A DISCOURSE of the Variation of the Cumpas, or Magneticall 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 Attractiue of R. N.
1581.
To the Trauelers, Seamen, and Mariners of England.
Having of late (gentle reader) received from the expert Artificer To. Norman,
his book entitled The new Attractiue (who of the great good will, and
affection he bears, has attributed in his dedication, that, which I
acknowledge not to be dew) in the which amongst other diverse virtues and
properties of the Magnes or Lodestone, he entreats of the declinyng 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 traueilers 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 variatio^ of the Cumpasse to be the cause of many errors and
imperfections in Navigations, and perceiuyng that all those that have as yet
gone about to give rules in that art, have left this (being a principal
poinct, and eue^ the ground of all the rest) untouched, or at least so
slightly handled the same, that little or no benefit could be gathred thereby:
I have here set down the su^drie ways to observe the same at all times and
places, that the inconvenience being known, might be considered of, and
avoided. Wherein, although my cheifest intent has 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, and wherewith few Seamen are
yet acquainted, I may seem to have missed 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 them selves can not presently apply to the
purpose, by others that are thoroughly instructed in these Mathematical
supputations, or by them selves 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 poincte incident to Navigations, and thereof to keep
continual notes and memorial. For these observations, there needs not many
troublesome Instruments, only for the variation, the new Instrument in the
end of this treatise I prefer before all other. And for eleuations, 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 does lighten very much the
conceiptes, for understanding diverse important poinctes, 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 Topographicall Instrument, for taking of
distances, and making descriptions upon the land. With these Instruments, and
the sailyng Cumpasse and Marine plat, (which are always to be understood the
principal, and most necessary Instruments for Navigations, for by them only
any voyage may be made, but without them no Navigations can be performed.)
the whole world may be traueled, discovered, and described. These are
sufficient for a perfect Mariner, and more then these were superfluous, only
the runnyng 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 and Traueiler's,
that desire to be cunnyng in their profession, first to seek knowledge in
Arithmetik and 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 him self And this not only in
Navigations, but in all Mechanical Sciences. As by the studious practice and
exercise in these arts, have attained to rare and singular knowledge: In
Architecture, Vitriuuius the Roman: In paintyng that famous German Albertus
Durerus: And in buildyng of Ships, Matthewe Baker our countrieman: And others
in other faculties as they have been most skillful 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 Banckes, Rocks, and Sholdes in the Sea, with the
deapthes and other necessary notes observed in his own travails
particularly, and effectually according to the truth, (which is the cheifest
part required in a perfect Mariner.) And not be always tied to the reports
of other, or to the Portugale, or Spanishe Marine platts, which are made by
the Cardmakers of those Countries, men that are no trauelers them selves, but
do all things therein, by information, and upon the credit of others, which
only committ to memory the form and manner of the Sea coasts, with making
some few notes of the liyng of one place from an other, which can never be
so perfect as the deferiptions that are made upon the present sight and vewe
of places, albeeit he be never so skillful and cunnyng, that shall so carry the
same by memory, how much less then by the unskillful. 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 imperfectly dooen, whereas
the Marine plattes ought to be described by such as can give reason, and
show observation of every particularitie contained in the same, as well for the
latitude of places, as the liyng by the Compass of the Capes, Headlandes,
Poinctes, I lands, Baies, Rocks, Sholdes etc. one from an other, and the
distances between them. The errors of those descriptions, I may not attribute
to the Cardmakers, but to the unskillful Seamen of those countries, for if they
were otherwise, as they have been accounted the most skillful 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 Countrieman Sir Francis Drake for valorous attempt, prudent
proceadyng, and fortunate performyng 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 collecte and gather Hydrographical, and
Geographicall descriptions of other men's trauatles or reports: as their
pains may be great, and deserve due commendation, so their doings may
bring commodity diuersely. And in this behalf Abrahamus Ortelius in his
Theatrum, has deserved immortal praise, for collectyng to gether, and
reducing into one commodious volume, the diverse plattes and descriptio^s, made
by diverse and sundry men. But amongst all those that have made Geographicall
descriptions, I can not a little marvel at Guilielmus Postellus, who being a
famous learned man, a great traueiler and Cosmographer, and Deane of the
Kings Professors in the University of Paris, in his universal Map. Anno
1580. besides that it is generally handeled after such a gross and confused
manner, that it might seem rather to have come from some rude unskillful, then
from him so famous a Doctor; has also in the imagined Countries about the
North Pole, so corrupted it with his fond dreams, and fantastical
inscriptions, attributyng to those supposed lands, diverse people, as the
Georgians and Hyperborians, and assignyng there to be the highest hills of the
world, and the people dwelling on the^, to have the continual light of the
Sun; Sueta Zemlia found by the Englishemen, An. 1550. the holy Land, the
place of the cheifest felicity, the Hyperborean feeldes, 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 goes, or opinion is conceived of any man for profound learning, and
smothe deliueryng of their conceiptes, or what so ever great promises are by
them selves made in these arts, to judge of them according to the works that
come from them, and not otherwise to be deceived.
