of the innumerable multitude of creatures in the Sea, and of her riches, I will not speak; but the artificial making and use of Shipping therein is admirable, as may appear in the whole Art of Navigation: A wonderful secret thereof is the variation of the compass; a cause thereof is imagined to be the hollowness of the Earth or depth of the Sea, and for that nature abhorring emptiness, the excellent virtue of the Loadstone doth always draw towards it in all places, where the needle being touched therewith shall draw most near; but I think it is a special gift of God, sent for man's use, but fare above his knowledge. Sea and earth are divided as the heavens: the beginning of the Equator or first Meridian being with Saint Michael and Saint Mary's Isles, the circle of the equator passing by Saint Thomas I'll, Abasa●●●, the famous I'll of Sa●●●ta, B●rni●, Pap●os, New Guinny, G●na●●s, and foe round, from Meridian to Meridian Eastward we count Longitude on any parallel: but Latitude is counted from the Equator towards the Poles of the World on the Meridian or colour. The Parallels sets out the Zones and Climates.  
Of creatures without life, or such as are in the earth only, are Gold, Silver, precious Stones, Minerals, Metals. But the Bodies which shall one day rise again, are chiefly to be considered. Of Creatures with life, some are fixed, as Plants, Herbs, Flowers, Spices, Trees, bearing fruit, without fruit. Gen. 2.19. Out of the ground God form beasts and fowls: and in another place it is said All flesh is grass. Therefore of moving things some are in the Earth, as Worms, Serpents, Moles, Co●●e●: some in the Air, as Fowls: in the Water, as Fish. Monsters on the face of the earth, as all manner of Beasts and Cattles wild and tame. The most excellent thing made is man (a shame for him to become so wicked) Being wonderfully made, Psal. 139. an epitome of the whole World; to seek and set forth God's glory: surely flesh and blood cannot set forth his glory, whose works do amaze the senses of the most learned. Therefore be not proud OH ye learned, nor vainglorious OH ye wise; but seek and set forth God's glory, that the unwise and unlearned, may see, mark, and learn: and be encouraged OH thou that art ignorant (of which I am chief) to see the works of the Lord in heaven and in earth, and his wonders in the deep Sea, Ps. 107.   

