CHAP. I. Containing some Geometricall Propositions, which will be of frequent use.

CHAP. II. Shewing how to divide a Circle several wayes which will be needful for many things.

THe first usuall division of a Circle, is into 24 equall parts, according to the 24 houres of a na­turall day; which is thus to be performed.

First, draw a line at pleasure, and crosse it in the midst with another line at right angles, To divide a Circle into 24 equall parts. then in the crossing of these two lines, set one foot of your com­passes, and open the other to what distance you please, and therewith draw the circle, which by the crosse lines of East and West, North and South in the figure, is divided into 4 equall parts or quadrants, each of them containing 6 houres apiece: set VI and VI, XII and XII to these 4 parts as in the figure; then keeping your compasses at the same distance wherewith you drew the circle, set one foot in the crossing of the line and the circle, at VI on the one side, and with the other make two marks in the circle, one above and the other below, so shall you mark out the houres of II and X: then removing your compasses to the VI on the other side, mark out the houres of II and X on that side: Again, keeping the compasses still at the same distance, [Page 10]

[diagram of compass points]

[Page 11] set the one foot in the crossing of the line and circle at the upper XII, and with the other you shall mark out the hours of VIII and IIII in the upper half of the circle, then re­move your compasses to the lower XII, and so mark out the hours of VIII and IIII, in the lower part of the cir­cle: thus the circle will be divided into 12 equall parts, and e­very one of these wil contain two hours apiece, and now it wil be easie for you to divide each of these parts into two, so you will have the 24 hours; lastly, you may divide each hour into four equall parts which will be quarters of hours, and so you may number them, and draw them out as in the figure.

Secondly, another usuall and necessary division of a circle is to divide a circle into 360 equall 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 equall 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 crosse 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 equall parts as before, then closing your com­passes divide each of these 12 parts into 3, so you shall have in all 36 parts, then 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 usuall division of a circle is into 32 equall parts To divide a circle into 32 parts. ac­cording to the number of the points of the compasse, which may be thus performed.

First, draw the line of East and West, and crosse 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 crosse 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 [Page 12] crosse 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, crosse the two upper arches at A and B: then set one foot where the line of South doth crosse the circle, and with the other crosse the two lower arches at C and D; then laying your ruler crosse-wayes to these crosses, draw the lines A D and B C: so the circle shall be divided into eight equall 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 compasse, which you may subdi­vide if you please into halues and quarters, and draw the lines, and by three or four letters expresse their names as in the fi­gure, which signifie as followeth.

• The names of the 32 points of the Com­passe.
  North
• North by West
• North-North-west
• North-west by North
• North-west
• North-west 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
• South-east by South
• South-east
• South-east 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 usefull 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.

First, set one foot of your compasses in the center of your divided circle, and open the other to the circumference of that circle which is divided in to degrees. Then with this distance, set one foot of your compasses in the point A, and with the other draw the arch BC. Then again set one foot of your compasses in that place of your divided circle, where the de­grees begin to be num­bred, (which is where the line of North and South doth crosse the circle) and open the other to 45 degrees of that circle; and with this distance set one foot in B, and with the other crosse the arch B C in the point C. Then lay your ruler by the point A and the crosse at C, and draw the line A C, so the angle at A made by these two lines A B and A C, wil be 45 degrees, as was required.

[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 shal represent the Southeast or the fourth Rumbe from the Meridian.

First, set one foot of your compasses in the center 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 crosse 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 doe this also by a Scale of degrees, and Rumbes, To make a Scale of Chords and Rumbs. which you may have upon a straight line on your ruler, which [Page 14] you may thus make. First, set one foot of your compasses in the center 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 straight line upon your ruler, and marke very well with some speciall marke, where this distance begins and ends, for this is your Radius or distance, which you must always take to draw your first arch withall, 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 crosse the line of North or South, open the other to 10 degrees in that circle, and then transferre that distance into your Scale, then again, take out the distance of 20 degrees out of the circle, and transferre 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 crosse 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, crosse the first arch at C, and then draw the line A C, as in the former example and figure.

CHAP. III. Shewing how to make a plain Chart, and many Propositions of sayling by it.

THe drawing of the plain Chart, and the way of sailing thereby, is the most plain and easie of all others. And though it be fit to be used, only in places neer 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.

CHAP. IIII. Shewing how to make a Sea-chart for any part of the World, which shall agree in all particulars with the Globe; with severall Propositions shew­ing the manner of using it.

The defects of the plain Chart. THe way of sayling by the plain chart, is very ea­sie 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 onely in places neer the Equinoctiall, 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 alwayes 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 lesse and lesse toward the Poles: so that though about the Equinoctiall, a de­gree of longitude is equall to a degree of latitude, viz. about 20 leagues: yet in the latitude of 60 deg. one degree of longi­tude is equall, but to half a degree of latitude, viz. about 10 leagues; and about the latitude of 75 deg. 30′, one degree of longitude is equall 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 performe some short voyages by it, yet you cannot use it in any long voyage without great errour: except you onely goe 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 Pro­blemes of sayl­ing. For example, suppose you were to sail from the Lyzard, to the Summer Islands, and should according to the common course first sail South-west, about 500 leagues, and then fin­ding your self to be in the latitude of 32 deg. 20′, you should then sail full West 782 leagues, and then you should finde your selfe directly South from the Summer Ilands, and about 2 leagues off them. Now by this reckoning upon the plain chart, these Islands should be distant from the Lyzard, 1189 leagues. Now admitting this reckoning outward to be true, and these places to be thus scituated upon the plain chart, let us suppose the reckoning homewards to be also kept upon the plain chart. And because in comming 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 halfe 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 Lyzard about 160 leagues, whereas you are already come unto the Lyzard, and so you will finde it, if you keep your rec­koning by this following Sea-chart.

