Untrodden Ground In Astronomy and Geology

Giving Further Details of the Second Rotation

and of the Important Calculations which can be Made by Aid of a Knowledge Thereof

by Major-General A. W. Drayson Rr.A.S.

Late R.A. Author Of "Practical Military Surveying", "The Last Glacial Epoch", "Common Sights In The Heavens", Thirty Thousand Years of the Earth's Past History," Etc.

London Kegan Paul, Trench, Trubjser & Co., Lt D . 1890

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PREFACE

IN the following pages some further details will be given of the second rotation of the earth, and the important facts that are revealed by a knowledge thereof.
The reader who wishes to become conversant with this hitherto unknown movement, is recommended to investigate this second rotation by itself; to omit, for the time being, all notice of the earth's daily rotation, and of its annual revolution round the sun, and to examine the details of a slow second rotation of a sphere. It will then be manifest that, whilst the poles of rapid rotation are carried over small arcs by the second rotation, every other point on the surface of the sphere will be carried over arcs by the same movement; but these arcs will differ not only in extent, but in direction, according to their positions relative to the poles of the second axis of rotation.
Thus every zenith of every locality on earth is differently affected during the year by the second rotation. Whilst the poles of rapid rotation [north and south poles] are carried over small arcs by the second rotation, every other point on the surface of the sphere will be carried over arcs by the same movement; but these arcs will differ not only in extent, but in direction, according to their positions relative to the poles of the second axis of rotation.
Thus every zenith of every locality on earth is differently affected during the year by the second rotation.
The zenith of the north pole of daily rotation is carried annually over an arc of about 20", which arc is traced nearly down a meridian of twenty-four hours right ascension; but a zenith 90° from the pole of second rotation is carried over an arc of about 40.9", the direction of the arcs being nearly in opposition to the daily rotation.
Two localities on the same meridian of terrestrial longitude, but differing in terrestrial latitude, may have their zeniths affected by the second rotation in a very different manner. One zenith may be carried directly north; the other zenith may be carried obliquely towards the east or west, These varied results will be most marked on those meridians which have about fifteen, sixteen, seventeen, nineteen, twenty, and twenty-one hours of right ascension. The detailed effect will be found fully explained in Chapter XIII.
By a knowledge of this second rotation results can be arrived at with minute accuracy by calculation, which results have hitherto been obtained, and then only for very short periods, by continued and laborious observation.
The effect of the second rotation as at present occurring, when traced back during 20,000 years, is to have produced a very great annual variation of climate in both hemispheres of the earth, more particularly in latitudes from 50° up to the poles, an arctic climate in winter reaching to about 54° latitude in each hemisphere, and these annual variations prevailing during about J 5,000 years, the date of the height of this period being about 13,544 B.C.
Whilst, therefore, a knowledge of this movement of the earth enables a geometrician to arrive by calculation at results hitherto unattainable by such means, it also gives a cause for those glacial effects, the history of which is written on the rocks themselves.
Those persons who will examine the simple details of the second rotation will probably perceive that it is merely an accurate description of a movement of the earth, which movement has hitherto been described in a slovenly and vague manner.

During the past two hundred years or more, theorists have been contented with the statement that the earth's axis had a slow conical movement, which caused the pole of the heavens to change its position in the heavens about 20" annually.
The pole of the heavens [North Celestial Pole] is merely the zenith of the pole of the earth, and that zenith is displaced annually 20" by this so-termed conical movement of the axis. But are we to be satisfied with this imperfect description, and fail to investigate how the zeniths of other localities on earth are affected by the same movement, which causes the zenith of the pole to move 20" annually? How much, and in what direction, are the zeniths of 51° north latitude and 70° north latitude displaced annually by "a conical movement of the axis " as regards meridians of six, twelve, and eighteen hours right ascension? It certainly appears remarkable that such important details should never have been examined or referred to, especially when astronomers make such use of the zenith for determining the declination of stars by their meridian zenith distance.




Physical astronomy and the laws of gravitation give no explanation as to why the axis of daily rotation of the various planets differ so much in the angle these make with the planes of the planet's orbits. The axis of Jupiter is inclined about 88° to the plane of its orbit ; that of Venus, about 15°; that of the earth, about 66.5°; that of Uranus, only about 12°. Neither does physical astronomy explain why the daily rotation of Jupiter and Saturn is more than twice as rapid as that of the earth.
All the sciences that are supposed to bear on astronomy have never yet defined or given any explanation as to how any zenith, except that of the pole of daily rotation, is affected by "a conical movement of the earth's axis." This movement of the earth has hitherto been undefined.
Whilst the axis of daily rotation is fixed in the earth, the axis of second rotation appears at present not fixed in the earth, but is fixed as regards the heavens, the north pole of this second axis being 29° 25' 47" from the pole of daily rotation, and having a right ascension of eighteen hours.

The north pole of the second rotation is shown on theStar Chart (above) of the sky at Giza, Egypt, on the December solstice.
Polaris (the pole of daily rotation) can be seen 29° 25' 47" from the pole of second rotation along the meridian that has a right ascension (RA) of 18 hours.
Notice Polaris, the pole of the ecliptic and the solar apex are all on the meridian of RA=18hrs. This meridian also passes through where the ecliptic crosses the galactic equator (line through the Milky Way), which is near the galactic centre. Since it is the December solstice, the equinoxes are on the horizon at High Noon and the sun will also be at the crossing point of the galactic equator with the ecliptic, but this happens only during the galactic alignment period (our era).

The earth rotates once during about 31,600 years round this axis, in opposition (nearly) to the daily rotation. Thus the conical movement of the two half axes of the earth is produced by the second rotation, just as the daily rotation causes a line from the earth's centre to a point on the earth's surface to trace a cone every twenty-four hours.
Exactly as a geometrician can calculate the length of arc over which each locality on earth is carried during a given time by the daily rotation, so can he calculate over what length of arc, and in what direction, each zenith is carried during a given time by the second rotation. We thus have exactness taking the place of that which hitherto has been vague and undefined, with the result that calculations can be made by this knowledge formerly considered impossible.
That this movement should not at once be comprehended by a geometrician, when proved as it is by calculations, is at least singular, but that certain persons should assert that it is " a vague theory with which they do not agree," whilst they fail to state what it is with which they do agree, is a most remarkable proceeding, and is one which must have been adopted in consequence of the problem not having been by these persons either examined or understood.
That this hitherto undefined movement of the earth, which fully explains the cause of the precession of the equinoctial points, of the change in polar distance and right ascension of the stars, of the decrease in the obliquity of the ecliptic, and also gives the date and duration of the last Glacial Period, and gives the detail changes of every zenith and meridian on earth, should be considered " a mere coincidence," is a statement exhibiting a condition of mind of a peculiar character. Especially is this the case when persons who make this statement assert that an undefined something, called " A conical movement," which fails to explain how any zenith except that of the pole of the earth is affected, which fails to supply them with the means of calculating the polar distance of a star from one observation only for a future date, which fails to give a cause for even the last Glacical Period, is a most profound truth, and is " exact astronomy."
The reader who can free his mind from the dogmatic and contradictory theories which have hitherto reigned triumphant in connection with this movement of the earth, and who will examine the effects of the second rotation, may revel in the number and accuracy of the new problems which he will be able to solve.
The opposition and arguments hitherto urged against the second rotation differ in no way from those brought against the daily rotation of the earth some three hundred years ago. Superstitious theories are put forward, with the firm conviction, apparently, that these can overcome calculation and recorded facts.
In the earlier chapters of this book, proof is given of the power of geometry to demonstrate the form and size of the earth. These proofs are original.
In the later chapters, some of the arguments supposed to be sufficiently strong by their utterers to controvert the second rotation are briefly dealt with.
Many socalled objections which have been advanced are so ridiculously absurd that it would be little short of an insult to an intelligent reader even to call his attention to their puerility.
The second rotation of the earth can stand against fair criticism, just as the daily rotation has stood against it. Those persons who consider that personal abuse and the parrot-like repetition of hitherto accepted theories will disprove facts, will occupy in the future a position corresponding to that in which the opponents of the daily rotation now luxuriate.

A. W. DRAYSON.
SOUTHSEA,
June, 1890.

CONTENTS.

I. THE FORM OF THE EARTH PROVED BY GEOMETRY ............................................................... 1
II. THE MOVEMENTS OF THE EARTH ............................................................................................. 17
III. THE KNOWN AND THE UNKNOWN AS REGARDS THE EARTH'S MOVEMENTS ... ... ... .. 28
IV. THE SECOND ROTATION OF THE EARTH ................................................................................ 43
V. THE PRECESSION OF THE EQUINOXES, AND THE DECREASE IN THE OBLIQUITY ......... 64
VI. SOME RESULTS OF THE SECOND ROTATION OF THE EARTH ............................................. 82
Vll. THE POLE OF THE HEAVENS AND THE POLE OF THE ECLIPTIC ......................................... 95
Vlll. THE UNION OF ASTRONOMY AND GEOLOGY ...................................................................... 102
IX. THE SO-CALLED PROPER MOTION OF THE FIED STARS ...................................................... 113
X. THE POLE-STAR .............................................................................................................................. 133
XL THE NAUTILUS CURVE ................................................................................................................. 143
XII. THE ZENITH AND THE MERIDIAN ............................................................................................. 163
XIII. THE MEASUREMENT OF TIME, AND RIGHT ASCENSION ................................................... 193
XIV. MODERN ASTRONOMICAL OBSERVATIONS ....................................................................... 220
XV. THE PLANE AND THE POLES OF THE ECLIPTIC .................................................................... 229
XVI. SOME EFFECTS OF THE SECOND ROTATION ...................................................................... 241
XVII. ANALOGY IN THE SOLAR SYSTEM ....................................................................................... 255
XVIII. OBJECTIONS OF THEORISTS ................................................................................................. 279

Untrodden Ground In Astronomy and Geology

CHAPTER I.

THE FORM OF THE EARTH PROVED BY GEOMETRY.

THE student of the history of the progress of astronomical science may probably perceive that, accuracy and truth, have in all past ages been compelled to force their way through the obstruction offered by ignorant superstition, vested interests, erroneous dogmatic theories, and an absence of a correct knowledge of geometry by those persons who were the authorities at various dates.
The success of such obstruction is indicated by the fact that, some five hundred years before Christ, the daily rotation of the earth was taught by Pythagoras and one or two other advanced reasoners, but was rejected and ridiculed by the authorities and their satellites during at least two thousand years, whilst the false theory of the earth being fixed in space was taught in all the schools of astronomy, and the reasoner who denied that the earth was immovable was declared to be not only ignorant of science, but to be actually a heretic.
The oft-repeated error, which has retarded the progress of true science, has been an indolent habit of hastily inventing theories which might explain some few phenomena, then to cease investigating, or to ignore facts which might prove these theories to be false.
When a theory had been accepted as correct, and taught by authorities during many years, it would cause a considerable amount of trouble if certain facts were accepted which would prove that the theories were erroneous.
The disinclination to re-examine whether any accepted theory is correct has been most marked in all past ages, and it has not been unusual for certain authorities at various dates to merely assert that the then accepted theories were highly satisfactory, in order to check inquiries relative to any novelty (however firmly based on facts) from being even examined.
The value of any theory may be best tested by carrying out computations based thereon. If the calculations can be arrived at correctly by aid of the theory, we may be tolerably certain that we are on firm ground. If, however, facts and calculations must be ignored and denied, in order that the theory should still be imagined correct, we may depend that this theory is untrue.
Among some of the earliest and most ignorant races on earth, it was supposed that the earth was a flat surface bounded by the sky. The learned historian Herodotus, writing on this question, states that " he cannot refrain from laughter when he hears men talking of the earth being round, as though made in a machine."

The following part of page 2 is not necessary. It has been left out. I continue on page 8

In order to picture in our own minds the geometry of the heavens, we must trace out on the sphere of the heavens the apparent course which the sun, and the various other celestial bodies, appear to trace during twenty-four hours, and during a year. An observer on the earth's surface in, say, latitude 51° N., and who faces the south, should realize that the equator of the earth, if produced to the heavens, forms an arch [the celestial equator]; the extremities of this arch cut the east and west points of the horizon, whilst the south point of the arch will be 39° above the south point of the horizon. The sun, consequently, on or about March 21 and September 21, rises in the east, attains a midday altitude of 39°, and sets in the west.
If the observer were in north latitude 50° at the above date, the sun would still rise in the east and set in the west, but would attain a midday altitude of 40°. When the sun is on the equator, a line joining the north and south limb of the sun will always be at right angles to that portion of the "equinoctial" (the term used to describe the trace of the equator on the sphere of the heavens) where the sun is at the time situated.
From this geometrical law we come to one or two interesting items, which at present seem in too many instances to be not only inexplicable, but never to have been observed by certain gentlemen who are regarded as astronomical authorities.

Take, for example, the following diagram (Fig. 4) to represent the horizon and the trace of the equinoctial on the sphere of the heavens from latitude 51° N., E being the east, S the south, and W the west points on the horizon. The sun would rise at E, reach M at midday, and set at W. We will suppose that when the sun is rising near E (Fig. 1), a spot is seen on the sun's surface as indicated at Fig. 1. Where will this spot appear on the sun's surface when the sun is near setting, as shown at Fig. 2?

The reader who wishes to solve this simple problem has merely to draw through the sun's centre a line at right angles to the equinoctial, as shown by the lines N 0, to find how the sun's " up " and " down " has changed during the twelve hours it has been above the horizon, and, if the spot has not moved on the sun's surface, it will appear on Fig. 2 in the position shown.
By the rotation of the earth on its axis, the "up" and "down" of an observer is changed ; it is not the sun that has altered its position. This apparent change in the position of a spot on the sun, and due to the daily rotation of the earth, can be observed between sunrise and sunset, on any day when there happens to be a spot visible on the sun. The fact, therefore, can be observed, and the cause has been explained in the last few paragraphs.
Another interesting phenomenon, and due to the same geometrical law, takes place with the moon, and is most remarkable when the moon is half full, and is over the equator, or, in astronomical language, has no declination. Under such a condition, the moon would rise in the east, reach the meridian at about 6 p.m., and set in the west at midnight. To an observer in 51° north latitude, the moon would appear to trace a curve on the sphere of the heavens, the greatest altitude of this curve above the horizon being 39°, when the moon was south, whilst the extremities of the curve cut the horizon at the east and west points.

We will take a date about March 21, when the sun is on the equator, and the moon half full, consequently 90° from the sun.
The line separating the illuminated portion of the moon from the darkened portion will, under the above conditions, be at right angles to that portion of the equinoctial on which the moon is at the time situated. Hence the moon will appear at various hours, during the time she is above the horizon, as shown in the following diagram (Fig. 5):-
[see Fig. 5, left ]
E, S, W represent the east, south, and west points of the horizon, to an observer in 51° north latitude. The curved line represents the trace of the equinoctial on the sphere of the heavens. The line separating the illuminated from the dark portion of the moon will always be at right angles to that part of the curved line on which the moon happens to be situated. Thus, shortly after the moon has risen, she will appear tilted as shown at 1. When the moon reaches the south-east, she will appear as shown by 2 ; when south, as shown by 3 ; when south-west, as shown by 4.
We may now note a singular fact which leads to some interesting results.
When the moon is at 2, the sun, also on the equinoctial, would be at 4, 90° from 2. And, although we know that the moon shines by the light reflected from the sun, it appears that if we drew a line at right angles to the line of light and shade in the moon, this line would pass considerably above the sun, not directly on to it. The moon, in fact, appears tilted, and not as though she were illuminated by the sun alone.

If we could see those rays of light which proceeded from the sun and illuminated the moon, those rays would appear curved, as shown in the following diagram (Fig. 6), where S is the sun, M the moon, E the east, W the west points on the horizon :
[see Fig. 6, left]

If an observer were at the equator of the earth, this apparent tilting of the moon would not occur, because the equinoctial traced from the east would pass as a straight line to the zenith, and from the zenith would pass as a straight line to the west point of the horizon. Hence that portion of the equinoctial between M and S which appears to an observer in 51° latitude as a curved line, would appear at the same time to an observer on the equator as a straight line.
If, again, a comet with a tail 90° more or less extended in the heavens from S to M (last diagram), this tail would appear curved to an observer in middle latitudes, but as a straight line to an observer at the equator.
This simple law of geometry seems to have been overlooked by the learned astronomers of the past, who in several instances have propounded the most verbose and profound theories to explain why a comet's tail should sometimes appear curved, and not straight. Even Sir John Herschel, in his " Outlines of Astronomy," Art. 557, states, " The tails of comets, too, are often somewhat curved, bending in general towards the region which the comet has left, as if moving somewhat more slowly or as if resisted in their course."
The geometrical law which has been explained relative to the trace of the equinoctial appearing as a great arch in the heavens to an observer in middle latitudes, but as a straight line to an observer at the equator, holds good for a great circle of the sphere [a meridian] cutting the east and west points of the horizon and passing through the pole of the heavens. This great circle would appear as an arch in the heavens to an observer in middle latitudes, but as a straight line to an observer at either pole. This great circle would coincide with the horizon to an observer at the equator, and would to him appear as a straight line.

The following diagram (Fig. 7) represents the trace on the sphere of the heavens of a great circle [a meridian] cutting the east and west points of the horizon, and passing through the pole of the heavens. The observer being in 51° north latitude, and facing the north, W represents the west, N the north, and E the east points of the horizon. P represents the pole of the heavens, the altitude N P being, as a geometrical law, always equal to the latitude, therefore in this case equal to 51°.

The great circle WPE (that part, at least, which is visible) appears to this observer as a great arch in the heavens. If a comet coincided with this great circle, this comet's tail would appear curved to the observer in 51° north latitude. To an observer at the north pole of the earth, however, the arc E P would appear as a straight line, drawn from the horizon to the zenith ; the arc P W also a straight line, drawn from the zenith to W. A comet, therefore, which coincided with E P W would appear to the observer at the north pole of the earth to have a straight tail, whereas to the observer in 51° north latitude the same comet seen at the same instant would appear to have a curved tail.
Owing to the rotation of the earth on its axis, causing all celestial bodies to appear to trace circles round the pole of the heavens as a centre, and owing to the fact that an imaginary line drawn from the horizon to the zenith appears as a straight line, it follows that under certain conditions a comet which appeared to have a curved tail would, six hours afterwards, appear to have a straight tail. Such a change during six hours I was fortunate enough to observe with the great comet of 1861. The geometrical law producing this apparent change is as follows.
Each celestial body appears during twenty-four hours to describe a circle in the heavens round the pole as a centre, therefore during six hours it traces one-fourth of this circle. A line drawn from the horizon to the zenith appears as a straight line to an observer on the earth's surface.

Let us now assume that a comet was seen by an observer in 51° north latitude at 8 p.m., with a tail 90° in length, and coinciding with that great circle on the sphere of the heavens which cuts the east and west points on the horizon and passes through the pole of the heavens. The head of this comet we will suppose 51° from the pole and to the west, the tail stretching to 39° from the pole and to the east. This comet would, under the above conditions, appear to an observer in 51° north latitude as shown, C P M (Fig. 8).

When six hours of the earth's rotation had occurred, that part of the comet at P would not have altered its position in the heavens. The head C would appear to be curved round to N (Fig. 9), the north point of the horizon, whilst the extremity of the tail M would be carried to the zenith of the observer, and the comet's tail would appear to this observer as a straight line, as shown in the following diagram (Fig. 9) :

The comet of 1861 underwent this change during the first night of its appearance. When first seen, the head was near the northern horizon, and the tail stretched in a straight line to the zenith. Six hours afterwards, the head was in the north-east, the extremity of the tail in the north-west, and the tail appeared curved.

The cause of this variation is a geometrical one, and is not due to any of those wonderful theories which the imagination of persons unacquainted with geometry have hitherto invented as supposed explanations.
Another example of this geometrical law is afforded by the "pointers" (alpha and beta Ursae Majoris) and the pole-star. The pole-star is about 1° 18' from the pole of the heavens, and during each rotation of the earth, appears to describe a circle round the true pole, the radius of which circle is 1° 18'; consequently, at intervals of twelve hours, the pole star changes its position from above to below the pole.

In order to illustrate the effect of the geometrical law relative to apparent curves and straight lines in the heavens the following diagram can be examined (Fig. 10) :
Suppose W the west, N the north, and E the east points on the horizon, E P W a portion of the great circle of the sphere [a meridian], passing through the pole P, and cutting the east and west points of the horizon. Suppose a and b two stars on this great circle, and c a star above the pole. The stars a, b, would appear to an observer to point exactly to the star c.

When, however, twelve hours of the earth's rotation has occurred, the stars a and b would have been apparently carried round to the position shown in the following diagram (Fig. 11), indicated by b and a; the star, however, would by the same rotation be carried to c below the pole, and the stars b, a, would appear to point above the star c, and not directly towards it, as they did on the former occasion.


The "pointers" a and b, Ursae Majoris, are so situated that when a south-west line can be drawn from the pole of the heavens through these two stars, they will appear to point directly towards the pole-star. When, however, the pointers are to the west of north, they will appear to point above the pole-star.
This is a fact as easily observed as is the apparent change in the position of a spot on the sun, or of the tilted moon when half illuminated, or of the curve of a comet's tail, and it is due to the same geometrical law.
It is, however, very remarkable that when, more than thirty years ago, I mentioned these facts to one of the most distinguished astronomers of the time, he stated that he had never remarked them; but, if they really occurred, he should attribute these effects to refraction. Had he said that he should have attributed them to electricity or witchcraft, his remarks would have been equally correct.

CHAPTER II.

THE MOVEMENTS OF THE EARTH.

ALTHOUGH the daily rotation of the earth and the annual revolution of the earth round the sun had been accepted as facts by the few advanced minds some five hundred years before Christ, yet the obstruction offered by ignorance and prejudice had prevented these astronomical truths from being generally received until about three hundred years ago, when Copernicus, and afterwards Galileo, revived the theory of the earth's two principal movements.
As soon as the facts could be realized, that the earth rotated every twenty-four hours, and was carried annually round the sun in an orbit either circular or very nearly approaching a circle, the diameter of which exceeded one hundred and eighty million miles, geometricians must have become aware that they were actually residing on an instrument the daily and annual movements of which would enable them to make a survey of the solar system, and of the universe.
That the earth is certainly an instrument, by aid of which we can determine the distance, size, and movements of other celestial bodies, ought to be fully comprehended by the geometrician, in order that he may understand how it happens that there is even now some untrodden ground connected with geometrical astronomy.
The great error that was committed by ancient astronomers was in attributing to other celestial bodies movements which were merely due to the motions of the earth itself. Thus the olden astronomers imagined that the whole of the celestial bodies revolved round the earth every twenty-four hours, whereas it is the earth that rotates every twenty-fours hours. They imagined that the sun moved among the stars, and described a circle among these every year, whereas it is the earth that describes the circle. It thus appears by no means an uncommon error for the mind unacquainted with geometry to attribute movements to external objects which are in reality due to an unknown movement of the earth itself. Having, however, found that in the history of the past this error has been committed, it requires that caution should be used lest we commit the same error, and hastily come to the conclusion that external bodies have some peculiar movement, which after all is due to an unknown motion in the instrument with which we make our observations.
In considering the earth as an instrument with which we make our observations, it is necessary to note certain peculiarities connected therewith. No matter whether the earth be a perfect sphere or a spheroid, we are justified in assuming that the sea-level of corresponding latitudes north and south of the equator is equidistant from the earth's centre. When we examine the distribution of land and water on the earth's surface, we at once perceive that there is an enormous preponderance of land in the northern hemisphere; consequently the centre of gravity of the earth must be north of the equator. There is also a very much larger quantity of land above the water between 15° west longitude and 160° east longitude than there is on the opposite side of the earth. The axis joining the north and south poles of the earth cannot, therefore, pass through the centre of gravity of the earth.
Having, therefore, a rotating and revolving sphere, in which the centre of gravity does not coincide with the centre of the sphere, a mechanician will naturally expect some other movement besides a rotation to take place, however slowly this other movement may occur.
Any apparent movement that takes place in connection with the celestial bodies is readily observed, even by men scarcely superior in intelligence to the savage. The rising and setting of the sun, the change from new to full moon, etc., being examples. To observe these facts is a simple proceeding ; to give the true cause for such changes requires intelligence.
Some hundred and forty years before Christ, an observer noted that at the period termed the vernal equinox, and when the sun consequently passes from the south to the north of the equator, the sun coincided with certain stars, with which it did not coincide some hundred or more years previously at the same time of year. In order that the sun should coincide with those stars with which it coincided formerly, it would be necessary to pass the vernal equinox two or three degrees.
Although these ancient observers were unprovided with telescopes, and were unacquainted with such accurate means of measuring small angles as we now possess, viz. verniers and micrometers, yet the size of the instruments used in former times enabled angles to be measured with sufficient accuracy to detect such a change as that just referred to.
When this fact, termed the "precession of the equinox", had been discovered, the ancient observers attempted to explain it as they explained all supposed movements of the stars, viz. by attributing to the sphere of the heavens a slow rotation during a long period of years. The earth at that date being supposed immovable, it would have been considered little short of blasphemy even to hint that the real cause was some movement of the earth, and not of the heavens.
When the daily rotation of the earth and its annual revolution round the sun were admitted as facts, another theory was invented to account for this precession of the equinoctial point.
It was known as a fact that the position of the pole of the heavens varied. The present pole-star, which is now little more than one degree from the pole of the heavens, was two thousand years ago fully ten degrees from this pole; consequently the axis of the earth (the direction of which produced to the heavens determines the position of the pole of the heavens) must have changed its direction. It was perceived that this change in direction of the earth's axis would also explain the precession of the equinoctial point. As at that date it was erroneously assumed that the obliquity of the ecliptic that is, the angular distance between the pole of the heavens and the pole of the ecliptic never varied, although the pole of the heavens had altered its position about 10° during about eighteen hundred years, it followed that the pole of the heavens must trace a circle on the sphere of the heavens round the pole of the ecliptic as a centre.

The next step made by theorists was to endeavour to explain what was the movement of the earth which caused the axis to change its direction, and they asserted that the earth's axis traced a cone during a period of about twenty-four thousand years, the pole tracing a circle during the same period on the sphere of the heavens round the pole of the ecliptic as a centre.
Before advancing further in this inquiry, attention must be called to the theory of the axis of a sphere describing a cone. When we assert that the axis of a rotating sphere describes a cone, no matter whether this cone is described in twenty-four thousand years or during twenty-four seconds, we must, if we give an accurate description of that which is imagined, state distinctly which pole of the sphere remains fixed, and which pole describes the base of the cone .
We must remember that the earth is the instrument with which we make our observations, and every minute movement of this instrument must be known before we can depend upon the observations made thereby. If we imagine that the south pole of the earth remain fixed, whilst the north pole describes the circle or base of the cone, and we happen to be wrong in this guess, we shall be committing the same error that the ancients committed, viz. we shall be attributing to the stars, etc., an independent movement which they do not possess, merely because the instrument by which we determine their positions moves in a manner which has not been suspected.
Hence a great oversight was committed when it was asserted that the earth's axis traced a cone, but it was not mentioned which pole remained fixed, and, remarkable as it may appear to those unacquainted with dogmatic theories, this statement of the earth's axis tracing a cone round the pole of the ecliptic as a centre has remained, during nearly three hundred years, unchallenged by geometricians. Although it has been well known, during the past hundred years at least, that the pole of the heavens was gradually decreasing its distance from the pole of the ecliptic, the supposed centre, yet the various writers on astronomy still continued to repeat that the pole of the heavens traced a circle round this supposed centre, from which it was known to decrease its distance ; at the same time, it was asserted that the earth's axis traced a cone, although it did not seem to occur to these learned theorists that, if exactitude were desirable, they ought at least to inform the ignorant public which pole it was that remained fixed, and which described the circle.
If a surveyor were supplied with an instrument with which to make observations, and he were told that this instrument rotated on an axis, but that the axis traced a cone, he would certainly like to know whether the upper or lower part of this axis remained fixed, in order that the cone should be traced.
If, also, he were told that the extremity of this axis traced a circle round a point as a centre, from which point it continually decreased its distance, he would be somewhat puzzled to know what sort of circle it could be that thus varied its distance from a supposed centre.
At about the commencement of the Christian era, the obliquity of the ecliptic was found to be about 24°. It is now found to be about 23° 27'. As the obliquity must be of the same value as the angular distance of the pole of the heavens [north celestial pole or earth's pole] from the pole of the ecliptic [sun's pole], it follows as an interesting inquiry, whether the radius of the circle which it has been stated the earth's axis traces round the pole of the ecliptic has a value of 24°, 23° 27', or some other value.
The important geometrical laws connected with the radius of this circle, and the errors which have been made in consequence of the true value of this radius not having been known, will be dealt with in future pages.
The statement that the earth's axis traces a cone, but no mention is made as to which end of the axis remains fixed, is so palpably deficient in detail, that no real investigator or competent reasoner can accept as satisfactory such a vague assertion.
The theory that the earth's axis traces a circle round a point as a centre, from which supposed centre it continually decreases its distance, is so directly in opposition to the well-known laws of geometry, that it is singular how such a belief can be accepted by any person who has learnt even the definitions in his Euclid. In order that the earth's axis traces a cone, the centre of gravity of the earth must be thrown out of its orbit, as this centre ceases to move round a uniform curve, and unless we know whether it be the north or south pole that remains fixed whilst this cone is said to be described, we cannot state how the centre of gravity of the earth, or any other parts of the earth, are displaced whilst this cone is described.
Now, if, instead of the whole axis of the earth describing a cone, the centre of gravity of the earth remained fixed as regards this movement, and the two semi-axes of the earth described cones, we have every detail of a change in the axis accounted for, although we have very different movements in other parts of the earth.
When, however, we have the two semi-axes describing cones, we have an exactly similar movement occurring during many thousand years that takes place every twenty-four hours, to a line joining the earth's centre with a given locality on the earth's surface. This line traces a cone every twenty -four hours, but the reason why this cone is traced is because the earth rotates every twenty-four hours. If, then, the earth has a second rotation during many thousand years round an axis directed to some point in the heavens, the two semi-axes of the earth would describe cones, and the whole axis would change its direction annually, as it is found to change it, but the centre of gravity of the earth would remain fixed as regards this movement.
In order to comprehend this movement, we will deal with it alone, neglecting for the time being the daily rotation of the earth.

Suppose N S the axis of daily rotation of the earth, C the centre of the earth. Draw an imaginary line, OCP, through the centre of the earth. Keep the points O and P on the earth's surface fixed, and give half a rotation to the sphere representing the earth. When half this rotation has been completed, the half-axis N C will occupy the position N' C; the other half of the axis C S will occupy the position S' C; and, whilst the whole axis will have changed its direction from N S to N' S', every point on the earth's surface will have described half a circle round the two points and P as centre. There will thus be two points on the earth's surface fixed as regards this movement, whilst the poles and every other point on the earth's surface will describe circles.
When a slow second rotation such as the above proceeds very slowly and is mixed up with the daily rotation, and partially concealed thereby, it presents, at first sight, some very complicated movements ; when, however, it is dealt with geometrically these apparent complications vanish, and we become acquainted with a mechanical movement, simple and defined, which explains many complications now known to exist. When compared with the present contradictory and vague theory of a circle being described round a movable centre, and an axis tracing a cone when it is not mentioned which end of the axis remains fixed, this second rotation yields most important results.

There appears to be to some minds a great difficulty in comprehending how two rotations can occur at the same time to a sphere such as the earth. Such minds confuse themselves by imagining that these two rotations can be resolved into a third rotation, which contains the other two. Such is not the case. The daily rotation of the earth takes places during twenty- four hours, whereas the second rotation takes upwards of 31,600 years to be completed. The daily rotation is performed round a permanent axis in the earth; the second rotation does not cause this axis to alter its position in the earth, but it causes this axis to change its direction as regards external objects.

In order to make the second rotation intelligible to those willing to comprehend it, the following model will be found of service.
Obtain a wooden sphere of any convenient size, through the centre of which drive an iron rod to represent the axis of daily rotation. Let the globe be supported by means of this rod by a circular arc which rests on a spindle and stand, the spindle pointing to the centre of the wooden sphere. The model will then appear as shown in the following diagram :
The axis of the sphere is represented by N S [N is missing from the diagram, but is at the other end of the axis where S is opposite], the arc by AR, and the spindle by 0. First, give to the sphere a slow movement by causing the arc to turn round. It will be found that the point P on the sphere does not alter its position during this movement, but the two semi-axes of the sphere will describe circles, during one complete revolution, round P as a centre. When half this revolution has been completed the axis N S will occupy the position N' S', the two semi-axes having described half a cone. When a complete revolution has been completed, it will be found that every point on the surface of this sphere has described a circle round the points P and as centres, and, in fact, that the sphere has described a rotation round an axis passing from P to O.

We may now realize the fact that we cannot cause the axis N S to occupy the position N' S' without giving to the sphere half a rotation, and we cannot cause the axis to return to its first position N S without giving to the sphere a complete rotation.
If we now give to the sphere a rapid rotation round N S, and at the same time cause the arc R A to turn round, we have the two rotations taking place at the same time, the one, corresponding to the daily rotation, taking place rapidly ; the other, the slow rotation which causes the change in direction of the earth's axis, occurring very slowly.
The second rotation is mixed up and concealed, as it were, by the daily rotation, but it nevertheless exists. The cause of the precession of the equinox, of the change in direction of the earth's axis, and of other effects, is not, as has hitherto been erroneously stated, a conical movement of the earth's axis in a circle round a point as a centre, from which point it continually decreases its distance, but is really due to a second rotation of the earth.
Immediately the fact is understood that a second rotation of the earth occurs, we have to deal with a problem by no means new. We have to deal with this second rotation in the same manner as we should deal with the daily rotation; we must find in what part of the heavens the pole of this second axis of rotation is situated, a proceeding not fraught with any great difficulty.

Any star which, for the time being, does not vary its distance from the pole of daily rotation is on the arc joining the pole of daily rotation with the pole of second rotation. As the pole of daily rotation is carried round an arc of a circle by the second rotation, various stars will fulfil such conditions, and we have merely to produce the axis joining the pole of daily rotation at various dates with those stars which do not vary their polar distance at various dates, and note where these arcs intersect, and this intersection will give us the position of the pole of the second axis of rotation.
For example, suppose O P Q (Fig. 14) the curve traced by the pole of the heavens during many years. When at O, the pole did not vary its distance from the star x ; the pole of second rotation was therefore on the arc Ox produced. When the pole was at P, the star y was found not to decrease its distance from P ; the pole of second rotation was therefore on the arc P y produced. When at Q the pole was not found to decrease its distance from the star z, and the pole of second rotation was therefore on the arc Q z produced. Where these arcs intersect, as at C, gives the pole of the axis of second rotation.
From an examination of these facts, it will be found that the pole of the axis of second rotation is 29° 25' 47" from the pole of daily rotation, and has a right ascension of 270° [which is 18 hours or RA=18h].
We can now advance to some most important geometrical laws hitherto overlooked or ignored by theorists. The neglect of these laws has caused endless confusion in the science of astronomy, and has compelled astronomers to employ numbers of observers, night after night, and year after year, in order, from these observations, to frame catalogues of stars.
To a vast multitude of stars, and to the whole solar system, have been attributed independent movements which do not occur, but which appear to occur in consequence of a movement of the earth taking place, which movement has been overlooked.

CHAPTER III.

THE KNOWN AND THE UNKNOWN AS REGARDS THE EARTH'S MOVEMENTS.

ATTENTION will now be drawn to those facts which are at present known to astronomers, and also to those which are not known.
From the observations of the past two thousand years, it is known that the pole of the heavens has a movement of about 20.09" annually, somewhere towards the first point of Aries. It is known that this change in position of the pole of the heavens is due to some change in the direction of the earth's axis of daily rotation, and it has been asserted that this change is in consequence of the earth's axis tracing a cone during about twenty-five thousand years.
Whether it is the south or north pole that remains fixed, in order that this cone be traced, has not been mentioned.
It is known that this change in the direction of the earth's axis (whatever it may be) is the cause of the shifting of the equinoctial point, termed the "precession of the equinoxes," and of the change in polar distance and right ascension of the stars.
So far we have approximated to the cause of known effects, but there is much more to be investigated before we can claim accuracy of detail. It has been asserted that the earth's axis traces a circle in the heavens round the pole of the ecliptic as a centre.
If this statement were true, then the pole of the heavens must always maintain the same distance from the pole of the ecliptic. The observations of the past two thousand years reveal the fact that the pole of the heavens has, during those years, continually decreased its distance from the pole of the ecliptic. It is, therefore, impossible that the pole of the heavens can trace a circle round the pole of the ecliptic as a centre, for if it did do so, the two poles would never vary their distance.
It is, therefore, one of the most remarkable exhibitions of mental imbecility to find men, who claim to be astronomers and geometricians, asserting in a parrot-like manner that the pole of the heavens traces a circle round the pole of the ecliptic as a centre, and yet admitting that these two poles have continued during the past two thousand years to decrease their distance.
If this singular contradiction stood alone it would be sufficiently remarkable; but we have another subject to consider in which mere vagueness assumes to occupy the position of exactitude.
It has been asserted that the earth's axis changes its direction annually to the amount of 20.09", and this is considered an ample explanation of the mechanical results which occur. Now, if an observer were located at the north pole of the earth, he would find that his zenith at the end of each year occupied a position in the heavens 20.09" distant from the point it occupied at the commencement of the year.
The north pole of the earth is only one point on the earth's surface, and the zenith of this point changes its position in the heavens annually to the amount of 20.09", and due to this movement of the earth's axis. How much does the zenith of 51° north latitude, 10° of north latitude, and 70° of north latitude change annually, and due to the same mechanical movement which causes the zenith of the pole to change 20.09" annually?
Until geometricians and astronomers can answer these questions accurately, and can define the direction and extent of the movement which takes place annually in the zenith of various localities on earth, they cannot claim to know what really occurs in connection with the known change in direction of the earth's axis.
Theorists have asserted that the joint action of the sun and moon on the protuberant equator of the earth causes the axis of daily rotation to change its direction 20.09" annually. So completely satisfied do these theorists appear to be with this explanation, that they repeat it as a cuckoo repeats its call, entirely omitting to notice that, whilst they have found from observation that the zenith of the pole of the earth varies its position 20.09" annually and towards the first point of Aries, no mention has ever been made of how the zeniths of other localities on earth vary annually.
Is the zenith of Greenwich affected in exactly the same manner as is the zenith of St. Petersburg? If not, what is the difference?
We may search in vain for any explanation of how various zeniths are affected by this change in direction of the earth's axis, both in the books written by assumed authorities, or in the Proceedings of the Royal Astronomical Society. We are favoured with numerous theories as to how long the sun will last, and numerous conjectures as to the exact composition of the stars, but it seems beneath the notice of these learned gentlemen to inquire as to how the earth, the actual instrument on which they make their observations, really moves.
It has been an accepted theory during more than two hundred years, that the joint action of the sun and moon produces a change in the direction of the earth's axis, and this theory is considered a sufficient explanation of what? Of the change in direction of the axis. But how do various zeniths change annually? What is the radius of the circle which the earth's axis traces? Is it 24°, 23°, or something greater or less than either of these values? Such trifling matters seem beneath the notice of philosophers who claim to know how long it will be before the sun is burnt out. Yet upon these items, viz. how each zenith is annually affected, and what is the true radius of the circle which the earth's axis describes, depend the most important results in astronomy.
It has hitherto not been known what the true radius of this circle really is. It has not been known how each zenith is affected, and the result has been that perpetual observation with expensive instruments has been necessary, in order to frame a catalogue of stars for even a few years in advance, and theories invented as to assumed "proper motions" in stars which have no foundation in fact.
Attention will first be directed to the importance of knowing the exact radius of the circle which the earth's axis traces on the sphere of the heavens. Unless this radius be known exactly, it is mere conjecture to talk about the "proper motion" of the stars, and the true radius of this circle has hitherto been a mere vague guess. There are laws of geometry which are rigid and true, and cannot be ignored; no imaginary theories can set these laws on one side, and the assertions of no assumed scientific authority can prevent these laws from being accurate.
Sir John Herschel, in his "Outlines of Astronomy," Art. 316, states, " It is found, then, that, in virtue of the uniform part of the motion of the pole, it describes a circle in the heavens around the pole of the ecliptic as a centre, keeping constantly at the same distance of 23° 28' from it."
In Art 640 of the same work, Sir John Herschel informs his readers that the pole of the heavens, in describing its assumed circle, around the pole of the ecliptic, its assumed centre, decreases its distance from this assumed centre 48" per century.
If two such contradictory assertions were made in connection with any simple problem or science, they would be at once ridiculed as too absurd to be meant in earnest, because the asserted conditions are in reality impossible.
As, however, the subject dealt with happens to be astronomy, the fact that a geometrical contradiction exists has been overlooked, and it is imagined that a very profound problem is meant when it is asserted that the pole of the heavens describes a circle round a .point as a centre, from which centre the circumference continually decreases its distance.
The theory invented to account for this wonderful movement, which, it must be remembered, is a geometrical impossibility, is that the joint action of the sun and moon on the protuberant equator of the earth causes the earth's axis to change its direction.

