A DISCOURSES of the Variation of the Cumpas, or Magnetical Needle.  
Wherein is Mathematically showed, the manner of the observation, effects, and application thereof, made by W. B.  
And is to be annexed to The new Attractive of R. N.  
1581.   

To the travelers, Seamen, and Mariners of England.  
Having of late (gentle reader) received from the expert Artificer Ro. Norman, his book entitled The new Attractive (who of the great good will, and affection he beareth, hath attributed in his dedication, that, which I acknowledge not to be dew) in the which amongst other divers virtues and properties of the Magnes or loadstone, he entreateth of the declining of the Needle touched there with from the plain of the Horizon, (a matter never before found, or written of by any). For the further behoof and benefit of all travelers and Seamen, I took occasion to enlarge the same with this discourse of the variation of the Compass, wherein I have handled the whole variety of that subject, both Practically, and Mathimatically, to the end I might partly satisfy both the vulgar, and also the learned sort. For, knowing the variation of the compass to be the cause of many errors and imperfections in Navigation, and perceiving that all those that have as yet gone about to give rules in that art, have left this (being a principal point, and even the ground of all the rest) untouched, or at least so slightly handled the same, that little or no benefit could be gathered thereby: I have here set down the sundry ways to observe the same at all times & places, that the inconvenience being known, might be considered of, and avoided. Wherein, although my chiefest intent hath been to pleasure those that shall have occasion to put the thing in practice by their own travail and experience, yet because some of the rules are deducted from the fountains of the Mathematical Sciences, and wrought by the doctrine of Sines and Triangles, which may seem strange in our English tongue, & wherewith few Seamen are yet acquainted, I may seem to have miss of my first good meaning, but I would wish them to choose that which is plain, and conformable to their capacities, and make their profit thereof, and for the rest understand, that of such observations as they themselves can not presently apply to the purpose, by others that are thoroughly instructed in these Mathematical supputations, or by themselves when they shall attain to the knowledge thereof, may be inferred such effectual matter as is by these rules and precepts promised. Wherefore I would have all Seamen to use such diligence in their travails, that no opportunity be omitted, when, or where any observation may be made, either for the variation, or latitude of places, or of any other necessary point incident to Navigation, and thereof to keep continual notes & memorial. For these observations, there needeth not many troublesome Instruments, only for the variation, the new Instrument in the end of this treatise I prefer before all other. And for elevations, a plain Astrolabe exactly made, and a cross staff, are sufficient. (The Globe were also a very good and necessary Instrument▪ for besides many pleasant conclusions that may be tried by it, it doth lighten very much the conceits, for understanding divers important points, but it is too troublesome [or otherwise not fit for every Mariner] to be carried to the Sea). Unto the which may be added the topographical Instrument, for taking of distances, and making descriptions upon the land. With these Instruments, and the sailing compass and Marine plat, (which are always to be understood the principal, and most necessary Instruments for Navigation, for by them only any voyage may be made, but without them no Navigation can be performed.) the whole world may be traveled, discovered, & described. These are sufficient for a perfect Mariner, and more than these were superfluous, only the running glasses, leads, lines, and such like appendances of the other excepted.  
But to have all these Instruments, and not to understand the grounds how to use them, were a great vanity. Therefore I wish all Seamen & Traueiler's, that desire to be cunning in their profession, first to seek knowledge in arithmetic & Geometry, which are the grounds of all Science and certain arts, of the which there is written in our English to gue, sufficient for an industrious and willing mind to attain to great perfection, whereby he may not only judge of Instruments, Rules, and precepts given by other, but also be able to correct them, and to devise new of himself And this not only in Navigation, but in all Mechanical Sciences. As by the studious practice and exercise in these arts, have attained to rare and singular knowledge: In Architecture, Vitriwius the Roman: In painting that famous German Albertus Durerus: And in building of Ships, Matthew Baker our countryman: And others in other faculties as they have been most skilful herein, so have they excelled. Having these helps and grounds with the Instruments before specified, a Mariner may be able to make description in platt of the coasts and Countries, and of the Banks, Rocks, and Sholdes in the Sea, with the depths and other necessary notes observed in his own travails particularly, & effectually according to the truth, (which is the chiefest part required in a perfect Mariner.) And not be always tied to the reports of other, or to the Portugal, or Spanish Marine platts, which are made by the Cardmakers of those Countries, men that are no travelers themselves, but do all things therein, by information, and upon the credit of others, which only commit to memory the form and manner of the Sea coasts, with making some few notes of the lying of one place from an other, which can never be so perfect as the deferiptions that are made upon the present sight and view of places, albeit he be never so skilful and cunning, that shall so carry the same by memory, how much less than by the unskilful. By this means the Cardmakers set down they know not what: as may appear by the descriptions of their own coasts, which are very grossly and unperfectly done, whereas the Marine plats ought to be described by such as can give reason, and show observation of every particularity contained in the same, aswell for the latitude of places, as the lying by the Compass of the Capes, Headlandes, Points, I lands, bay, Rocks, Sholdes etc. one from an other, and the distances between them. The errors of those descriptions, I may not attribute to the Card-makers, but to the unskilful Seamen of those countries, for if they were otherwise, as they have been accounted the most skilful of the world, those errors could not have continued as they do: true it is that for their great travails, they have been worthily famous above all other nations, till now at length our Countryman Sir Francis Drake for valorous attempt, prudent proceeding, & fortunate performing his voyage about the world, is not only become equal to any of them, but in fame far surmounteth them all. But those Cardmakers, and all other that collect and gather Hydrographical, and Geographical descriptions of other men's travatles or reports: as their pains may be great, and deserve due commendation, so their doings may bring commodity diversely. And in this behalf Abrahamus Ortelius in his Theatrum, hath deserved immortal praise, for collecting together, and reducing into one commodious volume, the divers plats and descriptions, made by divers and sundry men. But amongst all those that have made Geographical descriptions, I can not a little marvel at Guilielmus Postellus, who being a famous learned man, a great traveller and Cosmographer, and Deane of the Kings Professors in the University of Paris, in his universal Map. Anno 1580. besides that it is generally handled after such a gross and confused manner, that it might seem rather to have come from some rude unskilful, then from him so famous a Doctor; hath also in the imagined Countries about the North Pole, so corrupted it with his fond dreams, and fantastical inscriptions, attributing to those supposed lands, divers people, as the Georgians and Hyperborians, and assigning there to be the highest hills of the world, and the people dwelling on them, to have the continual light of the Sun; Sueta Zemlia found by the Englishmen, An. 1550. the holy Land, the place of the chiefest felicity, the Hyperborean fields, and therefore the felicity of the Moluccas, with many other ridiculous absurdities: That by the gross errors of this learned man in these matters, I am taught, that what so ever fame goeth, or opinion is conceived of any man for profound learning, and smooth delivering of their conceits, or what so ever great promises are by themselves made in these arts, to judge of them according to the works that come from them, and not otherwise to be deceived.  
For avoiding prolixity in this my Preface to so smala volume, I refer the gentle reader, to the work itself. Yet by the way it shall not be a miss, that I commend unto you, the table of the suns declination (or Regiment) made by R. N. which is calculated for the present time, and differeth not from the truth in any place above one minute, whereas in all other hitherto made and extant, there are great errors. Therefore, such as otherwise can not from time to time calculate their declinations, according to the place of the Sun to be given by the Ephemerides, and table of declination of Reinholdus may boldly use this Regiment for 20. years without any sensible error. And so wishing my travails in this trease may do such good as I meant, I commit the same to your gentle constructions, and your selves to the Almighty. At Limehouse the 26. of September. Anno 1581.  
William Borough.   

A Table of the Chapters contained in the treatis.  
¶ The first Chapter. 
OF the Variation of the Cumpas, or magnetical Needle.  ¶ The second Chapter. 
The manner how to use the Instrument of Variation.  ¶ The third Chapter. 
How to find the Variation of the Cumpas or Needle at any place, the elevation of the Pole, and situation of the meridian unknown.  ¶ The fourth Chapter. 
The elevation of the Pole, and place of the Sun given, how upon the Globe, to find the Variation of the Needle, by any one observation, either in forenoon or afternoon.  ¶ The fift Chapter. 
How to find the Variation by Arithmetical calculation, upon any one observation in forenoon or afternoon, the latitude of the place, and declination of the Sun being given.  ¶ The sixth Chapter. 
another way most general, how to find the Variation by one observation, either in forenoon or afternoon, the elevation of the Pole, and declination of the Sun being given.  The seventh Chapter. 
To find the elevation of the Pole, situation of the meridian, and variation of the Needle, at any place by the Sun, upon two observations, either in forenoon or afternoon.  ¶ The eight Chapter. 
Of the Pole of the Magnes.  ¶ The ninth Chapter. 
Of the point Respective.  ¶ The tenth Chapter. 
Of the inconveniences and defects in sailing, and in description of Countries, caused by the variation of the Cumpas.  ¶ The eleventh Chapter. 
Of the Instruments and rules of Navigation.  ¶ The twelfth Chapter. 
Of the application of the Variation, to the use of Navigation.    

