THE GEOMETRICAL SEAMAN: OR, THE ART OF NAVIGATION Performed by GEOMETRY. SHOWING How all the three kinds of Sailing, viz. By the Plain Chart, By Mercators' Chart, By a Great Circle. May be easily and exactly performed by a Plain Ruler and a pair of Compasses, without Arithmetical Calculation. By HENRY PHILLIPPES. LONDON: Printed by ROBERT and WILLIAM LEYBOURN, for GEORGE HURLOCK, and are to be sold at his Shop at Magnus' Church-corner, 1652. TO The INGENIOUS, INDUSTRIOUS, and Younger SEAMEN. AS it is chiefly for your profit that I writ; so I hope, that though I shall be condemned of others, yet I shall gain your pardon. I confess my labour may seem needless, after so many learned Authots, to set forth any thing of this subject. And it may be accounted a presumptuous folly for any to go about to teach Seamen in their Art, especially in these times, wherein there were never more, nor more skilful Seamen. But in answer hereto take notice of these considerations. First, That many Books of this subject, though they are plain and easy for all Seamen to understand, yet they are not so exact and perfect in all things as they ought to be. Secondly; That though Mr. Wright, Mr. Gunter, and Mr. Norwood, have very fully shown the errors of the foresaid Books, and have very perfectly reform them; yet their Books are not plain and useful for all men, because they require much knowledge in Arithmetic. Thirdly, Though there be many skilful Seamen, which are able to make use of these Authors; yet there are many others, who would willingly attain to a more perfect knowledge in this Art, and yet cannot, for want of Arithmetic. Now for the help of such as these especially, have I published this Book: wherein you shall find how all that which is to be performed by Arithmetic in the foresaid Authors, may be sufficiently exactly performed by Geometry with the Ruler and Compasses; so that any one that can but read and write a little, though he have no skill in Arithmetic, may hereby attain a true and perfect knowledge of the Art of Navigation, so that he shall be able to keep as true and certain an account of his Voyage in any part of the World, as one that works by the most exact rules of Arithmetic. Nay, I dare say, that one that can neither write nor read, yet being of an ingenious capacity; and having one to teach him according to this way, may attain sufficient skill in this Art. Fourthly, This may also be useful to those who can perform these things by Arithmetic, and the Doctrine of plain Triangles: For by this way they may perform them far more easily and readily, and almost as exactly. And though such men may know many of these things already, yet perhaps they may gain some knowledge hereby. Lastly, As by this way the knowledge of the Art is most easily and speedily attained, so it is more certainly and constantly retained and preserved, both in respect of the knowledge and practice thereof. First, the knowledge hereof is more certainly retained in your memory: for this is the reason why all Authors are forced to make use of some Geometrical figures; partly to explain their Rules, and partly to fix them in the memory, by making them visible to the eye; for we more surely remember any thing which we see done, than what we only hear. Now in this way, every thing is made much more visible to the eye, and therefore more easy both to learn and remember. Secondly, the practice of this knowledge will be more surely and constantly preserved: for Arithmetical calculations require many Tables, viz. of Sines, Tangents, Secants, Meridian, which cannot possibly be kept in memory; so that if by any cross accident a Seaman be deprived of his Books, he can make no use of his skill in Arithmetic. But if a man knows how to perform these things by Geometry, though all his Books and Writings be lost, yet having but a plain Ruler and a pair of Compasses, he may quickly recover his loss, and fall to his work as before. Upon these considerations, I have adventured to publish this little Book, wherein I have briefly and plainly laid down the whole Art (as to the Geometrical part thereof) beginning at the first principles, and proceeding by degrees to the highest conclusions. I have used the more figures to make every thing the plainer. And I have provided such figures, as serve not only for demonstration of the thing, but may serve for Instruments to work upon; or you may easily, by the directions given, make the like. To conclude, I have studied to make those things plainest, which have at any time most troubled myself to understand. So that I question not but that any one that is industrious, may here of easily and speedily attain, such a competent measure by knowledge in this Art, that by God's blessing upon his study and labour therein, he may obtain much credit and profit. Which is all the desire, of Your well meaning Friend Henry Phillippes. THE CONTENTS AND order of the whole BOOK. IN the two first Chapters is showed how to perform some Geometrical Principles; which are necessary to be known, because most of the following work is done thereby. The third Chapter shows the making and use of the plain Chart, being the common way of sailing. The fourth Chapter shows the making and use of the true Sea-Chart, being in many respects more perfect than the former. The fifth Chapter shows the way of sailing by a great Circle; being the most exact and best way of sailing that can be. The sixth Chapter shows many useful observations in all these kinds of sailing. The seventh Chapter shows another way of sailing by the arch of a great Circle. he last Chapter shows how to keep a perfect account of any Voyage by a little common Arithmetic, viz. Addition and Substraction. The names of such Books as are printed and sold by George Hurlock, at Magnus' Church corner. THe Seaman's Calendar. The enemy of Idleness, teaching how to indite Epistles & Letters in 4 Books, 83. Normans Art of Ten, or decimal Arithmetic, 4o. The Art of Navigation by Martin Curtis. Safeguard of Sailors, or great Rutter, by Robert Norman. A Table of Gauging all manner of Vessels, by John Goodwin, 8ᵒ. Path way to perfect sailing, by Richard Polter, 4ᵒ. Pitiscus his Doctrine of Triangles, with a Canon of natural Sins, Tangents, and Secants. Norwood's Doctrine of Triangles, with Logarithmes Norwood's Epitome, applied to plain and Mercators sailing. Norwoods' seaman's practice. The Navigator, by Captain Charles Saltonstall, 4ᵒ. Dary's description and use of a Universal Quadrant, 8o. Errata. Page 16. A is wanting in the figure. page 33 line last, read A F, page 56 line 25. for 28 read 58, page 62, 67, 69. some words doubled may b● left out. page 70, line last, read, thereby know whether, etc. page 80. last line of the Table, for 22 deg. read 33. page 88 in the Table, Col. 3. line 3. for 1, 80, read 1,00. Col. last line 3. for 0, 8, read. 0, 58. THE GEOMETRICAL SEAMAN. OR THE Art of Navigation performed by GEOMETRY. CHAP. I. Containing some Geometrical Propositions, which will be of frequent use. PROPOSITION 1. How to erect a Perpendicular line at the end of a line. The first Proposition What a perpendicular line is. A Perpendicular line, is a line that stands directly upright from another line. As in the figure, the line B C is a perpendicular line, to the line A B. Now in the Figure there are two ways of raising it, the one on the right side, and the other on the left. First, Let the line A B be given: and it is required to erect the line B C perpendicularly to it, at the end of the line in the point B. geometrical diagram Another way. The second way to perform this, is demonstrated on the left side of the figure. Let the line A B be given as before, and it is required to erect the line A D perpendicularly in the point A. To perform this, first, set one foot of your Compasses in any convenient point at pleasure, as at L, and open the other foot to the point A, and draw the arch N A M, then lay your ruler to the centre of this arch L, and the place where it crosseth the line A B, which is at N, and draw the line M L N, which doth cross the arch N A M in the point M. Lastly, laying your Ruler to this cross at M, and the point A, draw the line D M A, so you shall have your desire. PROPOSITION 2. To draw one line parallel to another line, at any distance required. The second Proposition. What is meant by a parallel line. A Line is said to be parallel to another line, when it is equally distant from it in every part thereof. Thus in the former figure the line P S is parallel to the line A B. Now the way to draw a parallel line is thus. Let the line A B be the first line given, and it is required to draw the line P S parallel to it, according to the distance P A. First, open your Compasses to the distance you have occasion to use, which in this example is P A, then setting one foot in A, with the other draw the arch at P, and then keeping your Compasses at the same distance, remove one foot to B, and with the other draw the arch at S: lastly, laying your ruler on the very edge of these two arches, draw the line P S, which will be parallel to A B, and so the proposition is performed. PROPOSITION 3. How to make a Geometrical Square. The third Proposition. A Geometrical square, is a square whose four sides are all of one and the same length. Now in the first figure, let the line A B be the side of such a Square, and it is required to make a Square of that length and breadth. First, How to make a Square. you must draw the line A B according to the length given, then erect the perpendicular line A D at one end thereof, as was showed before, then setting one foot of your compasses in the corner A, open the other to B, and keeping one foot still in A, with the other cross the perpendicular A D at D: then keeping your compasses at the same distance, set one foot in B, and with the other draw a short arch at C, then set one foot at the cross at D, and with the other cross the arch last drawn in the point C: now if you draw lines through these marks, from A to D, from D to C, and from C to B, so you shall make the Geometrical Square A B C D as was required. If you will try your work whether you have made it true or no, then set one foot of your compasses in A, How to try a Square. and open the other to the corner at C, then with that distance, set one foot in B, and turn the other to the corner at D, if both these opposite corners have the same distance, the Square is truly made, otherwise not. A long Square If you would make a long square, as the square A P S B, first, you may draw the line P S parallel to A B, and according to the length of the side of your square, then erect the perpendicular either at A or B, and draw the opposite side, parallel thereunto, according as the length of your square requires: and you may try the truth of this square, also by the opposite corners as before. PROPOSITION 4. To raise a perpendicular in the midst of a line. The fourth. Proposition. IN this second figure, let A B be the line given; and it is required to raise a perpendicular in the point C. geometrical diagram The second figure. First, set one foot of your compasses in the point C, and open the other to any distance at pleasure, and mark the given line therewith on both sides from C, at the points A and B, then setting one foot of your compasses in the point A, open the other to any distance you please beyond C, and draw a little arch above the line at F. Then with the same distance, set one foot in B, and with the other cross, the arch F with the arch D. Lastly, lay your ruler to this cross, and the point C, and draw the line G C, which is perpendicular to the line A B, in the point C as was required. PROPOSITION 5. From a point aloft, to let fall a perpendicular upon a line given. The fifth Proposition. LEt G be the point aloft, from whence it is required to let fall a perpendicular upon the line A B in the second figure. First, set one foot of your compasses in the point given, which is G, and open your compasses so wide that you may draw the arch A H B, which may cut the line A B in the points A and B, and the farther these two points are asunder so much the better, then keeping your compasses at this distance; set one foot in A, and with the other draw the arch E, then remove one foot to B, and cross the last arch at E, lastly, laying your ruler to the point G, and this cross at E, draw the line G C E, so you have performed the proposition. PROPOSITION 6. To draw a line squire wise to another line. The sixth Proposition. IN the second figure let A B be the line given, and it is required to draw the line G E squire wise to it, so that it may cross it at right angles. First, open your compasses at pleasure, and setting one foot in the line at B, with the other make two short arches, one above the line at D, and the other below the line at E. Then with the same distance, set one foot in A, and with the other cross the two former arches at D and E. Lastly, laying your ruler by these two crosses D and E, draw the line G E, which will cross the line A B at right angles as was required. PROPOSITION 7. To divide a line given into two equal parts. The seventh Proposition. IN the second figure let A B be the line given, to be divided into two equal parts. First, set one foot of your compasses at the one end of the line at A, and open the other to any distance above half the line, & therewith draw two little arches one above the line at F, and the other below the line at E, then remove your Compasses to B the other end of the line, and cross the two former arches at F and E, then lay your ruler to these two crosses F and E, and draw the line GC E, which will divide the line A B in two equal parts in the point C, so that A C is the one half, and C B the other. PROPOSITION 8. To raise a Perpendicular at the end of a line another way. The eighth Proposition. IN this figure, let the line given be A B, and it is required to raise a perpendicular at the end thereof at B. geometrical diagram Here you may note that if the three sides of a Triangle be made of these three numbers 3 4 5, or any other numbers that are proportionable thereunto, as 6 8 10, 9 12 15, 12 16 20, 30 40 50, it will have one right angle, which will be opposite to the greatest side, as in the Triangle D B E, the side E B is 3, the side B D is 4, and the side D E is 5, and the angle at B is a right angle. PROPOSITION 9 To make one angle equal, or like to another. The ninth Proposition. AN angle is the joining or crossing of two lines: What an angle is, with the general kinds of angles. if the two lines cross one another, or join one to another perpendicularly, than they are said to make a right angle, or angles: if two lines meet or cross one another any other way, they are said to make an obliqne angle or angles. Thus in the third figure, the lines D B and E B meeting in the point B, make a right angle. And in the second figure, the lines A B and G E, crossing one another in the point C, make four right angles, or quadrants. But in the third figure, the lines E D and B D, meeting in the point D, are said to make an obliqne angle. Now these obliqne angles, if they be less than a right angle, they are called acute or sharp angles: if they be more than a right angle, they are called obtuse or blunt angles. Now for example of the proposition, let the angle E D B be the appointed angle, and it is required to make the angle D B C like unto it. In this example, because the line D B is limited, and is common to both the angles, you shall need only to set one foot of your compasses in B, and open the other to the nearest distance of the line D E, which you may do by drawing the little arch which toucheth the line between 3 and 4: then remove your compasses to D, and draw the like arch at C, then lay your Ruler to the point B, and the very edge of this arch C, and draw the line B C, so shall the two angles be of one quantity or wideness, as was desired. In other cases this way will not serve, but this is sufficient for the present purpose, and I shall show you other ways to perform that in the next Chapter. PROPOSITION 10. To divide a line into any number of equal parts. The tenth Proposition IN the third figure, let the line B D be given, to be divided into four equal parts. First, from the end D, draw a line as D E, making an angle with the line D B at pleasure, then from the other end of the line B make the angle D B C equal to the former angle, as was showed in the last Proposition. Then from the point D set off with your compasses, such a number of any equal parts, as lacks one of the number desired, which in this example, therefore must be 3, set off therefore on the line D E three equal parts 1, 2, and 3; than you must with the same distance of your compasses set off 1, 2, and 3, from the point B, on the line B C, then draw cross lines from the last number in the one line, to the first in the other, that is from 3 to 1, from 2 to 2, etc. and these lines will divide the line B D into four equal parts as was desired. PROPOSITION 11. To bring any three points, not lying in a strait line, into a Circle. The eleventh Proposition IN this figure, let A B C be the three points given, and it is required to draw a circle through them all. geometrical diagram The fourth figure. Set one foot of your compasses in the middle point at B, and open your compasses to any distance you please, so it be above half the distance, between B, and either of the other marks (yea, it is no matter if need be, though it reach almost to, or quite beyond the nearest of the other marks) and draw the arch D E F G, then keeping your compasses at this distance, set one foot in A, and with the other draw the arch G F, which crosseth the former arch at G and F, then set one foot of your compasses in the third point C, and with the other draw the arch E D, which crosseth the first arch at E and D, then laying your ruler to the intersections of these arches; draw the lines G F H and D E H, which will cross one another in the point H, this cross at H, is the centre of the Circle: therefore setting one foot of your compass in this cross at H, open the other to any of the three points A B or C, and draw the circle; which if you have done well, will pass through all the three points A B C as was required. CHAP. II. Showing how to divide a Circle several ways which will be needful for many things. THe first usual division of a Circle, is into 24 equal parts, according to the 24 hours of a natural day; which is thus to be performed. diagram of compass points Secondly, another usual and necessary division of a circle is to divide a circle into 360 equal parts. To divide a circle into 360 degrees. For in all question of Astronomy, and in the calculation of all triangles, these parts are the measure of the angles; so that every arch in this respect is supposed to be divided into 360 equal parts, which are called degrees, and each degree is supposed to be divided into 60 lesser parts called minutes. To divide a circle after this manner, the ready way is thus. First, draw a line at pleasure, and cross it at right angles with another line, and draw a circle as before, then keeping your compasses at this distance, divide the circle from the four quarters into 12 equal parts as before, then closing your compasses divide each of these 12 parts into 3, so you shall have in all 36 parts, than you may easily with your pen divide each of these parts into 10 little parts, each of which stands for a degree, and so you may number them as in the middle circle of the figure. A third usual division of a circle is into 32 equal parts To divide a circle into 32 parts. according to the number of the points of the compass, which may be thus performed. First, draw the line of East and West, and cross it at right angles with the line of North and South, and draw the circle as before, then keeping your compasses at that distance, set one foot where the line of East doth cross the circle, and with the other draw two little arches one above at B, and the other below at D: then with the same distance of your compasses set one foot where the line of west doth cross the circle, and draw two little arches like the former at A and C: then with the same distance of your compasses, set one foot where the line of North doth cross the circle, and with the other, cross the two upper arches at A and B: then set one foot where the line of South doth cross the circle, and with the other cross the two lower arches at C and D; then laying your ruler cross-ways to these crosses, draw the lines A D and B C: so the circle shall be divided into