CAHP. FOUR The ground of the Art, the Instrument, Astronomycall propositions, with the Geographical and Hidrographicall description of the Earth and Sea.  
REason doth teach, and this 36. year's experience I have had, that to the gaining of these knowledges, quantity is to be considered, either Geometrically or Arithmetically, or most usually of both together. Arithmetic is the Art of numbering, and every number is expressed by certain Characters, figures, and cyphers, as: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. or: 2 ●. 3ᵈ. 30. 50. 20ls: But the figures and cyphers whether they be abstract or contract, have a double signification to express them, which is called their numeration, Psal. 90. ver. 12. OH Lord, teach us so to number our days, that we may apply our hearts unto wisdom. Figures do either signify themselves only, as 1. is one, 2. is two, &c. or are valued according to their place: Position of this kind is known as men read the Hebrew tongue. This Art giveth the numeral solution in all dimensions, and hath these kinds: Addition, Subtraction, Multiplication, Division; and these all have a diverse property in their use, either in working proportion, or otherwise. Another Arithmetic there is (not of numbers) but of parts of number, having like species, properties and passions, and fetch either their beginning from unity, but with a difference, for as number increaseth in multitude Infinitely: so do Fractions decreaseth infinitely but most commonly, as in this work, they are compound: for if I take in account of Time, Motion, Measure, Signs, Degrees, Minute, Seconds, &c. As in reckoning of the ☽ place, Thus: suppose the ☽ in the day of ♂ to be in the first degree of ♈; and a● 20. days old, I would know her place; say if the ☽ go in one day 12. degrees from the ☉, and the ☉ go each day one degree, Therefore the ☽ goes in 20. days 260ᵈ. which divided by 30ᵈ. giveth 8 ●: 20ᵈ. where ●. is the number 20, the Fraction, and his place the 20ᵈ. of Sagitarius: But where deg. are taken for numbers, there Minute's, Seconds, Thirds, are Fractions: as 15ᵈ. to an hour 4. to a degree, or 20ᵈ. is as much as one hour 20: or 20ᵈ. 50. 30. is as much as 1250 mile's ½: and generally all circles and parallels, are numbered with these kind of Astronomical Fractions or numbers.  
Geometry giveth a punctual termination to all dimensions, either in length, breadth, or thickness: which number many times cannot do; as in dividing a line by extreme and mean proportion. But number and parts do help Geometry to express the quantity, be it in length, breadth, or thickness. Linne●●●mall as angles right lined, of Spherical: & may serve to tell some distance 〈…〉 Instruments necessary in any work, are of necessity, and to this work which containeth so many Arts, it may be objected that one poor though of a good capassity shall be never able to attain. Say not so man, for the best instrument in any work is a willing mind: Again, shall I be ashamed to endeavour to do well, because some more learned than I, will despise my simplicity and weakness in knowledge? Not sure, the virtuous will commend it, God be thanked there are as many good Instruments as Arts, let every man be content with such as he hath, and God no doubt will bless the good endeavours of the godly honest. 2ˢ. 6ᵈ. on a pair of Compasses, and two pence on a strait ruler is not much: with these thou mayest begin in spending some spare hour's time to work thus: first make a circle, divid it in 360. parts, or ¼ into 90, and besides abundance of necessary conclusions, which the malicious ignorant will not believe, these following are not the lest, and are most meet to be known: Radius or any sign, cord, ark, Tangent, Secant, great circle, parallel: example: I did take the ☉ height, May the 26. 1627. and so found his declination 22ᵈ. 30. North, his place being in 15d d ♊. right ascension, 61ᵈ. 30. difference assentionall 28ᵈ. 15. height of the North pole 51ᵈ. 30. Amplitude 37ᵈ. 30 North. obliqne ascension 33ᵈ. 15.: semydiurnall ark 118ᵈ. 15. semynocturnall ark 61ᵈ. 45. his course from rising to his setting, 15 hours 40. length of twilight 8 hours 14. length of the night dark 0. hour. Likewise about December the 12. 1627. ☉ place 0 ♑ merid height 15ᵈ. declination 23ᵈ. 30. South, amplitude 40ᵈ. 30. south, right ascension 270ᵈ. obliqne ascension 298. deg. 30, difference assencional 28. d. 30. Likewise may the fourth 1629. suppose the son's place which is his longitude, found by the rules before going to be 53. d. or the 23. of ♉. in the latitude of 51. deg. 30. min. his merid. height is found by these rules to be 55. deg. declination 18. deg. north, amplitude 28. north, right ascension 50. deg. obliqne 27. deg. difference assentionall 23. deg. semydiurnall ark 7. hou. 32. min. semynocturnall ark 4. hou. 28. min. length of twilight 2. hou. 4. min. from noon to evening shut in 9 hou. 33. min. length of the day 19 hou. 6. min. night dark 4. hou. 54. min. Son rise at 4. hou. 28. m. setteth at 7 how. and 32. min. Sun above the Horizon 1● hou. 4 min. under the Horizon 8 hou. 56 min.  
And here remember, that, The works of the Lord ●p g●●●● sought out of all them that have pleasure therein. Ps. 111.2. The merciful and gracious Lord hath so done his maruilous works, that they aught to be had in remembrance, u 4. The Lord is high above the Heavens, Ps. 113.4. Who is like the Lord our God, that hath his dwelling so high, and y●● humbleth himself to behold the things that are in Heaven and in Earth, u 5. He taketh the simple out of the ●●st. u 6. The Heavens declare the glory of God, and the Fir●●●●●● showeth his handiwork, Ps. 19.1. As Arcturius Orio● pleads the hiden Chambers of the South, the great Laviathan that mo●eth in the water, but beyond all our Redemption in Christ, God commanded Noah to build an Ark, and to build Shipping were but vain except God bless it and be the Seaman's guide. Say then, is it not a great blessing we receive from him in guiding our Ships and Selves both by Sea and Land with such excellent Rules of Art: Surely the learned in the Geographical, Hidrographicall and Nautical Sciences must needs confess it, especially those that travel.  
Neither need any be so sottish as to think it a shame to spend ten min. of his idle time every day to some such good purpose, which may add him more comfort perhaps in distress, than all the friends in the world beside. By the Seal made on a strait Ruler and Compasses distance is had easily without measuring to them, for make a Circle and divide it in four equal parts with strait lines over the Center, then divide ¼ in two equal parts, and each of them in 3 then again these subdiuision● in 3 and lastly those into 5, so is one quarter of the Circle divided into 90 deg. by the same reason you may divide a right line as your Ruler into Foots, each Foot into Inches or Tenths, each 10 into ●● 3. 4. 5. 7. 11. 12. 16. 20 parts or what you please, and then begin to work thus, Imagine to stand on some Hill as at High 〈◊〉 from thence to all the highest Hills round about you cast your eye, and by your Circle divided you may take the quantity of the Angle from your eye to the two next Hills from you and nearest one to another. Suppose that Shutors' Hill and ●urfleet Hill do make an Angle of 1/● of the quadrant or 30 deg. Or Harrow Hill and Saint Alban's make an Angle of 40 deg. and the distance between Highgate and Suitors Hill to be 10 mil. more or less, and from Highgate to Harrow 12 mil. you shall go to Suitors Hill and observe the Angle between Highgate and Purfleet, be it right or obliqne, acut or obtuse make 2 lines, whereof one to be laid down by the Scale on your Ruler of the just distance between your 2 first stations, at the one end make the first Angle, at the other end make the other looking one towards the other, and the two lines continued will point you our Purfleet, do so by the other and it will point you out Saint Alban's, then by your Scale you may know their distance, having them distances you may easily found all the distances and angles of position or situation of all the Hills, Towns, Rivers, Borders, in any one County, and from thence ye may do the like by the rest, and so take a true survey of a Kingdom, and of all the Isles, & Rocks in the Sea, and Kingdoms adjoining, as Scotland with his Isles, and over the Sea, as Ireland with his Isles, likewise from Dover to Calais, and so take the Country and Kingdom of France, with his Rivers, Isles, Rocks, and every remarkable thing that cometh in view of your eye.  
But note, that if you measure such great distances, you must have regard to Longitude and Latitude, for if you shall measure from France to Spain, Portugal, and so over the mouth of the St●i●● of Guib●ater into Barbary, Guiny, by the Coast of Saint Thomas I'll, Monomotapa, Cape Good-hope, the Coast of Saint Latence, Mosamque, Prester john, the head of the ancient River Nilus, the Arabian, judean, Notalia, Graecian, Italian Countries, so come to Germany, Polonia, and all those North parts to the Pole if you can, or to Sweden, Moscovia, and the North part of Tartary, and th●● way discover all the North parts round about the Pole, and so if you can come into North America, if not, then come over Land near the Caspian Sea, Parthia, Persia, Ormuse, Gus●●● Goa, Cochin, and all the Isles to the South thereof, the Gulf of Bengala, Malaca, Sumatra, the Straight of Sunda, java maior, java minor, new Ginny, Papoos, Hiland, Timor, so to Beach, and to the discovery of the parts of Maletor Kingdom to the South Pole if you can, or by the Moluce●e, Bernie, Ca●●bia, Cochin China, japan China, to Cathay, from thence East a little North to the Straitss of Annian, and so into the most Westerly parts of North America, Portray all that new world, to be short, come by the Gulf of Mexico down to the Straight of Land called Nova Hispaniola to Peru, go over the straight of Ma●●●nica and discover all the unknown Land round about the South Pole.  
Which thing may be done by God's permission no doubt, and when thou hast done this most truly, with all the Coasts of the Seas, thou wilt say the circompherence of the Sea is also had● True, yet nevertheless ye shall never be able to lay the true semetry of both Earth and Sea on a plain superficies by this way.  
The like may you prove by the Hidrographicall description of the Sea, by the common Sea Card. And here note, that this may serve likewise to prove that the Earth and Sea make one round Body.   

