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DIRECTIONAL ASTROLOGYTO WHICH IS ADDED A DISCUSSION OF

PROBLEMATIC POINTS AND A COMPLETE SET OF TABLES NECESSARY

FOR THE CALCULATION OF ARCS OF DIRECTION

BY

SEP HARIALAUTHOR OF

‘ ' c o s m ic s y m b o l is m , ” “ a m a n u a l o f a s t r o l o g y , "“ a m a n u a l o f o c c u l t is m , " “ t h e k a b a l a o f n u m b e r s , ”

“ KABALIST#C ASTROLOGY," ETC.

LONDONW IL L IA M RIDER AN D SON, LIM IT E D

*915

PRINTED IN GREAT BRITAIN BY NEILL. AND CO., LTD.,

EDINBURGH.

PREFACEA word by way of introduction to this work may be necessary, inasmuch as it deals with a technical subject, and the scope and purport of it cannot very readily be apprehended by the casual reader. It is essentially a book for the astrological student. To the astronomer it is particularly informing in that it brings out the more scientific part of the subject and shows the mathematical basis underlying the ££ lucky hits ” to which many of our astrological exponents have undisputed claim.

The general scope of this work embraces all that is essential to the art of ££ directing ” as practised by Claudius Ptolemy and Titus de Placidus, and more recently by Sir John Wharton, Mr John Gadbury, Commander Morrison, R.N., and Mr A. J. Pearce, all of whom pursued the same general principles of astronomical directing, and differed considerably in their application of the celestial arcs to the measure of time. These points are reviewed and critically examined in the following pages.

vi DIRECTIONAL ASTROLOGY

An attempt having been made to bring the Arabian system of a day for a year into accord with the astronomical system of a degree for a year, some suggestions have here been made as to their rapprochement, the feeling being that, where credit is claimed for one system over another by exponents of either, the probability is that there is truth in both and hence there must be a co-ordinating factor. In the attempt to scientifically extend our horizon to include a prescience of coming events, we have primarily to remember that there are many ways up a mountain, but there is only one top. A study of these various methods may lead to the conclusion that they are all leading in the same direction. It is as if one should say there are three hundred and sixty paces from end to end of the path, and another should say that there are three hundred and sixty-five. Roth may be right according to their count and the measure of their tread, but the actual length of the path will remain the same whatever they make of it. This pathway is that which a man has to travel from his cradle to his grave; and there is nothing that concerns a man so vitally as that he should know its trend and gradient, its pitfalls and rocky eminences, in advance of his going, so that experience may be laid by the heels and made to serve instead of to subjugate. And in the direst extreme of human experience we have to remember that “ the wise man foreseeth the evil

PREFACE V l l

and obscureth himself, while the ignorant pass on and are hurt.”

I have used a well-known and thoroughly authenticated horoscope for purposes of illustration, and anybody following the rules here given in relation to that horoscope will have no difficulty in following them out in respect to any other horoscope. Particular care has been taken to define the principle underlying each operation, and to give a clean-cut rule of procedure. Unlike most authors, who proceed by befogging the mind of the student with technicalities and afterwards explaining them by means of an appendix, I have devoted the first chapters of my work to technical definitions which are essential to the proper understanding of the subject; and until these are clearly apprehended and understood, the student need go no further.

To save further expense and trouble, my publishers have completed my work by the insertion of a complete set of tables, which include tables of Right Ascension and Declination for every degree of the zodiac, together with the ascensional difference due to the latitudes of London, Birmingham, and Liverpool under the present obliquity of the Ecliptic ; also tables of Sines and Tangents, and tables of Proportional Logarithms. These are all that are essential to the present treatise, and in themselves constitute a very valuable addition to the volume. It is, of course, presumed that the

v iii DIRECTIONAL ASTROLOGY

student of “ Directional Astrology ” will have mastered the preliminary task of setting a horoscope for any given time and place with adequate precision, and hence that he is familiar with the use of an ephemeris. The present work is intended to replace and supersede P rognostic A s tr o n o m y , which is now out of print.

Beyond this I have nothing to say, save that I trust to have done my work efficiently and to have left no point on which a reader need question me. In such case the work may be regarded as com plete, and so I hope it will be found.

SE P H A R IA L.

CONTENTSOKAP. PAOB

1. A stronomical D efinitions . . . 12. E xample H oroscope . . . . . 1 13. D irections in Mundo . . . . . 184. D irections in the Zodiac . . . . 255. Z odiacal and Mundane Parallels . . 326. Order op D irecting . . . . . 377. E ffects of D irections . . . . . 408. Planetary Indicators and the Measure of

T i m e . . . . . . . . 499. Illustration . . . . . . . 57

10. Ptolemy and Placidus . . . . . 6211. D irections under Poles . . . . 7112. T he Part o f F ortune . . . . . 8113. L unar Parallax and Semi-diameter . . 8614. Lunar E quations . . . . . . 9415. Cuspal D istances . . . . . . 9916. Suggested Method o f True D irecting . 10717. Co n c l u s i o n ................................................................... 117

A ppendix— Tables of Sines, Tangents, etc. . 125Tables of Right Ascension and

Ascensional Difference . . 1 7 3Tables of Proportional Logarithms . 181

ix

Directional AstrologyCHAPTER I

ASTRONOMICAL DEFINITIONS

T he following definitions must be fully understood by the student before the more intricate part of the system of directing is undertaken.

Longitude is of two kinds : longitude in the Orbit, and longitude in the Ecliptic. The latter is the only one recognised and used in this system. It is defined as distance from the vernal equinox, Aries 0, measured on the plane of the Ecliptic or Sun’s path.

Latitude.— Celestial latitude is distance north or south of the Ecliptic.

Declination is distance north or south of the Equator. The Ecliptic lies in declination 23° 27' north and south.

Bight Ascension is distance from the vernal equinox measured on the plane of the Equator. Right ascension thus answers to geographical longitude in the same way as declination answers to geographical latitude.

Meridian Distance is the distance of a celestial1

2 DIRECTIONAL ASTROLOGY

body from the midheaven of a place ; that is to say, from its meridian, measured in right ascension.

Semiarc of a planet is half the time it remains above or below the horizon of a place, measured in degrees of right ascension. The diurnal semiarc is half the arc in right ascension of a planet above the horizon, and nocturnal semiarc is half the time it is (measured in right ascension) below the horizon. The diurnal semiarc taken from 180° will give the nocturnal semiarc, and the nocturnal semiarc taken from 180° will give the diurnal semiarc.

Horizontal Arc is the distance in right ascension from a body to the point of its rising or setting. The semiarc less the meridian distance is always the horizontal arc.

Oblique Ascension is the right ascension of a body increased or diminished by its ascensional difference, according as its declination may be south or north. In northern latitudes the right ascension is increased for a body having south declination and decreased for a body having north declination, but the reverse of this is the case in southern latitudes.

Ascensional Difference is the time (measured in right ascension) that a body is above or below the horizon more or less than six hours. If, therefore, its semiarc is more than 90° the excess of 90° is its ascensional difference. All bodies that are not exactly on the equinox (Aries 0 or Libra 0) have ascensional difference. For a planet in south declination the ascensional difference is added to

ASTRONOMICAL DEFINITIONS 3its right ascension to get its oblique ascension, and for bodies having north declination the ascensional difference is subtracted. The reverse of this gives the oblique descension. The O.A. plus or minus 180° gives the obi. descension of the opposite point.

Pole of Latitude.—The pole of a place is the same as its latitude. The pole of a planet is measured by a circle of position or small circle parallel to the meridian of a place. The pole of the ascendant is the same as the latitude of the place, and this diminishes as we reach the meridian, where it is 0.

Direction is the process by which we bring the body of a planet to the longitude or body of another in a different part of the heavens either by its rising or setting, and this direction of one body to another, or to the place of another, is measured in right ascension; that is to say, by the number of degrees which pass under the meridian of a place in the interval. All directions are taken in the prime vertical, or circle of observation—that in which a person stands upright facing south. Having the proportional distance of a planet between the meridian and horizon, we may bring another body to the same proportional distance along its own arc until it appears to be in the same relative position as the first body. This supposes that the position and influence of a planet is indelibly located in that part of the heavens in which it was found at the moment of birth. All arcs of direction are measured in right ascension.


Significators, in this scheme, are the Midheaven, Ascendant, Sun, and Moon. These are the bodies or positions that are directed or moved in the prime vertical in order to form conjunctions, oppositions, and various aspects -with other positions and bodies. They are called “ significators,” from the fact that they are found to signify certain things in the life of an individual; as, the Sun signifies male relationships, the Moon female relationships, the Midheaven honour and position, credit, etc., and the Ascendant the health and general play of events in the individual sphere of life. For further elaboration of this point refer to the Textbook of Astrology or The New Manual of Astrology.

Promittors.—These áre the planets Neptune, Uranus, Saturn, Jupiter, Mars, Venus, and Mercury. The Sun and Moon may also be classed as promittors when the Midheaven or Ascendant is directed to them.

Logarithms, invented by Baron Napier of Mer- chiston, first-class mathematician and astrologer, were designed for the purpose of simplifying calculations in spherical trigonometry. In this scheme the arc of 90° of a right sphere is made to equal 10 00000, which is called the radix. Then, having the logarithm of any arc, it may be multiplied into any other arc by simple addition of their logarithms ; and, similarly, arcs may be divided by one another by subtracting one logarithm from another. Napier thus emphasises the fact that multiplication is

ASTRONOMICAL DEFINITIONS 5merely the addition of a number to itself a given number of times, while division is merely subtraction a number of times. Then by means of a proportional circle we can multiply and divide any arc by simple addition and subtraction. The complement of an arc is what it lacks of 90°, and as this is equal to the radix 10, the complement of a logarithm is what it lacks of 10. Thus the logarithm of the sine of 32° is log. sine 9-72421, which is also the log. cosine of 58°, because 58 is the complement of 32, both together making 90. The arithmetical complement of the logarithm Is 0-27579, since this, added to the log. sine of 32°, makes 10-00000. Familiarity with the use of logarithms will readily establish their great value in all mathematical calculations connected with the sphere.

I may now ask the reader to take in hand an ephemeris for the current year, 1916, and turn to the 1st January, and the above definitions may then be illustrated.

Let us suppose that a birth took place at noon, Greenwich mean time, on that date in London. The ephemeris being calculated for mean noon at Greenwich, there will be no equation of time necessary. The Sun, Moon, and planets will be in the positions indicated in the ephemeris. The Sun’s longitude is seen to be Capricornus 9° 45' 14". The Sun never has latitude, inasmuch as it defines the Ecliptic, distance above or below which constitutes celestial latitude. All other bodies have


latitude except when they are on that point where their orbits cross the Ecliptic, that is, their nodes. The course of the Sun being across the plane of the Equator at an angle of 23° 27' it will attain that declination at the solstices ; that is to say, on the 21st June and the 22nd December. On the 1st January it is found to have declination 23° 6' south of the Equator, and, therefore, would be im mediately overhead at noon at a place which had geographical latitude 23° 6' south, and the Sun’s diurnal course around the Earth would follow this parallel of latitude. The Sun’s right ascension (R.A.) can be found in the tables (see Appendix) from its longitude.

Rule 1.—To find the R.A. of any body without latitude.

From the log. cosine of its distance from the nearest equinox subtract the log. cosine of its declination. Remainder is the log. cosine of its R.A. from the same equinox.

Example : The Sun is here 80° 15'from Aries 0 . . cos. 9-22878

Its declination is 23° 6' cos. 9-96370

Distance in R.A. fromAries 0 = 79° 23' cos. 9*26508

Therefore from 360° take 79° 23', and the R .A . of the Sun is thus found to be 280° 37'. Note that it is sufficient for our purpose to take the various quantities to the nearest minute of space.

ASTRONOMICAL DEFINITIONS 7Now take the Moon’s place in the ephemeris,

which is seen to be Scorpio 17° 54'. This is 47° 54' from Libra 0. The declination of the Moon is 22° 7'. Reference to the tables will show that the declination of Scorpio 17° 54' is 17° 10' only, and we therefore know that the Moon has latitude and is not on the Ecliptic at this time. The ephemeris shows it to have 5° 9' of south latitude. In finding its R.A., therefore, we have to take this latitude into account.

Rule 2.—To find the R.A. of a body having latitude.

Add the log. cos. of its distance from the equinox to the log. cos. of its latitude, and from the sum subtract the log. cos. of its declination. The remainder is log. cos. of its R.A. from the same equinox.

Example : Moon’s distance fromLibra 0=47° 54' . cos. 9-82635

Its latitude is 5° 9' . cos. 9-99824

Sum .Moon’s decimation,

22° 7' .

cos. 9-82459

cos. 9-96681

Its R.A. from Libra 0 = 43° 53' cos. 9-85778 R.A. Libra 0 =180° 0'

Moon’s R.A. =223° 53'


Note.— If we take the arithmetical complement of the log. cos. of the decimation and add it to the log. cos. of both the latitude and the longitudinal distance, we shall have the same result.

The R.A. of the other bodies is taken in the same manner, as they all happen to have some measure of latitude. Only when a body is in its node, and therefore coincident with the Ecliptic, does it have no latitude. In such case its R.A. is the same as that of the degree of the Ecliptic it holds.

We have next to find the meridian distances of the several bodies. To do this we have to find the R.A. of the Midheaven and Nadir, and take the nearest distance in R.A. of each body. Thus at noon on the 1st January 1916 the sidereal time is 18h. 39m. 16 secs. Convert this into degrees and minutes of the circle, thus : multiply the hours by 15 and call them degrees; divide the minutes of time by 4 and call them degrees and minutes of space; also divide the seconds of time by 4 and call them minutes and seconds of space.

Thus 18h. =270° 0' 0"39m. = 9° 45' 0"16s. = 0° 4' 0"

R.A. of M.C. =279° 49' 0*180° O' 0"

R.A. of I.C. = 99° 49' 0"The upper meridian is called the Midheaven

ASTRONOMICAL DEFINITIONS 9(medium cosli) and the lower meridian is called the Nadir (imaum cceli).

Having the R.A. of the M.C. and I.C., we are able to find the quantity of R.A. which separates the various planets from them, and this is the meridian distance of each of such planets.

Thus the Sun’s R.A. was found to he 280° 37', and that of the M.C. (to which it is nearest) is 279° 49'. The difference is 0° 48', which is therefore the meridian distance of the Sun.

The Moon is found to he in the South-west quarter of the heavens, and therefore nearer to the upper than the lower meridian. Its meridian distance must therefore he taken from this point.

The other bodies are taken in the same way according to which meridian (upper or lower) they are nearest in R.A.

The semiarcs of the planets and luminaries have next to be found.

Rule 3.— To the log. tangent of the latitude of place for which the figure is set, or the horoscope cast, add the log. tangent of the planet’s declination. The sum is the log. sine of the ascensional difference of that planet under the latitude of birth.

Thus :R.A. of M.C. R.A., Moon

=279° 49' =223° 53'

Meridian distance of M oon= 55° 56


U n i f o r m ly , a d d t h is a s c e n s io n a l d i f f e r e n c e t o 9 0 ° w h e n t h e p l a n e t ’ s R . A . is le s s t h a n 1 8 0 ° , a n d s u b t r a c t i t f r o m 9 0 ° i f t h e p l a n e t ’ s R . A . i s m o r e t h a n 1 8 0 ° . T h e r e s u lt is thte d i u r n a l s e m i a r c o f t h a t p la n e t . By s u b t r a c t in g t h i s f r o m 1 8 0 ° y o u w i l l h a v e t h e n o c t u r n a l s e m ia r c .

Finally, by taking the meridian distance of the planet from its semiarc (diurnal if above the horizon, and nocturnal if below), you will have the horizontal arc, or distance in R.A. from the horizon.

Next find the proportional logarithm of the semiarc of each body, and take its arithmetical complement. Add to this A.C. the proportional logarithm of the planet’s meridian distance. This is the constant log. of the planet for purposes of directing.

Enter all these elements into a single table, which is called the Speculum, an example of which will be found in the f oho wing pages. The scheme will now be ready for the practice of directing.

CHAPTER II

EXAMPLE HOROSCOPE

F or the purpose of illustrating the method of directing by proportional semiarcs, I have selected the horoscope of John Ruskin, whose Fors Clavi- gera, Mornings in Florence, and other world- renowned works have stamped him indelibly as artist and man of letters as well as an independent thinker of considerable virility.

He was born at 7.30 in the morning of 8th February 1819, in London.

It is an invariable rule in practice to use that semiarc and meridian distance which are related to one another. Thus the Sun in the speculum is just below the east horizon at the moment of birth, as may be seen by comparing its nocturnal semiarc with its distance from the lower meridian, which are 110° 1' and 108° 44' respectively. This shows the Sun to be 1° 17' below the horizon. But as by the diurnal rotation of the earth on its axis from west to east the Sun will be carried above the east horizon upwards towards the Midheaven, during the course of which it will pass the places

n


3 1C .

S p e c u l t j m .

Planets, Lat.j

Déclin. Bi.A. MeridLDist.

Semi-aro*

Hor.Arc.

o / o / O / o / © / © /Sun 15 13 S. 321 12 108 44 110 1 1 17Moon 5 1ST. 25 39 N. 120 17 50 21 52 51 2 30Mercury , 0 23 S. 21 34 S. 296 47 46 51 60 11 13 20Venus 5 ION. 18 10 S. 276 6 26 10 65 37 39 27Mars 0 55 S. 21 45 S. 299 6 49 10 59 53 10 43Jupiter . 0 21 S. 20 26 S. 302 37 52 41 62 3 9 22Saturn 1 56 S. 6 54 S. 348 54 81 2 98 45 17 43Uranus , 0 6 S. 23 24 S 262 49 12 53 57 1 44 8Neptune . 1 13 N. 22 14 S. 267 47 17 61 59 5 41 14

EXAMPLE HOROSCOPE 13of Mars, Mercury, Venus, Neptune, and Uranus, it will be convenient also to have the semidiurnal arc and the meridian distance from the Midheaven. For whenever we use the nocturnal arc we always use the corresponding meridian distance from the lower meridian, and whenever we use the diurnal arc we also use the corresponding meridian distance from the Midheaven or upper meridian. This point should not be forgotten. It cannot be overlooked if the constant log. of the planet is inserted in the speculum, because this embodies the proportion of the semiarc to the corresponding meridian distance.

Ruskin was agreeably disposed towards the idea of planetary influence in human life, for, although he confessed entire ignorance of the subject himself, he was always willing that others should have the full benefit of his experience and views, and he readily gave his birth data to those who sought it for the purpose of astrological calculations. His assertion that “ there is more in it than is generally supposed ” was doubtless the opinion he formed of the science from experience; and if it does not carry the weight of scientific criticism, it stamps Buskin, at all events, as a man of fearless integrity of thought.

In this horoscope we have a remarkable illustration of the principles of astrology. The Sun and Jupiter are rising in the humane sign Aquarius, while most of the planets are rising and above the


horizon. These are indications of success and distinction in the world. The conjunction of Mars and Mercury in opposition to the Moon indicated that asperity and outspokenness which characterised this man of genius and rendered him fearless in the expression of his views. His eccentricities may well he attributed to the meridian position of Uranus and Neptune, while Venus, in closest aspect to Jupiter, and well elevated, disposed to success in the pursuit of art, of which he became a foremost exponent. But, of course, these positions do not make character. They only afford the opportunity for its full expression. Character and environment together constitute destiny, and it is undoubtedly often the case that one or the other of them is a misfit. It is only when we get a strong innate character with appropriate celestial environment that we look for the expression of genius.

We may now proceed to use this horoscope to illustrate the principles of directing.

Take first the Midheaven. This is directed by right ascension, and the planets coming to the meridian will form arcs of direction to it. The aspects to Midheaven should be noted. Thus the semisquare aspect falls in Capricorn 26° 28', and the sextile aspect is Aquarius 11° 28', and planets coming to these points will form aspects in the zodiac to the Midheaven. The square aspect falls in Pisces 11° 28' ; and as Saturn is lower in the heavens than that point, it must come up to the

EXAMPLE HOROSCOPE 15

place of this aspect and form the zodiacal square to the Midheaven. And the times in which these aspects are formed by the several planets will be in the proportion of their semiarcs. These directions are in zodiac.

The other kind of direction is in mundo—that is, in the circle of observation or prime vertical. Thus a body that is on the cusp of the twelfth house is in mundane sextile to the Midheaven or upper meridian, and one that is on the cusp of the eleventh house is in mundane sextile to the horizon or Ascendant. A planet that is in the middle of the eleventh house will be half way between the Midheaven and Ascendant, and, therefore, in semisquare aspect in mundo, because the meridian and horizon are always at right angles to one another. If a planet is not thus situated at the moment of birth it will afterwards attain that position, and the number of equatorial degrees which pass under the meridian from birth to the time when the aspect is formed will be the measure of the arc of direction. The original position of a body, either in the zodiac or in mundo, is always that to which direction is made.

Mundane Directions are those that are made to the apparent place of a celestial body, or to its aspects, in the circle of observation.

ZodÂacal Directions are those which are made to the geocentric longitude of a body, or to aspects of that longitude, in the circle of observation or prime vertical.

All directions are formed by the rotation of the Barth upon its axis from west to east, by which the planets appear to rise, culminate, and set, pursuing a course that is from east to west. The lines or arcs traversed by the planets in this apparent motion are parallels of latitude of the same quantity and denomination as geographical parallels of latitude—that is, lines parallel to the Equator. The planets follow the parallel of declination in which they are found at the time of birth.

It is understood that the radical imprint of a planet is localised in that part of the heavens it occupied at the moment of birth ; and although the actual planets do thereafter change their declinations and semiarcs, as well as their meridian distances, the radical imprint of the planet remains ever the same, and is to be regarded as entirely distinct from the planet itself, which, of course, moves along its arc in the heavens.

In the process of directing we are, therefore, only concerned with the radix or root horoscope and the changes which thereafter take place in the heavens, not among the bodies themselves, but in their relations to the radix. All directions of this nature are formed within a few hours of the moment of birth.

Directions (whether in the zodiac or mundo) are of two orders. These are <e direct ” and “ converse/’

Direct directions are such as are formed by one

EXAMPLE HOROSCOPE 17body being carried by the motion of the Earth towards another body or aspect in the heavens that precedes it. Converse directions, however, are such as are formed in the opposite direction. Thus in the foregoing horoscope of Buskin, if we bring the Sun to the place of Jupiter, or Mars, or Mercury, or Venus, these would be direct directions, because that is the direct motion of the bodies in the heavens. But if we brought the Sun to the place of Saturn it would appear that we are carrying it backwards to a position that it held previous to the moment of birth. This, however, is not the case. The Sun is joined to the Earth by a direct ray which is called its earth-line, and it is this line which, by the rotation of the Earth on its axis, is carried down (bearing the solar imprint) to the place held by Saturn at the birth. This is a converse direction. But if we bring Saturn up to the place of the Sun it would be a direct direction.

Therefore all directions are formed by the one natural fact of the Earth’s rotation on its axis, and aspects that cannot thus be formed are not within the category of primary directions.

We may now pass on to illustrate the method of forming every kind of direction, direct and converse, in zodiac and mundo.

2

CHAPTER III

DIRECTIONS IN MTTNDO

The principle involved in this process is that which enters into the construction of the horoscope, wherein we take one-third of the Sun’s tropical semiarc as the extent of the house or division of the prime vertical. This principle enters into the construction of the tables of houses for various latitudes, the Sun’s extreme declination remaining a constant quantity.

But in every horoscope we have the various planets with different declinations, and therefore with different semiarcs ; and consequently we are dealing with arcs which, although parallel to the Equator and to one another, are not parallel to the circle of observation. Hence an equal division of the prime vertical into twelve parts or houses will not effect an equal division of the various planetary semiarcs, which cut the meridian and horizon at varying angles depending on their declinations. Nevertheless, it has been found in practice that one-third of the semiarc, great or small, is equal to a house-space under the pole of that planet.

18

DIRECTIONS IN MUNDO 19Suppose a planet to be exactly rising at the time

of birth. Let its semidiurnal arc be 66° 21'. This is an arc of right ascension. Therefore when it has traversed one-third of its arc from the horizon to the meridian, 22° 7' will have passed under the meridian, and that will be the arc of the planet’s direction to the cusp of the twelfth house. Another 22° 7' will bring it to the cusp of the eleventh house, and yet another arc of the same value will bring it to the meridian. When on the cusp of the twelfth house it will be in sextile aspect to the Midheaven, and when on the cusp of the eleventh it will be in sextile to the Ascendant, both directions being in mundo, as distinguished from similar aspects in the zodiac.

If the Sun or Moon happen to be exactly on the cusp of a house, then the planet coming to the cusp by one-third divisions of its semiarc will simultaneously form aspects in mundo to the Sun or Moon. But if they are not so placed, then we have to find their proportional distances from the nearest cusp or limit of a house, and bring the planet to the same proportional distance in order to form the aspect.

Rule.—To find the cuspal distance of a planet. Note the cusp to which it is nearest at the time of birth. The distance of that cusp from the horizon compared with the planet’s horizontal arc will give the planet’s cuspal distance.

Example.—In the specimen figure the Sun is


nearest to the cusp of the first house or ascendant, and therefore its horizontal arc, 1° 17', will be its cuspal distance. The Moon has a semiarc of 52° 51', and its horizontal arc is 2° 30', and as this is nearest to the cusp of the seventh house, that will also be its cuspal distance.

Now, as in all directions, the body to which direction is made is considered to remain stationary while the body directed is moved towards it by its natural motion in the heavens, we here direct the Moon to the sesquiquadrate aspect of the Sun, which it attains in the middle of the fifth house, that point being four and a half houses, or 135°, from the ascendant. The Sun, however, is not on the ascendant, and therefore we have to bring the Moon to a proportional distance from the middle of the fifth house. Thus :

As the semiarc of the Sun, 110° 1',prop, log...................................0-2L381

arith. comp............................ 9*78619Is to its cuspal distance, 1° 17', . 2*14693So is the semiarc of Moon, 52° 51', . 0*53223

To its proportional distance, 0° 37',prop, log................................ 2*46535

Now, as one-third of the Moon’s semiarc is 17° 37', that will be its house-space, and one-half will be 8° 48|', making for one and a half houses 26° 25|', and from this we subtract the above proportional

DIRECTIONS IN MUNDO 21distance, namely 0° 37', and there remains the arc of direction: Moon, 135°, Sun in mnndo, 25° 48J'.

Another example : Bring the Sun in the example horoscope to the mundane conjunction with Jupiter.

In order to effect this the Sun has to cross the horizon, its distance from which has been found to be 1° 17'. Thereafter we employ its diurnal arc and bring it to an equivalent distance from the horizon southwards as Jupiter is in the horoscope, by proportion of their semidiurnal arcs.

Jupiter's semiarc is 62° 3', and its meridian distance 52° 4:1', their difference 9° 22', which is the horizontal arc of Jupiter and therefore its distance from the cusp of the first house. Then we say :—

As the semiarc Jupiter (arith. comp.) is to its cuspal distance, so is the semiarc of the Sun (diurnal =69° 59') to its proportional distance from the same cusp southwards. This works out as follows :—

S.A. Jupiter, 62° 3' . . log. 0-46253

Arith. comp. 9-537471-283690-41028

Cusp, distance, 9° 22' S.A. Sun, 69° 59'

Sun’s prop, distance —10° 34' log. 1-23144Sun to horizon 1 ° 1 7 /

Arc of direction =11° 51' Sun conj. Jupiter m.


It should be observed that the arc of direction to the horizon must always be added when the planet or body has to cross the horizon in forming the direction. Here the proportion of the Sun’s arc to that of Jupiter gives a cuspal distance of 10° 34', and to this has to be added the distance of the Sun from below the horizon, making the arc altogether 11° 51'. When crossing the meridian to form a direction, no change of arc is necessary, but the arc to the meridian, which is the meridian distance of the planet, must be added to the arc formed on the other side of it.

It should be observed also that the body to which direction is made, and which is supposed to be stationary, supplies the first and second terms of the proportion, while that body which moves to form the direction supplies the third term and the resulting fourth term. In practice it will be found expedient to arrange all the mundane aspects in the order in which they are formed by each of the planets. The Midheaven and Ascendant remain stationary, and the Sun, Moon, and planets are the promittors that are moved to form directions upon them. Take one of these bodies at a time and make a list of the mundane directions it forms to the Midheaven, Ascendant, Sun, and Moon, calculate them, and arrange them afterwards in the order of their values. Always remember that the diurnal motion of the Earth upon its axis from west to east is the underlying cause of all

DIRECTIONS IN MUNDO 23

directions, and that the planet to which direction is made, remains still, while the other moves towards it. You cannot then go wrong in your application of the method.

