Saturday, August 18, 2012

Vâlmiki's change of Heart

  
Swami Vivekananda, 1894
There are two great epics in the Sanskrit language, which are very ancient. Of course, there are hundreds of other epic poems. The Sanskrit language and literature have been continued down to the present day, although, for more than two thousand years, it has ceased to be a spoken language. I am now going to speak to you of the two most ancient epics, called the Râmâyana and the Mahâbhârata.

They embody the manners and customs, the state of society, civilisation, etc., of the ancient Indians. The oldest of these epics is called Râmâyana, "The Life of Râma". There was some poetical literature before this — most of the Vedas, the sacred books of the Hindus, are written in a sort of metre — but this book is held by common consent in India as the very beginning of poetry.

The name of the poet or sage was Vâlmiki. Later on, a great many poetical stories were fastened upon that ancient poet; and subsequently, it became a very general practice to attribute to his authorship very many verses that were not his. Notwithstanding all these interpolations, it comes down to us as a very beautiful arrangement, without equal in the literatures of the world.

There was a young man that could not in any way support his family. He was strong and vigorous and, finally, became a highway robber; he attacked persons in the street and robbed them, and with that money he supported his father, mother, wife, and children. This went on continually, until one day a great saint called Nârada was passing by, and the robber attacked him.

The youthful sage Narada at the white-bearded Valmiki's hermitage
The sage asked the robber, "Why are you going to rob me? It is a great sin to rob human beings and kill them. What do you incur all this sin for?" The robber said, "Why, I want to support my family with this money." "Now", said the sage, "do you think that they take a share of your sin also?" "Certainly they do," replied the robber. "Very good," said the sage, "make me safe by tying me up here, while you go home and ask your people whether they will share your sin in the same way as they share the money you make."

The man accordingly went to his father, and asked, "Father, do you know how I support you?" He answered, "No, I do not." "I am a robber, and I kill persons and rob them." "What! you do that, my son? Get away! You outcast!" He then went to his mother and asked her, "Mother, do you know how I support you?" "No," she replied. "Through robbery and murder." "How horrible it is!" cried the mother. "But, do you partake in my sin?" said the son. "Why should I? I never committed a robbery," answered the mother. Then, he went to his wife and questioned her, "Do you know how I maintain you all?" "No," she responded. "Why, I am a highwayman," he rejoined, "and for years have been robbing people; that is how I support and maintain you all. And what I now want to know is, whether you are ready to share in my sin." "By no means. You are my husband, and it is your duty to support me."

The eyes of the robber were opened. "That is the way of the world — even my nearest relatives, for whom I have been robbing, will not share in my destiny." He came back to the place where he had bound the sage, unfastened his bonds, fell at his feet, recounted everything and said, "Save me! What can I do?

The sage said, "Give up your present course of life. You see that none of your family really loves you, so give up all these delusions. They will share your prosperity; but the moment you have nothing, they will desert you. There is none who will share in your evil, but they will all share in your good. Therefore worship Him who alone stands by us whether we are doing good or evil. He never leaves us, for love never drags down, knows no barter, no selfishness."

Vâlmiki composing the Râmâyana
Then the sage taught him how to worship. And this man left everything and went into a forest. There he went on praying and meditating until he forgot himself so entirely that the ants came and built ant-hills around him and he was quite unconscious of it. After many years had passed, a voice came saying,"Arise, O sage! "

Thus aroused he exclaimed, "Sage? I am a robber!" "No more 'robber'," answered the voice, "a purified sage art thou. Thine old name is gone. But now, since thy meditation was so deep and great that thou didst not remark even the ant-hills which surrounded thee, henceforth, thy name shall be Valmiki — 'he that was born in the ant-hill'." So, he became a sage.

And this is how he became a poet. One day as this sage, Vâlmiki, was going to bathe in the holy river Ganga, he saw a pair of doves wheeling round and round, and kissing each other. The sage looked up and was pleased at the sight, but in a second an arrow whisked past him and killed the male dove. As the dove fell down on the ground, the female dove went on whirling round and round the dead body of its companion in grief.

