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Gift of The People of the United States
Through the Victory Book Casnpaign
/ (A. L A. — A. R. C — U.S. 0.)
To the Armed Forces and Merchant Marine
Th» Old Corner Book
Store, Inc.
DIRECTIONAL ASTROLOGY
DIRECTIONAL ASTROLOGY
TO WHICH IS ADDED A DISCUSSION OF
PROBLEMATIC POINTS AND A COM
PLETE SET OF TABLES NECESSARY
FOR THE CALCULATION OF
ARCS OF DIRECTION
BY
SEPHARIAL
AUTHOR OF
"COSMtC SYMBOLtSM," " A MANUAL OF ASTROLOGY,"
"A MANUAL OF OCCULTtSM," " THE KABALA OF NUMBERS,"
"KABALtSTtC ASTROLOGY," ETC-
PHILADELPHIA
DAVID McKAY
604-608 SOUTH WASHINGTON SQUARE
1 92 1
rharvarpN
UNlVERSiT A
^J_IBRAR> J
PRtNTED tN GREAT BRtTAtN BY
NEtLL AND CO., LTD.,
EDtNBURGH.
PREFACE
A 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.
v
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 scientific
ally 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. Both
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 vii
and obscureth himself, while the ignorant pass on
and are hurt."
I have used a well-known and thoroughly
authenticated horoscope for purposes of illustra
tion, 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 under
standing of the subject ; and until these are clearly
apprehended and understood, the student need go
no further.
To save further expense and trouble, my pub
lishers 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 differ
ence 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
viii DIRECTIONAL ASTROLOGY
student of " Directional Astrology " will have
mastered the preliminary task of setting a horo
scope 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 Prognostic Astronomy,
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 complete,
and so I hope it will be found.
SEPHARIAL.
CONTENTS
CHAP. PAOE
1. Astronomical Definitions 1
2. Example Horoscope 11
3. Directions in Mundo 18
4. Directions in the Zodiac .... 25
5. Zodiacal and Mundane Parallels . . 32
6. Order op Directing 37
7. Effects of Directions 40
8. Planetary Indicators and the Measure of
Time 49
9. Illustration 57
10. Ptolemy and Placidus 62
11. Directions under Poles . . . .71
12. The Part of Fortune 81
13. Lunar Parallax and Semi-diameter . . 86
14. Lunar Equations 94
15. Cuspal Distances 99
16. Suggested Method of True Directing . 107
17. Conclusion 117
Appendix—Tables of Sines, Tangents, etc. . 125
Tables of Right Ascension and
Ascensional Difference . . 173
Tables of Proportional Logarithms . 181
iz
Directional Astrology
CHAPTER I
ASTRONOMICAL DEFINITIONS
The 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 de
fined 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.
Right 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 geo
graphical latitude.
Meridian Distance is the distance of a celestial
1
2 DIRECTIONAL ASTROLOGY
body from the midheeven 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 semi-
arc 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 semi-
arc 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 in
creased 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 3
its 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.
4 DIRECTIONAL ASTROLOGY
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 posi
tions 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 relation
ships, 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 Text
book of Astrology or The New Manual of Astrology.
Promittors.—These are the planets Neptune, Ura
nus, 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 calcu
lations 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 5
merely the addition of a number to itself a given
number of times, while division is merely subtrac
tion a number of times. Then by means of a
proportional circle we can multiply and divide any
arc by simple addition and subtraction. The com
plement 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 loga
rithm 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 mathe
matical 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 con
stitutes celestial latitude. All other bodies have
6 DIRECTIONAL ASTROLOGY
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. from
Aries 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 7
Now 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 ephe
meris 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 from
Libra 0 = 47° 54' . cos. 9-82635
Its latitude is 5° 9' . cos. 9-99824
Sum . . cos. 9-82459
Moon's declination,
22° 7' . 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'
8 DIRECTIONAL ASTROLOGY
Note.—If we take the arithmetical complement
of the log. cos. of the declination 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 sees. 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° 0' 0'
R.A. of I.C. = 99° 49' 0"
The upper meridian is called the Midheaven
ASTRONOMICAL DEFINITIONS 9
(medium cceli) 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 be 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 be in the South-west
quarter of the heavens, and therefore nearer to the
upper than the lower meridian. Its meridian dis
tance must therefore be taken from this point.
Meridian distance of Moon= 55° 56'
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 declina
tion. The sum is the log. sine of the ascen
sional difference of that planet under the latitude
of birth.
Thus :
R.A. of M.C.
R.A., Moon
= 279° 49'
= 223° 53'
10 DIRECTIONAL ASTROLOGY
Uniformly, add this ascensional difference to
90° when the planet's R.A. is less than 180°,
and subtract it from 90° if the planet's R.A. is
more than 180°. The result is the diurnal semiarc
of that planet. By subtracting this from 180°
you will have the nocturnal semiarc.
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 following pages. The scheme
will now be ready for the practice of directing.
CHAPTER II
EXAMPLE HOROSCOPE
Fob the purpose of illustrating the method of
directing by proportional semiarcs, I have selected
the horoscope of John Ruskin, whose Fors Glavi-
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
12 DIRECTIONAL ASTROLOGY
Speculum.
Planets. Lat. Deolin. E.A.Merid.
Dist.
Semi-
are.
Hor.
Aro.
O O 0 O / o / o /
Sun — 16 13 S. 321 12 108 44 110 1 1 17
IN. 17 21 52 51Moon 5 25 39 N. 120 50 2 30
23 S. 21 34 S. 47 46 51 60 11 13 20Mercury . 0 296
10 N. 18 10 S. 6 26 10 65 37 39 27Venus 5 276
55 S. 45 S. 6 49 10 59 53 10 43Mars 0 21 299
21 S. 26 S. 37 52 41 62 3 9 22Jupiter . 0 20 302
Saturn 56 S. 6 54 S. 348 81 2 98 45 17 431 54
Uranus . 6S. 23 24 S 12 53 57 1 44 80 262 49
13 N. US. 47 17 59 5 41 14Neptune . 1 22 267 51
EXAMPLE HOROSCOPE 13
of 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 Ruskin, at all events, as a man of fearless
integrity of thought.
In this horoscope we have a remarkable illustra
tion 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
14 DIRECTIONAL ASTROLOGY
horizon. These are indications of success and dis
tinction 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 be 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 HOEOSCOPE 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 direc
tions 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 Ascen
dant. 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.
Zodiacal 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.
16 DIRECTIONAL ASTROLOGY
All directions are formed by the rotation of the
Earth 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 ap
parent 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 declina
tions and semiarcs, as well as their meridian dis
tances, 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 " direct " and
" converse."
Direct directions are such as are formed by one
EXAMPLE HOROSCOPE 17
body 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 Ruskin, if we
bring the Sun to the place of Jupiter, or Mars,
or Mercury, or Venus, these would be direct direc
tions, 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 im
print) 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 19
Suppose 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 simul
taneously 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
20 DIRECTIONAL ASTROLOGY
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 0-21381
arith. comp 9-78619
Is to its cuspal distance, 1° 17', . 2-14693
So 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 21
distance, namely 0° 37', and there remains the arc
of direction : Moon, 135°, Sun in mundo, 25° 48£'.
Another example : Bring the Sun in the ex
ample 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 dis
tance 52° 41', 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
S.A. Sun, 69° 59' 0-41028
Sun's prop, distance =10° 34' log. 1-23144
Arith. comp. 9-53747
1-28369Cusp, distance, 9° 22'
Sun to horizon = 1° 17'
Arc of direction = 11° 51'
Sun conj. Jupiter to.
22 DIRECTIONAL ASTROLOGY
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 direc
tions 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 Uranus 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
24 DIRECTIONAL ASTROLOGY
same as the declination of the planet, but it is not
true in regard to any other pole or geographical
latitude. That is why we take the proportion of
the semiarcs in finding the cuspal distances of
planets. The test is this : If we take the oblique
ascension of a planet, that is, exactly one-third of its
semiarc from the horizon, it should have the same
oblique ascension as the cusp of the twelfth house,
but by adding 60 to the right ascension of the Mid-
heaven to get the oblique ascension of that house,
we shall find that if the planet has any other declina
tion than 23° 27' there is a difference between the
two results. It cannot, therefore, be truly said that
a planet is in mundane sextile aspect to the Mid-
heaven when it is one-third of its semiarc above
the horizon, inasmuch as its position in the prime
vertical does not then coincide with the cusp of
the twelfth house ; but it may be said to correspond
with that cusp on the general proposition that all
circles are equal to one another and therefore that
all quadrants are equal, and in practice it is found
that one-third of a semiarc corresponds with one-
third of the prime vertical, and this was allowed
by Placidus, who was the first exponent of this
system of mundane directions.
CHAPTER IV
DIRECTIONS IN THE ZODIAC
These are calculated on the same principle as
mundane directions, that is to say, by proportion
of the semiarcs ; 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 semi-
arc 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.
26
26 DIRECTIONAL ASTROLOGY
Example.—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 semi-
arc = 59° 9'.
Prop. log. meridian dist.
„ semiarc
Constant log. Aquarius
Prop. log. Sun's semiarc
. 52° 35' = -53442
. 59° 9' = -48332
. 0° 20' = -051 10
. 69° 59' = -41028
Sun's prop. dist. 62° 13' = -46138
Take from Sun's merid. dist. 71° 16'
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 27
this derive the semiarc. Subtract the propor
tional 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° 28', the
semisquare in Capricorn 26° 28', and the con
junction in Sagittarius 11° 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° 0'
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' -42932
Radical dist. of Sun from M.C. 71° 16'
Arc of direction, Sun sextile M.C. = 4° 17'
28 DIRECTIONAL ASTROLOGY
2. 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° 20' 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 dis
tance ;
which take from 71° 16' Sun's radical
distance,
remains 15° 53' the arc of di
rection 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'=arcof direction.
DIRECTIONS IN THE ZODIAC 29
These 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'. There
fore, 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 con
stant 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
30 DIRECTIONAL ASTROLOGY
stant log. due to these and add to it the log. of the
nocturnal semiarc of the Sun. The sum will be 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
semiarc.
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 of
the planet Neptune from 1800 to 1900 a.d. is published by
Messrs Foulsham & Co. Price Is.
DIRECTIONS IN THE ZODIAC 31
The 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° 56' =26° 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 con
junction with the Midheaven in mundo.
We may now pass to another series of directions.
CHAPTER V
ZODIACAL AND MUNDANE PARALLELS
In astrology the parallel of declination is deemed
of the same significance and value as the conjunc
tion, 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' of
declination. 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 PAEALLELS 33
direct 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° l'-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 same
side of the meridian, one south and the other
3
34 DIRECTIONAL ASTROLOGY
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 pro
portionate to their relative semiarcs.
Rule.—As half the sum of their semiarcs is to
half the sum of their meridian or horizontal dis
tances, 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 DIRECTIONAL ASTROLOGY
Arith. comp. 9-62439
Merid. dist. of Moon . 50° 21'
„ Saturn . 81° 2'
2)131° 23'
65° 41' prop. log. 43782
Semiarc, Moon . . .52° 51' „ 53223
Proportional dist., Moon . 45° 48' „ -59444
Radical distance . 50° 21'
Moon rapt, par., Saturn = 4° 33' arc of direction.
