Cardan's error when he seeks the distance of the
_centre of the earth from the centre of the cosmos by the_
motion of the stone of Hercules; in his
_book 5, On Proportions_.
One may very easily fall into mistakes and errors when one is searching
into the hidden causes of things, in the absence of real experiments, and
this is easily apparent from the crass error of Cardan; who deems himself
to have discovered the distances of the centres of the cosmos and of the
earth through a variation of the magnetick iron of 9 degrees. For he
reckoned that everywhere on the earth the point of variation on the Horizon
is always distant nine degrees from the true north, toward the east: and
from thence he forms, by a most foolish error, his demonstrative ratio of
the separate centres.
* * * * *
{170} CHAP. XII.
On the finding of the amount of variation: how great
_is the arc of the Horizon from its arctick or antarctick_
intersection of the meridian, to the point
_respective of the magnetick needle_.
Virtually the true meridian is the chief foundation of the whole matter:
when that is accurately known, it will be easy by a mariners' compass (if
its construction and the mode of attachment of the magnetick iron works are
known) or by some other larger horizontal versorium to exhibit the arc of
variation on the Horizon. By means of a sufficiently large nautical
variation compass (two equal altitudes of the sun being observed before and
after midday), the variation becomes known from the shadow; the altitude of
the sun is observed either by a staff or by a rather large quadrant.
On land the variation is found in another way which is easier, and because
of the larger size of the instrument, more accurate. Let a thick squared
board be made of some suitable wood, the surface of which is two feet in
length and sixteen inches in width: describe upon it some semicircles as in
the following figure, only more in number. In the centre let a brass style
be reared perpendicularly: let there be also a movable pointer reaching
from the centre to the outmost semicircle, and a magnetick versorium in a
cavity covered over with glass: then let the board be exactly adjusted to
the level of the Horizon by the plane instrument with its perpendicular;
and turn the lily of the instrument toward the north, so that the versorium
may rest truly over the middle line of the cavity, which looks toward the
point of variation on on the Horizon. Then at some convenient hour in the
morning (eight or nine for instance) observe the apex of the shadow thrown
by the style when it reaches the nearest semicircle and mark the place of
the apex of this shadow with chalk or ink: then bring round the movable
index to that mark, and observe the degree on the Horizon numbered from the
lily, which the index shows. In the afternoon see when the end of the
shadow shall again reach the periphery of the same semicircle, and,
bringing the index to the apex of the shadow, seek for the degree on the
other side of the lily. From the difference of the degrees becomes known
[Illustration] {172} the variation; the less being taken from the greater,
half the remainder is the arc of variation. The variation is sought by many
other instruments and methods in conjunction with a convenient mariners'
compass; also by a globe, by numbers, and by the ratios of triangles and
sines, when the latitude is known and one observation is made of the sun's
altitude: but those ways and methods are of less use, for it is superfluous
to try to find in winding and roundabout ways what can be more readily and
as accurately found in a shorter one. For the whole art is in the proper
use of the instruments by which the sun's place is expeditiously and
quickly taken (since it does not remain stationary, but moves on): for
either the hand trembles or the sight is dim, or the instrument makes an
error. Besides, to observe the altitude on both sides of the meridian is
just as expeditious as to observe on one side only and at the same time to
find the elevation of the pole. And he who can take one altitude by the
instrument can also take another; but if the one altitude be uncertain,
then all the labour with the globe, numbers, sines and triangles is lost;
nevertheless those exercises of ingenious mathematicians are to be
commended. It is easy for anyone, if he stand on land, to learn the
variation by accurate observations and suitable instruments, especially in
a nearly upright sphere; but on the sea, on account of the motion and the
restlessness of the waters, exact experiments in degrees and minutes cannot
be made: and with the usual instruments scarcely within the third or even
the halt of a rumbe, especially in a higher latitude; hence so many false
and bad records of the observations of navigators. We have, however, taken
care for the finding of the deviation by a sufficiently convenient and
ready instrument, by means of the rising of certain stars, by the rising or
setting of the sun, and in northern regions by the Pole Star: for the
variation is learned with greater certainty even by the skilful with an
instrument which is at once simple and less sensitive to the waves of the
sea. Its construction is as follows.