In a moment the poet became miserable, and looking round, he saw the hunter. "Thou art a wretch," he cried, "without the smallest mercy! Thy slaying hand would not even stop for love!" "What is this? What am I saying?" the poet thought to himself, "I have never spoken in this sort of way before." And then a voice came: "Be not afraid. This is poetry that is coming out of your mouth. Write the life of Rama in poetic language for the benefit of the world." And that is how the poem first began. The first verse sprang out of pits from the mouth of Vâlmiki, the first poet. And it was after that, that he wrote the beautiful Râmâyana, "The Life of Rama".

Friday, August 17, 2012

Update - SPX vs Heliocentric Latitude Composite

See also HERE

August 15th was the apex of a triangle; next day breakout to the upside. 
Next potential market CIT derived from this astro-composite is August 28th.

Update - Cycles

The 40 Year Cycle (SLTD) points to August 29 (+/- 4 CD) as THE MAJOR high of the year.
The 4 Year Cycle (LTD) points to a decline from today's high into August 24-27 (inverted).
The 1 Year Cycle (MTD) doesn't show a clear polarity.
The 4 Month Cycle (ITD)  should peak today, followed by a dip into August 21, and another brief rally into the major high on August 24-27.
The Lunar Mystery Cycle also suggests the topping process will last into August 24-27.

2012-08-23 (Thu)  = MER-MAR speed dif 00°59’ [Bayer Rule #02]

2012-08-24 (Fri)   = Bradley CIT [geo]
                          = First Lunar Quarter
                          = MOO 0 Node
                          = MER c.p. VEN [helio]
                          = NEP 0 EAR [helio] = SUN 180 NEP = Merriman Level 1 Cosmic Signature
                          = 60 CD from 6/25/2012 L
2012-08-25 (Sat)  = Tidal CIT
                          = JUP 90 EAR [helio]
                          = MER par PLU [helio]
                          = MAR 120 URA [helio]
2012-08-26 (Sun) = MAR 120 NEP = Merriman Level 1 Cosmic Signature
                          = MER @ 88 [helio] = Myles Wilson Walker-CIT
                          = MOO @ Max S-Dec
                          = 89 CD
from 5/29/2012 H
2012-08-27 (Mon) = MER 120 SAT + MER 120 NEP + MER 120 EAR [all helio]
                          = MOO Inc @ +3.00 degrees
                          = 34 CD from 7/24/2012 L + 55 CD from 7/3/2012 H 

www.fourmilab.ch

Thursday, August 16, 2012

August 15th = NR105

SPX on 2012-08-15 (Wed) = Open 1403.89, High 1407.73, Low 1401.83, Close 1405.53, Range = High - Low = 5.90 = narrowest range of the last 105 Trading Days !!! According to Toby Crabel, an expansion is due no later then after the narrowest daily range in 7 days (= NR7). Aug 15 (Wed) was another NR7-day + Inside Day (ID) = IDNR7 = the ideal setup for breakout move, a decent outside-day (OD). Best guess: a breakout up, even a fake one. Breakout days are often one day wonders, and don’t follow through from the first day move (Larry Williams calls them big Oops). Since we are moving into New Moon (Aug 17 09:56) and Moon in Leo (Aug 17 20:09), it is likely this will be the case (HERE & HERE). 

  • Stretch = 1.56 points = BUY above 1407 or SELL below 1404. Targets: 1417 + 1423. Support: 1402 + 1399 + 1393. 
  • Stretch for ESU12 = 4.43 = BUY above 1408 or SELL below 1399.Target: 1419 + 1431. Support: PP @ 1402 + 1399 + 1393.  
The 10 TD Hurst Cycle LOW is due Aug 16 +/-2 TD. Larger cycles point to a major higher-high around 1450 - 1500 by late August - early September (HERE & HERE & HERE). Other analysts even see a melt-up into October. Time will tell.