¶ Of the Variation of the Cumpas or Magnetical Needle.  
Chapter I.  
THE Variation of the Needle or Cumpas, is properly the ark of the Horizon contained between the true meridian of any place and the magnetical meridian of the same, and is denominated to be Esterly or Westerly, according to the position of the magnetical meridian to the Estwards or Westwards of the true meridian: And may be accounted either from the North part, or the South part thereof, but upon opposite points it hath contrary denominations.  
The magnetical meridian is to be understood a great circle passing by the Zenith and the Pole of the Magnes, dividing the Horizon into two equal parts crossing the same at opposite points: which intersections or crossings, are showed by the Needle or wires of the Cumpas touched with the Magnes or the loadstone.  
The Azimuth of the Sun is a great circle, passing by the Zenith and the true place of the Sun: crossing the Horizon at right Angles in opposite points, and dividing the same into two equal parts. And it is said to be given when the distance thereof from the true meridian is known.  
The Azimuths of the Sun upon equal elevations in forenoon and afternoon, have equal distances from the true meridian, so that the middle point of the whole difference of any two Azimuths observed upon equal elevations in forenoon and afternoon, is the true meridian.  
This difference of Azimuths is found upon the Instrument of Uariation, by adding together the Uariations of the suns shadow at equal elevations in forenoon and afternoon. The half whereof is the distance of the Azimuths from the true Meridian: the which compared with either of the same variations of the suns shadow, the difference shallbe the variation of the Needle from the true meridian.  
Or else subtracting the lesser variation of the suns shadow, from the greater (at equal elevations) the half of the remainder shall be the true variation of the Needle from the meridian.  
But the Azimuth of the Sun being otherwise given, and the variation of the shadow likewise given, the difference between them is the variation of the Needle.  
The Variation of the suns shadow I call, the horizontal distance between the Azimuth of the Sun and the magnetical circle, which are represented in the Instrument by the shadow of the line and the Needle.   

¶ The manner how to use the Instrument of Variation. ¶  
The second Chapter.  
FIrst you must place the Instrument upon some Stool, or other thing that is flat, so as it may stand level, and the Plummet in the Standard which is placed at the North end of the fixed Fly, may fall perpendicularly with the line in the same Standard.  
You must have regard that in removing the Instrument to the Sun as he goeth about, it may always stand level as aforesaid.  
You are then to consider, that the ●tring that reacheth from the South part of the Instrument, to the top of the Standard, is the chiefest string to give the suns shadow, which must be so directed by turning the Instruments South side to the Sun wards, that the shadow of the same may fall directly longest upon the line of South and North in the fixed Fly, for it ought not to cross or decline from the same line in any part, but if it do, you must seek to reform it by setting the Standard more upright, or removing it at the South end.  
Then must you also see, that the string that is fastened to the hoop of Brass that environeth the fixed Fly, may be so placed, that it agree justly with the shadow of the former line, and the line of South and North in the fixed Fly, in such sort that both the shadows may be as it were hidden in the said line of the Fly: which you may do aptly, by turning the said hoop, and removing the same line at either side of it, as you shall see cause.  
The Instrument being duly placed in form aforesaid, it differeth nothing from the Cumpas of Uariation, but only in this point, that whereas the Fly of the Cumpas of Uariation, is so turned by virtue of the Magnetical wires, that the North point thereof doth show the Pole of the Magnes or line of Uariation: In this Instrument, the North point of the Needle doth supply that, which the North point of the Cumpas should do. And the North point of the Fly which is fixed in the bottom of the Instrument, doth always answer to the shadow that the Sun giveth.   

¶ How to find the variation of the Needle or Cumpas at any place, the elevation of the Pole, and situation of the meridian unknown.  
¶ The third Chapter.  
WHen you would observe the variation in any place, you must begin in the forenoon, the sooner, the better, and the more effectual may your observations be, do thus.  
Take your Astrolabe and observe duly the height of the sun, for yourmore ease it shall be best for you to note the same, when it agreeth to be just upon a degree, without any consideration of minutes or fractions, and at the instant of the same height, turn your Instrument to the Sun, so as the shadow of the lines may fall justly upon the line of South and North in the fixed Fly.  
Then, when the Needle doth stand, look directly over the North point of the Needle, what degree and fraction, if there be any, doth answer unto the same in the fixed Fly, that is to say, how many degrees it is from the North of the fixed Fly, which you shall note diligently, and may say, that so many degrees etc. is the variation of the suns shadow from the North, as the North point of the Fly is from the North point of the Needle, either eastwards or westwards as you shall find the same. Thus may you observe divers times, upon several degrees of the suns elevation. And like as you do in the forenoon, so must you also observe the suns elevation in the afternoon, upon the same degree of height, and with the same side of the Astrolabe and Index turned towards the Sun, as it was in the forenoon, (for avoiding of error that may be in the Instrument) noting at every height what you find the variation. And when the Sun cometh to the meridian, it shall be good that you exactly observe his elevation upon the same, for knowing the true Latitude of the place: all which you shall set down in form following.  

¶ Example.  

¶ In Limehouse the sixteenth of October. Anno. 1580.  Fornoone. Fornoone. Afternoon. Afternoon. Afternoon. Elevation of the Sun. Variation of the shadow from the North of the Needle of the Needle to the Westwards. Elevation of the Sun. Variation of the shadow from the North of the Needle to the Eastwards. Variation of the Needle from the Pole or Axis. Deg. Degr. Min. Deg. D. M. D. M. 17 52 35 17 30 0 11 17½ 18 50 8 18 27 45 11 11 ½ 19 47 30 19 24 30 11 30 20 45 0 20 22 15 11 22 ½ 21 42 15 21 19 30 11 22 ½ 22 38 0 22 15 30 11 15 23 34 40 23 12 0 11 20 24 29 35 24 7 0 11 17 25 22 20 25 From N. to w. 0. 8 11 14  
The elevation of the Sun upon the meridian 25. d. 58′. the declination 12. d. 30′. which I add to the elevation, because the Sun hath South declination, and thereof amounteth 38. d. 28′. the elevation of the Equinoctial, the which I subtract from 90. d. the rest is 51. d. 32′. the elevation of the Pole Artik.  
Now are you to consider, that out of the greater variariation of shadow upon any degree of the suns elevation, is to be taken the lesser of the same degrees elevation, whether it be in the forenoon or afternoon, (except the same variations be both one way from the North of the Needle, which then are to be added) the half of the remainder is the variation of the Needle or Cumpas from the Pole or true meridian.  
In the former observations, I do find the greatest variation in the forenoon, for, at 17. d. elevation, the variation is 52. d. 35′. from North to West: And at the same elevation in the afternoon. I find the variation to be but 30. d. 0′. from North to East. I take the lesser out of the greater and find remaining 22. d. 35′. the half thereof is 11. d. 17′. ½. So much I say is the Pole Artik, and true meridian line that passeth to the Pole by our Zenith at London, to the Westwards of the North that the Needle showeth. And therefore the Needle or Cumpas varieth from the true north 11. d. 17′. ½. to the eastwards.  
Also at 25. d. elevation in the forenoon the variation is 22. d. 20′. from North to West: at the same elevation in the afternoon the variation is 0. d. 8′. from North to West. Now because the variations are both one way, (that is to the Westwards) I add them together (and so ought you to do as often as you find the variations so to agree) and I find that they amount to 22. d. 28′. the half thereof is 11. d. 14′. which is the variation.  
The variations of the Needle or Cumpas by the former observations, are set out towards the right hand against every degrees elevation; and conferring them all together, I do find the true variation of the Needle or Cumpas at Lymehouse to be about 11. d. ¼. or 11. d. ⅓. which is a point of the Cumpas just or little more. So that in a Cumpas whose wires are set directly under the flower de Luce, the North and by West, and South and by East points do show the true meridian.    

