1,00 Col. last line 3. for 0 8 read 0 58. THE GEOMETRICAL SEA-MAN OR THE Art of Navigation performed by GEOMETRY CHAP. I. Containing some Geometricall Propositions which will be of frequent use PROPOSITION 1. How to erect a Perpendicular line at the end of a line The first Proposition What a perpendicular line is A Perpendicular line is a line that stands directly upright from another line As in the figure the line B C is a perpendicular line to the line A B. Now in the Figure there are two wayes of raising it the one on the right side and the other on the left First Let the line A B be given and it is required to erect the line B C perpendicularly to it at the end of the line in the point B. Another way The second way to perform this is demonstrated on the left side of the figure Let the line A B be given as before and it is required to erect the line A D perpendicularly in the point A. To perform this first set one foot of your Compasses in any convenient point at pleasure as at L and open the other foot to the point A and draw the arch N A M then lay your ruler to the center of this arch L and the place where it crosseth the line A B which is at N and draw the line M L N which doth crosse the arch N A M in the point M. Lastly laying your Ruler to this crosse at M and the point A draw the line D M A so you shall have your desire PROPOSITION 2. To draw one line parallel to another line at any distance required The second Proposition What is meant by a parallel line A Line is said to be parallel to another line when it is equally distant from it in every part thereof Thus in the former figure the line P S is parallel to the line A B. Now the way to draw a parallel line is thus Let the line A B be the first line given and it is required to draw the line P S parallel to it according to the distance P A. First open your Compasses to the distance you have occasion to use which in this example is P A then setting one foot in A with the other draw the arch at P and then keeping your Compasses at the same distance remove one foot to B and with the other draw the arch at S lastly laying your ruler on the very edge of these two arches draw the line P S which will be parallel to A B and so the proposition is performed PROPOSITION 3. How to make a Geometricall Square The third Proposition A Geometricall square is a square whose foure sides are all of one and the same length Now in the first figure let the line A B be the side of such a Square and it is required to make a Square of that length and breadth First How to make a Square you must draw the line A B according to the length given then erect the perpendicular line A D at one end thereof as was shewed before then setting one foot of your compasses in the corner A open the other to B and keeping one foot still in A with the other crosse the perpendicular A D at D then keeping your compasses at the same distance set one foot in B and with the other draw a short arch at C then set one foot at the crosse at D and with the other crosse the arch last drawn in the point C now if you draw lines through these marks from A to D from D to C and from C to B so you shall make the Geometricall Square A B C D as was required If you will try your work whether you have made it true or no then set one foot of your compasses in A How to try a Square and open the other to the corner at C then with that distance set one foot in B and turn the other to the corner at D if both these opposite corners have the same distance the Square is truly made otherwise not A long Square If you would make a long square as the square A P S B first you may draw the line P S parallel to A B and according to the length of the side of your square then erect the perpendicular either at A or B and draw the opposite side parallel thereunto according as the length of your square requires and you may try the truth of this square also by the opposite corners as before PROPOSITION 4. To raise a perpendicular in the midst of a line The fourth Proposition IN this second figure let A B be the line given and it is required to raise a perpendicular in the point C. The second figure First set one foot of your compasses in the point C and open the other to any distance at pleasure and marke the given line therewith on both sides from C at the points A and B then setting one foot of your compasses in the point A open the other to any distance you please beyond C and draw a little arch above the line at F. Then with the same distance set one foot in B and with the other crosse the arch F with the arch D. Lastly lay your ruler to this crosse and the point C and draw the line G C which is perpendicular to the line A B in the point C as was required PROPOSITION 5. From a point aloft to let fall a perpendicular upon a line given The fifth Proposition LEt G be the point aloft from whence it is required to let fall a perpendicular upon the line A B in the second figure First set one foot of your compasses in the point given which is G and open your compasses so wide that you may draw the arch A H B which may cut the line A B in the points A and B and the farther these two points are asunder so much the better then keeping your compasses at this distance set one foot in A and with the other draw the arch E then remove one foot to B and crosse the last arch at E lastly laying your ruler to the point G and this crosse at E draw the line G C E so you have performed the proposition PROPOSITION 6. To draw a line squire wise to another line The sixth Proposition IN the second figure let A B be the line given and it is required to draw the line G E squire wise to it so that it may crosse it at right angles First open your compasses at pleasure and setting one foot in the line at B with the other make two short arches one above the line at D and the other below the line at E. Then with the same distance set one foot in A and with the other crosse the two former arches at D and E. Lastly laying your ruler by these two crosses D and E draw the line G E
