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 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 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 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 Difference of Longitude The course or Rumbe Distance or way sayled The true Longitude The
you must doe thus First from the point I set the first 40 double leagues upon the Rumbe N W by N which will end at R. Then from the point R draw the rumbe N E by E which is the line R Q and set thereon the 40 double leagues from R to Q thus you will finde Q to be the place you should be in according to your dead reckoning which is in 5 d. 5 10 and somewhat * 5d. 55. more of north latitude whereas by your observation you finde that you are but in 5 deg of north latitude now to know the true place where you are in respect of the longitude because you have sayled upon two rumbes draw the line I Q from I the first place you set sayl from to Q the place of your dead rekoning and then drawing the line F E G at 5 deg of latitude according to your observation of the latitude marke where it crosseth this line I Q which is in the point N and this is the true place you are in whose longitude is 6 deg and whose latitude is 5 deg north In like manner if you should sail upon 3 or 4 severall Rumbes before you can make an observation of the Latitude your best way will be to draw a line from the first place of your voyage to that present place according to your dead reckoning or at least from the last place where you made a fair observation and are thereby well assured both of the longitude and latitude thereof For otherwise you may be much mistaken in the longitude of your places As for instance if in the last example you should thinke you were in that place where the line of latitude F E G doth cut the last rumbe you sayled upon according to your dead reckoning viz. the line R Q by this account you would be but in O which is but in 5 deg 35 100 of longitude whereas you see by the other way which is the truth you are in 6 deg of longitude so that the difference is â— 100. which is very considerable in so small a space PROPOSITION 7. Being to sayl from one place to another but by reason of crosse winds or the coastings of the land you cannot sail thither upon the direct point of the compasse which lies between the two places but are forced to alter your course severall times yet how you should keep your account of your way so that you may know at any time what longitude and latitude you are in and how the place you are bound to bears from you and how farr you want to it 7. The manner of keeping your reckoning upon the Chart. This Proposition contains the use and practise of all the former FOr example suppose you were to sayle from the place I in the former Chart which is under the Equinoctiall and in 5 degrees of longitude unto the place H which hath 5 deg of longitude and 10 degrees of north latitude here the direct way from I to H lies full north But supposing that you cannot sail upon this point but are forced first to run N W by N 36 double leagues and then N E by E 36 doubled leagues more the question is what is the longitude and latitude of this place and how farre it is distant from the place H and upon what point of the compasse it lyes from it First from the point I draw the Rumbe N W by N and set off theron 36 double leagues from I to M. Then from this point M draw the Rumbe N E by E and set off thereon the 36 double leagues which you have sayled upon it from M to N thus you shall finde that N is the place wherein you are whose longitude is fix degrees and whose latitude is five degrees Now if you lay a ruler from this point N to the place you are bound to which is H and draw the line H N this line is the direct way to the place you are bound and by the help your circle or scale of Rumbes you shall finde that it lyes North by West or the first Rumbe from the meridian Westward Lastly if you set one end of your compasses in N and open the other to H and measure that distance in the sides of the Charts you will finde it to be about 5 degrees 1 â—â— or 51 double leagues and so much you want to the end of your voyage PROPOSITION 8. How to know the distance of any Cape Headland or Island from you which you can see at two distant places 8. To know the distance of any Cape from you SUppose that sayling on the Sea you espie an Island or Cape lying at the first sight just North-east from you and then sayling forward upon your way which lies full North to the distance of 5 leagues you then observe that the Island lies full East from you the question is to know the distance of this Isle from either of these two places In such questions as this you may suppose each degree in the former Chart to stand now but for a league These two following Propositions rather belong to the plain table then the chart and let the first place where you espied the Island be at A now because the Island lay North-east from this place draw the line A B which is N E from A. Then count the 5 leagues which you have sayled upon your course which was full north in the meridian line from A to F and because from this place the Island did lye ful East therefore from this point F draw the East line F E G and marke where this line doth crosse the former line A E of N E from A which is in the point E. This therefore must needs be the place of the Island whose distance if you take with your compasses and measure in the sides of the Chart you shall finde that the place E is distant from A 7 leagues and almost 1 15 part of a league and from F just 5 leagues * A double use of this proposition And by this means if you know the longitude and latitude of this Isle or Cape you may the more certainly know the truth of your account and if need be correct it Or if you knew not the place before you may set it down in your chart by its longitude and latitude which you finde it to be in according to the best account you can make by your observation PROPOSITION 9. By observing upon what Rumbes many places lye from you at two severall stations to finde the distances of those places and their true posture and bearing one from another 9. To finde out the true distance and bearing of many places The use of this Proposition AS in the former Proposition you did for one place so in this you may do for many And this will be of good use for hereby sayling in sight of any Coast you may finde out how the
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 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 â—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 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
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 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 Longitude Latitude Deg.  De. De. m. 100 parts 300 or difference of longit 00 40 00 these minutes are in 00 305 05 41 34 57 310 10 42 53 88 315 15 43 55 92 320 20 44 42 70 325 25 45 15 25 330 30 45 35 58 335 35 45 41 68 335 35 45 41 68 340 40 45 35 58 345 45 45 15 25 350 50 44 42 70 355 55 43 55 92 360 60 42 53 88 005 65 41 34 57 010 70 40 00 00 Note if you draw lines by every degree of longitude in the 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 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 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 The severall places where you alter your course The course you steere The dist or way sailed The Longitude The Latitude   Deg. P. Deg. m. Deg. m. P. 1 from N to a E N E 4 09 305 0 41 34 57 2 from a to c ½ 7 69 315 0 43 48 80 3 from c to e E b N 7 26 325 0 45 13 22 4 from e to f ½ 4 93 332 0 45 42 70 5 from f to g East 2
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 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 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 Lōg Latitude Deg D. m. 10 12 58 20 24 25 30 22 34 40 40 28 50 45 31 60 48 58 70 51 16 80 52 35 90 53 0 Secondly for the angle of * The difference of longitude being just 90 deg the angle of position is ready measured in the semidiameter E B being the distance B W which is 37 degrees but at other times must follow the rule position between the two places this is 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 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 Long. Latitude  Deg. Deg. m.