Note.—In all cases where the Midheaven (meri
dian) and Ascendant (horizon) are employed as
sectors, the Sun and Moon are employed as signi-
ficators. 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 VI
ORDER OF DIRECTING
The 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
38 DIRECTIONAL ASTROLOGY
(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.
(/) 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 Directions
Follow 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
OEDEE OF 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 know
ledge 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 : " Teach me the
number of my days, that I may apply my heart
to wisdom."
CHAPTER VII
EFFECTS OF DIRECTIONS
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 neces
sary 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 " effects "
of directions as if these latter had a direct dynamic
result upon the character and actions of an indivi
dual. 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 planet
ary configuration and an event which attends it.
They admit no possible intervention, amelioration,
or extenuation. Che sard sard, 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 41
are 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 born when they are born
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 illustra
tion 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
42 DIRECTIONAL ASTROLOGY
date their facts and figures to popular concepts
by a discreet use of a melange of spurious philo
sophy. They forever quote the effete adage : " The
wise man rules his stars, the fool obeys them " ;
and that other which says : " The stars incline
but 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 experi
ence, 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 " signs
of the times " as the driver of a locomotive regards
the signals, not as " 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
44 DIRECTIONAL ASTROLOGY
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 drink—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 accommo
date 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 under
stand why, in the following statement of the
" Effects of Directions," I have pursued the
common practice of attributing certain results or
EFFECTS OF DIRECTIONS 45
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 posi
tion, 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
46 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 associa
tions.
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, hin
drances, 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 sub
ject himself. It is thus associated more particularly
with the Midheaven.
EFFECTS OF DIRECTIONS 47
The Moon
denotes 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 horo
scope 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 Mid-
heaven, Ascendant, Sun, and Moon, are the signi-
ficators, because they signify such persons and
things in the life of the subject as are capable
48 DIRECTIONAL ASTROLOGY
of being affected by the conflict of human circum
stance.
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 signi
ficators.
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
PLANETARY INDICATORS AND THE MEASURE
OF TIME
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 (6) the nature of the planet involved.
In this light, therefore, it may be said that
Neptune in good aspect indicates events of a bene
ficial nature connected with the use of the faculties
or some special faculty, and frequently in connec
tion with a form of art ; benefits from unexpected
sources coming mysteriously to the subject ; unseen
49 4
50 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, am
bush ; an involved state of affairs ; nervous leak
age 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, advancement ;
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 commensu
rate good results, travelling, the executive powers
are stimulated, and much profitable work is done.
52 DIRECTIONAL ASTROLOGY
Benefits accrue from military men, business con
nected 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 con
ditions 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 domes
tic 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
PLANETARY INDICATORS 53
affectional nature is stimulated and the health
is good.
In evil aspect Venus denotes sorrows, disappoint
ments, bereavements, grief, and losses, domestic
and social troubles, and hurts associated with young
women or children. Venus was anciently known
as the Lesser Benefic, 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 connec
tion 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
54 DIRECTIONAL ASTROLOGY
quent loss. Hurts from women. Some public
affronts may be suffered. Changes are unfortun
ate, and best avoided.
The Measure of Time
In 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 incre
ment 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
PLANETARY INDICATORS 55
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 con
clusion that the secondary direction is, after all,
the important one and deserving of primary con
sideration. A very little experience will show that
it is not so, for, unless there are concurrent primaries
in operation, lunar or secondary directions fre--
quently 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
56 DIRECTIONAL ASTROLOGY
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 lati
tudes, 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 maturity
when the accumulated results of a man's labour
were heaped around him. (See Transits.)
The following illustrations will, however, suffi
ciently 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
ILLUSTRATION
In 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 work
ing out the following arcs of direction in the present
instance.
John Ruskin leapt into fame and became a
" lion " 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
58 DIRECTIONAL ASTROLOGY
Moon oppos. Venus mundo conv. 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 Ruskin'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 Cam
bridge in May 1867 under the appropriate direc
tions of
Ascendant trine Venus mundo 48° 2'
Ascendant sextile Moon zod. 48° 22'
He was elected Slade Professor of Fine 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 59
Ascendant square Saturn zod. con. 51° 59'
Moon square Mars mundo . . 52° 0'
Sun par. Uranus zod. con. . . 52° 0'
Ascendant square Jupiter mundo . 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'
Sun opposition Uranus mundo . 69° 14'
Moon rapt par. Saturn . . .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 ASTROLOGY
In 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 arc of 59° 14', and we find the appro
priate directions—
Midheaven square Moon zod. . 58° 58'
Ascendant opposition Moon . . 59° 19'
Saturn semisq. Ascendant . . 59° 22'
Midheaven conjunction Saturn zod. 59° 42'
These illustrations will doubtless serve for all
practical purposes, and they can be worked out at
leisure by those who wish to exercise themselves
in this art.
Other methods than that here illustrated must
claim our attention, inasmuch as they have con
sistently been advocated by various authors. There
are, moreover, several points which may be con
sidered as debateable, and these also have to be
considered before our work is rendered complete.
We must therefore pass on.
CHAPTER X
PTOLEMY AND PLACIDTTS
It is generally conceded that the system of direct
ing which has so far occupied our attention first
originated as a measure of time in the mind of
Claudius Ptolemy, the famous geographer, mathe
matician 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 informa
tion 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 born at Pelusium in Egypt, and
became a brilliant disciple of the Alexandrian
School. It appears that he was born 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 transla
tion is by Cooper. Placidus showed that Ptolemy
recognised two sets of directions arising out of
64 DIRECTIONAL ASTROLOGY
two 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. hi.,
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 ' 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 65
distances from both these, each equatorial degree being
taken 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 ex
position ; and as these semiarcs are regulated by
the latitude of the place of birth and the corre
sponding 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 converse
directions, whether in mundo or in the zodiac.
5
66 DIRECTIONAL ASTROLOGY
That he recognises the mundane position of a body
as distinguished from the apparent place of its
" degree " of longitude is obvious from his men
tioning both in the same sentence ; and we dis
tinguish ourselves between the mundane and zodi
acal conjunctions only by reference to the body of
the planet in the first instance and its longitude
in the other case.
To Claudius Ptolemy, therefore, may rightly be
accorded the honour of having set astrologers upon
the right track with regard to the correct measure
of time by reference to the equatorial degrees
separating one body from another, or one body
from the longitude or aspect of another, as seen
from the place of birth.
There is little doubt, from the illustrations of
his method that Ptolemy gives, that he uses the
" ascensional " times in all cases due to the latitude
of the place of birth ; and this method serves very
well not only for directions to the Ascendant and
Descendant, but also for intermediate positions
when the planets are in the same or different
quarters and on the same side of the meridian, for
then their arcs may be measured with great facility
and approximate accuracy from the Tables of
Houses alone.
Illustration
1. Bring the Sun to the place of Mars in the
horoscope of Ruskin.
PTOLEMY AND PLACIDUS 67
h. m.
The sidereal time on the Midheaven
when Mars' place rises is . . 15 49
That when the Sun rises is . 16 44
Difference in R.A. on the Mid-
heaven 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 44
That when Venus' place rises is . 14 35
Difference . .29
This 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 point
rises 17 30
S.T. on Midheaven when Sun rises . 16 44
Difference . 0 46
This gives an arc of 11° 30'.
4. Bring the Moon to the opposition of Venus
in zodiac.
68 DIRECTIONAL ASTROLOGY
The 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 Ascen
sion 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 direc
tion. Thus :
h. m.
S.T. on Midheaven when Venus long.
rises . . . . 14 35
S.T. on Midheaven when the 12th of
Aquarius rises . . . 16 30
Difference . 1 55
This gives an arc of 28° 45'.
5. Bring the Sun to the opposition of Uranus in
zodiac.
Take the opposite degree of the zodiac to that
held by Uranus, and bring the Sun to it by oblique
arc.
h. m.
S.T. when Gemini 23° 25' rises . . 21 21
S.T. when Sun rises . . . 16 44
Difference . 4 37
This gives an arc of 69° 15'.
6. Bring Sun to par. Uranus in zodiac direct.
PTOLEMY AND PLACIDUS 69
The decimation of Uranus is 23° 24', which
answers to that of Cancer 4°. Find the arc be
tween this and the Sun.
h. m.
S.T. on Midheaven when Cancer 4°
rises 22 6
S.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
" 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,
70 DIRECTIONAL ASTROLOGY
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 an
other, 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 XI
DIRECTIONS UNDER POLES
This method has been much advocated, and especi
ally by Mr R. C. Smith, the first of the almanac
writers under the pen-name of " Raphael." It con
sists 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 Planet
Take its R.A., declination, and semiarc.
Then say :
As the semiarc is to 90°,
So is its meridian distance
To 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
71
72 DIRECTIONAL ASTROLOGY
its 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 dis
tance (below) 108° 44', the semiarc 110° 1', and
the declination 15° 13'.
Semiarc 110° 1' . . prop. log. 0-21381
Arith. comp. . 9-78619
Quadrant of 90° . . . . 0-30103
Meridian distance . 108° 44' 0-21891
Difference . 88° 57' 0-30613
Asc. diff. under pole 19° 47' log. sine 9-52951
Sun'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' in
dicates the place of the circle of position from the
DIRECTIONS UNDER POLES 73
plane 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 meri
dian stands as equator and the Ascendant and De
scendant 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 R.A. according to the rule (see "Defini
tions," Chapter I.).
Example.—Direct the Sun in the example horo
scope 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—
74 DIRECTIONAL ASTROLOGY
Pole of Sun, 51° 13' . . log. tang. 10 09493
Dec. 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 ad
judged the most popular contribution to the theory
DIRECTIONS UNDER POLES 75
and practice of primary directions that I have been
able to make.
Example.—Direct the Sun under its own pole to
the opposition of Uranus in the zodiac.
The Sun's pole is 51° 13'. Therefore take in
hand the Tables of Houses or the Tables of Oblique
Ascension for latitude 51° 13' N.
The opposition of Uranus falls in Gemini 23° 25'.
h. m.
S.T. on Midheaven when Gemini
23° 25' rises . . . 21 21
S.T. when Sun's place rises . 16 43
4 38
This 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'.
h. m.
S.T. on Midheaven with Sun
rising . . . . 16 43
S.T. on Midheaven with Capri
corn 5° 49' rising . . 14 35
Arc of direction, Sun conj. Venus
zodiac =difference . .28
This is equivalent to 32° 0'.
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.
76 DIRECTIONAL ASTROLOGY
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 sub
sequent 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 experi
ence of the best artists, the import of primary
directions being accelerated or retarded by the
DIRECTIONS UNDER POLES 77
secondary 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 sub
tract 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 ascen
sion 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, there
fore, 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', equi
valent 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 58
3. Direct the Sun to Mercury in zodiac.
h. m.
Sun's rising as before . . . 16 41
Mercury's longitude rising . . . 15 39
Arc of direction = 50° 30' 1 2
DIRECTIONS UNDER POLES 79
4. Direct the Sun to Venus' longitude.
h. m.
