only 12 seconds hereof that is the Part Proportional answerable to the excesse of 9. sec. above 4 sec. Where Maginus after the Prutenick account maketh it to be in 0 degr 25 min. of Aries that is 33 minutes wanting of the truth found by Observation The like difference I have often found by many and diligent Observations especially for the space of the four years before mentioned the whole Catalogue of which Observations I thought good for thy further satisfaction herein to set down in a Table after I have first shewed with what Instrument and after what manner I Observed the same that if any error herein hath been committed it may the more easily appeare and be amended CHAP. XIX The description and use of a great Quadrant for Observation of the Sun on Land THe Instrument therefore wherewith I took those Observations was a Quadrant of more then six foot and a quarter semidiameter for the room wherein I was to use it could not well admit a greater quantity which by reason of his largeness was so exactly made and divided that both minutes and half minutes might therein be easily discerned The Limb and sides of the Quadrant were about two inches and a quarter in thickness the breadth of the Limb about four inches the breadth of the Sides about two inches and an half In the midst of the ends of one side of this Quadrant were two round holes made in either end one whereby the Quadrant was hanged like a gate on his hinges upon two round pins fitted to those holes and fixed in the ends of a copple of sockets put close upon a strong square post Perpendicularly erected and the upper end thereof fastned to the side of a principal rafter in an upper chamber where a window according to the Reclination of the Roof of the house was made between it and the next rafter in such sort that carrying your eye along by the circumference of the Quadrant you might by the Center thereof placed at the window see any part of the Heavens neer the Meridian from the Zenith to the Horizon The nether end of this post resting on the floor was put into the midst of a socket nailed to the floor which was so wide that on every side the post wedges might be put in to coyn it at pleasure this way or that way till the side of the Quadrant were found to stand exactly Perpendicular by the hanging of the plum-line all alongst most precisely upon a line Parallel to the Zenith line of the Quadrant To the Center of the Quadrant was fastned a strong Ruler of one inch in thickness two inches in breadth and almost six foot and an half in length carrying two Sights upon it viz. at either end one of equal breadth and length the end of the middle line of each Sight falling Perpendicularly upon the middle or fiducial line and plain Superficies of the Ruler Through the upper Sight placed at the Center was made a square hole as great as it could well be Through the midst of this Sight and hole was put a straight wyre erected Perpendicularly from the Fiducial line and plain of the Ruler and so much of it made flat and thin as was between the top and base of that square hole This wyer served for Observing the Stars the flat side whereof was to be turned towards the eye in Observing of great Stars and the narrow side or edge of it was turned to the eye-ward when smal Stars were to be Observed Through the midst of the nether Sight from the top of it to the Base thereof was made a narrow slit Perpendicularly erected likewise from the Fiducial line and plain of the Ruler and Quadrant When I Observed the Stars I looked through this slit Elevating and Depressing the Ruler till the wyer being first fitted to bigness of the Star did even cover the Star from my sight in such sort that I might see both edges of the Star alike on either side above and beneath the wyre The square hole in the Sight had a cover fitted to it like the cover of a box wherewith it was wholly covered when the Sun was to be Observed The nether end of the Ruler carrying the Sights was to be fastned with a screw-pin at any part of the Circumference of the Quadrant as need required With this Quadrant alwaies rectified by the Plumb line in time of Observation as before is shewed the height of the Sun was most easily and exactly Observed by turning the Quadrant this way or that away and Elevating or depressing the Ruler carrying the Sights till the top and sides of the shadow of the Upper Sight placed at the center fell upon the nether Sight placed at the center fell upon the nether Sight placed at the Circumference equidistantly from the top and sides thereof For then the upper edge of the Ruler sheweth precisely the height of the Sun desired in degrees and minutes upon the limb of the Quadrant saving that one whole degree was alwaies to be added thereto because the breadth of that part of the Ruler that lay upon the Limb of the Quadrant was made to be just equal to two degrees that is on either side one degree from the fiducial line Now for finding out the Meridian Altitudes of the Sun and Stars I first found the Meridian line thus with the quadrant rectified and used as before is shewed I Observed the height of the Sun in the forenoon and so warily letting the Quadrant stand immoveable and laying the side of a straight Ruler that was about seven foot in length close along to the perpendicular side of the Quadrant close by the end of that side of the Ruler touching the floor of the chamber I made a prick upon the floor Also laying the side of the Ruler to the perpendicular side and limb of the Quadrant I made in like manner another prick so far as conveniently I could from the former upon the floor close by the corner of that side of the Ruler By these two pricks I drew a right line which represented the intersection of the Suns Azimuth or of the continued plain of the Quadrant and of the plain of the floor in the time of Observation Likewise in the afternoon the Ruler of the Quadrant carrying the Sights being fixed in the same place where it was in time of Observation in the forenoon I Observed diligently till the Sun came to the same height that he had when I Observed in the forenoon which I did by following the motion of the Suns shadow with the Quadrant till the edges of the top and sides of the shadow of the upper Sight fell upon the nether Sight equidistantly from the top and Sides thereof Then carefully letting the quadrant stand immovable and drawing the line of intersection of the floor and Suns Azimuth in time of the afternoon Observation in like manner as I did in the forenoon setting one foot of the Compasses in the