For auoidyng prolixitie in this my Preface to so smala volume, I refer the
gentle reader, to the work it self. 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 differs 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
wishyng 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 magneticall 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 fowerth
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
fornoone or afternoon. The fifth Chapter. How to find the Variation by
Arithmeticall calculation, upon any one observation in fornoone or afternoon,
the latitude of the place, and declination of the Sun being given. The sixth
Chapter. An other way most general, how to find the Variation by one
observation, either in fornoone 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 fornoone or
afternoon. The eight Chapter. Of the Pole of the Magnes. The ninth Chapter. Of
the poinct Respective. The tenth Chapter. Of the inconveniences and defects in
saylyng, and in description of Countries, caused by the variation of the
Cumpas. The eleventh Chapter. Of the Instruments and rules of Navigations. The
twelfth Chapter. Of the application of the Variation, to the use of Navigations.
Of the Variation of the Cumpas or Magneticall Needle.
Chapter I.
THE Variation of the Needle or Cumpas, is properly the ark of the Horizo^
contained between the true meridia^ of any place and the magneticall meridian
of the same, and is denominated to be Esterly or Westerly, according to the
position of the magneticall 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 has contrary denominations.
The magneticall meridian is to be understood a great circle passing by the
Zenith and the Pole of the Magnes, deuidyng the Horizon into two equal parts
crossyng the same at opposite points: which intersections or crossynges, are
showed by the Needle or wiers of the Cumpas touched with the Magnes or the
Lodestone.
The Azimuth of the Sun is a great circle, passing by the Zenith and the true
place of the Sun: crossyng the Horizon at right Angles in opposite poincts, and
diuidyng 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 eleuations in fornoone and afternoon,
have equal distances from the true meridian, so that the middle poinct of the
whole difference of any two Azimuths observed upon equal eleuations in
fornoone and afternoon, is the true meridian.
This difference of Azimuths is found upon the Instrument of Uariation, by
addyng together the Uariations of the Suns shadow at equal eleuations in
fornoone 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 shall be the variation of the Needle from the
true meridian.
Or else subtracting the lesser variation of the Suns shadow, from the
greater (at equal eleuations) the half of the remayner 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 Horizontall distance between
the Azimuth of the Sun and the magneticall 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 Standerd which is
placed at the North end of the fixed Fly, may fall perpendicularly with the
line in the same Standerd.
You must have regard that in removing the Instrument to the Sun as he goes
about, it may always stand level as aforesaied.
You are then to consider, that the ^tring that recheth from the South part of
the Instrument, to the top of the Standerd, is the chiefest string to give the
Suns shadow, which must be so directed by turnyng the Instruments South side
to the Sun wards, that the shadow of the same may fall directly longst 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 Standerd more upright, or remouyng it at the South end.
Then must you also see, that the string that is fastened to the hoope of Brass
that enuironeth 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 turnyng the said hoope,
and remouyng the same line at either side of it, as you shall see cause.
The Instrument being duly placed in form aforesaid, it differs nothing from
the Cumpas of Uariation, but only in this poinct, that whereas the Fly of the
Cumpas of Uariation, is so turned by virtue of the Magneticall wiers, that the
North poinct thereof does show the Pole of the Magnes or line of Uariation: In
this Instrument, the North poinct of the Needle does supply that, which the
North poinct of the Cumpas should do. And the North poinct of the Fly which
is fixed in the bottom of the Instrument, does always answer to the shadow
that the Sun gives.
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 ye
fornoone, the sooner, the better, and the more effectual may your
obseruatio^s be, do thus.
Take your Astrolabe and observe duly the height of the Su^ne, for yourmore
ease it shall be best for you to note the same, when it agrees 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 does stand, look directly over the North poinct of
the Needle, what degree and fraction, if there be any, does 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 poinct of the Fly is from the North poinct of the Needle, either
Eastwardes or Westwardes as you shall find the same. Thus may you observe
diverse times, upon several degrees of the Suns elevation. And like as you
do in the fornoone, 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 toward the Sun, as it was in the fornoone, (for
auoydyng of error that may be in the Instrument) notyng at every height what
you find the variation. And when the Sun comes 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 followyng.
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 Westwardes. 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.
Degrees. Degr. Min. Degrees. D. M. D. M. 17 52 35 17 30 0 11 17 18 50 8 18 27 45 11 11
19 47 30 19 24 30 11 30 20 45 0 20 22 15 11 22 21 42 15 21 19 30 11 22 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 has South
declination, and thereof amounts 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 fornoone 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 remayner 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 fornoone,
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.. So much I say is the Pole Artik, and
true meridian line that passes to the Pole by our Zenith at London, to the
Westwardes of the North that the Needle shows. And therefore the Needle or
Cumpas varieth from the true North 11. d. 17.. to the Eastwardes.