Sun's rising as above . . . 16 41
Place of Venus rising . . . 14 30
Arc of direction = 32° 45' . .2 11
5. Direct the Sun to Neptune in zodiac.
h. m.
Sun's rising as before . . . 16 41
Neptune's long, rising . . . 13 59
Arc of direction =40° 30' . .2 42
6. Direct the Sun to Uranus in zodiac.
h. m.
Sun's rising as above . . . .16 41
Uranus' long, rising . . . 13 36
Arc of direction =46° 15' . .35
7. Direct the Sun to opposition of Moon in zodiac.
h. m.
Sun's rising as before . . . 16 41
Rising of Capricorn 27° 8', S.T. . . 15 47
Arc of direction = 13° 30' . .0 54
The various aspectr 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 DIRECTIONAL ASTROLOGY
up to the Sun's place. This involves knowing the
pole of Saturn.
We may also bring Saturn to the Ascendant
under its own pole. But if we were to bring the
Sun to Saturn under the Sun's pole, that would
be a prenatal direction, for the Sun cannot go back
from the position it has attained and sink below
the eastern horizon. We have therefore no alter
native but to regard these directions as invalid,
or to admit the thesis already suggested, that in
these directions, made contrary to the apparent
motion of the bodies in the heavens, we are dealing
with the locabsed impress of the planet at the
moment of birth, which impress is carried by the
Earth up the western heavens and down the eastern
heavens, so that the Sun's localised imprint is here
carried down to the place of Saturn. And this is
conformable to the theory of directions under the
poles of the significators.
CHAPTER XII
THE PART OF FORTUNE
For 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 ASTROLOGY
Its 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
denned 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. xii.) : " 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 PART OP 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' Moon to 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 Leo, and therefore close to the Moon.
Placidus says : " 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
84 DIRECTIONAL ASTROLOGY
add to the right ascension of the Moon 120° 17',
and the sum is the right ascension of the Part of
Fortune 119° 0'.
The right ascension of the imum cadi 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. Arc. Cusp. Dist.
119° 0' 49° 4' 52° 61' 3° 47' 3° 47'
These elements at once suggest that the pole
of the Part of Fortune can be found, and direc
tion made by the Part of Fortune in mundo and
THE PART OF FORTUNE 85
zodiac 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 demon
strable 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 " land, territory,
a field."
CHAPTER XIII
LUNAR PARALLAX AND SEMI-DIAMETER
Among the problems modernly confronting the
student of directional astrology, that of the hori
zontal parallax of the Moon is perhaps one of the
most important and at the same time most per
plexing.
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 in
considerable, 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 87
therefore 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 ob
served 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 ASTROLOGY"
Table op Anomaly.
Epoch 1800, Jan. Qi 0h 0m=98 20° 20'.
Years. Add Days. Add. Anom. Hor. Par.
• 0 • 0 i O
1 2 28 43 1 0 13 4 0 0 54
2 6 27 27 2 0 26 8 6 55
3 8 26 10 3 1 9 12 12 55
4 0 7 67 4 1 22 16 18 55
6 3 6 40 5 2 5 19 24 55
8 6 0 24 6 2 18 23 1 0 55
7 6 4 7 7 3 1 27 6 55
8 0 I5 64 8 3 14 31 12 65
9 3 14 38 9 3 27 36 18 55
10 6 13 21 10 4 10 39 24 56
20 1 9 46 11 4 23 43 2 0 56
40 2 19 32 12 5 6 47 6 56
50 0 2 63 13 6 19 51 12 56
60 3 29 18 14 6 2 66 18 57
70 10 12 39 16 6 16 58 24 57
80 5 9 3 16 6 29 2 3 0 67
90 11 22 24 17 7 12 6 6 58
100 6 18 49 18 7 26 10 12 58
Months. Add 19 8 8 14 18 59
20 9 21 18 24 59
January 0 0 0 21 9 4 22 4 0 59
February 1 16 1 22 9 17 26 6 69
March . j 60
April . 3 6 61 23 10 0 30 12 60
May . 4 7 48 24 10 13 34 18 60
June 0 22 49 25 10 26 37 24 60
July . 6 24 46
26 11 9 41 o 60
August 8 9 47 27 11 22 45 6 60
September . 9 24 48 28 0 5 49 12 60
October 10 26 46 29 0 18 63 18 61
November . 0 11 46 30 1 1 67 24 61
December 1 13 42 31 1 16 1 6 0 61
LUNAR PARALLAX AND SEMI-DIAMETER 89
Example.—Find the Moon's anomaly for 8th
February 1819, and the corresponding horizontal
parallax.
8 O '
Epoch 1800 . . . 9 20 20
Add 19 . . 9 27 59
February . . . 1 15 1
8 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
90 DIRECTIONAL ASTROLOGY
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
Log. cosine 50° 21'. . 9-80489
Log. cosine 46° 29'. . 9-83794
Log. cosine 63° 52'. . 9-64283
And as 90° is to 55', so is 63° 52' 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.
The semi-diameters of the Sun and Moon have
often been resorted to in order to accommodate a
directional arc to the date of an event. Allowing,
as is undoubtedly the case, that primary directions
have an orb of influence within the limits of which
LUNAR PARALLAX AND SEMI-DIAMETER 91
it 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, inas
much 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 cor
rection 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 ASTROLOGY
important 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 be 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 cen-
trality 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 1' 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
LUNAR PARALLAX AND SEMI-DIAMETER 93
approximately twenty-seven one-hundredths of
the parallax. Therefore multiply the parallax by
27 and divide by 100. Thus, when the parallax is
54', the semi-diameter of the Moon is 54 x 27 -f 100
=14$', and when the parallax is 60, the semi-
diameter is 60 x 27 -f 100 = 16'.
The Moon directed to the opposition of the
Ascendant in the example horoscope works out at
2° 30' ; but as the horizontal parallax of the Moon
is 55', its semi-diameter will be nearly 15', and
therefore the direction would read :
Asc. oppos. Moon in mundo, first contact 2° 15'
middle 2° 30'
„ „ last contact 2° 45'
thus giving a possible range of 30', or six months for
the duration of this indication. This may help
to account for the variability that has been noticed
in regard to lunar directions, and possibly we may
also have to consider taking the parallax into
account. The solar directions will be affected by
semi-diameter of the Sun, but not appreciably by
parallax.
CHAPTER XIV
LUNAR EQUATIONS
Under this head I propose to examine a problem
of some interest which appears to have escaped
general recognition, but which may very well be
considered with the questions of parallax and semi-
diameter 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 conjunc
tion with the nadir, which direction is known as
" Midheaven opposition Moon in mundo." It is
measured by the arc of the Moon's meridian dis
tance, 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, Ascen
dant, 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 95
in 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 Ruskin, whereas the arc M.C. opposi
tion Moon in mundo, 50° 21', exactly coincided
with the election of Ruskin to the Slade Professor
ship of Fine Art, a distinction which brought him
into the highest position in his sphere of life.
96 DIRECTIONAL ASTROLOGY
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 find
ing the pole, is 31° 32'
Its right ascension 120° 17'
Its oblique descension under its pole . 151° 49'
Add 180° 0'
Oblique ascension of opposition Moon =331° 49'
Then for Saturn's oblique ascension under the
same pole—
Pole of Moon
tang. . . 10 03712
Tang. Saturn's
decl. . . 9 08283
Ascl.diff. Saturn
sine . . 9- 11995= 7° 31'
R.A. of Saturn . . 348° 54'
OA. of Saturn . . 356° 25' under Moon's
pole.
O.A. of Moon's oppos. . 331° 49'
Arc. of Moon oppos. Saturn= 24° 36'
LUNAR EQUATIONS 97
This corresponds with Ruskin'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 R.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 Ruskin'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 Ruskin 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 in
constant, but why also they appear to have a more
7
98 DIRECTIONAL ASTROLOGY
durable 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 dura
tion ; and during this period, which naturally in
creases 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 arc of direction must
therefore be increased or decreased at the rate of
2' for every degree of the arc of direction. Thus an
arc of 39° 15' requires 1° 18J'.
CHAPTER XV
CTJSPAL DISTANCES
When 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 semi-
arc 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 Ruskin'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. Let us see.
90
100 DIRECTIONAL ASTROLOGY
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 ascen
sion 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.
Poles of Houses.
Lat. Cusps of 3, 5, 9, 11. Cusps of .2, 6, 8, 12.
0 0 / o /45 18 57 34 11
46 19 37 35 10
47 20 19 36 10
48 21 2 37 10
49 21 46 38 12
50 22 33 39 15
51 23 21 40 16
52 24 12 41 24
53 25 5 42 31
54 26 1 43 39
55 26 59 44 48
CUSPAL DISTANCES 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 Sun
under pole of 12th . . 13° 36'
Right ascension of Sun . . 321° 12'
Oblique ascension of Sun under
pole of 12th . . 334° 48'
Oblique ascension of cusp of the
12th 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
102 DIRECTIONAL ASTROLOGY
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 " 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 :
CUSPAL DISTANCES 103
O.A. of Sun 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 ASTROLOGY
And 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 R.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 arc 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 semi-
arc method now commonly in vogue.
But what appears of most vital importance as
CUSPAL DISTANCES 105
a 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 mathe
matics, 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 deal
ing with an approximation which, although useful
106 DIRECTIONAL ASTROLOGY
and facile, is not mathematically correct. Rather
than that bad habits should become popular, I
have undertaken a somewhat lengthy demonstration
of this point, which I consider to be now settled
beyond further debate.
CHAPTER XVI
SUGGESTED METHOD OF TRUE DIRECTING
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 con
clusions point. For 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
108 DIRECTIONAL ASTROLOGY
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 cal
culation 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 (astro
nomical) 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 N 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 HNZH will be the upper meridian and
HSNH 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° 0', 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 109
A 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 inter
mediate cusp of a house.
Rules for Directing
Rule 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
110 DIRECTIONAL ASTROLOGY
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. R.A. Declin. Pole. Ascen. Diff.
0 / O 4 o /
Sun . 321 12 15 13 51 13
10-09493
19 47
9-43458
Moon 120 17 25 39
9-68142
50 21
10-08147
35 24
Mercury 296 47 21 34 44 55 23 13
9-69688 9-99885
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 47 13 23 44
9-57119 10-03355
Saturn 348 54 6 54
9-08283
45 56 •
10-01423
7 11
Uranus 262 49 23 24
9-63623
3 13
8-74904
7 27
Neptune 267 47 22 14 3 48 9 20
9-61148 8-82147
SUGGESTED METHOD OP TRUE DIRECTING 111
Examples
Direct 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' declina
tion 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' 9-71125
Decl., log. tang. . 15° 13' 9-43458
Asc. diff. log. sine 8° 2' 9-14583
R.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'.
112 DIRECTIONAL ASTROLOGY
The declin. of Uranus
Pole of Moon
Asc. diff. under pole
Uranus' R.A. .
is 23° 24' tan. 9-63623
. 50° 21' tan. 0-08147
. 31° 28' sine 9-71770
. 262° 49'
O.D.Uranus . . 231° 21' underMoon'spole.