55 160 401.357 537.178 251.865.582  56 160 938.535 545.704 260.459.920 8.594.338 57 161 484.239 554.505 271 919.077 11.459.157 58 162 038 744 563.594 289 107.811 17.188.734 59 162.602.338 572.986 323.485.279 34.377.468 CHAP. III. The use of the two first columnes of the Table of Latitudes for graduating a Meridian in the general Sea-Chart BEfore you can make use of this Table for the true graduating or dividing of a Meridian of this Chart into his degrees or other parts of Latitude increasing from the Aequinoctial towards the North and South in such proportion as before hath been shewed there must be first some preparation made to that end which may be done after this manner Overthwart the midst of the plain superficies whereupon you will draw the lineaments of the Chart describe a right line representing the Aequinoctial circle which you shall divide into 360 parts or degrees and crosse the same squirewise with right lines by every fift or tenth degree Then take with your compasses the length of half the Aequinoctial that is 180 degrees and setting one foote of your compasses in the mutuall intersection of the Aequinoctial with the perpendicular or Meridian that passeth by either end of the Aequinoctial with the other foote make a prick in the same perpendicular or Meridian the space contained betwixt this prick and the Aequinoctial divide first into three equal parts and every one of these into other three so have you nine in all and againe every one of these into three so have you 27 parts and every one of these parts divide into four so have you 108 parts And againe if there be space enough divide every one of these into 10 or 100. So shall you have 1080 or 10800 parts which will bring you to the Latitude of 85 degrees and something more But if you would make your Chart to any greater Latitude you shall continue forth the foresaid perpendicular and divide it into so many more of the same parts as you shall find needful to attain to the Latitude you desire Then note every fift and tenth part with black lead and set figures at them beginning at the Aequinoctial and from thence proceeding Northwards and Southwards Then look what numbers in the second column are answerable to each degree or minute in the first column of this Table of Latitudes omitting alwaies four or five of the first figures towards the right hand and at the same numbers of parts in the perpendiculars make pricks on either side the Aequinoctial by which pricks draw right lines equidistant from the Aequinoctial for they shall be the Parallels of the true Nautical Planisphere or Sea-Chart Notwithstanding these Parallels are all oâ—â—hem a little further distant from the Aequinoctial then in truth they should be and so much the more the further they are from the Aequinoctial Which error might be something the lesse if the former Table had been first made to smaller parts then minutes But that were a matter more curious then necessarie the Table here before set down being so neere the truth that it is not possible by any rules or Instruments of Navigation to discover any sensible error in the Sea-Chart so farre forth as it shall be made according thereto The figure following containeth onely one part of the Nauticall Planisphaere from the Aequinoctial Northwards because the other part from the Aequinoctial Southwards must be altogether like and equal to this Herein first I drew the Aequinoctial line AC and divided it into 36 equal parts whereof every one is understood to contain ten degrees and I raised perpendiculars from every one of those parts which are the Meridians of the Nauticall Planisphaere every where aequidistant each from other Then I took half the length of the Aequinoctial with the compasses and setting one foot in the end of the Aequinoctial at C with the other foot I made a prick at D in the perpendicular or Meridian CD The space contained betwixt C and D I divided into 1080 parts understanding every one of the smaller parts or segments of the line CD to contain ten lesser parts in such sort as before hath been shewed and set figures to them as here you see for the readier numbring and finding out of any of those parts Then I looked in the former Table what number of equal parts of the Meridian answered to every tenth degree and casting away five of the first figures next the right hand because I conceive the space betwixt C and D to be divided only into 1080 parts I found out the parts answerable to the numbers remaining in the line CD and at those parts I made prickes by which I drew the Parallels As for example in the Table the number answerable to ten degrees is 60 casting away the five first figures towards the right hand therefore I look 60 in the line CD and by that part I draw the Parallel of ten degrees distance from the Aequinoctial Likewise the number answering to twentie degrees omitting the five first figures is 122 therefore by that number of equal parts I draw the parallel of twentie degrees Latitude from the Equinoctial c. And after this manner I drew all the rest as you may see in the former draught The Draught of the Meridians Parallels and Rumbs of the Nautical Planisphear truly made CHAP. IIII. 〈◊〉 way for graduating the Meridian of a general Sea-Chart OTherwise for the dividing of the Meridian of a general Sea-Chart into his degrees and other smaller parts of Latitude when the Chart hath not so great Latitude or breadth from the Equinoctial towards the North or South as hath the figure before set down you may go thus to work First find out what proportion the whole Longitude or lenght of the Chart from West to East must have to the whole breadth thereof betwixt the Parallels of the most Northerly and Southerly places that are to be set down therin which may be done after this manner Out of the second Column of the table of Latitudes take the numbers of equal parts of the Meridian answerable to the greatest North and South Latitudes that are to be set down in the Chart divide those numbers by 600,000 that is the number of equal parts of the Meridian answerable to one degree of the Equinoctial the Quotients will shew how many degrees of the Equinoctial the breadth of the Chart must be on either side the Equinoctial toward the North and South As for example in the generall Sea-â—hart to be adjoyned to this book the Latitude of the North 〈◊〉 of the New land found by the Hollanders about the yeer â—596 and by them called Gebrooken land lying Northwards â—rom Norway is about 80 degrees And the Latitude of Queen Elizabeths Iland first found by Sir Francis Drake lying to the â—outhwards of Magellanes streights is about 53 degrees The â—umbers of equal parts of the Meridian answerable to these Laâ—â—tudes found out in the foresaid Table of
