An Introduction to Yantra Magic Squares and Agrippa–Type Magic Matrices

An introduction to Yantra magic squares and Agrippa–type magic matrices

George P. H. Styan2

January 18, 2012

2 This beamer ﬁle is for an invited talk presented as a video on Thursday, 5 January 2012, at the International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India, 2–11 January 2012. This talk is based on joint research with Ka Lok Chu & Götz Trenkler and, in part, on Reports 2011-07 and 2012-01 (lecture notes) from the Department of Mathematics and Statistics, McGill University, Montréal. George P. H. Styan3 Yantra & Agrippa-type magic Acknowledgements: January 18, 2012 B1-01

This beamer ﬁle is for an invited talk presented as a video on Thursday, 5 January 2012, at the International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India, 2–11 January 2012.

This talk is based on joint research with Ka Lok Chu & Götz Trenkler and, in part, on Reports 2011-07 and 2012-01 (lecture notes) from the Department of Mathematics and Statistics, McGill University, Montréal.

Special thanks go to Christian Boyer for constructing the 22 × 22 Fermat–Boyer magic square and to Daniel J. H. Rosenthal for drawing the Jupiter Planet Yantra. In addition, we are grateful to Nicolas C. Ammerlaan and Thomas W. Ammerlaan for making the video, and to Oskar Maria Baksalary, S. W. Drury, and Evelyn Matheson Styan for their help. This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.

The handwritten image of the 14 × 14 Fermat magic square by Félix Vicq d’Azyr (1746–1794) is from MS #10556, Département de manuscrits, Bibliothèque nationale de France, Paris. My sincere thanks to Christian Boyer for the image and to the Bibliothèque nationale de France (BnF) for allowing us to reproduce it here.

George P. H. Styan4 Yantra & Agrippa-type magic Introduction B1-02

I want to tell you about

Yantra magic squares

and

Agrippa-type magic matrices

And to illustrate our ﬁndings with postage stamps

George P. H. Styan5 Yantra & Agrippa-type magic Gan. ita Kaumud¯ı (1356) by N¯ar¯ayan.a Pan.d.ita B1-03

George P. H. Styan6 Yantra & Agrippa-type magic Gan. ita Kaumud¯ı (1356) by N¯ar¯ayan.a Pan.d.ita: reprinted 2011 B1-04

George P. H. Styan7 Yantra & Agrippa-type magic yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM

sensuality, passion, sexual pleasure, love matters, luxuries, arts and music. It brings wealth, comfort, attraction to the opposite sex in the early part of life, a well-proportioned body, and the attractive features necessary for a sensuous nature. It makes its natives tender, gentle, and considerate. It inspires them to be poets, musicians, and seekers of truth and knowledge (secret yantrassciences). for surya chandra The nativemangal buddhayantras loves gurufor the surya shukra companychandra shani mangalrahu ofketu buddhathe navgraha members guru yantra shukra shaniof the rahu ketu navgraha yantra 11-09-10 11:26 AM 11-09-10 11:26 AM opposite sex, artists, and musicians. yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM

yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM Start using this Yantra on any Saturday Morning. yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM sensuality, passion, sexual pleasure, love matters,Rahu luxuries, Yantra arts: For the +ve influence of planet RAHU. and music. It brings wealth,The Sun comfort, rules our attraction life force, to the the power opposite of will, intellect,Online shoppingbrilliance, of astro-remedies Online shopping of astro-remediesOnline shopping of astro-remedies is temporarily closed. sex in the early part ofprosperity, life,is temporarily a well-proportioned success closed. in worldlyRahuis body, temporarilyis affairs,regarded and closed. the wealth, as similar personal, to Saturn conduct, in its effect. attractive features necessaryactivity, for cheerfulness, a sensuous nature.goodRahu fortune, isIt makes capable wisdom, its of giving ambition, power fame, when the rightly posited or exalted, natives tender, gentle, understanding and considerate. of the It inspires phenomenaland it themgives world,fame, to be success and the in knowledge politics, money, of and physical GO TOP beauty GO TOP GO TOP A 9 9 compositepoets, musicians, “Navagraha and medicine. Yantra”seekers of It magic truth also governs and square (i.e., knowledge our attraction). relationship (secret with temples andB1-05 holy × sciences). The native lovesplaces. the company of the members of the opposite sex, artists, and musicians. Rahu rules no sign of the zodiac but it has authority over Virgo. Start using this Yantra onStart any using Friday this Morning. Yantra on any Wednesday Morning. Start using this Yantra on any Monday Morning.

Shani Yantra: For the Guru+veOnline influence Yantrashopping of( Brihaspatiastro-remediesof planet SATURN. Yantra): For the planet JUPITER. is temporarily closed. Mangal Yantra: For the +ve influence of planet MARS. The Saturn gives its nativesJupiter integrity, rules wisdom, religion, spirituality, philosophy, fame, spirituality, wealth, and patience, ability to lead,progeny. authority, If favorable, long life,Mars this organizational planetgives GO TOP energy, gives name,determination, fame, success, great administrative ability, abilities, sincerity, honesty,honor, love wealth, of justice, progeny awarenessand and independent good of relationship right spirit. with It rules progeny, over courage,and bravery, patience, and wrong because it rulesit brings the inner benefits conscience, to whateverself-confidence, nonattachment, planets orleadership house it is associated abilities, physical strength, and asceticism. with. forcefulness. Start using this Yantra on any Sunday Morning.