eight equal parts; then closing your compasses, you may easily divide each of these 8 parts into 4, (for having divided one of them they will all fall out alike) and so you shall have the 32 rumbes or points of the compass, which you may subdivide if you please into halves and quarters, and draw the lines, and by three or four letters express their names as in the figure, which signify as followeth. The names of the 32 points of the Compass. North North by West North-North-west Northwest by North Northwest Northwest by West West-North-west West by North West West by South West-South-west South-west by West South-west South-west by South South-South-west South by West South South by East South-South-east Southeast by South Southeast Southeast by East East-South-east East by South East East by North East-North-east North-east by East North-east North-east by North North-North-east North by East To make an angle of any quantity. Having thus divided a circle into these three sorts of parts, it will be very useful to you in the dividing of any other circle, quadrant, or arch, and by this circle you may easily draw any angle of what quantity you please. For example, let A B be a line given, and it is required to draw another line from the point A, so that it may make an angle of 45 degrees. geometrical diagram In like manner supposing the line A B to be the Meridian or South line, and it is required to draw a line from the point A, Another Example. which shall represent the Southeast or the fourth Rumbe from the Meridian. First, set one foot of your compasses in the centre of your divided circle, and extend the other to that circle which is divided into Rumbes; and with that distance draw the arch B C. Then setting one foot of your Compasses in that point where the South line and the circle cross each other, open the other to the line of Southeast, and then set off that distance from B to C in this last figure, then draw the line A C which will represent the Southeast as was desired. You may do this also by a Scale of degrees, and Rumbes, To make a Scale of Chords and Rumbs. which you may have upon a strait line on your ruler, which you may thus make. First, set one foot of your compasses in the centre of the divided circle, and open the other to that circle, which parts the divisions of the degrees and rumbes, and set off this distance on a strait line upon your ruler, and mark very well with some special mark, where this distance gins and ends, for this is your Radius or distance, which you must always take to draw your first arch withal, it being the sixth part of a circle, or 60 degrees. Then setting one foot of your compasses where the circle, which is divided into degrees and rumbes doth cross the line of North or South, open the other to 10 degrees in that circle, and then transfer that distance into your Scale, than again, take out the distance of 20 degrees out of the circle, and transfer that likewise into your Scale, and so do for 30, 40, 50, 70, 80, 90 degrees. Always setting one foot in the place where the line of North or South doth cross the circle, and opening the other to the degree desired. And in like manner when you transfer these distances into your ruler, you must always set one foot of your compasses at the beginning of the line, and with the other mark the distance in the line. And thus also you may take out the distances of the Rumbes, and set them upon a line on your ruler, and so having made your Scale, you may draw out any angle by it, as well as by the circle, and it will be somewhat more ready. Example. Now if you would draw the foresaid angle of 45 degrees by this Scale, you must first set one foot of your compasses in the beginning of your Scale, and open the other to 60 degrees, which is the Radius of your Scale, and therewith draw the first arch B C, then setting one foot in the beginning of the Scale again, open the other to 45 degrees, and with that distance, setting one foot in B, cross the first arch at C, and then draw the line A C, as in the former example and figure. CHAP. III. Showing how to make a plain Chart, and many Propositions of sailing by it. THe drawing of the plain Chart, and the way of sailing thereby, is the most plain and easy of all others. And though it be fit to be used, only in places near the Equinoctial, or in short Voyages: yet it will serve, for a good introduction to that which follows, and this will not be lost labour, for the same kind of work (with some cautions) must be observed in all kinds of Sailing. The description and making of the plain Chart. The description of the plain Chart. First, make the square A B C D of what length, and breadth you please, and divide each side into as many equal parts as your occasion requires, and then draw strait lines through these parts crossing one another at right angles, and so making many little Geometrical squares, each of which you may suppose to contain one degree, in longitude and latitude, * According to account 20 Leagues are in one degree, & so each 10 part will be 2 leagus, but it is somewhat more, as you may see in the third proposition of this Chapter. Then on the four sides of the Chart, let each of these degrees be subdivided into 10 parts, so each part will contain about two leagues, which I therefore call double leagues. And this division of your Chart will be exact enough for the Seaman's use, so that you need not trouble yourself to divide the degrees into 20 parts or 20 leagues, especially because this way of account by decimals or tenth parts, is more easy and ready than any other. And if you keep your account by Arithmetic, you may suppose each of these parts to be subdided into 10, so every degree will contain 100 parts, which will very well agree with the Chart, better than the old division by 60 minutes, and is far more exact and easy. This short description, if you remember what hath been showed in the first and second Chapters, showing you how to draw and divide the lines, with the figure itself following, will I hope be very plain, so that I need not repeat those things before, but proceed to the uses of it. geometrical diagram The Figure of the Plain Chart. PROPOSITION I. Knowing the longitude and latitude of a place, to find out that very point upon the Chart, and so to set it upon the Chart. THe Longitude of places, is their distance East and West. What the longitude and latitude of a place is. The latitude of places, is their distance North and South. In the Globe, the longitude of places is accounted always from the West, Eastward; still increasing the number of degrees, until they come to 360, which is the whole compass of the Globe, so that you come to the first Meridian again. This account of the Longitude may begin at any place, but Geographers do commonly begin to reckon it from one of the Isles of the Azores. But it is the best way for the Seamen, not much to regard this, but to reckon by the difference of longitude, or (which is all one) by the difference of the Meridian's of the two places. The latitude of places is reckoned by their distance from the Equinoctial toward either of the Poles, so that it never exceeds 90 degrees. If the place lie between the Equinoctial and the North pole, it is said to have North latitude. If it lie between the Equinoctial, and the South Pole, it is said to have South latitude. Now though in some propositions, the Seaman reckons by the difference of latitude, yet in most of his accounts and observations he doth reckon the latitude of places by their true distance from the Equinoctial North or South. Now for example of the Proposition. By the longitude and latitude of a place to find the point thereof upon the Chart. Suppose a place to have 5 degrees of Longitude from the first meridian Eastward and to have five degrees of north latitude, and it is required to find out the point, where this place must be set upon the Chart. To perform this you must suppose the line A B in the Chart to be the first Meridian, and because the place proposed is 5 degrees from it to the Eastwards, therefore you must count 5 degrees from the line A B both in the top and bottom of your Chart, and laying your ruler there, draw the line H E I, then because the place hath 5 degrees of North latitude, you must suppose the lower line A D to be the Equinoctial line, and so accounting 5 degrees upward in both the sides of the Chart, lay your ruler there, and draw the line F E G. Now mark where these two lines do cross one another, which is in the point by E, and this is the point where you must set the place, or suppose it to be placed. PROPOSITION 2. The longitudes and latitudes of two places being known, to find the Rumbe, which you must steer your course upon, in sailing directly between them. SUppose the first place to be A, lying under the Equinoctial, and so having no degree of latitude, and likewise to be in the first Meridian, and so to have no degree of longitude. And let the other place be E, which hath 5 degrees of longitude, and 5 degrees of North latitude as before is said, and it is required to find the Rumbe between these two places. By the longitudes and latitudes of two places, to find the Rumbe. First, set the places A and E upon the Chart, according to their longitudes and latitudes, as is showed before, then laying your ruler by the two places, draw the line A E: this line shows the direct way between these two places, and if you would know what Rumbe it is, look back to your divided circle, and setting one foot of your compasses in the centre thereof, open the other to the circle of Rumbes, and keeping that distance, set one foot in the corner A, and with the other draw the arch K L, then setting one foot of your compasses, in the point where this arch doth cross the line A B, which is at K, open the other to L, which is the place where this arch doth cross the line A E, and with this distance return to your circle, and setting one foot of your compasses, in the point where the North line doth cross the circle of Rumbes, turn the other downward to the circle, the same way as it lies here, and it will point out the fourth Rumbe, which is North-east, and this is the Rumbe you must sail upon, from A to E. In like manner if you would know the Rumbe from E to A, you must set one foot of your compasses in the point E, as you did before in the point A, and draw the arch a b, & so by your circle of Rumbs you shall find that the Rumbe from E to A is South-west, which is opposite to the North-east. And this is a general rule, look what Rumbe you sail upon from one place to another, the Rumbe opposed to that, will carry you back again. You might have found out the Rumbe likewise by the Scale of Rumbes, and so you would have found it to be the fourth Rumbe from the Meridian, which must be either North-east or Northwest, Southeast, or South-west, which of the four it is, you may know by the situation of the places when you are a little versed therein, but till then the Rumbe will be found best by the circle. PROPOSITION 3. Knowing the Longitude and Latitude of two places, to know how fare they are distant one from the other. 3. By the longitude and latitude, to find the distance. LEt the two places be A and E, whose longitude and latitude is as aforesaid, and it is required to know how fare they are asunder in some known measure, viz. of miles, leagues, degrees and minutes, or degrees and tenths, or hundred parts. It is the common practice among Seamen to reckon the distance of places by leagues, accounting 20 leagues to a degree, The best way to reckon the distance is by tenths. and every league to contain about three miles, and so each mile to be the 60 part of a degree or one minute. But this way is very troublesome, and requires often reduction of one sort of parts to another. The decimal way of account is far more ready and easy, and therefore I have divided the degrees on the sides of the Chart only in to 10 parts which will be exact enough to this purpose, each of those parts will contain about two leagues, and therefore I call them double leagues or tenths of a degree. If you desire to be more exact when you use your pen you may suppose each of these to be subdivided into 10, and so make 100 parts in a degree, & then by adding to a cipher, or taking away the last figure, they will be reduced one to the other. Having thus determined the manner of the measure, the way to find the distance of the places is thus First, set the places upon the Chart according to their longitude and latitude at A and E, then setting one foot of your compasses in one of the places as A, open the other to E the other place, then measure this distance in one of the sides of your Chart, and your compasses will reach from the corner A to * 7 degrees, 7, 07. and one of the tenth parts very near, and that is the distance of the two places, or 71 double leagues or tenth parts, or 710 hundred parts ferè. The true quantity of a degree upon the earth. And here by the way, give me leave to tell you, that it is not enough for you thus to know the distances of places in degrees and parts, but it is necessary also to know, the just quantity or measure of a degree. And here in the common rule is much out, which accounts 5 foot to a pace, and 1000 such paces, that is 5000 feet to a mile, and 60 such miles, that is 300000 feet to a degree. Neither will the English mile which is somewhat more than this serve the turn, whose length is thus by the Statute. 16 feet and a half make a pole, 40 poles make a furlong, and 8 furlongs, that is 5280 feet, make a mile: and so 60 such miles do contain 316800 feet. See the Seaman's Practice But Mr. Norwood, by measuring the way from York to London found that a degree doth exactly contain 367200 English feet, and shows how this experiment agrees with former experiments made by others, if rightly considered: however his experiment is so full and punctual, that it may well pass for currant, rather than any others which differ from it. Now by this reckoning, if one degree contain 367200 feet, than the tenth part of a degree doth contain 36720 feet, and the hundreth part of a degree doth contain 3672 feet. Thus you shall know the true distance of places, knowing how many degrees and parts they are asunder. And the knowledge hereof is very considerable in the keeping of your dead reckoning by the Log-line, How your log-line ought to be marked, which you shall do well to rectify according to this experiment thus. If you keep your account by an half minute glass, then at every 30 feet length of your line, you must make a knot: and then so many of those knots as you veer out in half a minute, so many 100 parts of a degree the ship runs in an hour. As if you veer but one knot while the half minute glass is running, than the ship runs but 1/100, that is one hundreth part of a degree in an hour, if you veer 2 knots, than the ship runs 2/100 parts of a degree, and so if you veer 10 of those knots in half a minute, than the ship runs 20/100 or one tenth part of a degree in an hour. If the glass be out not just at a knot, then for every 3 feet from the last knot you may count a tenth part of an hundreth part more. And though by this reckoning there be but 360000 feet, allowed to a degree, whereas there should be 367200, these 7200 feet are thought fit to be abated, not only in regard of the rotundity of the number, to avoid fractions but for these considerations. First, because though he that veers the line be never so careful not to over-hale it, yet the log will be drawn thereby somewhat after the ship. Secondly, the Eddy which the ship makes, is subject to draw the log somewhat after it; or at least so to dead the water, that it will somewhat hinder the motion of the log. Thirdly, the wind and waves beating after the ship will drive the log somewhat forwarder than it should be. For these causes, the way of the ship may very well be somewhat more than the log line shows for. Besides if this were not so, yet it is the best way to have your reckoning run somewhat before your ship, that so you may not fall upon a place before you are ware of it. PROPOSITION 4. Knowing the longitude and latitude of the place from whence you came; the Rumbe you have sailed upon; and how fare you have sailed there on: to know the longitude and latitude of the place where you are, at that present. 4. By the Rumbe and distance, to find the difference of longitude and latitude. LEt the place from whence you have sailed be A, whose longitude and latitude suppose to be as aforesaid, ●00 deg. let the rumbe upon which you have sailed be North-east, and let the distance which you have sailed upon this rumbe from the place A, 7, 07. be almost * 71 double leagues, or tenths of a degree. Now it is required, to know what place you are now in, that is, what longitude and latitude you are come to. To perform this first set down the place A according to his longitude and latitude, then by your circle or scale of Rumbes, draw the line A L E C which is North-east from the point A, then setting one foot of your compasses in the beginning of your scale, which is on the sides of your chart, open the other almost to 71 double leagues, that is to * 7 deg. and one tenth part almost, 7, 07. and with this distance set one foot in the point A, and with the other cross the line A L E C in the point E. This cross at E is the place where you are now, and if you would know the latitude & longitude of this place, then lay your ruler (or rather a small thread) to the point E, so that it may cut the scale of degrees, on both sides the Chart at like parts, as the line F E G doth, which you see cuts the scale at just 5 degrees on each side, which shows the latitude of the place E to be 5 degrees North from the Equinoctial. So likewise to find out the longitude of the place, lay the thread by the point E, and like parts of the scale of longitude, both at the top and bottom of the Chart, as the line H E I doth, and it will show that the longitude of the place is just 5 degrees. So that the place where you now are which is E, hath 5 deg, of north latitude, and 5 deg. of longitude. Note. How to keep your dead reckoning upon the Chart. This is the way whereby you may best keep your dead reckoning upon your Chart. For the Seaman always knows what point of his compass he sails upon, and also by the log-line, or by experience, he may guess very well how far his ship goes in an hour, and by that, for any other time, which distance being thus set upon the Rumbe line in the Chart, shows him still whereabouts he is. And it will be very good, at all times to keep this dead reckoning as carefully as you can, yea, though you sail near the meridian, and so have no need of it for the present, yet than you may the better see how your dead reckoning agrees with your observations, and so gain experience to keep your dead reckoning more truly, in such courses, and against such times, as you shall have more need of it. For some times it may be close weather, for 3 or 4 days together, and in courses that lie near the East and West, you will be forced to stick to your dead reckoning, having no help to rectify it by the observation of the latitude. PROPOSITION 5. Knowing the longitude and latitude of the place, from whence you set sail: together with the Rumbe you have sailed upon; and by observation knowing the latitude of the place you are in: to know thereby the longitude of this place, and how fare it is distant from the place you came. LEt the place from whence you set sail be A, whose longitude and latitude suppose to be as before; let the Rumbe upon which you have sailed be North-east; and let the latitude of the place where you are, according to your observation be 5 degrees of North latitude. Now it is required to find out the longitude of this place, and how it is distant from the place at A. 5. By the Rumbe and the difference of latitude, to find the difference of the longitude and the distance. To perform this, first set down the place A in your chart, according to its longitude and latitude, and draw the line A L E C which is the North-east Rumbe from the point A, as is before showed. Then because by your observation you find yourself to be in 5 deg. of North-latitude, count 5 degrees on both the sides of your Chart, and laying your ruler thereto, draw the line F E G. Now mark well, where this line doth cross the line of the Rumbe A L E C, which is in the point E: for this point is the place where you are at the time of this observation. Now to know the longitude of this place, lay your ruler or a thread by the cross at E, so that it may cut like parts of the degrees of longitude, both at the top and bottom of your Chart, so you shall draw the line H E I, which shows that the longitude of the place is 5 degrees. Lastly, to find the distance of this place from A, set one foot of your compasses in A, and open the other to E, and measure this distance in your scale, you shall find it to be 71 double leagues, or 7 deg. one tenth almost. Note. Now this is the most certain way that the Seaman can keep his account by, and therefore if there be any difference between your dead reckoning and this: you must correct your dead reckoning by this, and not this by that. And therefore it concerneth the Seaman to be very careful in these two things, upon which the ground of this Proposition depends. First, that the ship be steered exactly upon the Rumbe supposed, and to this end, not only the Steers man must be careful to keep the ship to the Rumbe of the compass, which he is appointed, but you must be careful to observe the variation of the compass and allow for it. And secondly when you make observation of the latitude, you must do it with true and good large Instruments, and use the best diligence you can in observing by them, that so you may find your latitude as exactly as you can. PROPOSITION 6. How to rectify your account, when your dead reckoning differs from your account by observation. THis Proposition hath two cases, the first is when yond have kept your way only upon one point of the compass. 