Also at 25. d. elevation in the fornoone 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 Westwardes) 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
toward the right hand against every degrees elevation; and conferryng them
all together, I do find the true variation of the Needle or Cumpas at
Lymehouse to be about 11. d.. or 11. d.. which is a poinct of the Cumpas just
or little more. So that in a Cumpas whose wiers are set directly under the
flower de Luce, the North and by West, and South and by East poincts 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 fornoone or
afternoon.
The fourth Chapter.
IN the former declaration, the only way to try the variation, is by comparyng
of the several correspondent observations of the Suns elevation in the
fornoone, 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, then the former rule gives 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 dewly 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 placyng the Quadrant of
Altitude or moveable vertical, at the vertical poinct 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 toward 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 shows
the Pole of the world and Pole of the Magnes or Lodestone: But if they differ,
you shall subtract the lesser from the greater, the remayner shows the
variation. And if the variation upon the Instrument be greater then 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 fornoone or
afternoon. This precept needs no further demonstration, then the Instrument
it self, 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
fornoone, 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 toward the East of the Horizon, and
turn the Globe till the prick in the Eclipticke 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
nerest, about 41. from the meridian. And conferryng 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 fornoone from the North to the
West, or South to the East, and the variation upon the Instrument greater then
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.. from North to East, or South to West.
How to find the variation by Arithmeticall calculation upon any one
observation in the fornoone or afternoon, the Latitude of the place, and
declination of the Sun being given.
The fifth 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 attaynyng of the same arc. First it is
necessary to have the arc 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 semidiurnall arc, 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 arc 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 remayner 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, leaves the
second right sine, which you shall seek in the Table of sines, and thereby
findyng his arc, you shall subtract the same from the quadrant or 90. d. the
remayner is the arc of the forsaid parellell of the Sun, which is answerable
or correspondent in degrees and minutes, to the arc 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 Su^ne or any Star: so is the
whole sine, to the versed sine of the Semidiurnall arc. 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 Semidiurnall arc 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 Euclide.
LEt AMT. be the meridian circle. BDQ. the common section of the meridian and
Equinoctial their playnes, 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 center F. at the distance FL. which is the sine of the complement of
the declination. AB. the arc of the elevation of the Equinoctial. BO. the
first right sine thereof. AL. the arc of the meridian altitude. LX. the sine
thereof. AN. the arc 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 arkes and
sines. L R. the Semidiurnall 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 sines L S. and L K. L F. the whole sine in
respect of the arks and sines 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 fornoone 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. gives. 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. gives L K. 23952. the versed sine of the
arc I L. in his dew parts. The same subtracted from L F. 100000. the whole
sine, leueth K F. or I G. 76048. the second right sine of the same ark, which
is the first right sine of the arc I H. which arc you shall find in the
table of sines to be 49. d. 30. 24. the complement whereof to the quadrant is
40. d. 29. 36. the arc I L. of the parallel between the Sun and the
meridian, whose correspondent ark in the Equinoctial, is the arc 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 arc 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
shall be the sine of the ark of the Horizon contained between the Azimuth of the
Sun and the meridian, which is the arc 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, and 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
(...)e ark of the Suns parallel, E C. the correspondent ^rk of the
Equinoctial, which are given in the former work. M B. the elevation of the
Sun at the time of the observation. M H. the complement thereof. B D. the ark
of the Horizon intercepted between the Azimuth and the meridian, which is the
thing required to be found.
In this figure the Reader is to consider the manner of the sphericall
triangles, and to compare the sines 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 arc 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. gives. 63395.
73. 0. 90. 0. 41. 31. 22. H M. M G. H B. B D. Again if. 956 (...) give.
63395. -- 100000. gives. 66291.
Whose (...)B D. 41. d. 31. 22. is the Horizontall distance of the w(...)uth of
the Sun from the meridian, the thing that w(...)ught.
Now comparyng 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 fornoone, and the variation upon the Instrument
greater then the arc 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 comparyng the same arc 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 then the arc 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 fornoone, and that in the
afternoon, proceeds either of the imperfection of the Instrument, or
negligence of the obseruer. For in the rule there can be no error, being
grounded upon Geometrical demonstration, then which nothing can be more
certain.
The former precepts and examples do serve when the Sun does decline from
the Equinoctial either Northwards or Southwardes. But if the Sun be in the
Equinoctial, then the manner of the workyng 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 gives 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 arc: And entryng the table of sines
with it, you shall find his arc, which if you subtract from the quadraut or
90. d. leaves the arc 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 forsaid arc of
distance, what shall the whole sine give. Multiply and divide, the quotient
shall be the sine of the ark of the Horizon contained between the Azimuth of the
Sun and the meridian. Which arc being compared with the variation of the
Instrument in manner as before is showed, gives 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
turnyng your Instrument only to receive the shadow of the Sun, and looking
then to the North poinct 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
passes by the Poles of the world, but if it do differ from 90. d. the same
difference is the variation of the Needle.