¶ The elevation of the Pole and place of the Sun given how upon the Globe, to find the variation of the Needle by any one observation, either in forenoon or afternoon.  
¶ The fourth Chapter.  
IN the former declaration, the only way to try the variation, is by comparing of the several correspondent observations of the suns elevation in the forenoon, with those of the afternoon, so that if the Sun should be obscured, or by any other occasion like observation can not be made in the afternoon, than the former rule giveth not the desired purpose. Therefore I thought good to show, how by any one observation in the fore or afternoon, the elevation of the Pole and place of the Sun given, you may know the true meridian and the variation of the Needle from the same in any place, which thing may be done and aptly demonstrated upon the Globe, but most exactly calculated by the Table of Sines.  
To find out the variation upon the Globe, you must first set your Globe to stand duly according to the elevation of the Pole at the place proposed. Then seek in the Ephemerides for the true place of the Sun that day, and note it with some small prick in the ecliptik of the Globe. And placing the Quadrant of Altitude or movable vertical, at the vertical point or Zenith, take the elevation of the Sun observed by the Astrolabe or other Instrument at the time proposed, and note it justly upon the same quadrant of altitude. Then turn your Globe and quadrant towards that part of the Horizon that the Sun was in at the time of the observation, till the prick you made for the place of the Sun in the ecliptik, concur and agree justly with the elevation marked in the said quadrant of altitude. So shall you see the quadrant show you upon the Horizon, the Azimuth and distance of the Sun from the true meridian of that place, which you shall compare with the variation observed upon the Instrument at that instant of the suns elevation, And if they agree and concur just, then shall you be in the true and common meridian, which showeth the Pole of the world and Pole of the Magnes or loadstone: But if they differ, you shall subtract the lesser from the greater, the remainder showeth the variation. And if the variation upon the Instrument be greater than the true distance of the Azimuth from the meridian found upon the Globe, the same surplus is to be accounted for variation, upon the contrary side of the meridian: if it be less, it is to be accounted on the same side of the meridian that the variation is taken, whether it be in the forenoon or afternoon. This precept needeth no further demonstration, than the Instrument itself, the Globe I mean.  
But for example of the work, I take the first observation, in the former Chapter sperified, made at Lymehouse the sixteenth of October 1580. in the forenoon, which is 17. d. elevation, and variation 52. d. 35′. from North to West.  
First I set my Globe at 51. d. 32′. for the elevation of the Pole. Secondly I take the place of the Sun 2. d. 55′. m. and note it upon the Ecliptic. Thirdly I note upon the quadrant of altitude, the elevation of the Sun 17. d. This done, I move the quadrant of altitude towards the East of the Horizon, and turn the Globe till the prick in the Ecliptic for the place of the Sun, do agree justly, with the elevation noted upon the quadrant of altitude, and find the true Azimuth showed by the said quadrant upon the Horizon to be nearest, about 41. ⅔ from the meridian. And conferring the same with the variation found upon the Instrument 52. d. 35′. I find the difference 11. d. 15′. And because the observation is noted to be in the forenoon from the North to the West, or South to the East, and the variation upon the Instrument greater than the Azimuth found on the Globe, I account the same from the North to the East, or from the South to the West. So I conclude the variation at Lymehouse to be about 11. ¼. from North to East, or South to West.   

¶ How to find the variation by Arithmetical calculation upon any one observation in the forenoon or afternoon, the Latitude of the place, and declination of the Sun being given.  
¶ The fift Chapter.  
THE sum of the work, is to find the ark of the Horizon, between the meridian and the Azimuth of the Sun at the time of the observation, which being compared with the variation found in the Instrument, the difference is the variation of the Needle. For attaining of the same ark. First it is necessary to have the ark of the Equinoctial between the Sun at the time of the observation, and the meridian, which ark is thus found.  
Multiply the sine of the suns meridian altitude for the day proposed, by the whole sine, the product divide by the sine of the elevation of the Equinoctial (or the complement of the Latitude) the quotient is the versed sine or shaft of the semidiurnal ark, which you shall note for the first number.  
Then again multiply the sine of the suns elevation at the time of the observation, by the whole sine, and the produce divide by the sine of the elevation of the Equinoctial, the quotient subtract from the number you first noted, the rest is the versed sine of the ark of the distance between the Sun and the meridian in the parallel that it is in for the time proposed, in such parts as the Semidiameter of the Equinoctial is the whole sine: but it is necessary before you apply it any further, to reduce it into such parts as the Semidiameter of the parellell is the whole sine, which you may do thus: Multiply this remainder by the whole sine, the product divide by the sine of the complement of the declination (which is the Semidiamiter of the parallel) the quotient is the versed sine in his proportional parts.  
This versed sine thus reduced and subtracted from the whole sine, leaveth the second right sine, which you shall seek in the Table of sins, and thereby finding his ark, you shall subtract the same from the quadrant or 90. d. the remainder is the ark of the foresaid parellell of the Sun, which is answerable or correspondent in degrees and minutes, to the ark of the Equinoctial that you seek. The reason of the precept is this.  
As the right sine of the elevation of the Equinoctial, is in proportion to the right sine of the meridian altitude of the sun or any Star: so is the whole sine, to the versed sine of the semidiurnal ark. And again, as the right sine of the meridian altitude, is to the right sine of the elevation of the Sun or Star at the time of the observation: So is the versed sine of the semidiurnal ark of the same, to the excess or difference between the same versed sine and the versed sine of the distance from the meridian.  
For the better understanding of the premises, I have set down this figure following, and wish the Reader to consider of the same with the 4. Pro. of the 6. of Euclid.  

LEt AMT. be the meridian circle. BDQ. the common section of the meridian and Equinoctial their plains, which is also the diameter of both circles. ADT. the plain of the Horizon. LHP. the parallel of the Sun, which is described upon the centre F. at the distance FL. which is the sine of the complement of the declination. AB. the ark of the elevation of the Equinoctial. BOY. the first right sine thereof. AL. the ark of the meridian altitude. LX. the sine thereof. AN. the ark of the suns elevation at the time of the observation. N C. the sine thereof. B D. the whole sine in respect of the former arks and sins. L R. the semidiurnal ark of the parallel. R S. the first right sine thereof. S L. the versed sine of the same. L I. the ark of the suns distance from the meridian. I K. the first right sine thereof. I G. the second right sine, which is equal to K F. K L. the versed sine. N E. which is equal to K S. the difference of the 2. versed sins L S. and L K. L F. the whole sine in respect of the arks and sins of the parallel.  
Now as B O. is to L X. so is B D. to L S. And as L X. to N C. so is L S. to N E. Or else thus, as B O. to N C. so is B D. to N E.   

Example. The 16. October. 1580. in Lymehouse.  
The elevation of the Pole Artik 51. d. 32′. The declination of ehe Sun 12. d. 30′. The elevation of the Sun observed in the forenoon 17. d. 0′. The variation of the shadow upon the Instrument 52. d. 35′. from North to West.  
38. 28′. 90. 0′. 25. 58′.  B O. B D. L X. L S. If. 62205. give. 100000. — then. 43784. giveth. 70386.  
38. 28′. 90. 0′. 17. 0′.  B O. B D. N C. N E. Again if. 62205. give. 100000. 29237. shall give. 47001.  
Now out of. L S.— 70386. take. N E.— 47001. Rest. L K.— 23385.  
Then if L F. 97629. the sine of 77. d. 30′. the complement of the declination, give L F. 100000. then L K. 23385. giveth L K. 23952. the versed sine of the ark I L. in his due parts. The same subtracted from L F. 100000. the whole sine, leaveth K F. or I G. 76048. the second right sine of the same ark, which is the first right sine of the ark I H. which ark you shall find in the table of sins to be 49. d. 30′. 24″. the complement whereof to the quadrant is 40. d. 29′. 36″. the ark I L. of the parallel between the Sun and the meridian, whose correspondent ark in the Equinoctial, is the ark that was sought.  
Now haing 〈…〉 f the Equinoctial, you must work 〈…〉  
〈…〉 thereof, by the sine of the complement the declination, and divide the product by the whole sine, the quotient is the sine of an ark contained between the Sun and the meridian, making right angles with the meridian. This sine multiply by the whole sine, the product divide by the the sine of the complement of the suns elevation at the time of the observation, the quotient shallbe the sine of the ark of the Horizon contained between the Azimuth of the Sun and the meridian, which is the ark that was proposed to be found.  
LEt D H N P. be the meridian. D A K. the Horizon. E A N. the Equinoctial. M. the place of the Sun in the heaven at the time of the observation. L M O. the parallel. H M B. the Azimuth or vertical circle passing by the Sun. A M G. a great circle imagined to pass by the Sun, & to cross the meridian at right angles. I M P. a great circle passing by the Poles of the world, and place of the Sun at the time of the observation, commonly called the citcle of hours, or circle of declination. C M. the South declination of the Sun 〈◊〉. the complement thereof to the quadrant. M●●● the ark between the Sun and the ●●● of the former imagined circle. A M G  
In this figure the Reader is to consider the manner of the spherical triangles, and to compare the sins of their sides, according to the doctrine of Copernicus. in the 14. Chapter of his first book, and of Regiomontanus. his 25. and 27. propositions of his 4. book of triangles.  
As P C. is to C E. so is P M. to M G. but 3. of them are given, therefore the fourth shall be known.  
And as H M. is to M G. so is H B. to B D. the ark that is sought, which by the three first given is likewise given.   