which will crosse the line A B at right angles as was required PROPOSITION 7. To divide a line given into two equall parts The seventh Proposition IN the second figure let A B be the line given to be divided into two equal parts First set one foot of your compasses at the one end of the line at A and open the other to any distance above half the line therewith draw two little arches one above the line at F and the other below the line at E then remove your Compasses to B the other end of the line and crosse the two former arches at F and E then lay your ruler to these two crosses F and E and draw the line GC E which will divide the line A B in two equal parts in the point C so that A C is the one halfe and C B the other PROPOSITION 8. To raise a Perpendicular at the end of a line another way The eighth Proposition IN this figure let the line given be A B and it is required to raise a perpendicular at the end thereof at B. Here you may note that if the three sides of a Triangle be made of these three numbers 3 4 5 or any other numbers that are proportionable thereunto as 6 8 10 9 12 15 12 16 20 30 40 50 it will have one right angle which will be opposite to the greatest side as in the Triangle D B E the side E B is 3 the side B D is 4 and the side D E is 5 and the angle at B is a right angle PROPOSITION 9. To make one angle equall or like to another The ninth Proposition AN angle is the ioyning or crossing of two lines What an angle is with the generall kinds of angles if the two lines crosse one another or joyn one to another perpendicularly then they are said to make a right angle or angles if two lines meet or crosse one another any other way they are said to make an oblique angle or angles Thus in the third figure the lines D B and E B meeting in the point B make a right angle And in the second figure the lines A B and G E crossing one another in the point C make four right angles or quadrants But in the third figure the lines E D and B D meeting in the point D are said to make an oblique angle Now these oblique angles if they be lesse then a right angle they are called acute or sharpe angles if they be more then a right angle they are called obtuse or blunt angles Now for example of the proposition let the angle E D B be the appointed angle and it is required to make the angle D B C like unto it In this example because the line D B is limited and is common to both the angles you shall need onely to set one foot of your compasses in B and open the other to the neerest distance of the line D E which you may do by drawing the little arch which toucheth the line between 3 and 4 then remove your compasses to D and draw the like arch at C then lay your Ruler to the point B and the very edge of this arch C and draw the line B C so shall the two angles be of one quantity or widenesse as was desired In other cases this way will not serve but this is sufficient for the present purpose and I shall shew you other wayes to perform that in the next Chapter PROPOSITION 10. To divide a line into any number of equall parts The tenth Proposition IN the third figure let the line B D be given to be divided into four equall parts First from the end D draw a line as D E making an angle with the line D B at pleasure then from the other end of the line B make the angle D B C equall to the former angle as was shewed in the last Proposition Then from the point D set off with your compasses such a number of any equal parts as lacks one of the number desired which in this example therefore must be 3 set off therefore on the line D E three equall parts 1 2 and 3 then you must with the same distance of your compasses set off 1 2 and 3 from the point B on the line B C then draw crosse lines from the last number in the one line to the first in the other that is from 3 to 1 from 2 to 2 c. and these lines will divide the line B D into four equall parts as was desired PROPOSITION 11. To bring any three points not lying in a straight line into a Circle The eleventh Proposition IN this figure let A B C be the three points given and it is required to draw a circle through them all The fourth figure Set one foot of your compasses in the middle point at B and open your compasses to any distance you please so it be above half the distance between B and either of the other marks yea it is no matter if need be though it reach almost to or quite beyond the neerest of the other marks and draw the arch D E F G then keeping your compasses at this distance set one foot in A and with the other draw the arch G F which crosseth the former arch at G and F then set one foot of your compasses in the third point C and with the other draw the arch E D which crosseth the first arch at E and D then laying your ruler to the intersections of these arches draw the lines G F H and D E H which will crosse one another in the point H this crosse at H is the center of the Circle therefore setting one foot of your compasse in this crosse at H open the other to any of the three points A B or C and draw the circle which if you have done well will passe through all the three points A B C as was required 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 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 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 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 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 The description and making of the plain Chart. The description of the plain Chart. First make the square A B C D of what length and breadth you please and divide each side into as many equall parts as your occasion requires and then draw straight lines through these parts crossing one another at right angles and so making many little Geometricall squares each of which you may suppose to contain one degree in longitude and latitude * According to account 20 Leagues are in one degree so each 10 part wil be 2 leagus but it is somwhat more as you may see in the third proposition of this
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
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
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 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 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 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 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.
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
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
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