it followeth that so often as one of these parts is contained in the segment of the Rumb aforesaid in this planisphaere so many score leagues is the distance of the two places set at the ends of that segment Now it is manifest that by these three segments that is the segment of the Rumb betweene the two places the segment of the Meridian betwixt one of the places and the Parallel of the other that is the difference of Latitude and the segment of the Parallel contained betwixt one of these places and the Meridian of the other which is the difference of Longitude I say it is manifest that by these three segments a right angled Triangle is made because the segments of the Meridian and Parallel which are two sides of this Triangle include a right angle Againe it is plaine that taking with your Compasses so many degrees of the Equinoctial as are contained in the difference of Latitude then guiding one foote in the Equinoctial and carrying forwards the other Parallel wise till it crosse the Rumb of those two places in such sort that one foote of the Compasses being set in that crossing the other moved about may but only touch the Equinoctial and lastly drawing from that crossing a line perpendicular to the Equinoctial It is plaine I say that by this perpendicular and the two segments one of the Equinoctial betweene this perpendicular and the Rumb the other of the Rumb betweene the perpendicular and the Equinoctial by these segments I say and the said perpendicular there is comprehended another right angled Triangle which by the 14. e 4. c 3. e. 7 Ram. Is like to the former right angled Triangle because two angles of them both are equal that is the right angles and the angles of the same Rumb In the last of these Triangles the side perpendicular to the Equinoctial is proportional to the difference of Latitude and the segment of the Rumb betweene the end of this perpendicular and the Equinoctial is proportional to the segment of the same Rumb contained betwixt the two places Therefore by the 2 p 6. 17 p. 11 Eulc Because the line perpendicular to the Equinoctial containeth so many equal degrees of the Equinoctial as there are equal parts in the difference of Latitude that is so many as there are degrees in the difference of Latitude these equal parts also of the perpendicular and difference of Latitude are proportional Whereof it followeth that so oft as one of these equal parts of the difference of Latitude is contained in the segment of the Rumb betwixt the two places which before we shewed to be so oft as a degree of the Meridian in the Globe is contained in the segment of the Rumb betwixt the same places in the Globe so oft is one of the said equal parts of the perpendicular aforesaid that is a degree of the Aequinoctial contained in the segment of the same Rumb betweene the foresaid crossing or end of the perpendicular and the Aequinoctial Therefore look how many degrees of the Aequinoctial there are found in the segment of the Rumb of the two places so many score leagues is the distance of those two places which was to be demonstrated Thus have you a way infallible to find out the distance between any two places measured in their Rumb which because it is then onely their true distance that is the shortest space betwixt them upon the superficies of the Terrestriall Globe when both places lie North and South each from other or East and West having no Latitude as under the Aequinoctial whereas otherwise the segment of the Rumb betweene the two places is alwaies greater then the true distance yea sometime by halfe and more in places far Northward or Southward I thoâ—ght good also here to set down the way to find out the true distance of any two places according to the arch of a great circle drawn betweene them wherein I have been and yet am publikely charged with my promise and meane at this time to discharge my selfe thereof The true distance betweene two places is the arch of a great circle contained betwixt them which is thus to be found out If both places have no Latitude as when they are both under the Aequinoctial and one of them also no Longitude the Longitude of the other being lesse or not more then 180 degrees the Longitude is the distance But if the Longitude be greater then 180 degrees subtract it out of 360 the remainder is the distance If both places have either none or the same Longitude as when they are in the same semicircle of the Meridian betweeene the Poles and one of them onely have Latitude that Latitude is the distance But if both places agreeing in Longitude have Latitudes also of like denomination as both Northerly or both Southerly subtract the lesser Latitude out of the greater the distance remaineth If one place have Northerly Latitude and the other Southerly adde them together for the summe is the distance If one or both places have atitude Land differ also in Longitude in a great circle divided exactly into degrees with figures set to every fifth or tenth degree note the Longitudes of both places Now if one place only have Latitude draw a diameter from the Longitude thereof noted in the circle and with your Compasses take so many degrees and minutes in the same circle as that Latitude containeth then setting one foote of the Compasses in the Longitude of that place with the other make a pricke in the circle which may be called the point of Latitude From this point draw a line perpendicular crossing the diameter drawn from the Longitude of that place Take with your Compasses the distance of this crossing from the point of the other places Longitude noted in the circle and leaving one foote in the said crossing with the other make a pricke in the foresaid diameter take the distance of this pricke from the point of Latitude noted in the circle Then setting one foote of the Compasses in that point of the circle where the degrees begin to be numbred the other foote extended that way which the numbers proceed shall shew you in the circle the distance of the places Take for example the Citie of London and Saint Thomas Iland which lieth right under the Aequinoctial line in 32 degrees of Longitude The Longitude of London admit to be 22 degrees the Latitude 51 degrees 32 minutes Marke the Longitudes of Saint Thomas Iland and of London with A and B. From the Longitude of London because London hath also Latitude draw the diameter BC. Having taken with the compasses the Latitude of London in the circle set one foote in B and with the other make the prick E in the circle and draw the perpendicular EF crossing the diameter BC at F. Make FG equal to FA which is the distance of Saint Thomas Iland from the sine of Londons Latitude Then GE shall be the line