For the reformation hereof, Mr. Wright in his book of the Errours of Navigation, hath shewed how to make, and hath also made a table, by the continuall addition of the secants of every minute, which shews 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 easie, by reducing it into decimall parts. I shall here shew 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 limbe thereof into 90 degrees numbring them from B towards C, then divide the side of the Qua­drant A B into 10 equall parts, and draw straight 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 center 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 equall 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 equall to foure 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. then 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 center A to the line B D, thus then the line A I will be the secant of 30 degrees, [Page 34] 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, then it wil 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 wherof will be 115, and so by the continuall addition of these Secants you may make the Table following, which you shall finde 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 center, and then draw them under it. As for example, the distance between the lines 8 and 9 being equall 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 leasure, as in the following figure, which will be very ready upon any occasion.

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 needlesse and troublesome to make any use of them: but the chief reason is this. Because when you are to goe any long voyage, it wil be needfull for you first to make a generall Map of your whole voyage by the lesser line, where­by you may know the course and distance thereof in generall: 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 dayly 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 easie, To make a Sea Chart 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 equall parts or degrees, then 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 equall, (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 easie, lay a scroule of paper to the Meridian line which is ready drawn to your hand, and so mark out the degrees of latitude upon the scroule of paper, and then laying that scroule to the sides of your chart, you may trans­ferre 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.

Now though this be not a general chart of the whole globe, yet it may be called a generall chart in respect of others, which wil serve onely 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 wil serve as it now stands: but if the latitude be Southward, you must turne the bottome upward. If you have occasion in one chart, to set down both North and South latitude: then you must draw the like parallels of latitude below the Equinoctial, as these are above it.

Now I wil shew 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.

CHAP. V. Of sayling by a great Circle.

The praise of this way of sayling by a great Circle I Now come to shew how you may sail by the arch of a great Circle, which is the most exact way of sayling of all others, but in regard of the dif­ficultie that there is in the calculation thereof, it hath discouraged the Sea-man from looking after it: but I shall shevv you hovv this may plainly and ea­sily be performed by Geometry, vvhich I hope vvil be for the generall profit and ease of Seamen. For you must knovv that the distances of places found out as vvas before shevved in the use of the Sea-chart, First, it is the neerest way. is seldome the neerest distance betvveen any tvvo places, but it is onely their distance in the rumbe. So that if the tvvo places are not both under the Equinoctiall or both in one meridian, then there is somewhat a neerer cut betvven the tvvo places, then the rumbe points out: vvhich sometimes, especially neere the Poles, is very considerable.

But this is not all the benefit vvhich comes by this vvay of sayling, Secondly, it is the most con­venient way. but many times vvhen your course lies neer the East and West, this vvay is farre more convenient. For if you should sail full East or West, you must altogether depend upon your dead reckoning, having no vvay to help your self, by the ob­servation of the latitude, but novv if you sail by the arch of a great circle, betvveen tvvo such places, you not onely go the [Page 49] neerer vvay, but also may alter your latitude many degrees, vvhereby your account may be often rectified, So in the ex­ample of the Summer Ilands the distance by the rumbe is 3299 miles. The distance by the arch is 3204 miles, that is 95 miles lesse. as for exam­ple, suppose you vvere to sail from Spain to Virginia, both vvhich lye neer the parallel of 40 degrees, and suppose the difference of longitude betvveen tvvo such places in the parallel of 40, to be 70 degrees, the distance of these tvvo places measured in the parallel of 40 (vvhich is the rumbe that leads betvveen the tvvo 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 vvay of sayling: the chiefest is this, that in sayling between two such places by the arch of a great circle, you wil 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 wil alter the latitude 11 degrees 38/ [...]0, & so by the observation of the latitude you may rectifie your dead recko­ning very wel, which you cannot do, sayling in the parallel. Thus you see this way of sayling is not only the neerest but the best way.

Now concerning this way of sayling, there hath been but little written by any, Few have written of this subject. and therefore I shal be the more large in this. Captain Saltonstall in his Booke called the Navigator, hath said somwhat how to direct a parallel course, but for any other course he hath said nothing, and what hee sheweth is to be performed by Arithmetick. Master Norwood in his Book of Trigonometry, hath added as an appendix many Problemes of Sayling by the arch of a great circle whereby those; who both can, and wil take the pains, may by calculation finde out all things necessary in this way of Sayling. But those ways of cal­culation as they are very difficult to the unlearned, so they are tedious to those that have the best skil: and therefore I hope it will be wel accepted, if I here shew you how the same may be performed by Geometry, both plainly and speedily, and yet with as much exactnesse, as need be required.

The chiefe things to be known.And in the pursuance hereof, I shal keep as close to Master Norwood, as I can, both in his Propositions and Examples, that thereby you may see how neerly my plain lines wil approach to the exactnesse 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 finde out what longitudes and latitudes the arch of the great circle doth passe through be­tween the two places.

To finde the distance of two places.For the first of these, knowing the longitude and latitude of two places, to finde their distance in the arch of a great cir­cle, which is always the neerest distance. I might shew you how to perform this in the first place, but I here passe 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 litle way, as in sayling 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 neer 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 wil 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 neer. Lastly, all that trouble is needlesse. 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 shew that which follows, no way depends upon the true knowledge of this distance: it shal be sufficient therefore for the present, to tel you, that this way is always somewhat the neerest way.