CHAP. V Of certain considerations to be remembered, briefly set down.  
AS I conceive, there are these three things to be considered, without which failing cannot be performed. The first, is an exact observation of the Sun, Moon, and Stars. The second, is the perfect finding out of the variation of the Compass. Lastly, a true reckoning of the Ships way. These all do helps one another, and should be precisely had. Again, the Earth and Sea making one round Body, we may consider that to sail between 2 places, may be either by a Par●a●ll, a Rui●●, or a gr●●● Circle, as ye may perceive by the artificial Globe. Therefore if ye can draw some blank Charts to carry with you in your voyages, such whose Meridian's and Parallels bear such proportion one to the other in each Latitude, as those in the Globe, you shall do well.  
In the Globe you may perceive the degrees in the Equator to agreed and be equal with the degrees in the Meridian. Now the degrees in the Meridian's 〈…〉 where equal, but the degrees in the Parallels to the Equator are less and less, so that a degree of Longitude in the 60 deg. of Latitude is but ½ of that degree of Longitude in the Equator, and as the whole sign is to the number of parts in a degree of Longitude at the Equator, so is the fight of the compliment of any Latitude to a fourth number (by the rule of proportion of geometrical demostration) which number tells you how many parts of a degree of the Equator serveth a degree of Longitude in that Latitude, but in a plain Card where the Meridian's are all Parallels, there because the degrees of Longitude in each Latitude are equal, you must increase the degrees of the Meridian in each Latitude in such proportion as is above said, which is easy to be done with the Scale on your Ruler.  
This to look on to the eye will be strange, and to such as love to st●oke to their old errors, in the plain Card they will think it clean against reason, but let such describe some portion of the world in it, they may compare it with a Globe, and see it bear a true semetry, but in their plain Card it will appear monstrous. Here is now a view of the world from Center to Circomference, Here is an entrance to the means of obtaining to some ability in the use and practise thereof: But above all and in all things give unto the Lord the glory due unto his name, Ps. 29.2. And this indeed is the very thing wherein we aught to rejoice, and for which God made us, namely, to seek his glory, which is wonderfully seen: and for forth in the frame of the World, but most mightily in the part vns●●● but by Faith, with which eye thou must look into Heaven by viewing here on Earth God's holy Word, and harkening with the ear of Faith, to the Ministers of his Word, who do break the Bread of Life unto the faith 〈◊〉 ●●reable virgins strong meat unto men, & 〈◊〉 unto babes, so shalt thou be the better able to worship the Lord in the beauty of his holiness, Ps. 19.2.  
It appeareth, G●●. 1.14. that God made the Lights in the Firmament to divide the day from the night, and to be for signs and seasons, days and years. And u 15. to give light upon the Earth, according to the roundness of the Earth, and motion of the ☉ the light and darkness do differ, for the Sun's body being bigger than the body of the Earth, maketh that the Sun being at the Equator, although from Sun rising to Sun setting be just 12 hours, yet the light will appear from break of day to evening shut in 14 h. 24 m. and the darkness will continued 9 h. 36 m. Now if two men be one under the North Pole, another under the South Pole, they shall both see the Sun in their Horizon, if the ☉ decline towards the North, he will rise above the Horizon to him in the North, but to him in the South the ☉ setteth and when the ☉ is in the tropic of ♋ it is noon with him in the North, but with him in the South it is midnight (& contra.) By this it appeareth the time between Sun rising and Sun set to either of them is about 436. hours, on ●/● year, saving that the Lords loving kindness hath been of old, Psa 5, 6 to this Christian part of the world, in placing the Sun's Apoge in this part, by which we receive a double benefit, the one of light so much the longer by the Sun's slow motion, the other, that when the Sun is come so near our Zeneth, it hath pleased God to draw it up nearer Heaven and further from the Earth, that we be not burned with his heat, as it is written, The Sun shall not burn thee by day, nor the Moon by night, Ps. 121.6. Et 〈◊〉 in the South part of the World among the Heathen that know not God. But in the Poles the twilight lasteth so long as the Sun is within 18 deg. of the Horizon, which is about the first of May, so the dark night lasteth 1968 h. and the twilight is about 1200 h. that is 50 days with us. From these two differences in a right Sphere you may make to yourself a Theoric of the obliqne by considering the difference of light between any number of degrees from the Equator to the Poles as you please, or after this example: I did observe the height of the Sun this year 1627., and found him to be more than 62 deg. in Meridian height, proves that the Sun hath more than a ● deg. 30 min. declination, or else the Poles elevation to be less than 51 d. 30 m. here at London, which the learned have likewise proved in the●● most exact observations, but for the less learned, and the honest Seaman's use, the declination of 23 d. 30 m. will make no 〈◊〉. Now that in a Country of known Latitude you may finally the height of the Sun his Azimuth and declination in any Longitude, by which you may found the variation of your Compass at all hours of the day, or by the day of the month the Sun's Meridian height being given, is had the Latitude of the place, or height of the Pole, than the amplitude or azimuth as before, and to speak truly, these prepositions depend on one another, and what is said of the Sun, may likewise be done by the Stars being known, their Longitude, Latitude, and Declination being likewise had.  
If any man shall say, I have discovered my wants herein, he saith truly, if my malice my good will, let him know I care not, if any man will further my willing mind, to him I will sand this Boast for Patronage, and for ever pray for him, if any fleare and scoff at me, it were much better for his Soul that he were at prayers, and leave scoffing. Every good man may if please him, amend this, and so further me. If no ignorant man will profit by it, yet let such as are honestly minded, suffer this to live with me, because all that I have, or can do, is but to employ my time in such mean knowledge as I have, to God's glory, and the benefit of my Country. Wherein if knowledge and a purse did agreed with heart and good will, I would strive with the best Subject. In the mean time I will strive to do my best, as my duty doth bind me.   