6. How to perfect your account. The second is when you have been forced to sail upon two or more several points, before you could make any observation of the latitude. Case 1. If you have sailed only upon one Rumbe. In the first of these cases, if you have only sailed upon one point of the compass: as for example, suppose you have sailed from the point I which lies under the Equinoctial line in 5 degrees of Longitude; upon the third Rumbe from the Meridian N W by N, 40 double leagues, or tenths according to your dead reckoning: if you set this upon the Chart according to the Rules before showed, according to this your dead reckoning you will find yourself to be in the point R, which is in 2 deg. 8/10 of longitude, & in 3 deg. 3/10 of latitude. But at this time suppose by observation, you find that you are but just in 3 degr. of North latitude. Wherefore to find the true place where you are, do thus. First, lay your Ruler, to 3● deg. of latitude on both the sides of your Chart; and draw the line c M d, and mark where it doth cross the line of the Rumbe I R, which is at M, and this is the place where you must reckon yourself to be at the time of your observation, and not at R, as you supposed by your dead reckoning. Now if you examine the longitude and latitude of this place M by the former rules, you will find that it lies in 3 degr. of longitude, and in 5 deg. of north latitude, and from this place you must set off your next course, and distance, and not from R. But now for the second case. Case 2. When you have sailed upon divers Rumbes. If it so happen, that you are forced to sail upon two or more several points of the compass, before you can make an observation of the latitude to correct your dead reckoning by. As suppose you had sailed according to your dead reckoning, first N W by N, 40 double leagues, which is set down from I to R, (though in truth you had sailed but from I to M; which is but 36 double leagues) and then being forced to shift your course to N E by E, and should sail upon this rumbe likewise according to your dead reckoning 40 double leagues, and at this instant time, you find by observation that you are but in 5 degrees of north latitude: to know the true longitude of this place, you must do thus. First, from the point I, set the first 40 double leagues upon the Rumbe N W by N, which will end at R. Then from the point R draw the rumbe N E by E, which is the line R Q, and set thereon the 40 double leagues from R to Q: thus you will find Q to be the place you should be in, according to your dead reckoning which is in 5 d. 5/10 and somewhat * 5d. 55. more of north latitude, whereas by your observation you find that you are but in 5 deg. of north latitude: now to know the true place where you are, in respect of the longitude, because you have sailed upon two rumbes, draw the line I Q from I, the first place you set sail from, to Q, the place of your dead reckoning, and then drawing the line F E G at 5 deg. of latitude according to your observation of the latitude; mark where it crosseth this line I Q, which is in the point N, and this is the true place you are in, whose longitude is 6 deg. and whose latitude is 5 deg. north. In like manner, if you should sail upon 3 or 4 several Rumbes before you can make an observation of the Latitude, your best way will be to draw a line from the first place of your voyage to that present place according to your dead reckoning; or at least from the last place, where you made a fair observation, and are thereby well assured both of the longitude and latitude thereof: For otherwise you may be much mistaken in the longitude of your places. As for instance, if in the last example, you should think you were in that place, where the line of latitude F E G doth cut the last rumbe you sailed upon, according to your dead reckoning, viz. the line R Q, by this account you would be but in O, which is but in 5 deg. 35/100 of longitude, whereas you see by the other way which is the truth, you are in 6 deg. of longitude, so that the difference is ●/100. which is very considerable in so small a space. PROPOSITION 7. Being to sail from one place to another, but by reason of cross winds, or the coastings of the land, you cannot sail thither upon the direct point of the compass which lies between the two places, but are forced to alter your course several times: yet how you should keep your account of your way, so that you may know at any time what longitude and latitude you are in, and how the place you are bound to bears from you, and how far you want to it. 7. The manner of keeping your reckoning upon the Chart. This Proposition contains the use and practise of all the former. FOr example, suppose you were to sail from the place I, in the former Chart, which is under the Equinoctial, and in 5 degrees of longitude; unto the place H, which hath 5 deg. of longitude, and 10 degrees of north latitude: here the direct way from I to H lies full north: But supposing that you cannot sail upon this point, but are forced first to run N W by N 36 double leagues, and then N E by E 36 doubled leagues more, the question is what is the longitude and latitude of this place, and how fare it is distant from the place H, and upon what point of the compass it lies from it. First, from the point I, draw the Rumbe N W by N, and set off thereon 36 double leagues from I to M. Then from this point M draw the Rumbe N E by E, and set off thereon the 36 double leagues which you have sailed upon it, from M to N: thus you shall find, that N is the place wherein you are, whose longitude is fix degrees, and whose latitude is five degrees. Now if you lay a ruler from this point N, to the place you are bound, to which is H, and draw the line H N, this line is the direct way to the place you are bound, and by the help your circle or scale of Rumbes, you shall find that it lies North by West, or the first Rumbe from the meridian Westward. Lastly, if you set one end of your compasses in N, and open the other to H, and measure that distance in the sides of the Charts, you will find it to be about 5 degrees 1/●● or 51 double leagues, and so much you want to the end of your voyage. PROPOSITION 8. How to know the distance of any Cape, Headland, or Island, from you, which you can see at two distant places. 8. To know the distance of any Cape from you SUppose that sailing on the Sea, you espy an Island or Cape lying at the first sight, just North-east from you; and then sailing forward upon your way, which lies full North, to the distance of 5 leagues, you then observe that the Island lies full East from you, the question is to know the distance of this Isle from either of these two places. In such questions as this, you may suppose each degree in the former Chart to stand now but for a league, These two following Propositions rather belong to the plain table then the chart. and let the first place where you espied the Island be at A, now because the Island lay North-east, from this place, draw the line A B which is N E from A. Then count the 5 leagues which you have sailed upon your course which was full north, in the meridian line from A to F, and because from this place, the Island did lie full East, therefore from this point F, draw the East line F E G, and mark where this line, doth cross the former line A E of N E from A, which is in the point E. This therefore must needs be the place of the Island, whose distance if you take with your compasses, and measure in the sides of the Chart, you shall find that the place E is distant from A 7 leagues and almost 1/15 part of a league; and from F just 5 leagues. * A double use of this proposition. And by this means if you know the longitude and latitude of this Isle or Cape, you may the more certainly know the truth of your account, and if need be correct it. Or if you knew not the place before, you may set it down in your chart by its longitude and latitude which you find it to be in, according to the best account you can make by your observation. PROPOSITION 9 By observing upon what Rumbes many places lie from you at two several stations; to find the distances of those places, and their true posture and bearing one from another. 9 To find out the true distance, and bearing of many places. The use of this Proposition. AS in the former Proposition you did for one place, so in this you may do for many. And this will be of good use, for hereby, sailing in sight of any Coast, you may find out how the Points and Rivers and such like lie, and so make a Map thereof. The way to perform which is thus. First, observe well how the several places lie from your first station, which suppose to be A; and let these three places be observed by you viz. I M and E I bears from A full East, M and E, North-east, therefore draw the lines A I and A M E according to the bearing of the places from A. Secondly, you must observe your course which you sail upon, until you come to the second station, which suppose to be five leagues full North, let this be set down according to its place at F. Thirdly, this being your second station, observe how the three-former places bear from this place, and suppose you find, that E lies full east from this place, M lies Southeast by east, and I lies Southeast, from this second station. Then draw these lines F E, F M, F I, from the point F, and mark well where they cross the former lines, which will be in the said points E M and I, and thus these three places E M I are set down according to their true positions and distances, both from the two stations A and F, and likewise one from another, so that if you try by the former rules, and your comasses you shall find M lies from I, N W b N 3 leagues and 6/10, and E lies from M, N E, 2 leagues and 8/10. Thus you may easily describe the coast of a Country as you sail by it in sight of it, which will be both pleasant and profitable, especially when you light upon Coasts that have not been discovered. PROPOSITION 10. To draw a Rumbe line from any point assigned. IN this description of the plain Chart, I have purposely omitted the old custom, 10. To draw a Rumbe line from any point in the Chart. of pestering the Chart with so many Rumbe lines, to little or no purpose. For though they may seem to be of some use in this Proposition; yet it may better be performed, only by the lines of longitude and latitude with the help of the circle, or scale of Rumbes. For first, if your point assigned fall out in any of the meridian lines, than you may readily by the rules of the second chapter, pag. 13, 14. draw any rumbe line from that point. But if your point do not fall just upon, a meridian line, than you must, first, from the point assigned, draw a line parallel to any one of the meridian lines, as is showed by the second proposition of the first chapter, and then draw the Rumbe line from your point assigned in that line, as you did before. Thus you see these lines of longitude and latitude are of double use. For first, they readily show you the longitude and latitude of any place in some measure by the eye. And then they help in the drawing rumbe lines, from any point. Whereas to draw so many rumbes after the usual manner, is but a spending much time and labour to little purpose. CHAP. FOUR Showing how to make a Sea-chart for any part of the World, which shall agree in all particulars with the Globe; with several Propositions showing the manner of using it. The defects of the plain Chart. THe way of sailing by the plain chart, is very easy to understand, as you may see by what hath been said in the former chapter: But it hath this inconvenience, that it is fit to be used only in places near the Equinoctial, or in some short voyages. For it supposeth the degrees of Longitude to be all of one length, in every Latitude, and therefore it makes the degrees both of longitude and latitude, all of one length and breadth in all places. But you must know that though the degrees of Latitude, are always of one and the same breadth (viz. about twenty leagues) yet the degrees of longitude are not all of this length, but as you may see in the Globe, they grow less and less toward the Poles: so that though about the Equinoctial, a degree of longitude is equal to a degree of latitude, viz. about 20 leagues: yet in the latitude of 60 deg. one degree of longitude is equal, but to half a degree of latitude, viz. about 10 leagues; and about the latitude of 75 deg. 30′, one degree of longitude is equal but to a quarter of a degree of latitude, which is about 5 leagues. And therefore it is impossible to set three places upon the plain chart, which differ much in longitude and latitude, as they ought to be placed, that is according to their places on the Globe. But if you set them down, according to their longitudes and latitudes; then they will not stand in their true Rumbes and Distances one from the other: and if you strive to set them down according to their Rumbes and Distances; then their Longitudes and Latitudes will not fail out right. And therefore though you may make a shift to perform some short voyages by it, yet you cannot use it in any long voyage without great error: except you only go from one place to another, and so directly back again to the same place, from whence you came, and by the same course which you came. * This example is M. Norwoods' in his Problems of sailing. For example, suppose you were to sail from the Lizard, to the Summer Islands, and should according to the common course first sail South-west, about 500 leagues, and then finding yourself to be in the latitude of 32 deg. 20′, you should then sail full West 782 leagues, and then you should find yourself directly South from the Summer Lands, and about 2 leagues off them. Now by this reckoning upon the plain chart, these Islands should be distant from the Lizard, 1189 leagues. Now admitting this reckoning outward to be true, and these places to be thus situated upon the plain chart, let us suppose the reckoning homewards to be also kept upon the plain chart. And because in coming home, men keep to the Northwards, suppose that you steer away first N E half a point Easterly 200 leagues; then N E by E 100 leagues, next E N E half a point northerly 165 leagues; then E N E 130 leagues; then E N E half a point Easterly 88 leagues, then E by N 70 leagues; lastly, East 317 leagues; if you set down this reckoning upon the plain chart, you will be yet short of the Lizard about 160 leagues, whereas you are already come unto the Lizard, and so you will find it, if you keep your reckoning by this following Sea-chart. For the reformation hereof, Mr. Wright in his book of the Errors of Navigation, hath showed how to make, and hath also made a table, by the continual addition of the secants of every minute, which shows how much you are to lengthen the degrees of latitude in your Map, that so there may be a true proportion between the degrees of Longitude and Latitude in all places: which table Mr. Gunter hath abridged, and made it more plain and easy, by reducing it into decimal parts. I shall here show you how to do the same by Geometry, and how to make a line of Latitudes, or a Meridian line answerable to any line of longitudes. geometrical diagram The projection of the Meridi●n Line. First, make the Quadrants A B C of what largnes you please, and divide the limb thereof into 90 degrees numbering them from B towards C, then divide the side of the Quadrant A B into 10 equal parts, and draw strait lines from them parallel to A C, then take one of these parts from A to E, and subdivide it into 10 lesser parts, and draw lines from them parallel unto A C. Now you must note that the length of this one part A C is to be your radius, or the measure of one degree of longitude in your Chart, so that the whole line A B will be 10 degrees, and because these degrees of longitude are to be of one length in all latitudes, therefore the degrees of latitude must increase as the secants of the latitudes increase. Therefore if you would know how long one degree of latitude must be in the latitude of 30 deg. lay your ruler to the centre A, and the arch of the quadrant at 30 deg. and draw the line A G, now the radius being A E, this line A G is the secant of 30 deg. to that radius, and must be the length of one degree of latitude in a Chart for that latitude. So likewise the line A H which is the secant of 45 deg. to that radius, must be the length of one degree of latitude, in the latitude of 45 deg. and so for any other latitude: and note that the 10 intermediate lines may serve to divide each of these degrees into 10 parts. If you would examine the truth of this projection how it agrees with the Globe; The proof of of this projection. whereas in the Globe one degree of latitude is equal to two degrees of longitude in the latitude of 60 deg: so here A K the secant of 60 deg. is twice the length of A E the measure of one degree of longitude, and whereas in the Globe one degree of latitude is equal to four degrees of longitude in the latitude of 75 deg. 30′: so here A L the secant of 75 deg. 30′ is four times the length of A E, and so the proportion will hold in any other latitude. If you desire to make a table hereof, To make a Table of the Meridian line. than you may make the whole line A B to be your radius, or the length of one degree, which you may divide into 100 parts, and then the secants will be the lines drawn from the centre A to the line B D, thus then the line A I will be the secant of 30 degrees, whose length is 115, as you may measure in the scale A B if it were increased. And so the line A D is the secant of 45 degr. whose length is 141. Note. * And note here if you make the table for whole degrees only, than it will be the best way to draw the Secant line through the midst of the degree, as if you would know the length of the line which must reach from 29 degrees to 30 degrees, draw the line through 29 degrees 30 minutes, the length whereof will be 115, and so by the continual addition of these Secants you may make the Table following, which you shall find agrees very well with Master Gunters, only his radius is divided into 1000 parts and this but into 100 Another way to make a Meridian line. Also by the Quadrant, or the Table, you may make the two Meridian lines following. But if you make them by the Quadrant, then because the degrees would fall too close together if they were all drawn, under the first Radius A E, you may remove the Radius A E further from the centre, and then draw them under it. As for example, the distance between the lines 8 and 9 being equal to the distance A E, you may there draw the first 20 degrees. Then between the lines 7 and 8 being of the like distance, draw the next ten degrees which is to 30, and so do the rest as you see in the Quadrant. Then taking out these degrees one by one with your compasses, set them upon the Meridian line, of your Chart, or make a Meridian line, at your leisure, as in the following figure, which will be very ready upon any occasion. A Table of the Meridian line, as it is taken out of the Quadrant, as aforesaid, the Radius or the whole line A B being divided into 