[The current model for the Precession of the Equinox says that it is a slow, periodic conical motion (gyration) of the Earth's axis. It is due to the fact that Earth's axis of rotation is not perpendicular to the ecliptic but is inclined about 23°.5 and is thus affected by gravitational perturbations from other bodies in the solar system. The Moon and Sun pull harder on that part of the Earth's equatorial bulge nearest them than on that farthest away; this causes a torque which precesses the Earth's rotational axis. General precession is said to be the sum of the lunisolar (the combined gravitational effects of the sun, moon) and the planetary components acting on the Earth's equatorial bulge that causes the Earth's axis to sway clockwise in a slow circle (precess), like the motion of a spinning top. An observer on Earth, at the point of equinox changes his orientation to inertial space at the current rate of about 50.29 arc seconds annually (one degree every 71.5 years) . At this rate the entire precession cycle time required to traverse all twelve constellations of the ancient Zodiac, is 25,770 years, although evidence indicates it is declining.]

But in what manner does the earth's axis change its direction? In what manner are various zeniths affected, and what is the exact radius of the circle traced by the pole of the heavens? Not one of these important details has hitherto been defined by theorists.
The earth's axis traces a circle in the heavens round the pole of the ecliptic as a centre; at least, so it has been asserted. But as the course which the earth's axis traces decreases its distance from the pole of the ecliptic, the assumed centre, we at once arrive at the important question as to what is the radius of the circle which the earth's axis does trace. Unless the exact radius of this circle can be defined, the assertion that some stars have what is termed "a proper motion" is a mere guess, based on no sound evidence. The reader's special attention, therefore, is called to the following well-known law of geometry (Fig. 15).

Y, P, Q, X, are points on the circumference of a circle, the centre of which is E.
The radius of this circle is EP = EQ = EX.
We will now suppose X, Y two stars on the circumference of the circle traced by the earth's axis, and P the position of the pole of the heavens at a given date. The angle XPY will represent the angle measured at the pole between the stars X and Y, and, in astronomical language, would be the difference in right ascension between the stars X and Y.
The well-known law of geometry, to which attention will be called, is that if any points, such as Q, 0, etc., be taken on the circumference of the circle Y P X, the angles XOY, XQY, XPY, etc., will all be equal. In other words, any two stars on the circumference of the circle traced by the earth's axis will never vary in their difference of right ascension. The importance of this law must not be overlooked.
Suppose that on January 1, 1800, the difference in right ascension of two stars supposed to be on the circumference of the circle traced by the earth's axis was 100° , and on January 1, 1850, the difference was found to be 100° 1'. It would be at once assumed by theorists that these stars had a proper motion, either collectively or individually, of 1' during fifty years.
If, however, the two stars X and Y were not situated exactly on the circle traced by the earth's axis then they must change their difference of right ascension, and this change would not be due to any proper motion in the stars themselves, but to the fact that the radius of the circle which the pole is assumed to trace on the sphere of the heavens had been incorrectly estimated.
Sir John Herschel; in his " Outlines of Astronomy," states that the radius of this circle is 23° 28', and that it never varies; but it is known and admitted that this radius is now about 23° 27' 14", and that about the commencement of the Christian era it was 24°. It is known also to those persons who are acquainted with the second rotation of the earth, that the radius of this circle is 29° 25' 47", and that the pole of the ecliptic is not, and cannot be, the centre of the circle which the earth's axis traces. It follows, therefore, that to talk about stars having "a proper motion," when the radius of this circle is unknown, and when it is a mere guess as to what this radius really is, affords an example of unreasoning theory, even more remarkable than anything in connection with the past history of astronomy.
We find, however, even in the present day, men passing their time in repeating observations month after month, and publishing tables of the supposed proper motion of the stars (that is, a change in their right ascension or declination), which ought not to occur if the radius of the circle which the earth's axis traces were in reality that which theorists have imagined it to be. That this assumed radius was erroneous, they never seemed to even suspect; consequently confusion prevails, which to such persons is most .mysterious, and it requires hundreds of observers to be observing night after night, and numbers of computers to be correcting these observations, in order to frame a nautical almanac for even a few years in advance.
The practical result of not knowing the exact radius of the circle which the earth's axis traces on the sphere of the heavens is, that stars which never alter their relative positions have assigned to them an independent movement, merely because these stars do not vary their right ascensions in accordance with a theory relative to the radius of the circle traced by the pole being of a certain assumed value.
The radius of this circle is not that which has been assumed by theorists, and the endless observations, and the numerous pages of the supposed proper motions of stars, are worthless, inasmuch as the first assumption relative to the course traced by the pole of the heavens is incorrect.
It has been assumed by certain theorists that, when they state that "the joint action of the sun and moon on the earth's protuberant equator " produces a change in the direction of the earth's axis of daily rotation, they have given a complete and profound definition of this movement of the earth, and have also assigned a cause for it.
If any reader imagine that the preceding explanation fully explains this movement, let him take a small globe, and move this globe in accordance with the instructions given above. In the first place, he must elect which pole of the globe remains fixed, in order that the axis trace a cone, and why should one pole remain fixed any more than the other?
Secondly, he must define how each locality on earth is affected by this movement alone; he must not be contented with the fact that the zenith of the pole changes its position about 20'09" annually, but he must define how much other zeniths change annually, and also the direction in which these zeniths change, and due to this movement of the earth. Such details are at present untrodden ground to astronomers. We have very elaborate and no doubt profound theories to account for a movement of the earth, but what this movement really is has hitherto never been defined.

In order that the reader may clearly comprehend how vague and indefinite is the present condition of this movement, the following diagram is submitted for examination :
This diagram represents a projection of the northern hemisphere on the plane of the equinoctial [celestial equator]. P represents the north pole, QTR the equator, the circle ZAB a parallel of latitude of 51° north.
We will now take any date in the past when the pole was at P, and when a daily rotation of the earth caused the zenith Z to trace a circle round P as a centre during twenty-four hours, viz. the circle ZBA. A point Q on the equator, and on the same meridian of longitude as Z, would also, during one daily rotation, trace a circle, viz. Q R T, round P as a centre.
Any number of years afterwards, the pole is carried to O by that movement of the earth hitherto assumed to be accurately defined in all its details by the term "a conical movement of the earth's axis."
The zenith of 51° is, under these conditions, carried, during a daily rotation, round O as a centre, the circle being represented by the letter X, OX being equal to PZ. A locality on the equator will, during twenty-four hours, be also carried in a circle round O as a centre, this circle being represented by the letter Y, OY being equal to PQ.
We now have to consider the effects of this slow movement only, and must treat it as though independent of the earth's daily rotation, and define in what manner the zeniths Z, A, and B, and the zeniths Q, T, and R, have been affected by this movement only.
Whilst the zenith of the pole has been carried over an arc of about 20.9" annually, and in the direction of from P to O, nearly in the direction of the first point of Aries [the vernal equinox, where the the ecliptic and celestial equator cross], the zenith Z will have been carried over some arc and in some direction; also the zeniths Q, T, A, etc., will have been carried over some arcs and in some direction by the same movement which has carried the pole from P to O.
The problem to be now solved is, where will the zenith Z be situated at the instant that the pole has reached O, and when a given number of siderial rotations of the earth have been completed.
The zenith Z will have been transferred somewhere on to the dotted circle X; the zenith B will have been transferred somewhere on to the dotted circle X; the zeniths Q, T, and R, somewhere on to the dotted circle Y. But where on this circle, is the question.
Unless the exact value, and the exact direction of the arc over which each zenith is carried (and due to the same movement which causes the pole to move from P to 0), can be calculated and defined, it follows that the detail movements of the earth are even yet unknown.
All the theories that were ever invented in connection with this change in direction of the earth's axis are valueless, because these theories are supposed to explain some movement of the earth, but it has hitherto not been known what this movement really is.
Theories may be venerated as articles of faith, and observations may be repeated by the million, but nothing but confusion can occur unless it can be stated how each zenith on earth, and, consequently, each meridian, changes annually, and due to that movement of the earth which causes the zenith of the pole to move about 20.09" annually.
The amount and direction in which each zenith moves annually has hitherto never been defined by theorists, in spite of the fact that in all observatories the meridian zenith distance of stars is the item measured in order to determine the declination of these stars. That the pole changed its position about 20.09" annually was considered a sufficient explanation, and the changes in each zenith were overlooked.
How each zenith changes, and how important is a knowledge of this fact, will now be explained.
The pole of the heavens changes its direction in consequence of the second rotation of the earth, just as the zenith of a locality on earth changes its direction in con- sequence of the daily rotation of the earth. The pole of the second axis of rotation is located 29° 25' 47" from the pole of daily rotation, and has assigned to it a right ascension of 18 hours, equal to 270°.
Each zenith describes an arc of a circle round the second rotation pole as a centre, the amount and direction of this arc being dependent on the distance of the zenith from the pole of the second axis of rotation.
With this knowledge, the exact amount and direction in the change of position of any zenith can be simply calculated.

In the following diagram (Fig. 17), P represents the pole of daily rotation at a given date in the past; the circle ZBA, the course traced by the zenith of 51° north latitude during a daily rotation; QT is the equator; C, the position of the pole of the axis of second rotation PC = 29° 25' 47" ; E is the position of the pole of the ecliptic.
The second rotation of the earth is the cause of the change in direction of the earth's axis, and causes the pole to change its position in the heavens, the pole of daily rotation moving in a circle round the pole C of second rotation. Thus the pole P is carried to O round C as a centre, at the rate [found by observation] of about 20.09" annually. The zenith Z will also be carried round C as a centre by the second rotation, and, whilst the pole P is carried to O, the zenith Z will be carried to Z'. In like manner, the zenith Q will be carried to Q', B to B', A to A', etc.
Hence the direction in which each zenith is carried can be accurately defined, and we can advance from the mere vague statement that "the earth's axis has a conical movement," to such details as the direction and amount of movement of each zenith during the year, and due to the second rotation. In order to obtain the value of the arc over which each zenith is carried annually by the second rotation, we have a very simple problem, the direction in which each zenith moves being at right angles to the arc joining that zenith with the pole of second rotation.
The rate at which the second rotation occurs must first be found, and can be obtained as follows :

A point on the earth's surface, viz. the pole of daily rotation 29° 25' 47" from the pole of second rotation, is carried annually over an arc of 20.09". We have then :-

C P = 29° 25' 47", PO = 20.09", C A = 90°
to find the value of A B, the arc of the equator of slow rotation during one year.
Making use of the usual formula :-

OP = AB cosine A P, we obtain 40.9" for the rate of the second rotation annually.

[I understand this to mean that both the pole star and a point on the same meridian but at the equator of the pole of second rotation will move at a rate of 40.9" annually, but the arc OP (20.09") will be proportionally smaller than the arc AB (40.9"). ie., the pole star on its lattitude will produce the arc OP of 20.09" each year at the rate of 40.9"per year.]

[ His workings must have been thus:
A P = 90° - CP = 90° - 29° 25' 47" = 60° 34' 13"
O P = A B cosine A P or
A B = O P / (cosine A P) = 20.09" / (cosine 60° 34' 13") / = 20.09" / 0.491 = 40.9" ]

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[His method must be similar to the following spherical trigonometry method]

The result at the bottom of the diagram on the left is :-

arc(ab) / arc(AB) = Ca /OA = Ca /Oa = sin POa
or
ab / AB = Ca /OA = Ca /Oa = sin POa
[OA = Oa because they are both radii on the arc PA, meeting at O.]

This can be written as:
ab / AB = sin POa
ab = AB sin POa
AB = ab / sin POa

Substituting the values of fig.18 above into the diagram on the left, we have:

PO= ab = 20.09" and angle C P = angle POa = 29° 25' 47"

Therefore
AB = ab / sin POa = 20.09" / sin 29° 25' 47" = 40.88"

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All zeniths which are 90° from the pole of second rotation will be carried annually over arcs of 40.9" [A B = 40.9"].
It will be evident that, as each zenith varies its distance (owing to the daily rotation) from the pole of second rotation, the value of the arc traced annually by this zenith will vary considerably, both in amount and direction, according as this zenith is referred to various meridians of right ascension. It follows also that, as the distance of a zenith from the pole of second rotation will depend on the latitude of a locality on earth, the zenith of which is referred to, the zeniths of two localities will not be similarly affected, either in direction or in amount, if they differ to any great extent in latitude.
Take, for example, two localities on earth, one in 50° north latitude, the other in 60° north latitude, and calculate the value of the arc over which these zeniths are carried by the second rotation when referred to meridians of right ascension of eighteen and six hours. The zenith of 60° north latitude is 30° from the pole of daily rotation, but is only 30° - 29° 25' 47" = 34' 13" from the pole of second rotation, when referred to a meridian of eighteen hours right ascension. This zenith will, therefore, be displaced annually only to the amount of 36/100 (0.36) of a second, found in the following manner :

40.9" multiplied by the cosine of 89° 25' 47" = 40.9" x .009 = 0.36".

[His working would be:
90° - 34' 13" = 89° 25' 47"
O P = A B cosine A P
O P = 40.9" x cosine 89° 25' 47" = 40.9 x 0.009953 = 0.40" (nearly 0.36")

Doing it the alternative way (which is really the same) :
angle POa is the angle at the earth's center between the pole of second rotation and the lattitude in question.
Therefore POa = 34' 13"
AB = ab / sin POa
ab = AB sin POa = 40.9" x sin 34' 13" = 0.40" (nearly 0.36")]

The zenith of 50° north latitude is 40° from the pole of daily rotation, but is 40° - 29° 25' 47" = 10° 34' 13" from the pole of second rotation when referred to a meridian of eighteen hours right ascension.

[ab = AB sin POa = 40.9" x sin 10° 34' 13" = 7.50" (nearly 7.3")]

90° - 10° 34' 13" = 79° 25' 47"
This zenith will be displaced annually by the second rotation to the amount of :-
40.9" multiplied by the cosine of 79° 25' 47" = 40.9" x 0.183 = 7.3".

Referring to a meridian of six hours right ascension, the zenith of a locality in 50° north latitude will be 40° from the pole of daily rotation, and 69° 25' 47" from the pole of second rotation.

40° + 29° 25' 47"= 69° 25' 47"

This zenith, therefore, will be carried annually by the second rotation over an arc of :-
90 - 69° 25' 47" = 20° 34' 13"
40.9" multiplied by the cosine of 20° 34' 13" = 40.9" x cos 20° 34' 13" = 40.9" x 0.9362 = 38.3"

[ab = AB sin POa = 40.9" x sin 69° 25' 47" = 38.29" (nearly 38.3")]

A zenith of 60° north latitude, under the above conditions, will be 30° from the pole of daily rotation, and 59° 25' 47" from the pole of second rotation.

30° + 29° 25' 47"= 59° 25' 47"
This zenith will be carried by the second rotation annually over an arc of:
40.9" multiplied by the cosine of 30° 34' 13" = 40.9" x sin 59° 25' 47" = 40.9" x 0.861 = 35.2".

[ab = 40.9" x sin 59° 25' 47" = 35.21" (nearly 35.2")]

Hence the zeniths of two localities, one in 50°, the other in 60° north latitude, will be carried over arcs differing 7" in value when these zeniths are referred to a meridian of eighteen hours right ascension. Yet these zeniths will be carried over arcs annually differing only 42.3" when referred to a meridian of six hours right ascension. Can any geometrician or astronomer seriously assert that such important facts as these can be safely overlooked, and that, because a theory is believed in which is supposed to account for a change in the direction of the earth's axis, therefore no further investigations of the true movements of the earth need be undertaken?
The changes in the direction, and the amount of this change for various zeniths due to the second rotation, although hitherto untrodden ground, are more important than all the vague theories ever invented.

CHAPTER IV.

THE SECOND ROTATION OF THE EARTH.

A KNOWLEDGE of how various parts of the earth move in connection with the change in direction of the earth's axis, and consequently the simple means available by aid of which the direction and amount of change annually produced in each zenith by the second rotation, are in themselves important facts, inasmuch as we can substitute accurate details for mere uncertainty. If, however, this were all that could be achieved by a knowledge of the second rotation, it might not be considered of much practical value.
It would be indeed singular if nothing important could be arrived at from a knowledge of this movement of the earth, and it is therefore desirable that some few details should now be given of the calculations that can be made in consequence of, and based upon, the second rotation of the earth.
In the first place, we know that the course traced on the sphere of the heavens by the pole is a circle with a radius of 29° 25' 47", and that the variable circle, with a variable radius asserted to be 23° 28', is an erroneous theory. We know how each zenith is effected by this second rotation, and can therefore calculate every item connected with these changes of zenith. We know, also, that the assertion that a multitude of stars have a large independent movement of their own is erroneous, inasmuch as this statement has been based on the assumption that the pole of the heavens traces a circle round the pole of the ecliptic as a centre, and never varies its distance from this centre, whereas facts prove that, during the past two thousand years at least, the pole of the heavens has not moved round the pole of the ecliptic as a centre, but has gradually decreased its distance from this assumed centre. In other words, an incorrect radius has been imagined for the circle which the pole does trace, and consequently the true course of the pole of the heavens has hitherto been unknown.
If the true course of the pole were really known, no observations would be requisite in order to find what would be the polar distance of stars for any date in the future. This polar distance could be calculated with far greater accuracy than it could be observed, because the errors which may occur, owing to the uncertainty of refraction and of instrumental errors, are avoided.
As an example, the following problem is given for solution by astronomers.
On January 1, 1887, the mean declination of the pole-star was found, by observation, 88° 42' 2173", and its mean right ascension, Ih. 17m. 19 '6 3s. Without any reference to the annual variation in polar distance of this star, but by a knowledge of the true course of the pole of the heavens, calculate the mean north polar distance of this star for January 1, 1850, January 1, 1819, and January 1, 1950.
One of the first and most important results obtainable from a knowledge of the second rotation of the earth, is the ease with which the position of stars can be calculated for the future, quite independent of the present laborious system of perpetual observation. One accurate observation of the position of a star is sufficient to enable a geometrician to calculate the polar distance of this star for each year for a hundred years in the future or past.
For the information of those persons unacquainted with practical astronomy, it may be stated that any such calculation has hitherto been unknown; the method hitherto adopted being to find, by perpetual observation, how much a star increases or decreases its polar distance per year, and then to add or subtract this rate in order to approximate to this star's polar distance for two or three years in advance.
The very fact of such a mere rule-of-thumb system having been adopted proves that the true course of the pole cannot be known for if it were, a more accurate and less laborious process would be employed.
The means by which such important and hitherto unknown results can be obtained are very simple. From one accurate observation determining the mean polar distance of a star, and its mean right ascension at any given date, we can calculate the distance of this star from the pole of the second axis of rotation, and the angle subtended at the pole of second rotation by two arcs one drawn from the pole of second rotation to the star, the other drawn from the pole of second rotation to the pole of daily rotation, which is a constant quantity of 29° 25' 47". The third item to be known is the angle at the pole of second rotation formed by these two arcs, a value that can be calculated from one observation of this star. This angle varies, except under rare conditions, at the rate of the second rotation, viz. 40.9" per year the rate at which the second rotation carries the pole of daily rotation round its circular course.

The north pole of the second rotation is shown as the zenith (ie., in the center) of theStar Chart above. The stars on the chart circumference are on the equator of the pole of the second rotation. The pole of the ecliptic is 6° from the pole of the second rotation along the meridian that has a right ascension (RA) of 18 hours. The stars labelled in green are those that there are calculations for below.

To find the distance, therefore, of the pole of daily rotation from any star becomes a simple problem in spherical trigonometry. The endless observations, therefore, of astronomers, in order to arrive at this result, are quite unnecessary, and those individuals who now pass night after night, and year after year, in measuring the meridian zenith-distance of stars may test whether their observations are correct, and their instruments in adjustment, by means of such calculations as those given below. The observations themselves are quite unnecessary, except perhaps for amusement; but it seems a somewhat expensive amusement to spend many thousands per annum for incomes for those individuals who are employed night after night to make observations, which lead imperfectly to results which can be calculated with the greatest accuracy.
The method for calculating the polar distance of a star from one observation will now be described, and several important stars will be referred to as examples. The first star to which reference will be made is the pole-star [Polaris, which is also called alpha Ursa Minoris, and represented as the symbol 'a' in this calculation]. The calculations for this star (Fig. 19) are as follows:

C being the pole of second rotation; a, the star [alpha Ursa Minoris].
P, the pole of daily rotation on January 1, 1887;
PC is an arc of 29° 25' 47"
the pole P moves round C as a centre at the annual rate of 40.9".
Ca = 29° 52' 49.6".
The angle PCa = 2° 27' 5" for January 1, 1887.

Now, as the pole P moves round C as a centre at the rate of 40.9" annually, the distance that the pole was or will be from the star a can be readily calculated in the following manner. Take, for example, the date January 1, 1819. It is required to calculate the mean north polar distance of the pole-star for that date. Between 1887 and 1819 there are sixty-eight years, during which the second rotation has caused the angle at C to vary at the rate of 40.9" annually.
40.9" x 68 = 2781.2" = 46' 21.2".
As the date 1819 was earlier than 1887, the angle at C on January 1, 1819, was greater by 46' 21 2" than it was on January 1, 1887.
On January 1, 1819, the angle aCP was therefore 2° 27' 5" + 46' 21.2" = 3° 13' 26.2".
The two sides Ca and CP are constants, consequently we have two sides and the included angle of a spherical triangle to find Pa, the third side, which will be the polar distance of the pole-star for January 1, 1819.
The detail working of the method of finding the polar distance of this star for a date distant sixty-eight years will be given.

Log. cosine, 3 13' 26'2" = 9.9993121
Log. tangent, 29 25' 47" = 9.7513982
9 7507103 as tan. 29 23' 27'3'
+ 29 52' 49.6"
- 29 23' 27.3"
= 0 29' 22.3'
Log. cosine, 29 25' 47" = 9 9399977
Log. cosine, 29' 22.3" = 9.9999842
19.9399819
- Log. cosine* 29 23' 27.3" = 9.9401635
9.9998184 = Log. cos. 1° 39' 25" = P a

By this calculation the mean north polar distance of the pole-star for January 1, 1819, was 1° 39' 25". In the Nautical Almanack for 1822, the mean north polar distance of various stars for January 1, 1819, was given ; among these the pole-star for that date (1819) was recorded as 1° 39' 25".

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[I don't understand this method, but I got a similar answer using the spherical cosine rule

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

http://en.wikipedia.org/wiki/Law_of_cosines_(spherical)

We have two sides and the included angle of a spherical triangle to find Pa, the third side
Ca = 29° 52' 49.6"
PC = 29° 25' 47"
The angle aCP = 3° 13' 26.2"

cos(c) = cos(29° 52' 49.6") cos(29° 25' 47") + sin(29° 52' 49.6") sin(29° 25' 47") x cos(3° 13' 26.2")
......... = 52' 1.44” x 52' 15.45” + 29' 53.49” x 29' 28.88” x 59' 54.3”
......... = 45' 18.64” + 14' 39.84” = 59' 58.49”
c = acos(0° 59' 58.49"). Changing this to decimals:
c = acos(0.9995816) = 1.6574812024 = 1° 39' 26.93"

or, doing the whole calculation in decimals:
cos(c) = cos(29.8804444444)*cos(29.4297222222)+sin(29.8804444444)*sin(29.4297222222)*cos(3.2239444444)
......... = 0.8670668367*0.8709590373+0.4981918312*0.4913556303*0.9984173487
cos(c) = 0.9995816
c = acos(0.9995816) = 1.6574812024 = 1° 39' 26.93"

checking using the ordinary planar law of cosines for small spherical triangles:
a^2 = b^2 + c^2 - 2 * b * c * cos A

where b = 29° 52' 49.6", c = 29° 25' 47", A = angle aCP = 3° 13' 26.2"

a^2 = (29° 52' 49.6")^2 + (29° 25' 47")^2 - 2 x ( 29° 52' 49.62 x 29° 25' 47") x cos 3° 13' 26.2" = 2.7415

a = 1° 43' 41.47" (not as close)

a^2 = (29.8804444444)^2+(29.42972222222)^2-2*(29.8804444444*29.4297222222)*cos(3.2239444444)

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Thus it is possible, by a knowledge of the second rotation of the earth and of the true course traced by the pole of the heavens, to calculate the polar distance of a star to within a fraction of a second for any number of years in the past or future. Below are given results arrived at by calculation, and compared with recorded observation; these facts speak for themselves.

MEAN NORTH POLAR DISTANCE OP THE POLE-STAR
. Date .. .. Recorded observation .. .. .. .. Calculated.
1887 .. .. .. .. .. 1°17'38" .. .. .. .. .. .. .. .. .. .. 1°17'38"
1873 .. .. .. .. .. 1°22' 4.3" .. .. .. .. .. .. .. .. .. 1°22' 4.5"
1850 .. .. .. .. .. l° 29' 24.7".. .. .. .. .. .. .. .. .. .. 1°29'24"
1819 .. .. .. .. .. 1°39'25" .. .. .. .. .. .. .. .. .. .. 1°39'25"
1755 .. .. .. .. .. 2° 0' 18.9".. .. .. .. .. .. .. .. .. .. 2° 0' 20"

There is probably no star in the heavens which gives so severe a test of the true course of the pole of daily rotation as does the pole-star. Any person acquainted with geometry must know that the annual rate at which the pole decreases its distance from the pole-star is very variable, whereas with some stars the annual rate is nearly uniform.
Consequently, when it is proved that the polar distance of this star can be calculated for a hundred and thirty-two years to within 1", we may claim that a very severe test has been employed.
It has often been claimed, as a proof of the great labour performed at observatories, that over one hundred observations have been made of the pole-star during the year.
There is no denying the greatness of the labour, but the value of this may be questioned when it can be proved that the same results endeavoured to be arrived at by this perpetual observation can be calculated with ease and accuracy.
The reader who takes the trouble to work out the details of the second rotation, not only as regards the zenith of various localities, but also as regards the horizon, where the horizon is cut by various meridians, will find some singular changes in various meridians, not one of which changes has hitherto been known to routine astronomers. These changes, in the majority of instances, may cause a very slight apparent change in the rate of the second rotation for some stars, but the changes are due to a geometrical law.
As one of the principal labours of an observatory consists in that perpetual observation with the transit instrument at present considered necessary to obtain the annual rate of increase or decrease in the polar distance and right ascension of stars, so as to frame a catalogue of stars for two or three years in advance, it may be of interest to observers to be supplied with a list of a few stars, and the data by which their positions as regards their mean north polar distance can be calculated with minute accuracy, without any reference to the annual rate of change now found by continued observation. Such details as regards the pole-star have already been given, and it will be evident that, as the polar distance of this star can be calculated to within 1" for a hundred and thirty-two years, repeated observations of this star in order to predict its distance from the pole at future dates, is a proceeding quite unnecessary. So also is it with a multitude of other stars, and as the effect of refraction is always an item of some uncertainty as regards observations, calculation must give more accurate results, when it becomes known how these calculations ought to be made.

The star Beta Draconis is distant from the pole of second rotation 9° 17' 38" [Click here to see where it is on the star chart]; the angle at the pole of second rotation, between this star and the pole of daily rotation, was on January 1, 1887, 148° 8' 0".

This angle varies at the same rate as the second rotation, viz. 40.9" annually.
We have, therefore, a spherical triangle as follows (Fig. 20), CP.
From pole of second rotation C, to the pole of daily rotation P = 29° 25' 47".
C B, from pole of second rotation to star, 9° 17' 38".
The angle PCB, January 1, 1887 = 148° 8' 0'
variation in this angle annually, 40.9".
From the above data we can calculate the distance PB for any date, in the same manner as the mean polar distance of the pole-star has already been calculated, viz. finding the third side when two sides and the included angle are given.
Putting these items in a concise form as follows, they may the more easily be comprehended :

THE STAR BETA DRACONIS.
PC = 29° 25' 47".
CB = 9° 17' 38".
both constants
Angle PCB, January 1, 1887 = 148° 8' 0"
Annual variation in angle P C B = 40.9"

Calculate the polar distance P B for any other date.
The following results, obtained by calculation from the above data, are compared with the recorded observations at various dates :

Date .. .. Recorded observation .. .. .. .. Calculated.
1887 .. .. .. .. .. 37° 36' 53" .. .. .. .. .. .. .. .. .. .. 37° 36' 53"
1850 .. .. .. .. .. 37° 35' 8 2" .. .. .. .. .. .. .. .. .. 37° 35' 8"
1780 .. .. .. .. .. 37° 31' 47" .. .. .. .. .. .. .. .. .. .. 37° 31' 47"

________________________________________________________________________________

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

We have two sides and the included angle of a spherical triangle to find PB, the third side
CB = 9° 17' 38"
PC = 29° 25' 47"
The angle BCP = 148° 8' 0"

cos(c) = cos(9° 17' 38") cos(29° 25' 47") + sin(9° 17' 38") sin(29° 25' 47") x cos(148° 8' 0")

If you have a programmable calculator, just type in this formula
cos()*cos(29.4297222222)+sin()*sin(29.4297222222)*cos()
and insert the missing values thus:
cos(9.2938888889)*cos(29.4297222222)+sin(9.2938888889)*sin(29.4297222222)*cos(148.1333333333)
It will calculate the result to be in decimals
cos(c) = 0° 47' 31.68" = 0.7921329; convert this to degrees c = acos(0.7921329) = 37.6147165559 = 37° 36' 52.98" degrees .. .. .. .. .. 1887

You can copy and paste angles from clipboard (in this format: 29° 25' 47") into the pocket calculator program Calc98, and then convert them to degrees (to this format: 29.4297222222).

Information about downloading Calc98 at no cost can be found on:

Calculator Web site:

http://www.calculator.org/download.html

Then you can use the programmable calculator 'Math Calculator' [also known as Expression Calculator, or ExpCal] which is a free software, from

http://www.graphnow.com/math-calculator.html

Just paste the values into the formula above, then past the whole complete formula into the Math Calculator 'expression' area making sure there are no spaces (specially at the end) and press 'calculate'.

Angle PCB, January 1, 1887 = 148° 8' 0"
Annual variation in angle P C B = 40.9" per year
1887 - 1850 = 37 years
37 x 40.9" = 1513.3" = 0° 25' 13.30"
The angle BCP = 148° 8' 0" - 0° 25' 13.30" = 147° 42' 46.70" in 1850

Since CB and PC are both constants only the angle changes:

cos(c) = cos(9° 17' 38") cos(29° 25' 47") + sin(9° 17' 38") sin(29° 25' 47") x cos(147° 42' 46.70")
cos(c) = cos(9.2938888889)*cos(29.42972)+sin(9.2938888889)*sin(29.4297222222)*cos(147.7129722222)
It will calculate the result to be in decimals
c = acos(0.7924421) = 37.5856812108 = 37° 35' 8.45" .. .. .. .. .. 1850

Now for 1780 with, again, only the angle changing:

Angle PCB, 1780 = 148° 8' 0"
Annual variation in angle PCB = 40.9"
1887 - 1780 = 107 years
107 x 40.9" = 1° 12' 56.30"
The angle BCP = 148° 8' 0" - 1° 12' 56.30" = 146° 55' 3.70" in 1780 .. .. .. .. .. 37° 31' 47" .. .. .. .. .. .. .. .. .. .. 37° 31' 47" cos(c) = cos(9° 17' 38") cos(29° 25' 47") + sin(9° 17' 38") sin(29° 25' 47") x cos(147° 42' 46.70")
cos(c) = cos(9.2938888889)*cos(29.42972)+sin(9.2938888889)*sin(29.4297222222)*cos(146.9176944444)
c = acos(0.7930369) = 37.5297728406 = 37° 31' 47.18" .. .. .. .. .. 1780

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The reader who will take the trouble to investigate these facts will perceive that each star will have two " constants " which do not vary, viz. the distance of this star from the pole of second rotation, and the distance of the pole of second rotation from the pole of daily rotation, this last named distance being 29° 25' 47".
The angle formed at the pole of the second axis of rotation by the two arcs above named varies in consequence of the second rotation, but this variation can be calculated, and the angle obtained for any year. Hence the polar distance of the star can be calculated for any year when the above items are known.
The " constants " of a few stars will be found below, by aid of which the polar distance of these stars can be calculated without any reference to their " rate " found by observation, a fact which proves
(1) that the time, labour, and expense now devoted to the perpetual observation of these stars is unnecessary ; and
(2) that, unless the true course of the pole be known, such accurate calculations would be impossible. When these facts become known to the reader, he will be able to estimate the relative value of the present orthodox theory, and of the second rotation of the earth.
He may consider whether the theory that the earth's axis traces a cone round the pole of the ecliptic as the centre, from which centre it continually decreases its distance, is a clear description of how various parts of the earth move in accordance therewith. He may consider whether the theory that " the joint action of the sun and moon on the earth's protuberant equator " is a sufficient explanation to enable him to define exactly how each zenith is displaced during the year by this movement. If he be a reasoner, he may probably inquire why, if the exact movement of the pole of daily rotation and of other parts of the earth be known, it is necessary to spend many thousands of pounds per annum for the income of observers, whose principal occupation is to pass night after night with the transit instrument, in order to find the changes which annually occur in the polar distance and right ascension of stars, so that a catalogue of these stars can be made out for two or three years in advance. These and probably many other similar questions may occur to those persons who look to practical results rather than to dogmatic theories.
The following " constants," and the results obtained from these, are now submitted for investigation :
[Click here to see where the following stars are on the star chart];
We have two sides and the included angle of a spherical triangle to find PB, the third side

________________________________________________________________________________

CB = 21° 50' 12"; PC = 29° 25' 47"
The angle BCP = 31° 34' 27.3" on Jan. 1st 1887

cos(PB) = cos(21° 50' 12") cos(29° 25' 47") +
sin(21° 50' 12") sin(29° 25' 47") x cos(31° 34' 27.3")

cos(PB) = 0.808466+0.1827656*0.85196234 = 0.9641754
PB = acos(0.9641754) = 15° 22' 57.91"

In 1873 the angle BCP becomes:
1887 - 1873 = 14 years; 14 x 40.9" = 0° 9' 32.60"
BCP = 31° 34' 27.3" - 9' 32.60" = 31° 24' 54.70"
cos(PB) = 0.808466+0.1827656*cos(31.41519) = 0.9644404
PB = acos(0.9644404) = 15.3254 = 15°19'31.39"

1887 - 1850 = 37 years; 37 x 40.9" = 0° 25' 13.30"
BCP = 31° 34' 27.3" - 25' 13.30" = 31° 9' 14.00"
cos(PB) = 0.808466+0.1827656*cos(31.1538888889) = 0.9648733
PB = acos(0.9648733) = 15° 13' 52.54"

1887 - 1819 = 68 years; 68 x 40.9" = 0° 46' 21.20"
BCP = 31° 34' 27.3" - 46' 21.20" = 30° 48' 6.10"
cos(PB) = 0.808466+0.1827656*cos(30.8016944444) = 0.9648733
PB = acos(0.9654515) = 15° 6' 16.73"

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[Click here to see where it is on the star chart]

[Click here to see where it is on the star chart]



















[Click here to see where it is on the star chart]

We have above a few stars whose "constants" are given, by aid of which the polar distance of these stars can be calculated from one observation only. No reference need be made to the annual rate of variation in polar distance hitherto arrived at only by long and repeated observation. A true geometrical calculation, based on a knowledge of the second rotation of the earth, enables any person to arrive at results which hitherto have been unattainable.
When comparing the results as regards the mean polar distance of stars found by calculation with the recorded mean polar distances found by observation some fifty or a hundred years in the past, a cause of error in the ancient records is brought into notice. This error is due to the erroneous value given to refraction by the ancient observers. The following table shows the correction used by the olden observers for various altitudes, and these " corrections " can be compared with the tables of refractions now generally used. As, however, the amount of correction due to refraction varies with the height of the barometer and the thermometer, and also with the amount of moisture in the air, for which latter item no exact allowance can be made, there must and will ever be a small amount of uncertainty even in modern times as regards the true allowance to be made for refraction. Consequently calculation must by reasoners ever be considered more reliable and correct than mere instrumental observation. The following table will show how varied were the views of ancient observers as regards the value of refraction for different altitudes :

When we find that between even Halley and Bradley there was a difference of 3" for the allowance for refraction for 45 altitude, we need not trouble ourselves much when we find that the calculated polar distance of a star differs from the observed polar distance at the dates 1780 or 1755 even as much as 4" ; we leave it to the reader to conclude which is the more likely to be correct, calculations, or observations made with imperfect instruments and corrected by assuming an incorrect value to refraction.
When dealing with the second rotation of the earth, there are several geometrical problems requiring the greatest care before we can correctly calculate results by aid of a knowledge of this movement. Although the polar distance of a star has been referred to, this polar distance is deduced from the meridian zenith distance. We have, consequently, to determine how the zenith and how the meridian are affected by the second rotation, before we can make our calculations with certainty.
As an example of one among many of these problems, the following case will serve. It is a geometrical law that the zenith of a locality north of the equator of the earth is carried by the daily rotation over a less arc in a given time than is a locality on the equator. When we refer to the second rotation, this law holds good in its general principles, but the zenith of a locality. when this zenith is on a meridian of six hours right ascension, and has a latitude of 29° or thereabouts, will be carried annually by the second rotation over a greater arc than will the zenith of a locality on the terrestrial equator, the first-named zenith being nearly on the equator of slow rotation, whereas the zenith of a locality on the equator of daily rotation being 29° 25' 47" from the equator of slow rotation.

Again, when zeniths are near the pole of second rotation, very varied results will occur in the changes of these zeniths produced by the second rotation, both as regards amount and as to their direction. For example (Fig. 29), suppose C the pole of the second axis of rotation, P the pole of daily rotation, A, B, D, E, F, the . zeniths of various localities on earth. Whilst the pole of daily rotation is carried from P to round C as a centre, the zenith B is carried to B' round C as a centre, A to A', E to E', etc., the movements of all these zeniths being round C as a centre. These zeniths consequently differ considerably not only in the direction in which they are carried by the second rotation, but also in the length of the arcs over which they are carried.
In addition to the above-named problems, we have to consider how the equator of the earth is affected at various points by the second rotation, and also how that portion of the meridian between the zenith and horizon is affected by the second rotation.
Those gentlemen who are fully satisfied with the completeness of the theory that it is the joint action of the sun and moon that causes a conical movement of the earth's axis, have probably worked out all these calculations. If they have done so, they have kept them secret and have made no practical use of them, for they still continue employing numbers of observers in order to arrive at results for a few years in advance, which results could be calculated for one hundred years in advance with far greater accuracy. Although the mean rate of the second rotation is 40.9" per annum, the changes produced in the zenith and meridian under certain conditions causes this rate, as regards certain fixed stars which have no independent motion of their own, to appear to vary a fraction of a second of arc per year.
In order that the reader may understand the small amount in time represented by a fraction of a second of arc when referred to the daily rotation, and hence to the time of meridian transit of stars, he is referred to a table by which arcs are converted into time ; he will there see that 10" of arc = 7/10ths of a second in time, consequently 2" of arc = about 1/10th of a second in time.
Those who may have had considerable experience with chronometers will probably admit that it is rare to find such an instrument which can be relied on for 1/10th of a second per year. When, however, we come to calculations and to many years, such a minute difference as 1/10th or 9/10ths of a second of time can be dealt with.