CHAP. VI Of Shipping and going out of the Harbour.  
THese things afore spoken of, are most useful, and now I hold it a good method to examine your Ship whether she be for your turn or not, in all things well appointed for the Sea, if new, how well built, and strong, and how well fitted in her ge●re, if old, whether she be able to endure the surging waves of the Sea, to go another voyage, and likewise for her burden, proportioned according to the business ye use her for. But if ye wi●● build a new Ship in any proportion assigned, ye shall judge 〈◊〉 better how to have it done, by learning so much in Arithmetic, as to be able to extract the Square and Rub root, then may you do after this example next following, or otherwise at your pleasure. Let the proportion be as two to one, and suppose you have a Ship of 100 tun, in all things so well framed, that thou dost desire to have one of 200 tun, and like to the other to do this, first take the measures of your first Ship, which suppose to be these, the Keel 44 foot, at the Beam 20 foot, in Howl 9, foot, her Rake afore 13 foot, after 7 foot, here is given six numbers, and the proportion assigned, and seeing the proportion is to be doubled, therefore take each number & cube it, then double that cube number, and extract the cube root of that number, and you shall found the second Ship must have in Keel 55½ foot near, and Beam 25 and about ½, in Howl 11½, Rake afore 16●, after almost 9, and her burden will be 200 tun, or this may be found by that excellent Instrument of the memorable Mast. Edmund Gu●●●r, lately set forth by him in his life-time, and is called a Sector, his proportional Ruler and Crossestaffe is well known to be an excellent Instrument for these purposes following, and also for many other, Take in equal parts of the Sector 44, and fit it over in the Cubes at 44, the Sector so resting, take with your Compasses the distance over in 88 of the Cubes, and apply it to the equal parts, giveth 55½, as before, do so by the rest, and you shall found the like numbers.  
But say it be demanded to have her of any other proportion, namely, as 20 to 15, or 4 to 3, take the measures of the first, a before, and cube them, then say, if 15 give such a Cube, what Cube shall 20 give, it will give a number, whose Cube root in the number sought. But by the Sector take the numbers 〈◊〉 &c. in equal parts, and fit them over in 15 of the Cubes, then take the distance over in 20 of the same lines, and that applied to the equal parts, giveth the number sought.  
And now being fitted with a Ship for thy purpose and having launched, and going down the River take heed to the set of 〈◊〉 Tide, not only for running fowl of other craft, but lest ye bring yourself on ground on a Lee shore, on a Shelf, Sand, or Shoal, as Barking shelf, a Shoal against Gray's, Black shelf, the Piles below Tilberie, Milton shore, or in a dark night gape for a sho●e, and think to gain the point, but run your Ship clean out of the Thames as I see one had done, the 18 Decemb. 1627., and laid her on the March below Greene-hive in Kent. Going out or in any River whatsoever, note what mark ye see with the opening of any Point, as Yoke on a Wall, Tree, House, Hill, Wood, Wind-mill, Steeple, Castle, or Town, and learn to describe the Semetry of it, and note it in a Book, remembering to sound the depth, and note it likewise, and what grounds for ancoring good or bad, with the swiftness and indraft or outlet of the waters, ye may also describe the windings thereof by the Compass, with the lengths and breadths on a Paper, and be sure to note the magniticall Azimuth, for your variations, be it of the Son or Stars, which to be able to do, learn so much in Geometry, as to be able to describe the Sphere in plain, and to know the use of the Globes, or at lest to use your plain Scale, and keep your travice on your Card, & a journal or day's Book, and know both in River and at Sea, thou must make use of these two Globes, namely, the Scriptures where the good shine like Stars, and Christ jesus is the true Loadstone.   