100 parts. The Latitude The Meridian line The Sec to be ad. D. D. P. P. 0 0 00 1 1 00 100 2 2 00 100 3 3 00 100 4 4 00 100 5 5 00 100 6 6 01 101 7 7 02 101 8 8 03 101 9 9 04 101 10 10 05 101 11 11 07 102 12 12 09 102 13 13 11 102 14 14 14 103 15 15 17 103 16 16 21 104 17 17 25 104 18 18 30 105 19 19 36 106 20 20 42 106 21 21 49 107 22 22 56 107 23 23 64 108 24 24 73 109 25 25 83 110 26 26 94 111 27 28 06 112 28 29 19 113 29 30 32 113 30 31 47 115 30 31 47 31 32 63 116 32 33 80 117 33 34 99 119 34 36 19 120 35 37 40 121 36 38 63 123 37 39 88 125 38 41 14 126 39 42 42 128 40 43 71 129 41 45 02 131 42 46 36 134 43 47 72 136 44 49 10 138 45 50 50 140 46 51 93 143 47 53 38 145 48 54 86 148 49 56 37 151 50 57 91 154 51 59 48 157 52 61 09 161 53 62 73 164 54 64 41 168 55 66 13 172 56 67 90 177 57 69 71 181 58 71 57 186 59 73 49 192 60 75 46 197 60 75 46 61 77 49 203 62 79 58 209 63 81 75 217 64 83 99 224 65 86 31 232 66 88 72 241 67 91 23 251 68 93 85 262 69 96 58 273 70 99 43 285 71 102 43 300 72 105 58 315 73 108 91 333 74 112 43 352 75 116 17 374 76 120 16 399 77 124 45 427 78 129 07 462 79 134 09 502 80 139 58 549 81 145 65 607 82 152 42 677 83 160 10 768 84 168 95 885 85 179 41 1040 86 192 21 1280 87 208 70 1649 88 231 95 2325 89 271 70 3985 90 infinite geometrical diagram The lesser Meridian Line, for general Maps. The long Meridian Line, for particular Maps. I have made here two meridian lines and that for two reasons. First, because in the larger line after 80 degrees of latitude, the degrees grew so large, and increased so much, that it would be both needless and troublesome to make any use of them: but the chief reason is this. Because when you are to go any long voyage, it will be needful for you first to make a general Map of your whole voyage by the lesser line, whereby you may know the course and distance thereof in general: and then to make three or four other charts' by the greater line upon which, with your ruler and compasses you may set down your daily courses and distances more exactly. Also I have made these two lines in such proportion, that the one is the tenth part of the other, that so that they may both agree with the scale upon the Quadrant. Now the way to make one of these charts' is very easy, To make a Sea Chartley by these Meridian lines. and much after the manner of the plain chart. For first, you may draw the line of East and West A B of what length you please, and divide it into equal parts or degrees, than you may erect a perpendicular line either at one of the ends of the line or in any of the divisions toward the midst of the line, and then draw the other parallels of longitude parallel thereto, so far it is all one with the plain chart, but when you come to draw the parallels of latitude, you must not make them all equal, (though they must be all parallel) each to other; but you must either with your compasses, take them out of the Quadrant, or which is more easy, lay a scroll of paper to the Meridian line which is ready drawn to your hand, and so mark out the degrees of latitude upon the scroll of paper, and then laying that scroll to the sides of your chart, you may transfer the degrees of latitude into the sides of your chart, and through them draw the parallels, and set fit numbers to them, as in the figure. geometrical diagram The figure of a general Sea-Chart, containing almost an eighth part of the Globe. NORTH. Now though this be not a general chart of the whole globe, yet it may be called a general chart in respect of others, which will serve only for a lesser portion of the Globe. For this chart containeth almost an eighth part of the Globe, and may be fitted to set forth any part thereof. For if you change the numbers of the longitude, if the latitude be northward, it will serve as it now stands: but if the latitude be Southward, you must turn the bottom upward. If you have occasion in one chart, to set down both North and South latitude: than you must draw the like parallels of latitude below the Equinoctial, as these are above it. Now I will show you how the several Propositions which were performed by the plain chart, may be performed by this, and wherein they agree, and wherein they differ. PROPOSITION 1. Knowing the longitude and latitude of any place, to set it upon the Chart. 1. By the longitude and latitede, to find the point of any place in the Chart. THis must be done, as in the plain chart. For first laying your ruler by the longitude of the place, you must draw a little occult line as near the latitude of the place as you can guests, then laying your ruler to the latitude of the place cross that line you drew before with another little line, and so the crossing of these two lines will show you the point where the place must be supposed to stand. Example. Thus supposing the longitude of the Summer Lands to be 300 degrees, and the latitude thereof 32 degrees, 25 minutes, you will find that it must be set at S, upon the chart. PROPOSITION 2. The longitudes and latitudes of two places being known, to find the rumbe which you must sail upon, to go directly from the one place to the other. 2. By the longitude and latitude of two places to find the Rumbe. Example. SUppose the one place to be the Summer Lands, whose longitude and latitude we will suppose to be as is before set down, let the other place be the Lizard, whose latitude is about 50 deg. and let the longitude thereof be supposed to be 10 degr. so the difference of the longitude of the two places, will be 70 deg. (as Mr. Norwood both in his book of the Doctrine of Triangles, and his Seaman's Practice supposeth them to be, though as he saith in one place, he doth not think them to be so far distant) and it is required to find the rumbe. This Proposition must also be performed as in the plain chart. For first, the two places must be set upon the chart, according to their longitudes and latitudes, which will be at S and L, then draw a straight line from S to L, this represents the direct way between the two places, now to know what rumbe this is, open your compasses to the Radius of your scale of rumbes, and setting one foot of your compasses in S with the other, draw the arch R M, then setting one foot of your compasses in R, open the other to the crossing of the line, and the arch at M, and measuring that distance on your scale of chords or Rumbes, so shall you find it to be 71 deg. 21 min. or the sixth rumbe and somewhat above a quarter of a rumbe, from the Meridian. PROPOSITION 3. Knowing the longitudes and latitudes of two places to know how fare they are distant one from another. 3. To measure the distance of places. LEt the two places be as is before said S and L, it is required to find their distance. In the working this Proposition, there is some difference from the plain chart, for whereas there you measure the distance of places by one and the same scale of equal parts here you will have use of many scales, according to the latitude of the places. Mr. Gunter's way. Now the ordinary way prescribed by Mr. Gunter, to perform this is thus. Open your compasses to the distance of the two places; and then setting your compasses in the Meridian line, so that the one point of the compasses may stand just so much above the greater latitude, as the other doth below the lesser latitude, and so the degrees between them is the distance: this way may serve for small distances as Master Gunter useth it; but in greater distances it will not always hold true; and besides, it is somewhat troublesome to set the compasses just as much above the one latitude, as below the other. As in this example, if you take the distance S L in your compasses, and measure it so in the Meridian line, it will reach from about 16 degrees, to about 66 degree and an half, that is 16 degrees and an half above 50 degrees, the greater latitude, and 16 degrees and an half below 32 degrees, 25 minutes, the lesser latitude, and so the degrees intercepted between the points of the compasses are about 50 degrees and a half, whereas the distance of the two places is almost 55 degrees. But you shall find the distance more exactly, The way to measure the distances of places which differ in longitude and latitude. if you do thus. First, divide the space that is in the meridian line between the two latitudes into two equal parts, which in this Example will fall at 42 degrees, then take with your compasses half the length of the line L S, which is L M, and setting one foot of the compasses in 42 degrees, which is the middle point between the two latitudes, you shall find that the other point will reach down to 8 degrees and a half, then keeping the one foot still fixed in 42 degrees, turn the other foot upward, and it will reach to 63 degrees and a half; now the degrees of the meridian line between these two points are 55, which is the distance desired, which is 550 double leagues or tenths of a degree, or 1100 leagues. But if this distance were to be measured in the plain chart, the two places being set down therein according to their longitudes and latitudes aforesaid, their distance would be above 72 degrees, which is 340 leagues, more than the truth. If you would measure a parallel distance, How to measure the distance of places which differ only in longitude. as suppose the two places were L and T, both in the latitude of 50 degrees, and their difference of Longitude is 70, the way will be to take half the distance which is L Q, with your compasses, and setting one foot in 50 degrees of the meridian line, the other foot will reach to 22 degrees and a half downwards, and to 67 degrees and a half upward, and if you count the degrees between these two places, or else subtract the lesser from the greater, you shall find 45 deg. which is the distance of the two places. Another way to measure the distances of parallel places Yet this may more readily be performed by the Quadrant, Page 22, whereby the meridian line was made. First, if you would measure a parallel distance, as for example, suppose you would measure the distance of the two places T and L, being both in the latitude of 50 degrees, and their difference of longitude is 70 degrees. First, draw the line A P at 50 degrees in the quadrant, then take the distance of the two places T L with your compasses, and setting one foot in A, the centre of the quadrant, with the other foot, cross the line A P 50 at P, which is just in the midst between the fourth and fift parallel lines, which are drawn from the scale of the quadrant, now because the chart is of the least size, therefore you must account every of those greater parts, for 10 degrees, and so the distance will be found to be 45 degrees, as before. Now to know this exactly, these greater parts should each of them be divided with lines into ten parts, as the first of them is: or else you may make use of those lines in the first part thus. Having made the mark at P, as before; and seeing it doth not fall just upon one of the parallel lines you may set one foot of your compasses on the next parallel line at R, and turning the other towards the centre A, it will reach in the line P A to the middle line among the lesser diagonal lines, which shows it lacks 5 degrees of 50, and so it is 45 degrees. But if you would measure the distance of two places which differ both in latitude and longitude, To measure the distance of places which differ in longitude and latitude. as the two places L and S in the chart, you must do thus First, draw the line AM in the quadrant, pag. 32, by the arch of the one latitude 32 degrees 25 minutes, then draw the line A P 50, by the arch of the other latitude, 50 degrees, then in any of the parallel lines, divide the space between these two lines A M and A P 50 into two equal parts, which if you do in the eighth parallel, the half or midst will be at N,, then draw the line A N, which will be your scale to measure this distance by. Now if you take the distance S L in the chart, and set it on the line A N in the quadrant, it will reach from A to S, that is in the midst between the fift and sixth parallel lines, so that the distance is 55 degrees, for every one of these parallel lines stands for 10 degrees when the chart is drawn by the little meridian line. PROPOSITION 4. Knowing the longitude and latitude of the place from whence you came, and the rumbe you have sailed upon, and how fare you have sailed on that rumbe: to know the longitude and latitude of the place you are in. 4. By the rumbe and distance, to find the difference of longitude and latitude. The way of keeping your dead reckoning upon the Sea-Chart. THis Proposition shows the way how you must keep your dead reckoning upon your chart, which is good to be done always, but especially when you cannot have opportunity to observe the latitudes, or when your course lies near the East or West, so that observation of the latitude will do you little or no good, in keeping of your account of the way you have sailed. And that you may the more exactly keep this account, it will be needful to make your chart by the greater meridian line, and if your voyage be so long that one sheet of paper will not make a chart big enough, you may put it into two or three sheets, and keeping your daily accounts upon them, you may as often as you shall see cause, transfer your accounts out of these perticularr charts', into your general chart, and so you shall see the better how to direct your course in general. Thus supposing the voyage to be as is before mentioned between the Lizard and the Summer Lands, Example. the chart following drawn by the greater meridian line may contain a part of that voyage. And though this chart being straitened for room contain but a very little part of the voyage, yet a sheet of paper will contain one third part of the whole voyage, and it would be very ready and necessary for the Seaman's use, if such blank charts' were drawn to all latitudes, which might be done to 70 or 80 degrees in 4 or 5 sheets of paper so that by setting, fitting numbers of longitude to them, you might make them serve for most places in the whole World. geometrical diagram Having provided the blank chart, let the example to explain the Proposition be this. Having sailed from the Lizard, about the distance of 5 degrees, or 50 tenths on the fift Rumbe from the Meridian S W b W. I would know what longitude and latitude I am come into. Now to perform this, first you must set the Lizard according to its longitude and latitude aforesaid upon the chart at L, Latitude 50d. Longitude 10d then from L draw the line L M which is the rumbe you have sailed upon, then take five degrees which is the distance you have sailed, out of the meridian line; upon the side of your chart from 50 to 45, and set it from L to N, so the cross at N shall represent the place where you are, which you may readily see by the Map, to be about 47 degrees and 2/10 of latitude, and about 3 degrees 8/10 of longitude, that is 6 degrees 2/10 of longitude, distant from the Meridian of L. But if you should keep your account by the plain chart you would reckon yourself but 4 degrees, 2/10 of Longitude distant from the place L. And though this way there may be some small mistake, sometimes in over, sometimes in under reckoning, if you be not very careful to take your distance out of the meridian line, as near the latitudes as you can, yet if the distance be not great, and especially if the Rumbe be not far from the Meridian, your error will not be much. And you may the less regard it, because this being but your dead reckoning, you need not trust to it, but may correct it afterward, when you have opportunity to observe the latitude. But when your course lies near the East and West, Example of a parallel course. so that you are forced to trust to your dead reckoning, because you cannot correct it by observation of the latitude, than it will be the more needful, for you to be the more exact in setting off your distance, which you may do by the Quadrant: as for example, suppose you have sailed the distance of 5 degrees, or 50 tenthsful West from L, and would know what longitude you are then in. First, draw the line A R O in the Quadrant pag. 32, by the arch of 50 deg, which is the parallel latitude you have sailed in; then because you have sailed the distance of 5 degrees in this latitude, mark where the fifth parallel line in the Quadrant, crosseth the line A R O, which is at R, therefore take the distance A R out of the Quadrant, and transfer it into the chart from L to R, so shall R be the true point where you are, whose longitude you may see by the map to be about 2 degrees 2/10 being distant from the Meridian of L 7 degrees 8/1●, whereas by the plain chart you would think yourself to be but 5 degrees distant. PROPOSITION 5. Knowing the longitude and latitude of the place from whence you set sail, together with the rumbe you have sailed upon, and by observation knowing the latitude of the place you are in: to know thereby the longitude of this place you are in, and how fare you are distant from the place you came. 5. By the Rumbe and difference of latitude to find the point where you are. THis Proposition is the same with the fift proposition in the use of the plain chart, and is performed just after the same manner, but with far more truth in respect of the longitude or difference of meridians. For example, let the place from which you set sail be the Lizard, whose longitude and latitude is before set down, let the Rumbe upon which you have sailed be the fifth Rumbe from the meridian S W by W. and lastly, by your observation, you find yourself in the latitude of 45 degrees, the longitude and distance of this place from L is required. To perform this. First, having made the foregoing chart, set down the Lizard according to ' its longitude and latitude at L, then draw the line L M which is the fift Rumbe from the meridian, and lastly, because you find yourself to be in the latitude of 45 degrees, lay your ruler to the parallel of 45 in your chart, and draw the line D M. Now the point where this line D M crosseth the rumbe line L M, is the place which you are then in, whose longitude you see by the Map is 358 degrees 9/10: so that it differs from the meridian of L 11 degrees 1/10. But if you should perform this proposition upon the plain chart, you would account yourself to be but 7 deg. and an half from the meridian of L. Lastly, if you measure the length of the line L M, either by the meridian line on the side of the chart, or rather by the Quadrant, as was showed before, you will find that you have sailed from L; 9 degrees, or 90 tenth. Thus you see this, which is the most useful proposition of all, being the most certain way by which the Seaman can keep his account, is as ready performed, and just in the same manner, as in the plain chart. Neither shall the Seaman need to trouble himself, in keeping his accounts with more curious calculations: for considering he cannot observe the latitude so exactly but that he may miss therein 5 or 6 minutes: as also that the ship cannot be steered, so exactly, but that it may alter from the Rumbe supposed, many minutes, if not some few degrees: and seeing also, it will be easy to draw the lines of Rumbs and Latitudes, in a chart whose meridian line shall be about this size, more exactly than can be observed or steered. What profit will be gained then by more curious calculation? PROPOSITION 6. The rectifying of your dead reckoning, by your observation. 6. THis Proposition is to be performed here, as is showed in the plain chart, only you must measure your distances by the meridian line, near the latitude you are in. PROPOSITION 7. 