O.D. of Moon. . 155° 41' „
Arc of direction= 75° 40' Uranusconj. Moon
in 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 ascen
sion of the opposite point.
Oblique descension of Moon under Venus'
pole 134° 37'
Oblique descension of Venus under same
pole 105° 49'
Arc of Direction, Moon oppos. Venus
mundo 28° 48'
SUGGESTED METHOD OF 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 mun
dane 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
114 DIRECTIONAL ASTROLOGY
249° 56'
155° 45'
94'
249'
11'
56'
Right ascension of the Midheaven
Oblique descension of Moon under
its own pole
Moon's distance from Midheaven
westward ....
Added to R.A. of M.C.
Oblique ascension of the parallel,
eastward 344° 7'
Oblique ascension of Saturn under
Moon'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 under
its own pole, 51° 13' . . . 340°
Subtract 180 . . .180°
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'
59'
0'
SUGGESTED METHOD OF 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 Arcs
This 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 " 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 arc 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
Sun'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
116 DIRECTIONAL ASTROLOGY
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 chronocrater,
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" = 59' 15" in arc, but its
increase in longitude is 60' 43", and this being an
excess 1' 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
l\ 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 my
mind that the method of directing under the pole
of the significator 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 XVII
CONCLUSION
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 discrep
ancies are those due to incorrect use of words in
describing the facts. The term " 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
118 DIRECTIONAL ASTROLOGY
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 supple
mented this by a speculum drawn according to the
principles laid down, so that by mere inspection
of the same, and very little figuring, all directions
in mundo can be calculated. For 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 pro
portion 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
INCLUDING TABLES OF LOGARITHMIC
SINES, TANGENTS, ETC., TABLES OF
RIGHT ASCENSION, DECLINA
TION, AND ASCENSIONAL
DIFFERENCE, AND TER
NARY PROPORTIONAL
LOGARITHMS
9
TABLES OF LOGARITHMIC
SINES, TANGENTS, ETC.
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00 r-sOrt rt rt
rt NO »*- M 00 f-.NO
00 OO 00 00 oo oo 00 00 00 oo oo oo 00 00 00 oo oo oo
o o ooo oo 00 oO oo oo 00 00 OO 00 M
ON Os ON ONw
00 M ir
rt tvrt NO m -DO 00 vS
a NOQMO00
WO -* ON -qn-oo M00 fOO r-. usm rt H
r- rt r-rt O r-.rtrt-
NO M tSN Q ijam
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oWN M o •- -w-i rt t- us rt gM NO NO so3 r § 8 8 §88! 8 8 8 8 8 8 888 888 38 8 8 8 8 8 8 8
o goo 5 oo o 0
95U obbo o c> O O 0 o o o O o o 3 O O O O o o o o O O O 0 e-
£ rt"rt rt H rt. H rtrt rtrtrt rtrtrt rt rtrt rt rt rt - rt rtrt rt rtrt rt rtrtrt
Tan6ent.| ^- ON -*ON •-.
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us o soos
m so «rt ^- r—
r~, rt t— 00 -* ON
00-00 00-
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+- Os w Qrt •s Qrt r- r-os•^-•O ONrt rt rt
00 f-soom tI-sofs M t-sON ON OsOs os Os
ON rt t+- r-'ft JTr- r- oo
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Os Os ON OOs Os Os OOs Os Os 0
Os os osOs Os OsOs Os Os
ON ON ONON ON ONON bs ON
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Os os os . ON ON ONON ON ON
ON ON ONON ON ON
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m cs M rt m to m « to
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m m rt MmmrtMMrtMM
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rt m mM ro
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}
NO NOM OsNO
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13 rt rt J rt rt r rtrt rtrtrt rtrtrt-D rtrt rtrtrt t rt rt 0 rtrtrt rtrtrtc
J -4- os 0 vN ONO OO i^
us r- NO ~ r-^rt r-. « oo ON Osn-sNO 00
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rJ-Too--^III P-
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ON ON ONON ON ON
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Os Os Os OsON ON ON ON Os Os ■ ON Os ON Os Os Os Os Os ON ON ON ON Os Os os Os ON Os
O - rt i m * io so 1-.00 ON O « rt to tj- usNO r— 00 ON ort -rt rt" M
-rt rt rtrtrt
12
TABLES OF RIGHT ASCENSION,
DECLINATION, AND ASCENSIONAL
DIFFERENCE
VHaiqctNaexiuy
•qtoqjojsuieseq}aiv
■J)ICI,:>SVPna'ojpsa8tU.'seuyjoeeaSsp
suitjssqt0}o08TPPawqni°'VH»HJ0J
•33Q U90SV->H uopuoT ui,8u]uzj»g [OOdZ9A|1
0 0 0 t 0 t 0 t 0 J
0 0 0 0 0 0 0 0 0 0 0
T 0 fZ 0 99 0 08 0 T8 0 38
2 0 8f T 09 T 0 T 2 t *
8 T 2T 2 Qf T 08 T 88 T 78
f T 98 8 0* Z 0 2 f E 6
9 T 69 f 98 2 08 2 98 2 If
9 2 82 9 08 8 0 8 9 8 81
7 Z 9 92 8 08 8 7S 8 5*
8 8 OT 7 T2 f 0 \ 8 f 7T
6 8 fS 8 91 08 f 68 Bf
OT 8 89 6 TT 9 0 9 10 9 21
TT T2 OT L 9 08 9 Tt 9 89
2T 9* TT 2 9 0 9 2T 9 98
8T 9 8 TT 89 9 08 9 8f- 9 L9
n 9 18 2T 89 I 0 I n L 6Z
9T 9 99 8T 6f L 62 7 9f 8 T
9T 9 81 fl ff 7 69 8 9T 8 88
7T 9 If 9T Of 8 62 8 9t> 6 5
8T 7 f 9T 98 8 89 6 LI 6 18
6T 7 27 7T 28 6 82 6 8t 0T 8
O2 7 6* 8T 82 6 Li OT 81 0T 0*
T2 8 2T 6T fZ OT 72 OT 6* TT 2T
22 8 W O2 O2 OT 99 TT 61 TT
82 8 79 T2 17 TT 92 TT 6* ST 9T
fZ 6 61 ZZ 81 TT 99 2T O2 2T 94
92 6 Tf SZ OT 2T fZ 2T 09 81 7T
92 OT 8 fZ 9 2T 89 ST O2 8T 6*
72 OT W 92 8 8T zz 8T 09 n O2
8S OT 9* 92 0 8T T9 H O2 n 19
62 TT 7 92 79 n O2 fX 09 91. 22
08 n 6S 7s 99 fl 8? 9T 6T ex 89
mi
oiaaoogonasnnnvj. aoNaaaJHiQlanoisKaosy
•qtoqjojauiBS3q;axe
.JB?ao3svpireonipaGsqj,'sainexjoeaaSsp
aiiiaSeqi,o;o08tdpuoidjoogjo-yH«SS
-8»a -udssvW uopuoTa»t3uJtnjtg
[oodjven
0 0 t 0 4 0 4 O i 0 i
0 TT 65 75 99 n St 91 61 91 89
T TT 09 85 39 91 LX 91 6* 91 82
2 51 01 65 6f 91 9f 91 61 9T *9
8 3t 18 08 Lf 91 n 91 Sf 71 fZ
f 31 19 18 9f 91 Zf 7T LI 71 99
9 81 51 58 Sf 17 10 7T 9* 81 92
9 81 58 88 If 71 88 81 91 81 99
L 81 19 *e 68 81 9 81 ff 6T 92
8 fl TT 98 88 81 88 61 51 61 W
6 fl 08 98 78 61 0 61 05 >5
10 tl 6f 78 98 61 72 05 6 05 89
TT 91 8 88 *8 61 f<3 02 78 T5 2o
31 91 75 68 88 05 21 15 9 12 is
81 91 Qf Of 88 05 If 21 58 55 02
fX 91 8 If 58 15 fX 15 69 55 9f
91 9T 13 Zf 58 T2 Of 53 95 86 9T
91 91 88 2f 58 55 9 55 69 85
17 9T 99 n 58 55 18 82 05 fZ 51
81 71 51 Qf. 38 S5 99 85 9* fZ 68
61 71 65 9f 88 85 21 fZ 51 95 9
05 71 9f Lf 88 85 9f fZ 78 95 88
21 81 T 8* fB fZ 10 95 8 95 69
55 81 £1 6* 98 fZ *8 95 85 95 62
82 81 58 09 98 fZ 79 95 59 95 19
fZ 81 Lf 19 78 95 T5 95 91 72 9T
95 61 I 59 68 95 8? 95 Of 27 Tf
95 61 91 89 Of 95 9 75 f 85 9
75 61 08 W Zf 95 82 75 75 82 68
85 61 Sf 99 ff 95 6* 75 6* 85 89
62 61 75 99 Lf 75 TT 85 TT 65 91
08 05 10 79 6f 75 18 85 88 62 69
Gemini & Sagittarius Ascensional Difference
Deg. Declin. Rt. Ascen. London Birming'm Liverpool
Q o r 0 r 0 r 0 t o t
0 20 10 57 49 27 31 28 33 29 89
1 20 22 58 52 27 52 28 54 30 1
2 20 35 59 54 28 12 29 15 30 28
S 20 46 60 57 28 31 29 35 30 44
4 20 57 62 0 28 46 26 54 31 4
5 21 8 63 8 29 8 80 8T 31 24
6 21 19 64 7 29 25 80 32 31 43
1 21 29 65 10 29 42 80 50 32 2
8 21 39 66 14 29 59 31 7 32 20
9 21 49 67 18 30 15 81 23 32 87
10 21 58 68 22 30 30 31 40 82 54
11 22 6 69 26 30 45 31 55 33 10
12 22 14 70 30 30 58 32 9 33 26
18 22 22 71 84 81 11 32 23 33 40
14 22 29 72 39 81 24 32 87 33 54
15 22 36 78 43 31 36 82 49 84 7
16 22 43 74 48 81 48 33 1 34 20
17 22 49 75 52- 81 58 33 12 34 81
18 22 55 76 67 32 8 33 22 34 42
19 23 0 78 2 82 17 33 32 34 52
20 23 4 79 7 32 25 33 41 85 1
21 23 9 80 12 82 33 83 49 35 10
22 23 13 81 17 82 40 33 56 35 17
23 23 16 82 22 32 46 34 2 35 24
24 23 61 83 28 32 51 34 7 35 80
25 23 21 84 83 82 55 34 12 35 85
26 23 23 85 38 82 59 34 16 85 39
27 23 25 86 44 33 2 34 19 35 42
28 23 26 87 49 33 4 34 21 85 44
29 23 27 88 55 88 5 84 22 85 45
30 23 27 90 0 88 6 34 23 35 46
tS" For the R.A. of Sagittarius add 180° to the
same degree of Gemini. The Declin. and
Asc. Diff. are the same for both.