Chandra Yantra: For the +ve influence of planet MOON.

Online shopping of astro-remedies The moon rules peace of mind,Online shopping comfort, of astro-remediesgeneral well-being,is temporarily closed.and Online shopping of astro-remedies is temporarily closed. is temporarily closed. also the fortune of a person. It makes us moody, emotional, and sensitive. The Moon gives illumination, intuitive nature, GO TOP GO TOP GO TOP yantrassensuality, for surya chandra taste, mangal love buddhayantras of guru for poetry, surya shukra chandra shani fine mangalrahu arts, ketu buddha music, navgraha guru jewelryyantra shukra shani and rahu ketu navgraha yantra 11-09-10 11:26 AM 11-09-10 11:26 AM attractive appearance.

Start using this Yantra onStart any using Friday this Morning. Yantra onStart any using Sunday this Morning. Yantra on any Wednesday Morning.

Shani Yantra: For the Chandra+ve influence Yantra of: Forplanet theKetu SATURN. +ve Yantra influence: For ofthe planet +ve influence MOON. of planet KETU. Online shopping of astro-remediesOnline shopping of astro-remedies is temporarily closed. is temporarily closed. The Saturn gives its nativesThe moon integrity, rules wisdom, peace ofKetuspirituality, mind, bestows comfort, fame, spiritual general tendencies, well-being, asceticism, and and nonattachment patience, ability to lead,also the authority, fortune of long a person. life,to worldly organizational It makes desires us moody, and ambitions. emotional, Ketu and is also a half-planet that GO TOP abilities, sincerity, honesty,sensitive. love ofThe justice, Moon awarenessbestows gives wisdom, illumination, of right powers intuitive of discrimination, GO TOP nature, spiritual knowledge, and wrong because it rulessensuality, the inner taste, conscience, love and of nonattachment, poetry, the knowledge fine arts, of music, the self.jewelry It gives and psychic abilities to its and asceticism. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra attractive appearance. natives and makes them mastersPage of 5 theof 7 healing arts, natural George P.Start H. Styan using8 thisYantra Yantra & onStarthealing, Agrippa-type any using Thursday tantric this magic Yantra healing,Morning. on healingany Tuesday persons Morning. affected by evil spirits and ghosts.

Shukra Yantra: For theKetuBuddha +ve rules influence noYantra sign :of Forof the planet the zodiac +ve VENUS. but influence it has authority of planet over MERCURY. Pisces. Online shopping of astro-remediesOnline shopping of astro-remedies is temporarily closed. is temporarily closed. Venus governs the Mercury refined is attributes- the planet romance, of intellect beauty, and communication. It rules traders and businessmen. It rules our intelligence, speech, self- GO TOP GO TOP http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraconfidence, humor, wit, astrology, mathematics, and short Page 6 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra journeys. Page 4 of 7

Start using this Yantra onStart any using Thursday this Yantra Morning. on any Wednesday Morning. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7

Guru Yantra (Brihaspati Yantra): For the planet JUPITER. Navgriha Yantra: For the +ve influence of ALL THE NINE PLANETS.

Jupiter rules religion, philosophy, spirituality, wealth, and This yantra is beneficial for all the people and used to balance progeny. If favorable, this planet gives name, fame, success, the influences of all the planets. honor, wealth, progeny and good relationship with progeny, and it brings benefits to whatever planets or house it is associated with. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraOnline shopping of astro-remedies Page 3 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra is temporarily closed. Page 5 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7

GO TOP

Online shopping of astro-remedies is temporarily closed.

Feed Back | About Us | Contact Us | Terms of Use | Disclaimer | Privacy Policy | Security . Copyright © 2004-2010 AstrologyPlus.in- All rights Onlinereserved. shopping of astro-remedies is temporarily closed. Astrology Plus provides energized yantras crystal shri yantra sriyantra lockets astroremedies rudrakshas bead zodiac gems pandents, other astro remedies

GO TOP

Start using this Yantra on any Thursday Morning.

Shukra Yantra: For the +ve influence of planet VENUS.

Venus governs the refined attributes- romance, beauty,

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 4 of 7

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 7 of 7 Yantra magic squares for 9 planets B1-06

We study various properties of Yantra magic squares: “Yantra” is a “geometrical diagram used like an icon usually in meditation”.