Direction to the conjunction in mundo is effected by bringing the body of a planet to the body of another, and not to its zodiacal longitude merely, as is done in the case of the zodiacal conjunction.

Thus in the case of TJranus to conjunction M.C. in mundo, we take its meridian distance as the arc of direction, whereas in the zodiac we take the meridian distance of its longitude, Sagittarius 23° 25', and this will be the arc of direction.

In all cases we bring the body of the planet directed to the conjunction or aspect of another body in mundo, to form mundane directions, all such directions being formed in the prime vertical, and expressed in terms of right ascension.

It will be found convenient to have the constant log. of the cuspal distance of each planet in the speculum. Subtract the proportional log. of the semiarc from the proportional log. of the cuspal distance. This will give the constant log., to which we have merely to add the proportional log. of the semiarc of any other planet to find the proportional cuspal distance of that planet.

It has been customary to regard the semiarc of a planet as equal to the quadrant, and therefore one-third as equal to a house or 30°. This is true in regard to a prime vertical whose pole is the


s a m e a s t l i e d e c l in a t io n , o f t i i e p l a n e t , b u t i t is n o t t r u e in r e g a r d t o a n y o t h e r p o l e o r g e o g r a p h i c a l l a t i t u d e . T h a t is w h y w e t a k e t h e p r o p o r t i o n o f t h e s e m ia r c s in f in d in g t h e c u s p a l d i s t a n c e s o f p la n e t s . T h e t e s t is t h is : I f w e t a k e t h e o b l i q u e a s c e n s io n o f a p la n e t , t h a t is , e x a c t l y o n e - t h i r d o f i t s s e m ia r c f r o m t h e h o r iz o n , i t s h o u l d h a v e t h e s a m e o b l iq u e a s c e n s io n a s t h e c u s p o f t h e t w e l f t h h o u s e , b u t b y a d d in g 60 t o t h e r ig h t a s c e n s i o n o f t h e M i d h e a v e n t o g e t t h e o b l iq u e a s c e n s io n o f t h a t h o u s e , w e s h a l l f in d t h a t i f t h e p la n e t h a s a n y o t h e r d e c l i n a t i o n t h a n 2 3 ° 2 7 ' t h e r e is a d i f f e r e n c e b e t w e e n t h e t w o r e s u lts . I t c a n n o t , t h e r e fo r e , b e t r u l y s a i d t h a t a p la n e t is in m u n d a n e s e x t i l e a s p e c t t o t h e M i d h e a v e n w h e n i t is o n e - t h i r d o f i t s s e m ia r c a b o v e t h e h o r iz o n , in a s m u c h a s i t s p o s i t i o n i n t h e p r i m e v e r t i c a l d o e s n o t t h e n c o i n c i d e w i t h t h e c u s p o f t h e t w e l f t h h o u s e ; b u t i t m a y b e s a id t o c o r r e s p o n d w it h t h a t c u s p o n t h e g e n e r a l p r o p o s i t i o n t h a t a l l c ir c le s a r e e q u a l t o o n e a n o t h e r a n d t h e r e f o r e t h a t a ll q u a d r a n t s a r e e q u a l , a n d in p r a c t i c e i t is f o u n d t h a t o n e - t h i r d o f a s e m ia r c c o r r e s p o n d s w i t h o n e - t h i r d o f t h e p r im e v e r t i c a l , a n d t h is w a s a l l o w e d b y P la c id u s , w h o w a s t h e f ir s t e x p o n e n t o f t h i s s y s t e m o f m u n d a n e d i r e c t io n s .

CHAPTER IY

D I R E C T I O N S I N T H E Z O D I A C

T hese are calculated on the same principle as mundane directions, that is to say, by proportion of the sem iarcs; but instead of taking the actual body of the planet, or its position in the prime vertical, we take the longitude only and direct to that, and also to its aspects in the zodiac.

Thus in the horoscope of Ruskin the planet Neptune holds the longitude Sagittarius 28° 22', and therefore its zodiacal sextiles will fall in Aquarius 28° 22' and Libra 28° 22', its squares in Pisces 28° 22' and Virgo 28° 22', and so on.

The longitude of the planet, or its aspect if we are directing to it, remains stationary, and the actual body of the planet or luminary which is directed to it is moved along its own semiarc until it reaches the longitude or aspect to which direction is made.

Therefore we take the meridian distance and semiarc of the ecliptic degree held by a planet and use these as the first and second terms of a proportion, in which the semiarc of the body directed forms the third term.

25

26 DIRECTIONAL ASTROLOGYExample.— Direct the Sun to a conjunction with

Jupiter in the zodiac.Jupiter’s longitude is Aquarius 0° 20', and from

the tables we find this longitude to have R .A . 302° 31', from which take the R.A. of Midheaven, 249° 56', and we get its meridian distance, 52° 35'.

The same tables give the ascensional difference under London as 30° 51', which, taken from 90° as the declination is south, gives the diurnal semiarc—59° 9'.

Prop. log. meridian dist. ,, semiarc

52° 35' = -53442 59° 9 '= -48332

Constant log. Aquarius . 0° 2 0 '= -05110Prop. log. Sun’s semiarc . 69° 59' =-41028

„ Sun’s prop. dist. 62° 13': Take from Sun’s merid. dist. 71° 16'

-46138

Arc of direction . . 9° 3'The constant logarithm of a longitude, once ob

tained, should be reserved, as it will serve for all zodiacal directions made to the same point of the ecliptic by simply adding the log. semiarc of the body directed to it. We then have the proportional meridian distance, which, compared with its original distance, gives the arc of direction.

Uniformly, find the R.A. of the longitude to which direction is made; from this derive the meridian distance. Find its declination, and from

DIRECTIONS IN THE ZODIAC 27this derive the sexniarc. Subtract the proportional logarithm of the semiarc from that of the meridian distance, and derive the constant log. of the given longitude. To this constant log. add the log. semiarc of the body directed to it, and thus obtain the proportional distance of that body from the meridian at the point of direction. The difference between this and its radical meridian distance is the arc of direction.

Examples:—1. Direct the Sun to aspects of the Midheaven

in the zodiac. The aspects to which the Sun applies are the sextile in Aquarius 11° 287, the semisquare in Capricorn 26° 28', and the conjunction in Sagittarius ll°jj[28'.Aquarius 11° 28' has R.A. 313° 55'The Midheaven has R.A. 249° 56'

Merid. dist. of aspect = 63° 59' prop.log. -44921 Asc. diff. 23° 9'

90° O'

Semiarc 66° 51' prop. log. -43017

Constant log. of aspect in Aquarius 11° 28'=-01904 Add prop. log. Sun’s semiarc diurnal -41028

Sun’s prop. dist. from M.C. 66° 59' *42932Radical dist. of Sun from M.C. 71° 16'

Arc of direction, Sun sextile M.C. = 4° 17'

28 DIRECTIONAL ASTROLOGY2. The next aspect of the Sun to the Midheaven

in zodiac falls in Capricorn 26° 28', which is the semisquare aspect of 45°.

The R.A. of this longitude is 298° 29', and its meridian distance is therefore 298° 29' —249° 56' = 48° 33'. Its ascensional difference is 28° 40', which gives its diurnal semiarc = 61° 20'.

Proportional log. 48° 33'—prop. log. 61° 2 0 '= con - stant log. of aspect, -10150 To this we add the

prop. log. of Sun as before,namely, -41028

•51178—55°23' Sun’s propor.meridian distance ;

which take from 71° 16' Sun’s radicaldistance,

remains 15° 53' the arc of d irection Sun semisq. Mid- heaven.

3. The next aspect of the Sun to Midheaven in zodiac is the conjunction. For this the calculation is simply the difference of their right ascensions.

That of the Sun is 321° 12'That of the M.C. 249° 56'

Difference 71° 16 '= arc of direction.

DIRECTIONS IN THE ZODIAC 29These examples will doubtless serve for all cases

that may arise in the course of directing a planet to the longitude and aspects of another in the zodiac.

We may now consider converse directions in the zodiac. These are calculated in exactly the same manner as the direct directions; but instead of moving the directed body forward in the heavens, that is, from the Nadir to the Ascendant, from the Ascendant to the Midheaven, from the Midheaven to the Occident, and so on, we move it conversely against the natural diurnal motion of the celestial bodies in the heavens. Thus, in the example horoscope the Moon is in Cancer 27° 8'. Therefore, to bring Saturn to the square aspect of the Moon in the zodiac, we have to bring it to Aries 27° 8' by converse motion. We therefore find the meridian distance and semiarc of that point in the ecliptic, the meridian distance being taken from the lower meridian, to which it is nearest, and the semiarc being the nocturnal arc. Find the constant log. due to this point of the zodiac, and add to it the log. of the nocturnal semiarc of Saturn. From this we derive the proportional distance of Saturn from the lower meridian, and the difference between this and its radical distance is the arc of direction.

Similarly, we bring the Sun down the eastern heavens to form the converse zodiacal conjunction with Saturn. Here we take the meridian distance of Pisces 17° 5', and also its semiarc. Find the con-

DIRECTIONAL ASTROLOGY30stant log. due to these and add to it the log. of the nocturnal semiarc of the Sun. The sum will he the prop. log. of the Sun’s meridian distance at the conjunction, and the difference between this and the radical distance of the Sun from the same meridian will be the arc of direction.

The bodies of Jupiter, Mars, Mercury, Venus, Neptune, and Uranus are brought to the zodiacal conjunction with the ascendant conversely by the measure of their horizontal arcs, which are derived by subtracting the meridian distance from the semiare.

Thus Jupiter comes to the conjunction with the ascendant in zodiac conversely in an arc of 9° 21', Mars in an arc of 10° 43', Mercury in an arc of 13° 20', Venus in an arc of 39° 27', Neptune in an arc of 41° 14',1 and Uranus in an arc of 44° 8'. Similarly, the Moon is brought to an opposition of the ascendant in zodiac by an arc of 2° 30', which is the difference between its semiarc and meridian distance. This arc is much smaller than appears from its longitudinal position, and is due to the fact that the Moon has 5° of north latitude. A body with much north latitude sets much later and rises much sooner than does the degree of the ecliptic it holds. This is the radical difference between the mundane and zodiacal positions of a celestial body.

1 An ephemeris of the approximate longitude and latitude o f the planet Neptune from 1800 to 1900 a.o. is published b y Messrs Foulsham & Co. Price Is.

D I R E C T I O N S I N T H E Z O D IA C 31The Midheaven is directed to the conjunction

with these planets in the zodiac by an arc equal to the difference of the R .A . of the Midheaven and that of the longitude of the planet.

Thus Venus comes to the Midheaven with the R .A . of Capricorn 5° 49% which is 276° 25% and the difference between this and the R .A . of the Midheaven 249° 5 6 '= 2 6 ° 29' arc of direction of Midheaven conjunction Venus in zodiac.

Uranus comes to the Midheaven in the zodiac by an arc of 12° 53% Neptune by an arc of 18° 17% Venus by an arc of 26° 29' (as above), Mercury by an arc of 46° 44% Mars by an arc of 48° 41% Jupiter by an arc of 52° 36% and the Sun by an arc of 71° 16'. These arcs, it will be observed, differ from the meridian distances of the several bodies as given in the speculum by an increment which is due to the latitudes of the various bodies. The meridian distances in the speculum will be the same as the measure of their directions to conjunction with the Midheaven in mundo.

W e may now pass to another series of directions.

CHAPTER V

Z O D I A C A L A N D M U N D A N E P A R A L L E L S

I n astrology the parallel of declination, is deemed of the same significance and value as the conjunction, but its effects are more lasting, and if formed near the tropics, Cancer 0 or Capricorn 0, they will last for years together and characterise a whole period of the life.

A zodiacal parallel is formed by directing a body to the place held by a zodiacal degree which has the same declination as that held by a planet to which direction is made.

Example.—The Sun at birth has 15° 13' ofdeclination. On the principle that all parallels of declination, being at the same distance from the Equator, act magnetically in unison, any body coming to an ecliptic degree which holds the same declination as the Sun, namely, 15° 13', whether north or south of the Equator, will act as if in conjunction with the Sun. Reference to the tables will show that there are four points which have this same declination, namely, Aquarius 18° 46', Taurus 11° 15', and Scorpio 11° 15'. Therefore, if we

32

ZODIACAL AND MUNDANE PARALLELS 33direct any body to any of these four longitudes in the zodiac by the rules given for directions in the zodiac, we shall bring them to parallels of the Sun in zodiac. The process is exactly the same as if we were directing to an aspect in the zodiac.

A mundane parallel is formed by the direction of a body to the same distance on one side of the meridian or horizon as that radically held by another body on the other side of the same meridian or horizon. These can be readily computed by reference to their horizontal arcs.

Example.— Bring Saturn to the mundane parallel of the Sun. The Sun’s radical distance from the horizon northwards is determined by the difference of its meridian distance and semiarc, namely, 110° I' —108° 44' = 1° 17', and we therefore have to bring Saturn to the same distance above the horizon. The semiarc of Saturn is 98° 45', and its meridian distance 81° 2 '; its horizontal arc therefore is 17° 43\ Then say : As the semiarc Sun is to its horizontal distance, so is the semiarc Saturn to its proportional distance, which, being added to the first or radical distance of Saturn from the horizon, will give the arc of direction.

Some writers on this subject have repudiated the parallel in mundo formed upon the horizon, but without adequate reason being adduced in support of their objection. Yet the same writers have not denied the efficacy of parallels formed on the sameside of the meridian, one south and the other

3


north, as in the 4th and 9th houses, or the 11th and 2nd, 10th and 3rd, etc., forgetting that bodies so placed are at equivalent distances from the horizon !

The rule for parallels is the same as for aspects. As the semiarc of the stationary body is to its meridian distance, so is the semiarc of the moving body to its proportional distance, which, taken from its primary distance, or added if it passes into another quadrant in forming the aspect, will give the arc of direction.

Thus we may bring Saturn to a parallel with the Moon in mundo. The Moon here is 2° 30' from the west horizon, and below it. If we bring Saturn along its own arc until it reaches a proportionate distance below the east horizon, we shall have a mundane parallel formed on the same side of the horizon, but on opposite sides of the meridian. We could work this problem by reference to the meridian distances of the two bodies from the Nadir, and the result would be the same.

It should be observed that the Sun and Moon are regarded as significators in the formation of mundane parallels by the other bodies, and the meridian and horizon therefore become sectors, upon which the parallels are formed.

Another form of the parallel in mundo is what is known as the rapt parallel. This is formed by the motion of the Earth on its axis, whereby the various bodies are carried from east to west at

ZODIACAL AND MUNDANE PARALLELS 35

their several relative distances from one another until they come to the same distance on either side of the meridian or horizon. In this case both bodies move in the prime vertical at a rate proportionate to their relative semiarcs.

Buie.—As half the sum of their semiarcs is to half the sum of their meridian or horizontal distances, so is the semiarc of the body applying to the angle, to its distance from that angle at the formation of the parallel. This distance taken from its radical distance from the same meridian or horizon ■will give the arc of direction.

What we are actually doing is to bring the meridian or horizon to the mid-distance between the Sun and a planet, or between the Moon and a planet. And these mid-distances are of the greatest significance, whether in the zodiac or in mundo. Here we are considering them only in mundo.

Example.—Bring the Moon and Saturn to a rapt parallel. This is formed on the lower meridian.

Semiarc, Moon (nocturnal) 52° 51'„ Saturn „ 98° 45'

2)151° 36'

Half sum of semiarcs . 75° 48' prop. log. 37560

Arith. comp. 9*62439

36 D I R E C T I O N A L A S T R O L O G Y

Arith. comp. 9*62439 Merid. dist. of Moon . 50° 21'

,, ,, Saturn . 81° 2'

2)131° 23'

65° 4 1 'prop. log. 43782 Semiarc, Moon . . . 5 2 ° 51' „ 53223

Proportional dist., Moon . 45° 48' ,, *59444Radical distance . 50° 21'

Moon rapt, par., Saturn == 4° 33' arc of direction.

N o te .— In all cases where the Midheaven (meridian) and Ascendant (horizon) are employed as sectors, the Sun and Moon are employed as significators. They form aspects by their own apparent motions in the prime vertical, and the planets form aspects to the radical of the Sun and Moon by the same motion. This is the underlying principle of all parallels in mundo, and all rapt parallels. Remember that in mundane directions you are always dealing with the bodies themselves and not their longitudes.

CHAPTER VIO R D E R O E D I R E C T I N G

T h e student will do well to employ some definite method of noting the various directions, and of collating and tabulating his results, otherwise he is sure to overlook some that are important when considered in association with others that attend them, whether they be of the same or a contrary nature- Thus, if in a train or sequence of evil directions there should occur a good aspect of Jupiter to the Sun or Moon, the health and fortunes will be greatly sustained thereby, so that what would otherwise appear a fatal set of arcs, in the presence of this benefic arc of direction would lose that extreme significance, and, although sickness might supervene, the good direction would indicate a favourable crisis.

The following method is therefore suggested as inclusive of all legitimate directions.

1. Mundane Directions(a) Direct all the bodies to aspects and con

junctions with the Ascendant from east to west and from west to east.

37


(b) Direct each of the bodies to all the aspects and the conjunction with the Midheaven, both ways.

(c) Direct the Sun to other bodies and their aspects in mundo, both ways.

(d) Direct the Moon to other bodies and their aspects in mundo, both ways.

(e) Direct each of the planets separately to mundane aspects and conjunctions with the Sun.

(f) Do the same in regard to the Moon.(g) Direct the Sun to mundane parallels with the

Moon and planets.(h) Direct the Moon to mundane parallels with

the Sun and planets.(i) Direct the Sun to rapt parallels with the

Moon and planets.(j) Direct the Moon to rapt parallels with the

Sun and planets.

2. Zodiacal DirectionsFollow the same order as for mundane directions,

omitting classes {g), (h), (i), and (J) (mundane and rapt parallels), which are not formed in the zodiac.

Note that in zodiacal directions a body is always moved to a longitude to form a conjunction or aspect, never the reverse of this. Also that the meridian and horizon are fixed circles which do not move in regard to any particular locality. The Midheaven and Ascendant are those points where

ORDER OP DIRECTING 39

the ecliptic cuts through the meridian and horizon respectively.

All this long process of directing may appear to be very tedious. It certainly requires patience and method. But once done it lasts for a lifetime, which is a point to be considered. In possession of such a chart one may direct one’s course with wisdom and success, avoiding those dangerous shoals, sandbanks, and breakers which occur in the course of every life—or, if it be beyond the power of a man so to do, he can at all events divest evils of much of their power over him by adjusting himself to them, making provision against times of evil fortune and doubling his efforts when times of prosperity are shown. Thus may a man order his going and bring his life to a peaceful end. Sudden death cannot overtake the man who has knowledge of the time of that event years in advance ; and the keen edge of many afflictions, to which an all-wise Providence may dispose us for the greater ends of life, are dulled by a philosophic anticipation, so that, cutting less deeply, they leave the vital soul of man unhurt. Therefore, rather than pray that what is foreordained by the laws of life to the inscrutable ends thereof may be averted, let us rather pray with the Psalmist : ec Teach me the number of my days, that I may apply my heart to wisdom.”

CHAPTER V II

E F F E C T S O F D I R E C T I O N S

In order to complete this section of the work, which deals with that system of direction by semiarcs currently practised and approved, it will be necessary here to indicate the general effects of directions, so that the nature and import of events may be known as certainly as the time at which they are likely to transpire. I am here speaking of ie effects ” of directions as if these latter had a direct dynamic result upon the character and actions of an individual. I am disposed to classify astrologers in three main groups—fatalists, casuists, and idealists— according to the various views they take of the nature and purport of astrology. The Fatalists believe, or profess to believe, that there is a planetary configuration and an event which attends it. They admit no possible intervention, amelioration, or extenuation. Che sarà sarà, and that is the end of the matter. They argue a certain necessity of connection between character and environment as we find it and planetary positions at the moment of birth. As regards “ directions,” all of which

40

EFFECTS OF DIRECTIONS 41are formed within a few hours of the birth, they speak of them as “ seeds sown ” in the plastic soil of the human soul which spring up and bear fruit at the appointed time, as measured by the arc of direction. They are bom when they are bom by necessity of universal law, and they die when they die because fatal arcs of direction are then in force.

They speak of laws of Nature as if they were dynamic forces against which mankind cannot possibly contend. They forget that laws are only mental concepts induced upon our minds by an apprehension of the correlated successiveness of events, and that what we know about natural laws is an infinitesimal part of the possibly knowable. They speak of the bodies of this microscopic solar system of ours as if they were the be-all and end-all of existence. They forget that the continuity of matter is a fact only on the material plane, and that there are forces of an immaterial nature which transcend both matter and what we call the laws of material existence. The moral law is an illustration of this. It is spiritual in its origin and spiritual in its effects. If astrology teaches fatalism, its use is at an end and it becomes a suicidal science, since there is no object in knowing that which must inevitably take place. It would reduce man to an automaton and divest him of all moral responsibility.

The Casuists are those astrologers who accommo-


date their facts and figures to popular concepts by a discreet use of a melange of spurious philosophy. They forever quote the effete adage : 44 The wise man rules his stars, the fool obeys them ” ; and that other which says : 4 4 The stars inclinebut do not compel.” They put a premium upon the wisdom of experience and the will-power of a purposeful character, and promptly consign a man to destruction by telling him that his horoscope indicates he has neither one nor the other. They do not suggest to him that astrology, properly conceived and applied, is in itself the very concrete of experience, nor that the will-to-be and the will-to-do are functions of the human soul which rise superior to all circumstance, outlasting life itself.

The Idealists are those among astrologers who regard the intelligible universe as the expression of a Supreme Intelligence, who regard the planetary combinations merely as symbols, knowing that the causes of all effects are within man himself, the cogniser of all experience. They regard the 44 signs of the times 55 as the driver of a locomotive regards the signals, not as 44 causes ” of disaster, but as warnings against it, an open book to those who can read the signals, but of no value to those who cannot. They look upon the science of astrology as a wireless operator looks upon his code-book, merely as a means of interpreting the signals— a science evolved by man for the service of man.

My own view of the matter is that there is some-

EFFECTS OF DIRECTIONS 43

thing to say for the materialist side of the question, and a great deal more for the idealistic. There is not the shadow of doubt in my own mind as to the material fact of the interaction of the planetary bodies, nor as to the fact that this interaction is registered by an intervening body of the system only at certain angles. The Platonic dictum that “ God geometrises ” is nowhere better illustrated than in the law governing the interaction of bodies belonging to the same system. The physical effects of the syzygies, and especially of ecliptic conjunctions of the luminaries, are immediately appreciable. The law of the tides is a concrete example of the fact of interplanetary action. We cannot deny the dynamic effects of planetary action on the material plane, and we have every reason for including in this category the human organism, compounded as it is of cosmic elements and in direct physical relations with a material environment. But that does not warrant us in extending our views to include the action of physical bodies upon the immaterial part of us, the only part of us that is essential and distinctively human. The only thing that can directly affect the soul of man is the soul of another human being. There is continuity of action upon all planes of existence because there is a continuity of matter upon all planes, but we have no grounds for extending the range of action from one plane to another plane, except it be by mediation or agency. Else we


could say that a good soul must be possessed of a sound body, a beautiful soul of a comely body, and that our moral principles are derived from what we eat and d^ink—instead of which, what we eat and drink depends on our moral principles. There is sound philosophy in the words of Tennyson when he says that “ Soul to soul strikes through a finer element of its own.” It is capable of acting mediately through the physical body or immediately through its own essential being. These views will doubtless alter our viewpoint in regard to much that hitherto has been regarded as fundamental to a belief in astrology. The effort to accommodate the facts of astrology to the materialistic science of a generation agone has tended to this issue. Without in any way disposing of astrology as a physical science, it is high time that we learned to interpret the facts of that science in the light of the higher spiritual teaching to which we have access. Otherwise we shall debase the science and enslave our own souls. In such case it were better that our astrology had never been written. As a physical science, astrology has an immense future before it in this utilitarian age upon which we have embarked ; but as a fatalistic creed it is not worth an hour’s study.

These remarks will enable the reader to understand why, in the following statement of the “ Effects of Directions,” I have pursued the common practice of attributing certain results or


sets of conditions as accompanying the formation of “ directions ” or planetary combinations in the horoscope subsequent to the birth. They should not be regarded as inevitable “ effects ” of such directions, but rather as things signalled, as if we should hoist the red light to indicate “ danger ahead,” the green light for “ caution,” and the white light for “ road clear.” These signals do not cause disasters, but our ignorance of them, our inability to see them, or our wilful disregard of them may very well result in a catastrophe. Human science has harnessed many of the subtle and intangible forces of Nature and deployed them to the service of man. It may do the same with cosmic forces that are as universal as etheric action

The Midheaven

This point of the horoscope stands for dignity, influence, authority, and position, the worldly honour and credit of the subject, and for all that is associated with his social and communal status. Good directions, such as the sextile and trine of all planets, and the conjunction and parallel of Jupiter, Venus (and Mercury when well aspected at birth), are indications of an enhanced position, higher honours, social distinctions, increase of prestige, etc.

Evil directions, such as the semisquare, square, and opposition of all planets (including the Sun and Moon in this category), and the conjunctions

4 6 DIRECTIONAL ASTROLOGY

and parallels of Uranus, Neptune, Saturn, and Mars, indicate assaults upon the good name and credit of the subject, hurt to the business affairs, loss of position, rivalries, and unprofitable associations.

The Ascendant

This point of the horoscope indicates things personal to the subject, as health, general welfare, comfort, environment, changes, and the common relationships of life, that which affects him through collective influence, the public state of affairs, etc.

Good aspects (as above enumerated) tend to benefit the subject by a variety of means differing as the nature of the planet which is in aspect by direction.

Evil aspects signal bad health, obstacles, hindrances, incommodities, troubles and annoyances of various kinds, according to the nature and position of the planet directed.

The Sun,

when in a hylegliacal place (as defined by Ptolemy), has significance of the vital constitution and life of the subject. Generally it stands for the father and male representatives of a family, and for the honour, credit, and position of the subject himself. It is thus associated more particularly with the Midheaven.


The Moondenotes the health, changes of fortune, the mother and female representatives of the family the functional powers of the body, and, in its association with the Ascendant, public bodies, the populace, and public concerns generally.

If in a hylegliacal position, it indicates the vital organs and life of the subject.

Note.—Ptolemy defines certain parts of the horoscope as being vested with a vital prerogative, wherein the Sun has precedence by day and the Moon by night. It is a moot point whether other bodies, being in such positions (in the absence of the luminaries), may not be vested with the same prerogative, and again, whether the Sun or Moon, not radically in such a position, may become invested with such significance by coming to a hylegliacal place by direction after birth. Failing either the Sun or Moon, Ptolemy invests the Ascendant with the properties of hyleg or life- giver. But, whatever may be concluded in this debatable matter, it is certain that the Ascendant is most generally affected by evil directions at the time of a physical crisis, the afflicting planet generally indicating the nature of its cause.

The above points in the horoscope, the Midheaven, Ascendant, Sun, and Moon, are the significators, because they signify such persons and things in the life of the subject as are capable


of being affected, by the conflict of human circumstance.

All directions are made either (a) by the natural motion of the significators to the places and aspects of the planets, or (6) by the natural motions of the planets to the places and aspects of the significators.

The triangle (trine) and parts of it are good aspects, and indicate some advantage according to the position and nature of the planet directed. The cross (square) and parts of the square are evil aspects, and indicate similar disadvantages.

CHAPTER VIII

P L A N E T A R Y I N D I C A T O R S A N D T H E M E A S U R E

O E T I M E

The following definitions of planetary indications are necessarily only partial and incomplete, but they will serve doubtless to convey a more or less definite idea of the nature of events which, may be expected to attend directions formed by them with the various significators.

It should be observed that the house which a planet directed to holds in the horoscope of birth, or that which a planet which is directed arrives at when the aspect is complete, has chief significance in regard to the department of life in which the events will transpire, the nature of those events depending primarily on (a) the nature of the aspect and (b) the nature of the planet involved.