Tuesday, August 14, 2012

Geocentric Bradley Index Signal = August 13th (+/- 1 CD)

Donald A. Bradley (1950): Stock Market Prediction.
The Planetary Barometer and How to Use it.
Original

publication by the Llewellyn George Foundation for
Astrological Research; 43 pp.
In 1947 Donald Bradley wrote a small pamphlet titled Stock Market Predictions and sold it for the  astronomical price of $4.00, which in 1947 dollars was quite a bit. Bradley took the 12 planets and gave them a positive or negative rating. From this rating, he produced a sidereal graph that gave a trend for the stock market for the next year, and the next, and the next. In fact, it could go out hundreds of years. The pamphlet had only one year of the stock market (1946), but it  illustrated how well the correlation of his model to the actual price pattern worked for that year.

Unfortunately, no one ever followed up for years, and his work fell into obscurity until 1986 when Jim Twentyman and Larry Pesavento began looking at the Bradley model. They had all of the stock market data on the NYSE from 1876 (10 years after the civil war) through 1986. A 110-year sample size was statistically accurate enough to test Bradley’s theory. The results were quite amazing. Over the past 130 years (since 1876), the correlation is better than 70 percent. Remember that this model can be done years in advance. It is one small indication that the planets may have some say in what happens in the markets.

Although the
Bradley Index shows some excellent correlations between naturally occurring cycles and the stock markets, certain caveats must be mentioned:
(1) It has a tendency to invert (be a mirror image of) the regular model at certain times during the year. Sometimes, there are as many as three inversions in a year while, at other times, there are no inversions, and the model correlates quite well. Inversions can be a problem, but they also present an opportunity because the exact Bradley date (calendar date) has a very high probability of being near a significant trend change, plus or minus one day, better than 80 percent of the time.
(2) The second caveat relates to the fact that it is NOT a trading system; it is a guideline for what financial markets may or may not do between the Bradley dates. Practical experience has shown that, when the Bradley model is working, it behooves the trader to pay attention to the high correlation existing at that time.
However, when it is not correlating, it should be viewed with skepticism. So, it was never meant to be a trading system. In fact, it is still an unproven theory.

(above text is based on A Stock Market Model That "Shoots for the Stars" by Larry Pesavento, 2006)


2012-07-27 (Fri)
2012-08-13 (Mon)
2012-08-24 (Fri)

Monday, August 13, 2012

John H. Nelson - Cosmic Patterns


From 1946 to 1971 John H. Nelson, an astronomer and radio engineer working for RCA Communications Co. in New York, published an article in RCA Review describing a theory for predicting shortwave radio propagation over the North Atlantic. He compared planetary positions relative to the Sun with logs of propagation conditions maintained at RCA's receiving station at Riverhead, Long Island. Nelson found that disturbances occurred primarily in the presence of significant planetary configurations (HERE). Of nearly 1,500 forecasts that he made in 1967 he had an accuracy rate of 93.2%. He did not dare to admit his work was astrologically based. Instead he called it Astro Physics. But he was actually using all the planetary configurations and major angular aspects astrologers have been using ever since. This was a major accomplishment in the scientific realm to prove that astrology does work. 
In summation, after more than 25 years of research in this field of solar system science, I can say without equivocation that there is very strong evidence that the planets, when in certain predictable arrangements, do cause changes to take place in those solar radiations that control our ionosphere.  I have no solid theory to explain what I have observed, but the similarity between an electric generator with its carefully placed magnets and the sun with its ever-changing planets is intriguing.  In the generator, the magnets are fixed and produce a constant electrical current.  If we consider that the planets are magnets and the sun is the armature, we have a considerable similarity to the generator.  However, in this case, the magnets are moving.  For this reason, the electrical-magnetic stability of the solar system varies widely.  This is what one would expect.
John H. Nelson, Cosmic Patterns, 1974