CAHP. VII. Of the journal, observation, and projection.  
When you are come into the Sea, begin your journal on this wife: In the Name of God, Amen. The day's Book for the Voyage intended, for Saint Christopher's, and C●●●e de Mine. The 24 of May we came out of Portsmouth, and ●●cored in Stokes Bay. The 25 we weighed, and with a g●●e Basterly. The 26 we put through the Needles. The 27 we put in at Dartmo●th, the wind South South West, very fo●●e. And you may make a ●alender after such like order a● this.  
hear though I have set but some days of the mouth: yet I mean you should set down each day, and in the last space to note each day's variation, and which Pole is elevated in the end, say: This 31. of july, saying on an east wind by God's protection, we had sight of Matalena, and at noon our latitude found by the Sun pearing thorough the Vane of your Instrument 15. de. 3. mi. and by the North Star the next morning the like: so at noon the South East part of Matalena South, and the Meridian distance from the Lizard 1009 leagues, and diference of longitude 57 de. and dominico South end 58. de. and in your way note thus, for example in a book by itself: May the 24. from the 23. at noon, from South to East 10. de. 17. mi. latitude 23. de. 18. mi. ☉. magnitude, Azimuth 130. de. 5. mi. True Azimuth 117. de. 12. mi. the variation 12. de. 53. mi. as in chap. 4. and set down whatsoever is remarkable in your way, as well in the Sea as else for example being in the Latitude of 46. S. the body of the Isles of Babe North East, 7 leagues being to the North of the Main shoal that lieth in mid way between Mintain and Isle's Babe the West Land of Babe Iles North West ½. North 5. leagues from the South I'll of Babe towards the East by North. 20. leagues in Latitude 20. is a dry shoal that hath to the W. N. W. of it a shoal or Ledge that is 3. leagues off. Also from the dry shoal is on to the N. E. and from the dry shoal another to the E. S. E. from the dry shoal the pike of Pasamond Hill 31. de. Mag. E. to N. 42. miles, which I found by observation: for being 6. miles from shoal, my Angle of shoal and Hill 85. de. and the Angle from shoal to hill and Ship 87 de. from Hill to Shoal and Ship 8. de. by which the distance was found, and the Hill appeared thus.  
And for the soundings do thus, April 29. evening Bautum hill 17. de. 2. mi. from S. to W. 14. fathom Poloubaus and near point 26. from S. to W. 14. fathom, at the same time A. N.E. by ●. 3. leagues. 4. fathom.  
And now left any should say I have prescribed many things to others, and can do nothing myself; I will make an evident and plain demonstration of most of these things, wherein I shall make more easy the book of M. Thom●s Addison, sometime of Ratcliff▪ in his life time a good Seaman, and Mr. of the Ship called the Palsgrave, bound to the East-indieses in the year 1624. where he died before he had diuulged them: his widow not knowing what to do in the matter, and I hearing of the same, did purchase the said books at my own charge: Thinking it but my duty to quicken and raise unto life for the benefit of my Countrymen, according to the mind and intent of the Author, that which would have died, or at jest was like to have died in the shell.  
First, therefore as I have spoken of the cirkles of the material Sphere and Globe, so I would have understand them, though they be described in plano after this manner. This description is common: but I have described the parallel lines with pricks, the horizon and circle of depression with black, the almacanter with the other parallels are read, the sigments within are to show the hour lines as that which croseth the equator as it were at r●ct Angles: that which croseth the horizon: likewise to show the Azimuth and that which croseth the Ecliptic to show the Longitude and Latitude in the Heavens, and though there 3. inscribed sigments be something hard to do by Geometrical demonstration, and for so much as they are of singular use: Therefore when thou wilt find the true Azimuth, Longitude, or hour on a parallel; Take thy Sector (see chap. 6.) and open it to the distance of the Semidiameter of that parallel in the total Sine, then take the distance in the Sine given, and apply it in the parallel from the center it showeth the true place.  
Now I have given a rule before for finding the Sun's place in the Zodiac, see chap. 4. which we will for examples sake suppose to be the 30. day April 1629. which is by the Tables of Origanus 19 de. 21. m. 2. se. of ♉. and by my rule just 19 de. the point in this demonstration for that place is at B. and this point is found by the circle divided as is before spoken of, chap. 4. by taking fromward the poys of the Zodiac, the Sun's Longitude in degrees, which in this example is 49. de. having the day of the month, and the ☉. place found, the rest followeth most easily: For a parallel drawn to the equator thorough the point at B. showeth the declination at C. from A. right Ascension B. C. Amplitude A. D. Meridian height F. H. the difference Ascencionall C. D. and all the rest followeth, and are measured by degrees on the limb. By the same I may say if the declination be given, as from A. in the Center, to C. the ☉ parallel, which by Origanus for the 30. day of April 1629. ●● 17. de. 46. m●. and by this it is 17. de. 30. which is but little less; and that was by the neglect of the minute in the Sun's Longitude.  
I will give one demonstration more, and that shall suffice for this paper, for my purpose is to include all this work within these few sheets.  
This projection is of that chapter the 4. May the 4. 1629. ☉. place the 23. d. 12. m. 41. se. ♉. Origanus Tables, declination 18. d. 32. mi. N. ☉. Longi. 53. de. 12. mi. 41. se. but by this projection, Sun's place 23. de. and A. C. declination 18. de. 30. mi. Longitude 53. de. and at B. from A. A. D. the Amplitude 28. de. E. F. the Meridian height 55. de. B. C. right Ascension 50. de. C. D. difference Ascencionall 23. de. 0. mi. B. G. Oblige Ascension 27 de. o. mi. D. E. semydiurnall ark 7. ho 32. mi. D. E. semynocturnall ark 4. ho. 28. D. H. length of twilight 2. ho. 4. mi. H. E. time from noon to evening shut is 9 ho. 33. mi. that double is the length of the day. 19 ho. 6. mi. This last taken from 24. leaveth the length of the night dark 4. ho. 54. mi. Sun riseth at D. 4. ho. 28. mi. seateth at 7. ho. 32. mi. Time from rising to setting 15. ho. 4. from Sun setting, to his rising 8. ho. 56. mi. Height of the Pole N. R. 51. de. 30. mi. Though many more propositions may be wrought by this kind of projection, yet to make some good use of this mark, the last which is the height of the Pole found, so that if ye do but remember the day of the month; which you must needs do by your Calendar in the beginning of this chap. or his declination: take his amplitude Azimuth, or Meridional height, having with you your Compasses: Compass, crostafe, Sea-quadrant, Card, and some good Tables of the fixed Stars, that ye may use them for observation in the night: and by all means I would have you get the reckoning of your Ships way by that excellent way of Mr. Addisons of the 〈◊〉 glass, the Log-line, and also by what true way soever the experienced Seaman can bring to light; and then may you cast up your Traverse by that excellent way of the Arithmetical Navigation, by your Card and Traverse board, and by this Calendar, whose use is this; june the 23. being in Latitude 48. de. 14. found by observation, we failed until the 24. noon that is 24. ●●. there is the month, day, hour, in the 2 first columes upon a course of 33. de. from S. to W. 20. leagues, the wind being N. N. E. all this is in the 3. next columes: the Latitude then observed to be 47. de. 25. mi. Longitude in miles 88 from Lizard west and the variation 12. de. 53. ●●. depth 90. fathom, (if it were sound so) and these in the 2. last columes, and from july the 19 in Latitude 17. de. 40. mi. to july the 21. noon, that is 48. ho. we sailed on a course of 18. de. from W. to S. 74. leagues. The wind being at E. N E. which brought us into the Latitude of 16. de. 35. mi. and into Longitude 1834. miles: or 611 1/●. leagues W. from Lizard vad 10. de. 0. mi. Depths 60. fathom, S. W. 7. leagues a Rock, & S. by W. land thus. Note that if you get your height and course exactly, it will correct your way, height and way will correct your course: course and way will correct your height: but strive to do all as exact as possible you can, to do which, note what followeth.   

Propositions of Navigation, Arethmetically, Geometrically, and Instrumentally showed. Chap. 8.  

1. To find the leagues run on any course, the difference of Latitude and course being given.  
AS the sine of the course from the parallel is to the miles in difference of Latitude: so is the total sine to the leagues run.  
As the Tangent of the course, from the parallel is to the miles in a degree of Latitude; so is the Secant of the course to the way.  
As the total sine is to the miles in the difference of Latitude; so is the Secant of the course, from the Meridian to the miles run.  
As the sine of the course from the Meridian is to the miles in difference of Latitude; so is the Tangent of the course from Meridian to the way.  
Let the Logarithme of the course from the parallel be taken out of the Logarithme of miles in difference of Latitude, the remainder is the Logarithme of the miles in way.  
Instrumentally by the Sector, take in the equal parts the miles in difference of Latitude and set it over in the equal fine● of the course from the parallel, the Sector so opened, take the distance over in the total sign and that distance applied to the equal parts giveth the miles of way run.   