sines of their Latitudes if one be Northerly another Southerly are equal to the square of the line subtending the distance of the places 5. e 12. Ram. 47. pr. 1. Eucl. With no lesse facilitie also by help of the former Tables and the Canon of Triangles any two places being given there may Arithmetically and most exactly be found out first by their Longitudes and Latitudes the Rumb and distance measured in the Rumb secondly by their distance and Latitudes the Rumb and difference of Longitude thirdly by their Rumb and Latitudes the distance and difference of Longitude fourthly by their Longiâ—udes Rumb and one Latitude the other Latitude and Distance fiftly by the Rumb distance and one Latitude the other Latitude and the difference of Longitude or any other Nauticall or Geographicall probleme that by the Chart may mechanically be performed and the whole Art of Navigation Arithmetical as some call it may as easily be practised So as having only the Longitudes and Latitudes of the places by which and to which you are to Sail set down in a Table you may by Arithmetical Calculation only if you list take the pains without any Chart Map or Globe shew the Course and Distance from any place to other and so give most exact direction for the performance of an whole Voyage to any known place assigned how oft soever you have traversed or been tossed this way and that way by reason of scant violent or contrarie winds or any other occasion But seeing the first grounds of this Art that is the observations of the Latitudes but especially of the Courses at Sea can not but be far from such exquisite truth as is to be found in those Arithmetical operations how exact soever you be in the rest of the means you can look for no more truth in conclusion then such as is answerable to the first grounds and principles out of which the conclusion is gathered So as the Mariner shall not need to trouble himself any further herewith but only to cast up his accounts upon the Chart truly made as before is shewed which of all other is most fit and ready for his ordinary use Now therefore it may be sufficient only to shew how the former Problems may mechanically be performed upon the Nauticall Planisphaere before described First by the Longitudes and Latitudes of both places given the Rumb and Distance may thus be found Draw parallels by both Latitudes take the distance of those parallels according to which distance draw a parallel to the Equinoctial Then from the end of the difference of Longitude reckoned from the concurse of the Rumbs in the Equinoctial erect a perpendicular crossing the said Parallel A line drawn by this crossing from the concurse of the Rumbs is the Rumb of the two places Now to find out the Distance take so many degrees of the Equinoctial as the difference of Latitude conteineth and guiding one foot of the Compasses in the Equinoctial with the other foot carried parallel-wise at equall distance from the Equinoctial crosse the Rumb newly found out take the distance of this crossing from the concurse of the Rumbs and set both feet of the Compasses in the Equinoctiall for the degrees contained between them shew you the distance desired Secondly by the distance and latitudes knowing which place is more Eastwards or Westwards the Rumb and difference of Longitude is thus found Take with the compasses so many degrees and minutes of the Equinoctial as the difference of Latitude conteineth According to that distance draw a parallel to the Equinoctial take so many degrees of the Equinoctial with your Compasses as the distance given commeth to then one foot being set in the concurse of the Rumbs in the Equinoctial with the other crosse the parallel aforesaid A line drawn by that crossing from the concurse of the Rumbs in the Equinoctiall giveth you the Rumb desired Then both Laititudes being noted in the graduated Meridian therein take their difference with the Compasses and guiding one foot in the Equinoctiall with the other carried at that distance parallel-wise from the Equinoctial crosse the Rumb of the two places the distance of that crossing from the Meridian that commeth from the common meeting of the Rumbs in the Equinoctial taken with the Compasses and brought to the Equinoctial shall there shew you the difference of Longitude Or a perpendicular to the Equinoctial from that crossing shall point you out therein the difference of longitude Thirdly by the Rumb and Latitudes being both Northerly or both Southerly the distance and difference of longitude is thus found Take the difference of Latitudes in the Equinoctial according to that distance draw a parallel to the Equinoctial as before crossing the Rumb of the two places given take the distance of this crossing from the concurse of the Rumbs Then both feet of the compasses set in the Equinoctial will shew the distance of the places The difference of Longitude is found as before Fourthly by the longitudes Rumb and one Latitude knowing whether it be the lesser or greater to find the other Latitude and the distance doe thus From the point of concurse of the Rumbs in the Equinoctial count the difference of longitude from hence erect a perpendicular crossing the Rumb the distance of this crossing from the Equinoctial translated into the graduated Meridian setting one foot in the known Latitude and extending the other Northwards or Southwards according as the unknown Latitude is greater or lesser shall shew you the Latitude desired Now to find the distance work as before in the first Probleme Fiftly by the Rumb distance and one Latitude you may find the other Latitude and the difference of Longitude after this manner Take the distance given with the Compasses in the Equinoctial set one foot in the concurse of the Rumbs and with the other crosse the Rumb given from this crossing draw a perpendicular to the Equinoctial the length of that perpendicular taken with the compasses and brought into the Equinoctial shall shew you the difference of Latitude Thus having both Latitudes given the difference of longitude may also be found as before Prob. 2. Now in every one of these Problemes there may be some particular cases whereof some diversitie of working may arise yet such as can breed but small trouble to him that well doth conceive the reason of that is already set down in these five former Problems which are especialy to be applied to such places as are both on the same side of the Equinoctial and differ also both in Longitude and Latitude of which sort is the greatest number and in which the greatest use and most difficultie of working consisteth To prosecute every particularitie at large whereof some perhaps lesse acquainted with the reason of these Mathematical practises may be desirous would be now for me too long and tedious For some taste therefore of the use of this Nauticall Planisphere let thus much