John H. Nelson: Short-wave Radio Propagation Correlation with Planetary Positions.
In: RCA Review, March 1951

Sunday, August 12, 2012

Lunar Chaos Cycle

Hans Hannula a.k.a. Al Larson (1991): A Lunar Chaos Theory. In: Cycles, Vol. 42, No. 1, January/February 1991; pp. 14 - 18
 
Basic Physical Mechanisms: Development of a physical theory of cycles begins with an examination of the solar systems construction. It is composed of ten very important chunks of rock that orbit a ball of burning gas, the sun. The nine planets and our moon are the big rocks. For eons, these rocks have proceeded relentlessly on their courses, carefully balancing the forces they exert on each other and on the sun, and vice versa. 

"Fully 76% of the Venus events coincided with highs and lows,
as did 84% of the Mercury events.
"
To date, two mechanisms have been proposed that could explain the effects of this system on earthly events. Theodor Landscheidt has presented many correlations between the solar system's center of mass and the outburst of solar flares. His theory states that, as the planets rotate, they shift the center of mass of the combined planet/sun system around. Al times, this center of mass actually moves outside the surface of the sun. As it passes the sun's surface, a chaotic boundary condition exists, resulting in outbursts of large solar fares ... As the planets orbit the sun, they exert tidal forces on the sun's gases, much as the moon raises tides on the earth ...  Jupiter, Mercury, Venus, Earth, Mars, and Saturn are the most influential, in that order.
These gas swirls cause a number of solar effects, including sunspots, coronal holes, and solar ares. All these effects combine to vary the amount of radiation that leaves the sun. This solar radiation is carried toward the earth in two ways: 
  1. as direct radiation, such as sunshine and radio-waves, and 
  2. as particles carried by the solar wind. This flow of charged particles forms a torrent of energy that blasts spaceship earth, creating n bow wave and a wake, just as a boat going upstream would do.
This bow wave forms a magnetopause between the earth and the sun, and interacts with the earth’s magnetic field, both shaping it and adding energy to it. At the north and south poles, the charged particles follow the magnetic lines of force, and enter our atmosphere in what is called a Polar Cap Absorption Event. This leads to the auroral oval, producing our Northern and Southern Lights.

The bow wave also creates an envelope about the earth, called the magnetosphere. As the solar wind flows past the earth, the magnetosphere forms a teardrop-shaped envelope of trapped particles that ends in what is called the magnetotail. It is inside this envelope that the moon orbits.

As the solar radiation varies, so does the earth's magnetic field, atmospheric ionization, and temperature. Scientists have tracked down a host of relationships between these events and a variety of earthly phenomena such as climate, weather, crime rates, plant growth rates, frequency of thunderstorms, blood PH levels, psychiatric emergencies, etc. My own work has related these events to market action as well. I believe a third mechanism that involves the moon also is at work .

A Theory of Lunar Chaos: I believe I have discovered another lunar cycle, which I call the “lunar chaos cycle.” My theory is that, as the moon rides high and low and moves closer and further from the earth, it crosses the boundary between the ionized particles trapped in the moon's wake and the fast-flowing solar wind. The figure at left shows boundary crossings at two full-moon positions (1 and 2) and two new-moon positions (3 and 4). Such boundary crossings would lead to sharp disturbances in the earth's magnetic eld, which would affect those of us who live within it. A further perturbation can be theorized, as well- the perturbation of the nearby planets, Mercury and Venus. When the moon's balanced on the edge of the magnetopause, a chaotic balance point exists. Either interior planet can tug the moon into the solar wind, tipping the balance just as Lorenz's Butterfly Effect tips the balance in weather patterns.

A Simple Mathematical Model: To test this theory, I created a simple mathematical model. This model computes the degree of exact alignment of a planet (either Mercury or Venus) with the Earth and Moon, and when the Moon is above or below 3° inclination. This yields a lunar chaos input function for each planet (continue reading HERE).




Calculated and charted with Sergey Tarassov's Timing Solution.