7. THis Proposition is but for variety, and recollection of what was said before, and to be performed in like manner upon this Chart. PROPOSITION 8 & 9 8 & 9 THese two Propositions belong only to the plain chart or plain table, and is not so fit for this. In all these Propositions, there are these four principal things to be taken notice of. 1 the Longitude, 2 the Latitude, 3 the Distance, 4 the Rumbe. Any two of these being known; the other two may be known thereby. So that these Propositions might be varied many ways, as you may see in Mr. Gunter's Works. But what I have said already, being I hope sufficient to instruct you in the nature and use of the Chart, and these being the most necessary, I shall not further enlarge upon this in this place. CHAP. V Of sailing by a great Circle. The praise of this way of sailing by a great Circle I Now come to show how you may sail by the arch of a great Circle, which is the most exact way of sailing of all others, but in regard of the difficulty that there is in the calculation thereof, it hath discouraged the Seaman from looking after it: but I shall show you how this may plainly and easily be performed by Geometry, which I hope will be for the general profit and ease of Seamen. For you must know that the distances of places found out as was before shown in the use of the Sea-chart, First, it is the nearest way. is seldom the nearest distance between any two places, but it is only their distance in the rumbe. So that if the two places are not both under the Equinoctial or both in one meridian, than there is somewhat a nearer cut between the two places, than the rumbe points out: which sometimes, especially near the Poles, is very considerable. But this is not all the benefit which comes by this way of sailing, Secondly, it is the most convenient way. but many times when your course lies near the East and West, this way is fare more convenient. For if you should sail full East or West, you must altogether depend upon your dead reckoning, having no way to help yourself, by the observation of the latitude, but now if you sail by the arch of a great circle, between two such places, you not only go the nearer way, but also may alter your latitude many degrees, whereby your account may be often rectified, * So in the example of the Summer Lands the distance by the rumbe is 3299 miles. The distance by the arch is 3204 miles, that is 95 miles less. as for example, suppose you were to sail from Spain to Virginia, both which lie near the parallel of 40 degrees, and suppose the difference of longitude between two such places in the parallel of 40, to be 70 degrees, the distance of these two places measured in the parallel of 40 (which is the rumbe that leads between the two places being East and West) is 53 degrees 62/100, but their distance in the arch of a great circle is but 52 degrees, 08/100, that is 1 degree, 54/100 less. But this as said, is but the least part of the benefit that comes by this way of sailing: the chiefest is this, that in sailing between two such places by the arch of a great circle, you will first in the one half of the way raise the Pole 5 degrees 69/100, and then in the other half depress the Pole as much, so that in your whole Voyage you will alter the latitude 11 degrees 38/●0, & so by the observation of the latitude you may rectify your dead reckoning very well, which you cannot do, sailing in the parallel. Thus you see this way of sailing is not only the nearest but the best way. Now concerning this way of sailing, there hath been but little written by any, Few have written of this subject. and therefore I shall be the more large in this. Captain Saltonstall in his Book called the Navigator, hath said somewhat how to direct a parallel course, but for any other course he hath said nothing, and what he showeth is to be performed by Arithmetic. Master Norwood in his Book of Trigonometry, hath added as an appendix many Problems of Sailing by the arch of a great circle whereby those; who both can, and will take the pains, may by calculation find out all things necessary in this way of Sailing. But those ways of calculation as they are very difficult to the unlearned, so they are tedious to those that have the best skill: and therefore I hope it will be well accepted, if I here show you how the same may be performed by Geometry, both plainly and speedily, and yet with as much exactness, as need be required. The chief things to be known. And in the pursuance hereof, I shall keep as close to Master Norwood, as I can, both in his Propositions and Examples, that thereby you may see how nearly my plain lines will approach to the exactness of his calculations. Now if you observe him, there are these three things, which must be found out in every Example. First, the distance of the two places in the arch of a great Circle. Secondly, the angle of position from the one place to the other. Thirdly, to find out what longitudes and latitudes the arch of the great circle doth pass through between the two places. To find the distance of two places. For the first of these, knowing the longitude and latitude of two places, to find their distance in the arch of a great circle, which is always the nearest distance. I might show you how to perform this in the first place, but I here pass it by for these reasons. First, because Master Wright, Master Blundevile, and Captain Saltonstall, have all of them demonstrated it in their Books already. And secondly, because the chief benefit in this way of sailing doth not so much consist in saving of a little way, as in sailing the most convenient way: that is, so as you may alter your latitude most, and so your reckoning may be the more certain. For though near the Poles, the difference of the distance of two places, in the arch of a great circle, and in their rumbe, may be considerable; yet in most Voyages it is not: as in the forenamed Example of two places in the parallel of 40 degrees, the difference by calculation is found to be but one degree 54/100, which is scarce considerable in the whole Voyage, being 52 degrees. Thirdly, it will be somewhat difficult, & it requires great curiosity in drawing of those lines prescribed by them so exactly, that you may come to the knowledge of the distance any thing near. Lastly, all that trouble is needless. For though in calculation this distance must be found out first, that so you may find out the rest of the Propositions following: yet in this way I am about to show that which follows, no way depends upon the true knowledge of this distance: it shall be sufficient therefore for the present, to tell you, that this way is always somewhat the nearest way. For the second of these Propositions, which is to know the angle of position from the one place to the other. The angle of position is needless in this operation. Though this must be found out in calculation before you can proceed any further, yet in this work it is more needless than the former proposition, and therefore may be very well omitted. But now for the third Proposition, To find out the longitudes and latitudes by which the great circle doth pass. which is the finding out by what Longitudes and Latitudes the great circle must pass between the two places, this being the very end aimed at in all the work, may be thus attained. First, draw the following Quadrant A D B, and divide it into degrees; then consider of what length your Tangent line must be, and accordingly set off your Radius from A toward D the larger * You may make your tangent larger either by making your Quadrant larger, or by setting your Radius further from the Centre, Thus in the Quadrant the line D K is a larger tangent line, which though it reach but to 45 degrees, yet by lengthening of the line, you may set on the rest. the better, but in this Quadrant, the Radius is A R) and this Radius is always a tangent of 45 degrees. Then from the point R, draw the line R T parallel to the side of the Quadrant A B: this line R T is the Tangent line which you must divide into degrees, as you see in the figure by drawing strait lines from the Centre A to the limb of the Quadrant. Then transfer this line to the sides of the Quadrant A B and A D, and then setting one foot of your compasses in the centre A, open the other to the several degrees in the line A B or A D and draw the arches. Now you must know that these arches are the parallels of latitude; and the strait lines drawn from the Centre, are Meridian lines, or the lines of longitude. The arches of latitude you must number as in the figure: but the lines of longitude you may number as your occasion requires. geometrical diagram This is a projection of a part of the Globe in plano by Natural Tangents: You may if you please, when occasion requires, divide a Circle into four Quadrants, and draw the lines of Longitude from the Centre, and number them to 360, and likewise describe the Circles of Latitude round about the Centre, and you may make this Projection as large or as little as you will by the Table of Natural Tangents, if you lengthen or shorten your Radius. A Table of Natural Tangents The Radius being 1000 parts. D. Tan. D. Tangle. D. Tang. D. Tangent 1 017 24 445 46 1,036 69 02,605 2 035 25 466 47 1,072 70 02,747 3 052 26 488 48 1,112 71 02, 904 4 070 27 510 49 1, 150 72 03, 078 5 087 28 532 50 1, 192 73 03, 271 6 105 29 554 51 1,235 74 03,487 7 123 30 577 52 1, 280 75 03,732 8 141 31 601 53 1, 327 76 04, 011 9 158 32 624 54 1,376 77 04,331 10 176 33 649 55 1,428 78 04,705 11 194 34 675 56 1,483 79 05,144 12 213 35 700 57 1, 540 80 05, 671 13 231 36 727 58 1,600 81 06, 313 14 249 37 754 59 1,664 82 07,115 15 268 38 781 60 1, 732 83 08, 144 16 287 39 810 61 1,804 84 09, 514 17 306 40 839 62 1, 881 85 11,430 18 325 41 869 63 1,963 86 14,300 19 344 42 900 64 2, 050 87 19,081 20 364 43 933 65 2, 144 88 28, 636 21 384 44 966 66 2, 246 89 57,290 22 404 45 1000 67 2,356 90 Infinite 23 424 Rad. 68 2, 475 Let your Radius be of what length you please, first divide it into 10 equal parts, and then subdivide each of those parts into 10, so you shall have 100 parts in your line, than you may, if you can, divide each of these 100 parts into 10, so you shall have 1000, But this last division will be needless, for you may by your eye guess at the proportion ill part. Having thus fitted your Scale of equal parts, you may prick down the line of Tangents out of this Table. Note after you are passed 45 degrees in the Table, the Figure before the Comma, shows the whole Radius, or how many times the whole Radius is contained therein, and the three following Figures, the parts to be reckoned upon the Scale as before. You will find this Table necessary, either when you would make a large Tangent line to serve for places only near the Pole; Or when you would make a very little Tangent line that so you may bring in the degrees near the Equinoctial into your Quadrant. The flank being made will serve for many examples, so that the work will be very easy. Having thus drawn this blank Quadrant, you must set down therein the two places you are to sail between, according to their latitudes and longitudes, and then only by your ruler draw a strait line from the one place to the other, and this strait line will represent the great circle which passeth between the two places, and will exactly cross those degrees of longitude and latitude, which you must sail by. For the example Example. and proof hereof, I shall take Mr. Norwoods' example of a voyage from the Summer Lands to the Lizard, the latitude of the Summer-Ilands is 32 degrees 25 minutes, let the longitude thereof be supposed to be ●00 degrees, the latitude of the Lizard is near 50 degrees, the difference of longitude between the two places is supposed to be 70 degrees, so that the longitude of the Lizard will be 10 degrees. And it is required to know by what longitudes and latitudes the arch of a great circle drawn between these two places doth pass. The working of the example. First, let the line A B represent the meridian of the Summer Lands, upon which you must mark out their latitude 32 degrees, 25 minutes at B, and because the longitude thereof is 300: set down 100LS at the end of the line A B, so the Summer-Ilands shall be set down according to their longitude and latitude: then count still forward the degrees of the difference of longitude till you come to 70 degrees in the limb of the quadrant, and there draw the line A C 70, this line will represent the meridian of the Lizard, and upon this line you must mark out the latitude of the Lizard, which is 50 degrees at C, then lay your ruler to these two marks at B and C, and draw the strait line B C. This line B C will represent the arch of the great circle between these two places, and if you guide your eye along in this line, you may readily and truly perceive by what longitudes and latitudes you should sail, for mark well where this line crosseth the arches of latitude, and the lines of longitude, and that shows the true longitudes and latitudes of the arch of the great circle, according to your desire. The proof. Now the truth hereof will more evidently appear, if you compare the latitudes and longitudes which this line intersecteth with this table thereof calculated by Mr. * In the tenth Problem of sailing by the arch of a great circle. Norwood for every fifth degree of longitude. Longitude Latitude De. or difference of longitude. D. Deg. m. 310 00 32 25 305 05 35 52 300 10 38 51 315 15 41 24 320 20 43 34 325 25 45 24 330 30 46 54 335 35 48 07 340 40 49 04 345 45 49 47 350 50 50 15 355 55 50 31 360 60 50 33 005 65 50 23 010 70 50 00 Now you may hereby see, that the line B C in the point G doth cross the 305, or the 5 degree of longitude from B almost at the arch of 36 degrees of latitude, just as the table shows it should, at 35 degrees, 52 minutes of latitude. Again, the line B C doth cross the 310 or the 10 degree of longitude from B in the point h, almost at the arch of 39 degrees of latitude agreeing with the table which shows it to be in 38 degrees 51 minutes. And so in all the rest it so nearly agrees, that if you take any care in making of this blank Map to draw the arches of latitude, and the degrees of longitude truly, you shall not need to use any calculation, though you are well skilled therein, for the thing hereby may be much more exactly known, than the course of a ship can be steered. For the further explaining of this, take another example, An example of two places in one parallel. which shall be of a parallel course. Suppose two places to be situate in the parallel of 40 degrees of North latitude, and their difference of longitude to be 70 degrees, the one being in 300, the other in 10 degrees of longitude, and it is desired to know what longitudes and latitudes the arch of a great circle being drawn between these two places will pass through. To perform this, first in the line A B mark out the latitude of the one place which is 40 degrees at E. Then in that same arch count 70 degrees of longitude, from E to F, and there make a mark for the other place, thus the two places being set down upon the blank map according to their latitudes and longitudes, draw a strait line from E to F, and this will represent the great circle, which is to be drawn between the two places, and the intersections which it maketh with the arches of latitude, and the lines of longitude will show the true longitudes and latitudes by which this great circle ought to pass. Proof of the work, by its agreement with calculation. Now for the proof hereof though Mr. Norwood in his Book, hath not calculated the longitudes and latitudes of the arch of a great circle in such an example as this: yet his rules show how to do it, and according to them I have calculated this table, so that you might see the exactness of this way by its agreement with the table. Longitude Latitude Deg. De. De. m. 100 parts 300 or difference of longit. 00 40 00 these minutes are in 00 305 05 41 34 57 310 10 42 53 88 315 15 43 55 92 320 20 44 42 70 325 25 45 15 25 330 30 45 35 58 335 35 45 41 68 335 35 45 41 68 340 40 45 35 58 345 45 45 15 25 350 50 44 42 70 355 55 43 55 92 360 60 42 53 88 005 65 41 34 57 010 70 40 00 00 Note if you draw lines by every degree of longitude in the blank Map, as there is by every degree of latitude, you may then find out the latitude of the great circle for every degree of longitude. But this pains will be needless, yet the lines may be for some use, for if your two places differ more in latitude than they do in longitude, than it will be your better way to set down by what longitudes the great circle doth pass at every fourth or fift degree of latitude. Now that the longitudes and latitudes of a great circle thus found out will be exact enough for the Seaman's use, The longitudes & latitudes of the arch thus found out will be exact enough. if you be any thing careful and handsome in drawing of the lines of latitude and longitude true, observe what Mr. * See Master Norwood in his Problems of saling by a great circle. Prob. 9 latter end. Norwood saith to this purpose, his words are these. Having spoken before the calculation hereof: but notwithstanding all that hath hitherto been said, it may seem hard to direct a ship, and to keep such a reckoning as may be agreeable to this method of sailing. And indeed as it is in a manner impossible, so neither is it necessary that a ship should always persevere exactly in the arch of a great circle. It may suffice, and it is almost the same in effect, if a ship be so directed that she go near this arch. Which how to do he showeth in the next problem, wherein I shall follow him, only whereas he directs you to find out the longitudes and latitudes of the arch of the great circle, by calculation, I have showed you how to save that labour, and yet find it out sufficiently exactly for your use. Having therefore found but the longitudes and latitudes by which the great circle must pass, as is before showed, How to use the longitude and latitude being found out. you must likewise provide you a blank Sea-chart, drawing it either by the lesser or larger Meridian line, as is before showed. Then prick down in this chart the latitudes through which the arch of the great circle doth pass at every tenth degree of longitude Then if your chart be of the lesser size, you may with your compasses draw an arch of a circle through those pricks, and this arch will represent the great circle between the two places. But if your chart be of the larger size, and so your compasses be not large enough, to draw this circle; or else you are forced in regard of the length of the voyage to make two or three charts' for it, than you may prick down the longitudes and latitudes of the great circle for every fift degree of longitude, and with your ruler, draw little strait lines from one prick to another, and yet these lines will represent the great circle well enough. And thus the great circle being drawn upon the chart, you may easily by the former directions in the use of the chart, see what point you must steer upon at the beginning of your voyage, and afterward altering your course by half a point at a time, It is not good to steer upon quarter points, because they are not so visible in the Compass, neither is it good to alter your course too often. you may keep as near to the arch of the great circle, as either you need or can expect to do. Now because Mr. Norwood hath sufficiently explained this in the example of the Summer-Ilands, and the Lizard, I shall pass by that example, only setting it down upon the chart, and refer you to his directions, and show you the like in a parallel course. Suppose you were to sail from the coast of Virginia to the coast of Portugal between two places lying in the parallel of 40 degrees north latitude, and the difference of longitude between them is 70 degrees, the first place being in * These places are not set down according to their true Longitudes, it is only the difference of Long. which I respect. 300 degrees of longitude, and the second place in 10 degrees of longitude, and you would sail by the arch of a great circle, between these two places. geometrical diagram The several places where you altar your course. The course you steer. The dist. or way sailed. The Longitude The Latitude Deg. P. Deg. m. Deg. m. P. 1 from N to a E N E 4 09 305 0 41 34 57 2 from a to c ½ 7 69 315 0 43 48 80 3 from c to e E b N 7 26 325 0 45 13 22 4 from e to f ½ 4 93 332 0 45 42 70 5 from f to g East 2 09 335 0 45 42 70 6 from g to h East 2 09 338 0 45 42 70 7 from h to i ½ 4 93 345 0 45 13 22 8 from i to k, E by S 7 26 355 0 43 48 80 9 from k to l ½ 7 69 005 0 41 34 57 10 from l to P E S E 4 09 010 0 40 0 0 The Sum 52 12 You must not think to find these courses and distances which I have set down in this table, How to work upon a larger. Chart. can be so exactly found out by the former general chart, which is drawn by i lesser Meridian line, but if you draw two or three blank charts' by the larger meridian line, in two or three sheets of paper, you may then find them out easily, and as exactly as need be. In these several charts', you may set down your daily courses and distances, and then when you please you may prick down the sum of these reckon upon the general chart, and thereby the better see whereabouts you are in respect of your whole voyage. Thus you may easily, know the several parts, and the total sum of your voyage at any time. Or else you may keep account of such a voyage as this, and find out all your distances and courses upon one blank chart, A way to avoid drawing of many Charts. drawn in a sheet of paper, (or less if you please) as in the figure following. But I would not wish you to scant yourself to so small a chart as this is, this being so little, only in regard of the littleness of the book, and so the lines are broken off, oftener than otherwise you need to do. Now in this following chart, being fitted to the latitudes you must sail under, first set down your first place N according to the latitude thereof, which is 40 degrees, then prick down the latitude of the great circle, at the first fift degree of longitude, which is 41 degrees, 34 minutes at a, then laying your ruler from N to a, prick out the line N a, which will represent the arch of the circle from N to a. Then the latitude of the circle for the next 5 degrees, is 42 degrees. 