In this light, therefore, it may be said that Neptune in good aspect indicates events of a beneficial nature connected with the use of the faculties or some special faculty, and frequently in connection with a form of a rt; benefits from unexpected sources coming mysteriously to the subject; unseen

49 4

5 0 DIRECTIONAL ASTROLOGY

and intangible influences at work for the benefit of the subject; brilliant flashes and inspirations of the mind ; spiritual aid ; intuitive activity.

In evil aspect by direction it denotes chaotic and mysterious events adverse to the interests ; scandal, secret enmity ; undermining of the credit by misrepresentation and fraud ; treachery, ambush ; an involved state of affairs ; nervous leakage and depletion of energy ; wasting of tissue ; physical ennui and decline of the vital powers from inscrutable causes ; apprehension, fear, and dread of consequence ; danger of espionage ; loss by fraudulent concerns and false investments ; mental unrest and loss of faculty.

Uranus in good aspect denotes civic and governmental honours, preference, advancem ent; unexpected benefits arising out of public concerns and affairs ; ingenuity, inventiveness ; originality ; success in mechanical and engineering business ; strokes of good fortune coming from unexpected sources ; new associations and alliances.

In evil aspect this planet denotes the breaking down of existing relationships, lesions and fractures, partings and separations, loss of a sudden and unlooked-for nature; hurt by strikes and public demonstrations ; nervous lesion, paralysis ; breaks and dislocations.

Saturn in good aspect indicates favours from aged persons and benefits from old associations, long investments, time contracts, and a general

PLANETARY INDICATORS 51

state of stability and steadiness in the fortunes, congenial retirement and sequestration.

In evil aspect Saturn depletes the vital powers, causes physical hurts by falls and contusions, morbid diseases, colds and chills, inhibition of bodily functions ; loss of money and property; mental and nervous depression ; privations, obstructions, hindrances, and general misfortunes. Saturn is anciently known as the Greater Infortune.

Jupiter in good aspect denotes increase of fortune, opening up of new and lucrative opportunities, expansion of interests, advancement, progress, honours, confidence, good judgment, a general feeling of expansion and well-being, both physical and mental.

In evil aspect Jupiter denotes losses, errors of judgment, vanity or excessive confidence, disfavour of legal men and clericals, physical disabilities arising from congestion and surfeit, excess or over-indulgence, “ too much of a good thing,” too much confiance en soi, and consequent loss of esteem with others. It indicates a period of low finance, due to lavish expenditure, severe losses, or heavy investments. Jupiter is anciently known as the Greater Benefic, but it is certain that its evil aspects denote anything but a beneficial state of affairs.

Mars in good aspect denotes activity, new enterprises, great output of energy with commensurate good results, travelling, the executive powers are stimulated, and much profitable work is done.


Benefits accrue from military men, business connected with iron, steel, and fire. The muscular system is strengthened and there is a disposition to increased activity. Honours due to deeds of daring and chivalry. Women frequently marry under this aspect.

In evil aspect Mars denotes hurts by burns, scalds, fire, and steel, with loss of blood, abrasions and cuts, and also fevers and inflammatory conditions of the body or that part of it indicated by the position of Mars by direction. Loss by fire or theft, sometimes attended by violence. Sudden alarms and disasters of various sorts. Mars was anciently known as the Lesser Infortune.

The Sun in good aspect indicates increase of prestige, honours and emoluments, new friends and associations of a creditable character, general advancement and good fortune.

In evil aspect the Sun denotes losses, disfavour of superiors, troubles through male members of the family, the chief, overseer, or manager of a business ; loss by governing bodies ; ill-health due to fevers. Reverses of various sorts according to the house in which the direction is completed.

Venus in good aspect signifies social and domestic success, pleasures and enjoyments, gifts and presents, decorations ; the young court or marry, and the mature have children born or daughters engaged or given in marriage, and such events happen as cause pleasure and satisfaction. The


affectional nature is stimulated and the health is good.

In evil aspect Venus denotes sorrows, disappointments, bereavements, grief, and losses, domestic and social troubles, and hurts associated with young women or children. Venus was anciently known as the Lesser Benefie, and the less one has of it when in evil aspect the better for all concerned.

Mercury acts in terms of the planet to which at birth it is in closest aspect; but if not within orbs of an aspect with any planet, then in terms of the ruler of the sign it occupies. In good aspect it usually signifies activity, much occupation of a profitable nature, connected with writings, science, and business of a general nature. Travelling, profitable journeys, good news, gain in connection with the avocation or trade. An active time generally.

In evil aspect Mercury produces annoyances and disturbances, evil news, worry and anxiety, many short journeys to and fro to no purpose or profit, sleeplessness, irregular feeding, unrest.

The Moon in good aspect denotes pleasant and profitable changes, a change for the better in the general state of affairs, gain by public associations and concerns, favours from women of mature age, popularity.

In evil aspect it denotes loss by any of the above means, and a state of unrest both physical and mental which leads to neglect of duties and conse-

DIRECTIONAL ASTROLOGY54

quent loss. Hurts from women. Some public affronts may be suffered. Changes are unfortunate, and best avoided.

The Measure of TimeIn the foregoing system of primary directions

by proportion of the semiarcs, the measure of time is 1° = 1 year, and every 5' = 1 month.

Considerable discussion has been devoted to the question of time measurement in directions. Those who advocate the Arabian system of a day for a year have sought to bring primary directions into line with that system by equating arcs of direction made on the foregoing principle of semiarcs, by adding the arc to the Sun’s right ascension at birth, and then finding the day after birth at which the Sun attains this new right ascension. The count is made at the rate of one day for a year of life, and two hours for every month.

Others, again, have sought to apply a plus increment at the ratio of 365 to 360, seeing that the Sun moves through 360 degrees in 365 days, which is the same as taking the Sun’s mean motion 59' 8" as the value for 1 day = 1 year.

But it is obvious that none of these methods has any direct application to the system we are now concerned with, inasmuch as all the directions formed by the diurnal rotation of the Earth on its axis are formed within a few hours of birth so far as they apply to a life of ordinary length, and


they are measured in degrees of right ascension— that is to say, by the passage of the Equator under the local meridian in the prime vertical,— and therefore degrees of right ascension are the only uniform basis of measurement. It certainly does not seem consistent to measure arcs by one method and equate them in terms of another.

It should be observed, however, that primary directions in right ascension do not always coincide exactly with the events they are held to signify. Sometimes they are too short, and sometimes too long, but never more than a few minutes either way. Commander Morrison, R.N., was of opinion that the event signified was delayed or accelerated by reason of current transits in the horoscope at the time, and he further states that the chief effects may be expected to transpire when the lunar or secondary directions come into accord with them. This gives rather a wide margin of operation to the primary direction, and has led many to the conclusion that the secondary direction is, after all, the important one and deserving of primary consideration. A very little experience will show that it is not so, for, unless there are concurrent primaries in operation, lunar or secondary directions frequently pass with little or no result.

Primary directions and transits appear to answer to all the more important events in life.

At the same time we have to consider the duration of effects, and in regard to this it has been observed


that the process of formation of an arc of direction should be considered. For the longer a direction may be in forming, the longer ‘will those events endure which it signifies. Here we have Fitzroy’s old maxim again in evidence : “ Long foretold,long last : short notice, soon past.”

Thus an aspect to the Ascendant formed during the rising of a sign of short ascension such as Aquarius, Pisces, Aries, Taurus in northern latitudes, and the opposite signs to these in southern latitudes, will be speedily formed and over. On the other hand, a similar direction formed to the Ascendant when in a sign of long ascension will be formed more slowly, and will dissolve more slowly. In such case we might expect the signified event to begin to happen earlier and to end later than in the former case.

One finds in experience, however, that men frequently trace years of toil and suffering to a sudden disaster overtaking them in a moment. In my theory of transits, this could not happen in earlier years, but might easily occur at m aturity when the accumulated results of a man’s labour were heaped around him. (See Transits.)

The following illustrations will, however, sufficiently prove that there is adequate coincidence between arcs of direction and the events they are held to signify, to warrant the measure of time 1° = 1 year as scientifically valid.

CHAPTER IX

IL L U S T R A T IO N

I n the example horoscope given in these pages we have a singularly interesting subject. The chief events of the life are well defined and closely indicated by the attendant arcs of direction. Hundreds of horoscopes, whether pertaining to individuals in high walks of life or of modest position in the world, could be adduced to show that this coincidence of direction and event is not fortuitous, but regular and consistent, and as dependable as any astronomical formulary. The student will find pleasure and instruction in working out the following arcs of direction in the present instance.

John Ruskin leapt into fame and became a <c lion 55 in the world of art in the autumn of 1843 under the direction of

Sun sextile Midheaven mundo 24° 37'He was married on the 19th April 1848, and, while on his honeymoon, took a chill while sketching in Salisbury Cathedral and was seriously ill. This happened under the adverse directions—

57


Moon oppos. Venus mundo eonv. 29° 16'Moon square Venus mundo dir. 29° 16'

The nearness of these adverse arcs to the event of marriage proved unfortunate for such a domestic change. The danger of his choice of a wrong time and a wrong partner for marriage was radically indicated by the Moon’s opposition to Mars and Mercury, and nothing but constant bickering could have been expected from such indications.

The first serious break in Buskin’s health was in May 1840, for which we have the direction—

Moon oppos. Saturn zod. 21° 46'He received honours from the University of Cambridge in May 1867 under the appropriate directions of

Ascendant trine Venus mundo 48° 2' Ascendant sextile Moon zod. 48° 22'

He was elected Slade Professor of Pine Art on the 10th August 1869, and commenced his course of lectures under the following directions :

Ascendant trine Jupiter zod. 50° 35' Midheaven par. Jupiter zod. 50° 57' Ascendant par. Jupiter zod. 51° 14'Moon rapt par. Jupiter 51° 22'

In the following year his health gave way, and his mother died in December of that year, 1871. The arc for that year measured from 51° 53' to 52° 53', and within these limits we have the significant directions—

ILLUSTRATION 59Ascendant square Saturn zod. con. 51° 59' Moon square Mars mundo . . 52° 0'S u n p a r . U r a n u s z o d . c o n . . . 52° O'A s c e n d a n t s q u a r e J u p i t e r m u n d o . 52° 41'

followed by Moon par. Mars zod. 53° 3', close upon the death of his mother.

His health completely broke down again in 1888, under the directions—

Sun opposition Uranus zod. . . 68° 49'S u n o p p o s i t i o n U r a n u s m u n d o . 69° 14' M o o n r a p t p a r . S a t u r n . . . 69° 30'

Here the Sun is hylegliacal, and, being so heavily afflicted from angles of the horoscope, and the Moon also afflicted by Saturn, only disastrous illness and misfortune could have been signified.

Nevertheless, he survived this affliction, and further added to his reputation as a man of letters and exponent of fine art during some ten years, until eventually, with declining vitality laying him open to attack, he was afflicted by influenza and succumbed on 20th January 1900, the arc for that date being 80° 57'. The following significant train of directions was then in force :

Sun par. Uranus zod. . . . 80° 10'Ascendant square Saturn mundo .81° 2' Ascendant sesquiq. Sun zod. . 81° 11'Ascendant par. Uranus zod. . . 81° 27'Moon square Mars zod. con .. . 81° 30'

60 DIRECTIONAL ASTROLOGYIn view of these directions, it cannot be said

that we are not duly signalled by the celestial bodies, not only of the approach of evil times, when more than usual care and attention are due to health and fortunes, but also of those periods of good fortune when the sun smiles upon all our efforts and stimulates us to greater endeavours. The fault is altogether ours if we do not regard these portents. The beneficent Creator, having established these celestial bodies “ for signs and for seasons,” is ever faithful. He puts up the signals on every occasion. It is for us to apprehend and read them.

In King Edward VII.’s horoscope we have the attachment which led to his marriage indicated by

Venus conjunction Moon mundo . 19° 25'Moon conjunction Venus con. . 20° 7'

The attempt on his life by the maniac Sipido, when as King he was travelling in Germany, measures to an arc of 58° 25', and the appropriate direction was—

Sun opposition Neptune zod. 58° 21'

The death of the Empress Frederick (Princess Royal) in August 1901 was indicated by the direction—

Midheaven conjunction Saturn 59° 43'

The death of his royal mother, Queen Victoria,

ILLUSTRATION 61

requires an are of 59° 14', and we find the appropriate directions—

Midheaven square Moon zod. . 5 8 ° 5 8 ' Ascendant, opposition Moon . . 5 9 ° 1 9 'Saturn semisq. Ascendant . . 5 9 ° 2 2 'Midheaven conjunction Saturn zod. 5 9 ° 42'

These illustrations will doubtless serve for all practical purposes, and they can be worked out at leisure b y those who wish to exercise themselves in this art.

Other methods than that here illustrated must claim our attention, inasmuch as they have consistently been advocated by various authors. There are, moreover, several points which may be considered as debateable, and these also have to be considered before our work is rendered complete. W e must therefore pass on.

CHAPTER, X

P T O L E M Y AJSTI> P LA C ID US

I t is generally conceded that the system of directing which has so far occupied our attention first originated as a measure of time in the mind of Claudius Ptolemy, the famous geographer, mathematician and astronomer of Alexandria, who flourished in the second century of our era, and wrote a standard work on the subject of astrology called in the Greek Tetrabiblos, and in the Latin Quadripartite, being four books on the Influence of the Stars. He also wrote the Syntaxis and the Almagest, which, together with his work on astrology, have been translated into every language in Europe and into many Oriental languages also.

From the writings of Sir Isaac Newton we have evidence that there were many sources of information open to Ptolemy in the pursuit of astrological knowledge, and there is no reason to suppose that he did not avail himself of them fully, for none has ever suggested that astrology as a science was first promulgated by him. But it may certainly be

62

PTOLEMY AND PLACIDUS 63

affirmed that Ptolemy gave to the Western world the first scientific exposition of the subject. There are two Latin editions of the work and one in Greek. The best translation that we have is the paraphrase of Proclus from the Greek text rendered into English with extensive commentary by J. M. Ashmand, and recently published as a supplement to Coming Events. Ashmand has followed the Elzevir text, dated 1635.

The name of Claudius Ptolemy will be revived wherever astronomy and astrology are studied. It is enough for the purpose of this sketch to note that he was bom at Pelusium in Egypt, and became a brilliant disciple of the Alexandrian School. It appears that he was bom about the year 80 a .d ., flourished during the reigns of Adrian and Antoninus Pius, and died in the seventy-eighth year of his age.

Of Placidus de Titus, who first rendered a studied version of Ptolemy’s work on astrology, we have very little information. It appears that he was known as Didacus Placidus, and was a native of Bologna, became a monk, and was appointed mathematician to the Archduke Leopold William of Austria. He wrote in the early part of the seventeenth century a work entitled the Primum Mobile, in which he gives a thorough digest of the teaching of Ptolemy. The best English translation is by Cooper. Placidus showed that Ptolemy recognised two sets of directions arising out of

64 DIRECTIONAL ASTROLOGYtwo sets of planetary positions, one in the zodiac and the other in the world, i.e. in the prime vertical. To Placidus remains the credit of having elaborated that part of directional astrology which has regard to directions in mundo.

Ptolemy makes it clear in his chapter on the “ Number of the Modes of Prorogation ” (bk. iii., ch. xiv.) that “ when the vital prerogative is vested in the Ascendant, the anareta or killing planet may be brought to it by oblique ascension ; and if it be vested in the Midheaven or a body there situate, then direction is to be made by right ascension. If on the occidental horizon, the degrees of oblique descension are to be reckoned. But if not in either of these three places, but in some intermediate station, it should be observed that 4 other times ’ will bring the succeeding place to the preceding one, and not the times of ascension or descension nor of meridian transit as already declared.

“ For, if it be desired to calculate agreeably to nature, every process of calculation that can be adopted must be directed to the attainment of one object—that is to say, to ascertain in how many equatorial times the place of the succeeding body or degree will arrive at the position preoccupied at the birth by the preceding body or degree, and, as equatorial times transit equally both the horizon and the meridian, the places in question must be considered in regard to their proportionate

PTOLEMY AND PLACIDUS 65distances from both these, each equatorial degree being tahen to signify one year ”

Here Ptolemy makes it clear that he directs a body in the heavens to one that precedes it, or a body to a degree that precedes it, which direction is formed by the diurnal rotation of the Earth on its axis from west to east. He also makes it clear that he uses the proportionate distances of bodies from both the horizon and meridian as the basis of the calculation, and the arc of direction is the intervening degrees (equatorial) between them, at the rate of one equatorial degree for a year of life.

It is evident, therefore, that he takes a proportion of the semiarcs, or, as he calls them, “ the horary times,” of the planets involved. These arcs he describes as parallel to one another and to the Equator, but cutting the circle of the horizon at various degrees of obliquity.

Obviously, therefore, we have to take proportion of their semiarcs and meridian distances, exactly as we have been instructed in the foregoing exposition ; and as these semiarcs are regulated by the latitude of the place of birth and the corresponding ascensional differences of the planets, the positions of the bodies will have respect to the prime vertical and will be their apparent places in the plane of that circle. But it is important to note that Ptolemy says nothing concerning conversedirections, whether in mundo or in the zodiac.

5

66 DIRECTIONAL ASTROLOGYT h a t h e r e c o g n i s e s t h e m u n d a n e p o s i t i o n o f a b o d y

a s d i s t i n g u i s h e d f r o m t h e a p p a r e n t p l a c e o f i t s

“ d e g r e e ” o f l o n g i t u d e i s o b v i o u s f r o m h i s m e n

t i o n i n g b o t h i n t h e s a m e s e n t e n c e ; a n d w e d i s

t i n g u i s h o u r s e l v e s b e t w e e n t h e m u n d a n e a n d z o d i

a c a l c o n j u n c t i o n s o n l y b y r e f e r e n c e t o t h e b o d y o f

t h e p l a n e t i n t h e f i r s t i n s t a n c e a n d i t s l o n g i t u d e

i n t h e o t h e r c a s e .

T o C l a u d i u s P t o l e m y , t h e r e f o r e , m a y r i g h t l y b e

a c c o r d e d t h e h o n o u r o f h a v i n g s e t a s t r o l o g e r s u p o n

t h e r i g h t t r a c k w i t h r e g a r d t o t h e c o r r e c t m e a s u r e

o f t i m e b y r e f e r e n c e t o t h e e q u a t o r i a l d e g r e e s

s e p a r a t i n g o n e b o d y f r o m a n o t h e r , o r o n e b o d y

f r o m t h e l o n g i t u d e o r a s p e c t o f a n o t h e r , a s s e e n

f r o m t h e p l a c e o f b i r t h .

T h e r e i s l i t t l e d o u b t , f r o m t h e i l l u s t r a t i o n s o f

h i s m e t h o d t h a t P t o l e m y g i v e s , t h a t h e u s e s t h e

“ a s c e n s i o n a l 55 t i m e s i n a l l c a s e s d u e t o t h e l a t i t u d e

o f t h e p l a c e o f b i r t h ; a n d t h i s m e t h o d s e r v e s v e r y

w e l l n o t o n l y f o r d i r e c t i o n s t o t h e A s c e n d a n t a n d

D e s c e n d a n t , b u t a l s o f o r i n t e r m e d i a t e p o s i t i o n s

w h e n t h e p l a n e t s a r e i n t h e s a m e o r d i f f e r e n t

q u a r t e r s a n d o n t h e s a m e s i d e o f t h e m e r i d i a n , f o r

t h e n t h e i r a r c s m a y b e m e a s u r e d w i t h g r e a t f a c i l i t y

a n d a p p r o x i m a t e a c c u r a c y f r o m t h e T a b l e s o f H o u s e s a l o n e .

Illustration1 . B r i n g t h e S u n t o t h e p l a c e o f M a r s i n t h e

h o r o s c o p e o f R u s k i n .

PTOLEMY AND PLACIDUS 67h. EQ.

The sidereal time on the Midheavenwhen Mars’ place rises is . . 15 49

That when the Sun rises is . . 16 44

Difference in R.A. on the Midheaven in S.T. . . . 0 55

Divided by 4, this gives 13° 45' as the arc of direction.

The same arc of direction when exactly calculated by the semiarc method is 13° 49'.

2. Bring the Sun to the conjunction with Venus in zodiac.

h, m.The S.T. at sunrise (as above) is . 16 44That when Venus’ place rises is . 14 35

Difference . . 2 9This gives an arc of 32° 15'.

3. Bring Saturn to the place of Sun in zodiac. The declination of Saturn is 6° 54' S., and this

answers to the longitude of Pisces, 12° 37'.h. m.

S.T. on Midheaven when this pointrises . . . . . 17 30

S.T. on Midheaven when Sun rises . 16 44

Difference . 0 46This gives an arc of 11° 30'.

4. Bring the Moon to the opposition of Venus in zodiac.

68 DIRECTIONAL ASTROLOGYThe declination of the Moon is 25° 39', which

exceeds that of any degree of the zodiac owing to the Moon’s extreme latitude north added to the declination of its longitude. But reference to the Tables of Ascensional Difference and Right Ascension will show that its oblique descension answers to the twelfth degree of the sign Leo, which is the same as the oblique ascension of Aquarius 12°. Then the arc between the place and Venus in zodiac and Aquarius 12° will be the arc of direction. Thus :

h.* m.S . T , o n M i d h e a v e n w h e n V e n u s l o n g .

r is e s .............................................. 14 35S . T . o n M i d h e a v e n w h e n t h e 1 2 t h o f

A q u a r i u s r is e s . . . . 16 30

D i f f e r e n c e . 1 55T h i s g i v e s a n a r c of 28° 45'.

5. B r i n g t h e S u n t o t h e o p p o s i t i o n o f U r a n u s i n z o d i a c .

Take the opposite degree of the zodiac to that held by Uranus, and bring the Sun to it by oblique arc.

fcu m .S.T. when Gemini 23° 25' rises . . 21 21S.T. when Sun rises . . . . 16 44

Difference . . 4 37This gives an arc of 69° 15'.

6 . B r i n g S u n t o p a r . U r a n u s i n z o d i a c d i r e c t .

PTOLEMY AND PLACIDUS 69The declination of Uranus is 23° 24=', which

answers to that of Cancer 4°. Find the arc between this and the Sun.

h. m.S.T. on Midheaven when Cancer 4°

rises . . . . . 22 6S.T. on Midheaven when Sun rises

in Aquarius 18° 45' . . 16 44

Difference . 5 22

This gives an arc of 80° 30'.These examples will serve to show that without

recourse to the elaborations of a speculum or the use of proportional logarithms in the computation of proportional arcs, Ptolemy could, by the mere use of a table of ascensions under any latitude, find the time of an indicated event within an arc of 30' and even less, which, having regard to the approximations which are frequently adduced as cc arcs for the event ” when both are accurately known, show that they would serve for all practical purposes. I most frequently calculate arcs of direction in this manner, bringing out the results to the nearest quarter of a degree, which measures to three months of time. Ptolemy had constructed such tables, as appears from his Almagest, and this is obviously the method he used. In other words, he recognised no other directions than those that could be calculated by the difference of the oblique ascensions of the planets and of their longitudes,


taking the oblique ascension of their opposite degrees when the arc was formed by descension of a body.

A table of oblique ascensions such as that published by Worsdale enables the calculation to be made with even closer exactness. It has only to be remembered that when we are directing the body of a planet to the body or longitude of another, the longitude corresponding to its declination must be dealt with, and not the longitude of the body itself, as the above examples will sufficiently indicate.

CHAPTER X ID I R E C T I O N S U N D E R P O L E S

T his method has been much advocated, and especially by Mr R. C. Smith, the first of the almanac writers under the pen-name of “ Raphael.” It consists in directing a significator under its own pole instead of under the pole of the place for which the horoscope is cast.

To find the Pole of a PlanetTake its R.A., declination, and semiarc.Then say :

As the semiarc is to 90°,So is its meridian distanceTo the difference of its circle of position and

the meridian.And this difference, compared with its meridian distance, will give its ascensional difference under its own pole.

Then having this and also its declination, from the sine of its ascensional difference under

7i

72 DIRECTIONAL ASTROLOGYits own pole take the tangent of its declination, and the remainder will be the tangent of its pole.

Example.—In the horoscope of Ruskin find the pole of the Sun.

The R.A. of Sun is 321° 12', the meridian distance (below) 108° 44', the semiarc 110° 1', and the declination 15° 13'.

Semiarc 110° 1' . . prop. log. 0-21381

Arith. comp.Quadrant of 90°Meridian distance . 108° 44'

9-786190-301030-21891

D i f f e r e n c e 88° 57' 0-30613

Asc. diff. under pole 19° 47' log. sine 9*52951Sun’s declin. . . 15° 13' log. tang. 9-43458

Pole of Sun = 51° 13' log. tang. 10-09493

It is thus seen that the pole is measured along the tangent by its distance from the meridian or nadir, according as the body may be above or below the Earth at the time. At the meridian the pole would be 0, and at the horizon it would be the same as the latitude. Here “ pole” is the same as polar elevation. The difference 88° 57' indicates the place of the circle of position from the

D IR E C T IO N S U N D E R POLES 73plane of the meridian circle. Circles of position are small circles ■which are parallel to the great circle of the meridian and at right angles to the great circle of the horizon. They are like lateral circles of latitude in relation to which the meridian stands as equator and the Ascendant and Descendant as poles. Hence, if a planet be on the cusp of a house, it will have the same pole as that house.

Having calculated the poles of all the planets, and of the Sun and Moon, direction of one to another of them is thus made.

Rule.— Take the oblique ascension (or descension, as the case may be) of the promittor or body directed to under the pole of significator, and the difference of this from the oblique ascension (or descension) of the significator under the same pole is the arc of direction.

To find the oblique ascension of a body under the pole of another directed to it, to the log. tang, of its declination add the log. tang, of pole of the body directed, and the sum will be the log. sine of its ascensional difference under that pole. From this its oblique ascension can be found by referring it to its 3Ü.A. according to the rule (see c ‘ Definitions,5’ Chapter I.).

Example.— Direct the Sun in the example horoscope to the place of Venus in the zodiac.

The declination of Capricorn 5° 49' is 23° 20'. The Sun’s pole is 51° 13'. Then—


Pole of Sun, 51° 13'. . log. tang. 10-09493Dec. Venus long. . . log. tang. 9-63484

Asc. diff. of aspect 32° 28' log. sine 9*72977 R.A. of aspect . 276° 20'

O.A. of aspect . 308° 48' under pole of Sun.O.A. of Sun . . 340° 59' under its own pole.

Diff. . 32° 11' =arc of direction.

Applying this method to the hint I have already given as to the use of tables of oblique ascension, or tables of houses for various latitudes, we can calculate this arc perfectly well with a table of the houses for latitude 51° 13', which is the pole of the Sun. And we can calculate all the solar arcs by this means from the same table. Then if we find the pole of the Moon, and refer to the Table of Houses for equivalent latitude, we shall be able to take out all the directions of the Moon under its own pole. The directions of the Ascendant will, of course, be made under the pole of the place of birth, and those of the Midheaven by right ascension only. So that what appears at first a complex and exhaustive piece of work can readily be done by tables of houses, or tables of oblique ascension for various latitudes, in next to no time, as the saying is. And this, I think, may be adjudged the most popular contribution to the theory

DIRECTIONS UNDER POLES 75and practice of primary directions that I have been able to make.

E x a m p l e .— D i r e c t t h e S u n u n d e r i t s o w n p o l e t o

t h e o p p o s i t i o n o f U r a n u s i n t h e z o d i a c .

T h e S u n ’ s p o l e i s 51° 13'. T h e r e f o r e t a k e i n

h a n d t h e T a b l e s o f H o u s e s o r t h e T a b l e s o f O b l i q u e

A s c e n s i o n f o r l a t i t u d e 51° 13' NT.

T h e o p p o s i t i o n o f U r a n u s f a l l s i n G e m i n i 23° 25'.h. m.

S . T . o n M i d h e a v e n w h e n G e m i n i

23° 25' r is e s . . . 21 21S . T . w h e n S u n ’ s p l a c e r i s e s . 16 43

4 38This converted into arc of R.A. =69° 30'—arc of direction.

Example.—Direct the Sun under its own pole to Venus in the zodiac. Pole of Sun=51° 13'.

b.. m.S . T . o n M i d h e a v e n w i t h S u n

r i s i n g . . . . 16 43S . T . o n M i d h e a v e n w i t h C a p r i

c o r n 5° 49' r i s i n g . . 14 35

A r c o f d i r e c t i o n , S u n c o n j . V e n u s

zodiac = difference . . 2 8This is equivalent to 32° O'.