For the second of these Propositions, which is to know the [Page 51] 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 needlesse then the former proposition, and therefore may be very well omitted.

But now for the third Proposition, To finde 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 passe 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 Cen­ter, 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 straight lines from the Center A to the limbe of the Quadrant. Then transferre this line to the sides of the Quadrant A B and A D, and then setting one foot of your compasses in the center A, open the other to the severall 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 straight lines drawn from the Center, 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.

Let your Radius be of what length you please, first divide it into 10 equall parts, and then sub­divide each of those parts into 10, so you shall have 100 parts in your line, then you may, if you can, di­vide each of these 100 parts into 10, so you shall have 1000, But this last divisi­on will be needlesse, for you may by your eye guesse 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 past 45 degrees in the Table, the Fi­gure before the Com­ma, shews the whole Radius, or how many times the whole Ra­dius is contained therein, and the three following Figures, the parts to be reckond upon the Scale as before. You will finde this Table necessary, either when you would make a large Tangent line to serve for places onely neer the Pole; Or when you would make a very little Tangent line that so you may bring in the degrees neer the Equinoctiall into your Quadrant.

The flank be­ing made will serve for many examples, so that the work wil be very easie.Having thus drawn this blank Quadrant, you must set down therin the two places you are to sail between, according to their latitudes and longitudes, and then onely by your ru­ler draw a straight line from the one place to the other, and this straight line will represent the great circle which passeth between the two places, and will exactly crosse those degrees of longitude and latitude, which you must sail by.

For the example Example. and proof hereof, I shal take Mr. Nor­woods example of a voyage from the Summer Ilands 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 neer 50 degrees, the diffe­rence of longitude betvveen the tvvo places is supposed to be 70 degrees, so that the longitude of the Lizard vvil 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 passe.

The working of the exam­ple.First, let the line A B represent the meridian of the Sum­mer Ilands, upon which you must marke out their latitude 32 degrees, 25 minutes at B, and because the longitude there­of is 300: set down [...]00 at the end of the line A B, so the Summer-Ilands shal 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 limbe 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 marke out the latitude of the Lizard, which is 50 de­grees at C, then lay your ruler to these two markes at B and C, and draw the straight 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 marke well where this line crosseth the arches of latitude, and the lines of longitude, and that shews 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 [Page 55] which this line intersecteth with this table thereof calculated by Mr. In the tenth Probleme of sailing by the arch of a great circle. Norwood for every fifth degree of longitude.

Now you may hereby see, that the line B C in the point G doth crosse the 305, or the 5 degree of longitude from B al­most at the arch of 36 degrees of latitude, just as the table shewes it should, at 35 degrees, 52 minutes of latitude. Again, the line B C doth crosse 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 shews it to be in 38 degrees 51 minutes. And so in all the rest it so neerly 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 shal not need to use any calculation, though you are wel skil'd therin, for the thing hereby may be much more exactly known, then 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 shal be of a parallel course. Suppose two places to be scituate 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 passe through.

To perform this, first in the line A B marke out the latitude of the one place which is 40 degrees at E. Then in that same [Page 56] 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 blanke map according to their latitudes and longitudes, draw a straight 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 shew the true longitudes and latitudes by which this great circle ought to passe.

Proofe of the worke, 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 shew how to do it, and according to them I have calculated this table, so that you might see the exactnesse of this way by its agreement with the table.

Note if you draw lines by every degree of longitude in the blanck Map, as there is by every degree of latitude, you may then finde out the latitude of the great circle for every degree of longitude. But this paines wil be needlesse, yet the lines may be for some use, for if your two places differ more in la­titude then they do in longitude, then 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 [Page 57] found out will be exact enough for the Seamans use, The longitudes & latitudes of the arch thus found out wil be exact e­nough. if you be any thing carefull and handsome in drawing of the lines of latitude and longitude true, observe what Mr. See Master Norwood in his Problemes 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 rekoning 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 alwayes 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 shee go neer this arch. Which how to do he sheweth in the next probleme, wherein I shall follow him, onely whereas he directs you to finde out the longitudes and latitudes of the arch of the great circle, by calculation, I have shewed you how to save that labour, and yet finde it out sufficiently exact­ly for your use.

Having therefore found but the longitudes and latitudes by which the great circle must passe, as is before shewed, 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 shewed. Then prick down in this chart the latitudes through which the arch of the great circle doth passe 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, then you may prick down the lon­gitudes and latitudes of the great circle for every fift degree of longitude, and with your ruler, draw little straight lines from one prick to another, and yet these lines wil represent the great circle wel 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 [Page 58] at the beginning of your voyage, and afterward altering your course by halfe a point at a time, It is not good to steere 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 neer 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 passe by that example, onely setting it down upon the chart, and referre you to his directions, and shew 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.