53 minutes, or 88/100 parts, this must be set down at R, and then draw the pricked line from a to R, so you have the arch of the circle from N to R. Now if you would know what course you must steer: by your scale of rumbes you shall find that from N to a, the course is E N E, and the distance from N to a measured in the meridian line, is 4 degrees, 1/10, or 41 tenths of degrees. And here now because the rumb line doth run above the arch of the circle at a, I leave this course and alter my course half a point more towards the east. Also in regard of the shortness of the chart, I am forced to break off the arch of the great circle at a, and set down the latitude thereof in the first meridian again at a, and set down the latitude thereof in the first meridian again at a, drawing a line from a to a, then 5 degrees from this meridian that is 10 degrees, Take these Latitudes out of the Table. page 56. from the first place N, I set down the latitude of the great circle, which is 42 degrees, 53 minutes or 88 parts at b, and 5 degr. from b, that is 15 degrees from N, I set down the latitude of the circle, which is 43 degrees, 55 minutes, or 92 at c, and prick out the lines a b and b c, which represent the great circle, then by a scale of rumbes, I set off 6 rumbes and a half, which is the black line a c, which almost meets with the pricked circle at c, and the distance from a to c, is 7 degrees 7/10, as you may find by measuring it in the meridian line. And note though the rumbe line and the arch of the circle, do not here close exactly, yet it is no matter: for I have drawn it thus to even 5 and 10 degrees that it might agree with what hath been before said. geometrical diagram Here again because of the shortness of the chart I am forced to break off the circle, & the rumbe-line, & set them in the first meridian at c, then 5 deg. from the meridian at c that is 20 deg. from N, I prick down the latitude of the arch, which is 44 deg. 42 minutes, or 70 parts, at 20; and five degrees from this 20, I prick down the latitude of the arch in that longitude, which is 45 degrees, 15 minutes, or 25 parts, at c, then I draw the pricked lines from c to 20, and from 20 to e, which represent the arch, and I likewise draw the rumble line N by E from c to e, which doth very nearly concur with the arch at e, and the distance from c to e is 7 degrees, and almost ●/10 or as in the the table, 7 degrees, 26/100. Here again by reason of the shortness of the chart, I am forced to break off again, and setting the latitude of this point e, in the first meridian at e, 5 degrees from this I set down the latitude of the arch of the great circle belonging to that longitude, which is 45 degrees, 35 minutes, or 58 parts at 30, this meridian is 30 degrees distant from the first place at N. And then 5 degrees from this, which is 35 degrees from the first meridian at N, I set down the latitude of the arch, which is 45 degrees, 41 minutes, or 68 parts at g, then I draw the pricked lines from e to 30, and from 30 to g; this represents the arch, now at the point e, I altar my course half a point more to the Eastward, therefore by the scale of rumbes, setting off 7 points and a half from the point e, I draw the line e f, which is N by E, half a point to the East: and having sailed upon this point from e to f, the latitude will be 45 degrees, 42 minutes, or 70 parts, and the difference of longitude from e is 7 degrees, and the distance from E is 4 degrees, 9/10, but the difference of longitude from the first place at N, is 32 degrees. Lastly, because now I am as fare to the Northward as the arch of the great circle will allow me, I here at f alter my course half a point more, and so sail from f to g full East, so I have altered my longitude in all 35 degrees, and am come just one half of the voyage. Now to perform the other half, you must continue to do as you did before, first prick out the great circle, and then find out the rumbes you must sail upon from one point to another, which you may alter now and then half a point, and so you may lay the Pole in the same order and proportion that before you raised it, as you may see by the table before, page 61. CHAP. VI Showing some observations which may be of use in all these three kinds of sailing. HAving showed you how to sail, either by the Rumbe that leads from one place to another, or else by an arch of a great circle extended between two places, I shall now lay down some observations, which may be useful in either of these ways of sailing, for sometimes it is best to use the one way, sometimes the other, in some voyages it is best to sail by the Rumbe, in some voyages it is best to sail by the arch, in some voyages it is the best way to use both, and to keep neither to the rumbe nor to the arch exactly. In voyages to the West-Indies, though the nearest way, be by the arch of a great circle; and though the way by the direct Rumbe, lies very well: yet it is usual in these voyages to steer wide from both these nearer ways, viz first, to steer much more to the Southward than the course lies, until they come into the latitude of the place, and then to run their course West, until they arrive at their desired port. And this way is very good, especially when you sail unto a little loan Island. To get the benefit of the wind. For first, by sailing toward the Line, you shall gain the benefit of the Tradewind (as they call it) which doth most constantly blow between the North and East, between and near the Tropics. Secondly, hereby you may be sure not to overshoot the Island you would sail to, To avoid overshooting the place you go to which otherways may easily be done. For it is an hard matter in a long voyage, to steer your courses so exactly, and keep your account of your way so perfectly, as not to miss some few leagues: and beside if this could be done, yet the courses and distances can not be so exactly known, because the true longitudes of places one in respect of another is not so exactly found out, as is to be wished for. And if by either of these causes, when you shall come to the end of your reckoning, you shall chance not to be in sight of the Island, you will then be at such a loss, that you will not know which way to sail to find it, whether Eastward, or Westward, and so must be forced to wander at random until you have a sight of some known place, by which you may know how the Island bears from you. Therefore in sailing to such a place as this; it is the best way to be sure to get into the latitude of the place, a good while before you come to it, and then sailing near that latitude, you shall be sure not to pass by it without a sight of it. 2 To get a wind. In voyages from the West-Indies, the usual way is first to sail much more northerly, than the true Rumbe doth lie, and this likewise is to get the benefit of the wind, for as the wind lies most Easterly toward the Equinoctial, so it blows most westerly towards the Pole: also this way is the nearest way, because it lies near the arch of a great circle. But many Seamen not knowing so much, and especially keeping their reckoning upon the plain chart, this convenience might prove an inconvenience to them for they are many times at their journey's end 150 or 200 leagues before they are ware, and so might easily overshoot their port, and lose themselves, but that they sail to the main land, or great Islands that they cannot pass by. 3 The inconvenience of sailing in a parallel. Now as for these causes you sometimes stray from the rumb or arch which lies between the two places, so there is another consideration which may be a sufficient reason for a little wand'ring, sometimes out of the way, and that is the inconvenience that there is, in sailing far upon a course of East or West. Because you must always depend upon your dead reckoning which is subject to much mistake, having no way to correct it by observation. This parallel sailing makes the journey many times seem tedious. As a man that travails in an unknown way, thinks the miles and the way to be longer than indeed they are, whereas he that knows the road and how fare it is from place to place, goes on more cheerfully. Therefore the labour will not be lost if you go sometimes a little out of your way for this consideration, that so you may have the more certainty of your account. Indeed the way of sailing by the arch of a great circle doth very much help in this, How to avoid sailing in a parallel, partly. as I have showed at large in the former Chapter, but yet if you keep yourself in your yoyage too strictly to the arch, you must run much of your way in a parallel, or very near it. As in the example of the parallel voyage in the last chapter, the difference of longitude between the two places being 70 degrees, if you keep to the arch, you must first sail E N E, till you altar your longitude 5 degrees, then half a point more Easterly, till you altar your longitude 10 degrees more, than you must sail N by E till you altar your longitude 10 degrees more, that is in all 25 degrees, but afterward the 7 degrees which are set down half a point off the East, and the three degrees full East, is little better than a parallel course: then again this being the middle point of your voyage, you must sail 10 degrees more in the same proportional course, so that of the 70 degrees of the whole voyage you must sail 20 of them near the course of East and West. Now you shall see how easily this may be avoided, How to avoid sailing in a parallel, totally. and that several ways: first let the courses be continued as before till you come to 25 degrees difference of longitude, which is at e in the last * Page 63. chart, then if at this point you leave the great circle a little, and keep on your course still upon the 7 rumbe N by E, till you come to 35 degrees of longitude, your latitude will be 46 degrees, 36 minutes, or 60 parts, differing from the latitude of the arch 55 minutes, but your distance for these 10 degrees of longitude will be but 7 degrees 09/100, that is but 7/100, more than the other way, which makes but 4 miles, which makes 4 miles, which is so little that it is not to be regarded in respect of the distance in these 10 degrees, being 425 miles. Again if you begin sooner to swerve from the arch, yet the difference of your way will not be much, as you may see by this table which differs much from the other in the rumbes, latitudes, and longitudes, but yet it differs but little in the total sum of the distances, being but 20/100, which is but 12 miles. Difference of Longitude. The course or Rumbe. Distance or way sailed. The true Longitude. The Latitude. Deg. m. Deg. P. Deg. Min. Deg. Min. 10 0 E N E 08 10 310 0 43 6 10 0 ½ 07 46 320 0 45 16 15 0 E by N 10 60 335 0 47 20 15 0 E by S 10 60 350 0 45 16 10 0 ½ 07 46 360 0 43 6 10 0 E S E 08 10 010 0 40 0 The Sum 52 32 See the general Chart, page 58. Once * again if you try another way, as if you should sail from the first place N, in the latitude of 40 degrees, upon the 6 rumbe E N E, till you have altered your longitude 35 degrees, which is the half of your voyage, you will then find yourself to be at Q, in the latitude of 50 degrees, 12 minutes, and your distance or way sailed, will be 26 degrees 63/100. Then if from this point you alter your course, and sail upon the 6 rumbe Southward, again, E S E to P, your distance will be as before 26 degrees 63/100, and so you will come to the latitude of 40 again, being the place you desire, so that your whole voyage by this reckoning will be but 53 degrees 26/100, which though it be somewhat more than the distance by the arch, which is 52 degrees, 08/100, yet it is less than the distance of these two places, in the parallel which is 53 degrees, 62/100, and the loss of the way will be well recompensed, especially in sailing from the Indies, by the advantage of the wind, and by the certainty you may keep off your account, sailing always upon the sixth rumbe. By this also it appears, that the longitudes and latitudes of the arch of a great circle, and the courses and distances which you are to sail upon, in tracing thereof, may be sufficiently found out in the tracing thereof, may be sufficiently found out by the ways before showed, so that there needs no other calculation for them, since a small wand'ring from them, will not alter the length of the way in any considerable quantity. Thus also you may easily avoid sailing upon a course of East & West. Which way soever your voyage lies, To avoid sailing East or West at any time. it will be your best way to steer no nearer to the East or West, than the 7th rumbe from the meridian. For so you may by your observation of the latitude, correct your account which otherwise you cannot do, and in doing thus, you will not go much out of your way. For suppose you were to sail between two places, under the Equinoctial line, so that the direct and nearest way between these two places, lies full East and West, yet you may sail between these two places upon the seventh Rumbe, and go not much out of your way. For example, let the two places be distant 10 degrees, if you first sail the one half of your way upon the 7 Rumbe, that is till you have altered your longitude 5 degrees, you will raise the Pole very near one degree, viz. 994/1000, and your distance or way sailed will be but 5 degrees, 1/10 almost, viz. 5 degrees, 098/1000. Then if you altar your course, and so lay the pole as much upon the 7 Rumbe the other way; so you will come to the place desired, having run the same distance as before: thus in a voyage of 10 degrees, or 100 tenths or double leagues, you go not fully 2 tenths out of your way, which is not one in 50, which will be well recompensed, because in sailing thus upon the 7 Rumbe, you may by the observation of the latitude correct your account. For this rumbe raiseth or depress the Pole almost one degree in sailing of five, which will serve very well to correct your account by, and therefore I will not persuade you to go any further, out of your way. And if you direct your course according to this rule, this is the farthest that you need go out of the way. For in all other places, the arch of the great circle, which is the nearest way between two places, doth always lie somewhat between the Pole, and the parallel of East and West, and therefore in raising the Pole upon the 7 Rumbe, you will not go altogether so much out of your way, if you observe to incline your course always toward the Pole. Thus you see you need not sail directly East or West at any time unless it be when you fear you shall pass by some little loan Island, as is aforesaid. 4 Make use of the Sea chart and the Map, or Globe in plano, pag. 52. both together. In voyages near the Pole it is best to guide yourself as near the arch of the great circle as you can, having respect to the former considerations. And in these voyages your best way will be to keep your reckon, both upon the Chart and upoe the Map, so the one will help the defects of the other. As for example, the Rumbes from one place to another, are easiest to be found out upon the chart, but the distances of the places, and the arch between them will be found out more easily by the Map. For the places being set upon it according to their longitudes and latitudes, will be more composed and hold better conformity to the Globe, being in a manner the same with it, especially within 10 or 20 degrees from the Pole. Thus then if you are to sail upon discovery, and not to a certain place, your best way will be; by the rumbe which you sail upon, and by the latitude which you find by observation, to find out the longitude of the place you are in at any time by the chart, and then setting down this place, according to its longitude and latitude thus found out, upon the Map, you may more readily see its distance and position from any other place. But if you are to sail to a certain place, whose longitude and latitude is known, than it will be best, to set the place down first in the Map, and so to find out the longitudes and latitudes of the great circle between them, which being pricked down upon the chart, you may see how to steer your courses to the place appointed. 5 To find out if there be any current. In all your voyages it will be a good way to keep your accounts upon two blank charts', upon the one, you may keep your dead reckoning, upon the other, your corrected account. And the benefit hereof will be this: if you be careful to keep your dead reckoning outward, and homewards true, you may thereby whether there be any current between the two places, and which way that current sets, and how fast it runs. For if there be a current between the places you sail to, which way soever the current sets, your reckon outward and homeward will not agree, and indeed you can keep no good account of your way, till you know which way and how fast the current runs. For example: first, suppose the current runs East 12 miles a day, and you sail against this current according to your account, by your Log-line (which is the best measure of your dead reckoning) 60 mile's West in a day, your true pace or distance will be but 48 miles in a day, because the current will set you back 12 miles, which substracted from 60, there remains but 48. On the contrary, if you sail Eastward in this current, (and so sail with it) 60 miles a day according to your log-line, than your true motion will be 72 miles a day, because the current will set you forward 12 miles more than you seem to go. Secondly, suppose the current run East one mile an hour, if you sail in this current N E or S E, look how many hours you sail, so many miles the stream will set you more forward in your longitude, than you are ware of; and yet your latitude, will fall out according to your account. On the other side if you sail N W or S W in this current, the current will drive you backward so many miles, and yet your latitude will be according to your account. Thirdly, if the current run upon any rumbe between the meridian, and the East or West, than your true motion will differ from your dead reckoning, and also from your account by observation, both in longitude and latitude, so that until you know in some sort both which way and how fast the current runs, you can never keep a good account of your way and the only way to find out the current is to keep a good account by the log-line outward and homeward, and by setting this down upon your blank chart you may plainly see which way the current runs, and how fast, as Master Norwood hath very well demonstrated in the end of his Seaman's Practice. 6 To alter your course as seldame as you can. In most voyages, it will be good to keep your course constantly, (or as much as you can) upon one and the same rumb, for so your account will be more easily and certainly kept. For at every shifting of your course, the true point that you are in, cannot be so certainly known, but that you may misreckon somewhat both in the latitude and longitude thereof. For the latitude by which you have the most certainty of your place, may be mistaken 5 or 10 minutes, by any instruments ordinarily used, and this may cause 20 or 30 minutes error in the longitude, and this error, at every changing of your course, may as well chance to be increased, as to be balanced one time with another, whereas if you steer your course constantly upon one rumbe, as it will avoid the trouble of drawing so many rumbe lines, so there can be no greater error in your account, than there shall happen to be in the last observation of the latitude, which cannot be much. But if you are forced to shift your course in regard of the wind, then in the correcting of your account, observe the rule in the second case of the sixth Proposition of the plain chart, drawing a strait line from your first place to the place you are then in. 7 To gain sight of land when you can. It will not be amiss as often as you can, to get a sight of such Capes, Headlands or Islands, as lie near your course, which being standing marks, will give you certain knowledge, whereabouts you are, and so you may the better direct your course and perfect your account. 