By exact calculation we found it formerly to be 32° 11'. The difference is inconsiderable from the point of view of probable time of the event.


As to the merits and demerits of these divergent systems of directing, I leave my readers to decide for themselves. Experientia docet. I hold no brief for either system, my business being merely to represent and to simplify. This I think I may claim to have done.

The system of directing under the semiarcs in the prime vertical is that which was followed by Ptolemy. The system of directing under the poles of the planets is of considerably more recent origin, and dates to the seventeenth century only. It consists, as will be seen, in directing in the circle of observation due to the pole of the significator or planet directed. The difference is that which one may note as between the tables of houses for one latitude and another. Nothing is simpler or more demonstrable. I leave it at that.

But in general practice it will be found that equally close results may be obtained by simple proportion and the use of the tables. Take the following hint for what it is worth. I am quite satisfied in my own mind that what we call primary directions seldom or never operate exactly to time, and if we correct the observed time of birth by one direction for an event we shall find that subsequent directions are not on schedule time. We have to allow a latitude for the operation of these directions. Such being the case, and, in the experience of the best artists, the import of primary directions being accelerated or retarded by the

DIRECTIONS UNDER POLES 77secondary directions and transits, we do not need to observe scruples. Approximations are always valuable.

The following may be regarded as the via Icetitia in primary directing :—

Rule 1.—As the semiarc of the planet whose pole is required is to 90° of the prime vertical, so is the distance of the body in right ascension from the meridian (upper or lower as the case may be) to its proportional distance in the prime vertical.

Rule 2.— From the sine of their difference subtract the tangent of the planet’s declination. The remainder is the tangent of its pole.

Rule 3.— For all directions under the pole of that planet or significator use the Tables of Houses for that latitude which answers to its pole.

Rule 4.—Find the difference between the ascension of the body (by sidereal time or right ascension on the Midheaven) and that of the planet directed to. This will be the arc of direction.

Note.—If the planets involved or the positions involved are between the tenth and fourth westward, take the ascensional degrees of the opposite places.

Rule 5.—Direct the Midheaven by right ascension only, and the Ascendant by oblique ascension under the latitude of birth. Direct the Sun under its own pole and the Moon under its own pole. This completes the entire scheme of primary directing.

Example.—In the horoscope of Ruskin the Sun

78 DIRECTIONAL ASTROLOGY■was found, to have a pole equal to the latitude of 51° 13' N. (see p. 72). It must therefore be directed under the Ascendant of 51° 13'. Similarly, the Moon, whose pole is 47° 27', must be directed under the latitude of that degree. A significator on the Midheaven would thus be directed by right ascension only, as stated by Ptolemy (see p. 64).

For directions of the Sun to other bodies, therefore, we use the Tables of Houses for 51° 13'. Those for Taunton are 51° 1', which is deemed near enough.

1. Direct the Sun to Jupiter in the horoscope.h. m.

S.T. on Midheaven with Sun rising . 16 41 S.T. on Midheaven with Jupiter’s long.

rising . . . . . . 15 55

Arc of direction = 11° 30', equivalent to S.T. . . 0 46

2. Direct the Sun to Mars.h- m.

Sun rising as before, S.T. on Midheaven 16 41 Mars rising, S.T. on Midheaven . . 15 43

Arc of direction = 14° 30' . 0 583. Direct the Sun to Mercury in zodiac.

h., m.Sun’s rising as before . . . 16 41Mercury’s longitude rising . . . 15 39

Arc of direction = 50° 30' . 1 2

DIRECTIONS UNDER POLES 794. Direct the Sun to Venus5 longitude.

Sun’s rising as above Place of Venus rising

Arc of direction == 32° 45;

5. Direct the Sun to Neptune in zodiac.Sun’s rising as before Neptune’s long, rising

Arc of direction=40° 30'

6. Direct the Sun to Uranus in zodiac.Sun’s rising as above .Uranus’ long, rising .

Arc of direction=46° 15'

7. Direct the Sun to opposition of Moon

Sun’s rising as beforeRising of Capricorn 27° 8', S.T. .

Arc of direction = 13° 30'

h. m.. 16 41 . 14 30

. 2 11

h. m.. 16 41. 13 59

. 2 42

h. m.. 16 41 . 13 36

. 3 5

in zodiac.h. m.

. 16 41

. 15 47

. 0 54

The various aspects to these promittors can be picked up en route as we bring the Sun from the horizon to the Midheaven, which it reaches in an arc of 69° 59 '=70 years nearly.

We cannot direct Sun to Saturn by the diurnal motion of the Earth, and so we must bring Saturn

80 DIRECTION AL A ST R O L O G Y

up to the Sun’s place. This in volves know ing th e pole of Saturn.

W e m ay also bring Saturn to th e A scen d an t under its ow n pole. B u t if we were to bring the Sun to Saturn under the Sun’s pole, th a t w ould be a pre-natal direction, for th e Sun can n ot go b ack from the position it has attained and sink below the eastern horizon. W e have therefore no alternative bu t to regard these directions as in valid , or to adm it the thesis already suggested, th a t in these directions, m ade contrary to th e apparent m otion of the bodies in the heavens, w e are dealing with the localised impress of the p lanet a t th e m om ent of birth, which im press is carried b y th e E arth up the western heavens and dow n the eastern heavens, so that the Sun’s localised im print is here carried dow n to the place of Saturn. A n d th is is conform able to the theory of directions under th e poles of the significators.

CHAPTER X IIT H E P A R T O P E O R T T T H E

F or a considerable time there was much discussion as to the correct method of finding the place of the Part of Fortune. This, it should be explained, is one of the old Arabic points, which, like the Pomegranate, the Sword, and others, were regulated by the distances of the several bodies from the Sun in the zodiac, the particular point referred to being the same distance in zodiacal degrees from the Ascendant.

It was when astrologers came to apply this theory to the system of primary directions in vogue that the trouble arose as to the correct method of computing this point.

I think that the easiest expression of the case is this :— the Part of Fortune is a mundane point answering to the distance of the Moon from the Sun in the zodiac. Thus in the horoscope of Ruskin the Moon wants 21° 38' from the opposition of the Sun, and therefore the Part of Fortune will be 21° 38' below the western horizon in mundo.

81 6

82 DIRECTIONAL ASTROLOGYIts mundane position will therefore be 8° 22' inside the 6th house.

Its meridian distance will be 68° 22', and its pole 39° 13'. Under this pole we may direct it to aspects in the zodiac, and in mundo. It has been suggested that the Part of Fortune cannot be directed, but can only receive directions from other significators and the planets. This is surely nonsense. Any point in the heavens having been defined and located is carried by the motion of the Earth on its axis from its radical place to others successively in a direction that is contrary to the rotation of the Earth. Hence the Part of Fortune will here be carried down the heavens from the 6th to the 5th and from that to the 4th house successively, forming both mundane and zodiacal aspects under its own pole. The pole of the Part of Fortune and that of Saturn being near to one another, they must be near a mundane parallel, on the same side of the horizon.

There are, however, other suggested methods of taking the place of the Part of Fortune.

Ptolemy says (bk. iii., ch. x ii.): “ The Part of Fortune is ascertained by computing the number of degrees between the Sun and Moon, and it is placed at an equal number of degrees from the Ascendant in the order of the signs. It is in all cases, both by day and night, to be computed and set down, that the Moon may hold with it the same relation as that which the Sun may hold with

THE PABT OF FORTUNE 83

the Ascendant; and it thus becomes, as it were, a lunar horoscope or Ascendant.”

It is therefore clear that Ptolemy intended degrees of oblique ascension or descension, and not merely degrees in the zodiac, the relations of which, in regard to the horizon of any place, are continually changing.

Thus in the horoscope of Ruskin we have—O.A. of Sun . . 341° 13'O.D. of Moon 157° 26'

add 180° 0' 337° 26'

3° 47'Moonto oppos.Sun. O.D. of 7th . . 159° 56'

156° 9 'O.D. of Part of Fortune.

This gives us a position answering to the 10th degree of Deo, and therefore close to the Moon.

Placidus says : te Let the Sun’s oblique ascension taken in the Ascendant be subtracted always from the oblique ascension of the Ascendant, as well in the day as in the night, and the remaining difference be added to the Moon’s right ascension ; the sum will be the right ascension of the Part of Fortune, which will have the Moon’s declination.”

In the example horoscope the oblique ascension of the Ascendant is 339° 56', from which take the Sun’s oblique ascension 341° 13' (adding 360 for subtraction), and the remainder is 358° 43', which


add to the right ascension, of the Moon 120° 17', and the sum is the right ascension of the Part of Fortune 119° O'.

The right ascension of the imaum cceli being 69° 56', the meridian distance of the Part of Fortune will be 49° 4', and its semiarc will be that of the Moon, 52° 51', as it has the same declination as the Moon in all cases. Then semiarc 52° 51' — 49° 4 '= 3 ° 47', which is the same as we derived from the method of Ptolemy. For there we found the oblique descension of the Part of Fortune to be 156° 9'; and the oblique descension of the 7th being 159° 56', the difference is 3° 47'.

The method of Placidus appears preferable in that we derive at once the right ascension and meridian distance of the Part of Fortune.

The question is, however, whether either is true, and only directions made by the position as thus derived can settle the point in debate.

To enable the student to at once work out the primary arcs, we here append the speculum in the example horoscope :

R.A . Ner. Dist. Semiarc. Horiz. Axe. Cusp. D ist.

119° 0 ' 49° 4 ' 52° 51' 3° 47 ' 3° 47 '

These elements at once suggest that the pole of the Part of Fortune can be found, and direction made by the Part of Fortune in mundo and

THE PART OP FORTUNE 85zodiac to the planets, just as if it were a definite body.

The fact that it is merely a symbol, a point in space, does not in the least invalidate its significance in human affairs, as some impulsive students have suggested. For what else are the degrees of the zodiac known as the Midheaven and Ascendant ? They are points in space which bear a definite relationship to a particular place at a given time. They do not need to be identified with a star in the heavens in order to obtain a significance in the horoscope. Every tyro in astrology knows as an absolute fact that these points have a very demonstrable significance in a horoscope, and that transits of the major planets over these points, and the passing of these points by planets in direction, are attended by events which leave no shadow of doubt that they are an essential part of the signalling apparatus by which we are forewarned of coming events. And if these, why not the Part of Fortune ? Call it a “ myth ” if you like, but understand that a myth is a “ veil ” designed to hide a truth which a symbol is said to embody. The symbol handed down to us is identical with that used in China and also in Egypt to indicate £e land, territory, a fie ld /’

CHAPTER X IIIL U I T A R P A R A L L A X A N D S E M I - D I A M E T E R

A mong the problems modemly confronting the student of directional astrology, that of the horizontal parallax of the Moon is perhaps one of the most important and at the same time most perplexing.

The places of the planets as indicated in the horoscope are the geocentric longitudes. They are computed from the standpoint of an observer. But as the place of observation is on the surface of the Earth and not at its centre, the observed position of the Moon will not exactly coincide with its computed geocentric longitude. In the case of the Sun and planets, the distances from the Earth are so great as to render the parallax inconsiderable, that of the Sun being only 9", and the parallaxes of other bodies beyond it being proportionately less. But in regard to the Moon, its nearness to the Earth renders its parallax of importance if we are to regard the Moon as affecting us by its direct ray. The nearer the Earth it may be, the greater is the angle of parallax. It is

86

LUNAR PARALLAX AND SEMI-DIAMETER 87therefore greatest at the perigee and least at the apogee of the Moon.

As the amount of parallax depends on the Moon’s place in its orbit, we make use of the apogee as a point of departure, and the Moon’s distance from that point in its orbit where it is furthest from the Earth is called its anomaly.

By comparing the calculated place with the observed place it has been found that the difference of the two at the apogee is 53' 53", and at perigee 61' 23". It will be sufficient for our purpose if we call these 54' and 61' respectively. By the use of the “ Ready Reckoner ” the amount of the anomaly can always be found for any date or hour, and the corresponding parallax is set against it. The table is here repeated for convenience.

[Table

88 DIRECTIONAL ASTROLOGYT a b l e o f A n o m a l y .

Epoch 1800, Jan. 0d 0h 0m=9s 20° 20'.

Years. Add. Days. Add• Anom. Hor. Par.

8 o / S o / s ° /i 2 28 43 1 0 13 4 0 0 542 5 27 27 2 0 26 8 6 553 8 26 10 3 1 9 12 12 554 0 7 57 4 1 22 16 18 555 3 6 40 5 2 5 19 24 55

6 6 5 24 6 2 18 23 1 0 557 9 4 7 7 3 1 27 6 558 0 15 54 8 3 14 31 12 559 3 14 38 9 3 27 35 18 55

10 6 13 21 10 4 10 39 24 56

20 1 9 46 11 4 23 43 2 0 5640 2 19 32 12 5 6 47 6 5650 9 2 53 13 5 19 51 12 5660 3 29 18 14 6 2 55 18 57

70 10 12 39 15 6 15 58 24 5780 5 9 3 16 6 29 2 3 0 5790 11 22 24 17 7 12 6 6 58

100 6 18 49 18 7 25 10 12 58

Months. Add. 19 8 8 14 18 5920 8 21 18 24 59

January 0 0 0 21 9 4 22 4 0 59February 1 15 1 22 9 17 26 6 59A/Ts>.w"h 1 20 50i X l i a d L l l »

April . 3 5 51 23 10 0 30 12 60May 4 7 48 24 10 13 34 18 60June 5 22 49 25 10 26 37 24 60July . 6 24 46

26 11 9 41 5 0 60August 8 9 47 27 ‘ 11 22 45 6 60September 9 24 48 28 0 5 49 12 60October 10 26 45 29 0 18 53 18 61November 0 11 45 30 1 1 57 24 61December 1 13 42 31 1 15 1 6 0 61

LUNAR PARALLAX AND SEMI-DIAMETER 89Example.— Find the Moon's anomaly for 8th

February 1819, and the corresponding horizontal parallax.

S O /

Epoch 1800 . . . 9 20 20Add 19 . . 9 27 59

February . . . 1 15 18 days . . . . 3 14 31

Anomaly= 0 17 51

The Moon is therefore within 18° of its apogee or furthest distance from the Earth, and its parallax will therefore be near its minimum. Our table shows that the parallax due to this anomaly is 55'. This would be the difference between the Moon’s geocentric longitude and its observed position from the surface of the Earth if it were exactly on the horizon. At the meridian the parallax is 0, and at the horizon it differs, as stated, from 54/ to 61/ according to the distance of the Moon from the Earth, i.e. its place in its orbit.

Now, as the horizon is at all points 90° from the zenith or nadir, we can make one of these the apex of a triangle, of which the zenith distance of the Moon at transit is the perpendicular and the base its meridian distance. From these we may find the hypotenuse, which will be the Moon’s zenith distance at the time of birth.

Thus, in the example horoscope the latitude of the place is 51° 30' N., and the Moon has latitude


5° 1' N., which therefore must be subtracted, leaving 46° 29' as the zenith distance of the Moon at transit of the nadir. Its meridian distance is found from the speculum to be 50° 21'. Then

L o g . c o s i n e 50° 21'. . 9-80489L o g . cosine 46° 29'. . 9-83794

Log. cosine 63° 52'. . 9-64283

And as 90° is to 55', so is 63° 62' to 39', which is the Moon’s parallax, and by which amount she is apparently depressed further below the horizon than she is computed to be. This will affect its meridian distance, etc. The directions of the Moon, if operating dynamically by right lines of energy upon any part of the Earth instead of via the centre of the Earth, will hence be affected ; and it remains a problem worth some close study and consideration as to what view ought to be taken. It is sufficient here to have indicated the method of calculation. It is one of the factors in the vexed problem of “ the uncertain Moon,” which has frequently been charged with an inconstancy altogether absent from the directions of the Sun and planets.

T h e s e m i - d i a m e t e r s o f t h e S u n a n d M o o n h a v e

o f t e n b e e n r e s o r t e d t o i n o r d e r t o a c c o m m o d a t e a

d i r e c t i o n a l a r c t o t h e d a t e o f a n e v e n t . A l l o w i n g ,

a s is u n d o u b t e d l y t h e c a s e , t h a t p r i m a r y d i r e c t i o n s

h a v e a n o r b o f i n f lu e n c e w i t h i n t h e l i m i t s o f w h i c h

LUNAB PABALLAX AND SEMI-DIAMETER 91it may be said they begin to operate, attain their maximum, and pass off, there yet remains the fact that one would naturally expect the maximum to coincide with the most marked phase of a crisis in the life. This appears to be acknowledged, inasmuch as practitioners in the art of directing make use of arcs of direction, measured from the centres of bodies as determined by their longitudes, in order to correct approximate times of birth. This correction can only be legitimately made on the supposition that arcs of direction are close, if not exact, to the time of the events they are held to signify.

And unless there were this fundamental integrity of the system of direction advocated, unless there was a close agreement throughout a life between the arcs of direction and the events portrayed, there would be no use in making the calculations.

Our longitudes are geocentric and apply to the apparent centre of the bodies. The apparent diameter of the larger planets, on account of their great distance from the Earth, is inconsiderable. But when we come to the Sun and Moon, which are the chief significators, and the bodies that are directed to form the aspects of the promittors, we are concerned with orbs that have a visible diameter. The Sun on account of its immense size, and the Moon on account of its close proximity, appear to have a diameter of about half a degree, or from the centre to the limb about 15'. This becomes an

92 DIRECTIONAL ASTROLOGYimportant consideration when we are directing either of them to the aspect or conjunction of one of the planets, inasmuch as from first to last contact of the disc of the luminary with the said planet or aspect there will he an included arc of half a degree, and this means six months of time according to the Ptolemaic measure of 1° — 1 year. Hence it may well be that a direction is increscent for three months before it attains its actual centrality and maximum strength, and another three months may transpire before the effects wear off. And if to this we add the fact that directions formed at the tropics, i.e. near Cancer or Capricorn 0, are very slow in formation (as may be seen from the Tables of Declination), 4° of longitude including only I7 of declination, it will readily be understood that there is ample room for “ latitude ” in the timing of events.

It seems desirable, therefore, that a few cases of very well-observed birth-times should be taken, and the arcs of direction computed very closely; and then that these arcs should be compared with the course of events, so that an estimate of the value of the semi-diameters of the Sun and Moon may be made.

The apparent semi-diameter of the Moon is controlled by the same factor as the parallax, namely, its place in the orbit and consequent distance from the Earth. It may be useful to mention that the semi-diameter of the Moon is

L U N A R P A R A L L A X A N D SEMI-DIAM ETER 9 3

approxim ately tw enty-seven one-hundredths of the parallax. Therefore m ultip ly the parallax b y 27 and divide b y 100. T hus, when the parallax is 5 4 ', th e sem i-diam eter of the M oon is 54 x 27 -j -100 = 1 4 ^ ', and w hen the parallax is 60, the sem idiam eter is 60 x 27 — 100 = 16 '.

T he M oon directed to the opposition of the A scendant in th e exam ple horoscope works out at 2° 30 ' ; bu t as the horizontal parallax of the M oon is 5 5 ', its sem i-diam eter w ill be nearly 1 5 ', and therefore th e direction w ould read :

A sc . oppos. M oon in m undo, first contact 2° 15',, ,, m iddle 2° 30',, ,, last contact 2° 45 '

thus giving a possible range of 3 0 ', or six m onths for th e duration o f this indication. This m ay help to account for th e variability th at has been noticed in regard to lunar directions, and possibly we m ay also h ave to consider taking the parallax into account. T h e solar directions w ill be affected b y sem i-diam eter of the Sim , but not appreciably b y parallax.

C H A P T E R X I V

I/OTSTAB. e q u a t io n s

TJnjdbe. this head I propose to examine a problem of some interest which appears to have escaped general recognition, hut which may very well be considered with the questions of parallax and semidiameter as having some connection with the noted irregularity of primary lunar directions.

Take an illustration from the horoscope already submitted. We would direct the Moon to conjunction with the nadir, which direction is known as “ Midheaven opposition Moon in mundo.” It is measured by the arc of the Moon’s meridian distance, 50° 21', and is formed by the rotation of the Earth on its axis, by which the Moon is carried down the western heavens until it makes its meridian transit.

The theory underlying this direction is that there is a permanent significance and value attaching to the radical positions of the Midheaven, Ascendant, and other significators, which is unaffected by the subsequent changes taking place amongst the planets, either on account of their apparent motions

94

LUNAR EQUATIONS 95in the heavens or their real motions in the zodiac. But we have now to consider whether there may not be some value attaching to these subsequent motions of the bodies in the zodiac. These motions, within the narrow limits of time comprised in the formation of directions in a life of ordinary length, would not be appreciable in the case of the planets or the Sun, but in the case of the Moon there would be a quite appreciable increment owing to the velocity of that body in its orbit.

Thus the arc of 50° 21' cited above would occupy the interval of 3 hours 25 minutes, during which the Moon will have increased its longitude by about 1° 42', so that it would not actually make the meridian transit for another 7 minutes, although its radical place would then be exactly on the nadir. Its right ascension will be increased by about the same amount, and therefore the actual arc of direction from the time of birth until the bodily transit of the nadir would be about 52° 3'. So far as this case is concerned it is worthy of notice that this arc of the second distance of the Moon to the opposition of the Midheaven, and therefore to the mundane square of the Ascendant, coincided exactly with a period of serious illness and trouble in the life of Buskin, whereas the arc M.C. opposition Moon in mundo, 50° 21', exactly coincided with the election of Ruskin to the Slade Professorship of Pine Art, a distinction which brought him into the highest position in his sphere of life.


Obviously, therefore, the second distance of the Moon is by far the most appropriate.

Let us look at another direction from the same point of view. Direct the Moon under its own pole to the opposition of Saturn.The Moon’s pole is 47° 27', and its

ascensional difference under that pole, derived in the process of finding the pole, i s .......................................... 31° 32'

Its right ascension . . . . . 120° 17'

Its oblique descension under its pole . 151° 49'A d d ..................................................180° 0'

Oblique ascension of opposition Moon =331° 49'Then for Saturn’s oblique ascension under the

same pole—Pole of Moon

tang. . . 10*03712Tang. Saturn’s

decl. . . 9*08283

Ascl. diff. Saturnsine . . 9*11995 = 7° 31'

R.A. of Saturn . . 348° 54'

O.A. of Saturn . . 356° 25' under Moon’spole.

O.A. of Moon’s oppos. . 331° 49'

Arc. of Moon oppos. Saturn= 24° 36'

LUNAR EQUATIONS 97This corresponds with Buskin’s leap into public

estimation and fame, for which we have the arc of direction Sun sextile Midheaven in mundo. Most certainly the Moon to opposition Saturn could not be regarded as in the least degree akin to the nature of events then current in the life of the great artist.

But this arc took lh. 38m. 24s. to complete, and during that time the Moon had increased its B.A. by some 49 '; and as we are bringing Saturn up to the opposition of the Moon under the pole of the Moon, we shall have to curtail the direction by 49', which results in an arc of 23° 47'. This is nearly a year in advance of Buskin’s great advent, and may very well have coincided with a period of stress and indisposition.

The Moon to the opposition of Venus comes into force at about thirty years of age, or in the thirtieth year of life, when he married ; but by adding the increment due to the time of direction to the radical place of the Moon we get an arc which falls out a whole year later, when it is certain Buskin realised his disappointment.

The directions of the Sun during the course of sixty years would only be affected by an increment of 10', and they can always be relied upon; but the directions of the Moon are at present very unsatisfactory, and it has been thought that this question of second distances may serve not only to indicate why lunar primary directions are inconstant, but why also they appear to have a more

7

98 DIRECTIONAL ASTROLOGYdurable influence than those of the Sun. The suggestion is that from the time the direction is formed to the radical position of the Moon to the time that it is formed to the actual position of that body in the heavens, may be the extent of its duration ; and during this period, which naturally increases in length as the age increases, transits and other secondary indications may come up repeatedly to reinforce the portents of the lunar direction and bring them into play. Certain it is that there are many conditions affecting the directions of the Moon which arise out of its velocity, and to maintain its ancient reputation for inconstancy and fickleness it appears to have jealously guarded its secret even from the lynx eye of the practical astrologer. Whether we have succeeded in compassing the fickle goddess by this exposition remains to be decided by constant experiment conducted by several independent workers. In the cause of a scientific astrology this is worth carrying out, and it is to be hoped that qualified and unprejudiced students will communicate their experience.

It may assist the average student to know that all directions of the Moon to succeedent places will fall out sooner, while those to precedent places will fall out later, than indicated by the radical or first distance of the Moon, and the are of direction must therefore be increased or decreased at the rate of 2' for every degree of the arc of direction. Thus an are of 39° 15' requires 1° 18^'.

CHAPTER XV OTJSPAXj d is t a n c e s

W h e n giving instructions as to the method of directing bodies to aspects of the Ascendant and Midheaven in mundo, it is customary to affirm that one-third of a planet’s semiarc is equal to a house- space, so that a planet that is one-third of its semiarc above the horizon is held to be on the cusp of the 12th house, and when two-thirds of its semiarc above the horizon it is on the cusp of the 11th. But if this were actually the case, we should find that when on the cusp of a house the oblique ascension of an ascending planet is the same as the oblique ascension of the cusp of that house. Such is not the case.

Example.— Direct the Sun in Buskin’s horoscope to the sextile of the Midheaven in mundo. This aspect falls on the cusp of the 12th house.

The semiarc diurnal of the Sun is 69° 59', and one-third of this is 23° 20', to which add the Sun’s distance under the horizon, 1° 17', and we get the arc of direction—24° 37'. The Sun is then on the cusp of the 12th house presumably. Bet us see.

99


The R.A. of the Midheaven is 249° 56', to which if we add 60 we shall have the oblique ascension of the cusp of the 12th house, 309° 56'. Now, when the R.A. of the Midheaven is increased by an arc of 24° 37', the oblique ascension of the cusp of the 12th will be increased by the same amount, and will then be 334° 33', while the oblique ascension of the Sun is 341° 13'. Wherein lies the error ?

It lies in the fact that we are directing the Sun under the pole of the Ascendant, whereas we should direct it under the pole of the 12th house cusp. I here give a table of the polar elevation due to the various houses in several latitudes, from which, by proportion of their parts, we may derive the pole of any house for any minute of the included latitudes.

P oles oe H ouses.

Lat. Cusps of 3, 5, 9, 11. Cusps o f 2, 6, 8, 12.

O45

O *

18 57 O /34 11

46 19 37 35 1047 20 19 36 1048 21 2 37 1049 21 46 38 1250 22 33 39 1551 23 21 40 1952 24 12 41 2453 25 5 42 3154 26 1 43 3955 26 59 44 48

C U S P A L D I S T A N C E S 101

The pole of the 12th house for the latitude 51° 30' N. is seen to be 40° 51', and if we direct the Sun under this pole we shall have the

Ascensional difference of Sununder pole of 12th 13° 36'

Right ascension of Sun 321° 12'

Oblique ascension of Sun underpole of 12th 334° 48'

Oblique ascension of cusp of the12th house. . . . 309° 56'

Arc of direction . 24° 52'

This, although not exact, is certainly nearer, and seems to justify the method of directing under the poles of planets.

The fact, however, is that if we take a fixed pole for any house in a given latitude we shall always be in some degree of error, and for the simple reason that the semiarcs of the planets, being parallel to the equator, do not lie in the same plane as the prime vertical, which is the circle we divide into twelve equal parts to form the houses of the heavens. Therefore an equal division of the prime vertical will not result in an equal division of the semiarcs, and either we have to consider the poles of the houses as movable, or, as seems more consistent with the facts, we must regard the house-spaces as unequal. In other words, we shall find that the


time (measured by degrees of R.A.) that the Sun remains in successive houses is unequal, and the same is to be said of any other body. When, therefore, we take one-third of the semiarc of a planet as equal to one house-space, we are indulging in a free use of the metaphysical concept that ce all circles are equal to one another,” as defined by the doctrine of Correspondences. Against this I have nothing to say except that it is not mathematics.