Several important stars will now be referred to, their constants given, and the rate at which the angle at the pole of second rotation appears to vary in consequence of the movement of the zenith and meridian as regards these stars.
With the information thus given, the polar distance of any of these stars can be calculated for fifty years or more, without any further aid from observations, and without any reference to the annual variation in polar distance now obtained by observation. In each case the mean polar distance on January 1 of the year named is the item referred to.

These records might be multiplied by the hundred, showing how the polar distance of stars which have no independent motion of their own can be calculated with minute accuracy when the details of the second rotation are known. It may be of interest to watch how many more years official observers will continue, night after night, observing stars in order to obtain the means of framing a catalogue for a few years in advance, when in reality such a catalogue can be calculated independent of any further observations.
In order that the reader may understand how very simple this problem of determining the polar distance of a star becomes by means of a knowledge of the second rotation of the earth, the following items are given, which are sufficient to enable a person capable of working out a spherical triangle to calculate the mean polar distance of a star to one-tenth of a second without any reference to further observation.

From the above data (fig 33.) the polar distance of this star can be calculated for one hundred years or more, past or future.
To continue making observations night after night of this star is much the same proceeding as though a surveyor day after day measured inches on the earth's surface in order to find out how many inches there were in two miles.

Here is another star, viz. a Cephei (Fig. 34) :
Angle P C a, January 1, 1887 = 64° 42' 13'2
Annual variation in angle P C o = 40 - 823"
Find the polar distance of this star for any date, past or future.
The reader, if capable of working out a spherical triangle, is recommended to test these statements for himself ; he will, after such tests, not remain in doubt as to whether the polar distance of a star can be calculated to within a fraction of a second, when this star has no independent motion of its own. He will be able to prove that he can calculate this item, and he will realize the fact that those who cannot do so, but are compelled to be perpetually observing, cannot be acquainted with the true movements of the earth.

CHAPTER V.

THE PRECESSION OF THE EQUINOXES, AND THE DECREASE IN THE OBLIQUITY.

WHEN the science of geometry is again taken up, and geometricians realize the fact that dogmatic theories can no more put geometry on one side than these theories can ignore the multiplication table, the fact will become recognized that the statement made, and now accepted as correct and complete, in connection with the precession of the equinoctial point, is one of the most remarkable examples on record, of geometrical contradictions and incomplete reasoning.
We are informed by various writers, who copy one another, that the precession of the equinoctial point produced by "a conical movement of the earth's axis" amounts to 50.1" annually; therefore, write these gentlemen, "as the amount is 50.1" for one year, this is at the rate of 360° for 25,868 years, which is the period occupied by an entire revolution of the equinoxes."

[The Astronomical Almanac in 2002 gave 50.29164"/year as the value]

In order to make manifest the errors and unfounded conclusions now prevailing in connection with this problem, attention will be called to the cause which produces the precession, and also the geometrical laws affecting the rate per year.

At the period when the sun's centre is 90° from the poles of the earth, during the month of March, the vernal equinox takes place. The following diagram will show the course that the earth has followed in order to reach this position, and the cause which produces the precession.

The circle (Fig. 35) represents the earth; N, the north pole; C, the earth's centre; E R, the equator; T C, a portion of the earth's course round the sun, termed the ecliptic. At this period the sun is 90° from the pole N, therefore it is over the equator, the date being when the vernal equinox occurs.
If the axis of daily rotation of the earth were now slightly turned, so that the pole N were moved over towards the sun 20.09", then the pole N would not be 90° from the sun, but would be 90° less 20.09".
In order that the pole N should now be 90° from the sun, we should have to move the earth up the ecliptic C T, until it reached a point where T Q was 20.09", and where consequently the sun was 90°, from the pole N.
The distance that we have to move the earth up the ecliptic C T is dependent on three items :
(1) the amount of the change in direction annually of the earth's axis ;
(2) the exact direction in which this change takes place ;
(3) the value of the angle formed at the time between the plane of the earth's equator and the plane of the ecliptic, technically termed the obliquity of the ecliptic.
When we know the exact direction in which the pole moves, the exact amount of this movement annually, and the exact value of the obliquity of the ecliptic at the date, we can calculate the precession per year for that date ; but we must not assume that, because we find this precession of a given value for any one year, we can obtain the whole period by a mere rule-of-three sum. Why we cannot do so will be fully explained further on.
The calculation for obtaining the annual value of the precession, when we know the three items referred to, is very simple, and is as follows.

Suppose A B (Fig. 36) the amount of polar movement annually, say 20.09" ; A C B the obliquity of the ecliptic, say 23° 28'. The arc A C, which is the arc between two successive vernal equinoxes, can be calculated as follows, ABC being a right-angled spherical triangle :
Log. sine + Radius AB, viz. 20" = 15.9866049
Log. sine of angle ACB, viz. 23° 28' = 9.6001181
A C = 50.2" = 6.3864868

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Again, I dont understand his method, but I got a similar answer using the formula sin(b)=sin(B)sin(c), for a right-angled spherical triangle

angle ACB = 23° 28'; AB = 20"

sin(AB)=sin(ACB)sin(AC), or sin(AC)=sin(AB)/sin(ACB)
sin(AC) = sin(20") / sin(23° 28') = sin(0.0055555556) / sin(23.4666666667)

AC = asin(0.0002434931) = 0.0139511271 = 0° 0' 50.22"

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Consequently, under such conditions we should obtain an annual precession of 50.2". Now let us call attention to the important facts in connection with this problem which have hitherto been overlooked.
As long as the pole is carried annually over an arc of about 20.09", and nearly or exactly towards the first point of Aries, we obtain an annual precession of about 50.2", as shown by the above calculation. If, however, the radius of the circle which the pole traced in the heavens were only 10, or if it were 40, we should obtain exactly the same value, viz. 50.2" for the annual precession as long as the pole moved 20.09" annually, and the obliquity was 23° 28', or very close to this amount. If, however, the radius of the circle which the pole traced were 10 only, this circle would be completed in about 11,270 years found by the formula

PP'
______ = EQ
cos 80

If, however, the radius of the circle round which the pole traced its circle of 20'09" annually were 40, then this circle would be completed in about 43,200 years. Now, the time occupied by the pole of the heavens in completing its circle, is the time occupied by an entire revolution of the equinoxes.
This entire revolution is not to be assumed from the rate of the precession at any given date, but is due to, and must be calculated by, the time occupied by the pole in tracing a complete circle in the heavens, and consequently must be calculated on the knowledge of the radius of the circle which the pole really does trace. As this subject is at the present time in absolute confusion, some additional examples will be given, showing the manner in which it can be correctly dealt with, and the geometrical laws that bear upon it.
It is a geometrical law that the arc joining the pole of the earth's daily rotation with the pole of the ecliptic will give a small portion of the arc of what is termed the solstitial colure. This arc being produced each way until it cuts the ecliptic, will indicate those points on the ecliptic where the summer and winter positions of the earth occur.

In the following diagram, T Q I L represents the plane of the ecliptic ; E, the pole of the ecliptic; P, the position of the pole of the heavens at a date in the past. P E = 23° 28'. The arc joining P E and produced to T and I would give the position of the solstitial colure on the sphere of the heavens. The point I on the ecliptic would be the position which the earth would occupy at the period of the winter solstice to the northern hemisphere ; T would be the position which the earth would occupy on the plane of the ecliptic at the period of the summer solstice.
Let us now take P O as a portion of the arc traced by the pole of the heavens during many hundred years round the point C as a centre, and let us assume the radius C P of this arc to be only 10° and the pole P to be carried along this arc at the rate of 20" annually, equal to 1 in one hundred and eighty years.
The movement of the pole of 20" when the obliquity was about 23° 28' would, whilst the pole was moving from P to O, and when the obliquity P E and O E varied by only a few minutes, give an annual precession for the equinoctial point of about 50", found as before described. The course of the pole moving 20" annually round the small circle of which C, 10° only from P, is the centre, would be completed, as before shown, in 11,270 years.
Because the annual precession happens to be about 50" at a given date, it is an utterly incorrect assertion to state that therefore the whole revolution of the equinoxes will occupy 25,868 years.
The period during which an entire revolution of the equinoxes occurs depends on the radius of the circle which the pole of daily rotation traces in the heavens, and the annual rate at which the pole moves round this circle.
The annual value of the precession depends on the obliquity of the ecliptic at the date, and on the amount and direction of the polar movement. This annual rate at any particular date will not, however, give us any data by which to calculate the whole period of a revolution of the equinoxes. In order to calculate the whole period, we must know the true radius of the circle which the pole of daily rotation traces on the sphere of the heavens.
In order that this most important fact should be thoroughly understood by the reader, the last diagram will be referred to, and the point x, 40° from P, will be assumed as the centre of the circle traced by the pole P at the rate of 20" annually. The pole moving from P to O round x as a centre would trace on the sphere of the heavens a very slightly different course during some one thousand years from that which it would trace round C as a centre.
This slight difference would be indicated by small variations in the distances P E and E 0, which is the value of the obliquity, and also in small variations in the polar distances of some stars.
As long, however, as the obliquity P E varied but slightly from 23° 28', and the pole moved along its arc at the annual rate of 20", the annual rate of the precession would be about 50". But an entire revolution of the equinoxes would, with x as the centre 40 from P, occupy 43,200 years.
When, then, we find it stated by writer after writer who ventures to deal with this subject, that because the annual precession now takes place at the rate of 50.1" annually, therefore it will occupy 25,868 years to complete one revolution of the equinoxes, we may realize the fact that it is very easy to copy errors. Any person, however, who is acquainted with the laws of geometrical astronomy will at once perceive that, as regards this problem, there has been rather too free a use of gratuitous and erroneous assumptions.
The only conditions under which it would be possible to calculate the whole period of a revolution of the equinoxes from the rate found from one year, would be that the pole of the heavens traced a circle round the pole of the ecliptic as a centre, and at a uniform rate, and consequently, as a geometrical law, the distance between the pole of the heavens and the pole of the ecliptic never varied.
The distance between the pole of the heavens and the pole of the ecliptic must be, as a geometrical law, of the same value as the obliquity of the ecliptic. If, then, the pole of the heavens does trace a circle round the pole of the ecliptic as a centre, no variation can occur in the obliquity. We have, consequently, a very simple problem to investigate, viz. whether there has been, during the past two thousand years, any variation in the obliquity of the ecliptic. If there has not been, then the pole of the heavens traces its circle round the pole of the ecliptic as a centre. If there has been any variation, then the pole of the heavens cannot trace its circle round the pole of the ecliptic as a centre, but must trace its circle round some other point as a centre.
It may appear little short of marvellous to the reasoner who is unacquainted with the past history of astronomy, and is consequently unaware of the tenacity with which dogmatic theories, as regards the earth being a flat surface and being immovable, were clung to by the authorities in remote ages, when he realizes the fact that it has been known during more than two hundred years that the obliquity of the ecliptic has been found to decrease, during the past two thousand years at least. This fact has been long known to astronomers, and yet they continue to assert that the pole of the heavens traces a circle round the pole of the ecliptic as a centre. Persons who cling to this belief occupy, relative to geometry and astronomy, the same position as do those who assert that the earth is a flat surface and is not spherical in form, for they consider their preconceived opinions far more sound and true than the rigid laws of geometry.
Recorded observations prove that the pole of the heavens cannot trace a circle round the pole of the ecliptic as a centre, consequently all the theories and calculations based on the belief that it does do so are unsound and incorrect.
The pole of the heavens has decreased its distance from the pole of the ecliptic during the past two thousand years at least, consequently the distance between these two poles has decreased during the same period, and the pole of the ecliptic cannot be the centre of the circle which the pole of the heavens traces in its circular course. Let us now bring to bear on this hitherto confused and contradictory theory, the fact of the second rotation of the earth round an axis inclined to the daily axis of rotation at an angle of 29° 25' 47". By this second rotation, it has been already proved that we can calculate the polar distance of a star for one hundred years at least from one observation only, a proceeding hitherto considered impossible by theorists.
This second rotation of the earth will enable those who will take the trouble to examine it, to calculate the value of the obliquity with the same facility by which they could calculate the polar distance of a star, the two problems being, in fact, almost identical.
We find that the pole of the heavens decreases its distance from the pole of the ecliptic. We know the radius of the circle which the pole does trace, and we know the position of the pole of the second axis of rotation; with such data the value of the obliquity can be calculated for any date.
The following diagram will show the method of making this calculation, independent of any observations. We can therefore check the observations made at any observatory, and note whether these have been* correctly made, instead of being dependent on these, as is now the case. Let P be the pole of the axis of daily rotation, C the pole of second rotation;

P C = 29° 25' 47" (Fig. 39).

At the date 2295.2 AD, the pole P will have been carried to O round C as a centre.

The angle of second rotation at C varying at the rate of 40.9" annually.
E represents the position of the pole of the ecliptic, and at the date 2295.2 AD, CE, and O will be on the same meridian of right ascension.
The distance of the pole of the ecliptic E from any given point in the arc xPO will give the angular distance between the pole of the ecliptic and the pole of the heavens, when this latter pole is at that point. Thus E x will be the distance between the pole of the ecliptic and the pole of the heavens when this latter pole is at x, E P their distance when the pole is at P, and so on.
It being a geometrical law that the angular distance between the pole of the ecliptic and the pole of the heavens is always of the same value as the obliquity of the ecliptic, it follows that when the pole was at x, E x indicated the obliquity ; when the pole was at P, E P represented the obliquity ; and so on.
To find the value of the obliquity for any date now becomes a very simple calculation, inasmuch as C E is 6°, C P = 29° 25' 47", and the angle E C P a variable which can be obtained for any date. We have therefore two sides and the included angle, and we can therefore calculate the third side, which is the obliquity.
For example, suppose we wish to calculate the obliquity for any year, say January 1, 1887.

Subtract 1887 from 2295.2 and we obtain 408.2 years, during which the second rotation progresses at the rate of 40.9" annually. Multiply 408.2 by 40.9" and we obtain for the angle at C, January 1, 1887, 4° 38' 15.38". With this included angle and the two sides, viz. CE = 6°, and C P = 29° 25' 47", the third side PE, the obliquity, can be calculated. This example is worked out in detail below, so that the reader may test other cases for himself.

[I don't understand this method, but I got a similar answer using the cosine rule

a^2 = b^2 + c^2 - 2 * b * c * cos A

where b = CE = 6° = 21600"
…….. c = CP = 29° 25' 47" = 105947"
…….. A = angle at C = 4° 38' 15.38" = 4.6376055556°

a^2 = 21600^2 + 105947^2 - 2 * 21600 * 105947 * cos 4.6376055556

a = 23° 27' 15.7808925" (which is close to 23° 27' 14.22")]

By this calculation, independent of all observations, we find the mean obliquity for January 1, 1887 = 23° 27' 14.2". The mean obliquity for January 1, 1887, recorded in the Nautical Almanac was 23° 27' 14.22".
Any other date can be taken and the obliquity calculated quite independently; say, for example, the date January 1, 1800.
Without any reference to the former calculation or to any observations, we can, by a knowledge of the second rotation of the earth, and the true course traced in consequence of the second rotation by the pole of the heavens, calculate the mean obliquity of the ecliptic for January 1, 1800. This mean obliquity is nothing more than the angular distance of the pole of the heavens from the pole of the ecliptic.
The whole detail working of finding the obliquity for January 1, 1800, will be given, so that the reader may become conversant with the method, which is merely a repetition of the former example, 1800 being substituted for 1887.

When the reader has examined and knows how to calculate this problem, he will be able to accomplish more in a few minutes than all the observers and theorists have been able to arrive at since astronomy has been treated as a science. It is, therefore, quite worth the expenditure of a little time and thought, to master this simple problem.

To find the mean obliquity of the ecliptic for January 1, 1800.

1800 taken from 2295.2 leaves 495.2 years.

495-2 multiplied by 40.9" equals 5° 37' 33.68", which will be the angle at C, the pole of second rotation for January 1, 1800.

We then have two sides, viz. C E = 6 and C P = 29° 25' 47", and the included angle at C, to calculate the third side P E, which is the obliquity for January 1, 1800.

Here are the detail results of the calculation :