¶ The second part of the example.  
90. 0′. 40. 29′. 36″. 77. 30′.  P C. E C. P M. M G. If. 100000. give. 64935.— then. 97629. giveth. 63395.  
 73. 0′.  90. 0′. 41. 31′. 22″.  H M. M G. H B. B D. Again if. 956 〈…〉 give. 63395.— 100000. giveth. 66291.  
Whose 〈…〉 B D. 41. d. 31′. 22″. is the horizontal distance of the w 〈…〉 uth of the Sun from the meridian, the thing that which 〈…〉 ught.  
Now comparing the same with the variation found upon the Instrument at the instant of 17. d. elevation, which is 52. d. 35′. I find it to be less, and therefore subtract it, and so have I the difference 11. d. 3′. 38″. And because the observation was in the forenoon, and the variation upon the Instrument greater than the ark of the Horizon between the suns Azimuth and the meridian, therefore I conclude, that the variation is 11. d. 3′. 38″. from South to West, or North to East, which is the thing promised to be showed.  
But comparing the same ark of the Horizon 41. d. 31′. 22″. with the variation found at the correspondent elevation in the afternoon, which is 30. d. 0′. I subtract the lesser from the greater, and find the excess 11. d. 31′. 22″. which should be the variation. And because the variation found upon the Instrument is less than the ark of the Azimuth upon the Horizon, I account the variation on the same side of the meridian, which is from South to West, or North to East.  
This variety between the observation made in the forenoon, and that in the afternoon, proceedeth either of the imperfection of the Instrument, or negligence of the observer. For in the rule there can be no error, being grounded upon Geometrical demonstration, than which nothing can be more certain.  
The former precepts and examples do serve when the Sun doth decline from the Equinoctial either Northwards or southwards. But if the Sun be in the Equinoctial, than the manner of the working is more easy and brief. For if you multiply the sine of the suns elevation at the time of observation, by the whole sine, and divide the product by the sine of the elevation of the Equinoctial, which is the meridian altitude, the quotient giveth the second right sine of the distance of the Sun from the meridian, which is the first right sine of the complement of the same ark: And entering the table of sins with it, you shall find his ark, which if you subtract from the quadraut or 90. d. leaveth the ark of the distance of the Sun from the meridian. And having the same work thus. If the sine of the complement of the eluation of the Sun at the time of the observation, give the sine of the foresaid ark of distance, what shall the whole sine give. Multiply and divide, the quotient shallbe the sine of the ark of the Horizon contained between the Azimuth of the Sun and the meridian. Which ark being compared with the variation of the Instrument in manner as before is showed, giveth the variation required.  
But the Sun being in the Equinoctial, if the place where the observation is made, be likewise under the same circle▪ then is the variation most easily observed▪ for that the Equinoctial is the Azimuth of East and West, therefore turning your Instrument only to receive the shadow of the Sun, and looking then to the North point of the Needle, if you find the same to answer to the quadrant or 90. d. you shall be in the meridian of the Magnes, which passeth by the Poles of the world, but if it do differ from 90. d. the same difference is the variation of the Needle.  
But admitting the observer to be under the Equinoctial, and the Sun to have declination, than the proportion of the sine of the complement of the elevation at the time of the observation, unto the sine of the declination, shallbe such, as the whole sine, is to the sine of the ark of the Horizon included between the Azimuth of East and West, which is the Equinoctial itself, and the Azimuth of the Sun for the time of the observation, the complement whereof giveth the true meridian, which complement you may compare with the variation showed upon the Instrument, the difference is the variation.  
divers other cases might be proposed, and rules given for them, which for brevity I omit.  
But one thing I thought good to admonish you by the way, that whereas I have showed in the first part of this proposition the manner to find the two versed sins, the one of the semidiurnal ark, the other of the ark of the distance of the Sun from the meridian. By the first, the semidiurnal ark being found and 〈◊〉 into hours and minutes of time, is showed the just 〈◊〉 quantity of the day. And by the ark of the other likewise reduced, the hour of the day, or the time contained between the noonsteed and the instant of the observation. As in the same example. The versed sine of the semidiurnal ark LS. is given 70386. in such parts as the Semidiameter of the Equinoctial BD. is 100000. therefore I reduce the same into such parts as the Semidiameter of the parallel LF. is 100000. and find it to be 72095. which subtracted from the whole sine LF. 100000. there resteth SF. 27905. which is the second right sine of the semidiurnal ark LR. and the right sine of RH. 16. d. 12′. which is the complement of the semidiurnal ark LR. wherefore subtracting it from the quadrant LH. or 90. d. resteth 73. d. 48′. the semidiurnal ark LR. the same reduced into parts of time allowing 15. d. for an hour 15′. for a minute, and 15″. for a second of time, and for every degree 4. minutes of time, for every minute 4″. and for every second 4″. etc. I find the time of that ark from the point ascendent, to the meridian, which is half the day, to be 4. hours 55′. 12″. and consequently the whole day being the 16. of October above written, to be 9 hours 50′. 24″. long.  
This example may serve for a general precedent, whiles the Equinoctial is between the Sun & the elevated Pole, but if the Sun be between the elevated Pole and the Equinoctial, then will the versed sine fall out to be greater than the whole sine, and the semidiurnal ark to exceed a quadrant. Wherefore having reduced the same into his proportional parts, as before is showed, subtract from it the whole sine, the surplus is the sine of the excess of the semidiurnal ark above a quadrant, which being added to the quadrant, giveth the semidiurnal ark.  
By the other versed sine of the distance of the Sun from the meridian, which is LK. 23952. in such parts as the whole sine or Semidiameter LF. is 100000. subtracted from the whole sine, is given KF. 76048. the second right sine of the same ark of distance, and the first right sine of 49. d. 30′. 24″. which is the complement of the ark of the suns distance from the meridian: therefore subtracting the same from 90. d. resteth 40. d. 29′. 36″. the ark of the distance between the Sun and the meridian, which being reduced into parts of time as before, giveth 2. hours 41′. 58″. and the same (because it is in the forenoon) deducted from 12. hours the noonsteed, resteth 9 hours 18′. 2″. the just instant of the time of the day.  
But if this versed sine be found to be greater than the whole sine (as it may when the Sun is between the Equinoctial and the elevated Pole, and before the hour of six in the morning and after the hour of six in the evening) then doth the ark of distance consequently exceed a quadrant, the sine of this excess is the surplus of the versed sine above the whole sine. Whose ark added to the quadrant giveth the ark of the suns distance from the meridian, and reducing the same into parts of time, is given the instant of time of the observation.  
As by this means (the elevation of the Sun being precisely observed and Latitude known), the instant of time of the day is given more exactly, then by any Clock, Dial or other Instrument. So if there might be had a portable Clock that would continue true the space of 40. or 50. hours together (if longer time the better) then might the difference of longitude of any two places of known Latitudes, which conveniently may be traveled within that time, be also most exactly given. And in this sort traveling and observing from place to place, might the longitudes of any Country be perfectly described.    

¶ An other way most general, how to find the Variation by one observation either in the forenoon or afternoon, the elevation of the Pole and declination of the Sun being given.  
¶ The sixth Chapter.  
FOR the accomplishing of this proposition, you are to imagine a spherical triangle upon the superficies of the Globe, whose sides must be. First the portion or ark of the meridian between your Zenith and the Pole, which is the complement of the latitude. The second the ark of the vertical circle contained between your Zenith and the Sun, which is the complement of the suns elevation at the time of the observation. The third side is an ark of the circle of declination comprehended between the Sun and the elevated Pole, this ark is found by adding, or subtracting, the declination of the Sun, to or from, the quadrant or 90. d. which must be done with this consideration, that if you be on the same side of the Equinoctial that the Sun is, you are to subtract the declination from the quadrant. If on the other side, to add it to the same, so have you the three sides of the spherical triangle given. Then the substance of the work consisteth in finding the quantity of the angle of the same triangle at the Zenith, for the complement thereof to the Semicircle or two right angles, is the horizontal distance of the suns Azimuth from the meridian, which being compared with the variation of the suns shadow upon the Instrument, giveth the thing required.  

LEt FACE. be the meridian, wherein A. the Zenith, C. the Pole. AD. the vertical circle or Azimuth of the Sun passing by B. the place of the Sun at the time of the obsetuation. BD. the elevation of the Sun. . the complement of the elevation. AC. the complement of the latitude. BC. the ark of the circle of declination, or the chord of the same ark. FGE. the plain of the Horizon.  
Now from the three angles of the triangle ABC. let fall 3. perpendicular lines to the plain of the Horizon AG. CH. and BK. and by the 6. of the 11. of Euclid, these three lines shall be parallels.  
Then let fall a perpendicular line from C. upon AG. in the point L. from B. an other perpendicular upon the same line AG. at the point M. And from the same point M. erect a perpendicular line to N. which shallbe parallel and equal to LC. Then join B. and N. together. So have you a rightlined triangle. BMN. whose angle at M. is equal to the angle A. of the spherical triangle ABC. By the 4. definition of the 11. of Euclid, for the like reason is of obtuse angles as of acute or sharp. And the sides thereof BM. and MN are given BM. the sine of . and MN. equal to LC. the sine of CA And the third side BN. is found by subtracting the square of NC. from the square of the chord BC. as in the 47. of the first of Euclid.  
And in rightlined triangles, the three sides being given, the angles are also given, by the 44. 45. etc. of the first of Regiomontanus, and by the 7. proposition of the 13. Chapter of Copernicus his first book.   

For example I take the former observation of the 16. October 1580. and work as followeth.  
The elevation of the Pole CE. 51. d. 32′. the sine thereof CH. 78297. The elevation of the Sun BD. 17. d 0′. the sine thereof BK. 29237. The ark BC. 102 d. 30′. the chord thereof BC. 155976. The complement of the elevation of the Sun . 73. d. 0′. the sine thereof BM. 95630. The complement of the latitude AC. 38. d. 28′. the sine thereof LC. 62205. equal to MN. Now out of CH. 78297. subtract NH. equal to BK. 29237. Rest NC. 49060.  
Then out of the chord BC. squared.— 24328512576. Take the square of NC.— 2406883600. Rest the square of BN.— 21921628976.  
The root thereof is. 148059. the side BN.  