Now, just as we take the Sun’s oblique ascension under the pole of the Ascendant in order to find its distance from the horizon, so we must take its oblique ascension under the pole of the 12th house in order to find its distance from the cusp of the 12th, and its oblique ascension under the pole of the 11th to find its distance from the cusp of the 11th. Its right distance from the cusp of the 10th will be its arc to that cusp, since the meridian has no polar elevation. Thus :

The pole of the Ascendant is . 51° 30'The pole of the 12th house . . 40° 51'The pole of the 11th house . . 23° 46'

The Sun’s declination is 15° 13', log. tang. 9 64380, and if to this we add the tangent of the poles of the houses successively we shall have the sine of the ascensional differences of the Sun under these poles, which, added to its right ascension, will give its oblique ascension under those poles. These are:

C U S P A L D I S T A N C E S 103O.A. of Sim under pole of 1st house . 341° 13'O.A. of Sun under pole of 12th house 334° 48' O.A. of Sun under pole of 11th house 327° 59' R.A. of Sun under pole of 10th house 321° 12' Then, to find the arc of direction between the

Sun and any of these cusps, we merely subtract the oblique ascension of the one from the other. The oblique ascensions of the cusps are :

Of the Ascendant . . 339° 56'Of the 12th . . . 309° 56'Of the 11th . . . 279° 56'Of the Midheaven R.A. 249° 56'

Thus we have the following true arcs of direction of the Sun in mundo :

O.A. Sun under pole of Ascendant 341° 13'O.A. of the Ascendant 339° 56'

Arc of Sun to conjunction Ascendant 1° 17'O.A. of Sun under pole of 12th . 334° 48'O.A. of 12th house cusp 309° 56'

Arc of Sun to sextile Midheaven mundo 24° 52'O.A. of Sun under pole of 11th . 327° 59'O.A. of cusp of 11th . . . . 279° 56'

Arc of Sun to sextile Ascendant mundo 48° 3'R.A. of Sun under Meridian 321° 12'R.A. of Midheaven . . . . 249° 56'

71° 16'

104 DIRECTIONAL ASTROLOGYAnd in all these cases the Sun will have the same oblique ascension as the cusp of the house to which it is directed, at the time of direction being completed. This is what we argue for and obtain.

Also we may find the degrees of it.A . which pass under the meridian while the Sun passes from the cusp of one house to the next, and thus the house- space of the Sun at its present declination.

As the whole diurnal arc of the Sun is less than 90, the house-space will be less than 30°.

Subtract the arc of direction of Sun conjunct Ascendant from the are of direction Sun conjunct 12th=Sun sextile Midheaven. There remains 23° 35', the house-space of 12th house.

Subtract the direction of the Sun to the 12th from that to the 11th; there remains 23° 11', the house-space of the Sun in the 11th. Subtract the arc of direction Sun cusp of the 11th from the Sun conjunct Midheaven; there remains 23° 13', the house-space of the Sun in the 10th.

And the three house-spaces added together = 69° 59', which is the diurnal semiarc of the Sun.

Hence it appears that the mundane directions of planets must be taken in terms of the pole of the cusp to which they are directed. The cuspal distances of the planets must also be measured according to the same rule. This will affect all directions calculated by primary arcs on the semiarc method now commonly in vogue.

But what appears of most vital importance as

CUSPAL DISTANCES 105a legitimate conclusion drawn from this critique is that the correct method of directing to any body is by oblique ascension under the pole of that body, which is quite different from taking the direction under the pole of the body directed. At the same time, it appears to dispose of the semiarc method , except as a valuable approximation. For nothing can be more certain than that the cusps of the houses, measured in the prime vertical, are 30° distant from one another by oblique ascension.

These conclusions agree entirely with our mathematics, for we have seen that the house-space of the Sun in the 12th, due to its declination, is 23° 35 '; and if to this we add the Sun’s direction (from below) to the Ascendant = 1° 17', we have an arc of direction, Sun to conjunction cusp of 12th == Midheaven sextile Sun in mundo, 24° 52', which is exactly what we found the direction of the Sun to be by oblique ascension when taken under the pole of the 12th house.

This proves, if anything can, not only that the correct method of directing is under the pole of the planet or position directed to, but also that the house-spaces are variable and depend on the several declinations of the planets, and thus on their oblique ascensions and descensions, taken under the poles of the successive houses.

By the semiarc method, taking one-third of a semiarc as equal to a house-space, we are dealing with an approximation which, although useful


and facile, is not mathematically correct. Rather than that had habits should become popular, X have undertaken a somewhat lengthy demonstration of this point, which I consider to be now settled beyond further debate.

CHAPTER X V I

S U G G E S T E D M E T H O D O E T R U E D I R E C T I N G

As the result of this examination of the various methods of directing, both by semiarc proportions and by oblique ascensions under the poles, we may come to the conclusion that all the disparities which vitiate the present methods can be disposed of if we proceed along the lines to which our conclusions point. Eor this purpose we shall require a speculum containing :

1. The right ascension of a planet.2. Its declination.3. Its pole.4. Its ascensional difference under its own

pole.The first of these will, of course, be worked as usual. The declination will be that given in the ephemeris. The pole of the planet will be that derived in the usual way from the ascensional difference of its proportional place in the prime vertical taken under its own declination, as already shown. Its cuspal distance will be the difference between its oblique

107


ascension (or descension) taken under the pole of the cusp to which it is nearest and the oblique ascension of the cusp in the prime vertical. These are all the elements required for a complete calculation of all legitimate arcs of direction.

Directions must be made under the pole of the body to which we are directing another. The pole is the same as geographical latitude. It represents the latitude (geographical) or polar elevation (astronomical) at which the cusp of the house cuts into the circle of the prime vertical, or at which a circle of position cuts into it.

Thus in the following diagram let the great circle NZHS, etc., be the sphere of the Earth, of which NT is the north pole, S the south pole. Also let ZN be the great circle of the prime vertical at an elevation from the Equator of 51° 30' N., and H—H the horizon intersecting it at right angles. Then HNTZH will be the upper meridian and HSNTH will be the lower meridian, the points Z and N marking the zenith and the nadir. The cusps of the 10th, 11th, and 12th houses are shown by the great circles cutting through the prime vertical at different elevations, and these answer exactly to the geographical latitudes (north) of the same values. Thus the pole of the Ascendant is 51° 30', that of the 12th, 40° 51', that of the 11th, 23° 46', and that of the 10th, 0° O', as shown in the diagram, the ascensional difference being the arc in R.A. between N-S and H -H .

SUGGESTED METHOD OF TRUE DIRECTING 109A circle of position is thus seen to be one which

passes through a body and converges upon the

N

S

horizon north and south exactly like an intermediate cusp of a house.

Rules for DirectingRule 1.— Find the pole of the body or ecliptic

position to which direction is to be made. Find the ascensional difference under this pole of the body to be directed. Apply this ascensional difference to the right ascension of the body to be directed, and obtain its oblique ascension (or descension, as the case may require) under the

pole of the body to which direction is made. The difference between this and the oblique ascension of the body to which direction is made, taken under its own pole, will be the arc of direction.

Rule 2.—In mundane directions take the body of the promittor, i.e. body directed to. In zodiacal directions take its longitude.

Rule 3.—In directing to the aspect of a planet in mundo, its cuspal distance taken under its own pole must be directed to under the same pole.

This rule also serves for mundane parallels.Here is the Speculum required for Ruskin’s

horoscope.

Speculum

Planet. Bi.A< Déclin. Pole. Asoen. Diff.

Sun .o / O ✓ O / O t

321 12 16 13 9*43458

51 13 10*09493

19 47

Moon 120 17 25 39 9*68142

50 21 10*08147

35 24

Mercury 296 47 21 34 9*59688

44 55 9*99885

23 13

Venus 276 6 18 10 9*51606

27 13 9*71125

9 43

Mars . 299 6 21 45 9*60013

46 22 10*02066

24 44

Jupiter 302 37 20 26 9*57119

47 13 10*03355

23 44

Saturn 348 54 6 54 9*08283

45 56 10*01423

7 11

Uranus 262 49 23 24 9*63623

o 1 q8*74904

7 27

Neptune 267 47 22 14 9-61148

3 48 8-82147

9 20

SUGGESTED METHOD OE TRUE DIRECTING 111

ExamplesDirect the Sun in mundo to the conjunction

with Venus mundo. The pole of Venus is 27° 13', its ascensional difference under that pole is 9° 43', which added to its R.A., 276° 6' (as Venus’ declination is S.), gives its oblique ascension under its own pole=285° 49'.

The oblique ascension of Sun under the same pole is—

Pole, log. tang. . 27° 13' 9-71125Decl., log. tang.. 15° 13' 9-43458

Asc. diff. log. sine 8° 2' 9-14583R.A. of Sun. 321° 12'

O.A. of Sun . 329° 14' under Venus’ pole. O.A. Venus . 285° 49' „ „ „

Arc of direction = 43° 25' Sun conj. Venus in mun.

Note.— All the tangents being inserted in the speculum under the declinations and poles of the planets, they can be extracted as required.

Direct Uranus to the conjunction with the Moon in mundo.

The pole of the Moon is 50° 21', its ascensional difference under that pole is 35° 24', and its oblique descension 155° 41'.


The declin. of Uranus is 23° 24' tan. 9*63623 Pole of Moon . . 50° 21/ tan. 0*08147

Asc. diff. under pole Uranus’ R .A ..

31° 28' sine 9*71770 262° 49'

O.D. Uranus . . 231° 2 1 'underMoon’spole.O.D. of Moon. . 155° 41' „ ,, ,,

Arc of direction = 75° 4 0 'Uranus conj. Moonin mundo.

These directions take very much less time to calculate than to set out in writing, and with the speculum at hand they are readily figured out in a minute or two.

Direct the Moon to opposition of Venus in mundo.

Oblique descension of the opposition of Venus = 105° 49'.

This is taken under the pole of Venus, from Venus’ oblique ascension less 180°—oblique ascension of the opposite point.Oblique descension of Moon under Venus’

p o le ..................................... ......... . 134° 37'Oblique descension of Venus under same

p o le ........................................................105° 49'

Arc of Direction, Moon oppos. Venusmundo . . . . . 28° 48'

SUGGESTED METHOD OE TRUE DIRECTING 113

These examples will doubtless serve for all conjunctions in mundo. For zodiacal directions it will be necessary to find the declination of the degree of the ecliptic held by a planet to which direction is made, or of its aspect, and add the log. tang, of this declination to the log. tang, of its pole. This will give the ascensional difference under that pole. Apply this to the right ascension to get its oblique ascension or oblique descension under that pole. The difference between this and the oblique ascension (or descension) of the planet directed, taken under the same pole, will be the arc of direction.

Planets having the same pole are either in mundane conjunction or in mundane parallel. This gives us the hint as to the calculation of mundane parallels.

Find the oblique ascension or oblique descension of the planet on which the parallel is formed, taken under its own pole. Find the oblique ascension or oblique descension (as the case may require) of the planet forming the parallel, under the same pole. The difference will be the arc of direction.

Example 1. — Bring Saturn in the example horoscope to the mundane parallel of the Moon.

This direction is formed by Saturn coming up to the pole of the Moon on the other side of the meridian.

8


Right ascension of the Midheaven . 249° 56'Oblique descension of Moon under

its own pole . . . . 155° 45'Moon’s distance from Midheaven,

westward . . . . . 94° 11'Added to R.A. of M.C. . 249° 56'

Oblique ascension of the parallel,eastward . . . . . 344° 7'

Oblique ascension of Saturn underMoon’s pole . . . . 357° 18'Arc of direction= difference . 13° 11'

This arc of direction, when computed by the semiarc method, is seen to be 8' short of the actual figures, which throws the time out nearly two months. The arc by that method is 13° 3' as compared with 13° 11', the true arc.

Example 2.—Bring Uranus to the mundane parallel of Sun in mundo. Here the planet descends the western horizon until it comes to the same pole westward as the Sun holds eastward.

Oblique ascension of the Sun underits own pole, 51° 13 '. . . 340° 59'

Subtract 180 . . . 180° 0 'Oblique descension of aspect below

west horizon . 160° 59'Oblique descension of Uranus under

pole of Sun . . . . 230° 14'Arc of direction, Uranus parallel

Sun mundo . . . 69° 15'

SUGGESTED METHOD OP TRUE DIRECTING 115

This arc of direction by the semiarc method is found to be 70° 57', which shows an error of 1° 42', equal to one year and eight months of time.

Time Measure for ArcsThis remark brings me back again to the question

of the equation of time, so much in dispute among astrologers. I think there can be little doubt that the true method is ee a day for a year,” which is certainly the most ancient method, as it is also the most uniform. In twenty-four hours the Earth revolves on its axis and the Sun comes again to the same meridian, having in the interval increased its longitude by more or less than a degree according to its apparent place in its orbit, i.e. the season of the year. The mean rate of its motion is 59' 8". Then, as all our calculations are made in terms of equatorial degrees, we have to make a proportion 59' 8" to 60', and this gives 24h. 21m.=1 year 5-334 days = l year 5d. 8h. for each 1° in the arc of direction. Thus every 6° in the arc of direction will give an extra month, to be added to the time at the rate of 1° = 1 year, which is the measure of time used in the semiarc method. If we add 5' for every 6° of are it will come to the same thing approximately. The measure of a degree of R.A. for a year is due to Placidus. That of the Sim's mean motion, or 1° R.A. = 1 year 5 days, is due to Valentine Naibod. Both are a compromise with facts. The probability is that we ought to


take the measure according to the season of the year in which the birth takes place, and hence the Sun’s actual increase of R .A . on that date, since the Sun is in every natural sense the great ehronoerater, or time-maker. Thus, in the case of Ruskin, who was born on the 8th February, the Sun’s diurnal increase of R .A . is 3' 57" = 5 9 ' 15" in are, but its increase in longitude is 60' 43", and this being an excess I 7 35" over the mean motion in the zodiac, an arc of direction, at the rate of a day for a year, would measure to so much less, at the rate of about

minutes for every complete degree of the arc. It will thus be seen that the question of the validity of one method over another in primary directions does not rest entirely on the astronomical facts, but also upon the value we attach to the arcs of direction when obtained. As to the astronomy of the case, there is not the slightest doubt in m y mind that the method of directing under the pole of the signifLcator is the correct mathematical scheme. But as to the measure of time from arcs thus derived, this is a matter of experiment, and one needs to exhaust all the evidence before coming to a conclusion.

CHAPTER X V II

C O N C L U S I O N

In the foregoing pages I have endeavoured to set out and critically examine the methods of directing advocated by Ptolemy and Placidus as modernly represented ; and I have further sought to establish their validity on general principles. I have not been blind to their imperfections, and have clearly indicated my view of the semiarc method, derived from the principles laid down by these great- pioneers of a scientific astrology, when I speak of them as valuable approximations. The discrepancies are those due to incorrect use of words in describing the facts. The term et corresponding to ” should be more frequently used in the semiarc method in place of the term “ equal to.” It is admitted that in both systems—that of proportional semiarcs and that of direction under poles—we are concerned with the apparent places of the planets in the prime vertical, and therefore when we speak of planets as being directed to a conjunction we mean an apparent conjunction as seen from the place of birth, and not either in the zodiac or by

117


right ascension, but solely in the prime vertical or circle of observation, which coincides neither with the Equator nor the Ecliptic. Therefore, when we come to the test we find without doubt that the only way of doing this is to bring the directed body along its own arc or parallel of declination to the same pole as the promittor or body directed to. Also, it is apparent that as polar elevation is measured from the zenith in the plane of the prime vertical, planets having the same pole must be in mundane conjunction if on the same side of the meridian, or in mundane parallel if on opposite sides, which fact renders the calculation of mundane parallels a process of such extreme simplicity that I wonder it has never been pointed out before.

To correct the errors arising out of the methods of Ptolemy and Placidus, I have made a complete statement of the true doctrine of polar directions in the plane of the prime vertical, and have supplemented this by a speculum drawn according to the principles laid down, so that by mere inspection of the same, and very little figuring, all direction» in mundo can be calculated. Eor directions in the zodiac it will be necessary to have the pole of the aspect or position in the zodiac, which can be determined by the longitudinal distance from the cusp of the house taken in proportion to the degrees of the ecliptic included in that house from the Table of Poles of the Houses, and from this we get its oblique ascension or oblique de-

CONCLUSION 119

scension under its own pole, and direct to it as in mundane direction.

In effect, it will be found that with a set of tables of oblique ascension, and one of tables of poles, all directions can be correctly calculated in a fraction of the time usually devoted to them , even by the very facile but faulty method of proportion of semiarcs. I have fairly stated both cases, and criticised only where criticism was necessary to correct error. In this I have done no hurt to the cause of scientific astrology, and I conclude this treatise in the earnest belief that I have even done some small service.

TABLES FOR THE USE OF ASTROLOGICAL STUDENTS

IN C L U D IN G TABLES OF L O G A R IT H M IC SINES, T A N G E N T S , E T C ., TABLES OF

R IG H T ASCENSION, D EC LIN AT IO N , A N D ASCENSIONAL

DIFFERENCE, A N D T E R N A R Y PR OPORTIONAL

L O G A R IT H M S

TABLES OF LOGARITHMIC SINES, TANGENTS, ETC.

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TABLES OF RIGHT ASCENSION, DECLINATION, AND ASCENSIONAL

DIFFERENCE

A r i e s a n d L i b r a Ascensional DIFFERENCE

I>eg. ! Declin, Rt. Asceti. London Birxning’tn Liverpoolo o f o f o f o 9 o ro o O o O o O o O O 0i a 24 o 5 5 o 30 o 81 O 322 o 48 1 50 1 O 1 2 1 43 i 12 2 45 1 30 1 33 1 374 i 36 3 40 2 O 2 4 2 95 1 59 4 35 2 80 2 35 2 416 2 23 5 80 3 O 3 6 3 137 2 •Ì7 6 2 6 3 30 3 37 3 458 3 IO 7 21 4 O 4 8 4 179 3 34 8 16 4 80 4 39 4 49

XO 3 58 9 11 5 O 5 IO 5 2111 4 21 IO 7 5 30 5 41 5 53X2 4 45 11 2 6 O 6 12 6 2513 5 8 11 58 6 30 6 43 6 5714 5 31 12 53 7 O 7 14 7 2915 5 55 13 49 7 29 7 45 8 116 6 18 14 44 7 59 8 16 8 3317 6 41 15 40 8 29 8 46 9 518 7 4 16 86 8 58 9 17 9 3719 7 27 17 32 9 28 9 48 IO 820 7 49 18 28 9 57 IO 18 IO 40

21 8 12 19 24 IO 27 IO 49 11 1222 8 34 20 20 IO 56 11 19 11 4323 8 57 21 17 11 26 11 49 12 1524 9 19 22 13 11 55 12 20 12 46

i 25 9 41 28 IO 12 24 12 50 13 17

26 IO 3 24 6 12 53 i a 20 13 4927 1Q 24 25 3 13 22 13 50 14 2028 IO 46 26 O 13 51 14 20 14 5129 11 7 26 57 14 20 14 50 - 15 2230 11 29 27 55 14 48 15 19 15 53

ISS*’- For thè R.A. of Libra, add 180° to thè same degree of Aries- The Déclin, and Asc. Diff. are thè same for both.

l i t i

T a u r u s a n d S c o r p i o A s c e n s io n a l . D i f f e r e n c e

Declin* Ht, Aseen, London Birming'm Liverpoolo o t o 9 o * o 9 o 9

o i i 29 27 55 14 48 15 19 15 SB1 l i 50 28 52 15 17 15 49 16 232 12 IO 29 49 15 45 16 19 16 548 12 31 30 47 16 14 16 48 17 244 12 51 31 45 16 42 17 17 17 555 13 12 32 43 17 IO 17 46 18 25

6 13 32 33 41 17 38 18 15 18 557 13 51 34 39 18 5 18 44 19 258 14 XX 35 38 18 33 19 12 19 549 14 30 36 87 19 O 19 41 20 24

i o 14 49 37 35 19 27 20 9 20 53

11 15 8 38 34 19 54 20 37 21 2212 15 27 39 33 20 21 21 5 21 5113 15 45 40 S3 20 47 21 32 22 2014 16 8 41 32 21 14 21 59 22 4815 16 21 42 32 21 40 22 26 28 1616 16 38 43 32 22 5 22 58 23 4417 16 55 44 32 22 31 23 20 24 1218 17 12 45 82 22 56 23 46 24 3919 1 17 29 : 46 33 23 21 24 12 25 620 17 45 47 38 23 46 24 87 25 3321 18 1 48 34 24 IO 25 3 25 5922 18 17 49 35 24 34 25 28 26 2523 18 82 50 36 24 57 25 52 26 5124 18 47 1 51 37 25 21 26 16 27 1625 19 1 52 39 25 43 26 40 27 4126 19 16 53 40 26 6 27 4 28 527 19 30 54 42 26 28 27 27 28 2928 19 43 55 44 26 49 27 49 28 5829 19 57 56 47 27 11 28 11 29 1630 20 IO 57 49 27 81 28 33 29 89

JSs?" F o r R .A . o£ Scorpio add. 180° to' the sam e degree o f Taurus. T h e Declin. and A sc . Diff. are the sam e for both.

l i v

■Ge m i n i & S a g i t t a r i u s A s c e n s i o n a l D i f f e r e n c e

D e g . Declin. R t . Aseen. London Birming’m Liverpoolo o t o 9 o t o t o ro 20 I O 57 49 27 31 28 33 29 391 20 22 58 52 27 52 28 54 30 12 20 35 59 54 28 12 29 15 30 233 20 46 60 57 28 31 29 35 30 444 20 57 62 O 28 49 29 54 31 45 21 8 63 3 29 8 30 13 31 24e 21 19 64 7 29 25 30 32 31 487 21 29 65 I O 29 42 80 50 32 28 21 39 66 14 29 59 31 7 32 209 21 49 67 18 30 15 31 23 32 37

i o 21 58 « 8 22 30 30 31 40 32 5411 22 6 69 26 30 45 31 55 33 I O12 22 14 70 30 30 58 32 9 33 2618 22 22 71 34 31 11 32 23 33 4014 22 29 72 39 31 24 32 37 33 5415 22 36 73 43 31 36 32 49 34 716 22 43 74 48 81 48 33 1 34 2017 22 49 75 52- 31 58 33 12 34 3118 22 55 76 57 32 8 33 22 34 4219 23 O 78 2 32 17 33 32 34 5220 23 4 79 7 32 25 S3* 41 35 121 23 9 80 12 82 33 33 49 35 1022 ‘ 23 13 81 17 32 40 33 56 35 1723 23 16 82 22 32 46 34 2 35 2424 23 19 83 28 32 51 84 7 35 3025 23 21 84 33 32 55 34 12 35 35

26 23 23 85 38 82 59 34 16 35 392 7 23 25 86 44 33 2 34 19 35 422 8 23 26 87 49 33 4 34 21 85 4429 23 27 88 55 33 5 34 22 35 4530 23 27 90 O 33 6 34 23 35 ' 46

F or the R .A . o f Sagittarius add 180° to the sam e degree o f G em ini. T h e Declin. and A sc. D iff. are the sam e for both.

C a n c e r a n d C a p r ic o r n u s A s c e n s i o n a l D i f f e r e n c e

Deg. Oeclin. Rt. Aseen. London Birming'm Liverpoolo o t o 9 o o / o 9

o 23 27 90 O 33 6 34 23 35 46i 23 27 91 5 S3 5 34 22 85 452 23 26 92 11 33 4 34 21 85 448 23 25 93 16 33 2 34 19 35 424 23 23 94 22 32 59 34 16 35 395 23 21 95 27 32 55 34 12 35 35

6 23 19 96 32 32 51 34 7 35 807 23 16 97 38 32 46 34 2 85 248 23 13 98 43 32 40 33 56 85 179 23 9 99 48 32 33 33 49 35 lO

XO 23 4 100 53 32 25 33 41 35 1

11 23 O lO l 58 32 17 S3 32 34 5212 22 55 103 3 32 8 33 22 34 4213 22 49 104 8 31 58 83 12 34 3114 22 43 105 12 81 48 33 1 34 2015 22 36 106 17 31 36 32 49 34 716 22 29 107 21 81 24 32 37 83 5417 22 22 108 26 31 11 32 23 33 4018 22 14 109 80 80 58 32 9 83 2619 22 6 HO 34 80 45 31 55 83 lO20 21 58 111 38 80 30 81 40 82 5421 21 49 112 42 80 15 81 23 82 3722 21 89 113 46 29 59 81 7 82 2023 21 29 114 50 29 42 3 0 50 32 224 21 19 115 53 29 25 80 32 81 4325 21 8 116 57 2 9 8 8 0 18 31 2 426 20 57 118 O 28 49 29 54 31 427 20 46 119 3 28 31 29 85 30 4 428 20 35 120 6 28. 12 29 15 80 2329 20 22. 121 8 27 52 28 54 30 180 20 lO 122 11 27 31 28 83 29 8 9

« S ' For the R .A . o f Capricornus add 180° to the sam e degree o f Cancer. T h e D éclin, and. A sc. DÜF. are the sam e for both.

Jvi

L e o a n o A q u a r i u s A s c e n s io n a l D i f f e r e n c e

Deg. Declin. R-t* Aseen. London, j BIrming’m LiverpoolO 1 o #X o r O ' 9 o t o #o 20 i o 122 i i 27 31 28 33 29 391 19 57 123 13 27 11 28 11 29 162 19 43 124 16 26 49 27 49 28 533 19 80 125 18 26 28 27 27 28 294 19 16 126 20 26 6 27 4 28 55 19 - 1 127 21 25 43 26 40 27 4t 1e 18 47 128 23 25 21 26 16 27 167 18 32 129 24 24 57 25 52 26 518 18 17 130 25 24 84 25 28 26 259 18 1 131 26 24 10 25 3 25 59

i o 17 45 132 27 23 46 24 37 25 3311 17 29 133 27 23 21 24 12 25 612 17 12 134 28 22 56 23 46 24 3913 16 55 135 28 22 31 23 20 24 1214 16 38 136 28 22 5 22 53 23 4415 16 21 187 28 21 40 22 26 23 1616 16 3 138 28 21 14 21 59 22 4817 15 45 139 27 20 47 21 32 22 2018 15 27 140 27 20 21 21 5 21 5119 15 8 141 26 19 54 20 87 21 22So 14 49 142 25 19 27 20 9 20 5321 14 80 143 23 19 O 19 41 20 2422 14 11 144 22 18 33 19 12 19 5423 IB 51 145 21 18 5 18 44 19 252 4 13 32 146 19 17 38 18 15 18 5525- 13 12 147 17 17 IO 17 46 18 25

26 12 51 148 15 16 42 17 17 17 5527 12 81 149 13 16 14 16 48 17 2428 12 IO 150 11 15 45 16 19 16 5429 11 50 151 8 15 17 15 49 16 2330 . 11 29 152 5 14 48 15 19 15 53

JSS3* F or the JR.A. o f Aquarius add 180° to the sam e degree of L eo. T h e Declin. and Asc* Diff. are the same for both.

Ivii

V irgo an» Pisces A scensional D ifferen ce

3D eg. I>eclin. Rt. Aseen. London Birming’m Liverpoolo o 9 o f a # o t o 9

o l i 29 152 5 ! 14 48 15 19 15 53i i l i 7 153 3 14 20 14 50 15 222 : lO 46 154 O 13 51 14 20 14 51

/ & \ lO 24 154 57 i 13 22 13 50 14 204 lO 3 155 54 i 12 53 13 20 13 495 9 41 156 50 12 24 | 12 50 13 17

6 9 19 157 47 11 55 12 20 12 467 8 57 158 43 j 11 26 11 49 12 15a 8 84 159 40 j lO 56 11 19 11 439 8 12 160 36 lO 27 lO 49 11 12

1 0 7 49 161 32 9 57 lO 18 10 40

1 1 7 27 162 28 9 28 9 48 lO 812 4 163 24 8 58 9 17 9 3718 6 41 164 20 8 29 8 46 9 514 6 18 165 16 7 59 8 16 8 S315 5 55 166 11 7 29 7 45 8 1

16 ! 5 81 167 7 7 o 7 14 7 2917 5 8 168 2 6 30 6 43 6 5718 4 45 168 68 6 O 6 12 6 2519 4 21 169 58 i 5 30 5 41- 5 5320 S 58 170 49 J 5 O 5 lO 5 21

21 3 34 171 44 4 80 4 89 4 4922 3 lO 172 39 4 O 4 8 4 1723 2 47 173 34 3 30 3 37 3 4524 2 23 174 30 3 O 3 6 8 1325 1 59 175 25 2 30 2 35 2 41

26 ; 1 36 176 20 2 O 2 4 2 927 1 12 177 15 1 30 1 33 1 8728 O 48 178 lO 1 O 1 2 1 429 O 24 179 5 O 30 O 81 \°\o

3280 o O 180 O 0 O O o O

For the R.A. of Pisces add 180° to the same- degree of Virgo. The Declin. and. Asc. Diff. are the same for both.