But admittyng the obseruer to be under the Equinoctial, and the Sun to have
declination, then the proportion of the sine of the complement of the elevation
at the time of the observation, unto the sine of the declination, shall be such,
as the whole sine, is to the sine of the arc of the Horizon included between
the Azimuth of East and West, which is the Equinoctial it self, and the
Azimuth of the Sun for the time of the observation, the complement whereof
gives the true meridian, which complement you may compare with the variation
showed upon the Instrument, the difference is the variation.
Diverse 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
sines, the one of the Semidiurnall arc, the other of the arc of the distance
of the Sun from the meridian. By the first, the Semidiurnall arc being found
and (...) into hours and minutes of time, is showed the just (...) quantity of
the day. And by the arc 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 Semidiurnall 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 rests SF. 27905. which is the second right sine
of the Semidiurnall ark LR. and the right sine of RH. 16. d. 12. which is the
complement of the Semidiurnall ark LR. wherefore subtractyng it from the
quadrant LH. or 90. d. rests 73. d. 48. the Semidiurnall ark LR. the same
reduced into parts of time allowyng 15. d. for an hour 15. for a minut, and
15. for a second of time, and for every degree 4. minutes of time, for every
minut 4. and for every second 4. etc. I find the time of that ark from the
poinct 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 and 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 then the whole sine, and the Semidiurnall arc 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 Semidiurnall arc above a quadrant, which being added to the quadrant,
gives the Semidiurnall arc.
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
subtractyng the same from 90. d. rests 40. d. 29. 36. the arc of the
distance between the Sun and the meridian, which being reduced into parts
of time as before, gives 2. hours 41. 58. and the same (because it is in the
fornoone) deducted from 12. hours the noonsteed, rests 9. hours 18. 2. the
just instant of the time of the day.
But if this versed sine be found to be greater the^ 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 euenyng)
then does the arc of distance consequently exceed a quadrant, the sine of
this excess is the surplus of the versed sine above the whole sine. Whose arc
added to the quadrant gives the arc of the Suns distance from the meridian,
and reducyng 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 ye 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 fornoone or afternoon, the elevation of the Pole and declination
of the Sun being given.
The sixth Chapter.
FOR the accomplishyng of this proposition, you are to imagine a sphericall
tria^gle upon the superficies of the Globe, whose sides must be. First the
portion or arc 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 arc of
the circle of declination comprehended between the Sun and the elevated
Pole, this arc is found by addyng, or subtractyng, 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
sphericall triangle given. Then the substance of the work consists in findyng
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 Horizontall
distance of the Suns Azimuth from the meridian, which being compared with the
variation of the Suns shadow upon the Instrument, gives 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 eleuatio^ of the Sun. BA. 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
Horizo^.
Now from the three angles of the triangle ABC. let fall 3. perpendicular lines
to the plain of the Horizo^ AG. CHAPTER and BK. and by the 6. of the 11. of
Euclide, these three lines shall be parallelles.
Then let fall a perpendicular line from C. upon AG. in the poinct L. from B.
an other perpendicular upon the same line AG. at the poinct M. And from the
same point M. erect a perpendicular line to N. which shall be 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 sphericall triangle
ABC. By the 4. definition of the 11. of Euclide, 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 BA. 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 Euclide.
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
follows.
The elevation of the Pole CE. 51. d. 32. the sine thereof CHAPTER 78297. The
elevation of the Sun BD. 17. d 0. the sine thereof BK. 29237. The arc BC.
102 d. 30. the chord thereof BC. 155976. The complement of the elevation of the
Sun BA. 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 CHAPTER 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. gives 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. leaves 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 sines 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 nombers of the sines, 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. gives. 90376.
Whose ark is 64. d. 38. 45. And addyng these two arks together they give 138
d. 28. 23. the ark or quantity of the obtuse angle NMB. equal to the
sphericall angle BAC. And deductyng it from the Semicircle 180. d. there
rests 41^. d. 31. 37. the angle FAD. the Horizontall distance of the Suns
Azimuth from the meridian, and subtractyng that from 52. d. 35. the variation
found upon the Instrument from North to West in the fornoone, rests 11. d.
3. 23.. the variation of the Needle from the meridian, the thing that was
proposed to be found. And comparyng the same with the afternoones 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 workyng thereof, omitting the fractions in the divisions,
and neglectyng the proportional parts of the sines and arks.
In these examples I have used ye 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
aunswerable to the same, if the third Canon of the Hypothenusaes 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 Ioachimus
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 minut to minut: which if in
the mean time before I have finished, I shall not find it extant by any
other, I will publishe it for the commeditie of all such as shall have
occasion to use the same for Navigations and Cosmography.