[228]Let an instrument be made of the form of a true and meridional
mariners' compass of at least one foot in diameter (with a versorium which
is either nude or provided with a cardboard circle): let the limb be
divided into four quadrants, and each quadrant into 90 degrees. The movable
compass-box (as is usual in the nautical instrument) is to be balanced
below by a heavy weight of sixteen pounds. On the margin of the suspended
compass-box, where opposite quadrants begin, let a half-ring rising in an
angular frame in the middle be raised (with the feet of the half-ring fixed
on either side in holes in the margin) so that the top of the frame may be
perpendicular to the plane of the compass; on its top let a rule sixteen
digits in length be fastened at its middle on a joint like a balance beam,
so that it may move, as it were, about a central axis. At the ends of the
rule there are small plates with holes, {173} [Illustration] {174} through
which we can observe the sun or stars. The variation is best observed and
expeditiously by this instrument at the equinoxes by the rising or setting
sun. But even when the sun is in other parts of the zodiack, the deviation
becomes known when we have the altitude of the pole: that being known, one
can learn the amplitude on the Horizon and the distance from the true east
both of the sun and of the following fixed stars by means of a globe, or
tables, or an instrument. Then the variation readily becomes known by
counting from the true east the degrees and minutes of the amplitude at
rising. Observe the preceding star of the three in the Belt of Orion as
soon as it appears on the horizon; direct the instrument toward it and
observe the versorium, for since the star has its rising in the true east
about one degree toward the south, it can be seen how much the versorium is
distant from the meridian, account being taken of that one degree. You will
also be able to observe the arctick pole star when it is on the meridian,
or at its greatest distance from the meridian of about three degrees (the
pole star is distant 2 deg. 55 min. from the pole, according to the
observations of Tycho Brahe), and by the instrument you will learn the
variation (if the star be not on the meridian) by adding or subtracting,
_secundum artem_, the proper reduction [_prostaphæresis_][229] of the
star's distance from the meridian. You will find when the pole star is on
the meridian by knowing the sun's place and the hour of the night: for this
a practised observer will easily perceive without great error by the
visible inclination of the constellation: for we do not take notice of a
few minutes, as do some who, when they toil to track the minutes of degrees
at sea, are in error by a nearly whole rumbe. A practised observer will, in
the rising of sun or stars, allow something for refraction, so that he may
be able to use a more exact calculation.
Bright and conspicuous stars[230] which are
_not far distant from the equator which it
will be useful to observe at their rising and
setting: the amplitude at the Horizon on
rising being known from the altitude of the
pole and from the declination of the stars, by
means of a globe, or tables, or an instrument
whence the variation is perceived by
technical calculation._
{175}
_Right Ascension_ _Declination_
|Oculus Tauri | 62° 55' | 15° 53' N |
|Sinister humerus Orionis | 72° 24' | 4° 5' N |
|Dexter humerus Orionis | 83° 30' | 6° 19' N |
|Præcedens in cingulo Orionis | 77° 46' | 1° 16' S |
|Canis major | 97° 10' | 15° 55' S |
|Canis minor | 109° 41' | 5° 55' N |
|Lucida Hydræ | 137° 10' | 5° 3' S |
|Caput Geminorum australe | 110° 21' | 28° 30' N |
|Caput boreale | 107° 4' | 32° 10' N |
|Cor Leonis | 146° 8' | 13° 47' N |
|Cauda Leonis | 171° 38' | 16° 30' N |
|Spica Virginis | 195° 44' | 8° 34' S |
|Arcturus | 29° 13' | 21° 54' N |
|Cor Aquilæ | 291° 56' | 7° 35' N |
_An instrument for finding the amplitude at rising on the horizon._
Describe the circumference of a circle and let it be divided into quadrants
by two diameters intersecting each other at right angles at its centre. One
of these will represent the æquinoctial circle, the other the axis of the
world. Let each of these quadrants be divided (in the accustomed way) into
90 degrees; on every fifth or tenth of which at each end of each diameter
and on each side let marks (showing the numbers) be inscribed on the two
limbs or margins made for that purpose outside the circumference. Then from
each degree straight lines are drawn parallel to the æquator. You will then
