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
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
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
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
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
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
miles a day and you sayl against this current according to your account by your Log-line which is the best measure of your dead reckoning 60 miles West in a day your true pace or distance wil be but 48 miles in a day because the current wil set you back 12 miles which substracted from 60 there remains but 48. On the contrary if you sayl Eastward in this current and so sayl with it 60 miles a day according to your log-line then your true motion wil be 72 miles a day because the current wil set you forward 12 miles more then you seem to goe Secondly suppose the current run East one mile an hour if you sayl in this current N E or S E look how many hours you sail so many miles the stream wil set you more forward in your longitude then you are avvare of and yet your latitude vvil fal out according to your account On the other side if you sayl N W or S W in this current the current vvil drive you backvvard so many miles and yet your latitude vvil be according to your account Thirdly if the current run upon any rumbe betvveen the meridian and the East or West then your true motion vvil differ from your dead reckoning and also from your account by observation both in longitude and latitude so that until you knovv in some sort both vvhich vvay and hovv fast the current runs you can never keep a good account of your vvay and the onely way to finde out the current is to keep a good account by the log-log-line outvvard and homevvard and by setting this dovvn upon your blanck chart you may plainly see vvhich vvay the current runs and hovv fast as Master Norwood hath very vvel demonstrated in the end of his Sea-mans Practice 6 To alter your course as seldame as you can In most voyages it wil be good to keep your course constantly or as much as you can upon one and the same rumb for so your account wil be more easily and certainly kept For at every shifting of your course the true point that you are in cannot be so certainly known but that you may misreckon somewhat both in the latitude and longitude thereof For the latitude by which you have the most certainty of your place may be mistaken 5 or 10 minutes by any instruments ordinarily used and this may cause 20 or 30 minutes errour in the longitude and this errour at every changing of your course may as wel chance to be increased as to be ballanced one time with another whereas if you steer your course constantly upon one rumbe as it wil avoyd the trouble of drawing so many rumbe lines so there can be no greater errour in your account then there shal happen to be in the last observation of the latitude which cannot be much But if you are forced to shift your course in regard of the wind then in the correcting of your account observe the rule in the second case of the sixt Proposition of the plain chart drawing a straight line from your first place to the place you are then in 7 To gain sight of land when you can It vvil not be amisse as often as you can to get a sight of such Capes Headlands or Islands as lye neer your course which being standing marks wil give you certain knowledg vvhereabouts you are and so you may the better direct your course and perfect your account 8 To observe the variation of your Compass Lastly you must have an especial care of your compasse that it be every way perfectly and exactly made and likewise you must be as careful that you steer your course exactly upon that rumbe you reckon on to vvhich end not onely the Steers-man must be diligent to keep the ship to the course appointed but you must be frequent in observing the variation of the compasse vvhich may be so vvel performed by the Sea rings in use among Sea-men that no Instrument can be devised fitter for the purpose this variation being knovvn must be allovved for in your account that so you may knovv the true rumbe you sail upon vvithout vvhich there can be no true account kept CHAP VII Of sayling by a great Circle ALthough the way already proposed for the finding out of the arch of a great circle Another way to find out the longitudes and latitudes of the arch of a great circle between two places is the most easie and plainest way that can be yet because it is not so general as to take in all places but is onely to be used when the two places are both on the one side of the Equinoctiall as also it may seem somewhat defective because it doth not shew the distances of places I have therefore here added this second way partly for variety and partly to supply the defects of the former But as this way is more artificial so it is more difficult both in the demonstration and practise which cannot be avoided But you may the better bear with it because you wil seldome be forced to use this way but may very wel content your self with the other in most voyages This and all other questions of this nature concerning the resolution of any spherical triangle may very easily be performed by the Globe but because the Globe is a chargeable Instrument and so every one cannot have it therefore severall men have for severall uses invented severall wayes to project the Globe upon a Plane You may see the severall projections thereof in Mr. Gunters book of the Sector The fittest for this purpose wil be that of Gemma Frisius which is most used in Maps of the whole world the projection whereof is as followeth First draw the circle A D B C How to describe the Globe in plano and divide it into four parts or quadrants by the crosse diameters A B and C D then divide each quadrant into 90 degrees and number them as in the figure then if you keep one end of your ruler fixed at the point A and lay the other end to the several degrees in the lower Semicircle D B C so you shal divide the Diameter C D into its parts which are half tangents In the same manner if you keep one end of your ruler fixed in the point C and lay the other end to the severall degrees of the Semicircle A D B you may divide the diameter A B into halfe tangents Having thus divided the circumference and the diameters they must guide you in the drawing of the meridians and the parallels How to draw the Meridians and Parallels which are all parts of perfect circles and you may finde their centers by these three points First for the Meridians they all concurre in both the Poles A and B and their third point is their correspondent degree in the diameter C D. Then for the parallels two of their points are their degrees in the outward circle and