Log. cosine, 5° 37' 33.68" = 9.9979030
Log. cosine, 5° 37' 33.68" = 9.9979030

qq 19-9601490 - Log cosine, first arc = 9 9976372 9-9625118 = log. cos. 23 27' 55-3"= P E The mean obliquity, therefore, for January 1, 1800, was 23° 27' 55-3". The result arrived at by observers at that date was 23° 27' 551". Consequently from 1800 to 1887 the obliquity found by calculation has decreased 41*1". By the results of endless observations it has decreased 40.9", a difference of ^ of a second for eighty-seven years. It may be left to the judgment of the reader as to which is the more reliable, personal and instrumental observation, with the uncertainty always belonging to refraction, or a rigid mathematical calculation. By calculations similar to those made above, the follow- ing results have been arrived at for the mean obliquity : Date. Calculated obliquity. January 1, 1750 23° 28' 23" . 1800 23° 27' 55-3" 1850 23° 27' 30 9" 1900 23° 27' 8 8" . It will be evident to a geometrician that it is impossible that this decrease in the rate can be uniform. There cannot be the same amount of decrease between 1800 and 1850 as there will be between 1850 and 1900. The calculations agree with this law. For example, between 1800 and 1850 there was a decrease of 24'4", which gives a mean rate of 0-488" per year. Between 1750 and 1800 there was a decrease of 277", which gives a mean rate of 0'554" per year. Between 1850 and 1887 there was a decrease of 16 7", which gives a mean rate of 0'451" per year. A knowledge of this variation in the rate of the decrease in the obliquity is most important, especially when the cause of this variation is seen to be a geometrical law. Hitherto theorists have imagined the rate to be constant, a theory impossible according to the laws of geometry, and one that is contradicted by recorded observations. The reader now possesses the knowledge by which he can calculate the value of the obliquity for thousands of years, without any reference to observations, and he can calculate the rate per year of this decrease with the same ease. For example, suppose we wish to calculate the value of the obliquity for January 1, 890 A.D.; subtract 890 from 2295*2, and we obtain 1405'2 years, which, multiplied by 40-9", gives 15 57' 52-68" for the angle at C for 890 A.D. Adopting the same calculation as before, and we obtain for the obliquity for 890 A.D. 23° 42' 48". By substituting 990 for 890 and calculating as before, we obtain for 890 an obliquity of 23° 40' 30", which is at the rate of 2' 18" per hundred years for that date. It is a remarkable exhibition of the small amount of knowledge which has hitherto been possessed on this im- portant problem that theorists should assert that, because the obliquity between certain dates was found by repeated observations 48" for 100 years, therefore they were correct when they assumed it would amount to 480" for 1000 years, and 4800" for 10,000 years. When endeavouring to find what was the value of the obliquity arrived at by observers at distant dates, there are two sources of error which prevent these recorded observa- tions from being quite reliable. The first of these is the erroneous refraction by which the altitude of the sun was corrected; the second is that in ancient times the instru- ments used were not capable of measuring an angle with greater accuracy than 10', or at most 5'. In Ptolemy's catalogue of stars, he deals with nothing less than 5'. As regards the refraction, we will suppose an observer at Greenwich at the date 1690, and he attempted to find the obliquity. In the first place, at that date he did not know the difference between the mean obliquity and the apparent obliquity, the effect of the nutation being then unknown. Suppose he measured the greatest meridian altitude of the sun at the summer solstice, and the least meridian altitude at the winter solstice, and, having cor- rected these altitudes by the table of refractions that were used by Halley or Newton, he made the difference between the greatest and least altitude of the sun 46 57' 36". The half of this amount, viz. 23 28' 48", would be given as the value of the obliquity. The refraction used by Halley for the greatest altitude was nearly 2" too little, whilst the refraction used for the least altitude was 18" too little. Hence, if the correct refraction had been used, the difference between the sun's greatest and least altitude during the year 1690 would have been 16" more, that is, 46 57' 52", and the obliquity consequently would have been given as 23 28' 56", not as 23 28' 48". It may appear a startling announcement to make to the reader, that he would by calculation discover that an astro- nomer two hundred years ago either used an erroneous table of refraction, or made an error in his observations amounting to nearly 20". This announcement is, however, true. Take the date 1690 from 2295'2, and multiply by 40.9", and we obtain 6 52' 32*68" for the angle at C. Proceed as in the former calculations, and we obtain 23 28' 58" for the mean obliquity for 1690, within 2" of what the results ought to be by observation when the correct refraction was used. The obliquity recorded as having been found by observers about this date, even with their erroneous refractions, are as follows : La Hire, 1681 .. 23 28' 51" Picard, 1686 23 28' 50" Wertzelbaur, 1686 23 28' 53" A very excited discussion took place between the French and English astronomers relative to the mean obliquity of the ecliptic for the date 1775. The French astronomers asserted that they had, by the most perfect observations, ascertained it to be 23 27' 48". The English astronomers with equal confidence claimed that they had found it 23 27' 59". M. Le Lande, in his astronomical tables, gives 23 28' 0" for the obliquity for 1775. The second rotation of the earth and the true movement of the pole of the heavens being then unknown to the mere routine observer, each of the authorities held to their own opinions. The reader, by the knowledge he has gained from the preceding pages, can prove that the English were more nearly correct than the French, erroneous refraction (prob- ably that used by Bradley) causing them to be only 4" wrong, Bradley using 3' 30" instead of 3' 34" for the correc- tion due to an altitude of 15. Taking 1775 from 2295'2, and multiplying by 40.9", we obtain 5 54' 36" for the angle at C for January 1, 1775, and, w r orking as before, we obtain 23 28' 3'5" for the mean obliquity for that date; so the error due to refraction of 4" is very creditable to the English observers, for had they used a modern table of refraction they would have come within 1" of the correct obliquity at that date. An examination of the facts mentioned in the preceding pages will show that the very same movement of the earth which enables a geometrician to calculate the polar distance of a star for each year to one hundred years or more, also enables him to calculate the obliquity of the ecliptic for thousands of years. At the present time no theorist is able to calculate either of these items; he is compelled to depend on repeated observations, and when, by these observations, he has obtained an annual rate for the increase or decrease in the polar distance of a star, he adds or subtracts this rate, and so gives approximately the position of this star for two or three years in advance. By observation, also, he finds approximately what is the annual decrease in the obliquity, and then he subtracts this decrease, and so gives the obli- quity for two or three years in advance. No real method of calculation has hitherto been known, because the true movement of the earth has not been known. THE OBLIQUITY. 79 To add or subtract a constant quantity from year to year to the polar distance of a star in order to assign to this star a polar distance for some future date, is erroneous in principle, opposed to the elementary laws of geometry, and would give incorrect results in practice, and it is well known to observers that such is the case. When, however, the decrease in the obliquity is dealt with, a decrease due to exactly the same cause as that which produces the decrease in the polar distance of a star, this decrease is treated as a constant quantity, and from year to year the obliquity is decreased by exactly the same amount, viz. 0*476", an item extracted from certain solar tables. So utterly unacquainted with the geometrical law relative to the decrease which occurs when a curve approaches a point have the observers of the past proved themselves to be, that they have asserted that, as the obliquity decreased 47" per century, it would decrease 470" in 1000 years, and 4700' = 1 18' 20" in 10,000 years. Now, the rate in the decrease of the obliquity between 1800 and 1850 was 24*4", being at the mean rate of 0*488" per year, but no two consecutive years would have shown the same rate exactly. At the date 1800 the annual rate of decrease was greater than 0*488". At 1850 it was less than 0-488". Between 1850 and 1900 the mean rate will be 0'442". What are the practical results of adopting this erroneous system of subtracting a constant quantity from year to year, when the quantity is a variable, will be manifest to any geometrician. The error, although small for one year, will go on from year to year until it becomes too large to be ignored; then the accumulated error will have to be wiped out, and a fresh start made. This proceeding was adopted between 1861 and 1865, by altering the assumed rate from 0'457" per year to 0*557", and then returning again to 0*476". Consequently, between the mean obliquity for 1861 and that for 1865 there was made a difference of 2-23". The reader can now prove how and why this error oc- curred. He can calculate the mean obliquity for January 1, 1865, and he will find that between 1850 and 1865 the distance P E decreased 6*9", which gives a mean rate of 0*46" per year between these dates. Previous to 1850 the rate was nearly 0*48" per year. Consequently for many previous years, when only 0*457" was subtracted, a less amount was taken away than was correct, therefore an adjustment was necessary to set matters right. Between 1887 and 1900 the mean rate will be 0'415", consequently the present assumed rate obtained from solar tables is slightly too large, and though the present erroneous rate of 0'476", taken from solar tables, will cause an error of only y'Ju of a second per year, yet this error, if continued, will in twenty-five years require another adjustment to be necessary. The principle, however, of using a constant quantity for a cor- rection is so palpably erroneous, that it is surprising how such proceedings can be adopted by any person claiming to be acquainted with even the elementary laws of geometry. It being borne in mind that the rate of the decrease in the obliquity is a variable, just as is the rate in decrease in the polar distance of a star, the reader is not likely to fall into the same error as theorists have hitherto committed. The following rates may therefore be given to show how the obliquity decreases between certain dates : Between 1850 and 1865 1865 1887 1887 1900 1800 , t 1850 0-46" annually. 0-445" 0-415" 0-488" 0-554" These rates cannot be used in order to correct the observed obliquity of any one year for the year following, because THE OBLIQUITY. 81 these rates will not hold good for any two years in succes- sion. Thus, although the mean rate between 1800 and 1850 may be 0*488", this was not the actual rate at either date. The rate at 1800 was greater than 0'488", and at 1850 it was less. The reader who wishes to find the mean obliquity for any date need not adopt the erroneous system of subtract- ing a constant quantity from year to year, but he can work out each mean obliquity for any year, not only independent of all observation, but independent of any past recorded obliquity; thus he can calculate what was the obliquity at the date 2000 B.C. as easily as he can calculate what it will be at 2000 A.D., or at any other date. We may now refer to some of the important matters which can be calculated when we know that the earth has a second rotation, and when we are acquainted with the true move- ment of the pole of the heavens. First, we can calculate the varied effects produced by the second rotation on the zeniths of every locality on earth. Secondly, from one observation only we can calculate the polar distance of a star for one hundred years or more, without any reference to the annual rate found by repeated observation. Thirdly, we can calculate the value of the mean ob- liquity without reference to any observations, or any sup- posed rates now found by observations. Fourthly, we can calculate the value of the precession of the equinoxes for any year, by first finding the obliquity for that year, and then using the formula given on a previous page. These results now obtainable with minute accuracy by calculation, have hitherto been arrived at with uncertainty by long-continued and expensive observation. Several other important matters revealed by the second rotation will be dealt with in future pages. G 82 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. CHAPTER VI. SOME RESULTS OF THE SECOND ROTATION OF THE EARTH. REFERENCE will now be made to some of the methods hitherto considered essential, and the only means known, for obtaining some of the results which can be obtained by very simple calculation. Obtaining the zenith distance of the sun, and also the obliquity of the ecliptic, will first be described. In a large official observatory the instrument generally used is a large transit instrument, fixed in the plane of the meridian, and being capable of moving in a vertical plane. The latitude of the observatory having been correctly ascer- tained, the following items become known, viz. the altitude of the pole of daily rotation, which is of the same value as the latitude of the place of observation; the meridian alti- tude of the equinoctial, which is of the same value as the complement of the latitude. Hence, if the latitude of the observatory were 51 28' N., the altitude of the pole would be 51 28', and the meridian altitude of the equinoctial would be 90 - 51 28' = 38 32'. Each day that the sun crosses the meridian, its zenith distance or altitude is measured by the transit instrument, and this measured distance is corrected for semidiameter refraction, parallax, and any known instrumental errors, and the true zenith distance or altitude of the sun is then obtained, and its rdtitude above or below the equinoctial SOME RESULTS OF THE SECOND ROTATION. 83 will give the sun's declination north or south for the instant at which the observation was made. In order that the value of the obliquity should be found for any year, it is necessary that we know the greatest dis- tance that the sun reaches north and south of the equinoctial, and as this condition may not occur at the instant that the sun crosses the meridian of the observatory, the following method is adopted. The declination of the sun is measured for each day, for two or three days before and after the day on which the greatest north and south declination occurs, and by a simple calculation the greatest north and south declination can be arrived at; or, in other words, the differ- ence between the greatest and least altitude of the sun during the year is observed. This amount divided by two gives the obliquity. If, then, the difference between the greatest and least altitude were 47, the obliquity would be 23 30'; if the difference were 46 56', the obliquity would be 23 28', and so on. The obliquity thus found is not the mean obliquity. It is the mean obliquity affected by a small movement of the earth's axis termed the nutation, this nutation completing one complete cycle in about nineteen years. The amount of this nutation is only a few seconds, and, its effects being known, the mean obliquity can be calculated from the observed obliquity for any year. Practically, therefore, it may be said that the method of finding the value of the mean obliquity is to measure the greatest and least altitude of the sun during the year, to divide the difference by two, and make the necessary allowance for nutation. The results thus obtained are liable to be affected by the following items : First, by personal errors of observation. Secondly, by instrumental errors. 81 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Thirdly, by the uncertainty of ro/raction at the time of observation. Although the probability of the first two causes pro- ducing errors may be very slight, yet they exist. The prob- ability of error from the third cause, viz. refraction, is very great even in the present day; whilst fifty and more years in the past, the allowance made for refraction was very incorrect, as has been shown by the different values adopted by different observers. Sir Isaac Newton, for example, in his table of refractions, giving 3' 4" for the summer and 3' 28" for the winter refraction for 15 altitude; whereas, by the tables now in use, 3' 35 ' is the refraction used for 15 altitude for summer and winter. It may give an appearance of very great accuracy when accounts are published of official observations, by which it is claimed that altitudes of celestial objects have been cor- rectly ascertained to the one-hundredth of a second. Theo- retically this may have been done, but practically it is impossible as long as refraction exists, and there is a liability to the other sources of error named above. When calculation is employed, only clerical errors, such as incorrect addition or subtraction, or looking out an in- correct log., are possible, and such errors can be at once detected. When these observations are set in opposition to calculation, it is uncertainty struggling against certainty. Fortunately, however, observations have been made with fair accuracy in modern times^ and agree very closely with correct results obtained by calculation; Where differences do occur, it can be proved that these arise either from an in- correct use of refraction, or from the theories believed in by observers being opposed to the elementary laws of geometry. The repeated observations which are now considered necessary, as regards each star, are for the purpose of obtain- ing an annual rate for the increase or decrease in polar SOME RESULTS OF THE SECOND ROTATION. 85 distance and right ascension of each star; when this rate is obtained, and the approximate variation in this rate, a star- catalogue for a few years in advance can be framed from the results of these observations. This catalogue, however, is framed merely by adding or subtracting these rates from previous observations; no real calculation based on a know- ledge of the true movement of the pole of the heavens has ever been attempted, and for a very good reason, viz. that the true movement of the pole has hitherto been unknown. The reader who has made himself acquainted with the facts dealt with in previous pages, will know that he can calcu- late the polar distance of a star with accuracy for fifty or one hundred years without any reference to more than one observation of this star. Having found that, as long as observations fairly reliable can be obtained, the second rotation of the earth and the movement of the pole as herein defined are corroborated, we have merely before us the question of whether we have evidence of nature acting uniformly. It may perhaps be asserted by some theorists that although the earth now rotates from west to east, yet it formerly rotated from east to west. Or that the earth, which now revolves round the sun in a given direction, formerly revolved in the opposite direction. Such assertions are not at all of an unusual character to be made by a certain class of theorist, who delight in wonder-mongering speculations and assertions, in order to attract the attention of the unreasoning. That nature works uniformly is, however, accepted as a fact by the majority of reasoning people, and we may therefore examine what will be the results of one complete second rotation of the earth under conditions similar to those which will now enable us to arrive by calculation at re- sults hitherto considered unattainable except by perpetual observation. 86 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. One second rotation of the earth Around an axis inclined to the daily axis of rotation at an angle of 29 25' 47", and proceeding at a rate of 40.9" per year, will occupy, for 360 or one complete rotation, 31,686 years, during which a com- plete revolution of the equinoctial points will occur. During the half of this period, viz. 15,843 years, the obliquity of the ecliptic will vary 12, or double the distance of the pole of the ecliptic from the pole of second rotation. The date at which the least obliquity will occur is 229 5 '2 A.D., when this obliquity will be 29 25' 47" - 6 = 23 25' 47"; the greatest was 13,5478 B.C., when the obliquity was 29 25' 47" + 6 = 35 25' 47". At this latter date, the arctic circle each winter would reach to latitudes of 54 34' 13" in both hemispheres, and a large portion of England and the whole of Scotland would then have been within the arctic circle. This condition of the arctic circle would have been similar for both hemispheres of the earth, and there would have been a period of more than 15,000 years during which the arctic circle extended to more than 29 from the poles. In order that these facts may be clearly understood by those persons who will take the trouble to examine them, a familiar example of the effects of a daily rotation of the earth will be given. Thus during twenty-four hours an observer can perceive exactly similar changes occur in consequence of the daily rotation, that occur during 31,686 years in consequence of the second rotation. An examination of the following diagram is therefore suggested (Fig. 40). A projection of the northern hemisphere of the heavens is here given, on the plane of the equinoctial. P is the north pole of the heavens; S, a star distant 6 from the north pole; Z, the zenith of a locality on the earth, which locality is 29 25' 47" from the north pole of the earth. SOME RESULTS OF THE SECOND ROTATION. 87 Any time of year may be selected for this example, but in this diagram the 2 1st of March is the date when the day and night are of equal length (equinox). At midnight the zenith will be situated at Z, and we will assume the star S to be on the meridian and between the zenith and the pole at that hour. The zenith distance of the star S would at that hour be Z S, which is equal to P Z - P S; that is, to 29 25' 47" - 6 = 23 25' 47". The effect of the daily rotation is to cause the zenith Z to be carried in a circle round P as a centre during twenty- four hours. Consequently, as 360 of this rotation occupy twenty-four hours, 15 will occupy one hour, and 1 will occupy four minutes of time. We therefore know the rate at which the angle at P varies, we know that neither the pole P nor the star S alter their position during a few hours, and we know that the zenith Z moves uniformly in a circle round P as a centre, the radius of this circle being always of the same value, viz. 29 25' 47". Let us now suppose that, two hours after midnight, the zenith has been carried to B by the daily rotation, and that we wish to calculate B S, the zenith distance of the star S. 88 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Two hours of the daily rotation is equal to, 30, conse- quently the angle S P B is 30. We know that P S is 6, and that P B is 29 25' 47". Hence, with the two sides and the included angle, the third side B S can he calculated by the usual formula for spherical triangles. Several examples of the detail working of this problem have been given in the preceding pages. If at five o'clock a.m. we wished to find the zenith dis- tance of this star, the same process would be adopted, five hours, equal to 75, being the angle at P for this time. When the zenith had been carried by the daily rotation to F, no calculation would be requisite. The zenith distance of the star S would then be F P added to P S, equal to 29 25' 47" + 6 = 35 25' 47". Whilst the zenith was being carried by the daily rota- tion from Z to A, B, to F, etc., the zenith distance of the star S would increase, though not at a uniform rate. Whilst the zenith was carried from F to G, and on to Z, the zenith distance of the star S would decrease, but not at a uniform rate. During any hour of the day or night, however, the zenith distance of the star S could be calculated with the greatest ease, and this calculation could also be made even if the daily rotation occupied 24,000 years instead of twenty- four hours. During the night, however, when the stars are visible, this change in the zenith distance of a star can be observed, and those observers who care to test this problem may do so by using some instrument, and measuring the zenith dis- tance of a star, and comparing the results obtained by observation with those arrived at by calculation. It will be evident to the most average intellect that, when a rotation occurs round a fixed point, the figure de- scribed must be a circle; consequently each zenith during SOME RESULTS OF THE SECOND ROTATION. 89 twenty-four hours of the daily rotation must describe on the sphere of the heavens a circle, the common centre of all these circles being the pole of the heavens. Self-evident as this fact no doubt appears to the reasoner, or person gifted with common sense, yet attention is called to the fact for reasons which will be evident when the opinions of theorists as regards this problem are referred to. From the daily rotation of the earth and the movement of a zenith produced thereby, we will now advance to the second rotation of the earth, and the movement of the pole of the heavens caused thereby during one second rotation. The two problems are almost identical, and the results can be calculated with the same ease. The only difference being that the second rotation is partly in opposition to the daily rotation, and that we must treat the pole of the daily rotation as we treated a zenith in the former example. The daily rotation takes place during twenty-four hours; the second rotation takes place during 31,686 years. In the following diagram (Fig. 41), C represents the pole of the second axis of rotation; E, the pole of the ecliptic, 6 from C; P, the position to which the pole of the heavens will attain at the date 2295*2 A.D.; L M N P the course over which the pole of the heavens has been carried since the date 13,547'8 B.C. When the pole was at L, the distance to L E, which would be the measure of the obliquity, was C L = 29 25' 47", added to C E, which was 6; equal, therefore, to 35 25' 47". The rate at which the pole is carried round this circle by the second rotation is 31,686 years for 360. One-fourth of this circle, viz. from L to M, would occupy about 792 L years. Consequently, at the date about 5626 B.C. the pole \vas at M, and the distance E M was about 30, which gives the value of the obliquity at that date. 90 UNTEODDEN GROUND IN ASTRONOMY AND GEOLOGY. In 7921 years more the pole would be carried to P, at which date, viz. 2295'2 A.D.,the obliquity of the ecliptic will be 29 25' 47" - 6 = 23 25' 47". Thus from the date 5626 B.C. to 2295*2 A.D. the obliquity will have decreased about 6J, but it will not have decreased this amount at a uniform rate. Between M and N, the decrease was rapid; between and P, the decrease was small. To find the value of the obliquity for any date, we proceed exactly in the same manner as when we wished to find the zenith distance of a star for any hour of the night or day. We may select any date, say 500 A.D., for which we wish to know the obliquity. We subtract 500 from 2 29 5 '2, and multiply the remainder by 409", and thus obtain the angle at C ix>r the date 500 A.D.; then with the two sides, C E = 6 and C N - 29 25' 47", and the included angle at C, we can cal- culate N E, the obliquity; N, we will assume, representing the position of the pole of daily rotation at the date 500 A.D. From an investigation of this problem, any geometrician will perceive into what serious errors observers will fall who imagine that, by subtracting a constant quantity SOME RESULTS OF THE SECOND ROTATION. 9i annually from the mean obliquity found by observation at any date, they can assign a true value to the obliquity for a future date. The difference in rate is small between different years at present, as already shown thus between 1800 and 1850 the mean rate annually was 0*488", and between 1850 and 1865 it was - 0'46" annually but it is a serious error to imagine that the mean obliquity for 1000 or 2000 years in the past can be arrived at merely by adding 46" or 48" per century to the mean obliquity found by observa- tion for 1889. This problem cannot be dealt with in this mere rule-of-thumb manner, but must be treated like any other geometrical problem. The fact that the second rotation of the earth, as herein described, causes a variation of 12 in the obliquity of the ecliptic, and consequent extension of the arctic circle, may probably be of more interest to the general reader than other portions of the subject. An extension of 12 in the arctic and antarctic circles means a vast change of climate on earth, particularly affecting middle latitudes, although but slightly influencing the climate of tropical regions. From the date about 21,000 B.C. to about 6000 B.C., an arctic climate would prevail down to 60 latitude in each hemisphere during winter; whereas a tropical climate would prevail in summer in each hemisphere up to 30 at least. The midday altitude of the sun at the height of this period, viz. at about 13,000 B.C., would have been 12 greater during each summer, and 12 less during each winter than it is at present. Every person who has travelled in northern regions, or who has even crossed the Atlantic during summer, knows the effect of the sun's heat on arctic regions. It is during summer that icebergs are liberated and float southwards 92 UNTRODDEN GEOUND IN ASTRONOMY AND GEOLOGY. carrying their burthens of boulders, etc., which are deposited as the icebergs ground and melt. It is during summer that the masses of snow are melted, and vast floods of water rush over the country. Add 12 to the midday alti- tude of the sun during summer, and subtract 12 from the midday altitude in winter, and the effects which now occur, moderately only, in high northern regions, would have occurred with far greater force down to 54 latitude in each hemisphere. These conditions were comparatively only recent in the past history of the earth. They lasted from about 21,000 B.C. to about 6000 B.C., viz. whilst the pole was carried by the second rotation from E, to L and M, last diagram. It seems at least singular that, whilst the second rota- tion of the earth enables any person acquainted with it to calculate with the greatest accuracy results considered by modern theorists unattainable except by means of per- petual observation, this same movement, if continued during one entire second rotation, should prove that such a climate and such conditions must have prevailed on earth just previous to historic times. It is so singular, because geologists have during years asserted that their facts proved that just previous to his- toric times some such change of climate must have pre- vailed on earth in order to account for known effects. Modern theorists have denied that any change of climate can occur from astronomical causes, except of a very minute character. Their reasons for this denial will be examined further on. The reader may then be able to form an opinion as to the value of such assertions, and to really weigh the evidence advanced by such theorists, and to place this in the scale opposite to that in which the multi- tude of facts herein given may be placed. Attention may, however, be directed to the facts which SOME RESULTS OP THE SECOND ROTATION. 93 are brought into notice in this book. The observations of the past two thousand years prove that the earth's axis has continually changed its direction during these years. The rate of this change being very slow. An arc of only about 25 is the arc whose character is presented for analysis. The examination of this arc reveals the fact that it is the arc of a circle having a radius of 29 25' 47", and that the pole of daily rotation traces this circle in consequence of the earth having a second rotation. From a knowledge of this movement, the polar distance of a star can be calculated for one hundred years or more, a proceeding hitherto un- known in astronomy. A knowledge of the same movement enables us to calculate the precession of the equinoctial point with minute accuracy for any date. It also enables us to calculate the value of the obliquity of the ecliptic for any date, and shows that just previous to historic times there was a period of 15,000 years at least during which the arctic circle extended its limit from 6 to 12 more than at present, thus effecting vast changes in the climate of middle latitudes on earth. The same movement enables us to calculate how each zenith on earth is affected yearly and at various times of the year. Many other important results depending on this movement will be described in future pages. By means of the present orthodox theories, the polar distance of a star cannot be calculated for any remote date, a fudge rule of adding or subtracting a certain value found by observation being the only method hitherto known. The value of the obliquity of the ecliptic cannot be calcu- lated, the only means at present known being to subtract a constant quantity annually from the obliquity found by observation, with the result that it is necessary to fudge the records occasionally, to keep facts and theories in agreement. 94: UNTKODDEN GROUND IN ASTRONOMY AND GEOLOGY. It is asserted by theorists that no change greater than 1 21' can occur from astronomical causes in the extent of the arctic circle, and consequently that geologists must look elsewhere than to astronomical science for an explanation of their facts. CHAPTER VII. THE POLE OF THE HEAVENS AND THE POLE OF THE ECLIPTIC. WHEN individuals capable of reasoning possess the moral courage to free their minds from scientific dogma and the influence of mere authority, and realize the fact that astronomy, like all other sciences, must be dealt with by aid of common sense, of the exact laws of geometry, and by the careful investigation of facts, they may then perceive that the manner in which two problems in astronomy have been dealt with by theorists is most remarkable. These two problems are the changes or possible changes in" the obliquity of the ecliptic, and the so-termed proper motion attributed to nearly every star in the heavens. The first of these two problems will now be dealt with. When geology, after being ridiculed and opposed, had at length established itself as a science, it became acknow- ledged that the facts with which geologists were acquainted, plainly indicated that in former times climates in certain latitudes had existed quite different from those which now prevail, and which have prevailed, during the past 2000 years. Among the most remarkable and interesting of these changes, because probably it was the most recent, and its evidence the most easily seen, was the great boulder period, 96 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. or glacial epoch. The evidence existing throughout Europe and North America, and also in tne southern hemisphere down to about 50 latitude, of an arctic climate having pre- vailed just previous to historic times, was convincing. The evidence even showed certain details. It showed that this climate came on gradually; that the arctic climate crept down from the north, reached a maximum, and as slowly and gradually retired, until the present climatic conditions prevailed. There was distinct evidence that icebergs carry- ing enormous boulders, which now are liberated each summer in present arctic regions only, were formerly liberated in localities as far south as the north of England and corre- sponding latitudes in Europe and America. There was evidence of vast quantities of fresh water having flooded the country and produced those deep beds of gravel and sand now spoken of as " the drift," as though snow and ice in abundance were periodically melted, and thus produced these periodic floods with their consequent results. It was not the boulders and drift only which caused geologists to become convinced of the great change of climate which had occurred in comparatively recent times in middle latitudes; but the deposits of the flora and fauna in the drift was a mixture of an arctic and almost tropical character, as though the climate were sometimes arctic, and sometimes almost tropical. These facts being well known to geologists, they appealed to astronomers to ask whether there might not be some- thing in astronomy which had been overlooked or incor- rectly interpreted, and which might aid to give a solution of those mysteries, which had hitherto been admitted as unsolved problems in a grand science. This demand from geologists to astronomers was by no means unreasonable, and it would have had a firmer base THE INCLINATION OF THE AXIS. 97 had geologists been better acquainted with some of the known facts connected even with observational astronomy. The planet Venus, which is the nearest planet to the earth, revolves round the sun in an orbit which is inclined to that of the earth at an angle of about 3 23' only. Yet the axis of daily rotation of Venus is inclined to the plane of its orbit only 15. Hence the arctic circle, and an arctic climate, prevails each winter in Venus down to within 15 of her equator, whilst during the summer of each hemi- sphere a tropical climate prevails to within 15 of her poles. The orbit of the planet Uranus is inclined to that of the earth at an angle of about 46' only, yet the axis of this planet very nearly coincides with the plane of her orbit. Consequently the arctic circle in this planet extends nearly to the equator, and the tropics extend nearly to the poles. The most varied changes' of climate must prevail, therefore, on Uranus during its long year of about 30,686 "8 mean solar days. The planet Jupiter moves round the sun in an orbit which is inclined to that of the earth at an angle of about 1 19' only, but the axis of rotation of this planet is so nearly vertical to the plane of its orbit round the sun, that scarcely any change of climate takes place on Jupiter during its year. The variations from summer to winter which are experienced on earth cannot occur on Jupiter, the length of the day during midsummer scarcely varying from the length of day during midwinter, a uniform climate prevailing during each year in every latitude on that planet. When, then, geologists appealed to astronomers for some solution of the strange variations of climate revealed by geology, they ought to have been aware that the course or orbit which a planet during its year followed round the sun, was not a very important item as affecting the H 98 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. annual changes of climate in a planet. The important question should have been, what were the changes in the direction which the axis of this planet went through, or might go through, as regards the angle which it made with its orbit round the sun? How was it that whilst the orbit of the planet Yenus differed only 3 23' from that of the earth, yet her arctic and antarctic circles reached to within 15 of her equator? How was it that whilst the orbit of Jupiter differed only about 1 19' from the orbit of the earth, yet the arctic and antarctic circles on Jupiter did not extend even 3 from her poles? The question to be investigated was, what probable or even possible movement might take place in a great ro- tating sphere, which might cause its axis to make varied angles with its course round the sun? Was astronomical science acquainted with any change in the direction even of the earth's axis? If so, the subject for investigation was plainly indicated. The facts of geologists were numerous and strong, and naturally they appealed to astronomy for a solution, more especially when, as was probably known at the time, the condition of other planets was very different, as regards their annual changes of climate, to those which have pre- vailed on earth during the past 2000 or 3000 years. The problem for investigation was one of great interest, and requiring very careful geometrical handling. It was, in reality, to define the nature of the curve which the earth's axis had traced on the sphere of the heavens during the past 2000 years of which we have records. The manner in which this pole varied its distance from certain stars, especially from those near it, such as the pole-star, A Ursse Minoris, E Ursse Minoris, /3 Ursse Minoris, etc., would give co-ordinates by which the nature of this circle might be CURVE TKACED BY THE AXIS. 99 ascertained, and the problem was one well suited to a geometrician. That it was possible for two curves quite different in their general nature to be somewhat similar during a small portion of their course, the following diagram will prove. Suppose the circle LIT (Fig. 42) to be the plane of the ecliptic; E, the pole of the ecliptic; P, the pole of the axis of daily rotation, say 2000 years ago; C, the centre of the circle round which the pole P might be carried in the direction P Q S. Let E P - 24, C P = 11. The pole, in its movement from P to at the rate of 20" annually, would decrease its distance from E; there would consequently be a slight decrease in the obliquity owing to this movement, because a decrease in the obliquity is the same thing as a decrease in the distance of the pole of the heavens P from the pole of the ecliptic E. There would be a precession of the equinoctial point in consequence of this movement of the pole from P to 0. There would be a decrease in the polar distance of the stars in the direction towards which the pole was moving, and an 100 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. increase of their polar distance in the opposite point of the heavens. The small arc P O, if 10 in length, would occupy, for the pole to travel, some 1800 years at the rate of 20" annually. The curve from P to O, being a portion of the arc of a circle having C for its centre, would differ but little from an arc of a circle having x for its centre. E x we will take, for example, as 6; there would be a difference which a geometrician could easily discover, although a slight one. A very vast difference, however, would occur in the general results, according as C or x were the centre of the circle which the earth's axis described. If C were the centre, then, when the pole P had been carried round half its circle to S, the poles E and S would be only 2 apart. The obliquity of the ecliptic would then have been 2 only; the arctic and antarctic circles would have extended only 2 from the pole; the tropics would have extended only 2 from the equator; and an almost uniform climate during summer and winter would have prevailed in every locality on earth. With a? as a centre, and a radius of about 30, P, when half its circle had been described, would have been 36 from E. The arctic and antarctic circles would then have ex- tended 36 from the poles, and the tropics 36 from the equator, and extreme cold in winter and extreme heat in summer would have prevailed in middle latitudes on earth. As the question raised by geologists was one relative to the changes of climate which facts proved had occurred at remote periods, the inquiry as to the cause was narrowed into a single item. It was to investigate what had been the course of the pole of the heavens and the movement of the earth in recent times; what was the curve traced by the pole of the heavens; what was the radius of this curve; was the curve a circle? if so, where was its centre? Conse- INCLINATION OF THE AXIS. 101 quently, to determine the true character of this movement of the earth was the real problem submitted for solution. It was a known fact that there was nothing impossible in a planet which rotated daily round an axis and revolved annually round the sun, having its axis inclined only 15 to the plane of its orbit, as was proved by the planet Venus. There was nothing impossible in a planet having its axis inclined about 88 to the plane of its orbit, as was proved by the planet Jupiter. In fact, the angle which the daily axis of rotation made with the orbit along which the planet travelled, presented every variety from 90, as shown by Jupiter, to 2 or 3, as shown by Uranus. These variations were not due to the orbits of these planets being inclined to each other at very different angles, for it was known that these orbits differed from that of the earth in two of the cases mentioned less than 2. Conse- quently, this great variation in the angle which the planet's axis made with its orbit must be due to some movement or cause in the planet itself, not in its orbit round the sun. These facts being known, it will be evident in what manner the problem submitted by geologists ought to have been investigated; and it will now be stated how it was, and has been treated, 102 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. CHAPTER VIII. THE UNION OF ASTRONOMY AND GEOLOGY. WHEN geologists had brought pressure to bear on astro- nomers relative to the change of climate which might occur from astronomical causes, it was decided that the problem should be submitted to M. La Place, the French mathe- matician and theorist, for his investigation and decision. The manner in which M. La Place examined the case was as remarkable as it was inappropriate. Instead of endeavouring to investigate and define the true curve traced by the pole of the heavens over an arc of 10, of which we have correct and approximate evidence, according as we refer to modern or ancient observations, M. La Place set to work to discover how much the earth's orbit round the sun might vary. Instead of testing by the rigid laws of geometry, such as the change in polar distance of stars near the pole, what the true character of this curve really was, and if the curve were a part of a circle, giving the true radius of this circle, M. La Place, by theory, announced that the plane of the ecliptic (that is, the earth's orbit round the sun) could not vary from a mean quantity more than 1 21'. What this variation (assuming that it really occurred) had to do with the problem submitted by geologists, it is difficult to imagine, THE UNION OF ASTKONOMY AND GEOLOGY. 103 If the plane of the ecliptic varied as much as 3 23', then the earth's orbit and that of Venus would be in the same plane. Yet the arctic circles in Venus would even then reach more than 70 from her poles, whilst the arctic circles on earth would differ but slightly from their present limits. Having arrived at the conclusion that the orbit of the earth round the sun could not vary from a mean position (what the mean position was, this profound theorist did not consider worth mentioning) more than 1 21', he made the astounding assertion, that therefore the obliquity of the ecliptic could never vary more than 1 21', and therefore exact astronomy could give no help to geologists as regards any variation of climate in the past, as the limits of ex- tension of the arctic circle must be 1 21' from this " mean." Before mentioning the results which followed this so- termed investigation and its announcements, we must refer to the interesting assumption which must have been made by this theorist before he could put forward such assertions as those he made. A change of 1 21' in the plane of the ecliptic means the same thing as a change of 1 21' in the position which the pole of the ecliptic occupies in the heavens. That a change of 1 21' in the position of the pole of the ecliptic should cause a change of 1 21' in the obliquity, it must follow as a geometrical law that the pole of the ecliptic must either move away from, or come nearer to, the pole of the heavens by 1 21', because the distance between the pole of the heavens and the pole of the ecliptic is the exact measure of the obliquity. In order that this decrease of 1 21' should follow, as a matter of course, a change in the position of the pole of the ecliptic of 1 21', the pole of the heavens must trace a circle round some centre, from which centre the pole of the ecliptic varied its distance 1 21'. 104 UNTRODDEN GKOUND IN ASTRONOMY AND GEOLOGY. Where this centre was, and how far from the then posi- tion of the pole of the ecliptic, was considered not worth investigating or mentioning. Strange to say, however, whilst the assertion above mentioned was put forward as a finite and unanswerable truth, this same gentleman stated that the pole of the heavens always described a circle round the pole of the ecliptic as a centre. If the pole of the heavens did describe a circle round the pole of the ecliptic as a centre, no variation whatever in the position of this centre could cause a variation in the distance of the circumference from its centre. The circum- ference itself might present some peculiar curve on the sphere of the heavens, but the distance of the circumference of a circle never can vary its distance from its centre, no matter how much this centre alters its position as regards other objects. We can slowly describe, with a pair of compasses, a circle on a piece of cardboard whilst we are travelling in a railway train. The centre of this circle will alter its distance from surrounding objects, but it will never alter its distance from the circumference of the circle of which it is the centre. When, then, M. La Place, having investigated what might be the change in the position of the plane of the ecliptic as regards the fixed stars (an inquiry which had nothing to do with the problem submitted to him), arrived at the conclusion that, as the pole of the ecliptic could vary only 1 21' from some imaginary mean position, there- fore the obliquity could vary only 1 21', he made an assertion so utterly incorrect and devoid of any foundation in fact, that it is almost inexplicable how such an error could have been overlooked even sixty years ago. If the centre of the circle which the earth's axis traces THE UNION OF ASTRONOMY AND GEOLOGY. 105 be 6 from the pole of the ecliptic, we must have a variation of 12 in the obliquity without any change whatever in the plane, and hence in the pole, of the ecliptic itself. No sooner, however, had this erroneous statement of the theorist been made known, than hundreds of followers, with a blind submission and an unreasoning adherence to authority, joined in a chorus to the effect that it had been proved that no changes of climate ever had occurred on earth, and that exact astronomy was, therefore, unable to give any aid to geology. When we reflect on the particular problem that was submitted to the French theorist the facts that were at his disposal, and the evidence presented by geologists, and by the condition of other planets, and the actual movement of the earth which was known to occur and then find that his investigation was limited to an inquiry relative to the plane of the ecliptic, whilst he utterly ignored the movements of the earth itself, we believe that every reasoner must admit that it is one of the most remarkable exhibitions of dogmatic unreasonableness ever put before the scientific world. The result, however, was to stop inquiry in this direc- tion. Geologists, convinced of their facts, were hurrying along what appeared to them a clear and open road, by which they could reach the waters of truth, when suddenly M. La Place blocks the way, and " no thoroughfare " in this direction is the announcement. Checked in their attempt to reach truth by aid of astronomy, geologists seemed to lose their common sense, and immediately invented a series of extravagant and baseless theories, evolved from their imagination, and in which they were aided by wonder- loving lecturers and others, in the endeavour to account for their known facts. After the announcement had been made that theory 106 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. proved that the plane of the ecliptic, "by the joint action of all the planets," could not vary more than 1 21' from some imaginary mean position, and the consequent opinion arrived at, that no matter how the earth moved, yet no change greater than this 1 21' ever had or ever could occur, it was considered a proof of ignorance for any person even to hint at any greater change having taken place, even during millions of years. It was not considered in the least unscientific to assume that the earth, which, it was asserted, had once been a red- hot globe of fire, had cooled down till it had become so cold as to produce the glacial epoch, but was now warming up again. It was stated that the sun, which, it was asserted, was a great blazing body, kept going by shrivelling up, or by fuel in the form of meteors being shovelled into it, must have become hard up for this fuel, and the fire consequently had nearly gone out, and thus caused the cold of the glacial period; but that now, from probably having been stirred, it was again burning brightly. In order to leave no ground uncovered by speculations, it was also asserted that the whole solar system was rush- ing through space with enormous velocity, and therefore probably in this rush passed through climates which might be very hot or very cold, just as a traveller who might travel from the equator to arctic regions would encounter great changes of climate. These, and a multitude of other wonder-mongering theories, were considered really scientific probabilities; but to imagine that any change more than 1 21' in the obliquity could or had ever occurred, was stated to be a proof of ignorance. Had not the great French theorist, by the most ex- haustive analysis, proved that the plane of the ecliptic could THE UNION OF ASTRONOMY AND GEOLOGY. 107 vary only 1 21' from some mean position, and, no matter how the earth moved whilst travelling round this ecliptic, there could not by any means be produced a difference in the obliquity of more than 1 21'? The course which the pole of the heavens traced was thoroughly well known to theorists. It was a circle traced during 25,868 years round the pole of the ecliptic as a centre, from which centre it never varied its distance. How any variation in the obliquity could occur if the earth's axis did trace a circle round the pole of the ecliptic as a centre, was considered too trifling a matter to be even discussed. Divesting this problem of the verbiage and contradic- tions by which it has long been surrounded by theorists, a few practical facts will now be brought into notice. In order that any theorist can positively affirm that during all time there can be no change greater than 1 21' in the obliquity of the ecliptic, he must know with minute accuracy the following items : First, the exact course which the earth's axis has traced and will trace on the sphere of the heavens during millions of years must be known. If this course be a circle, the exact radius of this circle must be known, the exact position of the centre of this circle, and whether this centre ever has, or ever will alter its position. Second, the detail movements of the earth must be known, which take place in connection with the change in direction of the earth's axis. These two items are of the first importance. Third, what change has or will take place in the earth's orbit round the sun, by which change the position of the pole of the ecliptic will be altered, and the results produced by the first-named movement might be slightly modified. When M. La Place asserted that, because the plane of 108 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the ecliptic, by his theories, could \ary no more than 1 21', and that therefore the obliquity could not vary more than 1 21', he must have imagined that he knew with minute accuracy those items mentioned as first and second in the preceding paragraphs. If he did know these items, we have at once brought to our notice a neglect which was little short of unpardonable. If the true course of the pole on the sphere of the heavens had been (as it was claimed) correctly known, then, as this pole was carried uniformly round its known curve, the distance of this pole from various stars could be calculated for any date, without reference to observa- tions. If the course of the pole had been correctly known, not for millions of years (as was asserted), but only for one hundred years, then no further observations need be made in order to calculate by simple geometry the polar distance of a star for one hundred years, and without any possi- bility of error. Here was an opportunity for M. La Place and his followers to give a practical proof of the correctness of their theories. They might have said, " Observations are liable to error, both from personal and instrumental causes; refraction is also a very fruitful source of error. Thousands of pounds are spent annually at various observatories for the incomes of observers, who sit up night after night, with their eyes at a transit instrument, in order to find out the rate, and the variation in the rate, at which the pole moves towards, or away from, a star. All this labour and expense is unnecessary, because, we having by theory proved that the plane of the ecliptic can vary from a mean position only 1 21', everything as regards the movements of the earth must be known, and consequently to continue observing in order to find out what is known, would be as childish as to employ men to count how many times the second- THE UNION OF ASTKONOMY AND GEOLOGY. 