2 The difference of Latitude and Longitude given to find the leagues run.  
SQuare both sides, then add the 2. squares together and extract the square root is the miles in way run.  
Look which is the greater side, either the Lat. or the Lo●, 
This also giveth the Rombe.  and take his Log. out of the Log. of the lesser side, the remainder giveth the two acute Angles, then let the Log. of either of them be taken out of the Log of the side opposite: the remainder is the Log. of way I show not how it giveth the Angles but leave thee to inquire.  
Open the Sector to a rectangle: then take the miles in Longitude and set it on one side, the Sector from the Center in equal parts, and the miles in Latitude on the other side: then take the distance over with a pair of Compasses, and apply that distance to the equal parts, giveth the number of miles in way.   

3 To find the difference of Longitude, the course and difference of Latitude being given.  
AS the sum of the course from the parallel is to the difference in Latitude, so is the fine of the course from the Meridian to the difference in Longitude.  
As the total sine is to the difference in Latitude: so is the Tangent of the course from the Meridean to the Longitude.  
As the Tangent of the course, from the parallel to the miles in Latitude: so is the total sine to the miles in Longitude.  
As the Secant of the course from the parallel is to the difference of Latitude: so is the Secant of the course from the Meridian to the Longitude sought.  
Take the Logarithme difference of the Angle from the Meridian out of the Logaritme of the difference of Latitude in miles the remainder is the logaritme of Longitude in miles.  
Take the number of equal parts (in the Sector) which are the difference of Latitude, & fit that distance over in the lives of sins of course from the patrid all, and the instrument opened to that wideness: the distance over in the sine of the course from the Meridian: that distance applied to the equal parts, showeth the Longitude in miles.   

4 To find how many miles of the equator or Meridian is a degree of Longitude in any parallel of Latitude.  
AS the total sine is to the miles in a degree of the equator: so is the sine of the compliment of Latitude to the miles sought.  
As the Tangent of the Latitude is to a degree in the Meridian: so is the line of the Latitude to the miles sought.  
As the Secant of the L●●●tude is to the miles in the equator or Meridian: so is the total sine to the miles sought  
As the Secant of the compliment of Latitude is to the miles of Meridian: so is the Tangent of the compliment of Latitude to the miles, answering a degree.  
Let the logarithme of the compliment of Latitude be added to the logarithme of miles in a meridians degree; the total is the logarithme of miles sought.  
Instrumentally, take the number of equal parts in a degree of the Meridian or equator by the Sector, & fit them over in the sins total: then take the distance over in the sins of the compliment of Latitude, and apply that to the equal parts, showeth the miles that make a degree of Longitude in that parallel.   

5 To find the Longitude answering to a Meridian distance in any parallel of Latitude.  
AS the sine of the compliment of Latitude is to the Meridian distance: so is the total sine to the Longitude.  
As the sine of the Latitude is to the Meridian distance: so is the Tangent of the Latitude to the Longitude.  
As the total sine to the Meridian distance: so is the Secant of Latitude to the Longitude.  
As the Tangent of the compliment to the Meridian distance: so is the Secant of the compliment to the Longitude.  
Let the Logarithme of the compliment be taken out of the Logarithme of Meridian distance, the remainder is Logarithme of Longitude.  
Take from the scale of equal parts in the Sector the Meridian distance; and fit with your Compasses in the sins of the Latitudes compliment: then the distance taken between 90. and 90. applied to the same scale, giveth the Longitude.   

6 To find the sign of any Ark in any parallel.  
AS the total sine is to the compliment: so is the sine given in the great Circle to the sine in the Latitude.  
As the Tangent of Latitude to the sine of Latitude: so is the sine of the Ark in the great Diameter to the sine of the Ark in the lesser.  
As the Secant of Latitude to the total sine: so is the sine in the greater, to the sine in the lesser Semi diametor.  
As the Secant of the compliment is to the Tangent thereof: so is the sine in the greater to the sine in the lesser. 
This reduceth Longitud miles into degr:   
Ad to the Logarithme of the Ark given, the Logarithme of the compliment of Latitude, the total is the Logarithme of sine demanded.  
By the Sector let the distance from the Center to the Ark giuen be put over in 90. and 90. then the distance between the sins of the compliment of Latitude, applied from the Center, giveth the Ark whose sine is the demanded.   

7 By a sine given in a parallel, to find the Ark.  
AS the sine of the compliment of the Latitude is to the sine given▪ so is the total sine to the sine of the Ark.  
As the sine of the parallel is to the sine given: so is the Tangent of the parallel to the sine of the Ark sought.  
As the total sine is to the sine given; so is the Secant of parallel to the sine of the Ark sought.  
As the Tangent of the compliment is to the sine given: so is the Tangent thereof to the sine of the Ark sought.  
Out of the Logarithme of the sine given, ●●ke the Logarithme of the compliment of L●●i: the rest is the Logarithme of the sine.  
By Instrument let the distance of the sign giuen be taken from the Center and fitted in the sine of the compliment of Latitude; 
It reduceth Meridian degrees in to Longitude.  then the distance between 90. and 90. being set from the Center showeth the Ark.   

8 To find how many miles on Earth or Sea serveth to a minute of time in the Heaven, in any parallel of Latitude.  
AS the total sine is to 15. mile, which serveth a minute of time under the equator: so is the sine of the compliment of Latitude to the number sought.  
As the Tangent of the Lati●●●e is to the miles in the equator: so is the sine of the Latitude to the miles sought.  
As the Secant of the Latitude is to the miles in a minute at the equator: so is the total sine to the miles in a minute of that parallel.  
As the Secant of the compliment of Latitude, is to the miles in the equator: so is the Tangent thereof to the miles in a minute of that parallel.  
Add the Logarithme of the 15. miles, to the Logarithme of the compliment of Latitude, the sum is the Logarithme of the miles in that parallel.  
By the Instrument or Sector, take 15. the miles that serve in the equator to a minute of time out of the equal parts, and fit it over in the total sins: then the distance over in the sine of the compliment of Latitude applied to the equal parts, showeth the miles to a minute in that parallel.  
This chapter ha● shown how to find the leagues run in any course, or the Ships way, and to find the difference of Longitude, with things thereto pertaining, the next sheweh.    