Saturday, August 11, 2012

Chladni Patterns

HERE
In 1787 the German jurist, musician, physicist and astronomer Ernst Florens Friedrich Chladni published Entdeckungen über die Theorie des Klanges (Discoveries Concerning the Theory of Sound), laying the foundations for what came to be called acoustics, the science of sound. Chladni introduced a method to make sound waves visible and to show how sound actually does affect physical matter and has the quality of creating geometric patterns: With the help of a violin bow which he drew perpendicularly across the edge of flat plates covered with sand, he produced those patterns and shapes which today are known as Chladni Patterns (see also HERE & HERE & HERE & HERE). 

These Patterns show primarily two things (see below): areas that are vibrating and areas that are not. When a flat plate of an elastic material is vibrated, the plate oscillates not only as a whole but also as parts. The boundaries between these vibrating parts, which are specific for every particular case, are called node lines and do not vibrate. The other parts are oscillating constantly. If sand is then put on this vibrating plate, the sand (black in the illustration) collects on the non-vibrating node lines. The oscillating parts or areas thus become empty. The converse is true for liquids; that is to say, water lies on the vibrating parts and not on the node lines.  

more HERE

Friday, August 10, 2012

S&P 500 Inflation Adjusted Earnings near ATH

With second-quarter earnings largely in the books (over 84% of S&P 500 corporations have reported), the chart below provides some long-term perspective to the current earnings environment by focusing on 12-month, as reported S&P 500 earnings.

The chart also illustrates how earnings declined over 92% from its Q3 2007 peak to Q1 2009 low which brought inflation-adjusted earnings to near Great Depression lows. Since its Q1 2009 low, S&P 500 earnings have surged (up an inflation-adjusted 1129%) and currently come in at a level that is well above its dot-com bubble peak and fast approaching its credit bubble peak. It is interesting to note that the original run up in real earnings from Great Depression lows to dot-com highs took over 67 years. The current spike has taken 37 months. Also, current Q2 earnings times four would make for a new inflation-adjusted high for S&P 500 corporate earnings. Therefore, as long as earnings do not decline, S&P 500 companies are on pace to make a new all-time record in earnings by Q1 2013 at the latest (HERE).


 For every $20 bln the FED purchases in long term debt the stock market rises 1%
  • QE1 was $1.4 tln: expected rise 70%, actual rise 69%. 
  • QE2 was $600 bln: expected rise 30%, actual rise 29%. 
  • Op Twist was $400 bln: expected rise 20%, actual rise 22%. 
  • Op Twist expanded to $667 bln: expected overall rise 33%, actual rise yet to be determined (HERE).

Thursday, August 9, 2012

SPX vs George Bayer's Rule #27 - Mercury's Speed is 59' or 1°58'

MERCURY’S SPEED IN LONGITUDE GEOCENTRIC OF 59’ AND 1°58’
see also HERE
So let us take Raphael’s Ephemeris again, ... and follow the movements of Mercury from day to day in its longitude. Its speed ranges from 0 to 2.12. In another rule I told you already that 0 and 2.12 in Mercury’s speed bring changes of trend. 
Now we use Mercury’s speed when it is the same as the mean daily motion of the Sun in Longitude or double this amount. The daily motion of the Sun is 59’; double this amount is 1°58’.
Each time Mercury passes these values we can expect changes of trend and, mind you, often times they give big tops and big major bottoms ...
[George Bayer (1940): Stock and Commodity Traders´ Hand-Book of Trend Determination. Carmel, California; p. 35]
 
22.12.2011 01:12 (Thu) = Mercury Speed @ 00°58'59"
04.03.2012 15:25 (Sun) = Mercury Speed @ 00°58'59"
18.04.2012 00:21 (Wed) = Mercury Speed @ 00°58'59"
15.05.2012 20:50 (Tue) = Mercury Speed @ 01°58'00"
07.06.2012 09:24 (Thu) = Mercury Speed @ 01°58'00"
29.06.2012 20:44 (Fri) = Mercury Speed @ 00°58'59"
16.08.2012 03:49 (Thu) = Mercury Speed @ 00°58'59"