24 47 69 17 77 86 48 18 34 0 48 70 9 78 87 8 19 35 36 49 70 59 79 87 26 20 37 9 50 71 48 â—0 87 44 21 38 41 51 72 36 81 88 1 22 40 11 52 73 23 82 88 17 23 41 39 53 74 8 83 88 33 24 43 6 54 74 52 84 88 47 25 44 30 55 75 35 85 89 1 26 45 54 56 76 17 86 89 14 27 47 15 57 76 57 87 89 27 28 48 36 58 77 37 88 89 39 29 49 54 59 78 15 89 89 50 30 51 11 60 78 53 90 90 0 whose complement to a quadrant is the angle sought for ARZ According to this Diagram and demonstration was calculated the Table here following the first column whereof containeth the height of the Pole for every whole degree the second column sheweth the Inclination or Dipping of the Magnetical Needle answerable thereto in degrees and minutes CHAP. XV. Error in using the Crosse-staffe and how they may be avoided AFter the Chart and Compasse the Crosse-staffe may with good reason succeed as in the use whereof more error is committed then in any other Instrument of Navigation the two former excepted and that four severall waies First in neglecting the Paralax or Eccentricitie of the eye Secondly in not considering the height of the eye above the Water Thirdly and Fourthly in not regarding the Paralax and Refraction of the Sun For the first they count the height of the Sun and Stars in such sort as if the center of the eye or vertex of the visual cone in using the Staffe were even with the end thereof that is set to the eye Therefore how much the center of the sight is distant from the end of the Staffe so much are they deceived But how much the Eccentricitie or Paralax of the eye is it may be known after this manner Make two Transversaries the one twice so long as the other The longest of these two set fast at the further end of the Index the other of them move up or down upon the Index untill such time that your eye placed at the end of the Index in such sort as you use to place it when you observe you may see both ends of both Transversaries lie even together For then look how much the segment of the Index betwixt the two Transversaries exceedeth the segment from the shorter Transversarie unto the eye so much is the Parallax or Eccentricitie of your sight or the point wherein your eye wherein the visual beams concur is so much distant from the end of the Index As for example in this figure let the Transversarie HEI placed at E the end of the Index be double to the Transversarie FDG which is placed in such sort upon the Index that the visual lines AFH AGI of the eye placed at the end of the Index do passe straight on by FH and GI the ends of the Transversaries For in this figure A is the center of the sight or eye wherein the visual lines AFH AGI doe concurre B representeth the end of the Index placed at the corner of the eye and then AB is the Eccentricitie C signifieth the end of the Index set against the bone underneath the eye for observing of distances and then AC is the Eccentricitie which is thus demonstrated Secondly they increase the former error by not regarding the height of the eye above the Water Which although it be not so great a fault as the other yet it may deceive them by increasing the former error five or six minutes or more in a tall Ship For the higher the eye is above the water the greater is the angle contained betwixt the two visual lines whereof one toucheth the convex superficies of the Sea the other passeth on to the Sun or Stars And the lower the eye is the lesse is the foresaid angle and then onely it sheweth the true Altitude when the center of the sight is in the same line of levell with the superficies of the Water But if the eye be higher then the Water that angle is greater then the true Altitude and therefore subtraction must be made accordingly that you may have the true Altitude Now to find how much it is that should be subtracted at any height of the eye above the Water there be two waies the one without knowledge of the Earths semidiameter the other with knowledge of the same For the first you must have some such convenient place at the Water side where you may have a free and cleere prospect unto the Sea without impediment and where you may also have such provision made that you may place both your self and also an exact and large Water Levell in convenient manner to make exact observation at what height soever you desire above the superficies of the Sea till you come to the height of the tallest Ships that go upon the Seas that levell having the sight that you must look through at the end thereof next the eye so fitted that you may both easily and steadily set it higher then the fore sight that is the sight that is at the fore-end of the Levell so much as shall be needfull to lay the fore-sight precisely to the touching of the Sea and that you may also perfectly know how much the back-sight or sight at your eye is higher then the fore-sight above the line of Levell For by the difference of the heights of those sights above the line of Levell and the distance between them it may easily be found how much the visual line touching the roundnes of the Sea Dippeth under the line of levell or true Horizon from whence the height of the Sun and Stars is to be accounted thus As the distance betwixt the sights is to the difference of their heights above the line of levell so is the whole sine to the Tangent of the angle of Dipping which we desired to know This angle may otherwise be found the quantitie of the Earth semidiameter being first known which is to be done divers waies but they may be all reduced to two heads or kinds whereof the first requireth the certain measure of some arch of the Meridian to be first given which is also divers waies to be performed But the best and perfectest way of all others is to observe so axactly as is possible the Summer solstitiall Altitude of the Sun at two places so farr distant asunder and lying so neer North and South each from other with so direct and faire a way betweene them as conveniently may be chosen Suppose for example Portsmouth and Barwick or some other place in the furthest parts of Scotland for the further these places are each from other the more perfectly may this businesse be performed Then measure and plat down so truly as is possible all the way betweene those two places with all the turnings and windings ascents and descents that are therein out of which the arch of the great circle