Iv
Cancer and
CapricornusAscensional Difference
Deg. Declin. Rt. Ascen. London Birming'm Liverpool
e
0
O f o t o r o t O 9
23 27
23 27
28 26
28 25
23 23
23 21
90 0 88 6
33 5
83 4
33 2
82 56
32 55
84 23
84 22
84 21
34 61
34 16
34 12
35 46
35 45
35 44
35 42
35 89
35 35
1 91 5
2
8
4
6
92 11
93 16
94 22
65 27
6
7
8
9
10
23 19
28 16
28 8T
23 9
23 4
96 32 82 51
82 46
32 40
32 83
32 25
34 7
84 2
33 56
83 49
88 41
35 80
85 24
85 17
35 10
85 1
97 38
98 43
99 48
100 53
11
12
13
14
15
23 0
22 55
22 49
22 43
22 86
101 58 32 17
32 8
31 58
81 48
81 86
33 82
83 22
83 12
83 1
32 49
34 52
34 42
34 31
34 20
34 7
103 3
104 8
105 12
106 17
16
17
81
19
20
22 29
22 22
22 14
22 6
21 58
107 21 81 24
81 11
80 58
SO 45
80 30
32 87
32 23
82 9
81 55
81 40
33 54
33 40
33 26
33 10
82 54
108 26
109 30
110 34
111 38
21 21 49
21 89
21 29
21 19
21 8
112 42
118 46
80 15
29 59
29 42
29 25
29 8
81 23
81 7
80 60
80 82
30 13
82 37
82 20
32 2
31 48
81 24
22
23
24
25
114 50
115 53
116 57
26
27
28
29
30
20 57
20 46
20 35
20 22
20 10
181 0 28 49
28 81
28 12
27 52
27 31
29 54
29 35
26 15
28 54
28 83
31 4
30 44
80 28
80 1
29 89
119 3
120 6
121 8
122 11
IS* For the R.A. of Capricornus add 180° to the
same degree of Cancer. The Declin. and
Asc. Diff. are the same for both.
Ivi
sniHvnSyanyoht HOMHHaddIQTVKOISN33Sy
•qtoqjojsuicssqt3jb-jjiq
•osypua"niioecieqx-°8T!J°ssjSadsuits
aqi°;0O8TPP*snmsnby}o-yaeq*joj«sf
•8ea •U35BV-JH uopuoT ta.Sutuutg
O 0 1 0 i .O t 0 i 0 1
0 O2 01 212 TT 72 18 82 88 62 68
T 6T 57 82T 8T 72 TT 82 TT 62 91
5 6T St *2T 9T 95 6* 72 6* 82 89
8 61 08 92T 81 95 8S 72 72 82 63
6T 91 912 O2 95 e 7S t 82 9
5 61 X 27T T2 25 e* 62 0* 7s »
9 8T Lt 812 85 25 21 92 91 27 91
7 81 58 62T fZ fZ 57 92 25 92 T5
8 81 17 08T 52 tz *e 52 82 95 9Z
6 8T T T8T 92 *2 01 25 8 SS 69
10 7T 5* 28T 72 85 9* fz 78 25 88
TT 17 65 881 7S 85 21 fZ 2T 9Z 9
5T 7T 2T fSX 85 22 95 es 9* fz 68
81 9T 56 58T 85 22 18 es O2 fZ 51
tx 9T 88 981 82 22 9 22 89 85 ft
9T 91 15 78T 85 T2 Of 22 95 es OT
91 91 8 881 85 T2 fX 21 65 22 8*
17 5T 5t 681 72 O2 Lt I2 58 22 20
8T 91 27 OfX 72 20 21 T2 9 T2 19
6T 5T 8 XfX 95 61 w 20 78 T2 55
05. tx 6i> ZfX 25 61 72 O2 6 O5 86
15 tx 08 en 85 61 0 61 Xf O2 ts
55 tx TT ftx 22 81 88 61 zx 61 W
85 81 T5 9fX T2 81 9 81 ft 6T 92
tz 81 58 9fX 61 17 88 81 91 81 99
«5 81 2T 7H 7T 7T ox 17 9f 81 91
95 51 T5 8H 5T 91 Zf 17 LX LX 55
72 12 18 6H 8T 91 fX 91 Bt 7T fZ
85 51 OT O5T n 5T 9t 91 61 9T W
65 TT 09 15T 8 91 17 91 6* 91 855
08 TT 65 25T 9 n 8* 15 6T fiT 89
!!AI
Virgo and Pisces Ascensional Difference
Deg. Declln. Rl. Ascen. London Birming'm Liverpool
0 a r 0 i 0 1 0 i 0 1
0 11 29 152 5 14 48 15 19 15 58
1 11 7 153 8 14 20 14 50 15 22
5 10 46 154 0 8T 51 14 20 14 51
« 10 24 154 57 13 22 18 50 14 20
4 10 8 155 54 12 53 13 20 13 49
6 9 41 156 50 12 24 12 50 13 17
6 9 19 157 47 11 55 12 20 12 46
7 8 57 158 43 11 26 11 49 12 15
8 8 84 156 40 10 56 11 19 11 48
9 8 12 160 36 10 27 10 49 11 12
10 7 49 161 82 9 57 10 18 10 40
11 7 27 162 28 9 28 9 48 10 8
12 7 4 163 24 8 58 9 17 9 87
13 6 41 164 20 8 29 8 46 9 5
14 6 18 165 16 7 59 8 16 8 88
15 5 55 166 11 7 29 7 45 8 1
16 5 81 167 7 7 0 7 14 7 29
17 5 8 168 2 6 80 6 43 6 57
81 4 45 168 58 6 0 6 12 6 25
19 4 21 169 53 5 30 5 41. 5 58
20 S 58 170 49 5 0 5 10 5 21
21 3 34 171 44 4 80 4 39 4 49
22 8 10 172 89 4 0 4 8 4 17
23 2 47 173 34 8 80 8 37 8 46
24 2 23 174 30 8 0 3 6 8 18
25 1 59 175 25 2 SO 2 85 2 41
26 1 36 176 20 2 0 2 4 2 9
27 1 12 177 15 1 80 1 33 1 87
28 0 48 178 10 1 0 1 2 1 4
29 0 24 179 5 0 80 0 81 ,0 82
80 0 0 180 0 0 0 0 0 \o 0
IS" For the R.A. of Pisces add 810° to the same
degree of Virgo. The Declin. and Asc. Diff.
are the same for both.
Iviii
TERNARY PROPORTIONAL
LOGARITHMS
Ternary Proportional Logarithms
10°
0 Infinite
1 403342
1 373-i39
3 355630
4 343136
5 3 33445
6 325527
7 3-18833
8 3oJ3033
9 3-07918
10 303342
:t 2"99203
12 2-95424
13 2-91948
i4 288730
:J 285733
16 282930
«7 280297
: 8 2-77815
:9 2 7546720 273239
11 271120
22 2-69I00
2} 267170
»4 265321
»3 263548
26 2-61845
27 260206
28 258627
»9 2-57X03
30 255630
3« ».54»°6
32 2 52827
33 25M91
34 250194
33 2-48936
J6 2-477I2
37 246522
38 245364
39 2-44236
40 243136
4< 242064
4' 2-41017
43 2 39996
44 2-38997
43 2-38021
46 2-37067
47 236133
48 2-35218
49 234323
30 2-33445
S: 232585
3' 2-3t742
33 2-309I5
34 2-30103
33 229306
36 2-28524
37 2-27755
38 2-27000
39 2-26257
63 2-25527
2-213882207352-2009I2194572I8833
2-152612-146932-14I332I35802 t3O33
2I24942.X:9D12-114352-109142-10400
2-098932-093902-088942-084032-07918
2074382-069642064942 060302.05570
2051152046652-042202 03779203342
2-029tO2O24822-O20602-016392-01223
2-O08122-OO4O42-000001-996O0r99203
1-988IO1-98421t 98035t-97652197273
t 96897t 965241-96t541 95788195424
: 954241-95064t-94706t 94352t .940001 93651
1 93305t929621-92621I .922831-9I948
1-916151912851909571-906321-90309
1-899881-896701.893541-8904:1-88730
1-884201-881141 878091-87506187206
1869071-866u1-86316I .860241 85733
t- 85445I -85I581 848731 84590I 84309
I .840301-837521-834771832031 82930
1826601-823911-821241-8I8581-8I594
1-81332t.8I07I1808111 80554I 80297
I-8oo43179790t-795381792871 79039
1-78791t-78545178300178057177815
3°1 778151 775751 773351 770971 768611 76625
1 76391176158175927175696175467
1-752391750121-747871745621-74339
174117173896173676173457173'39
1-730231-7280?1725931723791 72167
1 71956t 717451 71 536171328171120
1-70914170709170504t -70301170099
1-698971-696961-694971692981-69100
1-68903t 6S7071-685121683181-68124
1-679321 677401-675491-67359167 1 70
1-66981t 667941-666071664211 66236
t.66o5I1-6586*I.656H51 655031 65321
4°
1-6532I1-65t4I1'64961I 647821646031 64426
1-642491 640731-63897I -637221-6^548
1 633751-632021'630301 628591-62688
1-625181 623491-621801-620121-61843
1616781-615I21-6I3471611821-61018
1608541-60691t 60529t .603671 .60206
1600451-598851-597261 59567t 59409
1-59»511-590941589381-587821-58627
t-584721583171-58164t .5801 11 57858
1-577061-575541-574031-57253157103
1 569531-568041-566561-565081 56360
1-562131 -560671-559211 55775155630
5°
I 55630I 554861-55342I 551981-55055154912
t 547701546291 544871-543471-54206
1-540661 539271-53788t 536491-53511
1-533741532361531001-529631 52827
1-52692t-525571 524221-522881-52154
1-520211-51888:-5t7551-516231-51491
1-51360'1-5I2291-5Jo981-50968I 50838
I -507081-505791-504511-503221-30194
1-500671 499401-498131-49687t-4956o
1-494351-493091-491841-490601-48936
1-488121486881-485651-484421-48320
t-48t97t 48076
:47954: 478,3t 47712
8°
1-477121 475921-474721 473521 472321 47113
1 410171-409141 408111-407081 406061 40503
1-4&994 1 4040141-46876 1-403001-46758 1 401981-46640 1-400971 46522 1 39996
1-464041 462881 461711 460551 45938
1 458241 457081 455931-45478145364
1452501-45 1 36t 45022144909t-447.96
1-44684t 445711-444591-44347t 44236
1-441251-44014: 43903i.437931-43683
t.435731-434631-433541-43245143136
1-430281-429201-428121-427041-42597
1-424901-423831-422761-421701-42064
1 419581-4185-14 1 7471-41642141538
1414J31413291 41225141121t.4JoJ7
1 398951 .397941-396941 395931 39493
1 393941-39294139195t 39096t 38997
1388991-388001-387021-38604t -38506
1-384091-383121-382151-381181-38021
t 37925t 378291-37733t 376371 37541
t 37446t-3735t137256t -371611-37067
1-369721-368781-367841-3669:: 36597
1-365041-364111-363181-36225.1-36I33
1 o36040
1-359481 358561 357651 35673
1-355821 35491t 354001 35309t 35218
8°
1352181-351281- 350381349481348581 34768
1 346791345891 345001-344111 34323
1-342341-341461 340581-33970133882
1-337941-33707I-336I91-335321-33445
1-33359t 332721331861-33099133013
t 32927t 328421327561-32671t 32585
1325001-3»4151-323311-322461-32162
l-320771-31993131909131826131742
t-3t6591 31 5751 314921314091-31326
1-312441-311611310791-30997130915
1 30833t-3075t1 306701 30588130507
1-304261 303451 302641 30183tJOt03
9°
1-301031-300231299421-298621-29782129703
1-296231-29544t 29464t 293851 .29306
1-292271-291481-290701-289911-28913
1-288351287571-286791-286011-28524
1-284461-283691-282921282 15t 28138
1-280611-279841-279081-278311-27755
1-276791-276031 275271 27451I- 27376
1273001 272251-27I501 270751-27000
1 269251-268501-26776.1-267011-26027
1-265531 264791-264051-26331126257
T26t84X-261101-26037t- 250641-25891
1-258181-257451-256721-256001 25527
lxi
Ternary Proportional Logarithms
11
:3M
»3
16
19
2223
»4»3
16
1282930
S<3*333433
*3738
39
4«
4:4»434443
4«474*4930
5I
3»333433
3«5758
5960
10°
t255271-I5+5J1-253831253111-23239:.25167
1-250951250241-240511-248811 24809
1 247381 246671 243961-245261*4455
1-243841-243141 242441-24173124103
1-240331 239631238941-238241-«734
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1 003631-003231-002821-002421-00202
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0-98693 0-964I30-98654 0 963760 98615 0-96339
0-98576 0-963020-98537 0-962650-98498 0-962280-98459 0-9619I0-98421 0-96254
0-98382 0-961170.98343 0-960810-96304 0-960440-98266 0-960070-98227 0-95971
0-98189 0 95934
0 98150 0 958970-98111 0-95861098073 0958240 98035 0-95788
0 97996 0-957510-97958 0-957150 97919 0-956780-97881 0-95642