First, you must by your blank Map finde out the longitudes and latitudes by which the arch of the great circle must passe, then having drawn the following Seachart, first prick down the first place therein at N according to its longitude 300 de­grees, The manner of working upon the generall Chart. and latitude 40 degrees. Then likewise prick down the second place at P, according to its longitude 10 degrees, and latitude 40 degrees, then at every tenth degree of longitude from N prick down the latitude of the great circle for that proper longitude thus at b, which is 10 deg. of longitude from N, you must make a prick at 42 degrees, 53 minutes of lati­tude, and then at d, which is 20 degrees of longitude from N, you must make another prick in 44 degrees, 42 minutes of la­titude, and so for every tenth degree of longitude. Then draw the circle N a b c d e, The course you must steere up­on. &c. to P, and this will represent the arch of the great circle between these two places. And hereby you may see how you must shape your course, viz. as It may seem to some that this Circle NO P cannot be the neerest way, but rather the straight line of N P, but this proceeds from this cause, be­cause the deg. of longitude grow still lesser toward the Poles, though they be made equall in the Chart, and therefore the deg. of latitude are made lar­ger to answer thereunto. But on the contrary, you see in the qua­drant that the line E F, is shorter then the circle E F, and so it is plainly to be seen upon the Globe. followeth. First, you must sail from N to a, upon the sixt rumbe from the Meridian which is E N E, 4 degrees, 09/100, or 41 tenths of a de­gree, so you will be in the point a, whose longitude is 305 de­grees, and the latitude thereof is 41 degrees, 34 minutes. Then you must sail from a to c, half a point more Easterly, [Page 59]

[geometrical diagram]

[Page 60] 7 degrees 69/100 or 77 tenths, or double leagues, so you will be in the point c, whose longitude is 315 degrees, and the latitude thereof is 43 degrees, 48 minutes. Then you may sail from c to e, upon the seventh rumbe from the meridian, which is E by N the distance of 7 degrees, 26/100, or 72 ½ tenths of a degree, and so you wil come to the point e, whose longitude is 325 degrees and the latitude 45 degrees 13 minutes. Then altering your course half a point more toward the East, you may sail from e to f, the distance thereof being 4 degrees 93/100, or [...] degrees, 9 tenths, and almost an half, so you wil come to the longitude of 332 degrees, and the latitude is 45 degrees 42 minutes. Then from f you may sail to g, ful East, the distance is 2 de­grees 09/100 or 2 degrees, 1 tenth, and the longitude wil be 335 degrees, and the latitude 45 degrees, 42 minutes, thus you vvil be come just the one half of your voyage. Novv to per­forme the other half, as you raised the Pole before by little and little, so novv in the same order you may depresse the Pole again, untill you come to the same parallel of 40 degrees, and so finish your voyage, vvhich you may more plainly see by this follovving table of the vvhole voyage.

You must not thinke to finde 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 generall chart, which is drawn by che 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 finde them out easily, and as exactly as need be. In these several charts, you may set down your dayly courses and distances, and then when you please you may prick down the summe of these reckonings upon the generall chart, and thereby the better see whereabouts you are in respect of your whole voyage. Thus you may easily, know the severall parts, and the total summe of your voyage at any time.

Or else you may keep account of such a voyage as this, and finde 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 your self to so small a chart as this is, this being so little, onely in [Page 62] regard of the littlenesse of the book, and so the lines are bro­ken off, oftner then 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, pricke 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 shal finde 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 halfe a point more towards the east.

Also in regard of the shortnesse 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 finde 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 shortnes 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 neerly concurre 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 shortnesse 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 alter 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 sayled upon this point from e to f, the latitude wil 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 farre to the Northward as the arch of the great circle will allow me, I here at f alter my course halfe 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 halfe 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 finde [Page 65] 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. Shewing some observations which may be of use in all these three kinds of sayling.

HAving shewed 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 shal now lay down some observations, which may be usefull in either of these wayes of sayling, 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 neerest way, be by the arch of a great circle; and though the way by the di­rect Rumbe, lies very wel: yet it is usual in these voyages to steer wide from both these neerer ways, viz first, to steer much more to the Southward then 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 lone Island. To get the be­nefit of the winde. For first, by sayling toward the Line, you shal gain the bene­fit of the Tradewind (as they call it) which doth most con­stantly blow between the North and East, between and neer the Tropicks. Secondly, hereby you may be sure not to over­shoot the Island you would sayl to, To avoyd over­shooting the place you go to which otherwayes 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 misse some few leagues: and be­side [Page 66] 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 shal chance not to be in sight of the Island, you wil then be at such a losse, that you wil not know which way to sail to finde it, whether Eastward, or Westward, and so must be forced to vvander at randome untill you have a sight of some knovvn place, by which you may knovv hovv the Island bears from you. Therefore in say­ling to such a place as this; it is the best vvay to be sure to get into the latitude of the place, a good while before you come to it, and then sayling neer that latitude, you shall be sure not to passe 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, then the true Rumbe doth lie, and this likewise is to get the benefit of the wind, for as the winde lies most Easterly toward the Equinoctial, so it blows most westerly towards the Pole: also this way is the neerest way, because it lies neer 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 jour­neys end 150 or 200 leagues before they are aware, and so might easily overshoot their port, and lose themselves, but that they sail to the maine land, or great Islands that they can­not passe 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 wan­dring, sometimes out of the way, and that is the inconveni­ence that there is, in sayling 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 sayling 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 then indeed they are, [Page 67] whereas he that knows the road and how farre it is from place to place, goes on more chearfully. Therefore the labour wil 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 shewed at large in the for­mer Chapter, but yet if you keep your self in your yoyage too strictly to the arch, you must runne much of your way in a pa­rallel, or very neer 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, til you alter your longitude 5 degrees, then halfe a point more Easterly, til you alter your longitude 10 degrees more, then you must sail N by E til you alter 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 ful East, is little better then 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 neer the course of East and West.