Xvili

TERNARY PROPORTIONAL LOGARITHMS

T E R N A R Y P r o p o r t i o n a l L o g a r i t h m s

t o° 1 ° 2 ° 3 ° 4 0 5 ° 6 ° 7 ° 8 ° 0 °o Infin ite 2*25527 i 95424 1 77815 1*65321 1 55630 1 47712 1 4IOI7 1 352x8 1*30x03I 4 03342 2*24809 I 95064 1 77575 1-65x41 1 55486 I 47592 1*40914 1*35128 1*300232 3*73239 2 24103 1*94706 1 77335 3-64961 i '55342 1*47472 I 40S1X 1*35038 1*299423 3*55630 2*23408 i ' 9+352 1 77097 i 64782 1 55198 1 47352 1*40708 1349 48 1*298624 3*43136 2*22724 1 94000 1 76861 1 64603 1*55055 1 47232 I 40606 1 34858 2 297825 3 33445 2*22051 1 93651 1 76625 3 64426 1 54912 J 47113 140503 1*34768 1 297036 3*25527 2*21388 1*93305 1 76391 1*64249 1*54770 1*46994 1C 404014 1*34679 1*296237 3 18833 2 20735 1*92962 1*76158 1 64073 1*54629 1*46876 1*40300 1*34589 1*295448 3*13033 2 2009r 1*92621 1*75927 1*63897 1*54487 1*46758 x 40198 1*34500 I 294649 3*07918 2*19457 1*92283 1*75696 i 63722 1 '54347 1*46640 1*40097 l ’344i i 1*29385JO 3 03342 2*18833 1*91948 1*75467 1*63548 1 54206 1 46522 1 39996 1 34323 1*29306

IX 2-99203 2*18217 1*91615 1*75239 1*63375 l 54066 1*46404 1*39895 2*34234 1*2922712 2*95424 2*17609 1*91285 1*75012 1*63202 1*53927 1 46288 1*39794 1*34146 1*291413*3 2-01948 2*17010 1*90957 1*74787 3 63030 1*53788 l 46x71 1*39694 2 34058 1*2907014 2-88730 2*16419 1*90632 1*74562 1 62859 1 53649 1*46055 1 39593 1*33970 1*28991*5 2*85733 2*15836 1*90309 1*74339 1*62688 1*53511 1 45938 1 39493 1*33882 1*28913

16 2 82930 2*15261 1*89988 1*74117 1*62518 1*53374 1 45824 1*39394 2*33794 1*2883537 2 80297 2*14693 1*89670 1*73896 1*62349 1*53236 I 45708 1*39294 1*33707 1*28757x8 2' 778i 5 2*14133 1*89354 1 73676 1*62180 1*53100 1 45593 1*39195 1 33619 1*2867939 2 75467 2*13580 1*89041 173457 1*62012 1*52963 1*45478 1 39096 2*33532 1*2860120 2*73239 2 13633 1*88730 1*73239 1*61845 1 52827 1*45364 1 38997 1*33445 1*28524

23 2*71120 2*12494 1*88420 i *73°23 1*61678 1*52692 1*45250 1*38899 2*33359 1*2844622 2*69100 2*11961 1*88114 l*728o7 i*6r5i2 i .*52557 1'45136 1*38800 1*33272 1 *2836923 2*67170 2* i i435 1*87809 1*72593 1*61347 1*52422 1*45022 1*38702 2*33186 1*28292«4 2*65321 2*10914 1*87506 1*72379 1*6x182 1*52288 1*44909 1*38604 1*33099 1*2821525 2*63548 2 10400 1*87206 1 72167 1*61018 1*52154 1*44796 1*38506 2*33013 1 2813b

26 2*61845 2*09893 1*86907 1*71956 1*60854 1*52021 1*44684 1*38409 2*32927 1*2806127 2*60206 2*09390 1*86611 1 71745 1*60691 1*51888 1*44571 1 38312 1*32842 1*2798428 2*58627 2*08894 1*86316 1*71536 1*60529 1*51755 1*44459 1*38215 1*32756 1*2790829 2*57103 2*08403 1*86024 1*71328 1*60367 I'5 l6 2 3 1*44347 i*38ri8 1*32671 1*2783130 2*55630 2*07918 1*85733 1*71120 1 60206 1*51491 1*44236 1*38021 1 32585 1*27755

33 2*54206 2*07438 1*85445 1*70914 1*60045 1*51360 1*44125 1*37925 1*32500 1*2767932 2 52827 2*06964 1*85158 x *70709 1*59885 '1*51229 1 *440 r 4 1 37829 2*32415 1*2760333 2*51491 2*06494 1*84873 1*70504 1*59726 1*5x098 1*43903 1*37733 2*32331 l 2752734 2*50194 2 06030 1*84590 1*70301 1*59567 1*50968 1*43793 1*37637 1*32246 1 2745135 2*48936 2*05570 1 84309 1*70099 1 59409 I 50838 1*43683 i*3754i 1 32162 1*27376

3« 2*47712 2 05115 l *84030 1*69897 i*5925i 1*50708 1*43573 1*37446 1*32077 1*2730037 2*46522 2*04665 1*83752 1*69696 1*59094 1*50579 1*43463 1*37351 2*31993 1*2722538 * 45364 2*04220 1*83477 1*69497 T'58938 1*50451 1*43354 1*37256 1*31909 1*2715039 2*44236 2*03779 1*83203 1*69298 1*58782 1*50322 1*43245 1*37x61 1*31826 1*2707540 2*43136 2*03342 1*82930 1*69100 1*58627 1*50194 1*43130 1*37067 2*31742 1*27000

4* 2*42064 2*02910 l *82660 1 *68903 1*58472 1*50067 1*43028 1-3697* 2*31659 1*2692542 2*41017 2*02482 1*82391 1 68707 1*58317 1*49940 1*42920 1*36878 1*31575 1*2685043 2 39996 2*02060 1 82124 1*685x2 1*58164 1*49813 1*42812 1*36784 1*31492 1*2677644 2*38997 2*01639 1*81858 1*683x8 I*58oiI 1*49687 1*42704 1-36691 1*31409 1*2670145 2*38021 2*01223 1*81594 1*68124 1*57858 1*49560 1*42597 1*36597 1*31326 1*26627

4 6 2*37067 2*00812 1*81332 1*67932 1*57706 1*49435 1*42490 1*36504 1*31244 1*2655347 2*36133 2*00404 1 81071 1*67740 1*57554 1*49309 1*42383 1 *36411 1*3x161 1*2647948 2*352x8 2*00000 1*80811 1*67549 1*57403 i*49i 84 1*42276* 1*36318 1*31079 1*2640549 2*34323 1*99600 t 80554 1*67359 1*57253 1*49060 1*42170 1*36225 1*30997 1*2633150 2*33+45 1*99203 1 80297 1 67x70 1*57103 1*48936 1*42064 ¿*36x33 1 30915 1 26257

51 2*32585 1*98810 1*80043 1*66981 1*56953 1*48812 1*41958 1*36040 1 30833 i **2 6 i 8 4

52 2 3174* 1*98421 1*7979° 1 66794 1*56804 1*48688 1*41853 i ' 35y48 1*30751 1*261x0

53 2*309x5 I 98035 1 79538 1 66607 1*56656 1*48565 1*41747 1*35856 1*30670 1*26037

54 2*30103 1*97652 1*79287 1 664.21 1*56508 1*48442 1*41642 1*35765 1*30588 1*2596455 2*29306 1*97273 X *79°39 1 66236 1*56360 x -48320 i 4x538 x*35673 1*30507 l 25891

56 2*28524 1*96897 1*78791 1*66051 1*56213 1*48197 1*41433 1*35582 1*30426 1*25818

57 3 'a 7 7 5 5 I 96524 1*78545 1*65868 1*56067 1 48076 1*41329 1*35491 1*30345 1 2574558 2*27000 1*96154 1*78300 1*656-55 1*55921 1*4795+ 1*41225 1 35400 1 30264 1*25672

59 2*26257 I 95768 1 78057 1 65503 1 55775 1 478x3 1*41X21 1 35309 1 30183 1*2560060 2*25527 1’95424 1 77815 1 6532t 1*55630 1 47712 1*41017 1*35218 1*30103 i 25527

lx i

T e r n a r y P r o p o r t i o n a l L o g a r i t h m s

1 0 ° 1 1 ° 1 2 ° 1 3 ° 1 4 ° 1 6 ° 1 6 ° 1 7 ° 1 8 ° 1 9 °0 2*25527 1*2x388 1*37609 1-14*33 1*10914 1*07918 2*05115 1*02482 1*00000 0*97652i 1*25455 1*21322 1*37549 x ■ 34077 3 *10863 i *07870 i *05070 1 *02440 0*99960 0*976142 i *25383 1*21257 1*17489 1*14022 X* 108x1 1*07822 2*05025 1 *02397 0*99920 0*975763 1*25331 1*2x193 1*17420 1-33966 2-X0760 1*07774 1*04980 1*02355 0*99880 0*975384 3*25239 1*2X126 1*17369 1*1391* I ’3O708 1*07726 1*04935 1*02312 C*99839 0*975005 1*25167 1*21060 1*17309 1*13855 1*20657 2*07678 2*04890 1*02270 0*99799 0*97462

6 1*25095 1*20995 1*17249 1*23800 1*20605 1 *07630 1*04845 1*02228 0*99759 0*974247 3*25024 1*20030 1*17189 1*13745 i* io 554 1*07582 1*04800 l*02l85 0*99719 0*973868 3*2495« 1*20865 1*17229 3 13690 1*10503 1*07534 1*04755 2*02143 0-99679 0*973489 1*24881 1*20800 1*17070 I ' i 3635 1*10452 1*07486 1*04710 1*02101 0*gg640 o '973io

10 1*24809 1*20735 1*17010 1*13580 1*10400 2*07438 1*04665 1*02059 0*99600 0-97273

21 1*24738 1*20670 1*16951 2-13525 1*20349 2*07391 1*04620 2*02017 0*99560 0*9723522 1*24667 1*20605 1*16891 1*13470 2*10298 1*07343 1*04576 1*01974 0*99520 0*9719723 i *24590 1*20543 3*16832 1*13415 3*10247 1 *07295 1*04531 I*OI932 0*99480 0*9713924 3*24526 3*20476 3*16773 1*13360 2*10197 i *07248 1*04486 1*01890 0*99441 0*9712225 3*24455 2*20432 1*16714 2-33306 1*10146 i *07200 2*04442 1 *01848 0*99401 0*97084

16 1*24384 3*20348 1*16655 1*23251 1*10095 1*07153 1*04397 l*Ol8o6 0*99361 0*9704717 1*34314 3 *20284 3*16596 1*13197 1*10044 1*07x05 1*04353 1*01764 0*99322 0*9700928 1*24244 1*20219 1*16537 1*13242 1*09994 1*07058 x *04308 1*0X723 0*992^2 0*9697219 1*24173 1*20355 1*36478 1*23088 2*09943 1*07011 1*04264 x*oi68r 0*99243 0*9693420 1*24103 1*20093 1*16419 1*13033 1*09893 1*06964 1*04220 1*01639 0*99203 0*96897

21 3 *24033 1*20028 1*16361 1*12979 1*09842 2*06916 1*04175 1*01597 0*99164 0*9685922 3*23963 1-19964 1*16302 1*12925 1*09792 1*06869 1*04131 1*01556 0*99124 0*96822«3 1*23894 1*19900 1*16243 1*12871 1*09741 1*06822 1*04087 1*01514 0*99085 0*9678424 1*23824 1*19837 1*36185 1*12817 3*09691 1*06775 1*04043 1*02472 0*99045 0*9674725 1*23754 1*19773 2*16127 1*12763 1*09641 2*06728 1*03999 2*01431 0*99006 0*96710

26 1*23685 x*i97 jo 1*16068 1*12709 1*09593 1-06681 1*03955 1*01389 0*98967 0*9667327 1*23616 1*19647 i *i 6oio 3*12655 X‘09540 1*06634 2*03912 1*01348 0*98928 0*9663528 1*23540 1*19584 1*15952 1*12601 1*09490 1 *06588 1*03867 1*01306 0*98888 0*9659829 1*23477 3*19520 1*15894 3*22548 1*09440 1*06543 1*03823 1*01265 0*98849 0*9656130 , 1*23408 1*19457 1*15836 1*12494 1*09390 1*06494 1*03779 1*01223 0*98810 0*96524

3X 1*23339 1*19395 1*15778 1*22440 1*09341 1*06447 1*03735 2*01182 0*98771 0*9648732 1*23371 1*19333 1*15721 1*32387 1*09291 1*06401 1*03691 1*01141 0*98732 0*9645033 1*23202 1*19269 1*15663 1*1*333 1*09243 1*06354 1*03647 1*01100 0*98693 0*9641334 1*23133 1*19206 1*25605 1*12280 1*09193 1 *06308 1 *03604 1*01058 0*98654 0*9637635 1*23065 1*19144 1*25548 1*12227 1*09142 1 *06261 1*03560 1*02027 0*98615 0*96339

3* 1*22997 1*19081 3*35490 - 1* 12173 1*09092 1*06215 1*03516 1*00976 0*98576 0*9630237 1*32928 3*190x9 1*15433 1*12320 1*09042 1 *06168 1*03473 1*00935 0*98537 0*962653« 1*22860 1*18957 1*15375 1*12067 1*08993 1*06122 1*03429 1*00894 0*98498 0*9622839 1*22792 1*18895 1*15318 1*12014 1*08943 1*06076 2*03386 1*00853 0*98459 0*9619140 3*23724 1*18833 1*15261 1*11961 2*08894 1*06030 1*03342 1*00812 0*98421 0*96254

41 3*22657 1*18771 1*15204 1*11908 3*08845 1'05983 1*03299 1*00771 0*98382 0*961174* 3*22589 1*18709 3*15147 1*21855 1*08796 1*05937 1*03256 1*00730 0*98343 0*9608143 1*22521 1*18647 1*15090 I*Xl8o2 1*08746 1*05891 1*03222 1*00689 0*98304 0*9604444 2*22454 1*18585 1*15033 2*12750 1*08697 1*05845 3*03269 1*00648 0*98266 0-9600745 1*22386 1*18523 i *J4976 1*11697 1*08648 1*05799 3*03126 1*00607 0*98227 o*9597i

46 1*22339 1*18462 2*14919 2*11644 1*08599 1*05753 1*03083 1*00567 0*98189 0*9593447 1*22252 1*18400 1*14863 1*11592 1*08550 1*05707 2*03039 1*00526 0*98150 0 *9589748 1*22185 1*18339 1*148 d6 1*11539 1*08501 1*05662 1*02996 1-00485 0*98111 0*9586149 3*22118 1*18278 1*14750 1*11487 1*08452 2*05616 1*02953 1-00445 0-98073 0*9582450 1*22051 1*18237 1*14693 1*21435 x *08403 1*05570 3*02910 1*00404 0*98035 0*95788

5* 1*2x084 1*18x55 1*14637 1*11382 3*08355 1*05524 1*02867 2*00363 0*97096 0 -9 5 7 5 152 i* 2 ig i8 1*18094 2*14581 1*11330 3*08306 1*05479 1*02824 1*0032*3 0*97958 0*9571553 1*2185*1 1*18033 2*14524 3*12278 2*08257 1*05443 1*02783 1*00282 0*97919 0-9567854 1*21785 1*17973 1*14468 1*12226 1*08209 1*05388 1*02739 1*00242 0*97881 0*9564255 1*2x718 1*17912 1*14412 1*11174 I -08x60 1*05342 1-02696 1*00202 0*97843 0*95606

56 3*21652 1*17851 *•14356 1*13122 1*08112 1*05297 1*02653 2*00162 0*97805 0*9556957 1*2x586 1*17790 2*14300 1*12070 2*08063 1*05253 1*02610 X-OQI2X 0*97766 0*955335« 3*23520 1*17730 114 2 4 4 3*130x8 1*08015 3*05206 1*02568 I*00080 0*97728 0*9549759 3*23454 1-17669 1*14189 1*10966 1*07966 1*05161 1*02525 1*00040 0*97690 0*9546060 1 1*21388 3*17609 2*14*33 1*30924 2*07918 1*05125 1*02482 2*00000 0*97652 0*95424

lxii

T ernary Proportional L ogarithms» 20° 21° 22° 23° 24° 25° 26° 27° 0GO 29°o 95424 93305 91285 §9354 87506 85733 84030 8239 X 808 ri 79287X 95388 93271 91252 89323 87476 85704 84002 82364 80786 792622 95352 93236 91219 89292 87446 85675 83974 82337 80760 792383 95316 93202 91186 89260 87416 85646 83946 82311 80734 792134 95380 93x68 91154 89229 87386 85618 83919 82284 80708 791885 95244 93133 9112 t 89197 87356 85589 83891 82257 80682 79163

6 95208 93099 9x088 89x66 87326 85560 83863 82230 80657 791387 95172 93065 91055 §9135 87296 85531 83835 82204. 80631 791138 95136 93030 91023 89103 87266 85502 83808 82x77 80605 790889 95*00 92996 90990 89072 87236 85473 83780 82150 80579 79063

IO 95064 92962 90957 89041 87206 85445 83752 82124 80554 79039IX 95028 92928 90925 89010 87176 854x6 83725 82097 80528 790x412 94993 92894 90892 88978 87x46 85387 83697 82070 80502 7898913 94956 92860 90859 §£947 871x6 85358 83670 82044 80477 7896414 94921 92825 90827 889I6 87086 85330 83642 82017 80451 7893915 94885 92791 90794 88885 87056 8530t 836x4 81991 80425 7891516 94849 92757 90762 88854 87026 85272 83587 81964 80400 7889017 94813 92723 90729 88823 86996 85244 8355g 8x938 80374 7886518 94778 92689 90697 88792 86967 852x5 83532 8x9x1 80349 7884019 94743 92655 90664 8876I 86937 85187 83504 81884 80323 7S81620 94706 9262 X 90632 88730 86907 85158 83477 81858 80297 78791

21 94671 92587 90599 88699 86877 85129 83449 81832 80272 7876622 94635 92554 90567 88668 86848 85101 83422 81805 80246 7874223 94600 92520 90535 88637 86818 85072 83394 81779 8022 X 7871724 94564 92486 90502 88606 86788 85044 83367 8175a 80195 7869325 94529 92452 90470 88575 86759 85015 83339 81726 80170 78668

26 94493 92418 90438 88544 86729 84987 83312 81699 80x44. 7864327 94458 92385 90406 88513 86699 84958 83285 81673 80119 7861928 94423 92351 90373 88482 86670 84930 83257 8x647 80094 7859429 94387 92317 90341 88451 86640 84902 83230 81620 80068 7857030 94352 92283 90309 88420 86611 84873 83203 81594 80043 78545

31 94317 92250 90277 88390 86581 84845 83175 81568 80017 7852132 94281 922X6 90245 88359 86552 84816 83148 81541 79992 7849633 94246 92X83 90213 88328 86522 84788 83121 81515 79967 7847234 94211 92149 90181 88297 86493 84760 83094 81489 79941 7844735 94176 921X5 90148 88267 86463 84732 83066 81463 79916 78423

36 94141 92082 901x6 88236 86434 84703 83039 81436 79891 7839837 94105 92048 90084 88205 86404 84675 83012 814x0 79865 7837438 94070 92015 90052 88175 86375 84647 82985 81384 79840 7834939 94035 9198X 90020 88144 86346 84619 82958 81358 79815 7833540 94000 91948 89988 88114 86316 84590 82930 81332 79790 78300

41 93965 919X5 89957 88083 86287 84562 82903 81305 79764 7827642 93930 9x881 89925 88052 86258 84534 82876 81279 79739 7835343 93895 91848 89893 88022 86228 84506 82849 81253 79714 7822744 93860 91815 89861 8799* 86199 84478 82822 81227 79689 7820345 93835 91781 89829 87961 86170 84450 82795 8I20X 79663 78x79

46 93791 91748 89797 87930 86140 84421 82768 81175 79638 7815447 93756 91715 89766 87900 86111 84393 82741 81x49 7g6ll 7813043 93721 91682 89734 87870 86082 84365 82714 8x123 79588 78x06

49 93686 91648 89702 87839 86053 84337 82687 81097 79563 78081

50 ‘93651 91615 89670 87809 86024 84309 82660 81071 79538 78057

5i 936x7 9X582 89639 87778. 85995 84281 1*633 81045 79513 7803352 93582 91549 89607 87748 85965 84253 82606 81019 79488 78009

53 93547 91516 89575 87718 85936 84225 82579 80993 79463 77984

54 93513 91483 69544 87687 85907 84197 82552 80967 79437 77960

55 93478 91450 89512 87657 85878 84169 82525 80941 79412 77936

56 93443 91417 89481 87627 85849 8414X 82498 80915 79387 7791357 93409 91384 89449 87597 85820 84114 82471 80889 7936a 77888

58 93374 91351 894 !7 87566 .85791 84086 82445 80863 79337 77863,

59 93340 9x318 89386 .87536 85762 84058 82418 80837 793x2 7783960 93305 91285 89354 87506 85733 84030 82391 80811 79287 77815

lxiii


t 30° 31° 32° 33° S 4° 35° 38° 37° 38° 39°o 7 7 8 1 5 7 6 3 9 * 7 5 0 1 2 7 3 6 7 6 7 2 3 7 9 7 1 1 2 0 6 9 8 9 7 6 8 7 0 7 6 7 5 4 9 6 6 4 2 1I 7 7 7 9 1 7 6 3 6 8 74990 73f 54 7 2 3 5 8 7 1 x 0 0 6 9 8 7 7 6 8 6 8 8 6 7 5 3 0 6 6 4 0 22 7 7 7 6 7 7 6 3 4 4 74967 7 3 6 3 2 7 2 3 3 7 7 1 0 7 9 6 9 8 5 7 6 8 6 6 8 6 7 5 IX 6 6 3 8 43 7 7 7 4 3 7 6 3 2 1 '74944 736*0 7 2 3 1 6 7 1 0 5 8 6 9 8 3 7 6 8 6 4 8 6 7 4 -93 6 6 3 6 54 7 7 7 1 9 7 6 2 9 8 74922 73588 7 2 2 9 4 7 1 0 3 8 6 9 8 x 7 6 8 6 2 9 6 7 4 7 3 6 6 3 4 75 7 7 6 9 5 7 6 2 7 4 74899 73566 7 2 2 7 3 7 1 0 1 7 • 6 9 7 9 7 6 8 6 0 9 6 7 4 5 4 6 6 3 2 8

6 7 7 6 7 1 7 6 2 5 1 74877 73544 7 2 2 5 2 70997 69777 6 8 5 9 0 6 7 4 3 5 6 6 3 1 07 7 7 6 4 7 7 6 2 2 8 74854 73523 7 2 2 3 1 7 0 9 7 6 6 9 7 5 6 6 8 5 7 0 6 7 4 1 6 6 6 2 9 1S 7 7 6 2 3 7 6 2 0 5 7 4 8 3 2 73501 7 2 2 0 9 7 0 9 5 5 6 9 7 3 6 6 8 5 5 1 6 7 3 9 7 6 6 2 7 39 7 7 5 9 9 7 6 1 8 1 7 4 8 0 9 73479 7 2 1 8 8 7 0 9 3 5 6 9 7 1 6 6 8 5 3 1 6 7 3 7 8 6 6 2 5 4

IO 7 7 5 7 5 7 6 1 5 8 74787 73457 7 2 1 6 7 7 0 9 x 4 6 9 6 9 6 6 8 5 1 2 6 7 3 5 9 6 6 2 3 6

31 77551 7 6 1 3 5 74764 73435 7 2 1 4 6 7 0 8 9 4 6 9 6 7 6 6 8 4 9 2 6 7 3 4 0 6 6 2 1 7xz 77527 7 6 1 1 2 74742 734*3 7 2 1 2 5 7 0 8 7 3 6 9 6 5 6 6 8 4 7 3 6 7 3 2 1 6 6 x 9 9*3 77503 7 6 0 8 9 74719 73392 7 2 1 0 3 7 0 8 5 2 6 9 6 3 6 6 8 4 5 4 6 7 3 0 2 6 6 1 8 0

77479 7 6 0 6 5 74697 73370 7 2 0 8 2 7 0 8 3 2 6 9 6 1 6 6 8 4 3 4 6 7 2 8 3 6 6 1 6 215 77455 7 6 0 4 2 74674 73348 7 2 0 6 1 7 0 8 1X 6 9 3 9 6 6 8 4 1 5 6 7 2 6 4 6 6 1 4 3

1 6 77431 7 6 0 1 9 7 4 6 5 2 73326 7 2 0 4 0 7 0 7 9 1 6 9 5 7 6 68395 6 7 2 4 5 6 6 1 2 517 77407 7 5 9 9 6 7 4 6 2 9 73305 7 2 0 1 9 7 0 7 7 0 6 9 5 5 7 6 8 3 7 6 6 7 2 2 6 6 6 1 0 61 8 77383 7 5 9 7 3 7 4 6 0 7 7 3 2 8 3 7 1 9 9 8 7 0 7 5 0 6 9 5 3 7 6 8 3 5 6 6 7 2 0 7 6 6 0 8 819 77359 7 5 9 5 0 7 4 5 8 5 7 3 2 6 1 7 1 9 7 7 7 0 7 2 9 6 9 5 x 7 68337 6 7 x 8 8 6 6 0 7 02 0 77335 7 5 9 2 7 74562 73239 7 1 9 5 6 7 0 7 0 9 6 9 4 9 7 6 8 3 1 8 6 7 x 7 0 6 6 0 5 1

2 1 773* i 7 5 9 0 3 74540 732X8 7*935 7 0 6 8 8 6 9 4 7 7 6 8 2 9 8 6 7 x 5 * 6 6 0 3 32 2 7 7 2 8 8 7 5 8 8 0 745*7 73196 7 i 9*4 7 0 6 6 8 6 9 4 5 7 6 8 2 7 9 6 7 x 3 2 6 6 0 1 423 7 7 2 6 4 7 5 8 5 7 74495 73174 7 1 8 9 2 7 0 6 4 7 6 9 4 3 7 6 8 2 5 9 6 7 1 1 3 6 5 9 9 624 7 7 2 4 0 7 5 8 3 4 74473 73*53 7 1 8 7 1 7 0 6 2 7 6 9 4 1 7 6 8 2 4 0 6 7 0 9 4 6 5 9 7 82 5 7 7 2 1 6 7 5 8 1 1 74450 73I3 I 7 x 8 5 0 7 0 6 0 6 6 9 3 9 7 6 8 2 2 1 6 7 0 7 5 6 5 9 5 9

2 6 77192 7 5 7 8 8 74428 73*09 7 x 8 2 9 7 0 5 8 6 6 9 3 7 7 6 8 2 0 1 6 7 0 5 6 6 5 9 4 X27 7 7 1 6 9 7 5 7 6 5 744°6 7 3 0 8 8 7 1 8 0 8 7 0 5 6 6 6 9 3 5 8 6 8 x 8 2 6 7 0 3 8 6 5 9 2 32 8 77*45 7 5 7 4 2 74383 7 3 0 6 6 7 1 7 8 7 7 0 5 4 5 6 9 3 3 8 6 8 1 6 3 6 7 0 1 9 6 5 9 0 429 7 7 1 2 1 7 5 7 x 9 7 4 3 6 1 73044 7 1 7 6 6 7 0 5 2 5 6 9 3 1 8 6 8 1 4 3 6 7 0 0 0 6 5 8 8 630 7 7 0 9 7 75696 74339 73023 7*745 7 0 5 0 4 6 9 2 9 8 6 8 x 2 4 6 6 9 8 1 6 5 8 6 8

3i 77Q74 7 5 6 7 3 743*7 7 3 0 0 1 7 1 7 2 4 7 0 4 8 4 6 9 2 7 8 6 8 1 0 5 6 6 9 6 2 6 5 8 4 932 7 7 0 5 0 7 5 6 5 0 7 4 2 9 4 7 2 9 8 0 7*703 7 0 4 6 4 6 9 2 5 8 6 8 0 8 6 6 6 9 4 4 6 5 8 3 133 7 7 0 2 6 7 5 6 2 7 74272 7 2 9 5 8 7 1 6 8 2 7 0 4 4 3 6 9 2 3 9 6 8 0 6 6 6 6 9 2 5 6 5 8 x 334 7 7 0 0 2 7 5 6 0 4 74250 7 2 9 3 6 7 1 6 6 2 7 0 4 2 3 6 9 2 1 9 6 8 0 4 7 6 6 9 0 6 6 5 7 9 435 76979 75581 7 4 2 2 8 7 2 9 1 5 7 1 6 4 1 7 0 4 0 3 6 9 * 9 9 6 8 0 2 8 6 6 8 8 7 6 5 7 7 6