To find the elevation of the Pole, Situation of the meridian, and Variatio^
of the Needle at any place by the Sun, upon two observations either in
forenoone 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, then the cheef matter that is by them required. For the common
precepts, which as yet have chiefly been given for the findyng 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 guardes and Pole
star. Therefore I have thought good, that as I have showed the way to know
the variatio^ upon any one observation, either in forenoone or afternoon, the
latitude of the place presupposed: so likewise, upon two observations by the
Sun, either in forenoone 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 nere the meridian, the lower elevation the nerer
it is taken to the Azimuth of East or West, or to the Horizon, the better, with
which eleuations, you are to note the difference of the Suns Azimuths or
variations found by the shadow upon the Instrument exactly, for without
that, the eleuations 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 them selves
equally into Semicircles. And also that the quadrant of altitude (or moveable
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 needs not any further regard to be had
for placyng 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 quadra^t 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 fastnyng the same you must regard that you force it not from
one side of the Horizon to an other, but that it rest equidista^t 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 nerer you set the same to the
meridian, the more conveniently, without impechement, will your trial be
made.) Then give a prick upon the Globe in the Azimuth, that the quadrant
shows 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, toward the
Azimuth of East or West, which shall be nerest the same) so many degrees as you
find the difference of Azimuthes between the two eleuations by the shadow of
the Sun, upon the instrument of Variation, and staiyng your quadrant of
altitude upon that poinct 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 reche to 113. d.. of the Equinoctial of
your Globe, (which is a quadrant, and the greatest declination of the Sun)
then you must consider which of the Poles of the world is elevated above your
Horizon, and whether your declination be toward, 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 subtracte 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 remainyng 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 arc 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
crossyng the former. The poinct of the intersection, or crossyng 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 poincte that
the same quadrant shows upon the Horizon, is the intersection of the meridian
and the Horizon, the Horizontall distance between this intersection, and the
Azimuth of the lesser observation, subtracted from the semicircle, or 180. d.
leueth the Horizontall distaunce 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 Horizontall 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 remainer shall be 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
forenoone.
The first elevation 21. d. 0. Variation 100. d. 30. from North to Weste. The
second elevation 50. d. 0. Variation 48. d. 0. from North to Weste. Difference
of the Azimuths 52. d. 30. The declination 16. d. 14. Northerly.
LEt I D B. be the Horizo^ 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 openyng your Cumpasses to 73. d. 46. of the Equinoctial (which is the
complement of the declination) and setting one end upon G. the poinct 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 vnaltred in E. the lesser elevation,
and with the other end describe a piece of a circle crossyng the former ark at
H. this intersection shall be the elevated Pole.
Then set the quadrant of altitude unto the poinct 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.. or there about. And the true meridian KAI. And from K. to D.
the Horizontall distance 90. d.. which subtracted from K I. 180. d. the
semicircle of the Horizon, rests the ark D I. 89. d.. 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, rests 11. d.. Therefore I say, the true
meridian showing the Pole Artik is 11. d.. to the Westwards of the magneticall
meridian showed by the Needle, and consequently the variation of the Needle 11.
d.. 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 then 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 consists in the exactnes of the Instruments, and the
orderly application and handlyng of them.
I might here have annexed the manner, how upon two observations of the Suns
elevation in fornoone or afternoon and difference of the Azimuths, to
calculate the premises more exactly by the table of Sines and doctrine of
sphericall 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 breefly set down the
ground and sum of the work, which is this.
The complements of your two eleuations, are two sides of a sphericall 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 eleuations) and the other angles may be given.
Then have you an other like triangle, whose three sides are these: the first,
one of the foresaied complements of elevation: the second, the arc of the
circle of declination, between the Sun at the instant of the same elevation,
and the elevated Pole. The third side is an arc 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 findyng 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 arc of the
circle of declination, and the third side of the first triangle, which angle
is diversely 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 Sphericall triangle two sides, and the angle by the same two sides
contained, given. And by the same 28. of the fowerth 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
fowerth Chapter shows 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, shows the Pole of the world directly,) passes at the Islands of the
Acores, or nere there about, but I find by great probability, that it is
somewhat to the Eastwardes of those Islands, and not to the Westwardes. From
which meridian I account the beginning of longitudes, and find our meridian
of London, to be from the same 23. d.. our latitude as before said 51. d. 32.
and the Veriation of the Compass or Needle 11. d.. from the North to the
Eastwardes. Now upon these grounds* I find by calculation, the Pole of the
Magnes, or the intersection of the two Magneticall meridians, upon the
superficies of the earth, to be from the Pole artike 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 Magneticall 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.
LOI. the magneticall meridian for London. B. for the place of London. HI. the
quantity of the angle of Variation at the end of the quadrants BH. and BY. C.
the intersection of the two magneticall meridians. CL. and CN. two quadrants of
the said magneticall circles, includyng the ark LN. the quantity of the angle
at C. PAM. the Semicircle of a meridian crossing the magneticall meridian of
London in the poinct O. at right angles.
Make out the quadrants IHK. and LNK. so shall they cross them selves with the
quadrant OAK. at the poinct K.