109 hand of a chronometer went round, during twenty-four hours." Had the assertions of these gentlemen been facts instead of mere theories, they might have given such proofs as the following : - " As we know the true course of the pole of the heavens, we will prove we know it. We will take the following stars near the pole, and demonstrate that by our knowledge we can, from one observation only (and without any reference to the variable change in polar distance hitherto found by aid of repeated observation), show how to calcu- late the polar distance of these stars for each year, for one hundred years or more. Here are a few stars only : a Ursae Minoris, X Ursse Minoris, e Ursse Minoris, )3 Ursse Minoris, a Draconis, /3 Draconis, X Draconis, and many others." M. La Place and his followers, however, gave no such practical proof that their theories were correct. They were fully aware of the immense amount of time and labour given to observations, to say nothing of the expense, all of which must be totally unnecessary if, as they asserted, they knew with exactitude the true course of the pole of the heavens. Yet they did not give the slightest hint of their profound knowledge, which, if possessed, would have rendered further observation quite unnecessary. Here in the present day, scores of years after this wonderful discovery, men are employed night after night to observe stars, in order that they may ascertain that which, it was claimed, was known with minute accuracy, and could therefore be arrived at by calculation, without any further reference to observation. Surely this was an unpardonable piece of neglect on the part of the French theorist and his followers. To possess such perfect knowledge, and yet to continue carry- 110 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. ing on laborious observations as though no such knowledge were really possessed, was an exhibition of remarkable inconsistency. Let the reader refer to Chapter IV. of this book, and he will see that the polar distance of the stars given, all near the pole, can be calculated with minute accuracy for each year for one hundred years or more, without any reference to observations. No matter whether the annual variation in polar distance of these stars be great or small, be variable or almost constant, yet by the same geometrical laws, can their mean polar distance be calculated either for past or present years. This calculation can be effected only when a know- ledge of the true movement of the pole, and of the second rotation of the earth, is possessed by a geometrician. Hitherto theorists have not known either of these facts, and hence even in the present day there is necessity for perpetual observation. When, then, M. La Place asserted that, because the plane of the ecliptic could vary only 1 21', therefore the obliquity could vary only 1 21', he made a statement utterly devoid of any foundation. When he stated that astronomy could give no help to explain the facts of geology, he made another statement devoid of truth. Astronomy can and does explain the facts of geology, but it is not the present orthodox astronomy of those theorists who assert that the centre of a circle can vary its distance from the circumference, and yet always remain the centre of the same circle. Nor is it the astronomy which asserts that no variation in the obliquity can occur, except by a change in the plane of the ecliptic. The planets Venus and Jupiter tell a different tale, even if geometry were unknown. T&E UNION OF ASTKONOM* 1 AND GEOLOGY. Ill So completely did the theory of M. La Place stop all inquiry relative to the cause of geological climates being due to movements of the earth, and so completely have theorists since then been as it were mental slaves to the assertions that were then made, that more may have been written on this subject than it really deserves, these errors and oversights being almost self-evident. Yet the history of astronomy proves how the strong* holds of error are obstinately clung to by incompetent reasoners, and when we read the remarks that have been made by certain authorities relative to the second rotation of the earth, it is not difficult to understand how it was that during many hundred years, the daily rotation of the earth was rejected and sneered at, by the astronomical authorities of the past. Some years ago, when standing on the banks of a lake in Nova Scotia (a locality well suited to the study of the evidence of the glacial period), I observed that the hard rocky shore was cut and marked by the glaciers and ice- bergs of the boulder period. In various inland localities were enormous boulders, which had been carried many miles from the parent rocks, and deposited in what was now a vast forest. My only companion was Paul, a Micmac Indian. Pointing to the boulders and the marks on the rocks, I said, " Paul, how do you account for all this? " Paul, without any hesitation, replied, " Long time ago, more winter in winter, more summer in summer. More winter make more snow, more icebergs; more summer melt snow quicker, float icebergs more than now. That what I think." This Indian hunter was so ignorant of science that he did not know even the multiplication table, yet his opinion was correct. 112 UNTKODDEN GBOUND IN ASTKONOMY AND GEOLOGY. When we compare the reasoning of this son of the forest with some of the assertions of modern scientists, it really seems possible, that the mere sing-song muttering s of the village softy may contain more truth and real science, than the long-incubated opinions of the over- crammed, dogmatic theorist. But we have not come to the end of this matter yet. Whilst M. La Place proved that no change greater than 1 21' could occur in the extent of the arctic circle, because the plane of the ecliptic could not vary more than that amount, and every follow-my-leader theorist bowed to the supposed infallibility and minute accuracy of this conclu- sion, we now are informed by another French theorist, M. Leverrier, that M. La Place committed an error; it is now stated that 4 52', not 1 21', is the amount that the plane of the ecliptic can vary. Will these theorists kindly state the exact course which the earth's axis traces on the sphere of the heavens, and prove their knowledge by calculations? CHAPTER IX. THE SO-CALLED PROPER MOTION OF THE FIXED STARS. THERE are few subjects which have attracted the attention of observers in modern times more completely than that termed somewhat peculiarly " the proper motion of the fixed stars." The manner in which a catalogue of stars for any future date, and for the use of surveyors, navigators, etc., is formed is as follows : By the aid of a transit instrument placed in the , plane of the meridian, the zenith distance of a star is measured, as often as it can be seen, on the meridian. The latitude of the observatory being known, it is also known what is the meridian zenith distance of the equator, and the zenith distance of the pole. For example, if the latitude of the observatory were 51 28' north, the meridian zenith dis-- tance of the equator produced to the heavens would be 51 28', and the zenith distance of the pole of the heavens would be 38 32', the pole being north, the equinoctial (or equator produced) being south, of the zenith. When, then, the meridian zenith distance of a star is measured with the transit instrument, the distance of this star from the equinoctial can be at once assigned, and this distance is the star's declination. For example, suppose a star were found to have a meridian zenith distance south of 10 28', the latitude of the observatory being 51 28', then I 114 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. this star would be 41 north of the equinoctial, and would be given 41 north declination, if a star were to pass the meridian 10 north of the zenith, then this star would be given a north declination of 51 28' + 10 = 61 28' north. Year after year similar observations are repeated in order to obtain the rate of change annually in the declination or polar distance of a star. Thus, for example, suppose the declination of a star deduced from its meridian zenith dis- tance were measured on the night of January 1, during several successive years, and were found as follows : January 1, 1850 30 10' 10-64" 1851 30 10' 0-64" 1852 30 9' 50-63" 1853 30 9' 40-61" The annual rate would be given to this star of - 10", with an increase in the rate of about two-hundredths of a second per year. It would be thus possible to fudge the declina- tion of this star for four or five years in advance with very fair accuracy, without knowing in the least why the rate varied from 10" to lO'Ol", and then went to 10'02". For example, we could predict the declination of this star for January 1, 1854, by subtracting 10'03" from the declination found for 1853, and we should thus assign the star a declination for January 1, 1854, of 30 9' 30'58". We might, for January 1, 1855, subtract 10'06" from that assigned for 1854, and so on. The reader who will carefully consider the means adopted to obtain the declination of a star, and the annual rate of change in this declination, will perceive that the results are derived from instrumental observation only, and are not obtained in consequence of any knowledge of those mechanical movements in the earth which produce the change. The declination of the star is found by measuring its meridian zenith distance with the transit instrument. SO-CALLED PEOPER MOTION OF THE FIXED STARS. 115 The annual variation in the declination is found by com- paring the declination found during successive years, due allowance being made for the nutation already referred to. If the earth were a fixed body having no daily rotation, but the" whole sphere of the heavens revolved every twenty - four hours, the declination of a star, and the annual varia- tion in this declination, could be obtained by exactly the same proceedings as are now adopted, and with the same ease and approximate accuracy. Thus by observation only we may obtain a fairly accu- rate knowledge of the changes which take place annually in connection with various celestial bodies, although we may not have the slightest idea of the true cause of these changes. As an example, a case may be mentioned which occurred more than two thousand years ago. Hipparchus, one of the ancient observers, determined by observation the longitude of the star Spica Virginis, and he found this star was 8 from the autumnal equinox. He found that one hundred and seventy years previously, two observers, viz. Timocharis and Aristyllus, had found the longitude of this same star 6 from the autumnal equinox. By the comparison of these longitudes, Hipparchus concluded that there was a preces- sion of the equinoctial point of 2 in one hundred and seventy years. Hipparchus did not know that the earth rotated on its axis during each twenty-four hours, or that it revolved round the sun during each year. He did not know that the earth's axis changed its direction, and thus produced this difference in longitude which he had found by observa- tion. He imagined that the earth was stationary. Yet had his observations, or those of his predecessors, been a little more accurate, he might, by merely adding a certain amount .annually to the then longitude of a star, have predicted 116 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. with fair accuracy what would be the longitude of this star at some future period. He would, by this rule-of-thumb method of adding a certain quantity annually, have given the longitude of a star for a future date by exactly the same method that the modern observer predicts the declination of a star for a future date, by adding or subtracting a given quantity for each year. It is by no means difficult to become acquainted with various changes which occur, and which can be discovered by observation, but it requires some amount of thought and reason to discover the cause of these changes. Even the ploughman is aware that the days in summer are much longer than they are in winter he knows this from observa- tion; but he is not aware that the cause of this change is due to the fact that the earth's axis is at present inclined about 66 33' to the plane of its orbit. That, if the earth's axis were at right angles to the plane of its orbit, the days during the year would be always of the same length, would be a truth which the ploughman could not com- prehend. It is, therefore, a very different thing to know from observation only that certain changes occur, an d to know the causes which produce these changes. The method of finding the declination or polar dis- tance of a star by observation has now been described, and the reader must bear in mind that when the declina- tion has been obtained, the polar distance is known, the decimation added to the polar distance being always 90. It will be remembered that the decimation of a star is- deduced from its meridian zenith distance, measured by aid of the transit instrument, and the annual rate of change in the star's declination is obtained by the comparison of stars' assigned declinations from year to year. SO-CALLED PROPER MOTION OF THE FIXED STARS. 117 The question of the so-termed " proper motion " of the stars may now be dealt with. To those persons unacquainted with technical astronomy, the term "proper motion " of the stars might be understood to mean that a group of stars seen in the heavens altered their form or shape relative to each other. For example, suppose three stars not far from each other formed at a distant period an exact equilateral triangle, but that in the present day they formed an isosceles triangle; if such a change had occurred, it would be a proof that these stars had altered their relative position as regards each other, and had, therefore, some independent motion of their own. It is not from such palpable changes that the conclusion has been arrived at that the stars have an independent movement of their own, but on account of their changes in right ascension and declination r being different from what it is assumed they ought to be, in accordance with the accepted theory of the movement of the pole of the heavens. It is quite possible, and even probable, that the stars have some independent movement among themselves, but the greatest caution is requisite, before we attribute to any stars such a motion, merely because their right ascension or declination changes in a manner not in accordance with the present accepted theories. In the first place, we must bear in mind that the declination of a star is deduced from its observed meridian zenith distance, and its right ascension is obtained by its meridian transit. The manner in which the zenith is affected by the second rotation of the earth, as shown in Chapter III., is very' varied, and is dependent not only in amount, but in direction, upon the distance of this zenith from the pole of 118 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the second axis of rotation. Two localities on the same meridian of terrestrial longitude may have their zeniths affected in quite different manners by the second rotation, one zenith being carried directly towards the pole, the other being carried obliquely towards the pole. The fol- lowing diagram will give some idea of these varied and important changes; and as these changes have hitherto been entirely unknown to astronomers, the reader's atten- tion is particularly called to this subject. P(Fig. 43) represents the pole of daily rotation on the sphere of the heavens; B and x, the zenith of two localities on earth, having the same terrestrial lon- gitude, but differing in latitude by the arc x B. B D is the arc traced on the sphere of the heavens, round P as a centre, by the zenith B during a portion of the daily rotation of the earth; x y is the arc traced on the sphere of the heavens, round P as a centre, by the zenith x during the same portion of the daily rotation as that which carried B to D. During one daily rotation of the earth, the pole of daily rotation may be taken as a fixed point in the heavens. We will now examine the effects which will be pro- duced during any given time by the so-termed change in direction of the earth's axis, which movement really is a second rotation of the earth, and the point C represents on the sphere of the heavens the pole of the second axis of rotation. The pole P is carried to Q, round C as a centre, by the second rotation, the arc P Q being taken of any length, SO-CALLED PROPER MOTION OF THE FIXED STARS. 119 as an illustration of the laws affecting this movement of the earth. At the date when the pole has reached Q, we take again a portion of the daily rotation of the two zeniths before referred to. No change whatever in the direction of the axis of a rotating sphere can cause any change in distance between the pole of this sphere and of each zenith. Consequently we take Q L = P A and Q N = P O, and the arc L D' will be the arc now traced by the zenith which formerly traced the arc A B D during a portion of the daily rotation. We now come to a most important item in connection with the changes of zenith important because it has hitherto never been examined by geometricians. The pole P is carried to Q by the second rotation of the earth, and the zenith B will be carried to B', the arc B B' being a portion of a small circle of the sphere having C for its centre. In like manner, the zenith at D will be carried to D' by the same movement that has carried the pole from P to Q and the zenith B to B'. The zenith which was at x will be carried by the second rotation to x' } the arc x x f being a portion of a small circle of the sphere the centre of which is C. Thus whilst the pole is carried from P to Q by the second rotation, and the zenith B is carried to B', the zenith x is carried to x f by the same movement of the earth. The zenith 7 will be carried to /, round C as a centre, and the whole effect of a portion of the second rotation as affecting the pole and these zeniths is, that whilst P is carried to Q, B is carried to B', D to D', x to x\ and 7 to' 7'. The meridian of right ascension which, when the pole 120 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. was at P, would pass through B and x, would now, when the pole was at Q, be represented By a great circle of the sphere passing from Q through B' and x! The zenith B is carried by the second rotation to B', or nearly north, taking P or Q as the north pole of the heavens; but the zenith x is carried to x', an arc which is oblique as regards the North Pole. Consequently the zenith x', in order to reach its former meridian, of which P x is a part, has to be carried by the daily rotation over a small arc; whereas the zenith B, which has been carried to B' by the second rotation, has by this movement passed the meridian, viz. P x, on which it was formerly situated. These varied changes, both in direction and in amount, over which various zeniths are carried by the second rota- tion, and the manner in which the meridian appears to change in consequence, afford some of the most beautiful yet simple problems ever submitted in connection with geometrical astronomy. To those persons unacquainted with the second rota- tion of the earth, the most mysterious supposed movements of the stars will take place near the pole of the axis of second rotation, and also where the distance of a star from the pole of second rotation is greater than the distance of this same star from the pole of daily rotation. This one branch of the second rotation is so varied in its effects, that many pages would be required to describe even a portion of these. A geometrician, however, will readily comprehend the principle, and can work out the details for himself, should he care to do so. From even the brief description which has already been given as to the varied changes produced on a zenith and meridian by the second rotation, and varying very much according to where the zenith may be situated in relation to the pole of second rotation, we see that it is a somewhat SO-CALLED PROPER MOTION OF THE FIXED STARS. 121 hasty assumption to make, that because the meridian zenith distance of certain stars varies in an apparently irregular manner, therefore these stars have a " proper motion " of their own. We ought first to examine whether it may not be a proper motion in our zeniths and meridians, and not in the stars. Another source of error in connection with the asserted proper motion of the stars, is due to the fact that the true radius of the circle which the earth's axis describes on the sphere of the heavens has hitherto been incorrectly esti- mated. It is a geometrical law that the relative right ascension of any two stars on the circumference of the circle traced by the pole of the heavens will never vary. If, as has been the case, it were assumed that the radius of this circle were 23 28', then any two stars on the circumference of this circle would never vary their relative .right ascensions. If by repeated observations it were found that the relative right ascension of these stars did vary, theorists would assert that one, or both of these stars, had a proper motion in right ascension. If, however, the radius of the circle traced by the earth's axis happen not to be 23 28', but of any other value, then the variation in relative right ascension of these two stars does not prove that they have any " proper motion," but it proves that the circle assumed to be traced by the earth's axis has been incorrectly estimated. To assume that a certain mechanical movement occurs in an instrument, and then to assert that all the apparent movements of distant objects which contradict this assumption are due to eccentric and varied movements of the distant objects, cannot be called a scientific method of investigation. It is, in fact, inverting the order in which inquiry should guide us to truth, inasmuch as a theory 122 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. is first invented, and then facts are denied, when these contradict the assumed truth of the* theory. It is at present assumed that the radius of the circle which the earth's axis traces is 23 28', whereas the most exact calculations prove that the radius of this circle is 29 25' 47". It follows, therefore, that there will be a point in the heavens where the circumference of these two circles are separated by the greatest distance, and at this point the greatest difference between the theoretical and actual position and changes of stars will be found. It may be interesting to note where this point in the heavens is situated, and the following diagram will show it : The point C is the centre of the circle which the pole traces in consequence of the second rotation of the earth. This circle is represented by O P Q x R, the radius being 29 25' 47". E is the centre of the circle A B D, which the earth's axis has been supposed to trace in consequence of " a conical movement," the radius E B being 23 28'. P is the position of the pole of the heavens at the date 1887. The arc P x will represent a portion of a meridian of eighteen hours right ascension. SO-CALLED PROPER MOTION OF THE FIXED STARS. 123 At the point R the circumference of the larger circle will be most distant from the circumference of the smaller circle. In what part of the heavens is this point R situated; and where, consequently, will the greatest differences be found between the theoretical and actual position of the stars? First, as regards the polar distance of R. The radius C P = 29 25' 47"; the radius C R = 29 25' 47", The distance P R will be nearly equal to twice 29 25' 47" that is, to about 58 51', which is about the polar distance of the point R. The point x has eighteen hours, or 270, of right ascension; and the point R is a few degrees short of 270 how many can be easily calculated, but, in round numbers, the right ascension of R would be about 265. A point in the heavens, having a north polar distance of about 58 51', and a right ascension of 265, is situated in the constellation Herculis, and is that point in the heavens where the true circle described by the pole of the heavens is at the greatest distance from the theoretical circle whose radius is assumed to be only 23 28'. It is singular that a point in the constellation Herculis almost coincident with the point mentioned above, has during many years attracted the attention of several theorists, in consequence of the stars in that region pre- senting the largest amount of difference between their theoretical and actual positions. Below will be seen the positions as assigned about 58 51 r north polar distance, 265 right ascension, by comparing the results of the curve traced by the second rotation with the imaginary curve, with radius 23 28'. From the discordances found in the positions of stars, the point at which this discordance was greatest has been located as follows : 124 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Xorth polar distance. Eight ascension. By 59 2' 261 11' ... .. Argelaiider. 62 24' 261 22' .. . . Strove. 75 34' 252 53' .. .. Luhndal. 55 37' .. . . 260 1' . . . . Galloway. That observers should have discovered that there was, about this point in the heavens, a marked difference be- tween the theoretical and actual position of stars might be expected, considering that near that point there was a difference of 12 between the real and imaginary circle traced by the earth's axis. Few persons, however un- acquainted with the tendency too often manifested to invent wonderful theories to explain simple geometrical laws, would have expected the conclusion arrived at by theorists to explain these discordances. The theory that was put forward, at once accepted as true, that was copied by writer after writer, and asserted to be a fact, was, that the whole solar system was rushing towards this point in the constellation Herculis at the rate of one hundred and fifty-four million one hundred and eighty-five thousand miles per year (154,185,000 miles), and consequently it was merely a matter of time as to when a grand crash would occur, and the solar system be thereby destroyed. Perhaps one of the most interesting items in connection with the history of astronomy, is to note the manner in which the general public and the tyro in the science, at once accept and repeat the most baseless theories, whilst they reject as absurd, conclusions which can be proved by exact geometry. No matter how absurd were the argu- ments employed to prove that the earth must be a flat surface, and could not be a globe, or that the earth must be a stationary body and could not rotate, yet these argu- ments were at once accepted as grand truths, whilst the real truth on these matters was considered a subject for SO-CALLED PROPER MOTION OF THE FIXED STARS. 125 ridicule only. Had this mental feebleness not existed, the theory of the earth being stationary, could not have held in a state of slavery the minds of the scientific world during upwards of thirteen hundred years. It is a fact that, at the point in the heavens in the con- stellation Herculis, there will be some marked changes in the positions of stars, but this is mainly due to the second rotation of the earth. Yet these changes never caused the slightest suspicion that the assumed circle traced by the pole might be incorrect. Had it not been proved that the earth's axis traced a circle round the pole of the ecliptic as a centre, from which it never varied its distance? Was it not known that the joint action of the sun and moon on the earth's pro- tuberant equator caused " a conical movement of the earth's axis "? To even question these theories was considered a proof of ignorance. Yet on so small a foundation as that great changes occurred in the observed right ascen- sions of stars near the constellation Herculis, the theory was invented that the whole solar system was rushing in that direction at an enormous rate, and that a grand crash must ensue. To question this theory was asserted to be a proof of one's ignorance of the grand discoveries of astronomy. It is always an unpleasant, and at the same time a most unpopular proceeding, to call attention to the errors which have been committed by those individuals, who have been long looked up to with veneration as authorities, and whose assertions or theories have been received with the greatest obedience. If, however, we desire to reach that which is true, we must overcome this unreasoning and blind submission to mere authority, otherwise we shall be committing the same errors as did the ancients, who denied that the earth could rotate, merely because all the great 126 UNTKODDEN GROUND IN ASTRONOMY AND GEOLOGY. authorities during more than thirteen hundred years had asserted that it was stationary, ancl could not rotate. There is probably no book, article, or paper written on the subject of the " proper motion " of the fixed stars which has tended so largely to promulgate error and mislead the mere follower as a paper by the late Professor F. Baily, in vol. v., "Memoirs of the Royal Astronomical Society," on the supposed method of finding the proper motion of the stars. So unquestioned was the assumed accuracy of the method put forward in this paper, that the gold medal of the Society was presented to its author. Scores of followers, accepting as perfection the method suggested, set to work to make calculations as to the proper motions of stars, and hundreds of pages were filled with lists of assumed " proper motions," all based on the method of calculation recommended by this authority. In order that the elementary error contained in the paper referred to may be perceived, the geometrical laws relative to a curve and a point must be studied. Suppose A B C D (Fig. 45) a curve of any description; but, for the sake of illustration, it will be taken as part of the circumference of a circle, the centre of which is y. Take any point, x, as a point of Fig- 4 5. reference, and this point might be a star. Take four points, A, B, C, D, equidistant from each other, and situated in the circumference referred to. The distances A x, B oj, C x, and D x, would represent the distances of x from four points in the circumference. Suppose a body or a point of reference moved along the curve A D. This point would decrease its distance from os whilst moving from A to D. But the rate at which SO-CALLED PKOPER MOTION OF THE FIXED STABS. 127 it decreased this distance would be a very variable rate, and in no two successive years would the rate be the same. Not only would the rate decrease, but it would not decrease uniformly; consequently, if we found the rate when the point was at A, and the rate when the point was at C, we should not obtain the rate when the point was at B by taking the mean of the other two rates. This law holds good for all cases where a curve ap- proaches or moves away from a point, and ought to be well known to every person acquainted with even the elementary laws of geometry. The manner in which the sines, tangents, and secants of an angle vary gives another illustration of this law. For example, the sine of 30 is O'oOOOOOO. The sine of 34 is 0*5591929. If we took the mean of these two sines, we should obtain 0'52959645. But this is not the sine of 32. The sine of 32 is 0-5299193. Again, the sine of 30 increases for each minute of angle 2519. The sine of 34 increases for each minute of angle 2411. The mean of these rates is 2465, but this does not give us the rate of increase for each minute at 32. :s ' Small as the differences may be under some conditions, such as when the curve is moving directly away from or towards a point, yet it is a geometrical law that there must be an unequal variation in the rate at which a curve varies its distance from a fixed point. Hence to compare the " rate " at one date with the " rate " found at another date, and to take the mean of this rate as a guide to the distance of a point from a curve, is a certain source of error. The method proposed in the paper referred to is as follows : " Thus if P denote the place of a star in Piazzi's Catalogue (either in right ascension or declination), and B the place of the same star in Bradley 's Catalogue, p the 128 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. annual precession in 1800,?r the annual precession in 1755, the annual proper motion // of a star for the first period (viz. 1755 to 1800) will be ,_P-B p + TT 45 2 "And it is in this manner," says Professor Baily, "that I have deduced the annual proper motion of the stars in the list subjoined to this memoir." With all due deference to this authority and his numerous followers, the laws of geometry prove that he will not obtain the proper motion of the stars by the method he has recommended. That which he will obtain is the difference between the mean rate at the middle period and the whole amount of the rate for the whole period, divided by the number of years between the two periods. It is most remarkable that so palpable an error in elementary geometry as that contained in the paper re- ferred to should have escaped the notice of the Council of the Royal Astronomical Society. It is equally as remark- able that the gold medal of the Society should have been given for the paper, and that followers by the score have copied this error and have framed tables of so-called " proper motion " which have no foundation in fact. The error, however, is of a similar kind to that made by more modern theorists, who subtract a constant quantity annually in order to obtain the mean obliquity. In both cases the geometrical law, that a curve cannot approach a point at a uniform rate, and that this rate does not vary uniformly, has been overlooked. By the method proposed by Professor Baily, nearly every star in the heavens would have assigned to it a " proper motion." Let us take some examples. In a catalogue of stars for January 1, 1780, the north polar distance of the star )3 Draconis was found to be SO-CALLED PROPER MOTION OF THE FIXED STARS. 129 37 31' 47", and its annual rate of increase in polar distance was found to be 3 '06" annually. In the Nautical Almanac for 1887, the mean north polar distance of this star is given as 37 36' 52'83", and its annual increase in north polar distance 2 "80" annually. Applying the method considered sound by Professor Baily and others, and substituting the items above, we should obtain the following results : 37 36' 52-83" - 37 31' 47" 3'06" + 2-80" motlon - 107 - 2 ~ That is, 2-93"- 2-858" = 0-072" annually. Consequently, it would be assumed that this star had a proper motion of 7 "2" in one hundred years. The fact really being that the rate 3'06" does not change to 2*80" uniformly, but has a "second difference," just in the same manner as the sine of an angle does not vary uniformly. The annual rate at which a star is supposed to increase or decrease its polar distance has hitherto been obtained by observation, and is consequently liable to be affected by all those causes of error to which reference has been made. To attempt to deduce from such uncertain data definite conclusions as to the proper motion of stars, when in addition the method proposed is geometrically incorrect, is a somewhat loose manner of proceeding, and must lead to errors and false conclusions. When, also, the curve traced by the pole of the heavens, which is the cause of the change in polar distance of the stars, has never hitherto been accurately defined, it is evident that this subject has not been definitely disposed of. Had it been so disposed of, the present system of perpetual observation must be quite unnecessary, and can only be considered a useless piece of extravagance. When we bring to bear on this question the knowledge of the second rotation of the earth, the confusion and incon- K 130 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. sistencies vanish. The polar distance of a star and its annual rate can be calculated for 1000 years past or future as easily and more correctly than it can now be arrived at by any known system for even ten years. The star ]3 Draconis will be taken as an example. The constants for this star are as follows : C = 9 17' 38". C P = 29 25' 47". The angle P C ft, January 1, 1887 = 148 8' 0". Annual variation in the angle P C ft 40*9". Find the polar distance of this star for 1000 years in the past, viz. January 1, 887 A.D., and the annual rate of change in polar distance at that date. The rate of the second rotation being 40*9" annually, the angle at C 1000 years ago would have been 40900" -11 21' 40" less than it was in 1887. Consequently, at 887 A.D. the angle at C was 148 8' 0" - 11 21' 40" = 136 46' 20". The supplement of this angle is 43 13' 40". We have now two sides and the included angle to find the third side P )3, the polar distance of the star at 887 A.D. The whole of the working of this question is given below. Log. cosine, 43 13' 40"= 9-8625110 Log. tangent, 9 17' 38"= 9-2139074 9-0764184 = log. tan. of first arc, 6 47' 59". 29 25' 47" 6 47' 59" 36 13' 46" = second arc. Log. cosine, 9 17' 38"= 9-9942612 Log. cosine, 36 13' 46"= 9-9066887 19-9009499 - Log. cosine, 6 47' 59" = 9-9969344 9-9040155 = log. cos., 36 42' 24". The polar distance of j3 Draconis, January 1, 887, was 36 42' 24". SO-CALLED PROPER MOTION OF THE FIXED STARS. 131 We can now find the approximate annual rate of change in polar distance at that date, as follows : Substitute 1050 years for 1000 years, and the angle C for 837 A.D. will be 136 12' 15", and its supplement, 43 47' 45"; proceed as in the last example, and calculate the polar distance of |3 Draconis for 837 A.D. The calculation gives 36 39' 17". The difference for fifty years was therefore = 36 42' 24" - 36 39' 17" = 3' 7"; that is, 187" for fifty years, or at the rate of about 374" annually at the date 887 A.D. We can now test these results with the observations recorded in the Nautical Almanac for 1887. The recorded north polar distance given in the Nautical Almanac for 1887 is Calculated for 887 = 37 36' 52-83" 36 42' 24" 54' 24-83" difference for 1000 years. 54' 24-83" = 3268-83" for 1000 years, or a mean rate of 3-268" per year. The annual rate given in the Nautical Almanac for 1887 for ]3 Draconis is 2-8" Annual rate calculated for 887 = 3*74' 2 | 6-54" 3-27" The difference between 3'27" and 3'268" is two-thousandth of a second only; but this difference even is not "proper motion " in the star, but is due to the fact that the annual rate does not and cannot vary uniformly. It is a remarkable exhibition of the tenacity with which men blindly follow a routine system which is erroneous, to find a multitude of observers continuing to measure year after year the meridian zenith distances of stars, and imagining that the slight unaccountable dif- 132 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. ferences which occur are due either to the proper motion of the stars, or to these stars having a parallax. How their zeniths and meridians are affected annually and semi-annually by "a conical movement of the earth's axis," is a question which they never seem to have imagined as of any importance. Hence we find pages of a scientific journal filled with the results of observations supposed to prove certain facts as regards distant stars, when these so-called facts have no foundation in truth. Thus errors are circulated far and wide, theories are based on these errors, and confusion reigns supreme. For men to base conclusions on the various meridian zenith distances of stars when they do not know how their zeniths or meridians move, and to refuse to examine how they do move, is a proceeding similar to that adopted by the followers of the system of Epicycles, who refused to examine the effects of the daily rotation of the earth. ( 133 ) CHAPTER X. THE POLE-STAR. PERHAPS one of the most important stars in the heavens, as regards astronomical observations, is the pole-star (a Ursse Minoris). This star, being at present very near the pole of the heavens, varies its altitude during each daily rotation of the earth only twice the amount of its polar distance, or less than 2^. This star, consequently, can always be seen from such latitudes as those in which England and Europe are situated, and, if the night be clear, the meridian transit of this star can be observed each night during the year. It would occur, during summer, that the meridian transit, both above and below the pole, might take place during daylight. Yet, in spite of all difficulties, the meridian transit of this star, either above or below the pole, can be frequently observed. It is sometimes claimed, as a proof of the important and valuable work carried on at an observatory, that more than one hundred observations have been made by observers of the pole-star during the year. Whether this work is as important as is imagined, will be left to the judgment of the reader who examines the facts in this chapter. On bringing to bear on the pole-star the knowledge of 134 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the second rotation, some interesting facts are revealed; and by aid of this the polar distance of this star will be calculated for the date 113 B.C. that is, 2000 years from 1887 A.D. The constants for the pole-star are as follows : C P (Fig. 46), from pole of second rotation to pole of daily rotation, 29 25' 47". C a, from pole of second rotation to Polaris, 29 52' 51". Angle a C P, January 1, 1887 = 2 27' 5". Annual variation in angle at C = 40.9". . ^ During 2000 years the angle at C would have been decreasing at the rate of 40.9" per year; consequently this angle 2000 years in the past would have been 22 43' 20" greater than it was on January 1, 1887. The angle C, at 113 B.C., was therefore 25 10' 25". With the two sides, C a = 29 52' 51" and C P - 29 25' 47", and the included angle at C = 25 10' 25", the side P a can be calculated, and will be found 12 23' 16", which was the polar distance of Polaris 113 B.C. Fifty years previous, viz. at 163 B.C., the angle at C was 25 44' 30", and, calculating as before, the polar distance of Polaris for 163 B.C. was 12 39' 46". Between 163 B.C. and 113 B.C. the polar distance of Polaris decreased 16' 30," being at the rate of 19'8" per year, between those periods. Comparing these items with those in the Nautical Almanac for 1887, we obtain the following : Polar distance Polaris 1887 = 1 17' 38-2" Polar distance Polaris 2000 years previous = 12 23' 16" Difference = 11 5' 37'8" The difference 39937'8" divided by 2000 gives 19'967" for the mean rate between these two periods, according to the difference in polar distance at the two dates. THE POLE-STAR. 135 The rate of decrease in polar distance 113 B.C. = 19*8" The rate of decrease 1887 A.D. in Nautical Almanack = 18-923" 2 1 38-723" Mean 19-361" We have here an excellent opportunity of testing the method recommended by Professor Baily for finding the proper motion of stars. We find 19' 967" the rate of change according to the difference in polar distance of the star Polaris at the two dates, but we find 19*361" as the mean of the rates at the two periods. According to the system recommended by this authority, the difference between these two items, viz. 0*606", would be assigned as proper motion to the pole-star annually. Here is an admirable opportunity for the theorist to step in and to invent some wonderful theory as to the cause of this supposed proper motion in the pole-star. Every person, of course, knows that the whole solar system is rushing towards the constellation Herculis; this movement has been accurately proved by authorities. Now, however, something equally as wonderful can be invented. As the pole-star has such a large proper motion, the solar system is probably also rushing towards the pole-star, and a grand collision must occur between the sun and the pole-star, with results fearful to contemplate. This theory is recommended to those gentlemen who lecture to audiences, with a view to astonishing these, for it is not more baseless and untrue than many statements on astronomy which, when made, are loudly cheered by the wonder-loving listeners. The pole-star has no " proper motion." The system practised in order to find the proper motion is incorrect. The difference, viz. 0*606" per year, found by taking 136 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the mean rate by the two systems, is an additional proof, were one necessary, of the true movement of the pole, and of the accuracy of geometry. We have now before us a problem worthy the attention of geometricians and astronomers, and of a very different class from that of merely adding or subtracting a quantity found by observation, in order to give the polar distance of a star for half a dozen years in advance. This is, to show why there is this difference of 0'606" as found above, and to prove that this is not " proper motion " in the pole-star. The course which the pole of the heavens traces on the sphere of the heavens, in consequence of the second rota- tion of the earth, is a circle, having for its centre a point 29 25' 47" from the pole of daily rotation. This rotation occurs at the rate of 40*9" annually. At the date 1887 A.D. the pole was at P (Fig. 47); at the date 113 B.C. the pole was at O; the pole-star being a fixed point at a, and C, the centre of the circle traced by the pole, being a fixed point at C. From 113 B.C. to 1887 A.D. the pole has been carried along the curve O P. When the pole was at O, the annual rate of decrease of the pole from a was 19*8". When the pole was at P, the annual rate of decrease of P from a was only IS'923", and the mean of these rates gives only 19*361". Whereas when the less polar distance is taken from the greater, and the remainder divided by 2000 (the number of years between the dates), we find that the pole must have approached the pole-star at a mean rate of 19'967" per year. It would be very easy, but very erroneous, to put this apparent inconsistency down to " proper motion; " but the true cause is so very simple, and proves so accurately the THE POLE-STAK. 137 course of the pole, that it is quite unnecessary to invent any fantastic theories to account for the facts. The truth is, that the pole, when at O, was not moving directly towards the pole-star, and when at P the pole was not moving directly towards the pole-star, but somewhere between these two dates the pole ^uas moving directly towards the pole-star, and was therefore decreasing its distance annually from the star, at a rate greater than that at which it decreased it either at O or P. When was the pole moving directly towards the pole- star, is now the question for solution. The problem is a simple one. The pole was moving directly towards the pole-star when an arc joining the pole-star and the pole was at right angles to the arc joining the pole of second rotation with the pole of daily rotation, and to discover the date of this event presents no diffi- culties. We have merely to investigate the case of a right- angled spherical triangle, as follows : C a (Fig. 48) is the hypotenuse = 29 52' 51"; from the pole of second rotation to the pole-star P, O P = 29 25' 47". The angle at P is a right angle when the pole is moving directly towards a. The value of the angle at C has now to be ". p calculated for these conditions, as follows : Log. tan., 29 25' 47" = 9-7513982 Log. cotan., 29 52' 51" = 0-2406491 9-9920473 = Log. cos. < C = 10 55' 53". At the date 1887 the angle at C was 2 27' 5", which subtracted from the above angle, leaves 8 28' 48" for the angle of second rotation between 1887 and the date when the pole was moving directly towards the pole-star. 138 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. 8 28' 48" = 30528", which, at th^rate of 40.9" per year, occupied 746*4 years. 746-4, taken from 1887, gives the date 1140'6 A.D. for the period when the pole of daily rotation was being carried directly towards the pole-star, and when, consequently, the annual decrease in polar distance of this star was greatest. The rate at which the pole, at the date 1140*6 A.D., ap- proached the pole-star was 40.9" x sine of 29 25' 47" = 20-09" per year. From the date 113 B.C. to the date 1140*6 A.D., the rate at which the pole increased its rate of approach to the pole-star was from 19'8" at 113 B.C., to 20'09" at 1140'6 A.D. From 1140'6 A.D. up to the present time the rate has decreased, though by no means uniformly, until it has. reached the rate at present found by observation of about 18-923" annually. From the present date the annual rate will decrease rapidly, until at the date 2102*76 A.D. the pole will not vary its distance from the pole-star for that year more than 1". The polar distance of the pole-star will then be 27' 4", the nearest approach of the pole to the pole-star. After the date 2102-76 A.D. the pole will increase its distance from the pole-star nearly in the same manner as it decreased this distance. We trust that the reader will now be able to work out such simple geometrical problems as the mean polar distance of the pole-star, for thousands of years, without any reference to observations. It may, therefore, fairly admit of question, whether it is a proof that very important work is carried on at official observatories, when it is stated that one hundred observa- tions at least have been taken each year of the pole-star alone. It will occupy observers nearly 215 years to find out THE POLE-STAR. 139 that which a geometrician acquainted with the second rotation of the earth can calculate during ten minutes. Let us note, however, the wonderful state of confusion into which theorists and the unreasoning followers of authority would have placed themselves by following the advice of the supposed best method of obtaining the proper motion of the fixed stars. At the date 113 B.C. the rate was 19*8", at 1887 it was 18-923", giving a mean of 19'361". Yet when the polar distance at 113 B.C. is compared with the polar distance at 1887 we obtain 19'967". Hence, according to a great authority, the difference, viz. 0'606", is to be put down as the annual proper motion of the star. This assumed annual " proper motion " is imagined to be arrived at by so accurate a method that theories can be based on it, and promulgated as though they were facts correctly proved by geometry. Whereas the very first step is erroneous, and the conclusions based thereon are as false as though we founded theories on the assumption of the earth being a flat surface. It is certainly very remarkable that in the present day, when we have so much talk about competitive examinations, and the advantages of a mathematical and geometrical training, that we should yet find men, claiming to be scientific, submissively copying the elementary errors of those predecessors who are regarded as authorities. It is equally as curious to find also that there are persons who are employed to write in papers, professing to teach science to the public, who seem to imagine that when, like parrots, they can say that " the joint action of the sun and moon on the earth's protuberant equator makes a shift (sic) in the earth's axis," that they have uttered something very profound. Such writers must imagine that the intellectual capacity of their readers is of a very low order, or they would not presume to string together such feeble nonsense. 140 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. From such proceedings, however, we find a probable solution of the mystery as to how it was possible that, during two thousand years after the daily rotation of the earth was suggested, this fact was sneered at and rejected by the so-called scientific authorities of the civilized world, whilst the dogmatic theory of Epicycles and a stationary earth was stated to be such a grand and difficult science, that those persons who questioned its accuracy were asserted to be ignorant and stupid. THE STAR URS^E MINORIS. This star will serve as a second example of the law relative to the variation in rate at which the pole of the heavens varies its distance from a star. The following diagram (Fig. 49) shows the relative position of the star, the pole of daily rotation, and the pole of second rotation at the date January 1, 1887. C P = 29 25' 47"; that is, from C, the pole of second rotation, to P, the pole of daily rota- tion. C = 22 1' 447", from C to , the star. P for January 1, 1887 = 7 46' 41 '6". The angle P C c January 1, 1887 = 5 34' 14'9". The annual variation in the angle P C c = 40 -9". When the pole P was, at some date in the past, situated at 0, so that the star was on the arc joining O and C, the pole would then be moving at right angles to the arc joining O and , and at this date there would for an instant be no increase or decrease in the polar distance of the star . The polar distance of this star would then be C C = 29 25' 47" - 22 1' 447" = 7 24' 2'3". At the date January 1, 1887, the polar distance of was found to be 7 46' 41-5"; the difference is 22' 39'2". THE POLE-STAB. 141 The date at which the pole was at O can be found by dividing 5 34' 14-9" by 40.9", which gives 490'4 years from 1887, that is at 1397'6 A.D. Consequently, during 490*6 years the polar distance of this star has increased 22' 39'2"; that is, if we took the rate as uniform, at a mean rate of 2 "76" annually. But at 1397'6 A.D. the annual rate was nothing, whilst in the Nautical Almanac for 1887 this rate is given as 5'390". Hence, to discover the supposed proper motion of this star, we should take the rate at 1397*6 A.D. from the rate at 1887 A.D.; divide by 2, and compare this result, viz. 2-69", with 276" found above, and the difference 0'07" would be attributed to the proper motion of the star. The error of this method is so palpable that it seems almost unnecessary to demonstrate it, yet reference to it cannot be avoided when we find that a council of learned gentlemen considered its discovery and the conclusions based thereon were so valuable as to deserve the gold medal of the society. The two stars, Polaris and e Ursse Minoris, to which reference has been made, serve to show the principle on which the polar distance varies, and how the rate varies. The same principle, however, holds good for other stars, although, in a multitude of instances, to a less degree than it does to those stars which have been dealt with. The method of finding the exact polar distance of a star for any date when we are acquainted with the second rota- tion of the earth, with the rate of this second rotation, and consequently with the true course of the pole, may possibly appear to superficial investigators a very easy and simple problem. There are, however, certain laws affecting this problem, and dependent on the present method of making observations, which must not be overlooked. ' It must be borne in mind that observers in the present 142 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. day do not measure the polar distance of a star with their transit instrument. That which they measure is the meridian zenith distance of the star, and they deduce the declination or polar distance of the star from the measured meridian zenith distance. The manner in which the zenith and meridian are affected by the second rotation of the earth becomes, there- fore, a most important item for inquiry, and it will be found that it is one which will fully repay those geome- tricians who will take the trouble to investigate it. ( 143 ) CHAPTER XL THE NAUTILUS CURVE. IN the preceding pages mention has frequently been made of the mean polar distance of a star, and the mean obliquity for some particular year. To the reader unacquainted with the details of astronomy, this term " mean " has no doubt been unintelligible. Until about a hundred and fifty years ago, observations were made so imperfectly, that it was imagined that the pole of the heavens appeared to move uniformly during the year over its small arc of about 20 '09" annually. Bradley, a careful observer, about a hundred and fifty years ago, used an instrument termed " a zenith sector," with which he could measure meridian zenith distances with great accuracy. By the aid of this instrument, Bradley found that the zenith distance of the star j Draconis varied its zenith distance during the year in a manner that had not been accounted for by any theorist. At first Bradley concluded that the cause of this change in the zenith distance of the star might be due to some movement of the earth, and hence of the zenith, which had hitherto escaped notice. He soon, however, gave up this idea, and stated that this change in zenith distance of the star must be due to the velocity of light and the movement of the earth. This theory was at once accepted, and was termed 144 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. " aberration/' a title by no means inappropriate if applied to something else, in addition to the stars. The effect of this " aberration " is to cause the stars to vary their distance from the zenith during the year, in a manner which will be described; hence, when the mean polar distance of a star is referred to, the term indicates the polar distance of the star if it were unaffected by aberration, and also the small displacement produced in the pole by the " nutation," or small elliptical movement of the pole, which occupies slightly less than nineteen years. In order to make clear the effect produced annually on the zenith distance of a star, and on its supposed polar distance, and also on its right ascension, the nautilus curve will be made use of. The nautilus curve is constructed as follows : Take a nearly straight line, O P, of any convenient length, and divide this line into twelve equal parts, shown by the numbers 1, 2, 3, etc. This line is actually 20'09" of the arc of a circle the radius of which is 29 25' 47". With O as a centre set off the angle A O P = 30, and with O as centre describe the arc P A. From A 1 set off the angle A 1 B = 30, and with 1 as a centre and radius 1 A describe the arc A B. From B 2 set off 30, and with 2 as a centre and radius 2 B describe the arc B C. From C 3 set off 30, and with C as a centre describe the arc C D. Proceed in the same manner, setting off twelve angles of 30 each, amounting to 360, and the nautilus curve as show in the diagram and defined by P, A, B, C, D E, etc. is constructed. From P to A is termed January, A to B is termed Feb- ruary, B to C, March, and so on, as written in the diagram. The distance from O to P must be taken as 20 '09", and THE NAUTILUS CURVE. 