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 The figure of a generall Sea-Chart containing almost an eighth part of the Globe NORTH Now though this be not a general chart of the whole globe yet it may be called a 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 PROPOSITION 1. Knowing the longitude and latitude of any place to set it upon the Chart. 1. By the longitude and latitede to finde the point of any place in the Chart. THis must be done as in the plain chart For first laying your ruler by the longitude of the place you must draw a little occult line as neere the latitude of the place as you can guess then laying your ruler to the latitude of the place crosse that line you drew before with another little line and so the crossing of these two lines wil shew you the point where the place must be supposed to stand Example Thus supposing the longitude of the Summer Ilands to be 300 degrees and the latitude thereof 32 degrees 25 minutes you wil finde that it must be set at S upon the chart PROPOSITION 2. The longitudes and latitudes of two places being known to finde the rumbe which you must saile upon to go directly from the one place to the other 2. By the longitude and latitude of two places to finde the Rumbe Example SUppose the one place to be the Summer Ilands whose longitude and latitude we wil suppose to be as is before set down let the other place be the Lyzard whose latitude is about 50 deg and let the longitude thereof be supposed to be 10 degr so the difference of the longitude of the two places wil be 70 deg as Mr. Norwood both in his book of the Doctrine of Triangles and his Seamans Practice supposeth them to be though as he saith in one place he doth not think them to be so far distant and it is required to finde the rumbe This Proposition must also be performed as in the plain chart For first the two places must be set upon the chart according to their longitudes and latitudes which will be at S and L then draw a strait line from S to L this represents the direct way between the two places now to know what rumbe this is open your compasses to the Radius of your scale of rumbes and setting one foot of your compasses in S with the other draw the arch R M then setting one foot of your compasses in R open the other to the crossing of the line and the arch at M and measuring that distance on your scale of chords or Rumbes so shall you finde it to be 71 deg 21 min. or the sixt rumbe and somewhat above a quarter of a rumbe from the Meridian PROPOSITION 3. Knowing the longitudes and latitudes of two places to know how farre they are distant one from another 3. To measure the distance of places LEt the two places be as is before said S and L it is required to finde their distance In the working this Proposition there is some difference from the plain chart for whereas there you measure the distance of places by one and the same scale of equall parts here you will have use of many scales according to the latitude of the places Mr. Gunter's way Now the ordinary way prescribed by Mr. Gunter to perform this is thus Open your compasses to the distance of the two places and then setting your compasses in the Meridian line so that the one point of the compasses may stand just so much above the greater latitude as the other doth below the lesser latitude and so the degrees between them is the distance this way may serve for small distances as Master Gunter useth it but in greater distances it wil not always hold true and besides it is somewhat troublesome to set the compasses just as much above the one latitude as below the other As in this example if you take the distance S L in your compasses and measure it so in the Meridian line it wil reach from about 16 degrees to about 66 degree and an halfe that is 16 degrees and an half above 50 degrees the greater latitude and 16 degrees and an halfe below 32 degrees 25 minutes the lesser latitude and so the degrees intercepted between the points of the compasses are about 50 degrees and a half whereas the distance of the two places is almost 55 degrees But you shall finde the distance more exactly The way to measure the distances of
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 This is a projection of a part of the Globe in plano by Naturall Tangents You may if you please when occasion requires divide a Circle into foure Quadrants and draw the lines of Longitude from the Center and number them to 360 and likewise describe the Circles of Latitude round about the Center and you may make this Projection as large or as little as you will by the Table of Naturall Tangents if you lengthen or shorten your Radius A Table of Naturall Tangents The Radius being 1000 parts D. TaÌ„ D. TaÌ„g D. Tang. D. TangeÌ„t 1 017 24 445 46 1,036 69 02,605 2 035 25 466 47 1,072 70 02,747 3 052 26 488 48 1,112 71 02 904 4 070 27 510 49 1 150 72 03 078 5 087 28 532 50 1 192 73 03 271 6 105 29 554 51 1,235 74 03,487 7 123 30 577 52 1 280 75 03,732 8 141 31 601 53 1 327 76 04 011 9 158 32 624 54 1,376 77 04,331 10 176 33 649 55 1,428 78 04,705 11 194 34 675 56 1,483 79 05,144 12 213 35 700 57 1 540 80 05 671 13 231 36 727 58 1,600 81 06 313 14 249 37 754 59 1,664 82 07,115 15 268 38 781 60 1 732 83 08 144 16 287 39 810 61 1,804 84 09 514 17 306 40 839 62 1 881 85 11,430 18 325 41 869 63 1,963 86 14,300 19 344 42 900 64 2 050 87 19,081 20 364 43 933 65 2 144 88 28 636 21 384 44 966 66 2 246 89 57,290 22 404 45 1000 67 2,356 90 Infinite 23 424  Rad. 68 2 475   Let your Radius be of what length you please first divide it into 10 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 â—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 â—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 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 Longitude Latitude De. or difference of longitude D. Deg. m. 310 00 32 25 305 05 35 52 300 10 38 51 315 15 41 24 320 20 43 34 325 25 45 24 330 30 46 54 335 35 48 07 340 40 49 04 345 45 49 47 350 50 50 15 355 55 50 31 360 60 50 33 005 65 50 23 010 70 50 00 Now you may hereby see that the line B C in the point G doth 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
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