Now have you ABC. a sphericall triangle, two angles whereof and the common
contaynyng side of them, are given. ABC. 11. d.. 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 sphericall triangle, not rectangle, whose two angles are given, and
their common contaynyng side, the other angle and sides shall be known, by the
31 of the 4. of Regiomontanus.
Wherefore the arc AC. the distance of the two Poles shall be given, which is
the thing required.
For as the sine of BH. is to the sine of HI. so is the sine of BA. 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. BA. AO. If. 100000. give. 19509. -- then.
62205. gives. 12135.
Now as AK. is to AH. (the sines I mean) so is KO. to OI. but the three first
are known AK. and AH. by their complements, and KO. the quadrant. Therefore
the 4. is given.
83.2. 51.32. 90.0. 52.4. KA. AH. KO. OI. If. 99261. give. 78297. -- then.
100000. gives. 78879.
And as BA. is to BO. (the complement of the arc OI. last found:) so is AE.
to EM. the quantity of the angle BAO.
38.28. 37.56. 90.0. 81.12. AB. BO.* AE. EM. If. 62205. give. 61474. -- then.
100000. gives. 98824.
Sohauyng 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. gives. 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. gives. 43410.
Which is the distance of the Pole of the Magnes from the Pole Artik: the thing
that was sought.
Of the poinct 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
rests now to be declared, of the poincte Respective, where it should be, by
the new property found of the declinyng of the Needle, at this place for
London 71. d. 50.
First it is to be considered, that as the Magneticall meridians do cross
them selves at their Pole, before specified: so do their plains likewise
cross in a right line, passing by the said Pole, and the center of the earth.
Then producyng a straight line, in the Magneticall plain of London, declinyng
from the plain of the Horizon 71. d. 50. where the same does cross with the
former common section of the two plains, there by reason should the poinct
Respective dee. Which intersection I find to be from the center 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 center of the
earth. Then QA. the axis of the world. QC. the common section of the
magneticall plains. BZ. the line of the Needles declination crossyng the said
common section at R. which is the poinct respective. QT. a straight line
crossing BZ. at right angles in X. QR. the distance of the poinct respective
from the center of the earth. RS. the distance there of from the axis. Now as
QV. is to QC. so is QX. to QR. But the three first are known. QV. 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. shall be given, by the 4. of the sixth of
Euclide.
30.56. 90.0. 18.10. QV. QC. QX. QR. If. 98750. give. 100000. -- then. 31178.
gives. 31572.
So have IQR. in such parts as the Semidiameter of the earth QC. is 100000.
which (being reduced into miles accompting 3436 4 / 11. for the Semidiameter of
the earth) do give 1084. miles and 10 / 11. which is the distance of the
poinct respective R. from the center of the earth Q.
Again, as QC. is to CY. so is QR. to RS. wherefore QC. and QR. being given as
before, and CY. the sine of the ark CA. likewise known, RS. shall be given.
25.44. QC. CY. QR. RS. If. 100000. give. 43410. -- then. 31572. gives. 13705.
Which being in the parts of the sines, I reduce into miles as before, and
find the same 470. miles and 10 / 11. which is the distance of the poinct
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 chartes 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 skale thereof,
and so wrested from the true courses that one place bears 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 plattes made for Newfoundlande, the course set down from
Silly to Cape Raso is due Wiste, which is found to be so by our common
sailyng Cumpasse, whose wiers are set at. a poinct 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., which is 3. d.. less then the
latitude of Silly.
To make a show of reformation of this error, (caused by the Variation and
setting of the wiers in the Cumpasse) or to give a light of that difference in
latitude, they have placed in the plat against that coaste, a new skale of
latitude, some upon the line of South and North, and some other have placed
the same upon the line of North Northeast, and South South-Weste, (because that
poincte of the Cumpasse shows the Pole nerest in that place) and have
furnished the degrees thereof, agreeably to the latitude of Cape Raso: and by
that means have had a double skale of latitude, one for the Easter costs, the
other for that Weste. But how far the same has been from reformyng the
error, or giuyng any help to Navigations, you may easily judge.
Others to avoid that error of the difference in latitude in that voyage and
course, have used Cumpasses whose wiers have been set directly under the
North poincte, and thereby saityng Weste from Silly, have fallen to the
Northwardes of Cape Raso about 50. leagues, and in latitude nere 49. d.
Some other have used in the same voyage to place a blank Fly upon their
sailyng Cumpasse, which they have removed from time to time, as they have
judged the variation has altered, by which way, albeeit they may seem to
keep them selves nearer the parallel, yet the same in Navigations works the
greatest confusion of all other, and therefore is to be utterly abolished.