145 a scale must be constructed in accordance therewith, as shown below the curve in the diagram. From this scale decimals of seconds can be measured. The production of the line P towards Q will be the a Fig 50. 10 iu 5 i ' ' ' "T 10 20" 50" 40" direction in which stars are situated which have twenty- four hours of right ascension. The direction P a?, at right angles to P O, will be the direction in which stars are situated which have six hours right ascension. L 146 UNTKODDEN GKOUND IN ASTRONOMY AND GEOLOGY. The line P produced will point to twelve hours right ascension, and P y to eighteen hours right ascension. This curve being very small, a star more than 1 from P will be so situated that lines or arcs drawn from the star to any part of the curve "will be nearly parallel to each other. These lines will not be quite parallel, as they must form an angle at the star, as will be shown, but they are nearly parallel. The more distant a star is from P, the pole of the heavens, on January 1 of any given year, the more nearly will the lines drawn from such star to any part of the curve be parallel to one another. The application of this curve will now be demonstrated. The pole-star, a Ursse Minoris, had on January 1, 1887, an apparent right ascension of Ih. 17m. 49 '23s., and a polar distance (apparent) of 1 17' 24", as stated to have been found by observation, and recorded in the Nautical Almanac for 1887. Ih. 17m. 49*23s. of right ascension converted into degrees and minutes will amount to about 19 27' 19". From P set off the angle Q P a = 19 27' 19", and the line or arc P a will be the direction of the pole-star. The distance of the pole-star from P on January 1, 1887, was found by observation to be 1 17' 24", which distance expressed in seconds is 4664". Hence if the pole-star's position were plotted on a diagram, on the same scale as that on which the nautilus curve is drawn, the pole-star would be in the direction of P a, and about thirty feet from P. It will then be seen that any lines or arcs drawn from a point thirty feet distant, and to any part of the nautilus curve, would be nearly parallel to each other. The observations that are now made in order to deter- mine the polar distance of a star are in reality meridian zenith distances, the polar distance being deduced from the THE NAUTILUS CUKVE. 147 zenith distance. Any changes, therefore, which may occur in any zenith, or in any meridian, or in any part of a meridian, would be transferred, as it were, by the present system, to a supposed movement in the pole of the heavens, or in the star itself. Let us now assume that these supposed movements are such as to cause the pole of the heavens to appear to move round the nautilus curve during a year, the course for each month being indicated on the diagram. The apparent effect as regards the pole-star's polar distance will now be examined, and the effects on its apparent right ascension will afterwards be dealt with. The apparent movement of the pole from P to A during January will cause the curve thus traced to slightly de- crease its distance from the pole-star. How much can be measured as follows : The curve being P A, join P A by a straight line A # P, and measure the greatest distance between the straight line and the curve. The greatest distance will be found at x, and by scale is ^ of a second, /^^ and would occur about the middle of Afig 51 January. On reference to the Nautical Almanac, 1887, page 310, it is recorded that the apparent polar distance of the pole- star found by observation, was as follows : January 1 1 17' 24" January 13 1 17' 23'2" January 21 1 17' 23 2" January 31 ] 17' 23*8" Thus between January 1 and the middle of the month, we have exactly -^ of a second decrease in the polar distance of Polaris, as indicated by the nautilus curve; whilst as the. curve approaches A the polar distance again increases. We can now examine the curve during the next month, February, viz. from A to B. 148 UNTKODDEN GROUND IN ASTRONOMY AND GEOLOGY. From an examination of the curve, it is evident that during February the polar distance of Polaris will increase, and the amount from the 1st to the 28th of February can be accurately ascertained as follows : From A draw A Z parallel to a P, and draw B Z at right angles to A Z, the dis- tance A Z measured by scale will give the increase of polar distance of Polaris during February. This distance by scale will be found 4'4" or 4'5". In the Nautical Almanac for 1887, the polar distance for Polaris is recorded as follows : February 1 1 17' 23'9" February 14 1 17' 25'6" February 28 . . . . 1 17' 28'5" During the first half of the month the decrease is only 17", during the second half it is 2'9", making a total of 4'6". Why the increase in the early part of the month is less than it is during the second half, will be evident by examin- ing the curve between A and B. An examination of the next month, March (from B to C), gives the distance for the increase in polar distance about 8'8 /r by scale. In the Nautical Almanac for 1887, the polar distance of Polaris is recorded as follows : March 1 .. .. 1 17' 28'7" ) , March 31 .. 1 17' 37'5" ) difference, 8-8 From an examination of the curve, it will be evident that during April the curve is moving more directly away from the pole-star than it is during either March or May; therefore between April 1 and May 1 there would be a larger increase in the polar distance than during thirty-one days in March. The amount by scale is about 9 "5". The amount given in the Nautical Almanac is THE NAUTILUS CURVE. 149 The next question to be investigated by aid of the curve is, at what date will the polar distance of Polaris be a con- stant? It will be a constant when the curve is moving at right angles to the arc joining the pole-star and the curve, a condition which will exist close to F in the curve. In the Nautical Almanac, 1887, the polar distance of the pole-star is given as a constant, viz. 1 IT 55*4?', from the 23rd to the 30th of June. From June 30 the polar distance of the pole-star will decrease, as shown by the curve, the amount in each case being capable of measurement as already explained. A very interesting fact will now be pointed out in con- nection with the nautilus curve, viz. that when this curve is moving directly towards the pole-star, the rate of decrease in polar distance will be at its greatest, and this will occur between I and J. But the distance between I and J is, by the construction of the curve, greater than is the distance between C and D. Hence the decrease for thirty days during October should be greater than the increase during thirty days in April. Measuring on the curve from I to J, we obtain about H'O" for the decrease. In the Nautical Almanac for 1887, the following is recorded : October 1 1 17' 33-9" ) , October 30 1 17' 23'0" } ^erence, 10-9" Whereas, during thirty days in April, the increase in polar distance is only 9'0". As the curve is carried round from J to K and on to P, the polar distance of the pole-star decreases, but not uniformly, the rate from day to day decreasing. When the curve approaches J near the end of October there will be found a point S in the curve, where a line drawn from P at right angles to P a cuts the curve. The 150 UNTKODDEN GROUND IN ASTRONOMY AND GEOLOGY. distance from this point S to the pole-star will be equal to the distance from P to the pole-star, the small distance P S, compared to the great distance P a and S a, causing P a and S a to be practically two sides of an isosceles triangle. The distance J S will be found by scale one-ninth of J S. Consequently, at about October 28, when the curve had reached S, the polar distance of the pole-star would be almost of the same value as it was on January 1 of the same year. In the Nautical Almanac for 1887, we find the polar distance of the pole-star recorded as follows : January 1 .. .. 1 17' 24" ) ,., October 28 . . . . P 17' 23T I difference & of a second - When the curve has reached L the 360 of the nautilus curve has been completed, and the arc P L indicates the apparent change in the position of the pole from January 1 to January 1 of the year following. In order to find how much nearer the point L is to the pole-star than the point P, we have merely to multiply the distance P L taken from the scale = 20 "09" by the cosine of the angle L P a, which angle is, for January 1, 19 19' 45", and we obtain 18'94" for the decrease in polar distance of the pole-star during the year 1887. In the Nautical Almanac for 1887, the decrease is recorded as 18'92", giving a difference of two-hundredths of a second. The reader will now probably comprehend that the effects of what is termed " aberration " are considerable during a year, and produce great changes in the supposed polar distance of a star near the pole. The other effect pro- duced by what is termed " nutation " is slight by com- parison, and at present need not be referred to. It will be evident that when an observer unacquainted THE NAUTILUS CURVE. 151 with the effect produced by aberration (viz. observers who made observations more than a hundred and fifty years in the past) states that he found the .polar distance of the pole-star of a certain value, but does not state the day of the month, but merely gives the year, his records are not of much value, inasmuch as, if he made these observations at the end of December, they would differ fully 50" from those made about the end of June of the same year. The effect on the apparent right ascension of the pole- star by the nautilus curve will now be briefly described. As the curve moves from P to A, the right ascension of the pole-star must decrease, and it will continue to decrease until a line or arc from the pole-star is a tangent to the curve. This, from an examination of the curve, it is evident, will take place near C in the early part of April. In the Nautical Almanac for 1887, April 7 is the date at which the right ascension alters its rate from decreasing to increasing. Again, it will be seen that about the middle of October, between I and J, a line from the pole-star to the curve will be again a tangent, at which date the right ascension of the star will vary from an increase to a decrease. In the Nautical Almanac for 1887, the change is recorded as occurring between October 13 and 16. Thus from January 1 until about April 7 the right ascension will increase; from April 7 till October 15 the right ascension will decrease. These dates will vary slightly each year, inasmuch as O P is really a curve, being a portion of the circle described by the pole of daily rotation round the pole of the second rotation as a centre. Another star, viz. X Ursse Minoris, will now be examined in connection with the nautilus curve. This star on January 1, 1887, had a right ascension of 152 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. 19h. 35m. 57s.; the direction of this star is plotted on the curve. It was distant by observation at that date 1 2' 12" from the pole. On examining the curve, it will be seen that the curve from P to A moves almost directly away from this star, the amount by scale being about 10". In the Nautical Almanac for 1887, the polar distance of this star is given as follows : January 1 . . . 1 2' 12" ) , -r o 10 ci, // difference. 10-2" January $1 . . 1 2' 22'2" 3 The increase in polar distance will continue until a line from the star to the curve is at right angles to the curve, which will occur at about one-third of the distance between C and I, or about April 10. The right ascension will slightly decrease through a portion of January; it will then increase. A line drawn from the star through P, and produced, cuts the curve about one-third of the distance from A to B, viz. about Feburary 10, at which date the right ascension will be the same as it was when the curve was at P. In the Nautical Almanac for 1887, page 310, we find the following : b. m. s. Right Ascension A Ursse Minoris, January 1 19 35 57-37 g February 10 . . 19 35 57-16 When a line from the star to the curve is a tangent to the curve, the right ascension will not vary during a day. From an examination of the curve, it will be seen that such will be the case just beyond F, or near the beginning of July. In the Nautical Almanac for 1887, we find the following recorded for X Ursse Minoris : h. m. s. Right Ascension, June 30 19 37 52-53 July 1 . . 19 37 52-62 July 2 19 37 52-63 July 3 19 37 52-53 THE NAUTILUS CURVE. 153 When a line from the star to the curve cuts the curve at right angles, there will at that date be no change in the polar distance of the star during two or three days. From an examination of the curve, it will be seen that this will occur about midway between I and J, or during October. In the Nautical Almanac for 1887, the following are the recorded polar distances of this star : October 15 1 1' 54'4" October 16 1 1' 54-4" October 17 1 1' 54'4" October 18 1 1' 54-3" November 1 1 1' 54-4" When the curve has reached L, on December 31, the polar distance of X Ursse Minoris will have decreased since January 1, 20'09" X cosine of the angle Q P A = 20'09" X cosine 66 = 8'171", which will be the approximate rate of decrease in polar distance of this star for the year 1887. In the Nautical Almanac for 1887, the rate given for this star's decrease in polar distance is 8*206", showing a differ- ence of thirty-five-thousandths of a second. The reader may take another star near the pole, viz. the star Cephei 51 Hev., the right ascension of which (apparent) on January, 1887, was 6h. 47m. 37'44s., and plot the direction of this star from P on the curve. 47m. 37'44s. correspond in round numbers to 11 44' 21". Consequently, by making at P an angle # P 51 Hev. 11 44' 21", we obtain the direction of this star. Every detail as regards the annual changes in Cephei 51 Hev. can be read off the nautilus curve; the dates at which the right ascension will be a constant, at which the polar distance will be a constant, etc., can be obtained at a glance. Any person who may not be provided with an obser- vatory or an expensive transit instrument can, by aid of 154 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the nautilus curve, measure the changes which occur annually in a star close to the pole, and can understand the principal causes of those differences in the polar distance or right ascension of such stars, and described by professional observers as produced by " aberration." Other stars near the pole may be selected and plotted on the diagram, such, for example, as S Ursse Minoris, X Ursse Minoris, and e Ursse Minoris, but an allowance has to be made on the annual effects when the star is at any great distance from the pole. The nautilus curve for the south pole will be in the opposite direction to that for the north pole, so that the curve appears as though seen through a transparent paper. The shape of the curve will be the same as for the northern pole, but merely reversed. The reader can construct such a curve for himself on any scale he may select. When a star is at a considerable distance from the pole of daily rotation, the annual changes as read by the nautilus curve will be the same in their general effect, though reduced in amount. Thus, for example, take the star 7 Draconis, which has a right ascension of about 17h. 54m. This star, it will be seen from an examination of the nautilus curve, will increase its polar distance rapidly during January, as the curve moves from P to A. It will increase its polar distance, but less rapidly, as the curve moves from A to B. At the end of March, viz. near C, a line from the star to the curve cuts the curve at right angles, at which date the polar distance will be a constant. The following extract from the Nautical Almanac for 1887 gives the polar distance of 7 Draconis on the first and other days of the month : January 1 38 29' 47'4" January 31 38 29' 57-2" February 10 38 29' 59-7" March 2 . 38 30' 3'2" THE NAUTILUS CUKVE. 155 March 12 38 30' 4'0" March 22 38 30' 4-2" > April 1 38 30' 3-7" chan S C8 ' April 21 38 30' 0-9" When a line from the star to the curve is a tangent to the curve, the right ascension of the star will not vary. An examination of the curve shows that this will occur about midway between E and F; that is, about the end of June. In the Nautical Almanac for 1887, the following are the recorded right ascensions of y Draconis : h. m. s. May 31 17 54 1-21 June 10 . . . f 17 54 1-34 June 20 17 54 1'41? l June 30 17 54 1-42 J ch ^g es - July 10 17 54 1-37 July 20 17 54 1-26 The right ascension will then decrease until about the middle of December, when a line from the star is again a tangent to the curve. Whilst, however, the polar distance of the star X Ursae Minoris varies during the year as much as 41 '3", that of j Draconis will vary only about 38 '2". The nearer we approach the equator of second rotation, the less will the change be in polar distance of a star thus situated. Hence, if we project the curve on the plane of the equator of second rotation, we obtain a very close approximation to the effects produced as regards polar distance. Consequently, a star with eighteen hours right ascension, and, say, 38 40' of north declination, would be about 68 5' from the equator of slow rotation; whereas a star with six hours right ascension and the same declination would be only 8 15' from the equator of slow rotation. The star with eighteen hours right ascension would vary its polar 156 UNTKODDEN GEOUND IN ASTEONOMY AND GEOLOGY. distance considerably during the. year; the star with six hours right ascension would vary it very slightly during the year. The two stars a Lyrse, with about 18h. 33m. right ascension, and Castor, with about 7h. 38m. right ascension, will serve as examples. The reader who is desirous of investigating the nautilus curve may test its effects as regards the general results on the following stars, the direction of which he can draw on the diagram : 6 Ursse Majoris a Urs8B Majoris A Draconis . . 7 Ursse Majoris a Draconis . . Ursse Minoris s Ursse Minoris ?7 2 Draoonis . . e Ursae Minoris a Cephei 2 Cephei . . 7 Cephei Right ascension approximate, h. m. 9 25 10 56 11 24 11 48 14 1 14 51 15 48 16 22 16 57 21 16 21 27 23 34 He will find, on reference to the nautilus curve, that the right ascension varies exactly in the manner that has been described, and that the polar distance also varies in the manner described, the amount of change in polar distance being decreased slightly as the star's place is nearer the equator of slow rotation. A comparison can be made of the changes in right ascension, etc., as found in the nautilus curve, and those recorded in the Nautical Almanac for each star as found by observation. The nautilus curve, although probably a novelty to the reader who is not acquainted with everything in the science of astronomy, ought to be thoroughly known to astronomers. About ten years ago, when I had tested the accuracy of this curve as a means of demonstrating the annual changes in THE NAUTILUS CURVE. 157 right ascension and polar distance of important stars, I thought, as it was quite new, it might be of interest to astronomers. I therefore drew the curves for the northern and southern hemispheres, and, adding a full written description with numerous examples, I forwarded these to the Royal Astronomical Society. Being aware that the pages of the monthly notices of the society were, to a great extent, filled with the recorded observations of various observers who did not seem aware how their zeniths or meridians were affected by that move- ment of the earth, vaguely defined as " a conical movement of the axis," I believed that attention might be directed to something novel, and certainly interesting. I found, however, that I had made a mistake that to submit for investigation an original subject was a most improper proceeding. Had I worked out the assumed proper motion of some hundred stars by means of the won- derful formula proposed by Professor Baily, my routine work would probably have been considered of some value, and would most likely have been printed and circulated; but the curve herein described was a novelty treated in a different manner. I was informed that my paper had been received, and was placed among the records of the society a polite way of describing " pigeon-holing " the document. I would not for a moment question the soundness of the proceedings of those who considered it desirable that my paper should be suppressed. It is probably due to some feebleness of intellect on my part that I am puzzled to comprehend why the gold medal of a scientific society should be given to a gentleman for a paper containing, as a base, a most elementary error in geometry, whilst another paper giving an original curve, by aid of which the changes in stars can be read off, was suppressed. 158 UNTEODDEN GEOUND IN ASTEONOMY AND GEOLOGY. If, some two thousand years^ ago, when the learned authorities all agreed that the earth was a flat surface, a geometrician had submitted to these gentlemen a paper giving such a proof of the spherical form of the earth as is contained in Chapter I. of this book, it is easy to predict the treatment which such a geometrical proof would have received. The audacity of any man who presumed to question the infallibility of the opinions of these ancient authorities must be put down at once, his paper suppressed, and his sound proofs treated as the learned Herodotus tells us, " only as subjects for laughter." If, some five hundred years ago, an individual had sub- mitted to the astronomical authorities a paper giving the results of his experiments with the pendulum, as a proof of the earth's daily rotation, those gentlemen would un- doubtedly have stated that they did not agree with the conclusions arrived at from such experiments, and they might probably suggest that the author of the paper could not understand the perfection of the Epicycles of Ptolemy, which proved that the earth could not move. Such proceedings in ancient times can be understood, but we have a far more difficult problem to deal with at the present day. We have now various learned societies, formed specially for the purpose of inquiring impartially on any problems connected with the special science for which the society was organized. The history of the past teaches us that erroneous theories were accepted as grand truths by all the scientific authorities of the whole world during more than five thousand years; and to these gentlemen it appeared that nothing more absurd and impossible than the daily rotation of the earth could be submitted to them. We have, however, in the present day, a defect as regards the so-called scientific training of students, which ought at once to be remedied. THE NAUTILUS CURVE. 159 In our schools and colleges, the science of geometry is taught, the earliest text-book usually being Euclid. From that book certain laws are made known relative to circles, ellipses, other curves, straight lines, and points. ^We there learn that the centre of a circle cannot vary its distance from the circumference; that a curve cannot increase or decrease its distance from a point at a uniform rate, and that the variation in the rate cannot be uniform. According to certain popular theorists, all this geometry is wrong. The centre of a circle can, and does, vary its distance from the circumference, and yet always remains the centre of the same circle; for does not the pole of the heavens trace a circle round the pole of the ecliptic as a centre, and yet decreases its distance from this centre about 46" per century? A curve can, and does, decrease its distance from a point at a uniformly increasing or decreasing rate; for was not the gold medal of the Royal Astronomical Society given for a paper on the supposed proper motion of the fixed stars, in which the assumed constant increase and decrease in rate was the base of the conclusions? The laws of geometry teach that the extension of the arctic circle and tropics on a planet are dependent on the angle which the axis of daily rotation of that planet makes with the plane of its orbit; that if, from any cause, say a slight alteration in the position of the centre of gravity of the planet (such as the transfer'of the waters of the oceans from one hemisphere to the other), the axis of the planet altered the angle it made with its orbit, then there would be a variation in the extent of the arctic circle and in the tropics. The movements of the planet itself appear to the geometrician the important question for investigation, not the movement of the plane of its orbit. 160 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. But the movement which does or may take place in a planet cannot in any way affect the angle which the axis of this planet makes with its orbit, for has not M. La Place stated that, as the plane of the ecliptic cannot vary more than 1 21', therefore no change greater than 1 21' in the extent of the arctic circle ever did or ever can occur? To the mere reasoner acquainted with geometry, it appears that if the axis of Jupiter has " a conical movement " round a point only 10 from the pole of daily rotation, there would be, during one complete conical movement, a variation of 20 in the arctic circle, although the plane of the orbit of Jupiter did not vary one second. But this must be an error of geometry, because the French theorist has asserted that no change greater than 1 21' can occur in the arctic circle, because the plane of the ecliptic cannot vary more than that amount. Hence we are led to conclude that no matter how a planet moves, yet any such movement produces no effect on the extent of the arctic circle or tropics, unless the plane of that planet's orbit moves; whereas geometry teaches quite a different law. To the mere reasoner, it appears by no means impro- bable that, because the obliquity of the ecliptic was the problem for inquiry, the great French theorist became a little mixed about the geometrical laws, and imagined that no change in this obliquity could occur except by a change in the ecliptic. He seemed to overlook the fact that no change whatever could occur in the obliquity, no matter how much the plane of the ecliptic varied, if the theory were correct that the pole of the heavens always traced a circle round the pole of the ecliptic as a centre. The subject, however, which requires most serious atten- tion is, whether students should continue to study Euclid and geometry. THE NAUTILUS CURVE. 161 Many years are now devoted to these subjects, which are supposed to give knowledge of sound laws, immutable and unchangeable. If, however, when we come to such a science as astronomy, these laws are to be treated with con- tempt; if a paper containing a geometrical impossibility receives the gold medal of a learned society; if one of the asserted movements of the earth contains another geometrical impossibility; if the accepted assertion of a theorist inter- feres with another geometrical law; then a knowledge of geometry must be a great detriment to a student who wishes to gain the favour of the present reigning scientific authorities. If he wish to be accepted as a learned man, he ought to ignore geometry and its laws, and should learn by rote the assertions of admitted authorities, and bow to these in the same manner in which the astronomers during 1400 years bowed to the assertions of Ptolemy, and his feeble-minded copyists. Any young man who may be desirous of gaining the approval of those persons, who, holding highly paid official scientific positions, have consequently considerable patron- age, would really ruin his prospects if he were more con- vinced of the accuracy of geometry than he was of the theories of these authorities. It is, consequently, a serious question for the consideration of those gentlemen who regulate public education and examinations, whether geometry should not now be struck out, and the theories of authorities be substituted in its place. On those persons who can prove the accuracy of the laws of geometry, the assertions and theories of those persons, which are opposed and contradicted by these laws, produce the same effect as when statements are read, " that it was utterly impossible for a steamship to cross the Atlantic; " " that if the earth rotated, all the water would M 162 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. be flung off; " " that if the earth were spherical in form, the people underneath would fall off, as it was impossible for a man to walk head downwards on the ceiling," etc. The gentlemen who made these objections were all authorities in their day, and no doubt meant well; but, unfortunately, they did not know. They succeeded for a time in burking truth, but only for a time, and as it has been in the past, so it is only reasonable to assume it will be in the future. We have, however, at the present time to make a selection: we must reject those theories, no matter from whom they emanate, which are contradicted by the laws of geometry, or we must reject geometry as a false and mis- leading science. The option is left to the reader. So baneful, however, is the influence of mere authority, that there are a multitude of unreasoning individuals who would more readily believe in the infallibility of the theories of M. La Place, however much these might be contradicted by the laws of geometry, than they would accept as correct the accurately proved problems in Euclid. ( 163 ) CHAPTER XII. THE ZENITH AND THE MERIDIAN. THE belief which has prevailed among observers during the past two hundred years appears to be, that it is only necessary to know how much the pole of the heavens changes its position in the heavens during the year, and then, as the meridian must, from its very name, pass through the pole, the meridian zenith distance of a star will enable the declination or polar distance of this star to be immediately deduced therefrom. This belief indicates that it is assumed that whatever change occurs in the direction and movement of the pole, must occur in each zenith and meridian. It also assumes that, no matter where a locality may be situated on earth, yet the zenith and meridian of this locality will be affected by "a conical movement of the earth's axis" (whatever such a vaguely defined movement may mean) exactly in the same manner as the zenith and meridian of any other locality is affected. These assumptions are not only with- out any foundation, but are erroneous, and they have been assumed mainly because it has been taken for granted that " a conical movement of the earth's axis " was a full and satisfactory explanation of that which really caused the pole 'of the heavens to change its position in the heavens. When it is remembered that the earth is the actual 164 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY, instrument with which observations are taken, it will be evident that it is essential that we know every detail of the movements that occur in this instrument. Let us refer to the following diagram (Fig. 52) as an example. N Q S E represents the earth; N and S, the north and south poles; E Q, a portion of the equator; N A, N B, N C, etc., various meridians of terrestrial longitudes. "The earth's axis has a conical movement," is the present accepted theory. If the axis does trace a cone, either the south point S or the north point N must remain fixed, whilst the pole which was not fixed described the base of the cone. Which pole remained fixed, and which pole described the base of the cone, has hitherto not been mentioned by theorists. It was, of course, thoroughly believed that the joint action of the sun and moon produced this conical movement of the axis, but perhaps a geometrician might imagine that, as a preliminary step, it would have been a more sound proceeding to define what the movement really was, than to invent a supposed cause for some vague move- ment. If the earth's axis trace a cone, as has been asserted^ then one pole must remain fixed, whilst the other gyrates. This is sound geometry, and as the truth of this science rests on a firmer base than the assertions of theorists, it must hold a stronger position. If neither pole remain fixed, whilst the two poles describe circles in the heavens, then the axis cannot trace a, cone, but will trace two cones, provided that the apex of each is within the earth, or a portion of a frustrum of a cone, provided the apex is outside the earth. From the manner in which this supposed movement has THE ZENITH AND THE MERIDIAN. 165 t>een described by authorities, and the diagrams given to explain to their readers what the movement really is, it appears to have been assumed that the south pole remains fixed whilst the north pole gyrates. A peg-top or teetotum is referred to as an illustration of the movement of the earth, whereas there is no similitude between the two. Taking, however, the theory as at present accepted by authorities, the following questions become of great and important interest. Refer to the last diagram, and suppose that the earth's axis has performed 10 of its conical movement, so that the pole N has moved over 10 of its circular arc. Then mark on the sphere the position of the various zeniths A, B, C, D, etc., as affected by this change in direction of the earth's axis of 10. Will the zenith D be carried directly up the meridian towards N, or will it be carried in any other direction? Will the zenith B be carried directly towards N, or obliquely in some other direction? In fact, mark on the sphere the exact positions of these zeniths as affected by " a conical movement of the earth's axis." It is almost unnecessary to state that the solution of this problem is beyond the knowledge of any professional astronomer. If he mark the position which he imagines any zenith will occupy, he at once gives a definite move- ment to the earth. If he assert that the zenith D moves up the meridian D N, he must claim that the south pole remains fixed during the conical movement. If he claim that the zenith D is carried to the left, he must claim that some other movement of the earth occurs in connection with the conical movement; and what is this other move- ment.? Are we to leave so important a problem as the move- ment of the earth the actual instrument with which we 166 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. observe vague and undefined, and then, by employing at great expense numbers of observers and computers, con- tinue to muddle on for a few years in advance, by aid of perpetual observations, in order to frame an astronomical almanac for two or three years in the future? Does not the very fact of the necessity of these perpetual observa- tions prove that the results for the future cannot be calcu- lated by the simple laws of geometry? The effects on each zenith and meridian of the second rotation of the earth will now be defined, and the reader will then perceive how very simple these changes become, but how hopelessly complicated they would appear to those persons who were unacquainted with this fact. Take a point, x, on the sphere (last diagram, Fig. 52) 29 25' 47" from the pole N, the point x being the pole of the second axis of rotation. The pole N of daily rotation is carried round x as a centre, and each zenith, A, B, C, D, etc., is carried round x as a centre. Whilst moving round a? as a centre, the zenith B is carried nearly parallel to the equator, E Q, but the zenith y on the same meridian is carried nearly at right angles to the equator, this zenith y being also carried round x as a centre. How, then, is the readjustment of this change to be effected? It is accomplished by the daily rotation of the earth a movement which affects the apparent right ascen- sion of stars, according to the present system of measuring right ascensions. This change in the meridian and zeniths due to the second rotation, and hitherto unknown to theorists, is one of the causes which produces effects hitherto erroneously attributed to " a proper motion " in the stars themselves. In each case a point on the earth's surface is carried over an arc round x as a centre; the pole N is carried over THE ZENITH AND THE MERIDIAN. 167 an arc of about 20 "09" annually, but a point on the earth's surface 90' from x will be carried annually over an arc of 40-9". The direction of the arc over which the pole N is carried is approximately towards the first point of Aries, but the direction in which a point D is carried is towards D', and the arc D D' is greater than the arc over which the pole N is carried, because the distance D x is greater than the distance N x. A point on the meridian N A, so situated that an arc from x to this point is at right angles to the meridian N A, will be carried directly up this meridian, consequently towards N, the position of the pole, whilst a point on the equator of this same meridian will be carried at right angles to the arc joining x to this point. As the centre of gravity of the earth remains fixed as regards both the daily and second rotation, the variations in position of different localities on earth, as produced by the second rotation, causes a marked difference in the position, not only of the zeniths, but also in the horizon of various localities. How varied are these differences may probably be better understood by means of a careful examination of the following diagram (Fig. 53). P represents the north pole of the earth at a date in the past; T S R Q V, the circle traced by the zenith of a locality, say, in 51 north latitude during one daily rota- tion. The circle F E B A G represents the circle traced during the same daily rotation by the zenith of a locality, say, in 68 north latitude. x represents the pole of the second axis of rotation, round which the pole P and the various zeniths describe circles during one second rotation. P T, P S, P R, etc., represent various meridians. 168 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Any number of years after th pole of daily rotation was at P, it is carried to P' round # as a centre, the direc- tion of P P' being nearly towards the first point of Aries that is, nearly towards 360 of right ascension. No movement of the earth can alter the angular distance between a given locality and the poles, as long as the axis remains fixed in the earth. With P' as the new position of the pole of daily rota- tion, we can trace two circles, represented by the dotted lines, to mark the circles traced during a daily rotation by the zeniths of the two localities before mentioned, the angular distance of P' from the dotted circles being equal to the angular distance of P from the continued circles. We can now trace out and calculate the exact effects on various zeniths produced by the second rotation of the earth. The pole P is carried to P' round x as a centre; the arc P x = F x = 29 25' 47". It must be remembered that if a person were located THE ZENITH AND THE MERIDIAN. 169 exactly at P, the north pole, his zenith would be transferred from P to P' on the sphere of the heavens. The zenith which was at T will be transferred to T', the arc T T' being traced round x as a centre. The zenith S is transferred to S', R to R', Q to Q', to 0', V to V, etc. The zenith F of the higher latitude is transferred to F', E to E', D to D', B to B', A to A', and G to G', each of these small arcs having the point a? as a common centre. The meridian which, when the pole was at P, traced the arc P F T, is transferred by the second rotation to P' F' T'. The meridian P E S is transferred to P' E' S'. The meridian P D R is transferred to P' D' R', thus intersecting the position of the former meridian. All the meridians from P S round by P to P Z will intersect each other in consequence of the second rotation; that is, the meridian of say fifteen hours for 1887 will intersect, near the pole of daily rotation, the meridian of fifteen hours for 1886. In the direction from hours to six hours, and from six hours to twelve hours, the meridian of one year will not intersect the meridian of the year previous. Between about sixteen hours right ascension and twenty hours right ascension the most marked changes will occur, in consequence of the second rotation. In order to understand these changes, the reader must bear in mind that the second rotation is in opposition to the daily rotation. For example, if a mere conical move- ment of the axis occurred, then whilst the pole P was carried to P', the zenith S would be carried up the meridian towards N. When, however, the second rotation is compre- hended, it will be seen that, whilst the pole is carried from P to P', the zenith S is carried to S', and it has to regain the former position by means of the daily rotation. The zenith D, however, is carried by the second rotation 170 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY, to D', over an arc which coincides with the meridian, RDP. In the immediate vicinity of x (the pole of second rotation), the changes in the zenith, both as regards direction and amount, will be very variable. All points on the earth's surface between x and P will be carried by the second rotation in the same direction as they are by the daily rotation, whereas all points on the earth's surface between x and the equator will be carried by the second rotation in a direction opposite to the daily rotation. There will also be remarkable changes in or near meridians of six hours right ascension. By the daily rotation, an arc of the equator or equinoc- tial joining two meridians will be greater as effected by the daily rotation than will be an arc on these two meridians joining two zeniths at a distance from this equator. A zenith, say 29 25' 47" north of the equator of slow rotation, would be carried annually by the second rotation over an arc, in opposition to the daily rotation, exactly equal to the arc over which a zenith 29 25' 47" south of the equator of slow rotation would be carried. But that part of the meridian of six hours right ascension which is 29 25' 47" south of the equator of second rotation, is on the equator of daily rotation. Hence, a zenith of a locality in latitude 59 51' 34" north would be carried annually by the second rotation over exactly the same arc as would that portion of the meridian of six hours right ascension which was on the equinoctial. Between the equator at six hours right ascension and the zenith of a locality 29 25' 47" north of the equator of second rotation, the meridian would be carried over an arc by the second rotation greater in amount than was this zenith on the equator of daily rotation. Hence, when we come to stars in this part of the THE ZENITH AND THE MEKIDIAN. 171 heavens, we encounter a very marked condition of confusion when their positions have to be determined by perpetual observation, and their right ascensions by their meridian transits. Consequently, theorists have attributed to these stars endless variations of proper motions, which so-called proper motions are in the zeniths and meridians, not in the stars. A very fair example of this law is afforded by the star a Aurigse (Capella). The constants for this star are as follows (Fig. 54) : P C = 29 25' 47". From C, the pole of second rotation, to P, the pole of daily rota- tion, C a = 73 2' 39 '5", a being the star. Angle P C a, January 1, 1887 = 9 21' 39'8". Annual decrease (apparent) in the angle P C a = 37'01". Kesults : Date. Mean polar distance by observation. By calculation. January 1, 1887 1873 . . . . 44 8' 3-04" . . . . 44 8' 3" 1850 . . . . 44 9' 39-4" . . . . 44 9' 39-5" 1843 . . . . 44 10' 7-42" . . . . 44 10' 8-6" 1819 . . . . 44 11' 53" . . . . 44 11' 52-2" 1780 . . . . 44 14' 43" . . . . 44 14' 44'8" The varied effects which are produced on the meridian zenith distance of a star as seen below the pole and above the pole, and also its time of transit across the meridian, afford some very interesting problems. For example, suppose a star to be near the meridian F P G, and near the pole P of daily rotation. Each year the zenith F is carried over a much longer arc by the second rotation, in opposition to the daily rotation, than is the zenith G. Consequently, when right ascensions are counted from an imaginary standard of time, such as the successive transits of the first point of Aries, we obtain some singular confusion in this item. 172 UNTKODDEN GKOUND IN ASTRONOMY AND GEOLOGY. The principal cause, however, .of the confusion which now exists, and which necessitates the continuous observa- tions now supposed to be essential, is due to the following fact. A meridian, although an arc of a great circle, appears to an observer as a straight line passing through the pole and the zenith, and cutting the equinoctial and horizon at right angles. The transit instrument used for measuring zenith distances and meridian transits moves in this imaginary straight line. If the movement of the earth were merely a change in the direction of the axis, without a second rotation, there N N would be less confusion than at present exists; but when the second rotation occurs, we have the changes in every meridian. As an example of one form of these changes, a meridian of six hours right ascension will be referred to, and reference made to a locality on earth in 51 north latitude. The circle N O S K (Fig. 55) represents the earth; N the north, S the south, pole; T R, the equator; Z, a locality in 51 north latitude; H, the southern point on the horizon 90 from Z, as seen on the sphere of the heavens. O Q K represents the trace of the ecliptic; E R, the trace of the equator of slow rotation. THE ZENITH AND THE MERIDIAN. 173 At the end of a year, if no change in direction of the earth's axis occurred, the meridian N Z T H S would, after say 366 complete sidereal rotations of the earth, coincide with its former position, viz. N Z T H S. But during one year a change does occur in the direction of the axis; the pole N is carried to N', and the pole S i& carried to S', the arcs N N' and S S' being each equal to about 20-09" for each year. The points N and S, although spoken of as the poles of the earth, are merely points on the earth's surface unaffected by the daily rotation. Both these points have a zenith, and this zenith changes its position in the heavens about 20-09" annually. But the north and south poles are not the only points on the earth's surface that have zeniths which are affected by some movement of the earth, which movement is inde- pendent of the daily rotation. In what manner is the zenith of Z affected? In what manner are the zeniths of E, Q, T, and H affected by that movement of the earth which causes the pole N to be transferred to N', and S to S'? Strange as it may appear to any reasoner, this is a question in astronomy which has never been dealt with, We have any number of visionary imaginings presented to us by theorists as regards the supposed cause of some move- ment of the earth, but what this movement really is has never hitherto been defined. Although, from the most simple observations, it is known that the pole N is carried to N' over an arc of about 20 f 09" annually, it has never even been hinted as to the amount or direction in which the zenith of latitude 51 north has been carried, or the amount and direction in which the zenith of any other locality is annually carried. Why is there this omission from a science claimed to be so exact as is astronomy? The answer is very simple, 174 UNTEODDEN GROUND IN ASTRONOMY AND GEOLOGY. Theorists do not know what the movement of the earth really is which causes the zeniths of the poles to vary their position in the heavens about 20*09" annually. If any theorist marked on the sphere the points to which the zenith Z and the zeniths E, Q, T, and H would be carried by the same movement which causes the zenith N to be carried to N', and S to S', he would at once commit himself to an exact movement of the earth, and as this exact move- ment has been hitherto unknown, it is more prudent to keep to such a vague assertion as " a conical movement of the earth's axis." This detail movement of the earth, unimportant as it may appear to the average reader, represents a very large sum of money when considered from a mere business point of view. During about a hundred and fifty years, observations to determine the future position of stars have been made with the transit instrument, in the plane of the meridian; a large staff of observers and computers have been employed for this work only. Several observatories in the United Kingdom carry on this same work. That such mere routine work must be totally un- necessary if, as was claimed, the true movements of the earth were really known, is an idea which never seems to have dawned on the mind of any reasoner. The amount of money paid at various official observatories, for persons engaged on this routine work, is understated when it is put down at 10,000 per annum. One hundred and fifty years of such work represents a sum of one and a half million pounds sterling, spent for making observations which must be unnecessary if the true movements of the earth were really known. Hundreds of pages of the journal, and of the proceed- ings of scientific societies, have been filled with lists of the supposed proper motion of the fixed stars; the gold medals THE ZENITH AND THE MERIDIAN. 175 of scientific societies have been given for these papers, amidst the cheers of the fellows; and yet, strange to say, in no single instance have any one of these distinguished gentle- men thought it necessary ever to examine or define how the zenith and meridian of their place of observation was affected by that same movement, which caused the zeniths of the two poles to trace an arc of 20'09" annually in a known direction. There are a number of individuals even in the present day who assert that they know how long the sun will last, how much fuel it consumes, what is the constitution of each star in the heavens, etc. It seems almost an insult to ask such gigantic intellects to descend from their thrones, and to define the actual movement of the earth produced by " a conical movement of the axis." So important, however, is this movement, so impossible is it to arrive at any sound knowledge by means of repeated observations unless the true movements of the earth are known, that, trifling as the request may appear, yet it may with justice be asked that some learned gentlemen mark on the sphere given in the last diagram, the amount and direction in which the zeniths of Z, E, T, and H are moved by that same motion of the earth which causes the pole N to be carried to N', and S to S'. In the history of astronomy, we have a precedent why such trifling matters as the shape and movements of the earth are beneath the notice of grand intellects. The ancient astrologers, for example, claimed to have discovered the lucky or baneful influence of every star and planet on each human being, according to the time of his birth. It would scarcely be expected that these ancient gentlemen could condescend to turn their thoughts on so small a thing as the earth; consequently, though they claimed to know all about the distant stars and planets, they imagined that the earth was a flat surface, and that it never moved. 176 UNTRODDEN GEOUND IN ASTRONOMY AND GEOLOGY, There may be, however, individuals in existence who can turn their attention to small matters, and for the informa- tion of these, the detail movements of the zenith and meri- dian will now be defined. In the following diagram (Fig. 56) N S R represents the earth; N the north, S the south, pole; T R, the equator; O Q R, the trace of the ecliptic; O E R, the trace of the equator of second rotation; N Z Q T H S, a meridian of six hours right ascension; Z, a locality in 51 north lati- tude; E, Q, and T, the points where the meridian intersects NN Fig- 5 6. S S' the equator of second rotation, the ecliptic, and the equator of daily rotation. H, the south point of the horizon projected on the- sphere of the heavens 90 from Z. The north pole of the second rotation impinges on the earth 29 25' 47" from N, on the opposite side of the sphere. The south pole of the second axis of rotation impinges on the earth at C, 29 25' 47" from the south pole S. The second rotation occurs at the rate of 40' 9" annually r consequently the point E during one year is transferred to E/ the arc E E' being 40.9". The pole N, distant 90 - 29 25' 47" from the equator of second rotation, is moved by the second rotation over an arc of 40-9" X cosine of 60 34' 13" = 20'09" from N to N'. THE ZENITH AND THE MERIDIAN. 177 The zenith of Z, a locality in 51 north latitude, and therefore 51 - 29 25' 47" = 21 34' 13" from the equator of slow rotation, is moved by the slow rotation over an arc of 40-9" x cosine of 21 34' 13" = 38", from Z to Z'. The point T on the equator, 29 25' 47" from E, is moved by the second rotation over an arc of 40 "9" X by the cosine of 29 25' 47" - 35-62" = T T'. The point H on the horizon for this latitude is distant from E, the equator of slow rotation, E T -f- T H = 29 25' 47" + 39 = 68 25' 47". The arc over which H is carried by the slow rotation is, therefore, 40*9" X cosine of 68 25' 47" = 15-03" only. The continuation of the meridian will pass through C> the pole of second rotation, and through S', the pole of daily rotation. We have, then, the following changes in the zeniths and meridian for six hours right ascension, and 366 daily rotations of the earth, as produced by the second rotation only : N N' = 20-09" Z Z' = 38" T T' = 35-62" H H' = 15-03" and E E' = 40-9" Hence, the zenith for six hours right ascension, and for latitude 51 north, is carried by the second rotation annually, in opposition to the daily rotation, over an arc of 2 -38" greater than is the equator of daily rotation. The meridian of one year also forms, with the meridian of a previous year, a curve, the most distant points of which are in latitudes 29 25' 47" north. The transit instrument which swept down the meridian N Z E T H C will, after 366 daily rotations of the earth, sweep down the meridian N' Z' E' T' H' and C', which again appears as a straight line. N 178 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. For this meridian to be carried by the daily rotation so that the point T comes again to T, the zenith 71 will by this daily rotation have failed to reach Z, because the equator is by the daily rotation carried over a greater arc than is the zenith of a locality in 51 north latitude; whereas by the second rotation the zenith of 51 is, for six hours right ascension, carried over a greater arc than is the equator of daily rotation for that meridian. Hence, as the right ascension of stars are determined by the time at which these stars transit the meridian, there appears by the present system a discordance or difference in the increase of right ascension in two stars, which difference is due, not to any proper motion in the stars themselves, but to the effect produced on the zenith and meridian by the second rotation. On meridians at or near eighteen hours right ascension, the effects of the second rotation is even more varied and remarkable than it is near six hours right ascension. The zeniths of some localities are carried almost directly towards the north pole, whilst the zenith of other localities on the same meridian are carried nearly at right angles to an arc joining these zeniths with the north pole. These varied movements of the zeniths and meridians cause a confusion in right ascensions and zenith distances which the mere observer is quite incompetent to unravel. The staff of our official observatories may be increased by scores of observers and computers, and with no other result than that which followed the adding of additional epicycles to the system of Ptolemy, viz. making confusion more complicated. The solution of the mystery is very simple, and can be easily arrived at by those persons who can move out of the erroneous grooves in which they have during so many years been content to travel. In the following diagram (Fig. 57), the changes pro- THE ZENITH AND THE MERIDIAN. 