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

" equinoctial " (the term used to describe the trace of the equator on the sphere of the heavens)

CHAPTER II. THE MOVEMENTS OF THE EARTH

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. The instruments used in former times enabled angles to be measured with sufficient accuracy to detect such a change.

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.
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. It was assumed that the obliquity of the ecliptic (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.

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 polar axis to wobble 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. The precessional cycle is also called the Great Year or Platonic or Plato's year and is divided into twelve ages with four seasons.

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'.
When, we have two semi-axes describing cones, we have a 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.

Suppose N S the axis of daily rotation of the earth, C the centre of the earth. Draw an imaginary line, OOP, 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 Fig- 12L. 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.
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. 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 follow- ing diagram : The axis of the sphere is represented by N S, the arc by A K, 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 O 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 due to a second rotation of the earth.
We must find in what part of the heavens the pole of this second axis of rotation is situated. Any star which 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.
[Drayso calles the North Celestial Pole the pole of the heavens, which is the zenith of the pole of the earth, and that zenith is displaced annually 20" by presession].
When at O, the pole did not vary its distance from the star x; the pole of second rotation was therefore on the arc O x 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°.

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

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. 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. 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.
Also, it has been asserted that the earth's axis changes its direction annually to the amount of 20.09". 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. 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 ?
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.
Is the zenith of Greenwich affected in exactly the same manner as is the zenith of St. Petersburg ? If not, what is the difference ? What is the radius of the circle which the earth's axis traces ? Is it 24°, 23°, or some- thing greater or less than either of these values ?
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 problem of science, they would be at once ridiculed, because the asserted conditions are in reality impossible. 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 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 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. The reader's special attention 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 circum- ference 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 X P Y 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 X O Y, X Q Y, X P Y, 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 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.

This diagram represents a projection of the northern hemisphere on the plane of the equinoctial.
P represents the north pole, Q T R the equator, the circle Z A B 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 Z B A.
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, O X being equal to P Z. 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, Y being equal to P Q.
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 20.09" annually, and in the direction of from P to O, nearly in the direction of the first point of Aries, 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 consequence 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 Z B A, the course traced by the zenith of 51 north latitude during a daily rotation; Q T is the equator; C, the position of the pole of the axis of second rotation P C = 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"
Zenith at 90° from the pole = C A = 90°
PO = OP = 20.09"
A P = 90° - CP = 90° - 29° 25' 47" = 60° 34' 13"

to find the value of A B, the arc of the equator of slow rotation during one year.
Making use of the usual formula :-

O P = A B cosine A P, we obtain

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"

for the rate of the second rotation annually.

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".

90° - 34' 13" = 89° 25' 47"
89° 25' 47" = 89.429722°
O P = A B cosine A P
O P = 40.9" x cosine 89° 25' 47" = 40.9 x 0.009953 = 0.4071" (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.

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"

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".

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 ?

CHAPTER IV. THE SECOND ROTATION OF THE EARTH.

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. 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.
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. To find the distance, therefore, of the pole of daily rotation from any star becomes a simple problem in spherical trigonometry.
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 a Ursa Minoris.

The calculations for this star (Fig. 19) are as follows,
C being the pole of second rotation;
P, the pole of daily rotation on January 1, 1887;
a, the star.
P C is an arc of 29° 25' 47", and the pole P moves round C as a centre at the annual rate of 40'9".
C a = 29° 52' 49.6".

The angle P C a = 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 a C P was therefore 2° 27' 5" + 46' 21.2" = 3° 13' 26.2".
The two sides C a and C P are constants, consequently we have two sides and the included angle of a spherical triangle to find P a, 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

[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 = c = PC = 29° 25' 47" =
…….. A = angle a C P = 3° 13' 26.2"

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

a = 1° 39' 20.68" (which is close to 1° 39' 25")]

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".

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"

The star Beta Draconis is distant from the pole of second rotation 9° 17' 38"; 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), C P.
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 P C B, January 1, 1887 = 148 8' 0'
variation in this angle annually, 40.9".

From the above data we can calculate the distance P B 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.
P C = 29° 25' 47".
C B = 9° 17' 38".

both constants

Angle P C B, 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"

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.
The following " constants," and the results obtained from these, are now submitted for investigation :

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.

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.

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 pro- duced by " a conical movement of the earth's axis " amounts to 501" annually ; therefore, write these gentlemen, " as the amount is 5O1" 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,"
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 A B, viz. 20" = 15-9866049
Log. sine of angle A C B, viz. 23° 28' = 9-6001181

A C = 50 2" = 6-3864868

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 2O09" 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