0 97843 0 95606
0 97805 0 955690-97766 0-955330 97728 0-954970-97690 0-95460
0-97652 0 954x4
lxii
lziii
Ternary Proportional Logarithms
130° 31° 32° 33° 34° 35° 36° 37° 38° 39°
0 77815 76391 75012 73676 72379 711-20 69897 68707 67549 66421
I 7779 1 76j68 74990 .73554 72358 71100 69877 6868868668
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4 77719 76298 749" 73588 72294 71038 69817 68629 67473 66347
J 77693 76274 74899 73566 72273 71017 69797 68609 67454 66328
6 77671 76251 74877 73544 72252 70997 69757 68590 67435 66310
7 77647 76228 74854 73523 72231 70976 69756 68570 67416 66291
8 77623 76205 74832 73501 72209 70955 69736 68551 67397 66273
9 77599 76181 74809 73479 72188 70935 69716 68531 67378 66254
Io 77575 76158 74787 73457 72167 70914 69696 68512 67359 66236
it 7755: 76135 74764 73435 72146 70894 69676 68492 67340 66217
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U 77479 76065 74697 73370 72082 79832 69616 68434 67283 66162
15 77455 76042 74674 73348 72061 7081 1 69596 68415 67264 66143
16 77431 76019 74652 73326 72040 70791 69576 68395 67245 6612517 77407 75996 74629 73305 72019 70770 69557 68376 67226 6610618 77383 75973 74607 73283 71998 70750 69537 68356 67207 66088
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11 773" 75903 74540 73218 71935 70688 69477 68298 6715 1 66033
22 77288 75880 74517 73196 71914 70668 69457 68279 67132 66014
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»4 77240 75834 74473 73153 71871 70627 69417 68240 67094 65978
2J 77216 758U 74450 73131 71850 70606 69397 68221 67075 65959
26 77192 75788 74428 73109 71829 70586 69377 68201 67056 65941
27 77169 75765 74406 73088 71808 70566 69358 68182 67038 65923
28 77H5 75742 74383 73066 71787 70545 69338 68163 67019 6590429 77121 75719 74361 73044 71766 70525 69318 68143 67000 65886
30 77097 75696 74339 73023 71745 70504 69298 68124 66981 65868
3: 77974 75673 74317 73001 71724 70484 69278 68105 66962 6584932 77050 75650 74294 72980 71703 70464 69258 68086 66944 65831
33 77026 75627 74272 72958 71682 70443 69239 68066 66925 65813
34 77002 75604 74250 72936 71662 70423 69219 68047 66906 65794
35 76979 75581 74228 72915 71641 70403 69199 68028 66887 65776
36 76955 75559 74205 72893 71620 70382 69179 6S008 66869 65758
37 76931 75536 74183 72872 71599 70362 69159 67989 66850 65739
38 76908 75513 74161 72850 71578 70342 69140 67970 66831 65721
39 76884 75490 74139 72829 71557 70321 69120- 67951 66812 65703
40 76861 75467 74t 1 7 72807 71536 70301 69100 67932 66794 65685
41 76837 75444 74095 72786 71515 70281 69080 67912 66775 65666
42 76815 75421 74072 72764 71494 70260 69061 67893 66756 65648
43 76790 75398 74050 72743 71473 70240 69041 67874 66737 65630
44 76766 75376 74028 72721 71453 70220 69021 67855 66719 65612
45 76743 75353 74006 72700 71432 70200 69002 67836 66700 65594
46 76710 75330 73984 72678 714" 70179 68982 67816 66681 65575
47 76696 75307 73962 72657 71390 70159 68962 67797 66663 6555748 76672 75285 73940 72636 71369 70139 68942 67778 66644 65539
49 76649 75262 739»8 72614 71349 701 19 68923 67759 66625 6552130 76625 75239 73896 72593 71328 70099 68903 67740 66607 65503
51 76602 75»i6 73874 72571 71307 70078 68884 67721 66588 6548452 76578 75194 73852 72550 71286 70058 6S864 67702 66570 65466
53 76555 75171 73830 72529 71265 70038 6SS44 67682 66551 65448
54 76531 75148 73808 72507 71245 70018 68825 67663 66532 65430
55 76508 75126 73786 72486 71224 69998 68805 676+4 66514 65412
56. 76485 75103 73764 72465 71203 69977 687S5 67625 66495 65394
57 76461 75080 73742 72443 71183 69957 68766 67606 66477 65376
58 76438 75058 73720 72422 71162 69937 C8746 67587 66458 65357
59 76414 75035 73698 72401 7iut 69917 68727 67568 66439 6533960 76391 75012 73676 72379 71 120 69897 68707 67549 66421 65321
lxiv
Ternary Proportional Logarithms
' 40° 4l° 42° 43° 44° 45° 46° 47° 48° 49°"
o 65321 64249 6320» 62180 61 182 60206 59251 58317 57403 56508I 65303 64231 63185 62164 61166 60190 59236 58302 57388 564932 65285 64214 63168 62147 61 149 60174 59220 58287 57373 564783 65267 64196 63151 62130 61133 60158 59204 58271 57358 564634 65249 64178 63133 62113 61116 60142 39189 58256 57343 56449
5 65231 64161 63116 62096 61100 60126 59l73 58241 57328 56434
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11 65123 64055 63013 61996 61001 60029 59079 58148 57238 56345n 65105 64038 62996 61979 609S5 60013 59063 58133 57223 5633113 65087 64020 62979 61962 60969 59997 59047 58118 57208 5631614 65069 64002 62962 61945 60952 59981 59032 58102 57193 56301
15 65051 63985 62945 61929 60936 59965 59016 58087 57I78 56287
16 65033 63967 62927 61912 60920 59949 59000 58072 57163 5627217 65015 63950 62910 61895 60903 59933 58985 58056 57148 5625718 64997 63932 62893 61878 60887 59917 58969 58041 57133 5624319 64979 63915 62876 61862 60871 59901 58954 58026 57118 56228
20 64961 63897 62859 61845 60854 59885 58938 58011. 57103 56213
21 £4943 63880 62842 61828 60838 59870 58922 57995 57088 56199
22 64925 63862 62825 6I8I2 60822 59854 58007 57980 57073 5618423 64907 63845 62808 61795 60805 59838 58891 57965 57058 5616924 64889 63827 62791 61778 60789 59822 58875 57949 57043 5615525 64871 63810 62774 61762 60773 59806 58860 57934 57028 56140
26 64853 63792 62757 61745 60756 59790 58844 57919 57013 56125
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36 64675 63618 62386 61579 60594 59631 58689 57767 56864 55979
37 64657 63601 62569 6I562 60578 59615 58673 57751 56849 5596438 64639 63583 6255a 61545 60561 59599 58658 57736 56834 5595039 64621 63566 62535 61529 60545 59583 58642 57721 56819 55935
40 64603 63548 625X8 6I3I2 60529 59567 58627 57706 5680i 55921
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59 64267 63220 62197 61198 6022» 59267 58333 574l§ 56522 5564560 64249 63202 63180 6U82 60206 59251 58317 57403 56508 55630
13
Ternary Pkoportional Logarithms
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55184 54332 53498 5267s 51874 51085 50310 49548 48799 48063
55109 54318 53484 52665 51861 51072 50297 49535 48787 48051
55155 54304 53470 52651 51848 51059 50284 49523 48775 48039
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54969 54122 5329: 52476 51676 50890 501 18 49360 48614 47881
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54927 54080 53250 52436 51636 50851 50080 49322 48577 47845
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54827 53983 53154 52342 51544 50760 49991 49234 48491 4776054813 53969 53M: 52328 51530 50747 49978 49222 48479 47748
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Ixvi
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44080 4342044069 43409
44058 4339844047 4338744036 43376
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4307143060430504303943028
4301743006429954298542974
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4274742737427264271542704
4269342683426724266142651
4264042629426184260842597
4258642575425654255442543
4210642096 .420854207542064
4205342043420324202242011
4200041990419794196941958
4194841937419274191641905
4M754146441454
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4137041360413504133941329
4131841308412984128741277
42533 41895 4126642522 41884 41256425H 41874 4124642500 41863 4123542490 41853 41225
42479 4184242468 4183242458 4182142447 4181 142436 41800
42426 41789
42415 4177942404 4176842394 4175842383 41747
4237242362423514234042330
4231942308422984228742276
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41 1624"524"42
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410174100740997409864097640965
4095540945409354092440914
4090440894408834087340863
4085240S4240832408214031 1
-4080140791407804077040760
4074940739407294071940708
4069340688406784066740657
4064740637406264061640606
4059640585405754056540555
4054440534405244051440503
4049340483404734046340452
4044240432404224041240401
71°
404014039140381403714036140350
4034040330403204031040300
4028940279402694025940249
4023940228402184020840193
4018840178401634015740147
401374012740U74010740097
4008740076400664005640046
4003640026400164000639996
39985
3997539965
39955
39945
3993539925399153990539895
3988539874398643985439844
39834398243981439804
39794
Ixvii
Ternary Proportional Logarithms
172°
• 39794397841
a 39774
5 39764
4 39754
> 39744
6 39734
7 39724
8 39714
9 39704
to 39694
:t 39684
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M 39653
:5 30643
:B 39633
17 39623
18 39613
:9 39603
20 39593
21 39583
22 3957323 39563
24 3955325 39543
26 3953327 39523
28 395:329 39503
30 39493
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32 39473
33 39464
343S 39444
3« 39434
37 39424
38 39414
39 39404
40 39394
41 39384
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44 3935443 39344
46 3933447 393^448 393H49 3930450 39^94
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53 39264
54 39254
55 39245
56 39235
57 3922558 39215
59 3920560 39:95
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39 :753916539155