3« 76955 7 5 5 5 9 74205 7 2 8 9 3 7 1 6 2 0 7 0 3 8 2 6 9 x 7 9 6 S 0 0 8 6 6 8 6 9 6 5 7 5 837 7 6 9 3 1 7 5 5 3 6 74*83 7 2 8 7 2 7*599 7 0 3 6 2 6 9 x 5 9 6 7 9 8 9 6 6 8 5 0 6 5 7 3 938 7 6 9 0 8 7 5 5 * 3 74*6* 72850 7X578 7 0 3 4 2 6 9 1 4 0 67970 6 6 8 3 1 6 5 7 2 139 7 6 8 8 4 7 5 4 9 0 74*39 7 2 8 2 9 7X557 7 0 3 2 1 6 9 1 2 0 * 6 7 9 5 1 6 6 8 x 2 6 5 7 0 34 0 7 6 8 6 1 7 5 4 6 7 74**7 7 2 8 0 7 71536 7 0 3 0 1 6 9 1 0 0 67932 66794 6 5 6 8 5

4 1 7 6 8 3 7 7 5 4 4 4 74095 7 2 7 8 6 7I 5I 5 7 0 2 8 1 6 9 0 8 0 6 7 9 1 2 66775 6 5 6 6 64 2 7 6 8 1 3 7542* 7 4 0 7 2 7 2 7 6 4 7*494 7 0 2 6 0 6 9 0 6 1 6 7 8 9 3 6 6 7 5 6 6 5 6 4 843 7 6 7 9 0 75398 74050 7 2 7 4 3 7*473 7 0 2 4 0 6 9 0 4 1 6 7 8 7 4 6 6 7 3 7 6 5 6 3 044 7 6 7 6 6 75376 7 4 0 2 8 72721 7*453 7 0 2 2 0 6 9 0 2 1 6 7 8 5 5 6 6 7 1 9 6 5 6 x 245 76743 75353 7 4 0 0 6 7 2 7 0 0 71432 7 0 2 0 0 6 9 0 0 2 6 7 S 3 6 6 6 7 0 0 6 5 5 9 4

4 6 76719 75330 73984 7 2 6 7 8 7*4* X 70X79 6 8 9 8 2 6 7 8 1 6 6 6 6 8 x 6 5 5 7 547 7 6 6 9 6 75307 73962 72657 71390 70*59 6 8 9 6 2 6 7 7 9 7 6 6 6 6 3 6 5 5 5 74 8 7 6 6 7 2 75285 73940 7 2 6 3 6 71369 70139 6 8 9 4 2 6 7 7 7 8 6 6 6 4 4 6 5 5 3 949 7 6 6 4 9 7 5 2 6 2 739*8 7 2 6 1 4 7X349 70119 6 8 9 2 3 67759 6 6 6 2 5 6 5 5 2 150 7 6 6 2 5 75239 7 3 8 9 6 72593 7 1 3 2 8 7 0 0 9 9 6 8 9 0 3 67740 6 6 6 0 7 6 5 5 0 3

5 1 7 6 6 0 2 7 5 2 1 6 73874 72571 71307 7OO78 6 8 8 8 4 6 7 7 2 1 6 6 5 8 8 6 5 4 8 452 7 6 5 7 8 75*94 73852 72550 7X286 70058 6 8 8 6 4 6 7 7 0 2 6 6 5 7 0 6 5 4 6 653 76555 75*7* 73830 7 2 5 2 9 7 1 2 6 5 7 0 0 3 8 6 8 8 4 4 6 7 6 8 2 6 6 5 5 x 6 5 4 4 854 76531 75148 7 3 8 0 8 72507 7X245 70O l8 6 8 8 2 5 6 7 6 6 3 6 6 5 3 2 6 5 4 3 055 7 6 5 0 8 7 5 x 2 6 73786 7 2 4 8 6 7 x 2 2 4 69998 6 S 8 0 5 6 7 6 4 4 6 6 5 x 4 6 5 4 x 2

56 76485 75*03 73764 72465 7 x 2 0 3 69977 6 8 7 8 5 6 7 6 2 5 6 6 4 9 5 6 5 3 9 457 7 6 4 6 1 7 5 0 8 0 73742 72443 71x83 69957 6 8 7 6 6 6 7 6 0 6 6 6 4 7 7 6 5 3 7 658 76438 75058 73720 7 2 4 2 2 71*62 «9037 6 8 7 4 6 67587 6 6 4 5 8 6 5 3 5 759 76414 75035 7 3 6 9 8 7 2 4 0 1 7i * 4* 6 9 9 x 7 6 8 7 2 7 6 7 5 6 8 6 6 4 3 9 6 5 3 3 96 0 76391 7 5 0 1 2 73676 72379 7 x 1 2 0 69897 6 8 7 0 7 67549 6 6 4 2 1 6 5 3 2 1


t 40° 41° 42° 43° 44° 46° 46° 470 OD 0 49°o 65321 64249 63202 62x80 61182 60206 59251 58317 57403 56508I 65303 6423I 63x85 62164 61166 60x90 59236 58302 57388 564932 65285 64214 63168 62147 61149 60174 59220 58287 57373 564783 65267 64196 63151 62x30 61133 60x58 59204 58271 57358 564634 65249 64X78 63133 62x13 6 n x 6 60x42 59189 58256 57343 564495 65231 64X6X 63116 62096 61x00 60126 59173 58241 57328 564346 652x3 64I43 63099 62080 61083 60x10 59157 58225 57313 564197 65195 64X25 63082 62063 61067 60094 59141 58210 57298 564048 65177 64XO8 63065 62046 6ib5i 60078 59126 58194 57283 563909 65159 64O9O 63047 62029 61034 6006 X 59110 58179 57268 56375

IO 65141 64073 63030 620x2 6x0x8 60045 59094 58164 57253 56360

XX 65123 64055 630x3 6x996 61001 60029 59079 58148 57238 5634512 65105 64038 62996 61979 60985 60013 59063 58133 57223 5633113 65087 64020 62979 61962 60969 59997 59047 58118 57208 5631614 65069 64OO2 62962 61945 60952 59981 59032 58x02 57193 5630115 65051 63985 62945 61929 60936 59965 59016 58087 57178 56287

i6 65033 63967 62927 619x2 60920 59949 59000 58072 57163 5627217 65015 63950 629x0 6x895 60903 59933 58985 58056 57148 56257x8 64997 63932 62893 61878 60887 59917 58969 5804X 57133 5624319 64979 63915 62876 61862 60871 59901 58954 58026 57118 5622820 64961 63B97 62859 61845 60854 59885 58938 580x1 57103 56213

21 -64943 63880 62S42 61828 60838 59870 58922 57995 57088 5619922 64925 63862* 62825 6x8x2 60822 59854 58907 57980 57073 5618423 64907 63845 62808 6x795 60805 59838 58891 57965 57058 5616924 64889 63827 62791 61778 60789 59822 58875 57949 57043 5615525 64871 638IO 62774 6x762 60773 59806 58860 57934 57028 56140

26 64853 63792 62757 6 X745 60756 59790 58844 57919 57013 5612527 64835 63775 62739 61728 60740 59774 58829 57904 56998 5 6 m28 64818 63757 62722 617x2 60724 59758 58813 57888 56983 5609629 64800 6374O 63705 6x695 60708 59742 58798 57873 56968 5608130 64782 63722 626S8 61678 60691 59726 58782 57858 56953 56067

3i 64764 63705 62671 61662 60675 59710 58766 57843 56938 5605232 64746 63688 62654 61645 60659 59694 58751 57827 56923 5603733 64728 63670 62637 61628 60642 59678 58735 57812 56908 5602334 64710 63653 62620 61612 60626 59663 58720 57797 56893 5600835 64692 63635 62603 6X595 60610 59647 58704 57782 56879 55994

36 64675 636X8 62586 6X579 60594 59631 58689 57767 56864 5597937 64657 636OX 62569 61562 60578 59615 58673 57751 56849 5596438 64639 63583 62552 61545 60561 59599 5865a 57736 56834 5595039 64621 63566 62535 61529 60545 59583 58642 57721 56819 5593540 64603 63548 625x8 6x512 60529 59567 58627 57706 56804 5592141 64586 63531 62501 6x496 60513 59551 5861X 57691 56789 5590642 64568 63514 62484 61479 60496 59536 58596 57675 56774 5589243 64550 63496 62468 61463 60480 59520 58580 57660 56759 5587744 64532 63479 62451 61446 60464 59504 58565 57645 56745 5586245 64514 63462 62434 6x429 60448 59488 58549 57630 56730 55848

46 64497 63444 62417 61413 60432 59472 58534 57615 56715 5583347 64479 63427 6 2 4 0 a 61396 604x6 59457 58518' 57600 56700 5581948 64461 634IO 62383 61380 60399 59441 58503 57584 56685 5580449 64443 63392 62366 61363 60383 59425 58487 57569 56670 5579050 64426 63375 62349 61347 60367 59409 58473 57554 56656 55775

51 64408 63358 62332 6x330 60351 59393 58456 57539 56641 5576152 64390 63340 62315 613x4 60335 59378 58441 57524 56626 5574653 64373 63323 62298 61297 60319 59362 58425 57509 56611 5573254 64355 6 3 3 0 6 62282 6x281 60303 59346 58410 57494 56596 5571755 64337 6 3 2 8 9 62265 6x264 60286 59330 58395 57479 56582 55703

56 64320 6327X 62248 6x248 60270 59314 58379 57463 56567 5568857 64302 63254 62231 6x231 60254 59299 58364 57448 56552 556745 8 64284 63237 62214 61215 60238 59283 58348 57433 56537 5565959 64267 6 3 2 2 0 62197 61198 60222 59267 58333 57418 5 6 5 2 2 5564560 1 64249 6 3 2 0 2 62x80 61x02 60206 59251 58317 57403 56508 5 5 6 3 0

hcv 13


t 6 0 ° 5 1 ° 5 2 ° 5 3 ° 6 4 ° 5 5 ° 5 6 ° 5 7 ° 6 8 ° 0 9 °o 5 5 6 3 0 5 4 7 7 0 5 3 9 2 7 5 3 1 0 0 5 2 2 8 8 5 1 4 9 1 5 0 7 0 8 4 9 9 4 0 4 9 1 8 4 4 8 4 4 2X 5 5 6 1 6 5 4 7 5 6 5 3 9 1 3 5 3 0 8 6 5 2 2 7 4 5 1 4 7 8 5 0 6 9 6 4 9 9 2 7 4 9 1 7 2 4 8 4 3 02 5 5 6 0 1 5 4 7 4 2 5 3 8 9 9 5 3 0 7 2 5 2 2 6 1 5 1 4 6 5 5 0 6 8 3 4 9 9 1 4 4 9 1 5 9 4 8 4 1 8

3 5 5 5 8 7 5 4 7 2 8 5 3 8 8 5 5 3 0 5 9 5 2 2 4 8 5 1 4 5 2 5 0 6 7 0 4 9 9 0 2 4 9 1 4 7 4 8 4 0 54 5 5 5 7 2 5 4 7 1 4 5 3 8 7 1 5 3 0 4 5 5 2 2 3 4 5 1 4 3 8 5 0 6 5 7 4 9 8 8 9 4 9 1 3 5 4 8 3 9 35 5 5 5 5 8 5 4 6 9 9 5 3 8 5 7 5 3 0 3 1 5 2 2 2 1 5 1 4 2 5 5 0 6 4 4 4 9 8 7 6 4 9 1 2 2 4 8 3 8 x

55543 5 4 6 8 5 5 3 8 4 3 5 3 0 1 8 5 2 2 0 8 5 1 4 1 2 5 0 6 3 1 4 9 8 6 4 4 9 1 1 0 4 8 3 6 9D 55529 5 4 6 7 1 5 3 8 3 0 5 3 0 0 4 5 2 1 9 4 5 1 3 9 9 5 0 6 1 8 4 9 8 5 1 4 9 0 9 7 4 8 3 5 67 35515 5 4 6 5 7 5 3 8 1 6 5 2 9 9 1 5 2 1 8 1 5 1 3 8 6 5 0 6 0 5 4 9 8 3 8 4 9 0 8 5 4 8 3 4 4o 55500 5 4 6 4 3 5 3 8 0 2 5 2 9 7 7 5 2 1 6 7 5 1 3 7 3 5 0 5 9 2 4 9 S 2 6 4 9 0 7 2 4 8 3 3 29

1 0 55486 5 4 6 2 9 5 3 7 8 8 5 2 9 6 3 52154 5 1 3 6 0 5 0 5 7 9 4 9 8 1 3 4 9 0 6 0 4 8 3 2 0I

55471 5 4 6 1 4 5 3 7 7 4 5 2 9 5 0 52141 5 1 3 4 6 5 0 5 6 6 4 9 8 0 0 4 9 0 4 7 4 8 3 0 755457 5 4 6 0 0 5 3 7 6 0 5 2 9 3 6 5 2 1 2 7 5 1 3 3 3 5 0 5 5 4 4 9 7 8 8 4 9 0 3 5 4 8 2 9 5

1 2 55442 5 4 5 8 6 53746 5 2 9 2 2 5 2 1 1 4 5 1 3 2 0 5 0 5 4 1 4 9 7 7 5 4 9 0 2 3 4 8 2 8 3x3 55428 5 4 5 7 2 5 3 7 3 2 5 2 9 0 9 5 2 1 0 1 5 x 3 0 7 5 0 5 2 8 4 9 7 6 2 4 9 0 1 0 4 8 2 7 1X415 55414 5 4 5 5 8 5 3 7 1 9 5 2 8 9 5 5 2 0 8 7 5 x 2 9 4 5 0 5 1 5 4 9 7 5 0 4 8 9 9 8 4 8 2 5 8

55399 5 4 5 4 4 5 3 7 0 5 5 2 8 8 2 5 2 0 7 4 5 x 2 8 1 5 0 5 0 2 4 9 7 3 7 4 8 9 8 5 4 8 2 4 61 6 55385 5 4 5 3 0 5 3 6 9 1 5 2 S 6 8 5 2 0 6 1 5 x 2 6 8 5 0 4 8 9 4 9 7 2 4 4 8 9 7 3 4 8 2 3 417 55370 5 4 5 1 6 5 3 6 7 7 5 2 8 5 5 5 2 0 4 7 5 x 2 5 5 5 0 4 7 6 4 9 7 1 2 4 8 9 6 0 4 8 2 2 2

55356 5 4 5 0 1 5 3 6 6 3 5 2 8 4 1 5 2 0 3 4 5 1 2 4 2 5 0 4 6 4 4 9 6 9 9 4 8 9 4 8 4 8 2 1 0*92 0 55342 5 4 4 8 7 5 ^ 6 4 9 5 2 8 2 7 5 2 0 2 X 5 1 2 2 9 5 0 4 5 1 4 9 6 8 7 4 8 9 3 6 4 8 x 9 7

55327 5 4 4 7 3 5 3 6 3 6 5 2 8 1 4 5 2 0 0 7 5 3 2 1 5 5 0 4 3 8 4 9 6 7 4 4 8 9 2 3 4 8 1 8 52 1 55313 5 4 4 5 9 5 3 6 2 2 5 2 8 0 0 5 1 9 9 4 5 1 2 0 2 5 0 4 2 5 4 9 6 6 1 4 8 9 x 1 4 8 1 7 32 2 55299 5 4 4 4 5 5 3 6 0 8 5 2 7 8 7 5 1 9 8 1 5 1 1 8 9 5 0 4 1 2 4 9 6 4 9 4 8 8 9 8 4 8 1 6 123 5 5 2 8 4 5 4 4 3 1 53594 5 2 7 7 3 5 1 9 6 7 5 1 1 7 6 5 0 3 9 9 4 9 6 3 6 4 8 8 8 6 4 8 x 4 9$*42 5 5 5 2 7 0 5 4 4 1 7 5 3 5 8 0 5 2 7 6 0 5 1 9 5 4 5 1 1 6 3 5 0 3 8 7 4 9 6 2 3 4 8 8 7 4 4 8 x 3 6

55255 5 4 4 0 3 53567 ‘ 5 2 7 4 6 5 1 9 4 1 5 1 1 5 0 5 0 3 7 4 4 9 6 1 1 4 8 8 6 1 4 8 1 2 42 o 55241 5 4 3 8 9 53553 5 2 7 3 2 5 1 9 2 7 5 1 1 3 7 5 0 3 6 1 4 9 5 9 8 4 8 8 4 9 4 8 1 1 227„ o 55227 5 4 3 7 5 53539 5 2 7 1 9 5 1 9 * 4 5 1 1 2 4 5 0 3 4 8 4 9 5 8 6 4 8 8 3 6 4 8 1 0 02 0 55212 5 4 3 6 1 53525 5 2 7 0 5 5 1 9 0 1 5 1 x 1 1 5 0 3 3 5 4 9 5 7 3 4 8 8 2 4 4 8 0 8 82930 55198 5 4 3 4 7 53511 5 2 6 9 2 5 1 8 8 8 5 1 0 9 8 5 0 3 2 2 4 9 5 6 0 4 8 8 1 2 4 8 0 7 6

5 5 1 8 4 54332 53498 5 2 6 7 8 5 1 8 7 4 5 1 0 8 5 5 0 3 1 0 4 9 5 4 8 4 8 7 9 9 4 8 0 6 331 5 5 1 6 9 54318 53484 5 2 6 6 5 5 1 8 6 1 5 1 0 7 2 5 0 2 9 7 4 9 5 3 5 4 8 7 8 7 4 8 0 5 132 55155 54304 53470 5 2 6 5 1 5 1 8 4 8 5 1 0 5 9 5 0 2 8 4 4 9 5 2 3 4 8 7 7 5 4 8 0 3 933 55141 5439° 53456 5 2 6 3 8 5 1 8 3 5 5 1 0 4 6 5 0 2 7 1 4 9 5 1 0 4 8 7 6 2 4 8 0 2 73435 55127 54276 53442 5 2 6 2 4 5 1 8 2 1 5 1 0 3 3 5 0 2 5 8 4 9 4 9 8 4 8 7 5 0 4 8 0 1 5

nfL 5 5 1 1 2 5 4 2 6 2 53429 5 2 6 1 1 5 x 8 0 8 5 1 0 2 0 5 0 2 4 6 4 9 4 8 5 4 8 7 3 7 4 8 0 0 33 0 55098 5 4 2 4 8 53415 5 2 5 9 7 51795 5 1 0 0 7 5 0 2 3 3 4 9 4 7 2 4 8 7 2 5 4 7 9 9 037-ft 5 5 0 8 4 54234 53401 5 2 5 8 4 5 x 7 8 z 5 0 9 9 4 5 0 2 2 0 4 9 4 6 0 4 8 7 1 3 4797830 5 5 0 6 9 5 4 2 2 0 53387 5 2 5 7 0 5 x 7 6 8 5 0 9 8 1 5 0 2 0 7 4 9 4 4 7 4 8 7 0 0 4 7 9 6 6394 0 55055 5 4 2 0 6 53374 5 2 5 5 7 51755 5 0 9 6 8 5 0 1 9 4 4 9 4 3 5 4 8 6 8 8 47954

55041 54192 5 3 3 6 0 5 2 5 4 3 51742 5 0 9 5 5 5 0 1 8 2 4 9 4 2 2 4 8 6 7 6 479424 1 5 5 0 2 6 54178 53346 5 2 5 3 0 51729 5 0 9 4 2 5 0 1 6 9 494X0 4 8 6 6 3 479304 2 55012 5 4 1 6 4 53332 5 2 5 1 6 51715 5 0 9 2 9 5 0 1 5 6 49397 4 8 6 5 1 4 7 9 1 843 54998 5 4 1 5 0 53319 5 2 5 0 3 5 1 7 0 2 5 0 9 1 6 5 0 1 4 3 49385 4 8 6 3 9 479064445 54984 54136 53305 5 2 4 8 9 5 1 6 8 9 5 0 9 0 3 5 0 1 3 1 49372 4 8 6 2 6 47893

54969 54122 53291 5 2 4 7 6 5 1 6 7 6 5 0 8 9 0 5 0 x 1 8 4 9 3 6 0 4 8 6 1 4 4 7 8 8 14 0 54955 5 4 x 0 8 53278 5 2 4 6 2 5 1 6 6 2 5 0 1 0 5 49347 4 8 6 0 2 4 7 8 6 9% 54941 54094 5 3 2 6 4 5 2 4 4 9 5 1 6 4 9 5 0 8 6 4 5 0 0 9 2 49334 4 8 5 9 0 47§5740 54927 5 4 0 8 0 5 3 2 5 0 5 2 4 3 6 5 1 6 3 6 5 0 8 5 1 5 0 0 8 0 49322 4 8 5 7 7 47845495 0 54912 5 4 0 6 6 53236 5 2 4 2 2 5 1 6 2 3 5 0 8 3 8 5 0 0 6 7 49309 4 8 5 6 5 47833

5 4 8 9 8 5 4 0 5 2 53223 5 2 4 0 9 5 1 6 1 0 5 0 8 2 5 5 0 0 5 4 49297 48553 4 7 8 2 15* 5 4 8 8 4 5 4 0 3 8 5 3 2 0 9 5 2 3 9 5 5 X 5 9 6 5 0 8 1 2 5 0 0 4 1 4 9 2 8 4 4 8 5 4 0 4 7 8 0 9

5 4 8 7 0 5 4 0 2 4 53195 5 2 3 8 2 5 1 5 8 3 5 0 7 9 9 5 0 0 2 9 4 9 2 7 2 4 8 5 2 8 4779753 5 4 8 5 5 5 4 0 1 1 5 3 1 8 2 5 2 3 6 8 5 1 5 7 0 5 0 7 8 6 5 0 0 1 6 49259 4 8 5 1 6 4778504-55 5 4 8 4 1 5 3 9 9 7 5 3 1 6 8 5 2 3 5 5 51557 5 0 7 7 3 5 0 0 0 3 49247 4 8 5 0 3 47772

cfi 5 4 8 2 7 5 3 9 8 3 53154 5 2 3 4 2 51544 5 0 7 6 0 4 9 9 9 1 49234 4 8 4 9 1 4 7 7 6 05 4 8 1 3 5 3 9 6 9 53141 5 2 3 2 8 5 1 5 3 0 5 0 7 4 7 4 9 9 7 8 4 9 2 2 2 4 8 4 7 9 47748

3 /cfi 54799 53955 53127 5 2 3 1 5 5 1 5 1 7 5 0 7 3 4 4 9 9 6 5 4 9 2 0 9 4 8 4 6 7 477363 °Kq 5 4 7 8 4 53941 53113 5 2 3 0 1 5 1 5 0 4 5 0 7 2 1 4 9 9 5 2 4 9 1 9 7 48454 47724DO 54770 53927 53100 5 2 2 8 8 5 1 4 9 1 5 0 7 0 8 4 9 9 4 0 4 9 1 8 4 4 8 4 4 2 47712

lxvi

T e r n a r i P r o p o r t i o n a l L o g a r i t h m s

t 60° 0 1 ° 62° 6 3 ° 64° 6 5 ° 06° 67° 08° 69° OO 71°o 47712 46994 46288 45593 44909 44236 43573 42920 42276 4x64a 4IO17 40401X 47700 46982 46276 45582 44898 44225 43562 42909 42266 41632 4IO07 4039155 47688 46971 46265 45570 44687 442x4 4355 X 42898 42255 4x621 40997 4038 X3 47676 46959 46253 45559 44875 44203 43540 42887 42244 4161X 40986 403714 47664 46947 46241 45547 44864 44x91 43529 42877 42234 4x600 40976 403615 47652 46935 46230 45536 44853 44x80 435x8 42866 42223 4x590 40966 403506 47640 46923 462x8 45524 44841 44169 43507 42855 42213 4x579 40955 403407 47628 46911 46206 45513 44830 44158 43496 42844 42202 41569 40945 403308 47616 46899 46195 45501 44819 44147 43485 42833 42x91 41559 40935 403209 47604 46S83 46x83 45490 44808 44136 43474 42823 42x81 41548 4O924 403x0

xo 47592 46876 46171 45478 44796 44125 43463 42812 42x70 4x538 409X4 40300

XX 47580 46864 46160 45467 44785 44x14 43452 42801 42159 41527 4O904 40289X2 47568 46852 46x48 45456 44774 44102. 4344X 42790 42149 4x5x7 40894 40279*3 47556 46S40 46137 45444 44762 44091 43431 42780 42x38 41506 40883 4026914 47544 46828 46125 45433 44751 44080 43420 42769 42x28 4x496 40873 4025915 47532 46817 461x3 4542X 44740 44069 43409 42758 42117 4x485 40863 40249

i6 47520 46805 46102 454X0 44729 44058 43398 42747 42106 4x475 40852 4023917 47508 46793 40ogo 45398 447x7 44047 43387 42737 42096 4x464 4084.2 40228x3 47496 46781 46078 45387 44706 44036 43376 42726 42085 ,.4x454 40832 4021819 47484 46769 46067 45375 44695 44025 43365 42715 42075 4x443 40821 4020820 47472 4675S 46055 45364 44684 44014 43354 42704 42064 4x433 40S1I 40x98

21 -47460 46746 46044 45353 44672 44003 43343 42693 42053 41423 ..40SQI 40x8822 47448 46734 46032 45341 44661 43992 43332 42683 42043 414x2 4079I 4017823 47436 46722 46020 45330 44650 43981 43321 42672 42032 41402 40780 40x6824 47424 467x0 46009 453x8 44639 43969 433X0 42661 42022 4I39I 40770 4015725 47412 46699 45997 45307 44627 43958 43300 42651 42011 41381 40760 40x47

26 47400 46687 45986 45295 446x6 43947 43289 42640 42000 41370 40749 4013727 47388 46675. 45974 45284 44605 43936 43278 42629 4x990 41360 40739 4012728 47376 46663 45962 45273 44594 43925 43267 42618 41979 41350 40729 4011729 47364 46652 45951 45261 44583 43914 43256 42608 41969 41339 40719 40x0730 47352 46640 45939 45250 44571 43903 43245 42597 41958 4x329 40708 40097

31 47340 46628 45928 45238 44560 43892 43234 42586 4x948 4x3x8 40698 4008732 47328 46616 459x6 45227 44549 4388t 43223 42575 41937 41308 40688 4007633 473X6 46604 45905 452X6 44538 43870 43212 42565 4x927 4x298 40678 4006634 47304 46593 45893 45204 44526 43859 43202 42554 4x916 41287 40667 4005635 47292 46581 45881 45193 44515 43848 4319X 42543 41905 41277 40657 40046

36 47280 46569 45870 45182 44504 43837 43180 42533 41895 4x266 40647 4003637 47268 46557 45858 45X70 44493 43826 43169 42522 4x884 41256 40637 4002638 47256, 46546 45847 45159 44482 438x5 43158 425x1 41874 4x246 40626 400x639 47244 46534 45835 45147 44470 43804 43147 42500 4x863 41235 40616 4000640 47232 46522 45824 45136 44459 43793 43X36 42490 4x853 4X225 40606 3999641 47220 46510 45812 45X25 44448 43782 43126 42479 41842 4x2x4 40596 3998542 47208 46499 45800 45X13 44437 43771 43115 42468 41832 41204 40585 3997543 47196 46487 45789 45X02 44426 43760 43X04 42458 4182X 4x194 40575 3996544 47185 46475 45777 45091 444x4 43749 43093 42447 4x811 41183 40565 3995545 47X73 46464 45766 45079 44403 43738 43082 42436 4x800 41x73 40555 3994546 47161 46452 45754 45068 44392 43727 43071 42426 4x789 4x162 40544 3993547 47149 46440 45743 45057 44381 437x6 43060 42415 41779 4x152 40534 3992548 47X37 4642% 4573X 45045 44370 43705 43050 42404 41768 41142 40524 399X549 47125 46417 45720 45034 44359 43694 43039 42394 41758 4x131 405X4 3990550 47113 46405 45708 45022 44347 43683 43028 42383 41747 4 I I 2X 40503 39895