Chapter the 9  

9 To find the course the way and difference of Latitude given.  
AS the miles in way (or run) is to the total sine: so is the miles in difference of Latitude to the course (or run) his fine from  〈…〉 
As the miles in difference of latit to the total sine, so the miles in way to the seca. of course from meridian.  
As the difference of latitude to the way, so the total sine to the secant of course from meridian.  
Square both difference of latitude and way, then subtract the lesser out of the greater, out of the remainder, 
This gives the longitude.  extract the square root, then as that root to the total sine, so the difference of latitude to tangent of course from parallel.  
Let the logarithme of way be taken out of the logarithme of difference of latitude, the remainder is logarithme of course from parallel.  
Take the equal parts in way, and fit over in the total sine, then take the difference of latitude, and fit over in equal sins, gives the course from parallel.   

10. To found the course, the way, and longitude given.  
AS the way to the total sine, so the longitude to course from the meridian.  
As the longitude to the total sine, so the way to the secant of course from parallel.  
As the longitude to the way, so the total sine to the secant of course from parallel.  
Square both longitude and way, then subtract the lesser out of the greater, out of the remainder extract the square root, 
This gives the latitude.  then as that root to the total sine, so the longitude to the tangent of course from the meridian.  
Let the logarithme of way be taken out of the logarithme of difference of longitude, the remainder is logarithme of course from the meridian.  
Let the equal parts in way be fitted over in the total sine, than the long●fitted over in equal sins, gives the course from the meridian.   

11. To found the course, the longitude and latitude gruen.  
AS the ½ of the 2 sides added, is to the difference of each side, so the tangent of ½ the 2 unknown angles, to the tangent of an ark, which being added to, or taken from ½ the 2 unknown angles, gives the course.  
Take the logarithme of the greater side, out of the logarithme of the lesser, the remainder is logarithme of the angle opposite to the lesser side.  
Open the sector to a rectangle, and take the longitude and latitude, and fit them on 2 sides from the centre; and from those 2 extensions take the distance with a pair of compasses, and fit it over in the total sine; Than take the longitude, and fit over in equal sins, gives the course from the meridian, but the latitude so fitted gives the course from parallel.   

12. To found the latitude, the course and way given.  
AS the total sine to the way, so the sine of course from parallel to the latitude sought.  
As the secant of course from parallel to the way, so the tangent of the same to the difference of latitude.  
As the secant of course from the meridian, to the way, so the total sine to the difference of latitude.  
As the tangent of course from the meridian, to the way, so the sine of the same course to the difference of latitude.  
Add the logarithme of way to the logarithme of course from parallel, the total is the logarithme of difference of latitude.  
Take the miles in way, and fit over in the total sine, than the distance over in the sine of course from parallel, gives the miles in difference of latitude.  
I have showed already, chap. 7. how the distance from Paramont hill was found; And as I conceive, it is easy to find the distance of places, if you do but know the use of my Table of sins, tangents and secants, proviso, note that the 3 angles of any rightlined triangle is equal to 2 rectangles or 180 degrees.  
For commonly at sea, you must observe the land, shore, or whatsoever it is you would know the distance from, what angle it maketh with your course; as admit 40 degrees, then reckon your ships way, suppose 1000 fathom, and then observe again, let your second angle be 100 degrees, then add 100 deg: & 40 degr: which subtract from 180 degr: remainder 40 degr: for the third angle; then have you 3 angles and one side given, by which means the other 2 sides are easily had, and consequently the distance.  
Now if the young beginner be troubled to find the sine of an angle, so big as 100 degr: let him note what the Table saith; subtract 90 degr: from 100 degr: the remainder 10 degr: subtract from 90 degr: rests So degr: or subtract 100 degr: from 180 degr: rests 80 degr: whose sine in my table is 985; the other two angles being 40 degr: and 40 degr: their sins are 643 and 643, and the distance 1000 fathom: Now to fetch the distance from shore say, as 643 the sine of 40 degr: at c, found as before, to 1000 fathom, the distance from a to b, the a places of observation, so 984 the sine of 80 degr: in the lesser quadrant, or 100 degr: in the great, (which is all one) to the distance from a to c 1531 ●6●/61● fathom. Another example. 996 sine of 84 degr: 50. 985 sine of 80 de. 262 sine of 15 d: 10. Let your first observation be at b 80 degr: the second at d 84 degr: 50, and by your reckoning the distance 254 fathom; now it is easy by the former rule to found the angle at c 15 degr: 10: then say if 262 give 254, what gives 985, facit c d 954 110/●●●: And as 262 to 254, so 996 to the distance b c. So that it is all one whether the angles be right, or obtuce, or acute, if they be all acute, there is no more to do but take the sins out of my table as in the figure KING, and no more, if one be a rectangle, than the other 2 are acute, but if an angle be obtuce, as in the figure L, the angle at b 100 degr: do by that angle whatsoever it be, as in the first example; here because the angles at a and c are equal, therefore the distance from b to c is 1000 fathom.  
But by the second example, it is but 965 77/●●●, therefore the angle at d was not truly taken; for what 965 77/●●● is in proportion to 1000, so is the angle at d to the true angle: which must needs be more than 84 degr: 50, and the angle at c less than 15 degr: 10 and consequently the side c d is more than 954 120/●●●. And further you may note, if you have observed so well that the angles be truly taken, then have you not reckoned your ship's way truly; and so you may see that one of these doth correct the other, as in the end of the 7 chapter.  
And you may fetch a distance by Geometrical protraction, as in chap. 4. suppose you would know the distance from c and b to f, in the figure m, first having drawn the angle b e f, then reckon your ships way from ● to b ●00 fathom, and having laid down so much by your scale and compasses, (see pag. 12) you shall observe to lay the line ● b right in the course your ship went, with b next your eye, and e towards the place of your first observation; then from b draw another line to f, and that line will intersect with the line a f at f, and then you may measure the distance; as in this example b f is 100 fathom, and e f is 140 fathom.  
The next thing that will trouble the young beginner, is the propositions in these 8 and 9 chapters, which are most easily performed, because the demonstration is so plain.  
First therefore as in all plain triangles, the sides are in proportion one to another, as the subtences of the angles opposite, or as the sins of the angles opposite to those sides. Example in the inscribed triangle: A the angle at b is in proportion to the side c d or ark c g d, as the angle c to the side b d, or ark b d e, and so is the angle d to the side b c, or ark b f c.  
So note that in all rectangled rightlined triangles, the two containing sides of the rectangle stand perpendicular one upon the end of another: Therefore if you put the lest side for radius or total sine, as in the figure b, the other side will be the tangent, and the subtending side of the rectangle will be the secant; or if you put the greater side for radius, the lest will be the tangent, and the subtence the secant, or you may put the subtence for radius, as in the figure c is done.  
It is manifest then that any side may be put for the total sine: First, if you put the subtending side for radius, as in the figure d, let a b be part of a meridian, and a c part of a parallel; then is the angle at b the course from the meridian, and the angle at c is the course from parallel.  
Than if the difference of latitude a b, and course, that is the angle at b or c be given, you may find the leagues run, or distance b c, as in the first resolution Arithmetical.  
Or if you put the longer side a b for radius, then say as the total fine a b to the difference of latitude a b, so the secant b c of course, b from the meridian, to the distance b c; if you put the lest side for radius, say as the tangent a b of course c from parallel, to a b the difference of latitude, so b c the secant of the same course, to b c the distance. Let this suffice for the first proposition chap. 8.  
But if you demand the difference of longitude, the latitude and course being given, you may do it by the first or second example Arithmetical in this chapter. page 29; or if you put the longer side b c for radius, say, as the course c to a b, so the course b to a c the longitude: but if you put the side a b for radius, say, as the total sine to the difference of latitude, so the tangent a c to the difference of longitude: if the lest side be put for radius, say, as the tangent a b to the difference of latitude, so the total sine to the difference of longitude. And this sufficeth for the Geometrical demonstrotion of the third proposition, chap. 8.  
And now you see how the proportional numbers do arise, it is easy to take them out of the Table, and work them Arithmetically; and what is said of these propositions may be said of all the rest by the very same reason; except the second proposition chap. 8, and the 11 proposition in this chapter page 27: the first of these is done as in Earlid 1.47. or thus Arithmetically.  
Let the line a be 3 leagues of longitude and the line b 4 leagues of latitude, first multiply 3 〈◊〉 3, it is 9, and 4 into 4 is 16, which added is 25, the square root is 5, for 5 times 5 is 25, and this is equal to the line c, the ships way.  
The Geometrical demonstration is this, set the line a perpendicular upon the line b, as in the figure f, then draw the line c, and it is 5 the ships way.  
There is another way of demonstration, by adding the sides of the squares into one; also if the way and longitude had been given, to subtract the one out of the other Geometrically, the remainder is the difference of latitude: if the way and latitude had been given, than the side of the remainder is the difference of longitude, Euclid 1.47. theor. 33. 6.31. theor. 21.  
The next should be of the 11 proposition, but I suppose he that understands that is more than an ignorant man, therefore whosoever will learn that, with the extraction of the square and cube root, or any thing else that is in my book, or any of those things that I have set out in any of my bills; let them come to my dwelling, and I will satisfy their demand so fare forth as God shall enable me: where youth may be taught, dieted and lodged for reasonable consideration.    