Now you shal see how easily this may be avoyded, How to avoid sailing in a pa­rallel, totally. and that several wayes: first let the courses be continued as before til 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 stil upon the 7 rumbe N by E, til you come to 35 degrees of longitude, your latitude wil 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 wil be but 7 degrees 09/100, that is but 7/100, more then 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 wil not be much, as you may see by this table which differs much from the other in the rumbes, [Page 68] latitudes, and longitudes, but yet it differs but little in the to­tal summe of the distances, being but 20/100, which is but 12 miles.

See the gene­rall 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, til you have altered your longitude 35 de­grees, which is the half of your voyage, you wil then finde your selfe to be at Q, in the latitude of 50 degrees, 12 mi­nutes, and your distance or way sayled, 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 wil be as before 26 degrees 63/100, and so you wil come to the latitude of 40 again, being the place you desire, so that your whole voyage by this reckoning wil be but 53 degrees 26/100, w ^ch though it be somewhat more then the distance by the arch, which is 52 degrees, 08/100, yet it is lesse then the distance of these two places, in the parallel which is 53 degrees, 62/100, and the losse of the way wil be wel recompensed, especially in sailing from the Indies, by the advantage of the wind, and by the certainty you may keep off your account, sayling alwayes upon the sixt 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 wayes before shewed, so that there needs no other cal­culation for them, since a smal wandring from them, will not alter the length of the way in any considerable quantity.

Thus also you may easily avoid sayling upon a course of East & West. Which way soever your voiage lies, To avoyd sail­ing East or West at any time. it wil be your best way to steer no neerer to the East or West, then the 7 ^th 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 vvil not go much out of your vvay. For suppose you vvere to sail betvveen tvvo places, under the E­quinoctial line, so that the direct and neerest way between these two places, lies ful East and West, yet you may sayl 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 wil raise the Pole very neer one degree, viz. 994/1000, and your distance or way sayled will be but 5 degrees, 1/10 almost, viz. 5 degrees, 098/1000. Then if you alter your course, and so lay the pole as much upon the 7 Rumbe the other way; so you wil 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 wil be wel 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 depresse the Pole almost one degree in sayling of five, which wil serve very wel to correct your account by, and therefore I wil not perswade 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, w ^ch is the neerest way between two places, doth always lie somewhat between the Pole, and the [Page 70] parallel of East and West, and therefore in raising the Pole upon the 7 Rumbe, you wil not go altogether so much out of your way, if you observe to incline your course alwayes to­ward the Pole. Thus you see you need not sayl directly East or West at any time unlesse it be when you feare you shal pass by some little lone Island, as is aforesaid.

4 Make use of the Sea chart and the Map, or Globe in plano, pag. 52. both together.In voyages neer the Pole it is best to guide your self as neer the arch of the great circle as you can, having respect to the former considerations. And in these voyages your best way wil be to keep your reckonings, both upon the Chart and up­oe the Map, so the one wil 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 wil be found out more easily by the Map. For the places being set upon it according to their longitudes and latitudes, wil 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 wil be; by the rumbe which you sail upon, and by the latitude which you finde by observation, to finde 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, then it wil be best, to set the place down first in the Map, and so to finde 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 wil be a good way to keep your ac­counts upon two blanck charts, upon the one, you may keep your dead reckoning, upon the other, your corrected account. And the benefit hereof wil 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, [Page 71] and which way that current sets, and how fast it runnes. For if there be a current between the places you sail to, which way soever the current sets, your reckonings outward and home­ward wil not agree, and indeed you can keepe no good ac­count of your way, til you know which way and how fast the current runnes.

For example: first, suppose the current runnes East 12 miles a day, and you sayl against this current according to your account, by your Log-line (which is the best measure of your dead reckoning) 60 miles West in a day, your true pace or distance wil be but 48 miles in a day, because the current wil set you back 12 miles, which substracted from 60, there remains but 48. On the contrary, if you sayl Eastward in this current, (and so sayl with it) 60 miles a day according to your log-line, then your true motion wil be 72 miles a day, because the current wil set you forward 12 miles more then you seem to goe.

Secondly, suppose the current run East one mile an hour, if you sayl in this current N E or S E, look how many hours you sail, so many miles the stream wil set you more forward in your longitude, then you are avvare of; and yet your lati­tude, vvil fal out according to your account. On the other side if you sayl N W or S W in this current, the current vvil drive you backvvard so many miles, and yet your latitude vvil be ac­cording to your account.

Thirdly, if the current run upon any rumbe betvveen the meridian, and the East or West, then your true motion vvil differ from your dead reckoning, and also from your account by observation, both in longitude and latitude, so that until you knovv in some sort both vvhich vvay and hovv fast the current runs, you can never keep a good account of your vvay and the onely way to finde out the current is to keep a good account by the log-line outvvard and homevvard, and by set­ting this dovvn upon your blanck chart you may plainly see vvhich vvay the current runs, and hovv fast, as Master Nor­wood hath very vvel demonstrated in the end of his Sea-mans Practice.

6 To alter your course as sel­dame as you can.In most voyages, it wil be good to keep your course con­stantly, (or as much as you can) upon one and the same rumb, for so your account wil 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 errour in the longitude, and this errour, at every changing of your course, may as wel chance to be increased, as to be ballanced one time with another, whereas if you steer your course constant­ly upon one rumbe, as it wil avoyd the trouble of drawing so many rumbe lines, so there can be no greater errour in your account, then there shal 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 sixt Proposition of the plain chart, drawing a straight line from your first place to the place you are then in.