A classic n n magic square (magic matrix) is an arrangement of the numbers × 1, 2,..., n2 in an n n array so that the numbers in every row, every column × and in the two main diagonals add up to the same magic sum

n(n2 + 1) m = . 2

4 9 2 When n = 3 the 3 3 classic Luoshu magic matrix 3 5 7 with m = 15. ×   8 1 6     George P. H. Styan9 Yantra & Agrippa-type magic Magic-square planetary talismans MB-01 Yantra magic squares for 9 planets B1-07 We study various properties of Yantra magic squares: We begin with 3 3“Yantra” magic squares is a “geometrical depicted on diagram “Planet used Yantras”: like an icon usually in meditation”. × We begin with 3 3 magic squares depicted on “Planet Yantras”: ×

1 Sun/Surya, 2 Moon/Chandra, 3 Mars/Mangala, 4 Mercury/Budha, 5 Jupiter/Brihaspati–Guru 6 Venus/Shukra 7 Saturn/Shani, 8 Rahu, 9 Ketu.

George P. H. Styan5 Yantra & Agrippa

George P. H. Styan10 Yantra & Agrippa-type magic The lunar nodes Rahu and Ketu B1-08

The lunar nodes Rahu and Ketu are the points where the moon’s path in the sky crosses the sun’s path.

We also consider “Navagraha Yantras”: 3 3 arrangements of the nine × Planet Yantras, usually with a Sun/Surya Planet Yantra in the centre.

George P. H. Styan11 Yantra & Agrippa-type magic Vaidika Pradishta Navagraha Yantra with Sun/Surya in the centre B1-09

Placing a Sun/Surya Planet Yantra in the centre, we construct 8 Navagraha Yantras, each in the form of a 9 9 composite fully-magic square. ×

George P. H. Styan12 Yantra & Agrippa-type magic Philatelic L1 of the 9 planets with Jupiter in the centre B1-10

George P. H. Styan13 Yantra & Agrippa-type magic The 9 planets in magic-square type L1 with Jupiter in the centre B1-11

6 1 8 L1 = 7 5 3 2 9 4     Venus Sun Rahu = Saturn Jupiter Mars  Moon Ketu Mercury     Dahomey Jaipur Laos =  Ciskei Ciskei Ciskei  . Italy Ketu Austria    

In the matrix L1 the numbers in the rows, columns, and two main diagonals all add up to same magic sum 15.

George P. H. Styan14 Yantra & Agrippa-type magic Planet Yantras each with a 3 3 magic square B1-12 ×

The only images of the 9 Planet Yantras depicting magic squares that we have found each depict a 3 3 magic square, respectively ×

Lk = L1 + (k 1)E, k = 1, 2, ..., 9, −

where L1 is the 3 3 double-ﬂipped Luoshu fully-magic matrix ×

6 1 8 0 0 1 4 9 2 0 0 1 L1 = 7 5 3 = 0 1 0 3 5 7 0 1 0 = FL0F. 2 9 4 1 0 0 8 1 6 1 0 0                

Here F is the 3 3 “ﬂip matrix” and L0 the “standard” Luoshu magic matrix. ×

George P. H. Styan15 Yantra & Agrippa-type magic yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM

The Sun rules our life force, the power of will, intellect, brilliance, prosperity, success in worldly affairs, wealth, personal, conduct, Online shopping of astro-remedies is temporarily closed. activity, cheerfulness, good fortune, wisdom, ambition, fame, the yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM understanding of the phenomenal world, and the knowledge of medicine. It also governs our relationship with temples and holy GO TOP sensuality, passion, sexual pleasure, love matters, luxuries, arts places. and music. It brings wealth, comfort, attraction to the opposite sex in the early part of life, a well-proportionedyantras for surya body,chandra andmangal the buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM Start using this Yantra on any Saturday Morning. attractive features necessary for a sensuous nature. It makes its yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM natives tender, gentle, and considerate. It inspires them to be yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM poets, musicians, and seekers of truth and knowledge (secret yantras for surya chandra mangal buddha guru shukra shanisensuality, rahu ketu navgraha yantra passion, sexual pleasure, love11-09-10 matters, 11:26 AMRahu luxuries, Yantra arts: For the +ve influence of planet RAHU. yantrassciences). for surya chandra The nativemangal buddhayantras loves gurufor the surya shukra companychandra shani mangalrahu ofketu buddhathe navgraha members guru yantra shukra shaniof the rahu ketu navgraha yantra 11-09-10 11:26 AM 11-09-10 11:26 AM and music. It brings wealth,The Sun comfort, rules our attraction life force, to the the power opposite of will, intellect,Online shoppingbrilliance, of astro-remedies opposite sex,sensuality, artists, passion, and sexual musicians. pleasure, love matters, luxuries,Online artsshopping of astro-remediesOnline shopping of astro-remedies and music. It brings wealth, comfort, attraction to the opposite is temporarily closed. sex in the early part ofprosperity, life,is temporarily a well-proportioned success closed. in worldlyRahuis body, temporarilyis affairs,regarded and closed. the wealth, as similar personal, to Saturn conduct, in its effect. sex in the early part of life, a well-proportioned body, and the yantras for surya chandra mangal buddha guru shukraattractive shani rahu features attractiveketu necessarynavgraha featuresyantrafor a sensuous necessary nature.activity, It makes for cheerfulness, itsa sensuous nature.goodRahu fortune, isIt makes capable11-09-10 wisdom, its 11:26 of giving AM ambition, power fame, when the rightly posited or exalted, natives tender,natives gentle, and tender, considerate. gentle, It inspires understanding and them considerate. to be of the It inspires phenomenaland it themgives world,fame, to be success and the in knowledge politics, money, of and physical GO TOP beauty GO TOP GO TOP poets, musicians,poets, and seekers musicians, of truth and and knowledge medicine. seekers (secret of It truth also governs and knowledge our relationship (secret with temples and holy The Sunyantras rulessciences). for surya our chandra The life nativemangal force, buddhayantras loves gurufor the the surya shukra company chandrapower shani mangalrahu of ketuof buddhathe navgraha will, members guru yantra shukra intellect, shaniof the rahu ketu brilliance, navgraha yantra (i.e., attraction).11-09-10 11:26 AM 11-09-10 11:26 AM prosperity,opposite success sex, sciences). artists, in worldly and musicians. The affairs, native wealth, lovesplaces. personal, the company conduct, of the membersOnline of shopping the of astro-remedies is temporarily closed. yantras for surya chandra mangal buddha guru shukra shaniopposite rahu ketu navgraha sex, yantra artists, and musicians. 11-09-10 11:26Rahu AM rules no sign of the zodiac but it has authority over Virgo. activity, cheerfulness,Start good using fortune, this Yantra wisdom, onStart anyambition, using Friday this fame,Morning. Yantra the on any Wednesday Morning. understandingThe Sun of rules the our life phenomenal force, the power world,of will, intellect, and brilliance, the knowledge of Start using this Yantra on any Monday Morning. prosperity, success in worldly affairs, wealth, personal, conduct, Online shopping of astro-remedies GO TOP medicine. It also governs our relationship with temples andis temporarily holy closed. places. activity, cheerfulness, good fortune, wisdom, ambition, fame, the understanding of Shani the phenomenal Yantra world,: For and the the knowledgeGuru+veOnline influence Yantrashopping of of( Brihaspatiastro-remediesof planet SATURN. Yantra): For the planet JUPITER. medicine. It also governs our relationship with temples and holy GO TOP yantras for surya chandra mangal buddhais temporarily guru shukra closed. shani rahu ketuMangal navgraha Yantrayantra : For the +ve influence of planet MARS.11-09-10 11:26 AM places. Start using this Yantra on any Saturday Morning. The Saturnyantras for surya gives chandra mangal its buddha nativesJupiter guru shukra integrity, shani rulesrahu ketu navgraha wisdom, religion, yantra spirituality, philosophy, fame, spirituality,11-09-10 wealth, 11:26 AM and yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra Start using this Yantra on any Saturday Morning. 