26.10.2012 10:56 (Fri) = Mercury Speed @ 00°58'59"
12.11.2012 09:20 (Mon) = Mercury Speed @ 00°58'59"
20.11.2012 22:40 (Tue) = Mercury Speed @ 00°58'59"
03.12.2012 22:57 (Mon) = Mercury Speed @ 00°58'59"
16.02.2013 06:24 (Sat) = Mercury Speed @ 00°58'59"


Calculated and charted with Sergey Tarassov's Timing Solution.

SPX vs George Bayer's Rule #38 - Latitude of Mercury

MERCURY LATITUDE HELIOCENTRIC *
* = latitude in heliocentric ecliptical coordinates
In the present rule we shall use the heliocentric Latitude of Mercury ... There are four values in this line which bring forth changes of trend. These values are:

0°00'  =      0.00 degrees
3°21'  =
+/- 3.35 degrees
6°42'  =
+/- 6.70 degrees, and 
7°00’ =
+/- 7.00 degrees

The inside values are effective on either side, on the way up also on the way down.

Some mighty fine tops and bottoms are produced when Mercury in this motion passes the above mentioned degrees. Check them on past or recent performance and you will say yes, it is so.

... During 1938, especially during the first half of the year practically all the changes registered on this one rule alone.
[George Bayer (1940): Stock and Commodity Traders´ Hand-Book of Trend Determination. Carmel, California; p. 41]


SPX vs George Bayer's Rule #40 - Latitude of Venus

VENUS HELIOCENTRIC LATITUDE* DIVIDED INTO PARTS
* = latitude in heliocentric ecliptical coordinates
This rule is intended to make you find the very small moves in Wheat lasting only a few days.
However, since we nearly always have to do with bottoms, tops, and gaps-either one and the changes explained here come one after the other, we must be in a position to watch the actual performance of Wheat at very close distance.

We turn to page 167 and up in the Nautical Almanac of the year 1937 or to page 187 in the year 1940. In the center of the page we have our column; Heliocentric Latitude of Venus.

We note the values vibrate between 3°23 and 0°, going from plus to minus to both extremes. 0° is the center of the vibration.

I have discovered that the following values within this vibration give changes, on the plus side as well as on the minus side. They miss occasionally one of the values, but, as said, we must be at the board to watch the sales and prices. My notes say: there are 17 values in the entire cycle movement, 8 on each side of the zero point. In my tests I found that one miss occurs (i.e. nothing happens) in one entire cycle.

Here is the rule: Changes of trend for minor moves occur at the following Latitude points of Venus when Venus passes over it;
  
0°00’ =       0.000 degrees
0°13’ = +/- 0.217 degrees
1°50’ = +/- 1.833 degrees
2°17’ = +/- 2.283 degrees
2°30’ = +/- 2.500 degrees
3°00’ = +/- 3.000 degrees
3°17’ = +/- 3.283 degrees
3°23’ = +/- 3.383 degrees
[George Bayer (1940): Stock and Commodity Traders´ Hand-Book of Trend Determination. Carmel, California; p. 42]


"It is the Thunderbolt that Steers the Universe !"

Heraclitus of Ephesus (c. 535 – c. 475 BCE)

The Beauty of Maths

Sequential Inputs of numbers with 8
1 x
8 + 1 = 9
12 x
8 + 2 = 98
123 x
8 + 3 = 987
1234 x
8 + 4 = 9876
12345 x
8 + 5 = 98765
123456 x
8 + 6 = 987654
1234567 x
8 + 7 = 9876543
1234567
8 x 8 + 8 = 98765432
1234567
89 x 8 + 9 = 987654321
......................................

Sequential 1's with 9
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 + 10 = 1111111111
......................................