7 To gain sight of land when you can.It vvil not be amisse as often as you can, to get a sight of such Capes, Headlands or Islands, as lye neer your course, which being standing marks, wil give you certain knowledg, vvhereabouts 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 compasse, 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 vvhich end, not onely the Steers-man must be diligent to keep the ship to the course ap­pointed, but you must be frequent in observing the variation of the compasse, vvhich may be so vvel performed by the Sea rings in use among Sea-men, that no Instrument can be devi­sed fitter for the purpose, this variation being knovvn, must be allovved for in your account, that so you may knovv the true rumbe you sail upon, vvithout vvhich there can be no true account kept.

CHAP VII. Of sayling 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 easie and plainest way that can be, yet because it is not so general as to take in all places, but is onely to be used when the two places are both, on the one side of the Equinoctiall: as also it may seem somewhat defective, because it doth not shew 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 wil seldome be forced to use this way, but may very wel content your self 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 severall men have for severall uses invented severall wayes to project the Globe upon a Plane. You may see the severall projections thereof in Mr. Gunters book of the Sector. The fittest for this purpose wil 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 crosse 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 shal divide the Diameter C D into its parts, which are half tangents. In the same man­ner, [Page 74] if you keep one end of your ruler fixed in the point C, and lay the other end to the severall degrees of the Semicircle A D B, you may divide the diameter A B into halfe 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 Meridians and Parallels. which are all parts of perfect circles, and you may finde their centers by these three points. First, for the Meridi­ans, they all concurre 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 finde the center, and so draw the arch as is shewed in the first chapter. But to save that labour, you must know, that the centers all lie in the diameter lines, which must be exten­ded beyond the circle, and then the centers 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 cen­ter E, so you shal have the centers of the meridians. Then if you set one foot of your compasses in that center, and open the other to the Pole A or B, it wil passe through the corre­spondent degree, or third point in the diameter C D: on the other side of the center, so the meridian wil 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 center E, and with the other, marke the diameter A B, both above and below, and these markes shal be the cen­ters of the parallels. Then set one foot of your compasses in these centers, and close your compasses, til 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 finde that the arches thus drawn, do passe exactly through their three respective points, in the circle and diameter, your work is true, other­wise not.

For example, if you would draw the meridians and paral­lels of 45 degrees, Example in the Meridians and Parallels of 45 degrees. first the tangent of 45 being doubled, makes [Page 75]

[geometrical diagram]

[Page 76] 90, which counted both ways from the center E doth reach to C and D, therefore setting one foot of your compasses in C or D, and opening the other to the Poles at A or B, draw the two like arches A f B, on either side the center. Then to draw the parallels of 45 degrees, keep your compasses at this distance, and setting one foot in the center E, with the other crosse the diameter A B, both above and below in the points G & H, these are the centers for the parallels of 45, therefore set one foot in the center G, and close the other, till it reach to 45 degrees in the outmost circle, and so draw the arch 45 R 45. Then set one foot of your compasses in the other H, & so draw the other parallel 45 P 45. thus these 4 arches are drawn.

And thus you may easily doe for any other degree under 45, but when you come to the degrees above 45, then 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 center therof. Or else when you draw the former meridians, you may remember to turne about the compasses, and marke the line C D without the circle; by these markes you shal divide the line into half tan­gents, and so you may finde out the centers as before.

How to help your self when your compasses wil not reach.But because some of these centers, wil fall so farre without the circle, that your compasses will not reach them; you may then bridle a thin ruler, that wil bend, with a double string like a crosse bow, and then by tvvisting the string together, you may by little and little, set it to what bent you please, till it shal cut the three points of your arch you would draw, and then vvith your pen, you may dravv your arch, vvhich if the ruler be all of one thicknesse, and so bend in all places alike, it vvil be very true. Your compasses vvil reach the centers ve­ry vvel, til you come to 60 degrees, but aftervvard you must be forced to use this or some such like way to help your self. 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 [Page 77] parallels through every degree, which you may do very wel 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 triall 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 centers so that they may be turned round, and the uppermost of these being drawn in fine thin paper, and a little oyled, you may easily see through it all the lines of the other. And thus you shal have an This wil be somewhat like Mr. Blagraves Mathematicall Jewell. Instrument whereby this and most other questions of spherical triangles may be resolved.

Having thus shewed the drawing of this projection, I shal now come to shew you the use of it in severall examples. The first example shall be the fore-mentioned 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 passe. Secondly, the angle of position from the first place to the second. Third­ly, the neerest 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 beares from the second. And herein you must regard how the se­cond place doth beare from the first If the second place lie West from the first, then 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, then you must set down the first place on the West-side of the circle, as it is here at S. Then from the point S through the center E, draw the diameter line S E K, and crosse it at right angles with the line M E N. Then accounting 70 degrees (which is the [Page 78] 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 finde the center 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 shal finde 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 passe (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 wil finde 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 crosse 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 wil reach almost to 49 degrees, which shews 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 finde it wil reach from A to V, now if you reckon the degrees of the parallels of latitude from A to this point V, you shal 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 wil shew the distance of that place. Thus the distances [Page 79] wil be found out as exactly as by any other Geometrical way, but in regard of the smalnesse of the projection, you may mi­stake some fevv miles or leagues.