11-09-10 11:26 AM yantras for surya chandra mangal buddha guru shukra shanipatience, rahu ketu navgraha yantra ability to lead,progeny. authority, If favorable, long 11-09-10 11:26 life,Mars this AM organizational planet gives GO TOP energy, gives name, determination, fame, success, great administrative ability, yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM yantras for surya chandraabilities, mangal buddha guru sincerity, shukra shani rahu ketu honesty, navgrahahonor, yantra love wealth, of justice, progeny awarenessand and independent good 11-09-10 of relationship right 11:26 spirit.AM with It rules progeny, over courage,and bravery, patience, sensuality,sensuality,yantras passion, for surya chandra passion, sexualmangal sexual buddha guru pleasure, pleasure, shukra shani loverahu ketu love matters, navgraha Rahu matters, yantra luxuries, YantraRahu arts :luxuries, For theYantra +ve artsinfluence: For theof planet11-09-10 +ve 11:26RAHU. influence AM of planet RAHU. and music. Itand brings wrong wealth,The Sun comfort,because rules our attraction life it force, rules it to the thebrings the power opposite inner of benefits will, intellect,conscience,Online to shoppingbrilliance, whateverself-confidence, of nonattachment,astro-remedies planets or leadership house it is associated abilities, physical strength, Online shopping of astro-remediesOnline shopping of astro-remedies and music. It brings wealth,Thesensuality, Sun comfort, passion,rules sexualour attraction life pleasure, force, love to matters, the the power luxuries, oppositeis temporarily artsof will, closed. intellect, brilliance, sex in the early part ofprosperity, life,is temporarily a well-proportioned success closed. in worldlyRahuis body, temporarilyis affairs,regarded andOnline closed. the wealth, as similar shopping personal, to Saturn ofconduct, astro-remedies in its effect.Online shopping of astro-remedies and asceticism.andOnline music. shopping It brings wealth,of astro-remedieswith. comfort, attraction to the opposite forcefulness. attractive features necessaryactivity, for cheerfulness, a sensuous nature.goodRahu fortune, isIt makes capable wisdom, its of giving ambition, power fame, when the rightly positedis temporarily or exalted, closed. sex in the early partStart ofprosperity, using sex life,is intemporarily thethis a early well-proportioned Yantra success part closed. of life,on in a any well-proportioned worldlyRahu Sundayis body, temporarilyis affairs,regarded body, Morning.and and closed. the the wealth, as similar personal, to Saturn conduct, in its effect. natives tender, gentle, understanding and considerate. of the It inspires phenomenaland it themgives world,fame, to be success and the in knowledge politics, money, of and physical GO TOP beauty attractive features necessaryactivity,attractive for features cheerfulness, a sensuous necessary for nature. a goodGO sensuous Rahu TOP fortune, nature. isIt makescapable It makes wisdom, GO its its TOP of giving ambition, power fame, when the rightly posited or exalted, poets, musicians, and nativesmedicine. seekers tender, of It truth also gentle, governs and and (i.e., knowledge considerate. our attraction). relationship (secret It inspires with themtemples to beand holy natives tender,sciences). gentle, The native understanding and lovespoets,places. theconsiderate. musicians, company and of of seekersthe the It members inspires of phenomenal and truth of andtheit themgives knowledge world,fame, to (secret be success and the in knowledge politics, money, of and physical GO TOP beauty opposite sex, artists, andyantrassciences). musicians. for surya chandra The nativemangal buddhayantras loves Rahugurufor the surya shukra rules company chandra shani no mangalrahu sign of ketu buddha the navgraha ofGO members guru the TOP yantra shukra zodiac shani of the but rahu ketuit has navgraha authority yantra over GO Virgo. TOP 11-09-10 11:26 AM 11-09-10 11:26 AM Start using this Yantra on any Wednesday Morning. poets, musicians,Start using this and YantraChandra medicine. seekers onopposite any Friday Yantrasex, of artists, It Morning. truth also :and For musicians. governs and the (i.e., knowledge+ve our influenceattraction). relationship (secret of planet with temples MOON. and holy Start using this Yantra on any Monday Morning. sciences). The native lovesplaces. the company of the members of the yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra Online11-09-10 shopping 11:26 AM of astro-remedies Guru Yantra (Brihaspati Yantra): For the planet JUPITER. opposite sex,Shani artists, Yantra : andFor the musicians. +veOnline influence shopping of astro-remediesof planet SATURN.OnlineRahu shopping rules ofno astro-remedies sign of theis temporarily zodiac butclosed. it has authorityOnline shopping over Virgo. of astro-remedies The moonis temporarily rules closed. peace ofMangal mind, Yantra comfort,: For the +ve general influence well-being, of planet MARS. and Start using this Yantra onStart Theany Sun using Friday rules our this lifeMorning. force,Yantra the poweris on temporarily of any will, intellect, Wednesday closed. brilliance, Morning. prosperity, success in worldly affairs, wealth, personal, conduct, Online shopping of astro-remedies is temporarily closed. The Saturn givesalso its the nativesJupiter fortune integrity, rules ofwisdom, religion, a person. spirituality, philosophy, It fame, makes spirituality, us wealth,moody, and emotional,is temporarily closed. and activity, cheerfulness, good fortune,Start wisdom, usingambition, thisfame, theYantra on any Monday Morning. patience, ability to lead,progeny. authority, If favorable, long life,Mars this organizational planet gives GO TOP energy, gives name, determination, fame, success, great administrative ability, understanding of the phenomenal world, and the knowledge of GO TOP sensitive.honor, The wealth, progeny Moonand and independent gives good relationship illumination, spirit. with It rules progeny, over courage, intuitiveand bravery, nature, patience, abilities, sincerity, honesty,medicine. love It of also justice, governs awareness our relationship of right with temples and holy GO TOP GO TOP it brings benefits to whateverself-confidence, planets orleadership house it is associated abilities, physical strength, GO TOP Shani Yantraand wrongyantras: becauseForsensuality, forthe it surya rules places.Guru+veOnline chandrathe taste,influence innerYantrashopping mangal conscience, love of buddha( yantrasBrihaspatiastro-remediesof ofnonattachment, planet guru for poetry, surya shukra SATURN. chandraYantra) shani fine mangalrahu arts,: ketuFor buddha music, navgraha the guru planet jewelryyantra shukra shaniJUPITER. and rahu ketu navgraha yantra 11-09-10 11:26 AM 11-09-10 11:26 AM and asceticism. with. forcefulness. attractive appearance.yantras for surya chandra mangalMangal buddha guru Yantrashukra shani rahu: ketu For navgraha the yantra +ve influence of planet11-09-10 MARS. 11:26 AM Start using this Yantra onis any temporarily Sunday Morning. closed. Start using this Yantra on any Saturday Morning. yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM The Saturn gives its nativesJupiteryantras for surya integrity, chandra rules mangal buddha wisdom, religion, guru shukra shani spirituality, rahu philosophy, ketu navgraha yantra fame, spirituality, wealth, 11-09-10 and 11:26 AM Start using this Yantra onStart any using Friday thisRahu Morning. Yantra Yantra: For onStart the any +ve usinginfluence Sunday this of planet Morning. Yantra RAHU. on any Wednesday Morning. patience, abilityChandra Yantra to : lead, Forprogeny.sensuality, the +ve authority, influence passion, If favorable, sexual of long planet pleasure, MOON. life,Mars this love matters, organizational planet gives GO luxuries, TOP energy, gives arts name, determination, fame, success, great administrative ability, and music. It brings wealth,The Sun comfort, rules our attraction life force, to the the power opposite of will, intellect,Online shoppingbrilliance, of astro-remedies OnlineOnline shopping shopping of astro-remedies of astro-remediesOnline shopping of astro-remedies sex in the early part ofprosperity, life, a well-proportioned successand independent in worldly body, affairs, and the wealth, spirit. personal,is temporarily It conduct, rules closed. over courage, bravery, patience, abilities, sincerity,The moon rules honesty, peacehonor, of mind,Online love wealth,shopping comfort, of of astro-remediesgeneral justice, progenyis temporarily well-being,is temporarily awareness closed. and closed.and goodRahu is temporarilyis of regarded relationshipOnline right closed.shopping as similar of astro-remedies towith Saturn progeny,in its effect. and attractiveis temporarily features closed. necessary activity, for cheerfulness, a sensuous nature.goodRahu fortune, isIt ismakes capable temporarily wisdom, its of givingclosed. ambition, power fame, when the rightly posited or exalted, also the fortune of a person. It makes us moody, emotional,self-confidence, and leadership abilities, physical strength, and wrong because it rulesit natives brings the tender, inner benefits gentle, conscience, understanding and toconsiderate. whatever of nonattachment, the It inspires phenomenaland planets it themgives world,fame, to be or success and house the in knowledge politics, it money, is of associated and physical GO TOP beauty Shani Yantra: For the Chandra+ve influence Yantra GO TOP of : Forplanet the Ketu SATURN. +ve GO Yantra TOP influence : For ofthe planet +ve influence MOON. of planet KETU. sensitive. The Moon poets, gives musicians, illumination, and medicine. seekers intuitive of It truth also nature, governs and (i.e., knowledge our attraction). relationship (secret GO TOP with temples and holy and asceticism. with. GOforcefulness. TOP GO TOP yantrassensuality, for surya chandra taste, mangal love buddhayantras sciences). of guru for poetry, surya shukra Thechandra shani fine native mangalrahu arts, ketu loves places.buddha music, navgraha the guru companyjewelryyantra shukra shani and of rahu the ketu members navgraha yantraof the Online shopping11-09-10 11:26of astro-remedies AM 11-09-10Online 11:26 shopping AM of astro-remedies Start usingattractive this Yantra appearance. on oppositeany Sunday sex, artists, andMorning. musicians. Rahu rules no sign of the iszodiac temporarily but it has authority closed. over Virgo. is temporarily closed. Start using this Yantra onStart any using Friday this Morning. Yantra on any Wednesday Morning. The Saturn gives its nativesThe moon integrity, rulesStart using wisdom, peace this Yantra ofKetuspirituality, on mind,any bestows Monday comfort, Morning. fame, spiritual general tendencies, well-being, asceticism, and and nonattachment Start using this Yantra onStart any using Friday this Morning. Yantra on any Sunday Morning. patience, ability to Start lead,also using the authority, this fortuneYantra on anyof long Wednesdaya person. life, Morning. organizational It makes us moody, emotional, and Guru Yantra (Brihaspati Yantra): For theto planet worldly JUPITER. desires and ambitions. Ketu is also a half-planet that Shani Yantra: For the +veOnline influence shopping of astro-remediesof planet SATURN. GO TOP GO TOP Chandra Yantraabilities,: For the sincerity,+ve influence honesty,is sensitive.temporarily of planet closed. love of TheMOON.Mangal justice, Yantra Moon : awarenessFor bestows the gives +ve influence wisdom, illumination, of rightof planet powers MARS. intuitive of discrimination, nature, spiritual knowledge, Shani Yantra: For the Chandra+ve influence Yantra of: Forplanet theKetu SATURN. +ve Yantra influence: For ofthe planet +ve influence MOON. of planet KETU. yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra The Saturn11-09-10 11:26 AM gives its nativesJupiter integrity, rules wisdom, religion, spirituality, philosophy,Online fame, shopping spirituality, of astro-remedies wealth, and sensuality,and passion, sexual pleasure,wrong love matters, luxuries, arts because it rulessensuality,Online theshopping inner of astro-remedies taste, conscience, love and of nonattachment, poetry, the