Sequential 8's with 9
9 x 9 + 7 =
88
9
8 x 9 + 6 = 888
9
87 x 9 + 5 = 8888
9
876 x 9 + 4 = 88888
9
8765 x 9 + 3 = 888888
9
87654 x 9 + 2 = 8888888
9
876543 x 9 + 1 = 88888888
9
8765432 x 9 + 0 = 888888888
......................................

Numeric Palindrome with 1's
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
......................................

Without 8
12345679 x 9 = 111111111
12345679 x 18 = 222222222
12345679 x 27 = 333333333
12345679 x 36 = 444444444
12345679 x 45 = 555555555
12345679 x 54 = 666666666
12345679 x 63 = 777777777
12345679 x 72 = 888888888
12345679 x 81 = 999999999
......................................

Sequential Inputs of 9
9 x 9 = 81
99 x 99 = 9801
999 x 999 = 998001
9999 x 9999 = 99980001
99999 x 99999 = 9999800001
999999 x 999999 = 999998000001
9999999 x 9999999 = 99999980000001
99999999 x 99999999 = 9999999800000001
999999999 x 999999999 = 999999998000000001
......................................

Sequential Inputs of
6
6 x 7 = 42
66 x 67 = 4422
666 x 667 = 444222
6666 x 6667 = 44442222
66666 x 66667 = 4444422222
666666 x 666667 = 444444222222
6666666 x 6666667 = 44444442222222
66666666 x 66666667 = 4444444422222222
666666666 x 666666667 = 444444444222222222
......................................



A Magic Square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. Building on the work of Ahmad ibn ‘Ali al-Buni, in about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia expounding on the magical virtues of seven magical squares of orders 3 to 9, each associated with one of the astrological planets.

Saturn = 15
4 9 2
3 5 7
8 1 6
Jupiter = 34
4 14 15 1
9 7 6 12
5 11 10 8
16 2 3 13
Mars = 65
11 24 7 20 3
4 12 25 8 16
17 5 13 21 9
10 18 1 14 22
23 6 19 2 15
Sol = 111
6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31
Venus = 175
22 47 16 41 10 35 4
5 23 48 17 42 11 29
30 6 24 49 18 36 12
13 31 7 25 43 19 37
38 14 32 1 26 44 20
21 39 8 33 2 27 45
46 15 40 9 34 3 28
Mercury = 260
8 58 59 5 4 62 63 1
49 15 14 52 53 11 10 56
41 23 22 44 45 19 18 48
32 34 35 29 28 38 39 25
40 26 27 37 36 30 31 33
17 47 46 20 21 43 42 24
9 55 54 12 13 51 50 16
64 2 3 61 60 6 7 57
Luna = 369
37 78 29 70 21 62 13 54 5
6 38 79 30 71 22 63 14 46
47 7 39 80 31 72 23 55 15
16 48 8 40 81 32 64 24 56
57 17 49 9 41 73 33 65 25
26 58 18 50 1 42 74 34 66
67 27 59 10 51 2 43 75 35
36 68 19 60 11 52 3 44 76
77 28 69 20 61 12 53 4 45


  

The order-4 magic square in Albrecht Dürer's engraving Melencolia I is believed to be the first seen in European art. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, and the corner squares (of the 4 x 4 as well as the four contained 3 x 3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3 + 8 + 14 + 9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle, the two sets of four symmetrical numbers (2 + 8 + 9 + 15 and 3 + 5 + 12 + 14), the sum of the middle two entries of the two outer columns and rows (5 + 9 + 8 + 12 and 3 + 2 + 15 + 14), and in four kite or cross shaped quartets (3 + 5 + 11 + 15, 2 + 10 + 8 + 14, 3 + 9 + 7 + 15, and 2 + 6 + 12 + 14). The two numbers in the middle of the bottom row give the date of the engraving: 1514. The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist. Dürer's image was distributed widely, and travelled as far away as India, where the Mughal miniaturist Farrukh Beg referenced the work in his 1615 miniature The Old Sufi.