subtending the distance of those two places Taking therefore the length of GE with the Compasses and setting one foote in H where the degrees begin the other stretched forwards in the circle will point you out the distance of Saint Thomas Iland and London 52 degrees of a great circle and about one halfe that is 1050 leagues or 3150 English miles If both places have Latitude do the like for both places as before you did for the one place having Latitude till you have crossed both diameters with perpendiculars then take with your Compasses the distance of those crossings Now if both their Latitudes be of one denomination that is both Northerly or both Southerly and equal set one foote of the Compasses where the degrees begin to be numbred in the circle and the other foote extended therein that way which the numbers succeede will shew you the distance As for example London and Cape Blanco neere the coast of New-found land have both Northerly and almost equal Latitudes of 51 degrees 32 minutes Having therefore drawn as well the diameters BC and DL from B determining the Longitude of London viz. 22 degrees and from the point of the Longitude of Cape Blanco which admit to be 331 degrees as also the perpendiculars or sines of both their Latitudes EF and KL as before was shewed crossing the diameters in F and L the distance FL taken with the Compasses and translated into the circle as in the former example will shew you the distance of Cape Blanco from London to be almost 31 degrees of a great circle that is 620 leagues or 1860 miles If the Latitudes be not both equal and also of one denomination leaving one foote of the Compasses in the crossing of the sine or perpendicular descending from the point of the greater Latitude with the other foote make a pricke in the same diameter wherein that crossing is then if the Latitudes be both of one denomination â—ake with the Compasses the length of the perpendicular or sine drawn from the point of the lesser Latitude and setting one foote in the point of the greater Latitude with the other make a prick in the perpendicular descending from it that is in the sine thereof Take the distance of this pricke from the former made in the diameter This distance transferred into the circle as in the first example will give you the distance of the places given As London and Hierusalem have both Northerly and unequal Latitudes Hierusalems Latitude being onely 32 degrees First therefore note in the circle both their Longitudes the Longitude of London viz. 22 degrees as before with B The Longitude of Hierusalem 68 degrees note with M Let the perpendicular or the sines of the Latitudes of London and Hierusalem EF and NO be drawn as in the former examples Make FP equal to OF and PQ equal to NO The space betwixt P and Q taken with the Compasses and then both feet set in the circle in such sort as in the first example was shewed shall containe between them the desired distance of Hierusalem from London 38 degrees and about ¾ that is 775 leagues which are 2325 miles But if the Latitudes be of divers denominations that is one Northerly and the other Southerly continue forth the perpendicular that crosseth the diameter wherein the foresaid prick was made till it be equal to both perpendiculars that is to the sines of both Latitudes The distance of the end of this continued perpendicular from the pricke aforesaid in the diameter taken with the Compasses and translated into the graduated peripherie of the circle as before will shew you how many degrees of a great circle are contained between both places To shut up this matter with one example suppose you would know how farre Cusco in Peru is from London Let the Longitude of Cusco be 295 degrees the Latitude 11 degrees Southerly The Longitude of London as before 22 degrees the Latitude 51 degrees 32 minutes From both these Longitudes noted in the circle with B and R draw the diameters as before BC and RV as also the perpendiculars or sines of their Latitudes EF and T S Make FX equal to FS the distance of those sines and EY equal to ST the sine of Cuscoes Latitude Take the distance XY between the feet of the Compasses and set them both in the circumference of the circle as in the first example so shall you find that there are betwixt London and Cusco almost 97 degrees of a great circle that is 1940 leagues or 5820 miles If you had rather keepe within the compasse of the circle make the perpendicular XZ equal to ST and proceede with EZ as you did before with XY Paste this upon the Margin of Letter N. fol. 65. so as it may ly open all the while the fore-going matter of the same Chap. is reading Also because all the sines of Latitude being perpendicular to the same plain of the Aequinoctial are Parallels by the 5. e 21. Ram. 6. pr. 11. Eucl. Therefore by the 11. e 2. Ram. or 35. d. 1. Eucl. FL is the line subtending the distance of London and Cape â—â—anco Again because FP whereto EF is perpendicular is made equal to FO the distance of the sines of London and Hierusalem to which distance EF is also perpendicular in the Globe and EQ also equal to NO Therefore FQ being the difference 〈◊〉 the sines of Londons and Hierusalems Latitudes there must needs be the same distance betwixt P and Q that there is between the tops of the sines of Hierusalems and Londons Latitudes in the Globe Lastly FX being equal to FS the distance of the sines of Latitude of London and Cusco in Peru and XZ perpendicular to FX and equal to ST the sine of Cuscoes Latitude as EF is the sine of Londons Latitude and perpendicular to the same line XF EZ to which XY is equal by the 6. c. 12. e 5 Ram. 33. pr. 1. Eucl. YE being equal and Parallel to XZ must needs be equal to a streight line extended within the Globe between the points of Latitude of Cusco and London Now out of this demonstration it were an easie matter if any list take the pains to be so curious to find out the distance of any two places Arithmetically by the doctrine of Triangles having alwaies two sides given which are the sines of the complements of the Latitudes of the two places as OP FP LP FP RP FP AP FP together with the angle contained between them that is the difference of their Longitudes whereby FA FO FL FS the distances of the sines of Latitude being found by 〈◊〉 2 3 4 5 Copernic de Triang planis the lines also subtending the distances of the places may most easily be found by the 3. Copernic de Triang plan For the squares of the distance of the sines and of the difference of the sines of their Latitudes if both be Northerly or both Southerly or of the sum of the