In our voiages from hence Eastwardes to S. Nicolas in Russia, and to the Narue
in Liuonia etc. the Marine plattes of the coasts are described by our common
sailyng Cumpas, with consideration of the variations at diverse places, whereby
the true meridians reformedly set down, declining from ye parallel meridians
of the plat, do necessarily widen. Northwardes, and straighten to the
Southwardes, 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 albeeit these plattes serve very well for those Nauigations, yet by
means of the variations considered, the sorme 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 Narue (being in latitude 59. d.. and in longitude from the meridian of
London 26. d. 10.) should be from S. Nicolas 9. d. 40. in longitude to the
Westwardes ( S. Nicolas being in latitude 64. d. 35. and in longitude from
London 35. d. 50.) In the sailyng 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 Eastwardes 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 and abodes of the most famous and learned men in all
ages, we see notwithstanding in the Marine plattes 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 defectes of the latitudes, have been very well reformed, by the
famous and learned Gerardus Mercator (whom I honor and esteem as the cheef
Cosmographer of the world) in his universal Mapp, which though he have made
with sailyng lines, and dedicated to the use of Seamen, yet for want of
consideration of the Variation, and partly by angme^tyng his degrees of
latitude toward the Poles, the same is more fit for such to behold, as
study in Cosmographie, by readyng aucthours upon the land, then to be used
in Navigations at the sea.
There is also in the same Universal Map, and likewise in all other modern
Mappes of the North parts of Europe, a great fault, by placyng 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
has grown by taking Wardhouse, and the Sea coasts, from thence to S.
Nicolas, Vaigats and the Ob etc. out of the Map of Anthony Ienkinsons
travail to Boghar and Persia. In the which I placed that border of the Sea
coaste, and for some causes went no further in that description then Wardhouse,
which is in latitude 70. d.. and in longitude from London 29. d. Wherefore to
accomplish the whole border of that coaste, he was forced to seek some other
description to join with it, and took as appears 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 Ienkinsons
Map, yet he was constrained to separate the^ the said distance of 20. d. in
lo^gitude (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 albeeit he had had the entire sailyng 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, Muscouia etc. I have
made a perfect plat and description, by mine own experience in sundry voiages
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 proceadyng of the Variation, there is yet
an other inconvenience, which oftentimes increases the former errors, and
that is, the diverse placyng of the wiers fixed to the Fly of the Cumpasse.
This variety ofsettyng the wiers, has caused great confusion in Navigations,
and in other accomptes 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 poincte of the Compass makes the full Sea, it is requisite to
be demanded, by what Cumpasse the observation was made, whereas if the wiers
had not been altered from the North poinct of the Fly, (which I wish had
never been any where) these doubts had been avoided.
It behooves therefore all men that will make Hydrographicall discriptions for
the use of saylyng, to have special regard of the Cumpas by which their
observations are made, and if they collect notes made by sundry Cumpasses of
diners setts, 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 Portingales and Spaniards have done, in compoundyng these
North parts of the world, with their own discoueries, without consideration
of the diverse sorts of the several cumpasses by which they were made.
Also it imports all Masters, Pilots and others by what name so ever that
shall give directions in Nauigatio^, to look circumspectly to the setting of
the wiers 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 our selves commonly in our
observations and Nauigations, with the Cumpas, whose wiers are set at. a poinct
from North to East, I mean in the discriptions that I shall make, to apply the
same agreeable to the said Cumpas, and would use the like without alteration
(and also the straight 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 Navigations.
The eleventh Chapter.
AMongst the rules and Instruments for Navigations, all such are vain and to
small purpose, wherein the true meridian is presupposed to be given* by the
magneticall Needle, without due consideration of the variation for that they
are all grounded upon false suppositions* Hereby it comes to pass that one
Michiel Coigner of Antwerp in his New instruction (as he terms it) of the
most excellent and necessary poincts of Navigations, wherein he shows the
making and use of a Nauticall Hemisphere, which he prefers before all other
Sea Instruments, is very childishly abused. For where as he pretends 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 promises to give by the same. The like may be said of al
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, entreting of saylyng upon the
poincts of the Cumpas, says, 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 poincts 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 saylyng upon the poinct of Northeast, he
shall describe a spirall line inclyning by little and little to wards the
Pole, as in his demonstration thereof in the same Chapter appears. 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 saylyng 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 poinct so ever, the Cumpas not
respectyng always the Pole of the world, as he supposes, but some other
poinct or poincts distant from the same, shall lead him accordyngly, whereby
he shall neither keep under one meridian, nor under one parallel of latitude,
neither make such a spirall line to the Pole of the world, as he demo^strateth.
His fault in setting down those rules is so much the greater, in that he
acknowledges in the Chapter next before the variation at Antwerp, to be about
9. d. from North to East according to Mercators position, of the Magneticall
Pole, which he also confirms by his own experience.
But it seems he has followed, that excellent Mathematician Petrus Nonius,
especially concerning the saylyng upon the poincts of East and West. For he,
in his first book of the rules and Instruments of Navigations, enforces him
self to prove and demonstrate, that in saylyng East or West, out of the
Equinoctial, the course is performed by pieces of great circles, and yet
describes a parallel. But how that may stand with the principles of
Geometry, I refer the judgment to the expert Mathematicians, for it is like
as a circle should be made of straight lines, which is impossible.