knowledge fine arts, of music, the self.jewelry It gives and psychic abilities to its and music. It brings wealth, comfort, attraction to the opposite sex in the early part of life, a well-proportioned body, and the progeny. If favorable, Online Mars this shopping planet gives GO TOP energy, gives of astro-remedies name, determination, fame, success, great administrative ability, attractive features necessary for a sensuouspatience, nature. It makes its ability to lead, authority,is temporarily long closed. life, organizationalis temporarily closed. natives tender, gentle, and considerate. It inspires them to be poets, musicians, and seekers of truth and knowledge (secret The Saturn givesyantrassciences).and for surya chandra The nativemangal itsbuddhayantras loves gurufor the surya asceticism.shukra companychandra nativesshani mangalrahu ofketuabilities,The buddhathe navgraha members guruOnline yantra shukra shani of the rahu moon ketu integrity, navgraha yantra sincerity,shopping rules11-09-10 11:26wisdom, AM peace honesty,11-09-10 11:26ofhonor, AM ofKetuastro-remediesspirituality, attractive lovemind, wealth, bestows of comfort, justice, progenyfame,is spiritual temporarilyappearance. awarenessgeneraland and independenttendencies, good well-being, of relationshipclosed. right spirit.asceticism, and with It rules progeny, and over nonattachment courage,and bravery, patience, http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraopposite sex, artists, and musicians. natives Online and shopping makes of them astro-remedies mastersPage of 5 theof 7 healing arts, natural The moon rulesyantras for surya chandra peace mangal buddha guru shukra shani rahu ketu navgraha yantra of mind,11-09-10 11:26 AM comfort, general well-being, and patience, ability to lead,andalso wrongthe authority, fortune because of long it arules itperson. life, bringsto the worldly organizational innerIt benefits makes conscience, desires to us whatever self-confidence,moody, and nonattachment, ambitions. planetsemotional, or Ketuleadership houseand is also it is associateda abilities, half-planet physical that strength, The Sun rules our life force, the power of will, intellect,is brilliance, temporarily closed. Online shopping of astro-remedies Start using this Yantra on any Tuesday Morning. prosperity, success in worldly affairs, wealth, personal, conduct, Start using this Yantra on any Thursdayis temporarily Morning. closed. is temporarily closed. GO TOP activity, cheerfulness, good fortune, wisdom, ambition, fame, the forcefulness. GO TOPhealing, tantric healing, healing persons affected by evil spirits abilities, sincerity,understanding of the phenomenal honesty, world, andandsensitive. the knowledge asceticism. of love ofThe justice, Moonwith. awareness gives illumination, of right intuitive nature, also the fortune ofmedicine. a It also governsperson. our relationship with temples and holy It makes GO TOP usbestows moody, wisdom, emotional, powers of discrimination, and spiritual knowledge, places. Start using this Yantra on any Sunday Morning. yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM and wrong because it rulessensuality,Start using this Yantrathe on any Saturdayinner Morning. taste, conscience, love of nonattachment, poetry, fine arts, music, jewelry and yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM and the knowledge of the self. It and gives psychicghosts. abilities to its yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM sensitive. The Moonsensuality, passion, sexual pleasure, love matters, givesRahu luxuries, Yantra arts: For the +ve influence of planet illumination, RAHU. intuitive nature, GO TOP and music. It brings wealth,The Sun comfort, rules our attraction life force, to the the power opposite of will, intellect,Online shoppingbrilliance, of astro-remedies Online shopping of astro-remediesOnline shopping of astro-remedies and asceticism. is temporarily closed. sex in the early part ofprosperity, life, a well-proportioned success inattractive worldly body, affairs, and the wealth, personal, conduct, appearance. is temporarily closed. Rahuis temporarilyis regarded closed. as similar to Saturn in its effect. natives and makes them masters of the healing arts, natural http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraattractive features necessaryactivity, for cheerfulness, a sensuous nature.goodRahu fortune, isIt makes capable wisdom, its of giving ambition, power fame, when the rightly posited or exalted, Page 5 of 7 natives tender, gentle, understanding and considerate. of the It inspires phenomenaland it themgives world,fame, to be success and the in knowledge politics, money, of and physical GO TOP beauty GO TOP GO TOP GO TOP poets, musicians, and medicine. seekers of It truth also governs andStart (i.e., knowledge our attraction). relationship (secret withusing temples and holy this Yantra onStart any using Thursday this Yantra Morning. on any Tuesday Morning. GO TOP yantrassensuality, for surya chandra taste, mangal Thesciences). love number Thebuddha native lovesplaces. the Sanskrit company of of the membersguruChandra “9” poetry, of the depicted shukra in Yantra two diﬀerentshani fine: For ways rahu the arts,healing, +veketu influence music, navgrahatantricB1-13 of healing, jewelry yantraplanet healingMOON. and persons affected by evil spirits 11-09-10 11:26 AM opposite sex, artists, and musicians. Rahu rules no sign of the zodiac but it has authority over Virgo. Start using this Yantra onStart any using Friday this Morning. Yantra on any Wednesday Morning. yantrasStart usingfor this Yantra suryaon any Monday Morning. chandra mangal Shukrabuddha guru Yantra shukra shani: For rahu the ketuBuddha navgraha+ve influence yantraYantra: Forof planet the +ve VENUS. influence of planet MERCURY.11-09-10 11:26 AM