innermost circle to the end of the foresaid brasse pointer so as the end thereof may enter thereinto Then set the line that is drawn from the end of the brasse pointer to the elevation of the Pole at the place of your Observation and so have you all parts of your Instrument rightly placed for Observation Now when you will make Observation with this Instrument hang the same by this Ring upon your finger as you do when you Observe the height of the Sun with the Astrolabe turning the foresaid brasse pointer Northwards till you find the midst of the shadow of the bead to fall upon the peripherie drawn round about through the midst of the concavitie of the Equinoctial ring which peripherie we called the parallel of the Sun and so keep the whole Instrument and Compasse as steady as you can till the flie of the Compasse remain quiet and still keeping in the mean time the midst of the shadow of the bead alwaies upon the foresaid parallel of the Sun and withall looking close by the graduated side of the Meridian directly down upon the midst of the Compasse and mark what degree and minute you see close thereby in the North part of the Flie for so much as the North point of the needle or wiers is from thence towards the East or West so much is the Variation Eastwards or Westwards And the shadow of the bead lighting upon the Parallel of the Sun sheweth the hour and time of the day The best time for the taking of these Observations is about the midst of the forenoon or afternoon because that about those times the height of the Sun altereth quickly and his Refraction also can breed no sensible error But because there be many that want both this Instrument and also the Globe and Astrolabe before-mentioned I have for their sakes thought it good to set down a way whereby the Declination and height of the Sun being given together with the Latitude of the place the Suns true Azimuth may be found with Ruler and Compasses onely after this manner Draw the circle ABCDEFGH representing the Meridian by the center hereof draw the diameter of the Horizon AF. From A the end of this diameter reckon the elevation of the Equinoctial AC from whence draw a line by the center which may be called the diameter of the Equinoctial AC from C the end of this diameter count the declination of the Sun CD thereby draw a Parallel to the diameter of the Equinoctial which may be called the diameter of the Suns Parallel DG Likewise from the diameter of the Horizon count the height of the Sun AB known by Observation and thereby also draw BE a Parallel to the diameter of the Horizon which may be called the diameter of the Suns Almicantar From I the intersection of these two Parallels draw IK a line perpendicular to the diameter of the Horizon Then setting one foot of the Compasses in L the midst of BE the diameter of the Suns Almicantar and stretching out the other foot to B the end of this diameter from thence draw therewith the arch BK till you come to the said perpendicular This arch resolved into degrees shall give you the true Azimuth of the Sun CHAP. XIIII To find the Inclination or dipping of the Magnetical needle under the Horizon First the angle OAR is given because of the arch OBR measuring the same 150 degrees and consequently the angle at R 15 degrees being equal to the equal sided angle at O both which together are 30 degrees because they are the complement of the angle OAR 150 degrees to a semicircle or 180 degrees Secondly in the Triangle ARS all the sides are given AR the Radius or semidiameter 10,000,000 RS equal to RO the subtense of 150 degrees 19,318,516 and AS equal to AD triple in power to AB because it is equal in power to AB and BD that is BO which is double in power to AB Or else thus The arch OB being 90 degrees the subtense therof OB that is the Tangent BD is 14,142,126 which sought in the Table of Tangents shall give you the angle BAD 54 degrees 44 minutes 8 seconds the Secant whereof is the line AD that is AS 17,320,508 Now then by 4 Axiom of the 2 book of Ptisc As the base or greatest side SR 19,318,516 is to the sum of the other two sides SA and AR 27,320,508 so is the difference of them SX 7,320,508 to the segment of the greatest side SY 10,352,762 which being taken out of SR 19,318,516 there remaineth YR 8,965,754 the half whereof RZ 4,482,877 is the sine of the angle RAZ 26 degrees 38 minutes 2 seconds the complement whereof 63 degrees 21 minutes 58 seconds is the angle ARZ which added to the angle ARO 15 degrees maketh the whole angle ORS 78 degrees â—1 minutes 58 seconds whereof 60 90 make 52 degrees 14 minutes 38 seconds which taken out of ARZ 63 degrees 21 minutes 58 seconds there remaineth the angle TRA 11 degrees 7 minutes 20 seconds the complement whereof is the Inclination sought for 78 degrees 52 minutes 40 seconds The sum and difference of the sides SA and AR being alwaies the same viz. 27,320,508 and 7,320,508 the product of them shall likewise be alwaies the same viz. 199,999,997,378.064 to be divided by the side SR that is RO the subtense of RBO. Therefore there may be some labour saved in making the Table of Magnetical Inclination if in stead of the said product you take continually but the half thereof that is 99,999,998,689,032 and so divide it by half the subtense RO that is by the sine of half the arch OBR Or rather thus As half the base RS that is as the sine of half the arch OBR is to half the sum of the other two sides SA and AR 13,660,254 so is half the difference of them 3,660,254 to half of the segment SY which taken out of half the base there remaineth RZ the sine of RAZ The Table of Magnetical Inclination First col Second col First col· Second col First col Second col Height of the Pole Magnetical Inclination Height of the Pole Magnetical Inclination Height of the Pole Magnetical Inclination· Degrees Degr. Min. Degrees Degr. Min Degrees Degr. Min. 1 2 11 31 52 27 61 79 29 2 4 20 32 53 41 62 80 4 3 6 27 33 54 53 63 80 38 4 8 31 34 56 4 64 81 11 5 10 34 35 57 13 65 81 43 6 12 34 36 58 21 66 82 13 7 14 32 37 59 28 67 82 43 8 16 28 38 60 33 68 83 12 9 18 22 39 61 37 69 83 40 10 20 14 40 62 39 70 84 7 11 22 4 41 63 40 71 84 32 12 24 52 42 64 39 72 84 57 13 25 38 43 65 38 73 85 21 14 27 22 44 66 35 74 85 44 15 29 4 45 67 30 75 86 7 16 30 45 46 68 24 76 86 28 17 32