But if you vvere to sayl from the Lizard to the Summer-Islands, The difference in sayling for­ward & back­ward by the arch. then you must first set dovvn the latitude of the Li­zard on the other side of the circle (as I noted before) & so the vvork vvil fal out much as it did before, for the longitudes and latitudes of the arch vvil be the same, only accounting them backvvards, the distance vvil be the same, viz. 53 degrees and a half, onely it must be measured in another meridian, accor­ding to the angle of position from the Lizard, vvhich vvil be about 81 degrees, so that in effect, all is the same, onely the an­gle of position, vvhich is of little use, but to finde out the scale of the distances. So that if you regard it vvel, one la­bour vvil serve to finde your vvay outvvard and homevvard.

I might here shevv you hovv to perform the parallel questi­on, but because such questions may vvith more ease and cer­tainty be performed by the former vvay, I shal not spend time about it, I shal onely instance in tvvo sorts of voyages, vvhich cannot be performed by the other projection, and in such ca­ses as these there vvil be some need of this vvay, and not else. First, vvhen one of the places is under the Equinoctial, and the other tovvard one of the poles. The other is vvhen 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 scituation. which lies under the Equinoctial, and hath about 35 degrees of longitude to the Straights of Magellan, which hath about 53 degrees of South latitude, and differs in longitude from the former place 9 [...] degrees to the West-ward: now it is re­quired to finde 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 Straights of Magellan, whereup­on you must mark out the latitude thereof, which is 53 de­grees [Page 80] at W, so you shal have these three points D W C, by which you may finde the center, 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 finde 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 shal finde for e­very tenth degree to be as in this table.

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 shewed 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 wil reach from C to 37 degrees, which is the angle required, & the scituation shews 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 wil reach from A to the Equi­noctial line, Likewise the distance D W needs no other measuring, but must needs be 90 degrees. which shews the distance to be 90 degrees.

The last example shal be of two places, the one being on the one side, and the other on the other side of the Equinocti­all. As, suppose you were to sail from the Summer-Isles, to the Cape of good Hope, Example of places in ano­ther scituation. 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 tvvo places.

To perform this first you must set down the first place ac­cording to the latitude thereof in the outmost circle at S, and [Page 81] 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 wil fal 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, finde the center which wil 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 finde them by seing how this arch doth crosse the cir­cles of longitude and latitude in the draught, which for every tenth degree of longitude is as followeth.

Then for the angle of position, you shal finde it thus, mark where the arch S X K, doth crosse 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 shal finde it wil 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 shal finde it wil 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 joyn them toge­ther, as I noted before, your work wil be somewhat more easie and readily performed. For then in any question of this na­ture, you need but turne the Pole of the upper paper, to the latitude of the first place in the under paper. Then marke out the second place according to its longitude and latitude, in the meridians and parallels of the under paper. Then mark what [Page 82] 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 shal 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 crosse the Equinoctiall 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 oblique 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 exactnesse, then any such Instrument wil afford: and likewise they are somewhat beside my present purpose; I shal not now speak of them.

CHAP. VIII. How to keep a perfect account.

ALthough the wayes I have already shewed 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 smal, you wil be sub­ject to some mistake, in finding out the several points, and this mistake being often committed may come to some considera­ble errour, I shal in this chapter shew you, how you may in­crease your scale, and by a little help of your pen finde 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 wil depend. [Page 83] 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 shal each degree by this means be divided into 100 very sensible parts, and be 10 times greater then the scale of your chart, by which means you shal keep your account as exactly as need be required. Then it wil 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 then the rest, and every fifth line you may prick out. Then making the corner at A the center, 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 wil 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 oyl it: or it wil be better, if you get a piecc of fine cleer horn, and make a scale of equal parts thereon, equal to the sides of the quadrat, and so fastening it to the center 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 shewed in many propo­sitions, but I shal onely shew 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.

The proposition for your dead reckoning is this. How to keep your dead rec­koning. By the rumbe and distance, to finde the longitude and latitude of the place you are in. As for example, suppose the place from whence you set sail to be A, which here is under the first Me­ridian, and under the Equinoctial, and therefore hath 00 de­grees of longitude or latitude. Now suppose you sayl from this place upon the first Rumbe from the Meridian N by E, to the distance of one degree or 100 parts; it is desired to know the latitude and longitude of this place. Or rather how farre [Page 84]

[geometrical diagram]

this place wil be Northerly from A, and likewise how farre it is Easterly from A

To perform this or any such like question, you must first [Page 85] 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 horne Index to the Rumbe in the arch, and finde out the distance you have sayled, upon the scale of your Index. Now marke very wel 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, wil shew 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 bottome, or B C at the top, wil 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 vvhich is one degree or 100 parts upon your Index, and marke vvhere this number ends, vvhich is in the point P, vvhere the line A P crosseth the arch B P. This is the ex­act point vvhere you are. Novv if you trace the line P B that runs through this point to the Meridian line, you shall there finde 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 bottome or top of the Quadrat, there you shal finde 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, vve vvil novv 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 dovvn 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 halfe, you shal finde that its place vvil be close by the [Page 86] 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 alter 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 sayled, which is 100 parts, and observing wel where this distance ends, which is where the line of the second rumbe doth cut the arch B P, which is in the crosse at 2: this is the point that shews you the difference of longitude and latitude of this place from the last. Now by the lines that passe through this crosse, you shal 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 adde the Northing and Easting of both the places toge­ther thus.

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 shal finde this place to be at the figure 2 neer the corner A.