8 To observe the variation of your Compass. Lastly, you must have an especial care of your compass, that it be every way perfectly and exactly made, and likewise you must be as careful, that you steer your course exactly upon that rumbe you reckon on; to which end, not only the Steersman must be diligent to keep the ship to the course appointed, but you must be frequent in observing the variation of the compass, which may be so well performed by the Sea rings in use among Seamen, that no Instrument can be devised fit for the purpose, this variation being known, must be allowed for in your account, that so you may know the true rumbe you sail upon, without which there can be no true account kept. CHAP VII. Of sailing by a great Circle. ALthough the way already proposed for the finding out of the arch of a great circle, Another way to find out the longitudes and latitudes of the arch of a great circle. between two places, is the most easy and plainest way that can be, yet because it is not so general as to take in all places, but is only to be used when the two places are both, on the one side of the Equinoctial: as also it may seem somewhat defective, because it doth not show the distances of places; I have therefore here added this second way, partly for variety, and partly to supply the defects of the former. But as this way is more artificial, so it is more difficult, both in the demonstration, and practise, which cannot be avoided. But you may the better bear with it, because you will seldom be forced to use this way, but may very well content yourself with the other in most voyages. This and all other questions of this nature, concerning the resolution of any spherical triangle, may very easily be performed by the Globe: but because the Globe is a chargeable Instrument, and so every one cannot have it: therefore several men have for several uses invented several ways to project the Globe upon a Plane. You may see the several projections thereof in Mr. Gunter's book of the Sector. The fittest for this purpose will be that of Gemma Frisius, which is most used in Maps of the whole world, the projection whereof is, as followeth. First, draw the circle A D B C, How to describe the Globe in plano. and divide it into four parts or quadrants, by the cross diameters, A B and C D, then divide each quadrant, into 90 degrees and number them as in the figure, then if you keep one end of your ruler fixed at the point A, and lay the other end to the several degrees in the lower Semicircle, D B C, so you shall divide the Diameter C D into its parts, which are half tangents. In the same manner, if you keep one end of your ruler fixed in the point C, and lay the other end to the several degrees of the Semicircle A D B, you may divide the diameter A B into half tangents. Having thus divided the circumference and the diameters, they must guide you in the drawing of the meridians and the parallels, How to draw the Meridian's and Parallels. which are all parts of perfect circles, and you may find their centres by these three points. First, for the Meridian's, they all concur in both the Poles A and B, and their third point is their correspondent degree in the diameter C D. Then for the parallels, two of their points are their degrees in the outward circle, and their third point is their correspondent degree in the diameter A B. By these three points you may find the centre, and so draw the arch as is showed in the first chapter. But to save that labour, you must know, that the centres all lie in the diameter lines, which must be extended beyond the circle, and then the centres are thus found out. The diameter C D being divided into half tangents as before, if for every degree you account two, beginning from the centre E, so you shall have the centres of the meridians. Then if you set one foot of your compasses in that centre, and open the other to the Pole A or B, it will pass through the correspondent degree, or third point in the diameter C D: on the other side of the centre, so the meridian will be drawn upon the one side. Then with the same distance of your compasses, you must draw the other answerable to that on the other side. Then keeping your compasses yet at the same distance, set one foot in the centre E, and with the other, mark the diameter A B, both above and below, and these marks shall be the centres of the parallels. Then set one foot of your compasses in these centres, and close your compasses, till the other foot reach to that degree of latitude in the outward circle and so draw that arch from side to side And if you find that the arches thus drawn, do pass exactly through their three respective points, in the circle and diameter, your work is true, otherwise not. geometrical diagram And thus you may easily do for any other degree under 45, but when you come to the degrees above 45, than you must extend the line C D, and laying one end of your ruler to the point A, and the other to the degrees of the upper semicircle, you may divide that part of the line without the circle as you did before that part which was within, into half tangents, and so doubling your degree find out the centre thereof. Or else when you draw the former meridians, you may remember to turn about the compasses, and mark the line C D without the circle; by these marks you shall divide the line into half tangents, and so you may find out the centres as before. How to help yourself when your compasses will not reach. But because some of these centres, will fall so fare without the circle, that your compasses will not reach them; you may then bridle a thin ruler, that will bend, with a double string like a cross bow, and then by twisting the string together, you may by little and little, set it to what bent you please, till it shall cut the three points of your arch you would draw, and then with your pen, you may draw your arch, which if the ruler be all of one thickness, and so bend in all places alike, it will be very true. Your compasses will reach the centres very well, till you come to 60 degrees, but afterwards you must be forced to use this or some such like way to help yourself. The larger you make your draught, and the more meridians and parallels you draw in it, so much the better it is, therefore if you can, make it so large that you may draw meridians and parallels through every degree, which you may do very well in a sheet of large paper in a lesser draught, you may draw every second degree, which is the least I would wish you to do. Lastly, to save time and labour in drawing of these blanks for every question: when you have made a little trial and know how to draw them, then draw two good large ones of one and the same size, which you may do very well by drawing the same lines in both, before you stir your compasses from their distances: then six the one of these to the other by their centres so that they may be turned round, and the uppermost of these being drawn in fine thin paper, and a little oiled, you may easily see through it all the lines of the other. And thus you shall have an * This will be somewhat like Mr. Blagraves' Mathematical Jewel. Instrument whereby this and most other questions of spherical triangles may be resolved. Having thus shown the drawing of this projection, I shall now come to show you the use of it in several examples. The first example shall be the voyage from the Summer-Islands, to the Lizard, the latitude of the Summer-Islands being 32 degrees, 25 minutes North, The use of this projection in finding out the great circle. and the latitude of the Lizard, being 50 degrees North, and their difference of longitude being 70 degrees, and it is required to know first the Latitudes and Longitudes by which the arch of a great circle drawn between these two places doth pass. Secondly, the angle of position from the first place to the second. Thirdly, the nearest distance between the two places. To perform this, first you must set down upon your draught, First example. the first place which is the Summer-Islands, according to the latitude thereof which is 32 degrees, 25 minutes in the outmost circle at S. * Note well which way the first place bears from the second. And herein you must regard how the second place doth bear from the first If the second place lie West from the first, than you must set down the first place on the East or right side of the circle: but if the second place lie Eastward from the first, as it doth in this example, than you must set down the first place on the Westside of the circle, as it is here at S. Then from the point S through the centre E, draw the diameter line S E K, and cross it at right angles with the line M E N. Then accounting 70 degrees (which is the difference of longitude of the two places) in the diameter C E from C to 70, mark that meridian arch, & thereupon mark out the latitude of the other place which is 50 degrees at L. Thus the two places are set down according to their latitudes and the difference of their longitudes at S and L. Now to help you to draw the arch of a great circle between these two places S & L, you have these three points S L & K, by which you may find the centre of the arch which is at M, in the line N M, therefore set one foot of your compasses in M, and opening the other to any of the three points, draw the arch S L K. This arch is the great circle that passeth through these two places, by which you shall find all the things desired. The longitudes and latitudes of the arch. As first, if you would know by what longitudes and latitudes this arch doth pass (which is the thing most needful to be known) if you trace the way of this arch, through the meridians and parallels of the draught, you will find them to agree with the former table hereof for every fifth or tenth degree. For at 10 degrees of longitude from S, the arch passeth through 39 degrees of latitude: at 20, through 43 ½, and so of the rest. The angle of position. Secondly, if you would know the angle of position from S to L, then observe in what point the arch S L K doth cross the line N M, which is at T, then take the distance N T, and measure it in the semidiameter C E from C toward E, and it will reach almost to 49 degrees, which shows the angle of position to be North-Easterly almost 49 degrees. The distance of the places. Thirdly, if you would know the distance of the two places, you must with your compasses take the distance of the two places S and L, and measuring it in that meridian, which agrees with the angle of position, viz. 49 degrees, you shall find it will reach from A to V, now if you reckon the degrees of the parallels of latitude from A to this point V, you shall have the distance which is 53 degrees and a half. Likewise you may measure any part of this arch S L, in this meridian A V, if you always set one foot in S, and open the other to the point required, and then set one foot in A, and the other will show the distance of that place. Thus the distances will be found out as exactly as by any other Geometrical way, but in regard of the smallness of the projection, you may mistake some few miles or leagues. But if you were to sail from the Lizard to the Summer-Islands, The difference in sailing forward & backward by the arch. than you must first set down the latitude of the Lizard on the other side of the circle (as I noted before) & so the work will fall out much as it did before, for the longitudes and latitudes of the arch will be the same, only accounting them backward, the distance will be the same, viz. 53 degrees and a half, only it must be measured in another meridian, according to the angle of position from the Lizard, which will be about 81 degrees, so that in effect, all is the same, only the angle of position, which is of little use, but to find out the scale of the distances. So that if you regard it well, one labour will serve to find your way outward and homeward. I might here show you how to perform the parallel question, but because such questions may with more ease and certainty be performed by the former way, I shall not spend time about it, I shall only instance in two sorts of voyages, which cannot be performed by the other projection, and in such cases as these there will be some need of this way, and not else. First, when one of the places is under the Equinoctial, and the other toward one of the poles. The other is when the one place hath North latitude and the other South. Suppose you were to sail from the Island of St. Thomas, Example of 2 places in another manner of situation. which lies under the Equinoctial, and hath about 35 degrees of longitude to the straits of Magellan, which hath about 53 degrees of South latitude, and differs in longitude from the former place 9● degrees to the Westward: now it is required to find out the arch of the great circle between these two places, and the longitudes and latitudes of this arch, with the angle of position, and the distances of the two places. To perform this, first set down the Isle of St. Thomas, which is under the Equinoctial, at the one end of the Equinoctial line at D, then accounting 90 degrees of longitude from D to E, there is the meridian of the straits of Magellan, whereupon you must mark out the latitude thereof, which is 53 degrees at W, so you shall have these three points D W C, by which you may find the centre, and draw the arch D W C, now this part of the circle from D to W, is the arch of the great circle, which lies between these two places; by which you may find all the other things required. As first for the longitudes and latitudes of this arch, they are found out by noting where it crosseth the circles of longitude and latitude in the draught, which you shall find for every tenth degree to be as in this table. Long Latitude Deg D. m. 10 12 58 20 24 25 30 22 34 40 40 28 50 45 31 60 48 58 70 51 16 80 52 35 90 53 0 Secondly, for the angle of * The difference of longitude being just 90 deg. the angle of position is ready measured in the semidiameter E B being the distance B W, which is 37 degrees, but at other times must follow the rule. position between the two places, this is showed by the arch DW crossing the Semidiameter E B, so that if you take the distance B W, and measure it in the diameter C D, it will reach from C to 37 degrees, which is the angle required, & the situation shows it to be South westerly. Lastly, for the distance of the two places, if you take the distance D W and measure it in the 37 meridian line (according to the angle of of position) it will reach from A to the Equinoctial line, Likewise the distance D W needs no other measuring, but must needs be 90 degrees. which shows the distance to be 90 degrees. The last example shall be of two places, the one being on the one side, and the other on the other side of the Equinoctial. As, suppose you were to sail from the Summer-Isles, to the Cape of good Hope, Example of places in another situation. the latitude of the Summer-Isles, is 32 degrees 25 North, and the latitude of the Cape of good Hope is 35 degrees south, and suppose the difference of longitude between these two places to be 90 degrees, and it is required to find out the arch of the great circle between these two places, according to the longitudes and latitudes thereof, the angle of position and the distance of the two places. To perform this first you must set down the first place according to the latitude thereof in the outmost circle at S, and draw the diameter S K, to which you must draw the line N M squirewise at right angles, then counting the difference of longitude which is 90 degrees from C, the meridian of the Cape of good hope will fall in the line A B, which you must mark out according to its latitude 35 degrees South at X, then by these three points S X K, find the centre which will be in the line M N extended, and so draw the arch S X K. Now first for the longitudes and latitudes of this arch you may find them by seeing how this arch doth cross the circles of longitude and latitude in the draught, which for every tenth degree of longitude is as followeth. Long. Latitude Deg. Deg. m. North Latitude decreasing. 0 32 25 10 26 57 20 19 40 30 11 18 40 02 05 50 7 19 South Latitude increasing. 60 16 7 70 23 48 80 30 05 90 35 0 Then for the angle of position, you shall find it thus, mark where the arch S X K, doth cross the line M N, which is at Z, then with your compasses take the distance Z M, and measuring it in the semidiameter C E, you shall find it will reach from C almost to 60, viz. 59 degrees 25 min. South easterly. Lastly, for the distance of the two places, if you take the distance of the two places in the arch S X, and measure it in this meridian, of 60, you shall find it will reach from A to Y, which is 107 degrees and a half, or more exactly 107 degrees, 34 minutes. If you make two of these draughts, and join them together, as I noted before, your work will be somewhat more easy and readily performed. For then in any question of this nature, you need but turn the Pole of the upper paper, to the latitude of the first place in the under paper. Then mark out the second place according to its longitude and latitude, in the meridians and parallels of the under paper. Then mark what circle of of the upper paper, passeth through these two places, this is the arch required. Now the longitudes and latitudes thereof, you may easily see by its crossing the meridians and parallels of the under paper. The distance of the two places, you shall know by counting the parallels in the upper paper between the two places. The angle of position is known by noting where the arch doth cross the Equinoctial in the upper paper. By this likewise most questions of Astronomy may be performed, as by the place of the Sun, to know the Declination of the Sun, his right ascension, and obliqne ascension, his amplitude at rising and setting, and by the height of the Sun to know the Azimuth and the time of the day, with many others but because some of these require more exactness, than any such Instrument will afford: and likewise they are somewhat beside my present purpose; I shall not now speak of them. CHAP. VIII. How to keep a perfect account. ALthough the ways I have already showed may be sufficient for the setting down of any reckoning upon your Charts: yet because it may be some trouble, if you often alter your rumbe, to draw a new rumbe line from every point: and besides the scale of the Chart being somewhat small, you will be subject to some mistake, in finding out the several points, and this mistake being often committed may come to some considerable error, I shall in this chapter show you, how you may increase your scale, and by a little help of your pen find out these points most exactly: and so you may set them down by the latitude, and their Easting or Westing, without drawing of the Rumbe lines, from point to point. First, for the increasing of your scale, which is the chief ground whereon the certainty of your reckoning will depend. In the figure following (which is the same with the plain chart before) The way to keep a perfect account is to work by a very large Scale. let each side of the square be supposed to contain but one degree of longitude or latitude (whereas before we supposed it to contain 10 degrees) so shall each degree by this means be divided into 100 very sensible parts, and be 10 times greater than the scale of your chart, by which means you shall keep your account as exactly as need be required. Then it will be necessary if you use this as an Instrument to draw lines through each of these 100 parts, crossing one another as those ten lines do; but for the better distinction, you may make every tenth line bigger than the rest, and every fifth line you may prick out. Then making the corner at A the centre, draw the arch of the quadrant B P 1 2 3 4 5 6 7 D, which you may divide either into 90 degrees, or else into the 8 Rumbes and their quarters, and it will be good to number them both ways, from B and D. Lastly, you must provide you a thin ruler or Index which must be divided as the side of the Quadrat is, into 100 equal parts. You may make this Index of a piece of thin paper, and oil it: or it will be better, if you get a piecc of fine clear horn, and make a scale of equal parts thereon, equal