CHAP. X. Of the motion of the Moon, and coming into the Harbour.  
I Might here have added many and manifold propositions, and chief those which do concern the great circle distance between any two places howsoever situate; but I will first see thy kind acceptance of this: which if you receive thankfully, I shall be ready to show thee many more to thy great satisfaction, with the Geometrical demonstration, which I have gathered into a book in Folio.  
And now I will finish this work with the Moon's motion; supposing you are come in sight of your wished port in safety: which I do hearty wish to all honest Seamen of my Country, that travel the seas to good ends.  
And admit you are in a strange coast, you may the better come to find the time of a full sea, at a spring tide, to bring your good ship over a bar or shoal: so I shall end my work where Mr. Addison began his: then having read this, you may read his which is hereto joined; giving you to understand, if you learn well these propositions of mine, they will make even the hardest things in his book easy for thy understanding.  
Let the full sea at your place of being be at a SAINT Moon on the day of conjunction as at a, and the Sun going in 4 degrees, as from a to c, the Moon removing in the mean time 48 degrees from the Sun, as to d, doth make 3 hours 12 minutes after noon for the time of high water; and in 16 days more the Sun going from c to e, that is ●0 degrees from a: in the mean time the Moon going from d to f, that is 260 degrees from a, which maketh 17 hours 20 minutes, then subtract 12 hours 0 minutes from 17 hours 20 minutes, the remainder 5 hours 20 minutes; out of which take 1 hour 20 minutes for the time of the Sun's motion in 20 days, remainder 4 a clock for the true time of full sea the Moon being 20 days old; if you perceive not the reason hereof inquire further, and so you may examine the truth hereof.  
Therefore to the ignorant and honest Seaman I say, join these things in practice with those things thy Master doth teach thee at sea: And to those things you learn at sea by experience join the practice of the things taught in this book, and you shall be able in short time, through God's grace, no doubt, to take charge thyself to thy great commendations.  
To conclude, you may do most of the things taught in this book as afore spoken of chapter the 4, with ruler and compasses; which if you practise and make yourself able to perform, (wishing to every good man that is willing to follow these rules, that I were at his elbow to make him a more plain demonstration) he shall be able in a short time, to do such service, which without those will not be performed in ten times so much time.  
And so I pray thee to accept of this my labour, for unto such as are ignorant and honest, that would learn, only do I writ this book: And be sure to read it all before you judge, so shall you see that such a thing there is, then read it over again, and you shall see what manne● of thing it is, so you may judge of the good will of the Author, and read it over the third time to thy singular profit.  
As in Ecclesiasticus chapter the 22. verse ●3, it i● 〈◊〉 neighbour, so I say to thee of thy friend to Navigation; 〈◊〉  
Be faithful to him in his poverty, that you may rejoice with him in his prosperity, abide steadfast with him in the time of his trouble, that you may be heir with him in (this his little book) his heritage: for a mean estate is not always to be contemned, nor the rich that is foolish to be had in admiration.  
Now I beseech Almighty God of his mercy in Christ jesus, that we may so profit in Christ's school, that we may be able thereby to pass the waves of this sea of glass, that we may all arrive at the haven of eternal happiness. Amen.   
FINIS.