Si 47101 46393 45£>97 45011 44336 43672 430X7 42372 41737 4XIII 40493 3988552 47089 46382 45685 45000 44325 43661 43006 42362 4x726 4IIOO 40483 3987453 47077 46370 45674 44988 443x4 .43650 42995 4235X 41716 4XO90 40473 3986454 47066 46358 45662 44977 44303 43639 42985 42340 41705 4XO80 40463 3985455 47054 46346 45651 44966 4429® 43628 42974 42330 4x695 4XO69 40453 39844

56 47042 46335 45639 44955 44280 43617 42963 423x9 4x684 4IO59 40443 3983457 47030 46323 45628 44943 44269 43606 42952 42308 4x674 4XO4.3 40432 3982458 47018 463IX 45616 44932 44258 43595 4294X 42298 41663 4XO38 40422 3981459 47006 46300 45605 44921 44247 43584 42931 42287 41653 41028 404x2 3980460 1 46994 46288 45593 44909 44236 43573 42920 42276 4x642 4IOI7 404OX 39794

borii


t 7 2 ° 7 3 ° 7 4 ° 7 5 ° 7 0 6 7 7 ° 7 8 ° 7 9 ° O 0 8 1 ° 8 2 ° 8 3 °O 3 9 7 9 4 39*95 38604 3 8 0 2 1 37446 3 6 8 7 8 3 6 3 1 8 3 5 7 6 5 352*8 3 4 6 7 9 3 4 * 4 6 3 3 6 1 91 3 9 7 8 4 39*85 38594 3 8 0 1 1 37436 3 6 8 6 9 3 6 3 0 9 3 5 7 5 5 3 5 2 0 9 3 4 6 7 0 3 4 * 3 7 3 3 6 x 1a 3 9 7 7 4 39*75 38585 3 8 0 0 2 37427 3 6 8 5 9 3 6 2 9 g 3 5 7 4 6 3 5 2 0 0 3 4 6 6 1 3 4 1 2 8 3 3 6 0 23 3 9 7 6 4 3 9 1 6 5 38575 3 7 9 9 2 374*7 3 6 8 5 0 3 6 2 9 0 3 5 7 3 7 35*9* 3 4 6 5 2 3 4 1 1 9 3 3 5 9 34 3 9 7 5 4 39*55 38565 3 7 9 8 3 3 7 4 0 8 3 6 8 4 1 3 6 2 8 1 3 5 7 2 8 35*82 3 4 6 4 3 3 4 * * 1 3 3 5 8 55 3 9 7 4 4 39*45 38555 3 7 9 7 3 37398 3 6 8 3 1 3 6 2 7 1 3 5 7 x 9 35*73 3 4 6 3 4 3 4 1 0 2 3 3 5 7 6

6 3 9 7 3 4 39*36 38545 3 7 9 6 3 37389 3 6 8 2 2 3 6 2 6 2 3 5 7 1 0 35*64 3 4 6 2 5 3 4 0 9 3 3 3 5 6 77 3 9 7 2 4 3 9 1 2 6 38536 3 7 9 5 4 37379 3 6 8 1 2 3 6 2 5 3 3 5 7 0 0 35X55 3 4 6 1 6 3 4 0 8 4 3 3 5 5 88 3 9 7 1 4 39**6 3 8 5 2 6 3 7 9 4 4 37370 3 6 8 0 3 3 6 2 4 4 3 5 6 9 1 35146 3 4 6 0 7 3 4 0 7 5 3 3 5 5 0Q 3 9 7 0 4 3 0 1 0 6 3 8 5 1 6 3 7 9 3 4 37360 3 6 7 9 4 3 6 2 3 4 3 5 6 8 2 35*37 3 4 5 9 8 3 4 0 6 6 3354*

IO 3 9 6 9 4 39096 3 8 5 0 6 3 7 9 2 5 3735* 3 6 7 ^ 4 3 6 2 2 5 3 5 6 7 3 3 5 x 2 8 3 4 5 8 9 3 4 0 5 8 33532

I I 3 9 6 8 4 3 0 0 8 6 38497 379*5 3734* 3 6 7 7 5 3 6 2 1 6 3 5 6 6 4 351*9 3 4 5 8 1 3 4 0 4 9 335241 2 3 9 6 7 4 3 9 0 7 6 38487 37905 37332 3 6 7 6 6 3 6 2 0 7 3 5 6 5 5 351*0 3 4 5 7 2 3 4 0 4 0 335X&13 3 9 6 6 4 3 9 0 6 6 38477 3 7 8 9 6 37322 3 6 7 5 6 3 6 x 9 7 3 5 6 4 6 35* o i 3 4 5 6 3 3 4 0 3 1 33500 .>4 3 9 6 5 3 39056 3 8 4 6 7 3 7 8 8 6 373*o 3 6 7 4 7 3 6 1 8 8 3 5 6 3 6 35092 3 4 5 5 4 3 4 0 2 2 3349815 3 9 6 4 3 3 9 0 4 6 38458 37877 37303 3 6 7 3 7 3 6 1 7 9 3 5 6 2 7 3 5 0 8 3 3 4 5 4 5 3 4 0 1 4 33489

2 6 3 9 6 3 3 39037 3844g 3 7 8 6 7 37294 3 6 7 2 8 36170 3 5 6 1 8 35074 3 4 5 3 6 3 4 0 0 5 3348o '17 3 9 6 2 3 39027 3 8 4 3 S 37857 37284 3 6 7 * 9 3 6 1 6 0 3 5 6 0 9 35065 3 4 5 2 7 3 3 9 9 6 3347*38 3 9 6 1 3 3 9 0 1 7 3 8 4 2 8 3 7 8 4 8 37275 3 6 7 0 9 3 6 1 5 1 3 5 6 0 0 35056 3 4 5 x 8 3 3 9 8 7 3346339 3 9 6 0 3 39007 38419 37838 37265 3 6 7 0 0 3 6 x 4 2 3 5 5 9 * 3 5 0 4 7 '3 4 5 0 9 3 3 9 7 8 334542 0 3 9 5 9 3 3 8 9 9 7 38409 3 7 8 2 9 37256 3 6 6 9 1 3 6 1 3 3 3 5 5 8 2 35038 3 4 5 0 0 3 3 9 7 0 33445

2 1 3 9 5 8 3 3 8 9 8 7 38399 3 7 8 1 9 37246 3 6 6 8 1 3 6 1 2 3 3 5 5 7 3 3 5 0 2 9 3 4 4 9 * 3 3 9 6 1 334372 2 3 9 5 7 3 38977 38389 3 7 8 0 9 37237 3 6 6 7 2 3 6 1 1 4 3 5 5 6 3 3 5 0 2 0 3 4 4 8 3 3 3 9 5 2 334282 3 3 9 5 6 3 3 8 9 6 8 3 8 3 8 0 3 7 8 0 0 3 7 2 2 7 3 6 6 6 3 3 6 1 0 5 3 5 5 5 4 3 5 0 1 x 3 4 4 7 4 3 3 9 4 3 334*92 4 3 9 5 5 3 3 8 9 5 8 38370 37790 372x8 3 6 6 5 3 3 6 0 9 6 3 5 5 4 5 3 5 0 0 2 3 4 4 6 5 3 3 9 3 5 334* *25 3 9 5 4 3 3 8 9 4 8 3 8 3 6 0 3 7 7 8 1 3 7 2 0 8 3 6 6 4 4 3 6 0 8 6 3 5 5 3 6 34993 3 4 4 5 6 3 3 9 2 6 33402

2 6 3 9 5 3 3 3 8 9 3 8 3835* 3777* 37199 3 6 6 3 4 3 6 0 7 7 3 5 5 2 7 34984 3 4 4 4 7 339*7 3339327 3 9 5 2 3 3 8 9 2 8 38341 3 7 7 6 1 3 7 * 8 9 3 6 6 2 5 3 6 0 6 8 3 5 5 1 8 34975 3 4 4 3 8 33908 333852 8 3 9 5 1 3 3 8 9 1 8 38331 37752 3 7 1 8 0 3 6 6 x 6 3 6 0 5 9 3 5 5 0 9 3 4 9 6 6 3 4 4 2 9 33899 333762 9 3 9 5 0 3 3 8 9 0 8 38321 37742 37* 7 1 3 6 6 0 6 3 6 0 5 0 3 5 5 0 0 34957 3 4 4 2 0 3 3 8 9 1 333673 0 3 9 4 9 3 3 8 8 9 9 3 8 3 1 2 37733 3 7 1 6 1 3 6 5 9 7 3 6 0 4 0 3 5 4 9 * 34948 3 4 4 1 1 3 3 8 8 2 33359

3 1 3 9 4 8 3 3 8 8 8 9 38302 3 7 7 2 3 37*52 3 6 5 8 8 3 6 0 3 1 3 5 4 8 1 34939 3 4 4 0 3 33873 333503 2 3 9 4 7 3 38879 3 8 2 9 2 377*3 3 7 1 4 2 3 6 5 7 8 3 6 0 2 2 3 5 4 7 2 34930 3 4 3 9 4 3 3 8 6 4 3334*33 3 9 4 6 4 3 8 8 6 9 38282 37704 3 7 1 3 3 3 6 5 6 9 3 6 0 1 3 35463 34921 3 4 3 8 5 33856 3333334 3 9 4 5 4 3 8 8 5 9 38273 37694 37*23 3 6 5 6 0 3 6 0 0 3 3 5 4 5 4 349*2 3 4 3 7 6 33847 3332435 3 9 4 S 3 8 8 4 9 3 8 2 6 3 37685 3 7 1 1 4 3 6 5 5 0 3 5 9 9 4 3 5 4 4 5 34903 3 4 3 6 7 33838 333*5

3« 3 9 4 3 4 38839 38253 37675 37*04 3 6 5 4 1 3 5 9 8 5 3 5 4 3 6 34894 3 4 3 5 8 3 3 8 2 9 3330737 3 9 4 2 4 3 8 8 3 0 3 8 2 4 4 37665 37095 3 6 5 3 2 3 5 9 7 6 3 5 4 2 7 34885 3 4 3 4 9 3 3 8 2 0 3329838 3 9 4 1 4 3 8 8 2 0 38234 37656 37085 3 6 5 2 2 3 5 9 6 7 3 5 4 1 8 3 4 8 7 6 3 4 3 4 0 3 3 8 x 2 3 3 2 8 939 3 9 4 0 4 3 8 8 1 0 3 8 2 2 4 37646 3 7 0 7 6 3 6 5 * 3 3 5 9 5 7 3 5 4 0 9 34867 3 4 3 3 2 3 3 8 0 3 3 3 2 8 14 0 3 9 3 9 4 3 8 8 0 0 38215 37637 37067 3 6 5 0 4 3 5 9 4 8 3 5 4 0 0 34858 3 4 3 2 3 33794 33272

4 1 3 9 3 8 4 38790 3 8 2 0 5 3 7 6 2 7 37057 3 6 4 9 4 3 5 9 3 9 3 5 3 9 1 34g49 3 4 3 * 4 33785 332634« 3 9 3 7 4 3 8 7 8 1 38195 3 7 6 1 8 3 7 0 4 8 3 6 4 8 5 3 5 9 3 0 3 5 3 8 1 3 4 8 4 0 3 4 3 0 5 33777 3325543 3 9 3 6 4 3 8 7 7 1 3 8 1 8 6 3 7 6 0 8 37038 3 6 4 7 6 3 5 9 2 1 3 5 3 7 2 3 4 8 3 * 3 4 2 9 6 33768 3 3 2 4 644 3 9 3 5 4 3 8 7 6 1 3 8 1 7 6 37599 3 7 0 2 9 3 6 4 6 7 359** 3 5 3 6 3 3 4 8 2 2 3 4 2 S 7 33759 3323745 • 3 9 3 4 4 38751 3 8 i 6 6 37589 37019 3 6 4 5 7 35902 3 5 3 5 4 3 4 8 1 3 3 4 2 7 8 33750 33229

4 6 3 9 3 3 4 38741 38156 37579 3 7 0 1 0 3 6 4 4 8 3 5 8 9 3 3 5 3 4 5 3 4 8 0 4 3 4 2 7 0 33742 3 3 2 2 047 3 9 3 2 4 3873* 38147 37570 3 7 0 0 1 3 6 4 3 9 3 5 8 8 4 3 5 3 3 6 34795 3 4 2 6 1 33733 3 3 2 1 14 8 3 9 3 1 4 3 8 7 2 2 38X37 37560 3 6 9 9 1 3 6 4 2 9 35875 3 5 3 2 7 34786 3 4 2 5 2 33724 3320349 3 9 3 0 4 3f 7*2 3 8 1 2 7 3755* 3 6 9 8 2 3 6 4 2 0 35865 353*8 34777 3 4 2 4 3 337*5 33*9450 3 9 2 9 4 3 8 7 0 2 3 8 r r 8 3754* 36972 3 6 4 * * 35856 35309 34768 3 4 2 3 4 33707 •33*86

5l 3 9 2 8 4 3 8 6 9 2 3 8 1 0 8 37532 3 6 9 6 3 3 6 4 0 1 35S47 35300 34759 3 4 2 2 5 3 3 6 9 8 33*7752 3 9 2 7 4 3 8 6 8 2 3 8 0 9 8 37522 36953 3 6 3 9 2 3 5 8 3 8 3 5 2 9 1 34750 3 4 2 x 7 3 3 6 8 9 33*6853 3 9 2 6 4 3 8 6 7 3 3 8 0 8 9 37513 36944 3 6 3 8 3 3 5 8 2 9 3 5 2 8 2 3474X 3 4 2 0 8 3 3 6 8 1 3 3 x 6 054 3 9 2 5 4 3 8 6 6 3 3 8 0 7 9 37503 36935. 3 6 3 7 4 3 5 8 2 0 35273 34732 34*99 3 3 6 7 2 33*5*55 3 9 2 4 5 3 8 6 5 3 3 8 0 6 9 37494 3 6 9 2 5 3 6 3 6 4 3 5 8 1 0 3 5 2 6 4 34723 34190 3 3 6 6 3 3 3 1 4 2

56 3 9 2 3 5 3 8 6 4 3 3 8 0 6 0 37484 3 6 9 1 6 3 6 3 5 5 3 5 8 0 1 35254 3 4 7 1 5 34*81 33654 33*3457 3 9 2 2 5 3 8 6 3 3 3 8 0 5 0 37474 3 6 9 0 6 3 6 3 4 6 35792 35245 34706 34*72 3 3 6 4 6 3 3 1 2 55 8 3 9 2 1 5 3 8 6 2 4 3 8 0 4 0 37465 3 6 8 9 7 3 6 3 3 6 35783 3 5 2 3 6 3 4 6 9 7 3 4 * 6 4 33637 331X759 3 9 2 0 5 3 8 6 1 4 3 8 0 3 1 37455 3 6 8 8 8 3 6 3 2 7 35774 3 5 2 :2 7 3 4 6 8 8 34*55 3 3 6 2 8 3 3 * 0 86 0 39*95 3 8 6 0 4 3 8 0 2 1 37446 36878 3 6 3 1 8 35765 3 5 2 1 8 34679 34*46 3 3 6 1 9 33099

l x v i i î


ì 8 4 ° 8 5 ° 8 8 *

í-00 8 8 ° 8 8 ° 0 0 ° 9 1 ° 9 2 ° 0 8 ° 0 4 ° 9 5 °o 33099 32585 32077 3X575 3X079 30588 30103 29623 29148 28679 282x4 277551 33091 32577 32069 3X567 3 io 7r 30580 30095 29615 29141 28671 28207 277472 33082 32568 32061 * 31559 31063 30572 30087 29607* 29x33 28663 28199 277403 33073 32560 32052 3X550 3X054 30564 30079 29599 29125 28656 28191 277324 33065 32551 32044 3X542 31046 30556 30071 295CH 29117 .28648 28184 277245 33056 32543 32035 3X534 31038 30548 30063 29583 29109 28640 28176 277x7

6 33048 32534 32027 3X525 3x030 30539 30055 29575 29101 28632 28168 277097 33039 32526 32019 31517 3x021 30531 30047 29567 29P93 28625 28161 277028 33O3O 32517 32010 3x509 3x013 30523 30039 29560 29086 28617 28153 276949 33022 32509 32002 3X501 31003 305X5 30031 29552 29078 28609 28145 27686

IO 33013 32500 31993 31492 30997 30507 30023 29544 29070 28601 28138 27679

i t 33005 32492 31985 3x484 30989 30499 30015 29536 29062 28593 28130 2767112 33996 32483 31977 3X476 30980 3049X 30007 29528 29054 28586 28122 2766413 32987 32475 31968 3x467 30972 30483 29999 29520 29046 28578 281x4 2765614 32979 32466 31960 3X459 30964 30475 29991 29512 29038 28570 28107 2764815 32970 32458 3*95* 3X45X 30956 30466 29983 29504 29031 28562 28099 27641

i6 32962 32449 31943 31442 30948 30458 29975 29496 29023 28555 28091 2763317 32953 32441 31935 3X434 30939 30450 29967 29488 29015 28547 28084 27626x8 32944 32432 31926 3x426 3093x 30442 29958 29480 29007 28539 2S076 2761819 32936 32424 31918. 31418 30923 30434 29950 29472 28999 28531 28068 2761020 32937 32415 3X909 31409 30915 30436 29943 29464 28991 28524 28061 27603

21 32919 32407- 31901 31401 30907 3O4I8 29934 29456 28984 28516 28053 2759522 329IO 32398* 31S93 31393 30898 30410 29926 29448 28976 28508 28045 2758823 32QO2 32390 31884 31384 30890 30302 29918 29441 28968 28500 28038 2758024 32893 32381 31876 3X376 30882 30393 29910 29433 28960 28493 28030 27572«5 32884 32373 3x867 3x368 30874 30385 29902 29425 28952 28485 28022 27565

26 32876 32365 3x859 31360 30866 30377 29894 294x7 28944 28477 28015 2755727 32867 32356 3x851 3X35X 30857 30369 29886 29409 28937 28469 28007 2755028 32859 32348 31842 31343 30849 30361 29878 29401 28929 28462 27999 2754229 3285O 32339 3x834 3X335 30841 30353 29870 29393 28921 28454 27992 2753430 32842 3233* 31826 3X326 30833 30345 29862 29385 28913 28446 27984 27527

32833 32322 31817 3X318 30825 30337 29854 29377 28905 28438 27976 275x932 32824 32314 /3X809 3 X3 X0 3o8r7 30329 29846 29369 28897 28431 27969 275x233 32816 32305 31801 3X302 30808 30321 29838 29361 28890 28423 27961 2750434 32807 32297 3X792 3X293 30800 303X3 . 29830 29354 28882 28415 27953 2749735 32799 32288 31784 3x285 30792 30305 29822 29346 28874 28407 27946 27489

36 3279O 32280 3X775 3X277 30784 30296 29814 29338 28866 28400 27938 2748137 32782 322>I 31767 31269 30776 30Í88 29806 29330 28858 28392 27930 2747438 32773 32263 3X759 31260 30768 30280 29798 29322 28851 28384 27923 2746639 32765 32255 31750 3x252 30759 30272 29790 293x4 28843 28376 279x5 2745940 32756 32246 31742 3x244 30751 30264 29782 29306 28835 28369 27908 2745 X

41 32747 32238 31734 3X236 30743 30256 29775 29298 28827 28361 27900 2744442 32739 32229 3i 725fc 3X227 30735 30248 29767 29290 28819 28353 27892 2743643 32730 3*221 3X717 31219 30727 30240 29759 29282 288x1 28346 27885 2742944 32722 32212 31709 3 r 2 i i 30719 30232 2975 X 29275 28804 28338 27877 2742145 327X3 32204 31700 31203 30710 30224 29743 29267 28796 28330 27869 27413

46 32705 32195 31692 31X94 30702 30216 29735 29259 28788 28322 27862 2740647 32696 32187 31684 31186 30694 30208 29727 29251 28780 283x5 27854 2739848 32688 32179 31675 31178 30686 30200 297x9 29243 28772 28307 27846 2739X49 32679 32X70 31667 31170 30678 30192 297x1 29235 28765 28299 27839 2738350 32671 32162 3x659 31161 30670 30183 29703 29227 28757 28292 27831 27376

5* 32662 32153 3x650 3X153 30662 30175 29695 29219 28749 28 284 27824 2736852 32654 32145 31642 3X145 30653 30167 29687 29211 28741 28276 27810 27360

* 53 32645 32136 3x634 31137 30645 30159 29679 29204 28733 28268 27808 2735354 32636 32128 31625 3x128 30637 30151 29671 29196 28726 28261 27801 2734555 32628 32120 31617 31120 30629 30X43 29663 29188 287x8 28253 27793 27338

56 326I9 3 2 m 3X609 31112 30621 30135 39655 29180 28710 28245 27785 2733057 326H 32103 31600 31104 30613 30127 29647 29172 28702 28238 27778 2732358 32602 32694 31592 3x095 30605 30119 29639 29164 28695 28230,. 27770 273X559 32594 32086 3X584 31087 30596 301x1 29631 29156 28687 28222 27763 2730860 32585 32077 3X575 3x079 30588 30103 29623 29148 28679 28214 27755 27300


t 96° 97° 98° 99° 100o101° 102° 103*104° 105° 106° 107°0 27300 26850 26405 25964 25527 25095 24667 24244 23824 23408 22997 2258gI 27293 26843 26397 25956 25520 25088 24660 24237 23817 23401 22ggo 225822 27285 26835 26390 25949 255*3 25081 24653 24229 23810 23395 22983 225753 27278 26828 26382 25942 25506 25074 24646 24222 23803 23388 22976 225694 27270 26820 26375 25934 25498 25066 24639 24215 23796 23381 22969 225625 27262 26813, 26368 25927 25491 25059 24632 24208 23789 23374 22963 22555

6 27255 26805 26360 25920 25484 25052 24625 24201 23782 23367 22956 225487 27247 26798 26353 25913 25477 25045 24618 24194 23775 23360 22949 225428 27240 26790 26346 25905 25469 25038 24610 24187 23768 23353 22942 225359 27232 26783 26338 25898 25462 25031 24603 24180 23761 23346 22935 22528

IO 27225 26776 26331 25891 25455 25024 24596 24*73 23754 23339 22g28 22521

i i 27217 26768 26323 25883 25448 25016 24589 24166 23747 23333 22922 2251512 27210 26761 26316 25876 25440 25009 24582 24159 23740 23326 22915 2250813 27202 26753 26309 25869 25433 25002 24575 24152 23734 23319 22908 22501*4 27195 26746 26301 25863 25426 24995 24568 24145 23727 233*2 22901 22494*5 27187 26738 26294 25854 254*9 24988 24561 24138 23720 23305 22894 22488

16 27180 26731 26287 25847 25412 24981 24554 24*31 237*3 23298 2288S 2248117 27172 26723 26279 25840 25404 24973 24547 24*24 23706 23291 22881 22474i8 27165 26716 26272 25832 25397 24966 24540 24117 23699 23284 22874 2246719 27157 26709 26265 25825 25390 24959 24533 24110 23692 23278 22867 2246120 27150 26701 26257 25818 25383 24952 24526 24103 23685 23271 22SÓO 22454

21 27142 26694 26250 25810 25376 24945 24518 24096 23678 23264 22854 2244722 27135 26686 26242 25803 25368 24938 24511 24089 23671 23257 22847 2244023 27127 26679 26235 25796 25361 24931 24504 24082 23664 23250 22840 2243424 27120 26671 26228 25789 25354 24923 24497 24075 23657 23243 22833 2242725 27112 26664 26220 25781 25347 24916 24490 24068 23650 23236 22826 22420

26 27105 26656 26213 25774 25339 24909 24483 24061 23643 23229 22819 2241327 27097 26649 26206 25767 25332 24902 24476 24054 23636 23223 22813 2240728 27090 26642 26198 25759 25325 24895 24469 24047 23629 23216 22806 2240029 27082 26634 26191 25752 253*8 24888 24462 24040 23623 23209 22799 2239330 27075 26627 26184 25745 253** 24881 24455 24033 23616 23202 22792 22386

31 27067 26619 26176 25738 25303 24874 24448 24026 23609 23*95 22785 2238032 27060 26612 26169 25730 25296 24866 24441 24019 23602 23188 22779 2237333 27052 26605 26162 25723 25289 24859 24434 240x2 23595 23181 22772 2236634 27045 26597 26154 25716 25282 24853 24427 24005 23588 S3175 22765 2235935 27037 26590 26147 25709 25275 24845 24420 23998 23581 23168 22758 22353

36 27030 26582 26140 25701 25267 24838 24413 2399* 23574 23161 22752 2234637 27022 26575 26132 25694 25260 24831 24405 23984 23567 23154 22745 2233938 27015 26567 26125 25687 25253 24824 24398 23977 23560 23147 22738 2233339 27007 26560 26118 25680 25246 24817 24391 23970 23553 23140 22731 2232640 27000 26553 26110 25672 25239 24809 24384 23963 23546 23*33 22724 22319

41 26992 26545 26103 25665 25231 24802 24377 23956 23539 23127 22718 2231242 26985 26538 26096 25658 25224 24795 24370 23949 23533 23120 22711 2230643 26977 26530 26088 25650 25217 24788 24363 23942 23526 23**3 22704 2229944 26970 26523 26081 25643 25210 24781 24356 23935 235*9 23106 22697 2229245 26962 26516 26074 25636 25203 24774 24349 23928 23512 23099 22690 22286

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Ixxi


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Ixxii

A SELECTION FROMRIDER’S PUBLICATIONSTHE N EW M ANUAL OF ASTROLOGY. In four books.

B o o k I .—-T h e Language o f the H eavens. B o o k I I .— T h e R ead in g o f a H oroscop e . B o o k I I I .— T h e M easure o f T im e. B o o k IV .— H in d u Astrology. W ith supplem ent including the Progressive H oroscop e , M eth od o f C alcu lating the Progressed H oroscop e , T h e Lunar Equivalent, T h e D iurnal H o ro sco p e and N ote on the Planetary Periods. W ith set o f Tables. B y c ‘ S e p h a r i a l . ” R ev ised and Enlarged Edition. D em y 8vo, cloth gilt, 2 6 4 pp., 1 0 s. 6 d. net.

THE K ABA LA OF NUMBERS. P a r t I .— A H a n d b ook dealing with the Traditional Interpretation o f N um bers and their Predictive Value. B y “ S e p h a r i a l , ” A uthor o f “ A M anual o f O ccu ltism ,” “ K abalistic A strology,” etc. 20 4 pp ., crow n 8vo, c lo th gilt, 2s. 6 d. net.

“ *The Kabala of Numbers5 Is an astonishing book that takes you back, in spirit, even further than Homer or the * Arabian Nights/”—Manchester Guardian*K ABALA OF NUMBERS. Part I I . By “ Sepharial, /

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handbook dealing with the occult and prophetic powers of numbers. , . . Wonderfully interesting.”—Sheffield Dazly Telegraph.YOUR FORTUNE IN YOUR NAME, OR KABALISTIC

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TH E T A R O T CARDSAN D TH EIR IN TER PR ETATIO N

THE TAROT OF THE BO H EM IAN S : T h e M ost A n cien t B ook in the W orld , for the E xclusive U se o f Initiates. B y P a p u s . Translated from the F rench b y A . P . M o r t o n . N ew edition, revised throughout, w ith in trodu ction by A . E . W a i t e . Crow n 8vo, ornam ental c lo th gilt, gilt tops, 38 4 pp., profusely illustrated, 6s. net.

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A SELECTION FROMRIDER’S PUBLICATIONSC O S M IC S Y M B O L IS M . By “ Sepharial,” A uthor o f “ A

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Contents.Part I .—T he Occult Sciences, comprising: Astrology—Palmistry—

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The Wellby Shilling Series of Scientific and Occult Handbooks

T h e volum es o f this series are designed to make an appeal to the intelligent reader who is too busy to devote his tim e to m ore exhaustive treatises.AM ANUALOFH YPN OTISM . B y H. E r n e s t H u n t , A uthor

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T h e favourable reception which this Primer on H an d-R eadin g has m et with has led to the Publishers bringing it out in a more attractive style, and uniform in size and scope with other volumes on similar subjects.ASTROLOGY. H ow to M ake and R ea d your own H o r o

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W IL L IA M R I D E R & S O N , L I M I T E D ,8 Paternoster Row, London, E.C.

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