cast his point in manner following and he shall errer as little as is possible Let him examine according to the ordinary running of his ship how much way she might make every day that he hath sailed and the leagues that shall amount in all the dayes let him take between the points of one compass and let him place one point thereof upon the place from whence he departed and taking another compass let him set one point thereof upon the graduation according to the heighth which he hath taken and the other point upon the next East and West Rumb Now let this compass run by his East and West rumb till the point coming from the graduation meet with the second point of the other Compass which he holdeth not upon the Chart and in the place where they meet he may say that there is his point and his ship And because in this case all Pilots do not use this point of imagination and heighth there grow great diversities among them concerning their distance from land when as in long voyages they confer and communicate their opinions one with another insomuch that one according to his conjecture judgeth himself to be 50 leagues from land another 100 another 200 and another thinketh he is hard by the land The reason is because some of them cast their point by traversing others by imagination onely and others by imagination and heighth who are alwayes more certain then the rest CHAP. XXIV What it is to increase or diminish in heighth THe Mariners call it increasing in heighth when they go further and further from the Equinoctial and diminishing in height when they approach neerer to the Equinoctial So that in our Navigation we either sail from a greater to a less altitude of the pole and then we go towards the Equinoctial and then the height is said to be diminished or we sail from a less to a greater height of the pole and then we go from the Equinoctial and are said to increase our heighth And hence it is that casting our point by imagination and afterwards having taken the heigth amending it by traversing either the heighth wherein we find our selves being taken by the Sun or star is greater or else it is less then that which we made account of by imagination And hence do arise four rules The first is that when in sailing we do increase the heighth if the point amended by traversing be of greater heighth then the point of imagination the ship hath gone more then the point of imagination shewed us The second when we increase our heighth if the point amended by traversing be in lesse height then the point found by imagination then hath the ship gone less way then we imagined The third is when we diminish our height in sailing if the point amended by traversing be in a greater heighth then the point found by imagination then hath the ship made less way then we ghessed by our imagination The fourth is when we diminish our heighth if the point amended by traverse be in less heighth then the point found by imagination then hath the ship made more way then we imagined CHAP. XXV How you may cast a traverse point without Compasses IF a Mariner chance to lose his Compasses he may cast his point of traverse after this manner Let him take two slender threds and putting the end of one of them upon the place from whence he departed let him stretch it in equal distance from the Rumb by which he hath sailed and putting another thred in equall distance from the next East and West rumb let him make it to pass by the degrees of heighth in which he findeth himself and where the two threds cross one another there is the point of the ship and alwayes the first thred if it hath not changed the course sheweth the way which the ship hath gone and the second the parallel wherein the ship is CHAP. XXVI Of another kind of casting a point by traverse EXamine the difference of the degrees of distance from the Equinoctial which are between the place from whence the ship set forth and the place where the ship is Then taking the heighths of both places very precisely and subtracting the lesser out of the greater that which remaineth is the difference which difference you must multiply by the leagues which answer to one degree in the rumb by which you have sailed and those leagues which the degrees and minutes of difference shall make you shall take between the points of a pair of compasses out of your scale of leagues and holding the said compass so open set one foot thereof upon the point from whence the ship departed and the other foot stretching towards the place whithre the ship hath sailed you must hold up a little from the Chart and you must set one foot of the other Compasse at the degree of the distance wherein your ship is from the Equinoctial when the said point is sought for and the other point you must place upon the next East and West Parallel And let this second Compasse run by his next East and West Rumb untill the point of the first Compasse lifted up being set downe that point of the second Compasse which came from the said degree doth meete therewithal and where those two points shall meet there is the true point of the ship CHAP. XXVII Of the leagues which in Navigation answer to each degree of Latitude in every Rumb IF we suppose as we have before said in the chap. of the quantity of the earth that the greatest circle thereof conaineth in compasse 6300 common Spanish leagues then unto every degree of the Meridian which is the greatest circle doe answer 17 Spanish leagus and an half so that sailing North and South if your heighth of the Pole or your distance from the Equinoctial be varied one degree you may say that you have gone seventeen leagues and an half but if you varie one degree and hold your course upon the first point then have you sailed 17 leagues and ⅙ And you have declined from the Meridian or right line which passeth by the place from which you departed three leagus and an half And if you sail upon the second point from the North or South till your heighth of the Pole be changed one degree you have then gone 19 leagues and ⅜ and are distant from your right line 7 leagues and ¼ And varying a degree upon the third point of the Compasse from North or South you have gone 21 leagues and are departed from your right line 11 leagues and ⅔ Sailing upon the fourth point of the Compasse there doe answer unto every degree 24 leagues and three fourths and you are distant from the right line or Meridian 17 leagues and an half Upon the fifth point you must allow for one degree 31 leagues and an half and then are you distant from your right line 26 leagues and ⅕ Upon the sixth point doe