179 duced by the second rotation on various zeniths for a meridian of eighteen hours right ascension will be de- scribed. The circle N Q S R represents the earth; N the north, S the south, pole; Q T R, the equator of daily rotation; Q E R, the equator of second rotation. N C Z T H S, a meridian of eighteen hours right as- cension. C, the pole of second rotation, 29 25' 47" from N. N. N. Fig-57. Z, a locality in 51 north latitude. T, a locality on the equator of daily rotation. E, a locality on the equator of second rotation. H, a point 90 from Z. The most visible effect of the second rotation is to cause the pole N to be carried to N', and S to S', each of these arcs being 2O09", and representing one year of the second rotation. The effects on other zeniths will be as follows : Z will be carried to Z' round C as a centre. The length of the arc Z Z' can be found as follows : EZ = ET-fTZ = 29 25' 47" -f 51 = 80 25' 47". The effect of the second rotation is greatest at the equator of second rotation, where 180 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. it is 40.9" annually; the arc Z Z' is therefore 40.9" X cosine of 80 26' 47" = 679". The arc T T' of the equator of daily rotation is equal to 40-9" x cosine of E T = 35'624". E E' = 40-9". H H' = 40-9" X cosine of E H = 40'32". Hence the meridian of eighteen hours right ascension is affected by the second rotation only by the same move- ment of the earth, it must be borne in mind, which causes the pole N to be carried to N', and S to S', as follows : C, the pole of second rotation remains fixed. Z Z' = 6-79" T T' = 35-62" E E' = 40-9" H H' = 40-32" S S' = 20-09" The meridian will, therefore, be displaced by the second rotation in the manner shown by the great circle passing from N' to C, Z', T', E', H', to S'. The arc of the equator intercepted between these two meridians will be T T' - 35 "62", whereas the arc inter- cepted between the two zeniths Z Z' = 6*79" only. Con- sequently, when the daily rotation brings the point T' to T, the zenith 71 will be carried past its former position, Z, by the daily rotation; consequently, it will appear as though this zenith had been displaced to a greater amount by the second rotation than it really had been affected. Such a star as y Draconis, with a right ascension in 1887 of I7h. 53m. 58 '88s., and a north declination at the same date of 51 30' 8'4", consequently near the zenith Z, serves well to illustrate this effect. The constants for this star are as follows (Fig. 58) : P C = 29 25' 47". C 7 = 9 6' 22-5" Angle P 7, January 1, 1887 = 174 4' 19-7". Annual rate of variation in angle C, 45". THE ZENITH AND THE MERIDIAN. 181 P The following are the results obtained by calculation compared with those recorded as having been obtained by observation at various dates for the mean polar distance : Date. Calculation. Recorded. January 1, ] 887 .. .. 38 29' 51-7" .. .. 38 29' 51-6" 1873 . . . . 38 29' 43'5" . . . . 38 29' 43'9" 1850.. .. 38 29' 29-4" .. .. 38 29' 29"4" 1780 .. . . 38 28' 43" . . . . 38 28' 46" For a period of 107 years, viz. from 1887 back to 1780, the polar distance of this star can be calculated, and a difference of only 3" exists between the calculation and the recorded observation at the earlier date. Considering the vague refraction used at the date 1780, and also the imperfection of the instruments then used, it is for the reader to judge which is the more likely to be correct, calculation, or observations made 107 years in the past; the difference, however, is so very slight as to serve as a fair example of the calculations that can be made when the second rotation of the earth is comprehended. Another star near the meridian of eighteen hours right ascension serves as an example of the changes in the meri- dian in this part of the heavens as effected by the second rotation. This star is Vega a Lyne. The constants for this star are as follows (Fig. 59) : P C = 29 25' 47". C a =22 29' 51'2". Angle P C , January 1, 1887 = 162 55' 6-8". Annual variation in this angle, 44-4". 182 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. In order/' to refresh the reader's memory as regards the method of calculating the mean polar distance for more than one hundred years, a calculation is given below for calcu- lating the mean polar distance of a Lyne for January 1, 1755. The interval between 1755 and 1887 is 132 years, which, multiplied by 44'4", amounts to 1 37' 40'8". This angle added to 162 55' 6'8" amounts to 164 32' 47'G", which will be the angle at C for 1755. With the two sides P C and C a, and the included angle at C, we can calculate P a, the polar distance of a Lyra* for 1755, by the usual formula, as follows : The supplement of 164 32' 47*6" is 15 27' 12 -4". Log. cosine, 15 27' 12-4" = 9-9840083 Log. tangent, 22 29' 51-2" = 9 6171719 9-6011802 = log. tan. first arc, 21 r -45' 41-1". 29 25' 47" 21 45' 41-1" 51 11' 28-1" = second are. Log. cosine, 22 C 29' 51-2" = 9-9656230 Log. cosine, 51 11' 28-1" = 9-7970765 19-7626995 - Log. cosine, 21 45' 41-1" = 9-9G78921 9-7948074 = log. cos. P a = 51 25' 51-2". In Bradley 's catalogue for 1755, the polar distance of Lyree is given, as observed by him, as 51 25' 49", showing , THE ZENITH AND THE MERIDIAN. 183 a difference of 2'2" only between Bradley 's observation, made 132 years in the past, and the result arrived at by calculation. Here again we must consider the class of instrument used by Bradley, and the uncertainty of the refraction he used to correct his zenith distances, and it is again left to the reader's judgment to consider which is the more dependable, calculation or observation. By the same means as shown above, the calculated polar distance of a Lyne, compared with that recorded as found by observation, is as folio ws : Date. Calculation. Recorded. January 1, 1887 . . . . 51 19' 16" . . . . 51 19' 16" 1850.. .. 51 21' 10-8" .. .. 51 21' 10 9" When it is borne in mind that hitherto no method has been known to astronomers by which they could calculate the polar distance of a star for even five years, but were obliged to employ a mere rule-of- thumb method of adding or subtracting an annual rate, found merely by perpetual observation, and then employing this for two or three years only, it may possibly be admitted that the real calculation given above is a considerable advance, and possesses the great advantage of simplicity, and renders unnecessary, perpetual observations. When a point on the. earth's surface is so situated that the arc from the zenith of this locality to the pole of the second axis of rotation forms a right angle with the arc drawn from the pole of daily rotation to the pole of second rotation, then this zenith will be carried by the second rotation directly towards, or away from, the pole of daily rotation according as the meridian of right ascension is greater or less than eighteen hours. A very slight difference in the polar distance of a star will cause a very great difference in the zenith of that locality in which the star is situated, the zenith being 184 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. carried in one instance directly towards that position in the heavens occupied by the pole of daily rotation; in the other case, the zenith is carried nearly at right angles to the arc joining that zenith with the pole of daily rotation. As these zeniths alter their positions considerably from year to year in consequence of the second rotation, a con- stant rate cannot be used according to the present system of determining polar distances from zenith distances. As an example of these apparent mysteries, two stars will be given, viz. Ursse Minoris and rf Draconis. The constants for Ursre Minoris are (Fig. 60) P C = 29 25' 47". C C = 20 25' 59-2". Angle P C C, January 1, 1887 = 18 41' 3". Annual variation in angle P C = 41". Fig 60. The results by calculation compared with recorded observation are, for polar distance Date. Calculation. Observation. January 1, 1887 .. .. 1151'31'3" .. .. 11 51' 31-3' ' 1850 .. .. 11 44' 48-1" .. ., 11 44' 48" For the star r? Draconis, the constants are as follows : P C = 29 25' 47". C 7j 2 = 11 44' 57". Angle P C if, January 1, 1887 = 73 32' 35-6 . I, Annual variation in angle, 40'72". P THE ZENITH AND THE MERIDIAN. 185 The following are the results obtained by calculation, compared with those recorded for the polar distance of this star : Date. Calculation. Recorded. January 1, 1887 .. .. 28 13' 48-4" .. .. 28 13' 48 4" " 1873 .. .. 28 11' 52-7" .. .. 28 11' 52'67" 1850 .. .. 28 8' 42-4" .. .. 28 8' 42'4" The difference in the effect of the rate of these two stars, as produced by the influence on the two meridians by the second rotation, amounts during thirty-seven years to about 0*28" annually. The amount 0'28" annually, measured on the equator of daily rotation and converted into time, is less than one-tenth of a second of time per year a rather small amount to be checked by any chronometer, but yet perceptible when an interval of several years is dealt with. The changes in the various zeniths and meridians in this part of the heavens, as produced by the second rotation, are so numerous and so varied, that calculations have to be made for each zenith or star as affected by this movement. These apparent complications are not due, however, to the assertion that the whole solar system is rushing towards the constellation Herculis, at the rate of upwards of one hundred and fifty-four million miles per year; nor are they clue to the assertion that the stars themselves are all rush- ing about. The effect of the second rotation of the earth on the meridians and zeniths in this part of the heavens, in a manner with which theorists have not been aware, is the true cause. When we come to meridians of twelve hours or hours of right ascension, we do not meet with those differences in the effects of the second rotation which are manifested from about fourteen to twenty-two hours of right ascension; the reason for this is almost manifest. Meridians at or near to hours and twelve hours of right ascension are displaced by the second rotation nearly 186 UNTEODDEN GROUND IN ASTRONOMY AND GEOLOGY. in opposition to the daily rotation,, and in such a manner that when the dally rotation carries these meridians over the small arc through which they were carried by the second rotation, the meridian of one year almost coincides with the position it previously occupied. The following diagram (Fig 62) will explain this fact: NEST represents the earth; N the north, S the south, pole; E Q T, the equator of daily rotation. C represents the position of the pole of the second axis of rotation; C', the opposite pole of second rotation; O Q R, the equator of second rotation; E - N = 29 25' 47". N Z Q H S represents the meridian of twelve hours right ascension; Z, a locality in 51 north latitude; N Z therefore = 39; H, a point 90 from Z; a, a point on the same meridian, in latitude 20 north. The effect of the second rotation on this meridian dur- ing one year is as follows : The pole N is carried over an arc of 20 "09", round C as a centre. Consequently, the pole N is carried down, as we may term it, and in the direction of hours right ascension, as shown by the arrow near N. The pole S will be carried to S', round C' as a centre; the arc S S' - 20'09". THE ZENITH AND THE MERIDIAN. 187 The point Q on the equator of second rotation will be carried to Q'; the arc Q Q' - 4O9". The zenith Z will be carried to Z', the arc Z Z' being a portion of a small circle having C for its centre. The value of the arc Z 71 can be calculated as follows : N Z = 39. N C - 29 25' 47"; angle at N a right angle. The hypotenuse C Z is therefore, in round numbers, 47 23'. The point Z is therefore, 47 23' from 0, and 42 37' from the equator of slow rotation. The extent of the arc Z Z' will therefore be 40.9" X cosine 42 37' - 27'69". The value of the arcs x %' and H H' can be calculated in the same manner, and the displacement of this meridian by the second rotation can be traced out in all its details, as shown by the great circle drawn from S to H', Q', %', and Z'. It will be evident that the displacement of this meri- dian by the second rotation is almost identical with the displacement caused by the daily rotation. Consequently, the daily rotation readjusts this meridian very nearly, although the zeniths H, Q, x, and Z will have been displaced 2O09" up this meridian during one year. The great displacements in the various zeniths and meridians which occur near eighteen hours right ascension, in consequence of the second rotation, do not occur near twelve hours or hours right ascension. Hence the ap- parent rates of stars near these meridians vary very slightly, as will be seen by the following examples. The stars X Draconis, a Draconis, a Ursse Majoris, 7 Ursre Majoris,?j Ursse Majoris, S Leonis, ft Leonis, a Canum Venaticor, Virginis, Spica Virginis, and ft Corvi, vary in their annual rates as affected by the second rota- tion, between 4076" per year and 411". Take, for example, the star ft Leonis, and calculate the polar distance of this star for any date, and it will be found 188 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. that calculation gives a rate whi^h is in accordance with the laws of geometry, whilst recorded observations, in many instances, give a sudden change in the rate, then a return to the before-mentioned rate, and so on. The following are the constants for |3 Leonis (Fig. 63) : P C = 29 25' 47 '. C = 78 49' 14-6". Angle P C j8, January 1, 1887 = 78 49' 43 6". Annual variation in angle = 41'0". Fig63, RESULTS OF POLAR DISTANCE, P Date. Calculation. January 1, 1887 . . . . 74 47' 46'8" . . 1850 .. .. 74 35' 23-5" .. 1755 .. .. 743'36-G" Recorded. 74 47' 46-8" 74 35' 22-6" 74 3' 35-6" The star y Ursse Majoris is another example. This star is north of the zenith of 51 north latitude, and the second rotation affects the meridian slightly less than it does the meridian on which (3 Leonis is situated. The constants for y Ursae Majoris are as follows (Fig. 64):- P C = 29 25' 47". C 7 = 46 11' 0-4". Angle P C 7, January 1, 1887 = 53 49' 3". Annual variation in angle P C 7 = 4(V8". Fig; 6 4*. THE ZENITH AND THE MERIDIAN. 182 The following is a comparison between the polar distance found by calculation and recorded at various dates : Date. Calculation. Recorded. January 1, 1887 .. .. 35 40' 37'2" .. .. 35 40' 37'2" 1873 .. .. 35 35' 57-2" .. .. 35 35' 57'4" 1850 . . . . 35 28' 16-8" . . . . 35 28' 16'8" 1830 .. .. 35 21' 35-1" .. .. 35 21' 36-5" 1755 .. .. 34 56' 36-7" .. .. 34 56' 35" In the various nautical almanacs professing to give the declination of the stars, this declination is given to the one- hundredth of a second, whilst the annual rate of change in this declination is given to the one-thousandth of a second. Any practical observer who is acquainted with the un- certainty of refraction, knows that this asserted minute accuracy is theoretical only, not really a fact. Whilst, however, this accuracy is claimed, we frequently corne across some amusing descrepancies, one of which may be mentioned in connection with the star j Ursse Majoris. In the Nautical Almanac for 1873, the mean declination of this star for January 1 is given as 54 24' 2*51"; in the Nautical Almanac for January 1, 1887, 54 19' 22'80"; giving a difference for fourteen years of 4' 3971". That is a decrease in the declination of 27971" in fourteen years, which is at the rate of 19*979" per year. If arithmetic is correct, this rate of 19'979" per year causes the declination given for January 1, 1873, to become 54 19' 22-80" on January 1, 1887. But in the Nautical Almanac for 1873 it is stated that the annual variation in declination for this star is 20'024", and in the Nautical Almanac for 1887 it is stated that the annual variation is 2O026". Which, then, are we to accept as correct, the recorded declinations as given in the Nautical Almanac and the rules of arithmetic, or the theories, formulae, and rate, also given as correct? It is claimed that the annual rate for 190 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. this star is known to the thousandth of a second. How is it that the star refuses to conform to this rate? The ex- planation of the theorist is, of course, very simple : the theory cannot move, so the star must do so, and y Urs&e Majoris must have a proper motion. As an example of the far greater accuracy to be obtained by calculation than can be reached by mere observation, it must be borne in mind that when the second rotation of the earth is understood, calculation can be made in order to find the polar distance or declination of a star for any date quite independent of the annual rate of change in these items found by observation. The geometrician may, as a matter of mere amusement, find what the mean rate was between any two dates, by dividing the difference between the polar distances found by calculation by the number of years between the two dates. For example, calculate the polar distance of this star for January 1, 1887, and January 1, 1850, and we obtain as follows : January 1, 1887 = 35 40' 37-2" 1850 = 35 28' 16-8" Difference for thirty-seven years = 12' 20'4" Being at the rate of 20'01" annually. Again, calculate the polar distance of this star from January, 1 1830, and compare this calculation with that obtained for 1850, and we have as follows : January 1, 1850 = 35 28' 16'8" 1830 = 35 21' 36-5" Difference for 20 years - G' 40-3" Being at the rate of 20-01" per year. The annual rate given for this star in star-catalogues for 1850 was 20'04", whilst in 1887 it was 20'026". Hundreds of similar examples could be given, proving THE ZENITH AND THE MERIDIAN. 191 that the minute accuracy claimed by theorists, such as giving the rates of stars to one-thousandth of a second, is theoretical only. The recorded observations alone prove this assertion to have no foundation in fact. A multitude of examples could also be given, proving that observations cannot compete in accuracy with calcula- tions that is, unless we are prepared to admit that arith- metic and geometry are not so much to be depended on as the theories of certain authorities. It will now probably be evident to the geometrician and reasoner, that the manner in which each meridian and each zenith is affected by the second rotation, is a problem of the greatest importance when observations are made by the transit instrument on the present system. In the whole history of modern astronomy, whether we examine the various books that profess to deal with the science of astronomy, or with the details of the calcula- tions, not one word is written as regards the changes that occur in connection with various zeniths and meridians, and produced by " a conical movement of the earth's axis." In the numerous papers published in the monthly notices of the Koyal Astronomical Society, the subject is never referred to. The most vague and baseless theories, if popular, are given many pages, but so important an item as how the zenith and meridian are affected, seems, if dealt with in a paper, to be considered only suitable for "a pigeon-hole." There are several problems connected with the second rotation which require to be worked out, more in detail, than one person alone could work out in a lifetime. But the most mysterious problem of all is to explain why so important a problem as the changes in the zeniths and meridians is considered beneath the notice of men claiming to be scientific. 192 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. At present the daily rotation ajid the second rotation are mixed up together, and confusion consequently ensues. To clear up this confusion, it is necessary to examine the effects of the second rotation by itself, and then to note in what manner the daily rotation is interfered with by the movements given to the earth by the second rotation. Hitherto the only notice taken of the second rotation is that the zenith of the two poles of daily rotation alter their positions in the heavens about 20-09" annually; all else has been ignored. CHAPTER XIII. THE MEASUREMENT OF TIME, AND RIGHT ASCENSION. IT is a most unusual proceeding to find an authority admitting that there is anything connected with his special subject which is not known. The very opposite proceeding is usually practised, by claiming that everything is known, and that the special science of which the individual may, for the time being, be a popular authority, has been exhausted. It speaks well, therefore, for the candour of the late Sir John Herschel that, in his "Outlines of Astronomy," in a note at the end of his book, he stated, as regards the measure- ments of time, that "the whole subject has fallen into confusion; " that it was necessary to fudge in 3m. 3'6Ss. of purely imaginary time between the end of the equinoctial year of 1833 and the beginning of 1834, in order to "cook " theories with facts. Considering that it is claimed by certain theorists that the rate of change in a star's right ascension is known to the one-thousandths of a second, it is rather startling to find that no less than 183*68 seconds had to be fudged in to make facts and theories agree. How great was the confusion that existed not even Sir John Herschel suspected, but when the fact is stated that hitherto no observer, theorist, or mathematician has known in what a daily rotation of the earth really consisted, it can be understood that there has been, and will be, a o 194 UNTKODDEN GROUND IN ASTRONOMY AND GEOLOGY. confusion, until this problem is known and is dealt with in a correct manner. The subject of time and its correct measurement is so vast, that only one item connected therewith can be dealt with in this book, but it can be stated, for the infor- mation of the reader, that there is connected with the measurement of time a problem hitherto unsolved, and which a geometrician who can free his mind from dogmatic theories will probably be able to solve. This problem will not be now dealt with, therefore it is open to any person for investigation and solution. The key to the solution may probably be found in the explanation which is given in this chapter, as to what constitutes a rotation of the earth. The time used by observers consists principally of two kinds, viz. siderial time and mean solar time. The former is the time which will be mainly dealt with in this chapter. A siderial day has been defined as the interval of time which elapses between two successive transits of the same star across a given meridian, and this day is divided into twenty-four hours. An apparent solar day in the interval of time between two successive transits of the sun across the same meridian. The reason why sun time varies from sidereal time will be understood from an examination of the following diagram (Fig. 65). The circle E F G H J represents the annual course of the earth round the sun, all details as regards the eccen- tricity of the orbit and its elliptical form being for the present omitted, as not necessary in the explanation which will now be given, of the difference between sun time and sidereal or star time. The earth moves round the sun in the direction from E to F, G, H, and J. MEASUREMENT OF TIME, AND EIGHT ASCENSION. 195 We will suppose that on March 21 the earth is at E, and the sun S and a star at an infinite distance pass the meridian at the same instant of time. On the next day the earth would have been carried in its orbit to F, and as the star referred to is at an infinite distance, a line drawn from F to the star would practically be parallel to a line drawn from E to the same star, but a line drawn from F to the sun would not be parallel to a line drawn from E to the sun. Consequently, the meridian would turn round by the daily rotation and come to the distant star before it turned over the small arc which brought this meridian to the sun. Hence the interval of time between two successive transits of the star would be less than the interval between two successive transits of the sun. When the earth has been carried round one-fourth of its annual orbit to G, a line from G to the distant star would practically be parallel to the line drawn from E to the same distant star, but a line from G to the sun S forms a right angle (90) with a line from G to the distant star. The daily rotation of the earth being from right to left, 196 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. as represented in this diagram, the star will come on the meridian 90 before the sun when the earth is at G. Now, as 24 hours of rotation represent 360 of daily rotation, 90 represent one-fourth, that is, six hours. The distant star, therefore, has gained six hours on the sun, owing to the movement of the earth from E to G. When the earth has been carried to H, a line from H to the distant star will be parallel to the lines drawn from E or G to the distant star, and the earth at H is between the sun and the star. The star, consequently, will come to the meridian 180 = 12 hours before the sun. The star, consequently, has gained half a day on the sun, in consequence of the movement of the earth from E to H. For the same reason, the star will gain another interval of twelve hours in consequence of the movement of the earth from H to J and on to E. It follows, therefore, as a geometrical law, that there must be one more transit of a distant star than there can be of the sun during the movement of the earth round the sun. If the earth rotated only one hundred times during its journey round the sun, there would be a hundred transits of a star, but only ninety-nine transits of the sun. The earth, in round numbers (for minute accuracy is not yet necessary for explanation), rotates yearly 365*25 times as regards the sun, but 366'25 times as regards a distant star. There being one sidereal day more during a year than there are mean solar days, we can, by simple arithmetic, find the proportion between solar and sidereal time as follows : Twenty-four hours equal 1440 minutes. Divide 1440 by 365*25, and we obtain 3m. 56'55s. Consequently, 24 MEASUREMENT OF TIME, AND BIGHT ASCENSION. 197 hours of mean solar time equal 24h. 3m. 56'55s. of sidereal time. Again, divide 1440 minutes by 366*25, and we obtain 3m. 5 5 "906s., which; subtracted from 24 hours, gives 23h. 56m. 4'09s. for the mean solar time, equivalent to 24 hours of sidereal time. It will be evident, from an investigation of the last diagram, that the interval between two successive transits of the sun does not give the true time of the earth's rota- tion round its axis. It gives the time of one rotation plus the small arc over which the earth must rotate in order to bring the sun again on the meridian. The interval between two successive transits of a star gives more nearly the measure of the time occupied by the earth in rotating, because the movement of the earth round the sun, owing to the immense distance of the stars, does not produce any apparent change in their relative positions. The difference between apparent and mean solar time is explained (after a fashion) in most works on astronomy. This subject will not be dealt with in this chapter, but untrodden ground will be passed over, this being an investigation of what constitutes a rotation of the earth. Sir John Herschel and his numerous copyists assert, " All the stars, it is true, occupy the same interval of time between their successive appulses to the meridian or to any vertical circle " (see Article 143, "Outlines of Astronomy "). Do they indeed? Geometry teaches quite a different thing. Geometry teaches that a star within the circle traced by the earth's axis, will come to the meridian once oftener during an entire revolution of the equinoxes, or during one complete circle in the heavens described by the earth's axis, than will a star outside this circle. Let this question be decided by facts, and it must be remembered that, no matter whether the time be long or 198 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. short, during which a movement occurs, the laws as regards this movement are equally sound. Suppose P Q R, (Fig. 66) the circle which the pole of the heavens describes among the fixed stars; C, the centre of this circle, the radius being of any length. Take as the position of the pole at some remote date in the past. The stars x, C, y, and S would transit a meridian simultaneously, whilst the star Z would transit a meridian twelve hours before these stars. After a long interval, the pole is carried to P. When the pole is at P, the star Z will transit a meridian before the star x by an interval of time measured by the time it occupies for the earth to rotate through the angle ZPx. The star x, consequently, instead of coming to the meridian twelve hours after the star Z, as was the case when the pole was at 0, comes to the meridian after the star Z by an interval of time measured by the time it occupies the earth to rotate through the angle Z P x. The stars C, y, and S will come to the meridian after the star x by an interval of time measured by the time it occupies the earth to rotate through the angles x P C, x P 7 , and x P S. MEASUREMENT OP TIME, AND RIGHT ASCENSION. 199 When the pole has reached the point Q, then the stars y, C, x, and Z will transit a meridian simultaneously, and each of these stars will therefore have gained twelve hours, or half a sidereal day, on the star Z. The star S, however, will, when the pole is at Q, transit twelve hours after the star Z, just as it did when the pole was at O. When the pole has reached R, the stars y, C, and x will transit before the star Z by an interval of time measured by the time it occupies the earth to rotate through the angles y R Z, C R Z, and x R Z. When the pole has again reached the point O, the stars ., C, and y will transit a meridian twelve hours before the star Z instead of twelve hours after the star Z. Con- sequently, if 9 represents the number of transits of Z, a star outside the circle described by the pole during any number of thousands of years, + 1 will represent the number of thousands of transits of a star within the circle described by the pole. The interval between one or one million transits of the star x will not give the same measure of time as will the interval between one or one million transits of the star y. There is a considerable amount of daily rotation to be per- formed after the star x is on the meridian and the pole is at P, before the star y comes on the meridian; whereas when the pole is at Q, the stars y and x will transit simultaneously. How, then, can an authority assert that each star occupies exactly the same interval of time to come to the meridian; or, in other words, that the interval of time between two successive transits of any star gives the same measure of time? The fact really is, that no two stars give exactly the same interval of time for their successive transits, and it is a geometrical law that a star within the circle described by 200 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the earth's axis will transit a meridian once oftener during the tracing of this circle than will a star which is outside this circle. But we have another remarkable assertion promulgated by so-termed authorities, viz. that the time occupied by the earth in performing a sidereal rotation is to be measured by the interval between two successive transits of the same star. Which star? Is it a star within, or without, the circle described by the pole? Is it a star near the circle de- scribed by the pole, or is it a star many degrees distant from this circle? As it is a geometrical law, that a star within the circle described by the pole transits a meridian once oftener than a star outside this circle, it is absolutely necessary that, before we can assert that a star has a proper motion, we must know whether this star is within or without the circle described by the pole; we must also know how many degrees the star is from this circle. Is the radius of this circle 23 27', 24, or of some other value? Even according to orthodox theories, this radius was formerly 24, but is now only about 23 27'. How about the star Vega (a Lyrse)? Is this star within, or without, the circle described by the pole? If without the circle, how many degrees distant is it from this circle? Will one million transits of the star a Lyrse give the same measure of time for the earth's rotation as one million transits of the pole-star (a Ursse Minoris)? Certainly not, even if the present accepted theories were correct. With these facts before us, what are we to think of the scientific knowledge of those persons who have asserted that " all the stars occupy the same interval of time between their successive appulses to the meridian," and that the actual time of a rotation of the earth is to be thus ac- curately defined. MEASUREMENT OF TIME, AND EIGHT ASCENSION. 201 The above is only one example of the slovenly manner in which this problem of time has been dealt with. Here is another. There is a movement of the earth occurring which causes the earth's axis of daily rotation to change its direction. Hitherto theorists have been contented to ac- cept, as a full and complete explanation of this movement, the assertion that it was " a conical movement of the earth's axis." Let us examine the following diagram, which represents the earth, the actual instrument by aid of which we measure time (Fig. 67) : Let N Q S T represent the earth; N the north, S the south pole; Q E T, the equator; N E S, a meridian of eighteen hours right ascension. We will first assume that no other movement takes place in the earth than its daily rotation round the axis N S. At the completion of 36G rotations of the earth, the meridian N E S would again coincide with N E S. But another movement does occur in the earth. During these 366 rotations, the pole N is carried to N', and the pole S is carried to S'. 202 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. In what manner will the meridian N' S' be affected by this other movement, which, although independent of, is yet mixed up with, the daily rotation? In what manner will the zeniths of localities, such as C, Z, E, and R, be affected by that movement which causes N to be carried to N', and S to S'? The reader who has understood the earlier chapters in this book, knows that the point C remains fixed as regards this movement, that Z is carried to 71, E to E', and R to R'. Consequently, after 366 rotations of the earth, the meri- dian which, during the first rotation, occupied the position N C Z E R S will occupy the position N' C 71 E' R' S', each of these points except C being displaced by the second rotation. The movement of Z to Z', E to E', and R to R' is in opposition to the daily rotation, consequently the daily rotation has to occur slightly to bring the point E' back to E. Hence we should have 366 daily rotations completed when the meridian occupied the position N' C 71 E'. But it would require 366 rotations plus the small arc of rotation represented by E' E to bring the meridian to the same point on the equinoctial that this meridian occupied at the time of the first rotation. Between the point C, however, and the pole N, a zenith x would be carried to x' in the same direction in which the pole N is carried, and the arc x x r is in the same direction as the daily rotation carries this zenith. Hence the second rotation retards the daily rotation from C down to S', but accelerates the daily rotation from C to N'. We find, consequently, that a star between C and N such, for example, as S Ursse Minoris varied its right ascension annually, in 1887 as much as 19'437s.; whereas another star, X Draconis, varied its right ascension only + l'392s., the latter star being south of the point C, and the daily rotation affecting the readjustment of the locality MEASUREMENT OF TIME, AND RIGHT ASCENSION. 203 under the star X Draconis more rapidly than it does under the star 8 Ursse Minoris. Can it, however, be asserted that the interval of time between two successive transits of these two stars gives the same measure of time, when, as it is proved by their annual changes in right ascension, there is for 366 transits a difference of more than twenty seconds? By which of these stars are we to measure the time it occupies the earth to rotate? By which of the other stars are we to measure the time that it occupies the earth to- rotate? It is difficult to find any two stars which give the same interval of time for their successive transits when long periods of time are dealt with, and this result must follow as a geometrical law, due to the movement of the pole of the heavens in a circle round a point as a centre, which pjoint has been hitherto in an unknown position in the heavens. Every star within the circle traced by the pole of the heavens transits a meridian once oftener, during the time it occupies the pole to trace this circle, than does a star outside this circle. Stars both within and without the circle traced by the pole will vary their relative rates of transit across a meridian. There is but one point in the heavens in each hemisphere which will give a uniform standard measure of time, the successive transits of which point will give the exact measure of the earth's rotation on its axis; that point is the centre of the circle which the earth's axis traces, or, in other words, the pole of the axis of second rotation. All stars between this pole of second rotation and the pole of daily rotation will give by their successive transits an interval of time less than that given by the successive transits of the pole of second rotation, hence the annual 204 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. variation in right ascension of sil\ stars so situated will in almost every case be a minus quantity, increasing in amount the nearer the star is to the pole of daily rotation. As examples of this change, the following stars may be referred to for 1887 : Star. Annual variation in right ascension. e Ursa? Minoris . . - 6'3433s. 5 Urste Minoris 19 -4370s. 7 Urs^e Minoris 64'4299s. Stars which are south of the point C will all increase their right ascensions annually, but not uniformly, because those stars near the pole of second rotation, where the zenith is but slightly affected by this second rotation, will have their right ascension less affected than will those stars farther from the pole of second rotation. As examples of such effects, the following stars may be given for 1887 : Distance from Star. pole of second rotation. Variation in right ascension. $ Draconis . . . . 9 17' 38" . . . . + 1-351 5s. 7 Draconis .. .. 9 6' 22'5" .. .. + 1 -3923s. As examples of stars which most increase their annual right ascension, we must look for stars near the equator of second rotation, viz. stars which have somewhere near eighteen hours right ascension, and a south declination of near 29 25' 47". As examples of such stars we find the following for 1887 : Star. South declination. Right ascension. Annual variation. 6 Ophiuchi . . . . 24 53' . . . . 17h. 15m + 3'6778s. ft Sagittarii .. .. 21 5' .." .. 18h. 7m + 3'5838s. a Sagittarii . . . . 25 29' . . . . ISh. 21m + 3-"020s. Stars in this part of the heavens, but north of the equator of daily rotation, will be at a greater distance from the equator of slow rotation, and consequently will not be so much affected by the second or slow rotation as will those stars that are south of the equator of daily rotation, MEASUEEMENT OF TIME, AND RIGHT ASCENSION. 205 and consequently are nearer the equator of second rotation. The following stars will serve as examples : Star. North declination. Eight ascension. Annual variation. a Ophiuchi .. Ophiuchi . . H Herculis . . 12 38' . . . . 4 36' . . 27 47' . . . . 17h. 49m. . . . . 17h. 37m. . . .. 17h. 42m. ... . . 2-7792s. . . 2-9598s. . . 2-3440*. a Lyrae . . 38 40' . . . . 18h. 33m. . . . . 2-0304s. It will be seen, by an examination of the various declinations of these stars, that the nearer a star happens to be to the equator of second rotation the greater will be its annual variation in right ascension, and the nearer a star approaches the pole of second rotation the less, conse- quently, will be its annual change in right ascension. At the pole of second rotation there will be little or no annual change in right ascension, but north of this pole of second rotation the annual change of right ascension will alter from plus to minus. There will be another condition under which a star will vary its annual right ascension by a very minute quantity that is, where an arc from the star to the pole of second rotation is nearly at right angles to the arc from the star to the pole of daily rotation. Under these conditions, that part of the meridian over which the star is situated is carried towards or away from the pole of daily rotation, and the position of this part of the meridian is but slightly altered as regards the daily rotation. For example (Fig. 68), take N C S as a meridian of eighteen hours right ascension; C, the pole of the axis of second rotation. N Z R S, a meridian short of eighteen hours say, for example, of seventeen hours right ascension. There will be a point in this meridian, uch as Z, where an arc drawn from Z to C will be at right angles to an arc drawn from Z to N. This is the same thing as stating that an arc 206 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. drawn from C to Z is at right angles to that part of the meridian on which Z is located. Whilst the second rotation carries the pole N to N' round C as a centre, it carries Z to 71 round C as a centre. The arc Z Z' coincides with the meridian N Z; consequently a star which is situated above Z will have its right ascen- sion altered very little, if any, by the second rotation. A point 7 on the same meridian would be carried to y' round C as a centre by the second rotation; this star, con- sequently, which was in the zenith of y, will have increased its right ascension by just the amount of daily rotation required to bring y' to the meridian N' Z y. As examples of this simple law, two stars are given below; their right ascensions, omitting seconds, are given, and their north polar distances, and the rate at which they varied their right ascensions annually at the date 1850, as found by observation. Star. North polar distance. lligbt ascension. Variation. 22Draconis .. .. 24 G' ,. 17h. 8m. .. + O'loOs. 65HerculisS .. . . 64 58' .. ]7h. 8m. .. + 2'459s. The first-named star occupies a position near Z on the last diagram, where its annual variation in right ascension is very slight. The second star occupies a position near y, MEASUREMENT OF TIME, AND RIGHT ASCENSION. 207 where the zenith is considerably affected by the second rotation, and where, consequently, the annual variation in right ascension is considerable. By taking a point farther down this meridian, such for example as R, we shall find that R R', the arc of displace- ment produced by the second rotation, is greater than yy f , because R is at a greater distance from C, the pole of second rotation; consequently, a star in the zenith of R would have a greater rate for its annual variation in right ascension than would the star above y. The star 40 Ophiuchi , 110 50' from the north pole, and with a right ascension in 1850 of about I7h. 12m., serves as an example, the annual variation in right ascension at that date being recorded as 3'592a The same law holds true as regards a meridian such as N x S (Fig. 68), which we will suppose a meridian of 19h. 30ni. right ascension. A star near the zenith of x will have its right ascension altered very slightly annually, because this zenith x will be carried to x r round C as a centre, the arc x x' nearly coinciding with the meridian N a? S. That part of the meridian which is at will be carried to 0' round C as a centre; consequently, a star at the zenith of O will show a large variation annually in right ascension. The two stars given belo\v serve as examples of this law : Star. North polar distance. Right ascension. Variation. 58 Draconis TT . . . . 24 3-i' , . 19h. 19m. . . + 0'332s. VulpeculsB .. .. 65 38' ... 19h. 22m. .. + 2'495s. The reader who comprehends geometry can now make an interesting calculation. He can calculate at what point on each meridian a star will very slightly change its right ascension annually. He can then trace out a curve or 208 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. irregular figure joining these points. All stars with few exceptions within this curve will decrease their right ascensions annually. All stars outside the curve will increase their right ascensions annually. The method of calculating this curve is very simple, and is as follows. The angular distance between the pole of daily rotation P and the pole of second rotation C, viz. 29 25' 47", is the hypotenuse of a right-angled spherical triangle, the right angle being on any particular meridian, say seventeen hours of right ascension. The pole of second rotation is assigned a right ascension of eighteen hours. Hence for a meridian of seventeen hours we have the following spherical triangle, PCD : P C = 29 25' 47". P D C, a right angle. The angle D P C = Hi. = 15. Calculate P D, the polar distance of that point on a meridian of seventeen hours where an arc from the pole of second rotation is at right angles to this meridian. Using the common formula, we have Log. tangent, 29 25' 47" = 9-7513982 Log cosine, 15 0' 0" = 9-9849438 9-7363420 = log. tan., 28 35' 14" = P D. For a meridian of 16h. substitute 2h. = 30 for Ih. and we obtain 26 2' 20" for the polar distance of the point at which an arc from the pole of second rotation cuts the meridian of 16h. at right angles. For a meridian of 15h. we obtain 21 44' 53" for the polar distance of this point. For a meridian of 14h. we obtain 12 37' 45"; for 13h. 8 18' 26"; for 12h. 30m., 4 12' 41"; for 12h., 0. For 19h., the point will be 28 35' 14" from P; for 20h., 26 2' 20"; for 21h., 21 44' 53", and so on. MEASUREMENT OF TIME, AND RIGHT ASCENSION. 209 Hence we can construct this curve as follows (Fig. 69) : Describe a semicircle to represent that half of the sphere from twelve to twenty-four hours right ascension. P Fig-69 Take P as the north pole of daily rotation; C, the pole of second rotation; P C consequently = 29 25' 47". Set off the various meridians at 15 interval Ih. The numbers 12, 13, 14, etc., represent meridians of twelve hours, thirteen hours, fourteen hours right ascension. From the calculations already given, we can mark on each meridian that point at which an arc from C will cut each meridian at right angles, thus : P Q = 8 18' 26" = P Z P R = 12 37' 45" = P Y P S = 21 44' 53" = P X P T = 262' 20" = P W P U = 28 35' 14" = P V P C = 29 25' 47" The following are the geometrical laws appertaining to this curve. All stars outside of this curve will increase their right ascensions annually, the rate of increase being least the nearer these stars are to the curve; this law holding true from twelve to twenty-four hours right ascension. Stars within this curve will as a rule decrease their right ascensions, except under conditions named below. The nearer a star is to the pole of daily rotation, the greater will be the annual rate of decrease in the right ascension of this star. 210 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Whilst every point on this curve will be carried directly towards that point where the pole was located, it may not be carried directly towards that point where the pole at the end of one year will be located. The pole P is carried annually over an arc of 20'09" in the direction of twenty- four hours right ascension (nearly). A point a few degrees within the curve at eighteen hours right ascension will be carried annually over a small arc round C as a centre far less than 20'09". Consequently, the meridian may, under certain conditions, cause stars near this curve, although within it, to increase very slightly, instead of decreasing, their right ascension annually. The calculations of these details are too long and too technical to be here elaborated. Some few examples, how- ever, of the general effects may be given. The star )3 Ursse Minoris is near the meridian marked P S, having about 14h. 51m. right ascension. The distance of this star from the pole P is about 15 15', whereas P S is more than 21. Hence the annual rate of variation in this star will be minus. The star 4949 (British Association Catalogue) Draconis, which has a right ascension of about 14h. 55m., and is distant from the pole P about 23 28', will have an annual variation in its right ascension plus in value, because outside the curve. Another star, viz 4982 (British Association Catalogue) Ursse Minoris, with about fifteen hours right ascension, but only about 6 53' from the pole P, will have a large rate of decrease in its annual right ascension. Below are given the various annual rates of change in right ascension of these stars for the date 1850, as found by observation : ft Ursse Minoris ,. 0-273s. 4949 Draconis + 0'935s. 4982 Ursa* Minoris .. -4-797s. MEASUKEMENT OF TIME, AND EIGHT ASCENSION. 211 A star, such for example as /3 Bootis, about 49 1' from the pole P, increases its right ascension annually about 2'264s., and this star has a right ascension of about 14h. 56m. It might be interesting to inquire whether a true rota- tion of the earth is to be measured by the interval between two successive transits of j3 Bootis, or of 4982 Ursae Minoris. Taking even 366 rotations of the earth, there would be a difference of 7s. in this rate, according as we elected to take one or the other star. It is not true, therefore, that the daily rotation of the earth can be measured by the interval of time which elapses between two successive transits of any star, or that all stars by their successive transits give the same interval of time. If a star be selected, this star may give a longer or shorter interval of time by two successive transits than the earth occupies in completing one rotation. Here, then, we have one cause for that confusion to which Sir John Herschel candidly called attention; but there are other causes, which have yet to be made known. One of the most important elementary matters to be first dealt with is to separate what is a true rotation of the earth round its axis of daily rotation, and then what is the effect of the second rotation in retarding this daily rotation in some parts, but accelerating it in other parts. The daily rotation will be accelerated between the poles of second rotation and of daily rotation, but retarded between the poles of second rotation and other points of the earth's surface. The equator of second rotation cuts a meridian of six hours right ascension at a point in latitude (terrestrial) 29 25' 47" north. This point on the earth's surface has its daily rotation retarded annually by the second rotation by an arc of 40.9". The same locality referred to a meridian of eighteen hours right ascension will be twice 29 25' 47" = 212 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. 58 51' 34" from the equator of second rotation, and will be retarded by the second rotation only by an arc of 40-9" X cosine of 58 51' 34" - 21149" annually. These two arcs, being referred to the equator of daily rotation, will be increased as follows : Z Z' = 40-9" Q Z =29 25' 47" 409" Again, if Z Z' = 21-149" ^ Q = Cos. of 29 25' 47" = ^ 2 An arc of the equator of 46;9" converted into time, as indicated by the daily rotation equals 3'ls.; an arc of the equator of 24*2" converted into time equals l*6s. Hence the daily rotation required to readjust, as it were, the effects of the second rotation for a meridian of six hours right ascension amounts to 3'ls. annually, whilst the daily rotation required to readjust the effects of the second rotation for a meridian of eighteen hours right ascension amounts to l'6s. annually. Hence there is a difference of 3'ls. l'6s. = l'5s. in the apparent effect of the daily rotation during a year between the meridians of six hours and eighteen hours right ascension, as regards a latitude of 29 25' 47" north. It follows that the earth would have to perform T5s. more of its daily rotation to bring a star with six hours right ascension on the meridian, than it would have to perform to bring a star on the meridian which had eighteen hours right ascension, these stars in each case having a north decimation of 29 25' 47". Although there are no two stars situated exactly in the positions referred to, we yet find that there are two sufficiently near to the points to serve as examples. These MEASUKEMENT OF TIME, AND RIGHT ASCENSION. 213 stars are as shown below, their right ascensions, declinations, and annual variation in right ascension being extracted from a star catalogue of 1850. Star. Declination north. Right ascension. Annual variation. K Auriga .. .. 29 32' 51" .. 6h. 5m. 49'14s. .. + 3'828s. o Herculis .. .. 28 44' 43" 18b. 1m. 41'53s. + 2'341s. Difference .. l'487s. From this example, a geometrician may perceive how each case may be worked out, and the simple calculation can be made. Take, for instance, a star with six hours right ascension, and 10 north declination; this star would be 19 25' 47" from the equator of second rotation. Then take a star with eighteen hours right ascension, and 10 north declination; this star will be 39 25' 47" from the equator of second rotation. Work out the effects of the second rotation on these two stars, and it will be found that the annual variation in right ascension of the star having six hours right ascension will exceed that of the star having eighteen hours right ascension by about 0'5s. Again, take two stars on these meridians, with 10 south declination, and the opposite effect would be mani- fested, viz. the star with six hours right ascension would be 39 25' 47" from the equator of second rotation, whereas the star with eighteen hours right ascension would be 19 25' 47" from the equator of second rotation. Hence the annual variation of the star with six hours right ascension would be less by about 0*5s than would that of the star with eighteen hours right ascension, but with the same declination. Although no stars are situated exactly in the positions to serve as examples of this law, the two stars, given below give from their positions the correct results, viz. Star. Declination south. Right ascension. Rate. 6 Leporis .. 14 55' .. .. 5h. 59m + 2 718s. 6279 Sagittarii .. 14 39' .. .. 18h. 20m + 3'419s. Difference 0'701s. 214 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. When the effects of the second rotation come to be examined as regards meridians 01" twelve and twenty-four hours right ascension, none of those hitherto mysterious variations occur which take place in connection with six and eighteen hours right ascension. The reason why these variations do not occur is very simple, and can be readily understood after an examination of the following diagram : P E S Q (Fig 71) represent the earth; P the north, S the south, pole; E Y Q, the equator of daily rotation; P Z X Y S, a meridian of twelve hours right ascension; C, the position of the pole of axis of second rotation. fig 71 The various points O, Z, X, and Y are carried by the second rotation annually over small arcs round C as a centre, consequently is carried to 0', Z to Z', X to X', and Y to Y'. But these changes are very similar to those which are produced by the daily rotation failing to complete one rotation. Let the daily rotation be completed, and 0', Z', etc., would by the daily rotation be brought almost exactly into the meridian P, S. Their zeniths, in each case giving a difference in zenith distances of stars, but no great variation in the annual rate of right ascension. The same law holds good for a meridian of twenty-four MEASUKEMENT OF TIME, AND RIGHT ASCENSION. 215 hours right ascension, each zenith being carried south over small arcs round C as a centre. Hence, when we examine the annual variation in right ascension of a star within some 20 or 30 of the pole, and of one some 80 or 90 of the pole, we find but a very slight difference in the annual rate of change in right ascension. Here are some examples : Star. 8 Ursee Minoris North polar distance. Right ascension. . . 32 8' . . 12h. 8m. . . Annual rate. 3-016s. 7 Corvi 6 Comae . . 106 42' . . . . 74 16' . . 12h. 8m. 12h. 8m. . . 3-077s. 3-0548. 2 Canuin Venaticor . . 48 30' 12h. 8m. 3-0348. 7? Virginia . . 89 50' . . 12h. 12m. .. 3-067s. When we turn to twenty-four hours right ascension, we find that the same uniformity in rate must occur in the annual variation in right ascension, stars very near the pole of daily rotation being affected by those changes already described in connection with the last curve given; for example (1850). B.A.C. Star. North polar distance. Right ascension. Annual variation. 8334 Cassiopese . . . . 29 37' 23h. 54m. + 3-0098. 29 Piscium . . 93 52' 23h. 51m. + 3-075s. 22 Andromedse . . . . 44 46' Oh. 2m. + 3-093s. 7 Pegasi . . 75 39' Oh. 5m. + 3-084s. a Andromedse .. 