to the sides of the quadrat, and so fastening it to the centre A, you may turn it to any Rumbe in the arch, and very plainly see all the lines of the Instruments through it. The use of this Instrument might be showed in many propositions, but I shall only show you how to use it in two very necessary propositions, by the one of which you may keep your dead reckoning, by the other your true reckoning by observation. geometrical diagram To perform this or any such like question, you must first consider which way your course is in general, and lay the Quadrat so that the sides A B or A D, How to work the proposition upon the instrument. the one of them may represent the Meridian; and the other the line of East or West as in this example, the side A B must represent the Meridian, and the side A D the line of East. Then lay your horn Index to the Rumbe in the arch, and find out the distance you have sailed, upon the scale of your Index. Now mark very well the point, where your distance upon the scale ends, and what lines of the Quadrat meet in this very point, and if by your eye, you trace these lines to the sides of the Quadrat, the figures there, will show you the Northing or Southing, Easting or Westing as the question requires. As in this example, because your course in general is North-Easterly, let the side A B represent the North Meridian line, so the side A D, at the bottom, or B C at the top, will represent the lines of easterly longitude or distance. Then because your course is N by E, which is the first rumbe from the North Meridian Eastward, lay your horn Index upon the first rumbe from the line A B, which is the line A P. Then count your distance which is one degree or 100 parts upon your Index, and mark where this number ends, which is in the point P, where the line A P crosseth the arch B P. This is the exact point where you are. Now if you trace the line P B that runs through this point to the Meridian line, you shall there find that you are to the Northward of the place A 98 parts of a degree divided into 100 parts, likewise if you trace the line that runs through this point P to the bottom or top of the Quadrat, there you shall find that you have altered your longitudes 19/●●● parts and a half. Having thus found out the exact Northing and Easting H●w to set the Northing and Easting found out, upon the Chart. of this place from A, we will now to avoid the inserting of another figure) suppose this last to be a plain chart again, hahaving each side divided into ●0 degrees as before, and if you set down this place by the first proposition of the plain chart, according to the latitude and longitude thus found out, viz. latitude 00 degrees 9● parts, and longitude 0● degrees 19 parts and a half, you shall find that its place will be close by the corner A, at the figure 1. Likewise for your better proceeding afterwards, you may write down the longitude and latitude of this place 1, as you may see in the table following. Another example in the same voyage- Now suppose that you altar your course at this place 1, and sail from hence N N E, which is the second rumbe from the Meridian one degree, or 100 parts more. And it is desired to know the longitude and latitude of this second place. To know this lay the Index upon the second rumbe from the Meridian line A B, and count upon it your distance sailed, which is 100 parts, and observing well where this distance ends, which is where the line of the second rumbe doth cut the arch B P, which is in the cross at 2: this is the point that shows you the difference of longitude and latitude of this place from the last. Now by the lines that pass through this cross, you shall see the difference of latitude to be 92 parts, and the difference of longitude to be 38 parts. Now if you would know the true longitude and latitude of this place, in respect of the first place A, that so you may set it down upon your chart; you must add the Northing and Easting of both the places together thus. From A to 1 N by E 100 parts is, North 98. East 19 ½ From 1 to 2 N N E 100 parts is, N. 92. E. 38 The Sum of both is 1, 90 57 ½ So that the latitude of this second place is 1 degree 90 parts of North Latitude, and 0 degrees 57 parts and a half of longitude. And then setting it down according to the first proposition, of the plain chart, you shall find this place to be at the figure 2 near the corner A. And thus you shall find out the point of any place in your chart more certainly than other ways it is possible for you to do, and with less trouble, as you may see by the table following of the rest of the places, 3 4 5 6 7 in the chart. Note this well. But here you must observe that as in these places where the latitude and longitude doth still increase, you make use of Addition, so if your course lies so, that the longitude and latitude both decreaseth, you must make use of Substraction both in longitude and latitude. And sometimes the course may lie so, that the longitude may increase, and the latitude may decrease, than you must add the one, and subtract the other. your best rule to guide you in this, when to add and when to subtract is to observe your course in general, which way it tends, which you may easily see by your Chart. As for example Example. being come to the place 7 in the chart, whose latitude is 4 degrees, 57 parts, and the longitude thereof is 4 degrees, 57 parts, and from this place you sail S by W, that is upon the first Rumbe from the Meridian to the Westward, 100 parts, and it is desired to know what longitude & latitude you are then in. To perform this, first you may see by your course in general, which is South-westerly from the place 7, that both the longitude and latitude of the place you sail to must needs decrease, and therefore the difference of longitude and latitude which you shall find out by the Instrument, must be substracted from the longitude and latitude of the place 7, that so you may have the true longitude and latitude thereof. Now having considered this; by your Quadrat you shall find that the course being the first rumb from the Meridian, and the distance being ●0 parts, the difference of latitude will be 98 parts, and the difference of longitude 19 parts and a half: this substracted from the longitude and latitude of 7, shows the longitude and latitude of the place 8 where you are. The place 7 hath latitude 4 deg. 57 parts. long. 4 deg. 57 p. Subst. the differ of latitude 0 98 long. 0 19 ½ Rest for the place 8 latit. 3 59 long. 4 37 ½ Thus likewise you must do to find the longitude of the other places 9 10 11 12 13 A. * A good way for young beginners to avoid mistakes. And it will be a great help to direct you when to add, and when to subtract, if you join four of these Quadrats together, which you may very well do, if in a large sheet of paper, you draw a large circle with your compasses, and dividing it into four parts, make each of them like to this, and set the names of the points of the compass to the rumbes, as in the figure of the compass page 10. By this means without any trouble you shall plainly see the Easting and Westing, the Northing and Southing of your course. How to set down these distances in a table. But now for the better setting down of these distances, you must divide your table into four columns, for the four principal quarters, East, West, North, South, and set down therein according to their titles, the Easting or Westing, Northing or Southing of your course, which you find by the Instrument, and then in casting up your accounts, you must add altogether that you find under one column, but if you have any sums in contrary columns, as East and West, or North and South, you must subtract the sum of the East from the sum of the West; and the sum of the North column from the sum of the South, so often as you would find the longitude and latitude of the place where you are, as you may see by this following table of all the voyage outward and homeward. Course. dist. d pa. Nor d. pa. Sou, d. pa. East d. pa. west d pa Lat. d. pa. Lon d. pa The place A 0, 00 0, 00 Compa e this table and the Chart well together. from A to 1 N by E 1, 00 0, 98 0, 19 0, 98 0, 19 from 1 to 2 N N E 1, 80 0, 92 0, 38 1, 90 0, 58 from 2 to 3 N E by N 1, 00 0, 83 0, 55 2, 73 1, 13 from 3 to 4 N E 1, 00 0, 71 0, 7● 3, 44 1, 84 from 4 to 5 N E by E 1, 00 0, 56 0, 83 4, 00 2, 67 from 5 to 6 E N E 1, 00 0, 38 0, 92 4, 38 3, 59 from 6 to 7 E by N 1, 00 0, 19 0, 98 4, 57 4, 57 from 7 to 8 S by W 1, 00 0, 98 0, 19 3, 59 4, 38 from 8 to 9 S S W 1, 00 0, 92 0, 38 2, 67 4, 00 from 9 to 10 S W by S 1, 00 0, 83 0, 55 1, 84 1, 44 from 10 to 11 S W 1, 00 0, 71 0, 71 1, 13 2, 73 from 11 to 12 S W by W 1, 0 0, 56 0, 83 0, 57 1, 90 from 12 to 13 W S W 1, 00 0, 38 0, 92 0, 19 0, 98 from 13 to A W by S 1, 00 0, 19 0, 98 0, 00 0, 00 Thus I hope you see by this table, how that by adding or substracting, the Northing, Southing, Easting or Westing, to, or from the longitude and latitude of the place before, you may still find out the longitude and the latitude of the place you are in, for the keeping of your dead reckoning, as exactly as need be required. Now the other Proposition which is useful in the rectifying of your account, according to your observation is this. How to find out the true point of your longitude, by observation of the latitude. By the rumbe and the difference of latitude, to find the longitude of the place where you are, and the distance of this place from the other. Now the way of working this Proposition upon the Quadrat is thus, First, you must suppose one of the sides to be the meridian line, and the other to be the line of East or West as before, then lay your Index to the degree of the Rumbe, on which you have sailed, and keep it there. Then count upon the Meridian line from A, the difference of latitude of this place you are in (according to your observation) from the place you came last, and note where this number ends, and mark well the line that is drawn from this place, and trace it to your Index, and this point where this line crosseth the Index, is the point where you are. Now if you count the parts upon your Index from A to this point, so you shall see the true distance you have sailed in that Rumbe. And if you trace the line that runs through this point to the line of longitude at the top or bottom of the Instrument, there you shall see the difference of your longitude, or rather how fare Eastward or Westward it is from the former place. For example, Example. suppose you sailed from the place A. (whose longitude and latitude let be as before) N by E, that is upon the first Rumbe from the Meridian, so long, till by observation you found you had altered your latitude 98 parts of a degree divided into 10 parts: and you desire to know how far you have sailed in the Rumbe; and how far you are to the Eastward of the place A. First, lay your Index upon the line A P, which is the first Rumbe from the Meridian, and keep it there, then count the difference of the latitude, which is 98 parts, in the meridian line from A to B, then mark the line that is drawn from this part, and observe well where it crosseth the Index or Rumbe line, which is in the point P, and this is the very point where you are. Now if you count the parts upon your Index from A to this point, you shall find your distance sailed to be 100 parts or one degree. And if you mark the line that runs through this point to the line of longitude, you shall there see the difference of longitude which is 19 parts and a half to the Eastward from A, thus the proposition is fully resolved. But in this proposition you must take notice, that if you sail upon several Rumbes before you observe the latitude, than first you must find out the Rumbe which lies from your first place, to the place where you are according to your dead reckoning (as is showed in the sixth proposition of the Plane Chartley) and then laying your Index upon this Rumbe, work as aforesaid. How to work when the question is of more than 100 parts. Note that in working of either of these propositions by the Quadrat, if the distance or difference of latitude be more than a degree or 100 parts, than you may first find out what belongs to one degree, and then what belongs to the odd parts, and adding them together, you may find the truth as exactly, as if your Instrument were large enough to show both at once. But if your distance or difference of latitude be four or five degrres, you may then very well work the proposition upon your chart, without using the Quadrat, and need not fear error, if you be any thing careful, and work by a true scale of rumbes and equal parts. Hitherto I have showed you how to work these two propositions upon the plane chart, How to perform these two propositions in all latitudes. or in places near the Equinoctial: But because the degrees of latitude in the true Sea-chart must not be all of one and the same length, therefore neither can the true latitude nor longitude of places be set down by one and the same scale of equal parts. Therefore I shall now show you, how when you have found out the Northing or Southing, Easting or Westing of any place, as is before showed by the Quadrat, how you shall set it down in any latitude that it may show the true longitude thereof. For you must know, that though the Northing and Southing, doth show the true difference of latitude in all latitudes: yet the Easting and Westing found out by the Quadrat, doth not show the true difference of longitude in those places, which are any thing distant from the Equinoctial, I shall therefore show you how to perform these two Propositions in all latitudes. For example of the first Proposition. First, for the dead reckoning. Suppose you were at the place H, in the following figure, whose longitude is under the first Meridian, but it lies in 45 degrees of North latitude, and suppose you should sail from this place N by E to the distance of 100 parts, as before said, and it is desired to find the latitude and longitude of this place. First, you must find out the Northing and Easting by the quadrat as is before showed. The first way to perform this by the Chart itself. Now the Northing of this course being 98 parts, must be added to the latitude of the former place 45 degrees, so the latitude you are in will be known to be 45 degrees, 98 parts, then lay your ruler to this latitude in the sides of your chart, and draw a short line a 1, then consider how much the Easting of your place is, which in this example is 19 parts and a half, take this distance with your compasses out of the scale of latitude, about the middle latitude between these two places, and then setting one foot of your compasses in the Meridian of H at a, with the other cross the line a 1 at 1, and so you shall set down the place according to its true longitude which is 00 degrees, 28 parts from the Meridian of H, as you may see by the chart: and not 19 parts, and a half, as before it was under the Equinoctial, or by the plain chart. geometrical diagram Thus in this example, the middle latitude between the two places being 45 degrees, 50 parts, draw the line A c D by this latitude, & then counting the Easting of this course, which is 19 parts & a half in the side a b, the line b c drawn from thence will cross the former line of the middle latitude at c, then setting one foot of your compasses in this point at c, & opening the other to the centre A, this distance A measured in the scale A B will show 28 parts, which is the true difference of the longitude of the place 1 from the Meridian of H. And if thus you will make trial, you shall find out the longitudes and latitudes of all the other places, 2 3 4 5, etc. as they are set down in the table following. Course. dist. d. pa. Nor d. pa. Sou, d pa East d. pa. west d pa Latit. de. pa. (*) d. pa. Lon. d. pa The place H 45, 00 0, 00 from H to 1 N by E 1, 00 0, 98 0, 19 45, 98 0, 28 0, 28 from 1 to 2 N N E 1, 00 0, 92 0, 38 46, 90 0, 74 0, 82 from 2 to 3 N E by N 1, 00 0, 83 0, 55 47, 73 0, 82 1, 64 from 3 to 4 N E 1, 00 0, 1 0, 71 48, 44 1, 06 2, 70 from 4 to 5 N E by E 1, 00 0, 56 0, 83 49, 00 1, 24 3, 94 from 5 to 6 E N E 1, 00 0, 38 0, 92 49, 38 1, 41 5, 35 from 6 to 7 E by N 1, 00 0, 19 0, 98 49, 57 1, 50 6, 85 from 7 to 8 S by W 1, 00 0, 98 0, 19 48, 59 0, 30 6, 55 from 8 to 9 S S W 1, 00 0, 92 0, 38 47, 67 0, 57 5, 98 from 9 to 10 sweet by S 1, 00 0, 83 0, 83 47, 84 0, 82 5, 16 from 10 to 11 S W 1, 00 0, 71 0, 71 46, 13 , 102 4, 14 from 11 to 12 sweet by W 1, 0 0, 56 0, 83 45, 57 1. 19 2, 95 from 12 to 13 W S W 1, 00 0, 8 0, 92 45, 19 1, ●1 1, 64 from 13 to 14 W by S 1, 00 0, 19 0, 98 45 00 1, 39 0, 25 frō● to H West. 0, 7 0, 17 45, 00 ●, 25 0, 00 This table agrees with the former in most things, the chief difference is in the longitude, Note the difference of this Table from the former. which must be thus found out. By the Easting or Westing of your course, you must find out the difference of longitude thereof according to the middle latitude, and set it in the table under the last column, but one, noted thus (*) and this difference added or substracted as it ought to be in the last column, shows the true longitude of the place where you are. This will be your best way to keep your dead reckoning. How to perform the second proposition in all latitudes. Now for the second proposition, which is by the Rumbe, and the difference of latitude; to find the difference of longitude, and the distance. To perform this, first knowing the latitudes of the two places, draw a line in your Quadrant of latitude, by the middle latitude of the two places. Then count the difference of latitude in the scale of the quadrant of latitude, A B, and where the number ends, draw or observe the line drawn from it, where it crosseth the line of the middle latitude, then taking this distance in your compasses, measure it in the scale A B (increased if need be) then work with this number upon the quadrat as is showed before page 89, as if this number were the true difference of latitude between the two places, so you shall find out the true difference of longitude in degrees and parts. And for the true distance, if you observe the parts of the Index of the quadrat at these points, and then measure them in the line of the middle latitude, the line that runs from this point to the scale of the quadrant, will show the true distance. For example, let the place from which you sail be H, in the latitude of 45 degrees, let the Rumbe you sail upon be N by E and let the difference of latitude be 98 parts; and it is required to find the true difference of longitude, and the distance of this point. First in the quadrant of latitude, draw the line A Z, by the middle latitude of the two places; which is 45 degrees, 49 parts, then in the scale of the quadrant A B (counting the whole scale or Radius for one degree) reckoning the difference of latitude 98, and observe the parallel line that comes from this where it crosseth the line of the middle latitude which is at Z. Then setting one foot of your compasses in this point Z, open the other to the centre A, and measure this distance in the scale A B lengthened, and you shall find it to be 140 parts: thus the true difference of latitude 98 parts is enlarged to 140 parts, which is the proportion it ought to have in this latitude. Then working with this difference of latitude enlarged upon the quadrat, as was showed pag. 89. & 90, first, for 100 parts, I find the difference of longitude to be 20 parts, and the distance upon the Index is 102 parts, than likewise for the 40 parts; I find the difference of longitude to be 8 parts, and the distance on the Index is almost 41 parts. These two sums added together makes the true difference of longitude to be 28 parts, and the distance 143 parts almost. Now, lastly, this distance measured in the line of the middle latitude, in the quadrant of latitude, will reach from A to D, and the line that runs from this place up to the scale A B shows the true distance sailed, which is one degree or 100 parts. Thus having found out by observation at any time the true latitude of the place where you are, if it shall differ from your latitude by dead reckoning, you may find exactly in what point you are in respect of the longitude as well as the latitude. And though some of these things may seem at the first a little difficult; yet if you consider well, all that hath been said before and so understand the first principles of the Work, you shall find, that a little practising of these questions, and some such like, which you may propose to yourself, will make all very plain and easy. FINIS.