So are the three sides of the triangle given.  
BN. 148059.  
MN. 62205.  
BM. 95630.   
Now to find the angle M. I subtract from the square of BM. the bigger side, which is. 9145096900. the square of MN. the lesser side, which is. 3869462025. Rest 5275634875. which divided by the base BN. 148059. giveth 35631. which number I take out of the said base rest. 112428. the half thereof. 56214. is IN. the lesser case or shorter part of the base divided by the perpendicular line MI. falling upon the same from the obtuse angle M. which subtracted from the whole base BN. 148059. leaveth IB. 91845. the greater case or longer part thereof.  
Now it is manifest that these two cases or parts of the base BY and IN. are the sins of the two sharp angles IMB. and NMI. made of the obtuse angle M. by the perpendicular falling from the same angle to the base, and the arks of them joined together, are the quantity of the obtuse angle NMB.  
Therefore to reduce them to the numbers of the sins, first for the greater case BY. making BM. the whole sine, say.  
BM. BM. BY. BY. If. 95630. give. 100000.— then shall. 91845. give. 96042.  
The ark thereof is 73. d. 49′. 38″. Again for the lesser case, making MN. the whole sine, say.  
MN. MN. IN. IN. If. 62205. give. 100000.— then. 56214. giveth. 90376.  
Whose ark is 64. d. 38′. 45″. And adding these two arks together they give 138 d. 28′. 23″. the ark or quantity of the obtuse angle NMB. equal to the spherical angle BAC. And deducting it from the Semicircle 180. d. there resteth 41●. d. 31′. 37″. the angle FAD. the horizontal distance of the suns Azimuth from the meridian, & subtracting that from 52. d. 35′. the variation found upon the Instrument from North to West in the forenoon, resteth 11. d. 3′. 23. ″. the variation of the Needle from the meridian, the thing that was proposed to be found. And comparing the same with the afternoons observation, you shall find it 11. d. 31′. 37″. the cause of this difference I have declared in the former Chapter.  
If the Reader be delighted with variety of demonstration of this matter, let him peruse the 34. proposition of the 4. of Regiomontanus, and the 13. proposition of the 14. Chapter of the first book of Copernicus.  
But whereas you see this calculation to differ from the former in some odd seconds, the reason thereof is not as it might be taken the different nature of the rules, but in working thereof, omitting the fractions in the divisions, and neglecting the proportional parts of the sins and arks.  
In these examples I have used the abridged table of 100000. the whole sine, which though it give some ease in the working, yet it is not so exact as that of 10000000. of Erasmus Reinholdus. Unto the which, with his Canon foecundus answerable to the same, if the third Canon of the hypothenusas were annexed, we should have an entire table for the doctrine of triangles, that might worthily be called The table of tables. Which thing though Georgius joachimus Rheticus, have well begun and framed it orderly from ten minutes to ten: yet is it left very rawly for such as desire the exact truth of things. I have therefore for mine own ease and use, calculated the complement of this table, and almost ended it, for the whole quadrant from minute to minute: which if in the mean time before I have finished, I shall not find it extant by any other, I will publish it for the commeditie of all such as shall have occasion to use the same for Navigation and Cosmography.    

¶ To find the elevation of the Pole, Situation of the meridian, and Variation of the Needle at any place by the Sun, upon two observations either in forenoon or afternoon.  
¶ The seventh Chapter.  
WHereas in the three last Chapters, the grounds of the calculations consist in the elevation of the Pole to be given, which thing to know is no less difficuit, than the chief matter that is by them required. For the common precepts, which as yet have chiefly been given for the finding thereof, depend only upon the observation of the meridian altitude of the Sun or Stars, or else upon certain false and gross rules of the guards and Pole star. Therefore I have thought good, that as I have showed the way to know the variation upon any one observation, either in forenoon or afternoon, the latitude of the place presupposed: so likewise, upon two observations by the Sun, either in forenoon or afternoon, to set down the way and manner how to find the elevation of the Pole, situation of the meridian, and variation of the Needle in any place by the Globe.  
But this you must always regard, that your two observations may have convenient distance of time between them, the greater the better: So as the higher elevation be not taken near the meridian, the lower elevation the nearer it is taken to the Azimuth of East or West, or to the Horizon, the better, with which elevations, you are to note the difference of the suns Azimuths or variations found by the shadow upon the Instrument exactly, for without that, the elevations only are in vain.  
First it is requisite, that your Globe be so fitted, that the meridian circle and the Horizon do cross each other at right angles, and divide themselves equally into Semicircles. And also that the quadrant of altitude (or movable vertical) be placed duly upon the meridian circle at the Zenith, so as being turned circularly, it may touch the Horizon equally in every part. These things being duly considered, there needeth not any further regard to be had for placing of the Globe, only this you may respect in setting the Pole at adventures above the Horizon between it and the Zenith, that the meridian circle may cut the Horizon in just degrees, so may your quadrant of altitude be placed at your Zenith justly upon a degree also.  
Then must you fasten your Globe to the Horizon, so as it may remain immovable, but in fastening the same you must regard that you force it not from one side of the Horizon to an other, but that it rest equidistant in the same. And having your Globe thus disposed, it is ready for you to apply your observations upon, which you shall thus do.  
First, take your highest elevation, and note it upon your quadrant of altitude, and place the end of the said quadrant upon the Horizon at 10. 15. or 20. d. from the meridian circle, (but the nearer you set the same to the meridian, the more conveniently, without impeachment, will your trial be made.) Then give a prick upon the Globe in the Azimuth, that the quadrant showeth at the degree of the elevation, noted upon the quadrant, then again note the lesser elevation upon the quadrant of altitude, and remove the same upon the Horizon (from that place where it was first fixed, towards the Azimuth of East or West, which shallbe nearest the same) so many degrees as you find the difference of Azimuthes between the two elevations by the shadow of the Sun, upon the instrument of Variation, and staying your quadrant of altitude upon that point of the Hozizon: note also your lesser elevation in the same Azimuth upon your Globe. This done you must have a pair of Calliper compasses, such as may conveniently reach to 113. d. ½. of the Equinoctial of your Globe, (which is a quadrant, and the greatest declination of the Sun) than you must consider which of the Poles of the world is elevated above your Horizon, and whether your declination be towards, or from that Pole, that is to say, whether the Sun be between the elevated Pole and the Equinoctial, or the Equinoctial between the Sun and the Pole. If the Sun be between the Pole and the Equinoctial, then are you to subtract the declination from 90. d. If the Equinoctial be between the Sun and the Pole, you must add the declination to 90. d. And take the same remaining or collected number of degrees etc. with your compasses upon the Equinoctial. And set the one end of your compass at the prick made upon your Globe, for the highest observation, and with the other end describe an ark or piece of a circle, upon the same side of the meridian that your prick is on, from the meridian to the Horizon. Then again with your compass unaltered, setting the one foot in the prick for the lowest observation, describe an other piece of a like circle crossing the former. The point of the intersection, or crossing of these two circles, is the elevated Pole, to the which if you remove the quadrant of altitude, you shall find what the elevation thereof is. And the point that the same quadrant showeth upon the Horizon, is the intersection of the meridian and the Horizon, the horizontal distance between this intersection, and the Azimuth of the lesser observation, subtracted from the semicircle, or 180. d. leaveth the horizontal distance of the same Azimuth from the true meridian. So have you the elevation of the Pole, and situation of the meridian.  
Now if you compare the horizontal distance of the Azimuth of the Sun, from the meridian at the time of the observation, with the variation by the suns shadow found upon the instrument, at the time of the same observation, and taking the one out of the other, the remainder shallbe the true variation, which you are to account, as in the latter end of the third Chapter is showed. So have you given the elevation of the Pole, the meridian, and Variation of the Needle, the things proposed to be showed.  