And thus you shal finde out the point of any place in your chart more certainly then other ways it is possible for you to do, and with lesse 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 stil 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 lye [Page 87] so, that the longitude may increase, and the latitude may de­crease, then you must adde the one, and substract the other. your best rule to guide you in this, when to adde and when to substract 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 shal finde 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 shal finde that the course being the first rumb from the Meridian, and the di­stance being [...]0 parts, the difference of latitude wil be 98 parts, and the difference of longitude 19 parts and a half: this sub­stracted from the longitude and latitude of 7, shews the lon­gitude and latitude of the place 8 where you are.

Thus likewise you must do to finde the longitude of the o­ther places 9 10 11 12 13 A. A good way for young begin­ners to avoid mistakes. And it wil be a great help to direct you when to adde, and when to substract, if you joyn four of these Quadrats together, which you may very wel 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 compasse to the rumbes, as in the figure of the compasse page 10. By this means without any trouble you shal plainly see the Easting [Page 88] 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 finde by the Instru­ment, and then in casting up your accounts, you must adde al­together that you finde under one columne, but if you have any summes in contrary columns, as East and West, or North and South, you must substract the summe of the East from the summe of the West; and the summe of the North co­lumne from the summe of the South, so often as you would finde 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.

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 stil finde 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 finde 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 marke wel 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 shal see the true distance you have sayled in that Rumbe. And if you trace the line that runnes through this point to the line of longitude at the top or bottome of the Instrument, there you shal see the difference of your longitude, or rather how farre 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, til 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 [Page 90] line from A to B, then marke the line that is drawn from this part, and observe wel 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 shal finde your distance sailed to be 100 parts or one degree. And if you marke the line that runs through this point to the line of longitude, you shal 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, then first you must finde out the Rumbe which lies from your first place, to the place where you are according to your dead reckoning (as is shewed in the sixt proposition of the Plane Chart) and then laying your Index upon this Rumbe, worke as aforesaid.

How to work when the que­stion is of more then 100 parts.Note that in working of either of these propositions by the Quadrat, if the distance or difference of latitude be more then a degree or 100 parts, then you may first finde out what be­longs to one degree, and then what belongs to the odde parts, and adding them together, you may finde the truth as exact­ly, as if your Instrument were large enough to shew both at once. But if your distance or difference of latitude be four or five degrres, you may then very wel work the proposition up­on your chart, without using the Quadrat, and need not fear errour, if you be any thing careful, and work by a true scale of rumbes and equal parts.

Hitherto I have shewed you how to worke these two pro­positions upon the plane chart, How to per­form these two propositions in all latitudes. or in places neer 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 shal now shew you, how when you have found out the Northing or Southing, Easting or Westing of any place, as is before shew­ed by the Quadrat, how you shal set it down in any latitude that it may shew the true longitude thereof. For you must [Page 91] know, that though the Northing and Southing, doth shew the true difference of latitude in all latitudes: yet the Easting and Westing found out by the Quadrat, doth not shew the true difference of longitude in those places, which are any thing distant from the Equinoctiall, I shal therefore shew you how to performe 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 degres 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 finde the latitude and longitude of this place.

First, you must finde out the Northing and Easting by the quadrat as is before shewed. The first way to perform this by the Chart it selfe. 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 wil 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 crosse the line a 1 at 1, and so you shal 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.

But if you think this scale in the sides of your chart wil be somewhat to smal to finde out the difference of longitude ex­actly, you may have recourse to the Quadrant of latitudes, A second and better way, by the Quadrant of Latitudes, pace page 32, and thus help your self. Suppose the whole side of the quadrant A B to stand but for one degree, so you shal have in a degree 100 very sensible parts, and then proceed thus in your work. Consider the latitude of the two places, and draw your line through the middle latitude between them, or lay a thred or ruler to it: then count the Easting or Westing of your [Page 90]

[geometrical diagram]

[Page 95] place in the side A B, and where it ends observe the line that runs from that place, where it crosseth the line of the mid­dle latitude. Then setting one foot of your compasses in this crosse, open the other to the center A, and measure this di­stance in the scale A B, so you shal have the true longitude of the place.

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 wil crosse the former line of the middle latitude at c, then setting one foot of your cōpasses in this point at c, & opening the other to the center A, this distance A measured in the scale A B wil shew 28 parts, w ^ch is the true difference of the longitude of the place 1 from the Meridian of H. And if thus you wil make trial, you shal finde out the longitudes and latitudes of all the other places, 2 3 4 5, &c. as they are set down in the table following.

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 finde 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 columne, shews the true longitude of the place where you are. This wil 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 finde 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 shewed before page 89, as if this number were the true diffe­rence of latitude between the two places, so you shal finde 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, wil shew 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 finde 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 [Page 95] 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 center A, and measure this distance in the scale A B lengthened, and you shal find it to be 140 parts: thus the true difference of latitude 98 parts is inlarged to 140 parts, which is the proportion it ought to have in this latitude. Then working with this difference of latitude inlarged upon the quadrat, as was shew'd 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, then likewise for the 40 parts; I finde the difference of longitude to be 8 parts, and the di­stance on the Index is almost 41 parts. These two summes 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, wil reach from A to D, and the line that runs from this place up to the scale A B shews 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 shal differ from your latitude by dead reckoning, you may finde exactly in what point you are in respect of the longitude as wel as the latitude. And though some of these things may seem at the first a little difficult; yet if you consider wel, all that hath bin said before and so understand the first principles of the Worke, you shal finde, that a little practising of these questions, and some such like, which you may propose to your selfe, wil make all very plain and easie.