6L44' Oh. Om. 38s. + 3-086s. When we examine the results of the second rotation as affecting the annual rate of variation in right ascension of two stars having nearly the same north declination, the one star being near twenty-four hours, the other being near twelve hours, right ascension, we shall not find those great changes or differences in this rate which occurs with stars having six and eighteen hours right ascension. The reason for this is evident. The zeniths of twenty-four hours and twelve hours right ascension are nearly at the same distance from the equator of slow rotation, no matter what the latitude of each locality may be, as long as these latitudes are in each case the same; but with six hours and eighteen hours 216 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. the same latitudes are at very different distances from the equator of second rotation, and are consequently very differently affected. Hence we find that two stars having nearly the same declination or polar distance, but the one being near a meridian of twenty-four hours right ascension, the other near a meridian of twelve hours right ascension^ will vary their relative annual rate very slightly. Take for example, the two stars given below for 1850 : Star. North polar distance. Right ascension. Annual variation. ft Cassiopese .. 31 40' 39" .. Oh. 1m. 12s. .. + 3'149s. 8 Ursse Miuoris . . 32 8' 1" . . 12h. 7m. 58s. . . + 3-016s. Now, if these two stars had six and eighteen hours right ascension, the effect of the second rotation would cause their annual rates to vary as much as T5s. Again, take the two stars given below, viz. : Star. North polar distance. Eight ascension. Annual variation. 7 Pegasi . . . . 75 40' . . Oh. 5m. 31s. . . + 3-084s. 4125 Comse .. .. 74 16' .. 12h..8m. 22s. .. + 3'054s. Here we have a difference in the annual rate of only three-hundredths of a second per year. When, however, we examine the difference in rates for stars with about six and eighteen hours right ascension, the results are very different. Example Star. North polar distance. Right ascension. Annual rate. v Orionis . . . . 75 13' 7" . . 51i. 59m. Os. . . + 3'428s. 6049 Herculis . . 73 14' 12" . . 17h. 53m. 22s. . . + 2'667s. Here we have a difference of seventy-six-hundredths of a second in the rate. The cause of these variations will be evident from an examination of the diagrams in this chapter. It can be at once seen that the second rotation does not produce similar results on the two meridians of six and eighteen hours right ascension, but it does produce similar effects on twelve and twenty -four hours right ascension. Another and most important result of the second rotation is, that a locality 29 25' 47" from the equator of MEASUREMENT OF TIME, AND EIGHT ASCENSION. 217 daily rotation is retarded annually by the second rotation by an arc of daily rotation amounting to 40 '9" for this latitude, when referred to a meridian of six hours right ascension, and this arc of 40.9" is directly in opposition to the daily rotation. This same locality is retarded by the second rotation over an arc less than 40.9" when referred to twelve hours right ascension, and to be found as follows : C P = 29 25' 47" P Z = 90 - 29 25' 47" = 60 34' 13" Angle C P Z = 90 Therefore Z, the hypotenuse = 64 39' 45" Fig- 72 The point Z, therefore, is 25 20' 15" from the equator of second rotation, and the length of the arc Z Z', over which the second rotation carries this point, is 40.9" X cosine of 25 20' 15" = 36-9". But this arc Z Z' is not in direct opposition to the daily rotation; it is only partially in opposition. The daily rota- tion will carry the point Z' to the meridian P Z over a small arc, found as follows : Z x =2009" Zr jZ Z Z' = 36-9" \y Angle at x = 90 Z Calculate Z' x F* 73. 218 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. 71 x will be about 30". Hence, whilst the daily rotation has to readjust the point on the meridian of six hours by a rotation over an arc of 40.9", it has for twelve hours to perform a rotation of only 30" to reach the same meridian it occupied the year previously. Here we have an interesting fact on which geome- tricians may exercise their skill, inasmuch as this result of the second rotation has something to do with the assumed variation in the eccentricity of the earth's orbit round the sun, inasmuch as the zenith of such a locality as that named appears to separate annually from six hours to twelve hours by a small arc. We may now gather together the principal facts con- nected with the measurement of time and of right ascen- sions, and note what these prove. First, it is not true, as has been stated, that "all the stars occupy the same interval of time between their successive appulses to the meridian." A star within the circle described by the pole of the heavens will transit once oftener whilst this circle is being described than will a star outside the circle. Hence it is necessary, for this purpose alone, to know exactly the radius of the circle which the pole does describe. Secondly, the time occupied by the earth in making one rotation can be measured by one item only, viz. by the successive transits of that point in the heavens which is unaffected by the second rotation, and which conse- quently is the pole of the axis of second rotation. Thirdly, the measurement of sidereal time from that point on the equinoctial termed the first point of Aries is a cause of confusion, inasmuch as this point shifts annually, and does not shift uniformly; nor will it shift uniformly until the date 2295*2 A.D. Neither does this MEASUREMENT OF TIME, AND RIGHT ASCENSION. 219 point shift to an equal amount that the line of the solstices shifts, for the reason given in the latter part of this chapter. Fourthly, the differences that occur in the relative rates for the annual variation in right ascension of stars in meridians of six hours and eighteen hours right ascension, and which differences do not occur with stars on twelve and twenty-four hours right ascension, are due to the geometrical laws which follow the effects of the second rotation. Fifthly, the hitherto unaccountable changes in stars' right ascension which take place between meridians of fifteen hours and twenty-one hours, are due to the effects of the second rotation, as already explained. These changes are not due to the assertion that the whole solar system is rushing towards the" constellation Herculis at the rate of one hundred and fifty-four million miles per year, and that sooner or later a general crash will occur, and the whole universe will be destroyed. Such remarks are promulgated by those persons who, either from ignorance or laziness, are not acquainted with the laws of geometry and mechanics. 220 UNTRODDEN GEOUND IN ASTRONOMY AND GEOLOGY. CHAPTER XIV. MODERN ASTRONOMICAL OBSERVATIONS. THERE is probably no performance so likely to impress a visitor, or even a board of visitors, with the supposed perfection of modern astronomy as to attend at a large observatory, and to witness some important observation being made. The huge transit instrument, looking much like a piece of heavy artillery, revolves noiselessly on its pivots, and could be moved by even a child. The various adjustments of the instrument can be readily made; and, as far as the machinery is concerned, everything is perfect. Even the observer is reduced to a sort of machine, his personal error of observation having by repeated trials been ascertained and consequently allowed for. When the meridian transit of the sun occurs, various members of the staff occupy positions which enable them to read the micrometers, by which hundredths and thou- sandths of a second are ascertained. By long practice the observers become so expert that, although great haste is used, there is no apparent hurry; and when it is announced that the results of this transit have been arrived at to the ten-thousandth of a second, a visitor would probably ex- claim, " Wonderful ! " and he would no doubt be convinced that modern astronomy was now in such a state of perfec- tion, that to question the minute accuracy of any portion of it was a proof of ignorance only. MODERN ASTRONOMICAL OBSERVATIONS. 221 It must be admitted that instrumental observation, as far as it goes, is very near perfection; but there is a serious item to be considered in connection with astronomical science, and even as regards this instrumental observation. More than seventeen hundred years ago there was a very able observer named Ptolemy, who used, for deter- mining the position of the celestial bodies, various instru- ments. These instruments were of three different kinds. One was a meridional armillse, consisting of two rings, with which he measured the zenith distance of the sun and the obliquity of the ecliptic. The exterior of these two rings was graduated in degrees, each degree subdivided to as small an amount as was possible. Another instrument was a quadrant with a very large radius, and was used to measure zenith distances. The third instrument has been termed " the parallactic rods," which consisted of a pillar placed vertically on a base. At the upper extremity was a joint, on which turned a long alidade carrying two sights. This instrument was also used to determine zenith distances. Another instrument also used by Ptolemy was an astrolabe, used for measuring the distance between the sun and the moon, or between the moon and a star. In order that observations should be made under all conditions, the ancients dug caves, the entrance to the cave being opposite the meridian. Thus Strabo relates (" Geogr.," lib. xvii.) that in his time, at Heliopolis, caves were shown which had been used by observers for ascertaining the position of celestial bodies by day. There were also at Cnidus, in Asia, similar caves used for the same purpose. All these arrangements, and the vast instruments used at the time, were no doubt as perfect as the mechanical skill at that date could accomplish. They have been sur- passed in accuracy by the moderns, it is true, but they 222 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. accomplished with moderate accuracy only that which at the present day is accomplished with great accuracy, viz ascertaining by observation the positions of the celestial bodies. The ancient observers could foretell with very fair accuracy when an eclipse would occur, when a star would come to the meridian, when a star would be occulted by the moon, etc., etc.; and yet these observers believed that the earth was stationary, and neither rotated every twenty- four hours on its axis, nor revolved annually round the sun. If a visitor, or a board of visitors, had gone to the observatory of Ptolemy, had examined his grand instru- ments, seen how he made his observations, and verified the fact that he could predict the day and hour at which an eclipse or an occultation of a star would occur, these persons would undoubtedly have asserted that the science of astronomy was perfect, and that any person who doubted this perfection must be very ignorant. Yet two important items were unknown at that date, viz. the daily rotation and annual revolution of the earth round the sun. It is a fact, therefore, that in spite of the accuracy which may be arrived at by perpetual observation as regards the predicting the positions of celestial bodies for a few years in advance, yet the causes which produce the movements may be unknown. It follows, therefore, that although the instruments and observations of the^ ancients were admirable, considering the means at their disposal, and the instruments and observations at the present time are very near perfection, yet in each case it is a question of observation only, and does not give any proof that the true movement of the earth is known at the present day any more than it was seventeen hundred years ago. It may appear a strange assertion to those persons who MODERN ASTEONOMICAL OBSERVATIONS. 223 take everything for granted, yet it is a fact, that if the earth did not move, whilst the various celestial bodies revolved daily round it, every item now obtained by observation could be equally as well obtained by the same means, and a Nautical Almanac could be framed on exactly the same lines as one is now framed. Hence, although instrumental observations may be carried out with great perfection, we must not make the mistake of imagining that therefore the science on which these observations are made is perfect. The ancients did make this mistake, and consequently during more than fourteen hundred years their system of astronomy was wrong, and they considered it a duty to ignore or persecute all those who, being gifted with reasoning powers, asserted that it was more probable that the earth rotated each twenty-four hours, than that the heavens revolved round the earth during the same period of time. It will probably be admitted that, had instruments been, in the days of Sir Isaac Newton, in as perfect a condition as they now are, this philosopher would soon have made himself acquainted with their mechanical details, and could, as easily as any modern observer, have read by aid of his instruments the supposed zenith distance of a star to the one-thousandth of a second. What would have been the result? A visitor might be struck and impressed with the most wonderful accuracy thus obtained, and would probably imagine that perfection had been reached. Unfortunately, however, Sir Isaac Newton framed a table of refractions for various altitudes, and he imagined that in summer the refraction for certain altitudes varied from that in winter as follows : Altitudes. Summer. Winter. 15 3' 4" 3' 28" 20 2' 17" 2' 33" 25 1' 46" 2' 0" 30 1' 26" 1' 30" 224 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. Hence it would follow that, whilst he imagined he was by his instruments giving results accurate to the one- thousandth of a second, he was wrong more than 31" for an altitude of 15 measured in summer, the refraction for 15 being now taken as 3' 35". The effect produced by refraction varies according to the state of the atmosphere, and must ever bring an element of uncertainty to bear on observations; it is not too much to admit that this uncertainty may amount to 1". Conse- quently, whilst the instruments used for observations may give readings to one-thousandth of a second, refraction may cause these readings to give results erroneous to 1". The minute accuracy, therefore, which is claimed by observers is theoretical only, as it cannot exist as long as refraction affects the position of a star as seen with a telescope. But does the minute accuracy which is claimed actually exist? For example, in the Nautical Almanac for 1873, the mean right ascension for the star & Ursre Minoris for January - -226d. is given as 18h. 13m. 18'497s. In the Nautical Almanac for 1887, the mean right ascension for the same star for January 0' + '165d. is given as 18h. 8m. 45'996s. Consequently, between these dates the mean right ascension of this star has decreased 4m. 32'501s. = 272'501s. during fourteen years that is, at the mean rate of - 19'4643s. per year. Yet in the Nautical Almanac for 1873, it is stated that the annual variation in right ascen- sion for this star is - 19'3924s., and in 1887 it is - 19'4370s. These discrepancies occur in a multitude of instances, proving that the minute accuracy claimed is theoretical only. It is, of course, very easy to explain such matters by asserting that the star itself moves; but the fact remains that, whilst it is claimed that accuracy exists to MODERN ASTRONOMICAL OBSERVATIONS. 225 the fourth place of decimals, facts prove that even the second place of decimals does not agree. Is it, however, probable that accuracy to the ten- thousandth of a second can be practically arrived at, when, as stated by Sir John Herschel, it was found necessary to add more than three minutes of purely imaginary time, in order to cook the accounts? The statement of Sir John Herschel, that the whole subject of " time " is in confusion, is corroborated by facts, and it will remain in confusion until theorists become more practical, and really investigate the true movement of the earth as a geometrical problem. As long as authori- ties are contented to accept as highly satisfactory the vague definition that the earth's axis, by the joint action of the sun and moon, has " a conical movement," they will be compelled to depend entirely on continued instrumental observation for their results. When, however, they really investigate the second rotation of the earth, they will dis- cover that observations are a clumsy and inaccurate method of arriving at results. Among the numerous " visitors " and other persons con- nected with observatories, it is a most remarkable fact that these individuals seem to be fully convinced that astronomy is in a most perfect condition, because very fairly accurate observations can be made. If reason were brought to bear on this question, the inquiry ought to be, Why are observa- tions any longer necessary? No instrument that man ever made can be so accurate and uniform in its movements as is the earth itself. Some changes may occur at long intervals of time, but these changes must be very slow, and one or two observations per year would be sufficient to check any irregularity, and reveal in what direction this was occurring. Yet we find that the mere routine observer usually claims it as a proof of the importance of the work Q 226 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. carried out at his observatory, that upwards of one hundred observations per year have been made of various important stars, not at one observatory only, but at scores of obser- vatories. What is the use or object of all this labour, when every item can be calculated to within a fraction of a second for fifty or a hundred years, when the second rotation of the earth is comprehended? It would be a somewhat puerile proceeding to employ a number of persons to count how many times the second-hand of a chronometer rotated during each minute, hour, and day of the year, and to claim as most important the labours of individuals so employed. A chronometer correctly constructed will cause the second-hand to rotate sixty times during each minute, 3600 times during each hour, and 86,400 times during each day of twenty -four hours. The polar distance and right ascension of each star can be calculated with equal facility. If we want to know what the mean polar distance of the pole-star, for example, was or will be for any date past or future, the formula is given in Chapter X. If we want to know what the apparent polar distance of this star will be for any day of the year, we can measure this by aid of the nautilus curve given in Chapter XI. To claim, therefore, that it is necessary to make some hundred observations annually of any star, is a proof, not of the efficiency of astronomy, but of the incom- petency of observers to arrive at the results by any other means. It seems, also, little short of marvellous, that although from the time of Ptolemy, on to that of Bradley, and down to the present date, instruments have been used to measure meridian zenith distances, yet it has never been considered worth while to investigate in what manner various zeniths were affected by " a conical movement of the earth's axis." MODERN ASTRONOMICAL OBSERVATIONS. 227 Observers have carefully noted how the zenith of the pole of daily rotation has been affected annually by this movement; but how other zeniths were affected did not seem to trouble them. Observation, perpetual observation alone, was the end of everything; good instruments, a large staff of highly paid observers, night after night devoted to the transit instrument and chronometer, day after day devoted to correcting the observations, and astronomy was claimed to be in the most perfect condition, in spite of the fact that the mean polar distance of not one single star could be calculated by any sound geometrical law for even ten years in advance or in the past, the only method known being to approximate to this position by means of adding or subtracting some annual rate, found by repeated observation only. That instrumental observations can now be made with great accuracy, is no more a proof of the perfection of astronomy than it was in the time of Ptolemy. In fact, the necessity for maintaining a large staff, whose duty it is to continue night after night making these observations, is a proof that there is something not yet known connected with the movement of the earth. We have not far to search in order to find what this something really is. When we find that a vague movement of the earth termed merely a conical movement of the axis is accepted as satis- factory although no mention is made of which pole remains fixed, and which gyrates, or how each zenith is affected by this conical movement, it is easy to understand why perpetual observation is assumed to be the only available means by which star-catalogues for the future can be framed. The reader may now comprehend how great will be the difference in the real value of instrumental observation when the second rotation of the earth, with all its details, becomes understood. Hitherto it has been by means of 228 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. these observations only, that Ijjie position which a star would occupy at a date only four or five years in advance could be assigned. Now, however, there is a means by which the position of this star can be calculated for a hundred years or more, without reference to more than one observation. Instead, therefore, of being dependent on instrumental observation, we can actually check these, and can decide whether instruments have been in adjustment, observers have been careful, and whether the correct refrac- tion has been used at various dates. Take, for example, any star, say )3 Ursae Minoris, the mean right ascension of which star for 1887 is January 0* + -165d. = 14h. 51m. 2'486s. ) The mean decimation = 74 37' 1-88" J recorded observatlon - From this data alone, calculate the mean declination of this star for any other dates say January 1, 1850, 1819, 1755, and 1950 and show what errors, if any, were made by observers at former dates. To the mere routine observer, this calculation could no more be accomplished than a Zulu could work out a quad- ratic equation. A geometrician who understands the second rotation of the earth can solve the problem without diffi- culty. The solution has been already given herein, and in "Thirty Thousand Years of the Earth's Past History'' (Chapman and Hall). ( 229 ) CHAPTER XV. THE PLANE AND THE POLES OF THE ECLIPTIC. THE plane of the ecliptic is the course along which the earth travels annually round the sun. It may be defined as that circle in the heavens which the earth would trace annually among the fixed stars if an observer were located in the sun. It is also the course which the sun's centre appears to trace among the fixed stars when seen by an observer on the earth's surface. The poles of the ecliptic are two points in the heavens, distant 90 from all parts of the ecliptic. These poles bear to the ecliptic the same relation which the poles of the earth bear to the equator, or the poles of the heavens bear to the equinoctial. Theoretically, it is a very simple problem to determine exactly the position which the plane, and hence the poles, of the ecliptic occupy in the heavens. Practically, it is one which presents difficulties which, it appears, are so great that since astronomy has been a science it has never been accomplished. The four earliest catalogues of stars with which we are acquainted are those formed by Ptolemy, 140 A.D.; Ulugh Beigh, 1463 A.D.; Tycho Brahe; and Hevelius. In these catalogues the latitudes of stars are given, that is, their least angular distance from the plane of the ecliptic, and 230 UNTKODDEN GEOUND IN ASTRONOMY AND GEOLOGY. their longitudes, which is the angular distance or arc of the ecliptic intercepted between two circles of latitude, one passing through the star, the other passing through the vernal equinox. It must be remembered that, no matter what minor instruments were used by the ancients, yet they employed for their observations the one great instrument which the moderns use, viz. the earth itself; and it was as necessary in ancient days as it is at present, that the exact move- ments of this great instrument should be correctly known. The determination of the position of the pole of the ecliptic by the ancients appears to have been conducted in a very vague manner. Ptolemy, for example, gave the position of this pole as 24 from the pole-star, 14 20' from /3 Draconis, and 7 from S Draconis. When this position is assigned, it differs from that which Ptolemy gave it by three other stars, for he assigned its position as 14 30' from y Draconis, 24 30' from a Draconis, and 15 20' from Draconis. Tycho Brahe gave the pole of the ecliptic 23 58' from the pole-star, thus indicating that from the time of Ptolemy to that of Tycho Brahe there had been by these obser- vations a decrease of only 2' in the distance of the pole of the ecliptic from the pole-star, although during the same period there had been a decrease of nearly 30' in the distance of the pole of the heavens from the pole of the ecliptic. Upon checking the various latitudes of stars as assigned by various ancient observers, and hence attempting to fix the position which they imagined the pole of the ecliptic occupied, we encounter a confusion only, leading to the conclusion that they could not with accuracy fix the posi- tion of this pole. When we come to modern times, we find equally as THE PLANE AND THE POLES OF THE ECLIPTIC. 231 interesting a subject for reflection. The modern observer starts from a theoretical assumption. He assumes that the course which the pole of the heavens traces on the sphere of the heavens is a circle round the pole of the ecliptic as a centre. He concludes that when the pole in its course neither increases nor decreases its distance from a star, that therefore the pole of the ecliptic must be on the arc joining that star and the pole of daily rotation, and at such a distance from the pole of daily rotation as the obliquity of the ecliptic happens to be at the time. This conclusion might be true, supposing that during the past thousand or more years there had not been the slightest variation in the obliquity of the ecliptic, because there being no change, would prove that the pole of the heavens did not vary its distance from the pole of the ecliptic, and hence that the pole of the heavens must trace a circle round the pole of the ecliptic as a centre. When, however, as is known, the obliquity decreases, and has decreased during two thousand years at least, then it is impossible that the pole of the heavens can trace a circle round the pole of the ecliptic as a centre, and the method adopted to determine the position of the pole of the ecliptic is erroneous. In order to demonstrate the very serious error which both the ancient and modern observers have made in this matter, the following diagram (Fig. 74) can be examined. The circle N P Q R is the circle traced by the pole of the heavens during one second rotation round C as a centre, the radius C O = C P = C Q = 29 25' 47". E is the position of the pole of the ecliptic; C E = 6. R is the position which the pole of the heavens will reach at the date 2295'2 A.D. When the pole of daily rotation at a remote date in the, past was at 0, a star, x on the meridian joining and C, 232 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. would neither increase nor decrease its polar distance; that is, for a brief period would not vary its distance from x. The modern theorist, therefore, would assume that the pole of the ecliptic must be on the arc O x C; and, as the angular distance O E would be the measure of the obliquity at that date, he would place the pole of the ecliptic at A, the distance O A being equal to O E. The pole of the heavens, continuing its uniform move- ment round C as a centre, would after many years reach P, at which date a star y on the arc joining P and C would not increase or decrease its polar distance for a short period. The theorist would now assume that the pole of the ecliptic was on the arc P y C, and distant from P as much as the obliquity was at that date. P E would represent the obliquity at that date, con- sequently P B would be made equal to P E, and the pole of the ecliptic would be assumed to be at B, on the arc PC. In the same manner, when the pole of the heavens was at Q, the pole of the ecliptic would be assumed to be at D, on the arc Q C, the distance Q D being made equal to Q E, the obliquity at that date. We can now examine how, by this erroneous system, the pole of the ecliptic has been assumed to move. At a remote THE PLANE AND THE POLES OF THE ECLIPTIC. 233 E, a parallel of latitude of 20; F, the equator; and N x D S, a meridian of twelve hours right ascension. The pole N is carried over an arc of 1 during 180 years, directly down (as we may term it) the meridian on the opposite side of the sphere, by some movement of the earth hitherto assumed to be accurately defined by calling it " a conical movement of the earth's axis." Will some or any learned theorist 'mark on this diagram where the points A, x, B, D, C, E, and F, will be carried by that same movement, which has caused the pole to be carried 1 down the meridian on the opposite side of the sphere? What is the length and what the direction of the various arcs over which these points are carried, is a fair question for inquiry. OBJECTIONS OF THEORISTS. 291 It may be an interesting question for the student or the reader to put to those gentlemen who claim to tell us what it is impossible that the earth under certain assumed conditions can do, but who have hitherto failed to state what the earth does do and has done as regards its movements during the past two thousand years. Considering that the detail movements of the earth, which accompany the change in the zeniths of the poles of 2O09" annually, have never been defined or even hinted at by theorists, it is scarcely likely to add to the scientific reputation of certain gentlemen to find them treating this important problem in such a jaunty, offhand manner as they too frequently have treated it. In a work lately published entitled " Discussions on Climate and Cosmology," the author, Dr. James Croll, de- votes seventeen lines to the subject of that movement of the earth to which reference has been made in this and in my former works. In the first two sentences of the paragraph, Dr. Croll writes as follows : " The theory of a change in the obliquity of the ecliptic has been appealed to. This theory for a time met with a favourable reception, but, as might have been expected, it was soon abandoned." I am at a loss to understand what information this writer wishes to convey to his readers by these two sentences. If it were stated that some three hundred years ago Copernicus, and afterwards Galileo, suggested that the rising and setting of the various celestial bodies would be explained by a daily rotation of the earth, but that " this theory, as might have been expected, was soon abandoned," we should not be disposed to attribute any great intel- lectual capacity to those learned gentlemen who so soon abandoned this theory. Does Dr. Croll mean it as complimentary, or the reverse, 292 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. that the variation in the obliquity was so soon abandoned, whilst at the same time the actual movements of the earth had never been defined? Any person who has carefully examined the details of the second rotation of the earth, and has proved that most important and accurate calculations can be made by aid of a knowledge of this movement, also that hitherto no such calculations have been possible, and no details of the movements of the earth have ever been given, will attribute the "soon abandoned" to the same cause as that which led to the daily rotation of the earth being abandoned. It is, however, doubtful whether this is the exact meaning which Dr. Croll intended to convey to his readers. The remainder of the paragraph which this writer devotes to the variation in the extent of the arctic circle is very interesting, he states. "The researches of Mr. Stockwell of America, and of Professor George Darwin and others in this country, have put it beyond doubt that no probable amount of geo- graphical revolution could ever have altered the obliquity to any sensible extent beyond its present narrow limits. It has been demonstrated, for example, by Professor George Darwin that, supposing the whole equatorial regions up to latitude 45 north and south were sea, and the water to the depth of two thousand feet were placed on the polar regions in the form of ice and this is the most favourable redis- tribution of weight possible for producing a change of obliquity it would not shift the arctic circle by so much as an inch " " Climate and Cosmology," p. 4. The obliquity of the ecliptic, found by repeated observa- tions at the early part of the Christian era, was at least half a degree greater than it is at present. According to modern observations, the decrease at present is about 46" per century. OBJECTIONS OF THEORISTS. 293 One degree of meridian for latitude 66 is stated to be about 365,800 English feet. Half a degree, therefore, is 182,900 English feet, equal to two million one hundred and ninety-four thousand eight hundred English inches (2,194,800). To be able to state that not one inch out of these two million and more inches, which it is known the arctic circle has " shifted " during even so short a period as eighteen hundred years, can be accounted for on the supposition of an equatorial ocean two thousand feet in depth, and extending 45 each side of the equator, being turned into ice, and placed on the polar regions, is a valuable contribution to exact science. This "proof" is valuable in more ways than one. It proves that our old earth is utterly independent of, and uninfluenced by, those dynamic laws which influence every other rapidly rotating body. If a gyroscope top be spun, and we find that the axis of rapid rotation changes its direction in a certain way, we shall find that this direction changes almost immediately in another way when we alter the position of the centre of gravity by means of the smallest weight attached to the surface of the top. If, however, we could cause this top to rotate rapidly during one hundred years, a thousandth part of the weight would, during a long period, produce somewhat similar results to those produced by a heavier weight almost instantly. It is, therefore, a very important item which is referred to by Dr. Croll, viz. that it is proved that, if the position of the centre of gravity of the earth be altered several thousand feet, yet the direction of the axis of rotation would " not " shift " by so much as an inch." This is the theory which is put forward as a proof of what? Probably that the obliquity can vary only between 294 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. very narrow limits; but what are these limits? and by what laws is this narrowness limited? Without following these theoretical conclusions further, or even hinting that they are anything but quite correct, an important suggestion is at once presented to us. It has been stated how, on certain assumptions, and on the theories based thereon, it is impossible that the earth ever did or ever can move in some particular manner. Would it not be a proceeding of more practical value if these theorists were to rigidly define how the earth does move and has moved even during the past 180 years. The poles of the axis of daily rotation have, during this period, traced arcs of about 1; the north pole having traced an arc of 1* nearly towards the first point of Aries. What is the length of the arcs which the zenith of Greenwich has traced owing to this movement? and what is the direction of these arcs as regards six hours, twelve hours, and eighteen hours right ascension during 180 years? What are the lengths and directions of the arcs which the zenith of 62 north latitude has traced on the sphere of the heavens, owing to the same movement of the earth, during the past 180 years, for meridians of six, twelve and eighteen hours right ascension. Considering the present distribution of land and water on the earth's surface, it would be a useful practical problem to demonstrate where the centre of gravity of the earth was at present situated, and whether it coincided to one inch with the centre of the sphere. According to Dr. Croll, it has been proved that, if we took 45 on each side of the equator, amounting to 90 in all, and a depth of water of two thousand feet, and con- verted this water into ice, and piled it all within a few degrees of one or both poles, we should not alter the direc- tion in which the earth's axis now moves by one inch. OBJECTIONS OF THEORISTS. 295 Let us assume that we pile this mass of ice within 5 of the north pole. 90 piled into 5 would give eighteen times the depth of two thousand feet, equal to thirty-six thousand feet deep of ice, giving extra weight around the north pole, and withdrawing this weight from the equatorial regions. Would this condition produce no change in the position of the centre of gravity of the earth, and hence, as a mechanical law, causing the earth's axis of rapid rotation to change its direction, in a manner slightly different from that in which it now changes it? If it be asserted that, no matter how you change the position of the centre of gravity of a rotating sphere, yet you can by this change produce no alteration in the direction in which the axis of this sphere points, I must at once claim that the laws of dynamics and actual facts prove the very opposite to be the truth. But why do certain learned theorists occupy their time in attempting to prove what cannot occur? Why do they not state what does occur and has occurred during the past two thousand years of which we have records? The learned Sizzi proved to his own satisfaction, and also to that of the school of science to which he belonged, that it was impossible that Jupiter could possess satellites. Would it not have been a more practical proceeding if he had looked through a telescope and seen them? Would it not be a more practical proceeding if theorists were to look at the second rotation of the earth, and prove for themselves that a knowledge of this movement enabled them to calculate for any length of time the polar distance of a star from one observation only? Surely such an investigation would be more useful than inventing vague theories, in order to endeavour to prove that something could not occur? Would it not be a more practical proceeding to define 296 UNTRODDEN GKOUND IN ASTRONOMY AND GEOLOGY. accurately every detail connected with the movement of the earth, which accompanies the charige in position of the zenith of the pole, instead of leaving this movement as it is at present, vague and undefined, and inventing theories to account for something, when it has never been defined what this something really is? With regard to this movement of the earth, we have to deal with facts, not with vague theories; we have to define how each portion of the earth moves during each year, not how the poles only of the earth move. Certain gentlemen who have distinguished themselves by rushing in where more prudent persons would hesitate to tread, have stated that the exact geometrical investigation to which I have given many "years of research is " an absurd theory." I have examined carefully the various books and articles which I have written on this movement of the earth, and also the notes and reports of various lectures that I have given on the same subject. I can find no theory put forward by me in any one of these works. That which I have done is to examine in detail the movement of a globe, which accompanies a change in direc- tion of the axis of rapid rotation. I define how each point on the surface of this globe moves annually during thousands of years. I then prove that, by a knowledge of this move- ment, the polar distance of a star can be calculated for dis- tant dates with minute accuracy, without reference to more than the one observation which determines the polar distance and right ascension of this star for one particular date. Where is the " absurd theory "? If this rigid geome- trical proof can be termed a theory, then it is a theory that two sides of a plane triangle are greater than the third side, and it is also a theory that the three triangles of an equilateral triangle are equal to each other. OBJECTIONS OF THEORISTS. 297 As another example of the unreasoning arguments put forward as regards this geometrical investigation being " some theory," the following may be given. Suppose an observer to be located in 51 north latitude and provided with an instrument by which he could measure zenith distances. By aid of this instrument he would find that his zenith had traced on the sphere of the heavens an arc which was 9 25' 48" in length during one hour of time. A geometrician might now inform this observer that if he had been located on the equator, instead of at 51 lati- tude, his zenith would have traced on the sphere of the heavens an apparently straight line 15 in length during one hour. Also if he had been located in 70 latitude, his zenith would have traced an arc 5 7' 48" in length during one hour. A geometrician would know these facts in con- sequence of being acquainted with the daily rotation of the earth. What opinion should we form of the intellectual or scientific qualifications of a writer who stated that this assertion of the zeniths of different latitudes tracing arcs of very different values was an " absurd theory " which is not worth considering? A knowledge of the second rotation of the earth enables a geometrician to make similar exact calculations in con- nection with the second rotation that can now be made in connection with the daily rotation. It can be stated that whilst the zenith of the locality in south latitude 29 25' 47", and on a meridian of eighteen hours right ascension, is carried annually by the second rotation over an arc of 40.9" in the exactly opposite direction to that in which the daily rotation carries this zenith, yet the zenith of a locality on the equator is carried annually by the second rotation for the same 298 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. meridian over an arc of only 35*62", in opposition to the daily rotation. The zenith of a locality in latitude 51 28' north, such as Greenwich, is carried annually by the second rotation for the same meridian of right ascension over an arc of only 6 "47". The zenith of Edinburgh is carried over an arc of 3*3" only, whilst that of Petersburg, in north lati- tude 59 56', is carried over an arc of only 0'65" annually by the second rotation, these arcs in each case being for eighteen hours right ascension in direct opposition to the daily rotation. The daily rotation does not readjust these zeniths exactly in the relative positions which they before occupied, the daily rotation being round the pole of the heavens as a centre; the second rotation being round the pole of second rotation as a centre, this pole of second rotation being 29 25' 47" from the pole of daily rotation. A point on the equator is, for eighteen hours right ascension, carried over an arc of 35*62" in opposition to the daily rotation; consequently, when 35*62" of the daily rotation has occurred, this point occupies, as regards its zenith, exactly the same position it occupied previous to its displacement by the second rotation. This 35*62" of daily rotation for the equator produces for the zenith of Greenwich a daily rotation represented by an arc of 22*17". But the zenith of Greenwich has been displaced by the second rotation by an arc of 6*47" only during the year, whilst that of Petersburg has been displaced only 0*65". Will those gentlemen who have thought it desirable to term such investigations as the above " absurd theories," kindly inform those who may be desirous of knowing the actual facts connected with the earth's movements, how much the lengths of localities on the equator, in south latitude 29 25' 47", in north latitude 51 28', and in north latitude 60 are displaced annually as regards a meridian OBJECTIONS OF THEOKISTS. 299 of eighteen hours right ascension by their theory of " a conical movement of the earth's axis/' which movement makes " a shift " " of this axis? " It is a mere waste of time to discuss scientific questions with individuals who ignore even the elementary laws of geometry and dynamics, and who claim that this and the other has been proved by certain persons whom they put forward as " authorities." In a work published in 1875, termed, " Climate and Time," the author, Dr. James Croll, makes the following statement : " The polar regions owe their cold, not to the obliquity of the ecliptic, but to their distance from the equator. Indeed, were it not for the obliquity, those regions would be much colder than they really are, and an increase of obliquity, instead of increasing their cold, would really make them warmer. ... It therefore follows that, although the arctic circle were brought down to the latitude of London, so that the British Islands would become a part of the arctic regions, the mean temperature of these islands would not be lowered, but the reverse. The winters would no doubt be colder than they are at present, but the cold of winter would be far more than compensated by the heat of summer." Having had considerable experience of the climates of various countries on the earth's surface, I believe that the meridian altitude of the sun and the length of time it remains above the horizon is the great cause of the various temperatures in different localities, and that the equator and the distance of localities from the equator is an effect, not a cause. The annual variations of climate in Quebec and many other parts of Canada may be taken as an example. The latitude of Quebec is about 46 48' north. In winter I 300 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. have seen the thermometer at 30 F. below zero. I have seen freshwater lakes during twenty-four hours frozen ten inches thick with ice. If this locality were deprived of 12 more of the sun's meridian altitude, it would be equiva- lent to placing the locality 12 farther north, and giving to the locality a temperature some 15 or 20 more of cold, when, consequently, more ice and more snow would accu- mulate during winter. In summer, at Quebec, the ther- mometer rises to 90 F. in the shade; the ice and snow are rapidly melted, and the rush of water, bearing icebergs, etc., is very great. Increase the cold during winter by 15 or 20, and the heat in summer by even a greater increase, and what would be the result? A vast accumulation of ice and snow rapidly melted and scattered over the country. I fail, however, to perceive why, because Dr. Croll imagines that an enormous increase in the eccentricity of the earth's orbit occurred, he should assert that a rigid investigation of the actual movements of the earth is an "absurdity." I can assure this gentleman that, if he will examine the geometrical laws connected with the second rotation of the earth, he will find that the assumed variation is even, in its details, based on a geometrical error. But what have such theories to do with the actual movements of the earth? State how the earth does move and has moved during the past two thousand years even, is a fair question. State over what length of arc and in what direction the zeniths of location 10, 30, 50, and 90, from the north pole have been carried during even the past 180 years by " a conical movement of the earth's axis " as regards six, twelve, and eighteen hours right ascension. Because a gentleman invents a theory to account for some effects, which theory is based on a geometrical error, why should he state that to rigidly examine and define those movements which have never hitherto been defined OBJECTIONS OF THEOKISTS. 301 is an "absurdity "? 'When this and any other theorist can calculate the polar distance of a star for a hundred years or more from one observation only, then they may claim to have some knowledge of the earth's movements. As, how- ever, such a calculation is unknown to them, it really appears that they are expressing their opinions on a problem which they have not examined or even understood. It is no part of the work of a geometrician to join in the disputes of geological theorists. There are certain gentlemen who assert that, during the last glacial period, the earth from middle latitudes to the poles was covered by a cap of ice several thousand feet in thickness which was never exposed to any great heat. Theories have been invented to account for this imaginary cap of ice. As stated in my late work, " Thirty Thousand Years of the Earth's Past History," both Sir John Lubbock and Professor Tyndall state that heat is as necessary as is cold to produce glaciers. Consequently, this cap of ice is a mere theory, on which learned gentlemen differ. It really appears difficult to account for vast masses of floating ice being carried over the land unless by the aid of heat. To prove what the movements of the earth really are and have been during many hundred years is the work of a geometrician. If these movements do not agree with the speculations of theorists, it is so much the worse for the theories. Certain theorists have stated that, if the obliquity of the ecliptic were so great as to cause the arctic circle to reach, as it does in Venus, to within 15 of the equator, the mean annual temperature of all localities in middle and high latitudes would be much greater than it is at present. In the first place, this is a mere assertion. We know so little as to the effects which would be produced by the melting of vast masses of ice and snow, that positive 302 UNTRODDEN GKOUND IN ASTRONOMY AND GEOLOGY. statements are mere assumptions. Whether all the ice formed each winter would be melted each summer is a question which no man can answer positively. Secondly, in the St. Lawrence river, under present conditions, I have seen masses of ice of twenty feet cube formed, not during one winter, but during one month of intense cold. Give several degrees more cold, which would result from 12 more of obliquity, and much larger icebergs would be formed. The mean annual temperature for any locality is no criterion of the climate. There are several localities where the mean annual temperature is about 60 F., the heat in summer being about 70 F., that in winter about 50 F. When the obliquity was about 35, there might be localities where the mean annual temperature was 60 F., the winter cold being some 70 below zero, the summer heat some 130F. Water frozen at a temperature considerably below zero takes a long time to thaw, and masses of ice thus thawed would produce fogs, as they do on the banks of Newfoundland, and thus veil the sun, and prevent its full power from being felt. The conclusions formed as to what must happen under certain conditions are mere theories, which may or may not be near the truth; thus, although reference is made to the supposed conditions which theorists assume must follow certain variations in the obliquity, it is necessary to remem- ber that the statements put forward as facts are merely guesses. At p. 417, " Climate and Time," the author makes the following remark : " But even supposing it could be shown that a change in the obliquity of the ecliptic, to the extent assumed by Mr. Belt and Lieut.-Colonel Drayson, would produce a glacial epoch, still the assumption of such a change is one which physical astronomy will not permit." This hitherto is quite correct. Physical astronomy and OBJECTIONS OF THEORISTS. 303 its theories " will not permit " any change in the obliquity beyond very narrow limits, and for the following reasons : Physical astronomy does not deal with geometry. Physical astronomy has never yet defined how the earth has and does move. It has never even referred to the detail movements of various zeniths and meridians which accom- pany the change in direction of the earth's axis. Physical astronomy has been content to accept " a conical move- ment of the earth's axis " as a full and complete definition of all the detail movements which are produced by the second rotation. Physical astronomy cannot calculate the position of any star even for one year from one observation of this star. It claims to be able to state what is the density of the various planets, such as Mercury, Venus, Mars, Jupiter, etc. The proof as to the accuracy or otherwise of these speculations does not exist. But physical astronomy omits to give any explanation as to why the axes of daily rotation of these planets are inclined at such very different angles to the planes of their orbits. It makes no mention of what sort of conical movement these axes make, or whether they make any movement. Let us now refer to something more practical, and capable of being tested as to its accuracy. In the Nautical Almanac for 1887 the mean right ascension and the south declination for January 1, 1887, of the star j3 Corvi are recorded as found by observation at that date as follows : Mean right ascension. Mean south declination. 12h. 28m. 26-99s 22 46' 17'85" Let physical astronomy, or any branch of astronomy hitherto known, calculate the mean declination of this star for January 1, 1850, and January 1, 1780, without reference to any other observation of this star. 304 UNTRODDEN GROUND IN ASTRONOMY AND GEOLOGY. It is almost needless to state that astronomers hitherto have not been competent to make such a calculation. When the second rotation of the earth is understood, the problem is easily solved, and the solution is as follows : From the above observation we obtain the distance of this star from C, the pole of second rotation = 106 19' 15'2". We know that the distance between the pole of second rotation and the pole of daily rotation is 29 25' 47". From the one observation given above, we can calculate the angle at C for January 1, 1887, this angle being 107 33' 52". This angle is increased annually by the second rotation at the rate of 40.9". At the date 1780 the angle, consequently, was less than it was in 1887 by 40*9" X 107 = 1 12' 56". The angle at C for January 1, 1780, was 106 20' 56". With the two sides, viz. C P=29 25' 47", C j3- 106 19' 15-2", and the included angle P C /3-P C /3 = 106 20' 56". The third side P ]3, the polar distance of the star /3 Corvi, can be calculated, and will be found to be 112 10' 38"; from which take 90, the distance of the equator from the pole, and we obtain 22 10' 38" for the south declination of /3 Corvi for January 1, 1780. In a catalogue of stars now before me, the south decli- nation of this star for January 1, 1780, is recorded as 22 10' 38". This is not the kind of test that physical astronomy cares to deal with, nor has it hitherto defined the detail movements of the earth. The assumption that physical astronomy cannot " permit " this or that, is a similar claim to that of the theorists who would not " permit " Jupiter to possess satellites, nor the earth to have a daily rotation. Further discussions on such classes of arguments and objections are unnecessary. It is a mere waste of time to have to point out the errors of pompous assertions, and to prove the feebleness of objections urged against the second OBJECTIONS OP THEORISTS. 305 rotation of the earth, and the facts proved thereby, when, as is the case, exactly similar objections were brought against the daily rotation in former times. The reader will find it a more useful occupation to examine the details of the second rotation, and to prove for himself that by a knowledge of this movement he can solve problems hitherto entirely beyond the powers of present theorists, who claim infallibility, than to devote his time and attention to replying to objections which have no foundation other than the beliefs and assertions of certain persons put forward as " authorities." It has been truly remarked by Seneca, that men would rather hold fast to an error they know to be false, in order to keep up the appearance of not having been deceived, than avow they were in the wrong. Thus we find writers asserting that " a conical move- ment of the earth's axis round the pole of the ecliptic as a centre " is a statement accurate and detailed of the move-- ments of the earth. This assertion is accepted as complete, although it has never been stated which pole remains fixed, how each zenith and meridian is affected, in what direc- tion the centre of gravity of the earth is thrown out of its orbit by the conical movement of the axis, etc. It may, however, be of interest to the geometricians and astronomers of the future if the so-called arguments and proofs of objections against the second rotation of the earth be collected and preserved as specimens of the scientific reasoning of the present day. We possess examples of the arguments used in former times against the daily rotation of the earth, and against the possibility of Jupiter possess- ing satellites; it will be curious to compare the ancient with the modern objections, and to discover, if possible, where they differ,