Example of two observations made at Limehouse the 29. of Julie 1581. in the forenoon.  
The first elevation 21. d. 0′. Variation 100 d. 30′. from North to West. The second elevation 50. d. 0′. Variation 48. d. 0′. from North to West. Difference of the Azimuths 52. d. 30′. The declination 16. d. 14′. Northerly.  
LEt I D B. be the Horizon of the Globe. C A B. the meridian circle. F G A. the Azimuth of the greater elevation showed by the quadrant of altitude upon the Horizon at F. 10. d. from the meridian circle of the Globe C. F G. the greater elevation marked upon the Globe at G. F D. the difference of the Azimuths upon the Horizon. 52. d. 30′. E. the prick of the lesser elevation marked upon the Globe in the Azimuth A E D.  
Then opening your Cumpasses to 73. d. 46′. of the Equinoctial (which is the complement of the declination) and setting one end upon G. the point of the greater elevation, describe with the other end, an ark or piece of a circle at H.  
This done, set one foot of the Cumpas unaltred in E. the lesser elevation, and with the other end describe a piece of a circle crossing the former ark at H. this intersection shall be the elevated Pole.  
Then set the quadrant of altitude unto the point H. and it will show the meridian to cross the Horizon at K. So shall you have the elevation of the Pole K H. 51. d. ½. or there about. And the true meridian KAI. And from K. to D. the horizontal distance 90. d. ¾. which subtracted from K I. 180. d. the semicircle of the Horizon, resteth the ark D I. 89. d. ¼. the distance of the Azimuth of the first observation from the meridian I. which distance compared with the variation found upon the Instrument at the first elevation 100 d. 30′. and deducted from the same, resteth 11. d. ¼. Therefore I say, the true meridian showing the Pole Artik is 11. d. ¼. to the Westwards of the magnetical meridian showed by the Needle, and consequently the variation of the Needle 11. d. ¼. from the North to the East.  
In this example the declination is subtracted from the quadrant, because the Sun is between the Equinoctial and the elevated Pole, but if the Equinoctial were between the elevated Pole, and the Sun than should you add the declination to the quadrant, and with that distance taken upon the Equinoctial with your Cumpasses, proceed as in the former example  
These examples that I have showed, and such like experiments to be done upon the Globe, are easy to be conceived, and the reasons very manifest: but the truth of the matter consisteth in the exactness of the Instruments, and the orderly application and handling of them.  
I might here have annexed the manner, how upon two observations of the suns elevation in forenoon or afternoon and difference of the Azimuths, to calculate the premises more exactly by the table of Sines and doctrine of spherical triangles: but that it is a very tedious way, and my meaning is rather to give the Reader a proof of the pleasant use of these calculations (which I think I have sufficiently done in the former Chapters) then to cloy him at the first with the hard and painful practice of many examples. Notwithstanding, for the satisfaction of some, I will briefly set down the ground and sum of the work, which is this.  
The compliments of your two elevations, are two sides of a spherical triangle not rectangle. The angle by these two known sides contained at the Zenith, is given by the difference of the Azimuths or variations upon the Instrument. Wherefore by the 28. of the 4. of Regiomontanus the third side (which is the ark comprehended between the two elevations) and the other angles may be given.  
Then have you an other like triangle, whose three sides are these: the first, one of the foresaid compliments of elevation: the second, the ark of the circle of declination, between the Sun at the instant of the same elevation, and the elevated Pole. The third side is an ark of the meridian between the Zenith and the Pole: which is the complement of the elevation of the Pole, or latitude of the place. The two first sides are always given. For finding the third side, it is necessary to know the angle that the two given sides contain, which is the difference of two angles, whereof one is an angle of the first triangle given, the other an angle contained between the ark of the circle of declination, and the third side of the first triangle, which angle is diversly found, and being found and subtracted from the other angle, or that from it, the difference is the angle of this other triangle: And so have you in the Spherical triangle two sides, and the angle by the same two sides contained, given. And by the same 28. of the fourth of Regiomontanus the third side is found, the complement whereof is the elevation of the Pole.  
And the elevation of the Pole, and declination of the Sun being given, the fourth Chapter showeth by one observation, to find the Variation of the Needle.    

Of the Pole of the Magnes.  
¶ The eight Chapter.  
FIrst it is to be understood, that by experience of travelers, it is found and confirmed, that the meridian common to the Pole of the world, and the Pole of the Magnes, (that is to say, where the Compass, or Needle touched with the Magnes, showeth the Pole of the world directly,) passeth at the Islands of the Açores, or near there about, but I find by great probability, that it is somewhat to the eastwards of those Islands, and not to the Westwards. From which meridian I account the beginning of longitudes, and find our meridian of London, to be from the same 23. d. ½. our latitude as before said 51. d. 32′. and the Veriation of the Compass or Needle 11. d. ¼. from the North to the eastwards. Now upon these grounds▪ I find by calculation, the Pole of the Magnes, or the intersection of the two Magnetical meridians, upon the superficies of the earth, to be from the Pole arctic 25. d. 44′. and in longitude 180. d. that is to say. 25. d. 44′. in the former common meridian, on the other side of the Pole.  
It may be happily that some of you will be desirous to know the manner how this Magnetical Pole is found out, that you may apply the same to like purpose hereafter. Therefore I thought good to set down an example of the former calculation.  

LEt A. be the Pole Artik. PEF. the Equinoctial.  
DAG. the common meridian of the Pole Artik, and Pole of the Magnes. EAF. the meridian of London.  
Make out the quadrants IHK. and LNK. so shall they cross themselves with the quadrant OAK. at the point K.  
Now have you ABC. a spherical triangle, two angles whereof and the common containing side of them, are given. ABC. 11. d. ¼. the angle of Variation at London. BAC. 156. d. 30′. the complement of the angle DAE. (the difference of the longitudes) to two right angles. And the side AB. 38. d. 28′. the complement of the latitude of London.  
And in a spherical triangle, not rectangle, whose two angles are given, and their common containing side, the other angle and sides shallbe known, by the 31 of the 4. of Regiomontanus.  
Wherefore the ark AC. the distance of the two Poles shallbe given, which is the thing required.  
For as the sine of BH. is to the sine of HI. so is the sine of . to the sine of AO. and three of them being given the 4. is found.  
90 0′. 11. 15′. 38. 28′. 6. 58′. BH. HI. . AO. If. 100000. give. 19509.— then. 62205. giveth. 12135.  
Now as AK. is to AH. (the sins I mean) so is KO. to HEY. but the three first are known AK. and AH. by their compliments, and KO. the quadrant. Therefore the 4. is given.  
83. 2′. 51. 32′. 90. 0′. 52. 4′. KA. AH. KO. HEY. If. 99261. give. 78297.— then. 100000. giveth. 78879.  
And as . is to BOY. (the complement of the ark HEY. last found:) so is A to EM. the quantity of the angle BAO.  
38. 28′. 37. 56′. 90. 0′. 81.12. AB. BOY.▪ A EM. If. 62205. give. 61474.— then. 100000. giveth. 98824.  
Sohaving EM. 81. d. 12′. the quantity of the angle BAO. I subtract the same from EG. 156. d. 30′. the quantity of the whole angle BAC. rest MG. 75. d. 18′. the quantity of the angle CAO. to the which is equal the opposite angle PAD. And as AP. is to PD. so is AK. to KN.  
90. 0′. 75. 18′. 83. 2′. 73. 46′. AP. PD. AK. KN. If. 100000. give. 96726.— then. 99261. giveth. 96011.  
The complement of which ark KN. is NL. 16. d. 14′. the quantity of the angle ACB. And as NL. is to NC. so is AO. to AC. Wherefore I say.  
16. 14′. 90. 0′. 6. 58′. 25. 44′. NL. NC. AO. AC. If. 27954. give. 100000.— then. 12135. giveth. 43410.  
Which is the distance of the Pole of the Magnes from the Pole Artik: the thing that was sought.    

¶ Of the point Respective.  
¶ The ninth Chapter.  
Having showed in the former Chapter, upon the grounds therein specified, the place of the Pole of the Magnes, upon the superificies of the earth: there resteth now to be declared, of the point Respective, where it should be, by the new property found of the declining of the Needle, at this place for London 71. d. 50′.  
First it is to be considered, that as the Magnetical meridians do cross themselves at their Pole, before specified: so do their plains likewise cross in a right line, passing by the said Pole, and the centre of the earth. Then producing a strait line, in the Magnetical plain of London, declining from the plain of the Horizon 71. d. 50′. where the same doth cross with the former common section of the two plains, there by reason should the point Respective dee. Which intersection I find to be from the centre of the earth 1085. miles (after the rate of 60. to a degree in the Equator, and 3436 4/11. for the Scinidiameter of the earth) and the distance of the same from the axis of the world 471. miles.  

LEt the circles be as in the last demonstration. And Q. the centre of the earth. Then QA. the axis of the world. QC. the common section of the magnetical plains. BZ. the line of the Needle's declination crossing the said common section at R. which is the point respective. QT. a strait line crossing BZ. at right angles in X. QR. the distance of the point respective from the centre of the earth. RS. the distance there of from the axis. Now as QU. is to QC. so is QX. to QR. But the three first are known. QU. the second right sine of the ark CT. 9 d. 4′. (the difference of the ark BT. 71. d. 50′. And BC. 62. d. 46′.) Then QC. the Semidiameter or whole sine, and QX. the second right sine of the ark BT. Wherefore QR. shallbe given, by the 4. of the sixth of Euclid.  
30. 56′. 90. 0′. 18. 10′.  QU. QC. QX. QR. If. 98750. give. 100000.— then. 31178. giveth. 31572.  
So have IQR. in such parts as the Semidiameter of the earth QC. is 100000. which (being reduced into miles accounting 3436 4/11. for the Semidiameter of the earth) do give 1084. miles and 10/11. which is the distance of the point respective R. from the centre of the earth Q.  
Again, as QC. is to CY. so is QR. to RS. wherefore QC. and QR. being given as before, & CY. the sine of the ark CA likewise known, RS. shallbe given.  
 25. 44′.   QC. CY. QR. RS. If. 100000. give. 43410.— then. 31572. giveth. 13705.  
Which being in the parts of the sins, I reduce into miles as before, and find the same 470. miles and 10/11. which is the distance of the point respective R. from the axis of the world QA.    