prepare a rule or alhidade equal to the diameter of that circle and divided
throughout into the same parts into which the diameter of the circle
representing the axis of the world is divided. Let there be left a small
appendage attached to the middle of the rule, by which the middle of the
fiducial line itself of the rule may be connected with the centre of the
circle: but to every fifth or tenth part of that rule let numbers be
attached proceeding from the centre toward each side. This circle
represents the plane of the meridian; its centre the actual point of east
or west, _i.e._, the common intersection of the horizon and æquator; all
those lines æquidistant from the æquator denote the parallels of the sun
and stars; the fiducial line of the rule or alhidade represents the
horizon; and its parts signify the degrees of the horizon, beginning from
the point of setting or of rising. {176}
[Illustration]
Therefore if the fiducial line of the rule be applied to the given latitude
of the place reckoned from either end of that diameter which represents the
axis of the world; and if further the given declination of the sun or of
some star from the æquator (less than the complement of the latitude of the
place) be found on the limb of the instrument; then the intersection of the
parallel drawn from that point of the declination with the horizon, or with
the fiducial line of the rule or alhidade, will indicate for the given
latitude of the place the amplitude at rising of the given star or the sun.
* * * * *
{177} CHAP. XIII.
The observations of variation by seamen vary, for the
_most part, and are uncertain: partly from error and inexperience_,
and the imperfections of the instruments; and partly
_from the sea being seldom so calm that the
shadows or lights can remain quite
steady on the instruments_.
After the variation of the compass had first been noticed, some more
diligent navigators took pains to investigate in various ways the
difference of aspect of the mariners' compass. Yet, to the great detriment
of the nautical art, this has not been done so exactly as it ought to have
been. For either being somewhat ignorant they have not understood any
accurate method or they have used bad and absurd instruments, or else they
merely follow some conjecture arising from an ill-formed opinion as to some
prime meridian or magnetick pole; whilst others again transcribe from
others, and parade these observations as their own; and they who, very
unskilful themselves, first of all committed their observations to writing
are, as by the prerogative of time, held in esteem by others, and their
posterity does not think it safe to differ from them. Hence in long
navigations, especially to the East Indies, the records by the Portuguese
of the deviating compass are seen to be unskilful: for whoever reads their
writings will easily understand that they are in error in very many things,
and do not rightly understand the construction of the Portuguese compass
(the lily of which diverges by half a rumbe from the needles toward the
west), nor its use in taking the variation. Hence, while they show the
variation of the compass in different places, it is uncertain whether they
measure the deviation by a true meridional compass or by some other whose
needles are displaced from the lily. The Portuguese (as is patent in their
writings) make use of the Portuguese compass, whose magnetick needles are
fixed aside from the lily by half of one rumbe toward the east. Moreover on
the sea the observation of the variation is a matter of great difficulty,
on account of the motion of the ship and the uncertainty of the deviation,
even with the more skilful observers, if they use the best made instruments
hitherto known and used. Hence there arise different opinions concerning
the magnetick deviation: as, for instance, near the Island of St. Helena
the Portuguese Rodriguez de {178} Lagos measures half a rumbe. The Dutch in
their nautical log fix it at a whole rumbe. Kendall, the expert Englishman,
with a true meridional compass admits only a sixth part of a rumbe. A
little to the East of Cape Agullias Diego Alfonso makes no variation, and
shows by an Astrolabe that the compass remains in the true meridian.
Rodriguez shows that the compass at Cape Agulhas has no variation if it is
of Portuguese construction, in which the needles are inclined half a rumbe
to the East. And there is the same confusion, negligence, and vanity in
very many other instances.
* * * * *