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3913639126391 163910639096
3908639076390663905639046
3903739027390173900738997
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3874138731387223871238702
38692386823867338663
38653
3864338633386243861438604
3820538195381863817638166
3815638147381373812738118
3810838098380893807938069
3806038050380403803138021
3757937570375603755137541
37532375223751337503
37494
37484
3747437465
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3696336953369443693536925
3691636906368973688836873
3640136392363833637436364
3635536346363363632736318
35801 3525435792 3524535783 3523635774 3522735765 35218
lxviii
Ternary Proportional Logarithms
/84° 85° 88° 87° 88° 89° 90° 91° 92° 93° 94° 95°
0 33099 32585 32077 31575 31079 30588 30103 29623 29148 28679 28214 27755
t 33091 32577 32069 31567 31071 30580 30095 29615 29141 28671 28207 27747
2 33082 32568 32061 31559 31063 30572 30087 29607 29133 28663 28199 27740
3 33073 32560 32052 31550 31054 30564 30079 29599 29125 28656 23191 27732
4 33065 32551 32044 31542 31046 30556 30071 29591 29117 .28648 28184 27724
5 33056 32543 32035 31534 31038 30548 30063 29583 29109 28640 28176 27717
6 33048 32534 32027 31525 31030 30539 30055 29575 29101 28632 28168 27709
7 33039 32526 32019 31517 31021 30531 30047 29567 29093 28625 28161 27702
8 33030 32517 32010 31509 31013 30523 30039 29560 20086 28617 28153 27!&
9 33022 32509 32002 31501 31005 30515 30031 29552 29078 28609 28145 27686
10 33013 32500 31993 31492 30997 30507 30023 29544 29070 28601 28138 27679
it 33005 32492 31985 31484 30989 30499 30015 29536 29062 28593 28130 27671
12 32996 32483 31977 31476 30980 30491 30007 29528 29054 28586 28122 27664
13 32987 32475 31968 31467 30972 30483 29999 29520 29046 28578 28 1 14 27656
14 3»979 32466 31960 31459 30964 30475 29991 29512 29038 28570 28107 27648
15 32970 32458 3 195 1 31451 30956 30466 29983 29504 29031 28562 28099 27641
16 32962 32449 31943 31442 30948 30458 29975 29496 29023 28555 28091 27633
17 32953 32441 31935 31434 30939 30450 29967 29488 29015 28547 28084 2762618 32944 32432 31926 31426 30931 30442 29958 29480 29007 28539 28076 27618
:9 3293S 32424 319*8. 31418 30923 30434 29950 29472 28999 28531 28068 2761020 32927 32415 31909 31409 30915 30426 29942 29464 28991 28524 28061 27603
21 32919 32407- 31901 31401 30907 30418 29934 29456 28984 28516 28053 2759522 32910 32398- 31893 31393 30898 30410 29926 29448 28976 28508 28045 2758823 32902 32390 31884 31384 30890 30302 29918 29441 28968 28500 28038 2758024 32893 32381 31876 31376 30882 30393 29910 29433 28900 28493 28030 27572
»5 32884 32373 31867 31368 30874 30385 29902 29425 28952 28485 28022 27565
26 32876 32365 31859 31360 30866 30377 29894 29417 28944 28477 28015 2755727 32867 32356 31851 31351 30857 30369 29S86 29409 28937 28469 28007 2755028 32859 32348 31842 31343 30S49 30361 29878 29401 28929 28462 27999 2754229 32850 32339 31834 31335 30841 30353 29870 29393 28921 28454 27992 2753430 32842 32331 31826 31326 30833 30345 29862 29385 28913 28446 27984 27527
3: 32833 32322 31817 31318 30825 30337 29854 29377 28905 28438 27976 2751932 32824 32314 /31809 31310 30817 30329 29846 29369 28897 28431 27969 27512
33 32816 32305 31801 31302 30808 30321 29838 29361 28890 28423 27961 27504
34 32807 32297 31792 31293 30800 30313. 29830 29354 28882 28415 27953 27497
35 32799 32288 31784 31285 30792 30305 29822 29346 28874 28407 27946 27489
36 32790 32280 31775 31277 30784 30296 29814 29338 28866 28400 27938 27481
37 32782 32271 31767 31269 30776 30:88 29806 29330 28858 28392 27930 2747438 32773 32263 31759 31260 30768 30280 29798 29322 28851 28384 27923 27466
39 32765 32255 31750 31252 30759 30272 29790 29314 28843 28376 27915 27459
40 32756 32246 31742 31244 30751 30264 29782 29306 28835 28369 27908 27451
41 32747 32238 31734 31236 30743 30256 29775 29298 28827 28361 27900 2744442 32739 32229 31725, 31227 30735 30248 29767 29290 28819 28353 27892 27436
43 32730 3*221 31717 31219 30727 30240 29759 29282 288 1 1 28346 27885 27429
44 32722 32212 31709 31211 30719 30232 29751 29275 28804 28338 27877 27421
45 32713 32204 31700 31203 30710 30224 29743 29267 28796 28330 27869 27413
46 32705 32195 31692 31194 30702 30216 29735 29259 28788 28322 27862 27406
47 32696 32187 31684 31 186 30694 30208 29727 29251 28780 28315 27854 2739848 32688 3»t79 31675 31178 30686 30200 29719 29243 28772 28307 27846 27391
49 32679 32170 31667 31170 30678 30192 2971 1 29235 28765 28299 27839 27383
50 32671 32162 31659 31161 30670 30183 29703 29227 28757 28292 27831 27376
51 32662 32153 31650 3"53 30662 30175 29695 29219 28749 28284 27824 2736852 32654 32145 31642 31145 30653 30167 29687 29211 28741 28276 27816 27360
53 32645 32136 31634 31137 30645 30159 29679 29204 28733 28268 27808 27353
54 32636 32128 31625 31128 30637 30151 29671 29196 28726 28261 27801 27345
55 32628 32120 31617 31120 30629 30143 29663 29188 28718 28253 27793 27338
56 32619 32m 31609 31112 30621 30135 29655 29180 28710 28245 27785 27330
57 32611 32103 31600 31104 30613 30127 29647 29172 28702 28238 27778 2732358 32602 32694 31592 31095 30605 301 19 29639 29164 28695 28230. 27770 27315
59 32594 32086 31584 31087 30506 301 1 1 29631 29156 28687 28222 27763 2730860 32585 32077 31575 31079 30588 30103 29623 29148 28679 28214 27755 27300
Ixix
1 107°96° 106°97° 105°98° 104°99° 103°100° 102°101°
Ternary Proportional Logarithms
o 27300 26850 26405 25964 25527 25095 24667 24244 23824 23408 22997 22589
1 27293 26843 26397 25956 25520 25088 24660 24237 23817 23401 22990 22582
2 27285 26835 26300 25949 25513 25081 24653 24229 23810 23395 22983 22575
3 27278 268:18 26382 25942 25506 25074 24646 24222 23803 23388 22976 22569
4 27270 26820 26375 25934 25498 25066 24639 24215 2379S 23381 22969 22562
5 27262 26813, 26368 25927 25491 25059 24632 24208 23789 23374 22963 22555
6 27255 26805 26360 25920 25484 25052 24625 24201 23782 23367 22956 22548
7 27247 26798 26353 25913 25477 25045 24618 24194 23775 23360 22949 22542
8 27240 26790 26346 25905 25469 25038 24610 24187 23768 23353 22942 22535
9 27232 26783 26338 25898 25462 25031 24603 24180 23761 23346 22935 22528
10 27225 26776 26331 25891 25455 25024 24596 24173 23754 23339 22928 22521
n 27217 26768 26323 25883 25448 25016 24589 24166 23747 23333 22922 22515
12 27210 26761 26316 25876 25440 25009 24582 24159 23740 23326 22915 22508
13 27202 26753 26309 25869 25433 25002 24575 24152 23734 23319 22908 22501
M 27195 26746 26301 25861 25426 24995 24568 24145 23727 23312 22901 22494
:5 27187 26738 26294 25854 25419 24988 24561 24138 23720 23305 22894 22488
16 27180 26731 26287 25847 25412 24981 24554 24131 23713 23298 22888 22481
17 27172 26723 26279 25840 25404 24973 24547 24124 23706 23291 22881 22474
: 8 27165 26716 26272 25832 25397 24966 24540 24117 23699 23284 22874 22467
19 27157 26709 26265 25825 25390 24959 24533 24110 23692 23278 22867 22461
20 27150 26701 26257 25818 25383 24952 24526 24103 23685 23271 22860 22454
21 27142 26694 26250 25810 25376 24945 24518 24096 23678 23264 22854 22447
52 27135 26686 26242 25803 25368 24938 24511 24089 23671 23257 22847 22440
23 27127 26679 26235 25796 25361 24931 24504 24082 23664 23250 22840 22434
24 27120 26671 26228 25789 25354 24923 24497 24075 23657 23243 22833 22427
25 27112 26664 26220 25781 25347 24916 24490 24068 23650 23236 22826 22420
26 27105 26656 26213 25774 25339 24909 24483 24061 23643 23229 22819 22413
27 27097 26649 26206 25767 25332 24902 24476 24054 23636 23223 22813 22407
28 27090 26642 26198 25759 25325 24895 24469 24047 23629 23216 22806 22400
29 27082 26634 26191 25752 25318 24888 24462 24040 23623 23209 22799 22393
30 27075 26627 26184 25745 2531 1 24881 24455 24033 23616 23202 22792 22386
3: 27067 26619 26176 25738 25303 24874 24448 24026 23609 23195 22785 22380
32 27060 26612 26169 25730 25296 24866 24441 24019 23602 23188 22779 22373
33 27052 26605 26162 25723 25289 24859 24434 24012 23595 23181 22772 22366
34 27045 26597 26154 25716 25282 24852 24427 24005 23588 23175 22765 22359
35 27037 26590 26147 25709 25275 24845 24420 23998 23581 23168 22758 22353
36 27030 26582 26140 25701 25267 24838 24413 23991 23574 23161 22752 22346
37 27022 26575 26132 25694 25260 24831 24405 23984 23567 23154 22745 22339
38 27015 26567 26125 25687 25253 24824 24398 23977 23560 23147 22738 22333
39 27007 26560 261 18 25680 25246 24817 24391 23970 23553 23140 22731 22326
40 27000 26553 261 10 25672 25239 24809 24384 2396? 23546 23133 22724 22319
41 26992 26545 26103 25665 25231 24802 24377 23956 23539 23127 22718 22312
42 26985 26538 26096 25658 25224 24795 24370 23949 23533 23120 2271 1 22306
43 26977 26530 26088 25650 25217 24788 24363 23942 23526 231 13 22704 22299
44 26970 26523 26081 25643 25210 24781 24356 23935 23519 23106 22697 22292
45 26962 26516 26074 25636 25203 24774 24349 23928 23512 23099 22690 22286
46 26955 26508 26066 25629 25196 24767 24342 23921 23505 23092 22684 22279