¶ Of the inconveniences and defects in sailing, and in description of Countries, caused by the Variation of the Cumpas. ¶ The tenth Chapter.  
IN all sea charts generally, which are made without consideration of the variation, are committed great errors and confusion. For, either the parts in them contained, are framed to agree in their latitudes by the scale thereof, and so wrested from the true courses that one place beareth from an other by the Cumpas, or else in setting the parts to agree in their due courses, they have placed them in false latitudes, or abridged, or over stretched the true distances between them.  
In the Marine plats made for Newfoundlande, the course set down from Silly to Cape Raso is due wist, which is found to be so by our common sailing compass, whose wires are set at ½. a point from North to East, notwithstanding Silly being in latitude 50. d. little more. Cape Raso in Newfoundland is found to be but in 46. d. ⅓, which is 3. d. ⅔. less than the latitude of Silly.  
To make a show of reformation of this error, (caused by the Variation and setting of the wires in the compass) or to give a light of that difference in latitude, they have placed in the plat against that coast, a new scale of latitude, some upon the line of South and North, and some other have placed the same upon the line of North north-east, & South South-Weste, (because that point of the compass showeth the Pole nearest in that place) and have furnished the degrees thereof, agreeably to the latitude of Cape Raso: and by that means have had a double scale of latitude, one for the Easter costs, the other for that West. But how far the same hath been from reforming the error, or giving any help to Navigation, you may easily judge.  
Others to avoid that error of the difference in latitude in that voyage and course, have used Cumpasses whose wires have been set directly under the North point, and thereby saiting West from Silly, have fallen to the northwards of Cape Raso about 50. leagues, and in latitude near 49. d.  
Some other have used in the same voyage to place a blank Fly upon their sailing compass, which they have removed from time to time, as they have judged the variation hath altered, by which way, albeit they may seem to keep themselves nearer the parallel, yet the same in Navigation worketh the greatest confusion of all other, and therefore is to be utterly abolished.  
In our voyages from hence eastwards to S. Nicolas in Russia, and to the narve in Livonia etc. the Marine plats of the coasts are described by our common sailing Cumpas, with consideration of the variations at divers places, whereby the true meridians reformedly set down, declining from the parallel meridians of the plat, do necessarily widen. northwards, and straighten to the southwards, contrary to the true form and nature of meridians. And yet notwithstanding, that is the best means hitherto known, to reform in plat, the errors that else would grow, by the strange variations that way.  
And albeit these plats serve very well for those Navigations, yet by means of the variations considered, the form of those coasts is so distorted from the right shape it should bear, being truly described upon the Globe or otherwise in plain, according to the true latitude and longitude: That whereas the narve (being in latitude 59 d. ¼. and in longitude from the meridian of London 26. d. 10′.) should be from S. Nicolas 9 d. 40′. in longitude to the Westwards (S. Nicolas being in latitude 64. d. 35′. and in longitude from London 35. d. 50′.) In the sailing plat it is brought to be in the meridian of Colmogorod, (which is in latitude 64. d. 20′. and in longitude from London 37. d. 45′.) which is 1. d. 55′. to the eastwards of the meridian of S. Nicolas.  
In the Mediterranean Sea, and in the coasts thereof, where, in great reason should be the perfectest descriptions of the world, for that in those parts have been the seats & abodes of the most famous and learned men in all ages, we see notwithstanding in the Marine plats of those parts, gross errors committed, through want of knowledge of the variation and the use thereof, in which they have not accounted of 3. 4. or 5. degrees error in the latitude of places.  
But those defects of the latitudes, have been very well reform, by the famous and learned Gerardus Mercator (whom I honour and esteem as the chief Cosmographer of the world) in his universal Map, which though he have made with sailing lines, and dedicated to the use of Seamen, yet for want of consideration of the Variation, and partly by angmenting his degrees of latitude towards the Poles, the same is more fit for such to behold, as study in cosmography, by reading authors upon the land, then to be used in Navigation at the sea.  
There is also in the same Universal Map, and likewise in all other modern Maps of the North parts of Europe, a great fault, by placing two Wardhouses distant one from the other above 20. d. in longitude, whereas indeed they are but one thing, and no such distance between them. This error hath grown by taking Wardhouse, and the Sea coasts, from thence to S. Nicolas, Vaigats and the Ob etc. out of the Map of Anthony jenkinsons travail to Boghar and Persia. In the which I placed that border of the Sea coast, and for some causes went no further in that description than Wardhouse, which is in latitude 70. d. ⅓. and in longitude from London 29. d. Wherefore to accomplish the whole border of that coast, he was forced to seek some other description to join with it, & took as appeareth the Map of Olaus Magnus of the North countries, wherein he found likewise Wardhouse, but falsely placed, in latitude about 19 d. too much, and in longitude as much too little, the which, although he might take to be the same specified in Master jenkinsons Map, yet he was constrained to separate them the said distance of 20. d. in longitude (or to leave there so much superfluous room) otherwise he should have thrust the South parts of those countries togethers, and confounded the whole description.  
And albeit he had had the entire sailing plat, that we use for those parts, yet if he had not known the secret effect of the Variation in the making thereof, he might have fallen into the like absurdity or worse. But of those coasts and of the inward parts of the countries Russia, Muscovia etc. I have made a perfect plat and description, by mine own experience in sundry voyages and travails, both by Sea and Land to and fro in those parts, which I gave to her Majesty, in Anno 1578.  
Besides these and like imperfections proceeding of the Variation, there is yet an other inconvenience, which oftentimes increaseth the former errors, and that is, the divers placing of the wires fixed to the Fly of the compass.  
This variety ofsetting the wires, hath caused great confusion in Navigation, and in other accounts of Sea causes, for when it is said, that from such a headlande, to such a place is such a course, or at such a place the Moon upon such a point of the Compass maketh the full Sea, it is requisite to be demanded, by what compass the observation was made, whereas if the wires had not been altered from the North point of the Fly, (which I wish had never been any where) these doubts had been avoided.  
It behoveth therefore all men that will make Hydrographical descriptions for the use of sailing, to have special regard of the Cumpas by which their observations are made, and if they collect notes made by sundry Cumpasses of divers sets, they ought to reduce all the varieties unto some one certain, and to give notice of the same, in their platt: And not to make a confused mingle mangle by joining together all varieties of observations, notes and reports, as the Portugese's and Spaniards have done, in compounding these North parts of the world, with their own discoveries, without consideration of the divers sorts of the several cumpasses by which they were made.  
Also it importeth all Masters, Pilots and others by what name so ever that shall give directions in Navigation, to look circumspectly to the setting of the wires of the Cumpas by which they shall sail, that the same Cumpas be correspondent, to the lines of the Sea Card that they shall use: that is to say, that it be of the same set for the Uariation, that the Cumpas was of, by which the Card was made.  
And seeing we have in this our Country acquainted ourselves commonly in our observations and Navigations, with the Cumpas, whose wires are set at ½. a point from North to East, I mean in the descriptions that I shall make, to apply the same agreeable to the said Cumpas, and would use the like without alteration (and also the strait lines in Sea Cards) if I should sail round about the world to make the description thereof, but always with regard of the several variations of every place where the same should be observed.   