It appears in the discourse that he has made of those matters, that he had
not a right judgment of the nature of the Cumpas in saylyng (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 poincts of South and North, and by the poincts of East and West to
show in the Horizon the vertical and Equinoctial East and West (being a
great circle) and yet in saylyng East or West, except in the Equinoctial, it
should perform but a parallel.
But it is to be understood, that albeit the poincts, 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 poincts of East and West the vertical
circle of East and West, each crossyng other at right angles, and likewise of
the other poincts. (The reason whereof is, because the Cumpas lies every where
level with the Horizon, so as a perpendicular line descendyng from the center
thereof at right angles with the plain of the same, will always fall upon the
center 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 poincts thereof, is likewise in great circles: Yet in saylyng by the
Cumpas, the poincts of South and North only, describe great circles generally,
which are the meridians, and 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 parallells. And the saylyng upon any other poinct of the
Cumpas, from any place, describes a spirall line, according to the angle it
makes with the meridian. And hereby in saylyng upon the poincts of East or
West, out of the Equinoctial, (the* North poinct always respectyng the Pole)
the course performs a parallel, according to the distance of the center of
the Cumpas from the Pole. The manner thereof you may perceive by fastnyng a
small thread or Virginal wier at the Pole of a Globe, or center of a circle,
which shall represent a moveable 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 wier, and the North poinct
thereby always respect the North Pole: then in turnyng the thread about the
Globe or circle, upon the Pole or center, if the center of the Fly be out of
the Equinoctial (between it and the Pole) albeit the points of East and West
crossyng 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 cariyng
the same Fly upon the thread or moveable meridian, about the Pole or center,
you shall by the center of the same Fly, describe but a parallel, according
to the distance thereof from the Pole of the Globe, or center of ye circle, not
unlike the circular motion of a Horse drawyng in a mill, who though he look
forth straight 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 center 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 crossyng the same Axis at right angles, represents the vertical
East and West, which is the Equinoctial, imaginyng to descend from the center
of the Cumpas a line, to fall perpendicularly, and at right angles with the
Axis of the world (which shall be at the center of the earth) and in saylyng
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 poinct of the same line at the center of
the Cumpas, the circumference of the Equinoctial, upon the superficies of the
Sea: So being from the Equinoctial on either side, imaginyng the line of
South, and 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 center of your
Cumpas to the Axis of the world, making right angles with the same Axis. In
saylyng 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 poinct
thereof at the center of the Cumpas, the circumference of the parallel upon
the superficies of the Sea: which parallel should be represented by the
poincts 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 does the vertical circle
of East and West showed by the Cumpas, decline from the parallel circle 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 shows not always
the Pole of the world, but varieth from the same diversely, and in sailing
describes circles accordyngly. Which thing if Petrus Nonius and the rest that
have written of Navigations, had jointly considered in the tractation of their
rules and Instruments, then might they have been more available to the use of
Navigations, but they perceiuyng 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 appears having some small suspicion
of the matter, reasons very clerkly, that it is not necessary that such an
absurdity as the Variation, should be admitted in such an excellent art as
Navigations is) they have all thought best to pass it over with silence. But I
hope such as intend hereafter to write of Navigations, will either frame their
rules, precepts, and Instruments, with regard of the Variation, as herein I
have showed, or else ease them selves of that travail, for as good none as
unprofitable.
Of the application of the Variation to the use of Navigations. The twelfth
Chapter.
Upon the Hypothesis of the Pole of the Magnes on the superficies of the earth,
and the poinct Respective in the body thereof, according to the former
calculatio^s, might be inferred many pleasant conclusions, both for the
longitude and latitude of places.
But touching the poinct Respective by the declinyng of the Needle, seeing this
is the first and only experiment that has 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 seems 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 Navigations.
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 poinct 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 trauelles to the Northest parts, I have found this position of the
magneticall Pole clean reuersed: for where as the angle of Variation from hence
Eastwardes 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
diverse 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 diverse other Countries, I reserve, with intent (if it be
possible) to find some Hypothesis for the saluyng of this apparent confused
irregularitie.
At Ratisbona or Regensburg in Bauaria, being in latitude 48. d. 52. and in
longitude 36. d. 20. where, by the former position of the magneticall 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.. as I gather by his placyng of the
magneticall Pole at 16. d. 22. from the Pole Artik upon his observation made at
that place: which confirms the retrograde quality in the Variation from
hence Eastwardes, as aforesaid.
Which strange variety, I have here plainly proposed, to the end that the
learned sort might consider thereof, and sharpnyng their wits, see what
probable causes and grounds they can assign for the same. For, considering it
remains always constant without alteration in every several place, there is
hope it may be reduced into method and rule.
As for that Westwardes, because it carries proportion, and has some apparent
regularitie, 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
Hydrographicall Plat, wherein all such errors and defects as have been hither
to used, shall be reformed, 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 does 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 doutes 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 remouyng 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 streate. Anno. 1581.