47 26947 26501 26059 25621 25188 24760 24335 23914 23498 23086 22677 22272
48 26940 26493 26052 25614 25181 24752 24328 23908 23491 23079 22670 22265
49 26932 26486 26044 25607 25174 24745 24321 23901 23484 23072 22663 22259
5° 26925 26479 26037 25600 25167 24738 24314 23894 23477 23065 22657 22252
51 26917 26471 26030 25592 25160 24731 24307 23887 23470 23058 22650 22245
52 26910 26464 26022 25585 25152 24724 24300 23880 23464 23051 22643 22239
53 26902 26456 26015 25578 25145 24717 24293 23873 23457 23044 22636 22232
54 26895 26449 26008 25571 25138 24710 24286 23866 23450 23038 22629 22225
55 26887 26442 26000 25563 25131 24703 24279 23859 23443 23031 22623 *22l8
56 26880 26434 25993 25556 25124 24696 24272 23852 23436 23024 22616 22212
57 26872 26427 25986 25549 251 17 24689 24265 23845 23429 23017 22609 2220558 26865 26419 25978 25542 25109 24681 24258 23838 23422 23010 22602 22198
39 26858 26412 25971 25534 25102 24674 24251 23831 23415 23004 22596 22192
60 26850 26405 25964 25527 25095 24667 24244 23824 23408 22997 22589 22185
Izx
Ternary Proportional Logarithms
1 108° 109° 110° 111° 112° 113° 114° 115° 116° 117° 118° 119°
0 22185 21785 21388 20995 20605 20219 19837 19457 19081 I8709 18339 179731 22178 21778 21381 20988 20599 202I3 19830 19451 19075 I8702 18333 17966
» 22171 21771 21375 20982 20593 20207 19824 19445 19069 I8696 18327 17960
3 22165 21765 21368 20975 20586 20200 19818 19439 19063 18690 18321 I7954
4 22158 21758 21362 20969 20580 20194 1 981 1 19432 19056 I8684 18315 17948
J 22151 21751 21355 20962 20573 20187 19805 19426 19050 I8678 18308 I7942
6 22145 21745 21349 20956 20567 20181 19799 19420 19044 18672 18302 17936
7 22138 21738 21342 20949 20560 20175 19792 19413 19038 I8665 I8296 179308 22131 21732 21335 20943 20554 20168 19786 19407 19032 I8659 I829o 17924
9 22125 21725 21329 20936 20547 20162 19780 19401 19025 I8653 I8284 1791810 221 18 21718 21322 20930 20541 20155 19773 19395 190I9 I8647 I8278 17912
tS 22m 21712 21316 20923 20534 20149 19767 I9388 19013 1 864 1 18272 1790612 22105 21705 21309 20917 20528 20143 19761 19382 I9007 I8634 I8266 17900
13 22098 21698 21303 209tO 20522 20136 19754 19376 19000 18628 18259 17894
»4 22091 21692 21296 20904 20515 20130 19748 19369 18994 18622 18253 17887
15 22084 21685 21289 20897 205C9 20123 19742 19363 I8988 18616 18247 17881
:6 22178 21678 21283 20891 20502 2o"7 19735 19357 18982 18610 18241 1787517 22071 21672 21276 20884 20496 201" 19729 19351 18976 18604 18235 17869
:8 22064 21665 21270 20878 20489 20104 19723 :9344 I8969 I8597 18229 17863
:9 22058 21659 21263 20871 20483 20098 19716 19338 18963 18591 18223 17857
20 22051 21652 21257 20865 20476 20091 19710 19332 18957 18585 18217 17851
»i 22044 21645 21250 20858 20470 20085 19704 I9325 1 895 1 18579 18210 1784522 22038 21639 21243 20852 20464 20079 19697 19319 18944 18573 18204 17839
23 22031 21632 21237 20845 29457 20072 I969I t93t3 18938 1S567 18198 17833
»4 22024 21626 21230 20839 20451 20066 19685 19307 18932 18560 18192 17827
>5 22018 21619 21224 20832 20444 20060 19678 I9300 18926 18554 18186 17821
26 2201 1 21612 21217 20826 20438 20053 I9672 19294 18920 18548 18180 1781527 22004 21606 212tt 20819 20431 20047 I9666 19288 18913 18542 18174 17809»8 21998 21599 21204 20813 20425 20040 19659 19282 18907 18536 18168 17803
»9 21991 21592 21 198 20806 20418 20034 19653 19275 1 890 1 18530 18162 17797
30 21984 2I586 2XX91 20800 204I2 20028 19647 19269 18895 18523 18155 17790
3» 21978 21579 21184 20793 20406 20021 19640 19263 18888 18517 18149 17784
3» 21971 21573 21 178 20787 20399 20015 19634 19257 18882 185" 18143 : 7778
31 21964 21566 2U7t 20780 20393 20009 I962s 19250 18876 18505 18137 17772
34 21958 21559 21 165 20774 20386 20002 19621 19244 18870 18499 18131 17766
35 21951 21553 21158 20767 20380 19996 196I5 19238 18864 18493 I8t25 17760
36 21944 21546 2tt52 20761 20373 19989 19609 19231 18857 18487 I8It9 :775437 21938 21540 21 145 20754 20367 19983 I96o2 19225 1885I 18480 I8II3 17748
3» 21931 21533 2tt39 20748 2036I :9977 19596 I9219 18845 18474 18107 17742
39 21924 21526 2tt32 20741 20354 19970 19590 19213 18839 I846s 18100 17736
40 21918 21520 21126 20735 20348 19964 19584 19206 18833 1846» 18094 17730
4: 21911 21513 2ttt9 20728 20341 19958 19577 19200 18826 I8456 1808818082
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4» 21904 21507 2ttt2 20722 20335 19951 t957t 19194 18820 18450 17718
43 21898 21500 21106 20715 2Q328 :9945 19565 19188 18814 18443 18076 17712
44 2 1 891 21493 21099 20709 20322 19938 19558 19181 18808 18437 18070 17706
43 21884 21487 21093 20702 20316 19932 19552 19175 18802 18431 18064 17700
46 21878 21480 21086 20696 20309 19926 19546 19169 18795 18425 18058 17694
47 21871 21474 2t08o 2069O 20303 t99:9 19539 19163 18789 18419 18052 1768848 21864 21467 21073 20683 20296 19913 19533 191 56 18783 18413 18046 17682
49 21858 21460 21067 20676 202Q0 19907 19527 19150 18777 18407 18040 17676
s» 21851 21454 2t060 20670 20284 19900 19520 19144 18771 18400 18033 17669
51 21844 21447 21054 20664 20277 19894 19514 19138 18764 18394 18027 17663
5» 21838 81441 21047 20657 20271 I9888 19508 19131 I8758 18388 18021 17657
53 21831 » 1 434 2104t 2065t 20264 I988I 19502 19I25 18752 18382 18015 17651
54 21824 21427 21034 20644 20258 19875 I9495 19"9 I8746 18376 18009 17645
55 21818 21421 21028 20638 20251 19869 19489 t9"3 I8740 18370 18003 17639
3« 21811 21414 2t02t 2063t 20245 19862 19483 19106 18733 18364 :7997 17633
57 21805 21408 21015 20625 20239 19856 I9476 19100 18727 18357 17991 17627
58 21798 2140X 21008 206I8 20222 19849 19470 19094 18721 18351 17985 1 762 1
59 2179: 21395 2t00t 20612 20226 19843 19464 19088 187I5 18345 17979 17615
60 21785 21388 20995 20605 20219 19837 19457 19081 18709 18339 17973 17609
lxxi
Ternary Proportional Logarithms
r 120° 121° 122° 123°
0 17609 17249 16891 16537
i :7603 17243 16885 16531
I "7597 17237 16879 16525
3 »7S9< 17231 16871 16519
4 17585 17225 16868 165 13
S :7579 17219 16862 16507
6 17573 t7»is 16856 16501
t : 7567 17207 16850 16496
B 17561 17201 16844 16400
9 17555 17195 16838 16484
10 »7549 17189 16832 16478
ii 17543 17183 16826 16472it :7337 1 7177 16820 16466
13 :753t 17171 16814 16460
14 17525 17165 16808 16454
13 :75t9 17159 16802 16449
:6 »75:3 17153 16796 1644317 17507 17147 1 6791 16437
18 :7501 17141 16785 16431
:9 17493 17135 16779 16425
to 17489 17129 16773 16419
91 17483 17123 16767 164132» «7477 17117 16761 16407
»3 :747t 17m 16755 16402
«4 17465 17105 16749 1639625 :7459 17099 16743 16390
»t 17453 17093 16737 16384
n . 17447 17087 16731 1637828 17441 17082 16725 16372
*9 17435 17076 16720 16166
30 17429 17070 16714 16361
3« 17423 17064 16708 16355
3» 17417 17058 16702 16349
33 174" 17052 16696 16343
34 17405 17046 16690 16337
33 17399 17040 16684 16331
J« 17393 17034 16678 16325
37 17387 17028 16672 1632038 17381 17022 16666 16314
39 17373 17016 16660 16308
4« 17369 17010 :6655 16302
4: 17363 17004 16649 16296
4< 17357 16998 16643 16290
43 17351 16992 16637 16284
44 17345 16986 16631 16279
45 17339 16980 16625 16273
46 17333 16974 16619 16267
47 : 7327 16968 16613 1626I48 17321 16963 16607 16235
49 17315 16957 16602 16249
JO 17309 16951 16596 16243
5» 17303 16945 16500 1623852 17297 16939 16584 16232
53 17291 16933 16578 16226
54 17285 16927 16572 16220
55 17279 1692 1 16566 16214
56 17273 16915 16560 1620857 17267 16909 16554 16203
5* 17261 16903 16549 16197
59 17255 16897 16543 1619160 17249 16891 16537 16185
M463
15467 15124 14784 1444615461 15118 14778 14440
15790 15444 15101 14761 14423
15433 15090 14750 14412
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159461 594 1159351592915923
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1577315767157611575515749
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1571515709157031569715692
1568615680156741566915663
156571565 1156461564015634
1562815623156171561115605
:559915594155881558215576
155711556515559
:555315548
1554215536155301552515519
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1539815393153871538115375
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t534115335153301532415318
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1525515250152441523815232
1522715221152151521015204
151.98t5t92151871518115175
15170151641515815153
15M7
150841507915073
1506715061
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1502715022150161501015005
1499914991149881498214976
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t49M14908149021489714891
1488614880148741486914863
1485714852148461484014835
1482914823148181481214806
1474414738147331472714722
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M69914693
1468814682146761467114665
1465914654146481464314637
1463114626146201461414609
1460314598
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14575145691456414558
:4553
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t45:914513145081450214496
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