¶ Of the Instruments and rules of Navigation.  
¶ The eleventh Chapter.  
AMongst the rules and Instruments for Navigation, all such are vain and to small purpose, wherein the true meridian is presupposed to be given▪ by the magnetical Needle, without due consideration of the variation for that they are all grounded upon false suppositions▪ Hereby it cometh to pass that one Michiel Coigner of Antwerp in his New instruction (as he termeth it) of the most excellent and necessary points of Navigation, wherein he showeth the making and use of a Nautical Hemisphere, which he preferreth before all other Sea Instruments, is very childishly abused. For where as he pretendeth by it, to give the elevation of the Pole and the hour and instant of the time of the day, by any one observation in any place, besides that it is of all other that hitherto have been used at Sea, the most tedious and unfit for that purpose, it is also by reason of the variation not considered, ●iere false and erroneous. For, the true meridian which is the ground of his purpose, is as far to seek as the thing he promiseth to give by the same. The like may be said of all other Instruments made upon the same ground whether they serve for the Sea or Land.  
The same Author in the 4. Chapter of his book, entreating of sailing upon the points of the Cumpas, saith, that in sailing South or North, he shall pass by the Poles of the world, and keep under one meridian, till he come to the place from whence he first departed. And upon the points of East and West out of the Equinoctial, he shall sail under a parallel, till he return to the place from whence he went. But in sailing upon the point of North-east, he shall describe a spiral line inclining by little and little to wards the Pole, as in his demonstration thereof in the same Chapter appeareth. But for want of due consideration of the variation, his rules, reasons and demonstrations, and such others hitherto given for like purposes, are frivolous and false.  
For if he direct his sailing by the Cumpas (as of necessity he must, being the only Instrument for that purpose) it is manifest, that whether he sail North or South, East or West, or by what other point so ever, the Cumpas not respecting always the Pole of the world, as he supposeth, but some other point or points distant from the same, shall lead him accordingly, whereby he shall neither keep under one meridian, nor under one parallel of latitude, neither make such a spiral line to the Pole of the world, as he demonstrateth. His fault in setting down those rules is so much the greater, in that he acknowledgeth in the Chapter next before the variation at Antwerp, to be about 9 d. from North to East according to Mercators' position, of the Magnetical Pole, which he also confirmeth by his own experience.  
But it seemeth he hath followed, that excellent Mathematician Petrus Nonius, especially concerning the sailing upon the points of East and West. For he, in his first book of the rules and Instruments of Navigation, enforceth himself to prove and demonstrate, that in sailing East or West, out of the Equinoctial, the course is performed by pieces of great circles, and yet describeth a parallel. But how that may stand with the principles of Geometry, I refer the judgement to the expert Mathematicians, for it is like as a circle should be made of strait lines, which is impossible.  
It appeareth in the discourse that he hath made of those matters, that he had not a right judgement of the nature of the Cumpas in sailing (admitting the same to show the Pole without Variation) for if he had, he would never have entered into such a Labyrinth as he did. But he thought it a great absurdity that the Cumpas in every Horizon should show the meridian and Poles of the world by the points of South and North, and by the points of East and West to show in the Horizon the vertical and Equinoctial East and West (being a great circle) and yet in sailing East or West, except in the Equinoctial, it should perform but a parallel.  
But it is to be understood, that albeit the points, or lines of the Cumpas do always in every Horizon represent great circles in the Heavens, the points of South and North the meridian, and the points of East and West the vertical circle of East and West, each crossing other at right angles, and likewise of the other points. (The reason whereof is, because the Cumpas lieth every where level with the Horizon, so as a perpendicular line descending from the centre thereof at right angles with the plain of the same, will always fall upon the centre of the earth, and consequently be the Semidiameter of a great circle.) So that where so ever the Cumpas be carried, these circles are supposed to be carried about with it, and the view of every thing in the Horizon represented by the points thereof, is likewise in great circles: Yet in sailing by the Cumpas, the points of South and North only, describe great circles generally, which are the meridians, & the points of East and West, describe a great circle in the Equinoctial only: in all other places out of the Equinoctial, they describe but parallels. And the sailing upon any other point of the Cumpas, from any place, describeth a spiral line, according to the angle it maketh with the meridian. And hereby in sailing upon the points of East or West, out of the Equinoctial, (the▪ North point always respecting the Pole) the course performeth a parallel, according to the distance of the centre of the Cumpas from the Pole. The manner thereof you may perceive by fastening a small thread or Virginal wire at the Pole of a Globe, or centre of a circle, which shall represent a movable meridian to be carried about the Globe or circle, and fix upon the same, a small Fly of a Cumpas, so as the line of South and North be answerable to the thread or wire, and the North point thereby always respect the North Pole: then in turning the thread about the Globe or circle, upon the Pole or centre, if the centre of the Fly be out of the Equinoctial (between it and the Pole) albeit the points of East and West crossing the same line and moneable meridian at right angles, do show the vertical East and West upon the Globe, which is a great circle, yet in carrying the same Fly upon the thread or movable meridian, about the Pole or centre, you shall by the centre of the same Fly, describe but a parallel, according to the distance thereof from the Pole of the Globe, or centre of the circle, not unlike the circular motion of a Horse drawing in a mill, who though he look forth strait in a right line, yet being fastened to the beam of the mill, is forced to make his course in a circle, whose Semidiameter is the length of the beam contained between the Horse, and the centre of the mill, or mispost.  
And as in the Equinoctial, the line of South and North in the Cumpas (by supposition representing the meridian) is parallel to the Axis of the earth (which is the common section of all the meridian plains) and the line of East and West crossing the same Axis at right angles, representeth the vertical East and West, which is the Equinoctial, imagining to descend from the centre of the Cumpas a line, to fall perpendicularly, and at right angles with the Axis of the world (which shall be at the centre of the earth) and in sailing East or West by the Cumpas, the imagined perpendicular line being carried about with the same (making always right angles with the Axis) shall describe the plain of the Equinoctial equidistant from the Poles of the world, and at right angles with the Axis, and the point of the same line at the centre of the Cumpas, the circumference of the Equinoctial, upon the superficies of the Sea: So being from the Equinoctial on either side, imagining the line of South, & North in your Cumpas to represent always the Axis of the world, and to lie parallel with it, the line of East and West most cross the same Axis always, at right angles: and supposing a line to fall from the centre of your Cumpas to the Axis of the world, making right angles with the same Axis. In sailing East or West, that imagined line being carried about with the Cumpas (always at right angles with the Axis) shall describe the plain of a parallel, equidistant to the plain of the Equinoctial, and the point thereof at the centre of the Cumpas, the circumference of the parallel upon the superficies of the Sea: which parallel should be represented by the points of East and West of the Cumpas, if the line of South and North of the same-were parallel is the Axis of the earth, as was supposed: but it is not. And therefore as they decline one from the other, so doth the vertical circle of East and West showed by the Cumpas, decline from the parallel circled every where.  
The angle of which declination, is always equal to the latitude of the place, or distance of the parallel from the Equinoctial.  
But as I have already sufficiently declared, the Cumpas showeth not always the Pole of the world, but varieth from the same diversly, and in sailing describeth circles accordingly. Which thing if Petrus Nonius and the rest that have written of Navigation, had jointly considered in the tractation of their rules and Instruments, than might they have been more available to the use of Navigation, but they perceiving the difficulty of the thing, and that if they had dealt therewith, it would have utterly overwhelmed their former plausible conceits, with Pedro de Medina (who as it appeareth having some small suspicion of the matter, reasoneth very clerkly, that it is not necessary that such an absurdity as the Variation, should be admitted in such an excellent art as Navigation is) they have all thought best to pass it over with silence. But I hope such as intend hereafter to write of Navigation, will either frame their rules, precepts, and Instruments, with regard of the Variation, as herein I have showed, or else ease themselves of that travail, for as good none as unprofitable.   

¶ Of the application of the Variation to the use of Navigation. ¶ The twelfth Chapter.  
Upon the Hypothesis of the Pole of the Magnes on the superficies of the earth, and the point Respective in the body thereof, according to the former calculations, might be inferred many pleasant conclusions, both for the longitude and latitude of places.  
But touching the point Respective by the declining of the Needle, seeing this is the first and only experiment that hath been made of it, I can not infer any further matter thereof, then that which I have already set down, until by observations in other places, we find how it will hold.  
And as for the Variation, if it were generally regular and certain, as in some part it seemeth to be: (that is to say, from hence West wards to Meta Incognita, New foundland, Florida, and that part of the coast of America) then might there be given by it general rules most certain and commodious for the use of Navigation.  
And by the same Hypothesis of the Pole of the Magnes at 25. 44′. from the Pole of the world, the greatest variation of the Needle in the Equinoctial, should be (at 90. d. of Longitude) 25. 44′. from North to East. And consequently the greatest variation in the parallel of 70. d. should be (at the longitude of 128. d. 51′.) from North to East 81. d. 14′. And in the meridian of 180. d. of longitude between the two Poles, (the Pole Artik I mean, and the supposed Pole of the Magnes,) there should the North point of the Needle or Cumpas respecting his own Pole, show the South, and the South point, the North Pole of the world.  
But in my traveles to the Northeast parts, I have found this position of the magnetical Pole clean reversed: for where as the angle of Variation from hence eastwards in the parallel of 70. d. should increase and grow wider, till it came to 81. d. 14. from North to East as before. At the Island Vaigats being in longitude from London 58. d. and in the same parallel of 70. d. where, by the Hypothesis, the variation should be 49. d. 22′. from North to East, I find the Needle to vary 7. d. from North to West. And the like effect I have found by divers observations in sundry other places of the East parts. Which observations with many more that I have caused to be made, and daily procure to be done in divers other Countries, I reserve, with intent (if it be possible) to find some Hypothesis for the salving of this apparent confused irregularity.  
At Ratisbona or Regensburg in Bavaria, being in latitude 48. d. 52′. and in longitude 36. d. 20′. where, by the former position of the magnetical Pole at 25. d. 44′. the Variation should be 16. d. 44′. from North to East. Gerardus Mercator found the same to be only 11. ¾. as I gather by his placing of the magnetical Pole at 16. d. 22′. from the Pole Artik upon his observation made at that place: which confirmeth the retrograde quality in the Variation from hence eastwards, as aforesaid.  
Which strange variety, I have here plainly proposed, to the end that the learned sort might consider thereof, and sharpening their wits, see what probable causes and grounds they can assign for the same. For, considering it remaineth always constant without alteration in every several place, there is hope it may be reduced into method and rule.  
As for that westwards, because it carrieth proportion, and hath some apparent regularity, I will apply the same to the general commodity of all such as shall travail that ways: which if I should here particularly decipher, it would require a volume, whereby (contrary to my first intent) I should far exceed the bounds of an addition, I will therefore abridge it to a Hydrographical Plat, wherein all such errors and defects as have been hither to used, shallbe reform, which shall be easy for the meanest capacities to conceive, and serve more effectually in use, then if I should have expressed the same by multitudes of rules in writing. Therefore for this matter I refer you to the same, the which you shall look for very shortly.   

¶ A new Instrument for the Variation.  
BEcause I have found some imperfections in the first Instrument for the Variation (which notwithstanding doth far excel the Cumpasses of Variation heretofore used for that purpose) I have here set down the form of a new Instrument, wherein all scruple of doubts and defects that might grow by the other is quite avoided. Which being once exactly placed with the Needle upon the line of South and North, will serve without removing for a whole days observation, the Index only being carried about with the Sun, to give the degrees of Azimuth upon the Instrument by the shadow of the line thereof, and is otherwise to be used according to the prescript rules of the former Instrument.  
These Instruments are made by Robert Norman, and may be had at his house in Ratclif.   

Imprinted at London for Richard Ballard, and are to be sold at his shop at Saint Magnus' cornar in Themes street. Anno. 1581.