Shani Yantra: For the Guru+veOnline influence Yantrashopping of( Brihaspatiastro-remediesof planet SATURN. Yantra): For the planet JUPITER. and ghosts. Mangal Yantra: For the +ve influence of planet MARS. Ketu rules no sign of the zodiac but it has authority over Pisces. is temporarily closed. Online shopping of astro-remedies attractive appearance.The Saturn gives its nativesJupiter integrity, rules wisdom, religion, spirituality, philosophy, fame, spirituality, wealth, and Online shopping of astro-remedies patience, ability to lead,progeny. authority, If favorable, long life,Mars this organizational planetgives GO TOP energy, gives name,determination, fame, success, great administrative ability, Online shopping of astro-remedies abilities, sincerity, honesty,honor, love wealth, of justice, progeny awarenessand and independent good of relationship right spirit. with It rules progeny, over courage,and bravery, patience, Online shopping of astro-remediesis temporarily closed. Online shopping of astro-remedies and wrong because it rulesit brings the inner benefits conscience, toThe whateverself-confidence, nonattachment, planets moon orleadership house it is associated abilities, rules physical strength, peace of mind, comfort, general well-being, and and asceticism. with. forcefulness. is temporarilyBuddha closed. Yantra : For the +ve influence of planet MERCURY. MostStart using thisof Yantrathe on any Sunday Planet Morning. YantrasShukra with 3 Yantra3 magic squares: For that the we have +ve found influence show of planet VENUS. is temporarily closed. is temporarily closed. also the fortune× of a person.Ketu It makesrules no us signmoody, of theemotional, zodiacOnline andshoppingbut it has of astro-remedies isauthority temporarily over Pisces. closed. theChandra Yantra numerals: For the +ve influence in Sanskrit, of planet MOON. with the number “9” depicted in twoOnline diﬀerent shopping ways. of astro-remedies Online shopping of astro-remedies The moon rules peace of mind,Online shopping comfort, of astro-remediesgeneral well-being,is temporarily closed.and Online shopping of astro-remedies is temporarily closed. sensitive.is temporarily closed.The Moon gives illumination, intuitive is temporarily nature, closed. GO TOP also the fortune of a person. It makes us moody, emotional, and Venusis temporarily governsclosed. the Mercury refined is attributes- the planet romance, of intellect beauty, and communication. It rules Start using this Yantrasensitive. The Moon giveson Start illumination, any intuitive nature, using Friday GO TOP this Morning. Yantra on any Sunday Morning. GO TOP GO TOP Start using this Yantra on any Wednesday Morning. yantrassensuality, for surya chandra taste, mangal love buddha of guru poetry, shukra shani fine rahu arts, ketu music, navgraha jewelryyantra and 11-09-10 11:26 AM GO TOP yantras for surya chandra mangal buddha guru shukra shani rahu ketu navgraha yantra 11-09-10 11:26 AM GO TOP attractive appearance. yantrassensuality,Venus for surya chandragoverns taste, mangal love buddha theyantras of Mercuryguru for refinedpoetry, surya shukra chandra shani fine is attributes- mangalrahu the arts, ketu planetbuddha music, navgraha guru romance, ofjewelryyantra shukra intellect shani and rahu beauty, ketu and navgraha communication. yantra It rules 11-09-10 11:26 AM 11-09-10 11:26 AM Start using this Yantra onStart any using Friday this Morning. Yantraattractive onStart any using Sunday this Morning. Yantra on any Wednesday appearance. Morning. traders and businessmen. It rules our intelligence, speech, self- Shani Yantra: For the Chandra+ve influence Yantra of: Forplanet theKetu SATURN. +ve Yantra influence: For ofthe planet +ve influence MOON. of planet KETU. traders and businessmen. It rules our intelligence, speech, self- GO TOP Online shopping of astro-remediesOnline shopping of astro-remedies GO TOP is temporarily closed. is temporarily closed. GO TOP The Saturn gives its nativesThe moon integrity, rules wisdom, peace ofKetuspirituality, mind, bestows comfort, fame, spiritual general tendencies, well-being, asceticism, and and nonattachment GO TOP patience, ability to lead,also the authority, fortune of long a person. life,to worldly organizational It makes desires us moody, and ambitions. emotional, Ketu and is also a half-planet that GO TOP abilities, sincerity, honesty,sensitive. love ofhttp://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraThe justice, Moon awarenessbestows gives wisdom, illumination, of right powers intuitive of discrimination, GO TOP nature, spiritual knowledge, confidence, humor, wit, astrology,confidence, mathematics, and humor, short wit, astrology,Page 6 of 7 mathematics, and short and wrong because it rulessensuality, the inner taste, conscience, love and of nonattachment, poetry, the knowledge fine arts, of music, the self.jewelry It gives and psychic abilities to its http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 6 of 7 and asceticism. attractive appearance. natives and makes them masters of the healing arts, natural http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Start usingPage 5 ofthis 7 Yantra onStart any using Friday this Morning. Yantra onStart any using Sunday this Morning. Yantra on any Wednesday Morning. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraStart using this Yantra onStarthealing, any using Thursday tantric this Yantra healing,Morning. on healingany Tuesday persons Morning. affected by evil spirits Page 4 of 7 and ghosts. journeys. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraShukra Yantra: For theKetuBuddha +ve rules influence noYantra sign :of Forof the planet the zodiac +ve VENUS. but influence it has authority of planet over MERCURY. Pisces. Page 4 of 7 Online shopping of astro-remediesOnline shopping of astro-remedies Shani Yantra: For the +ve is temporarilyinfluence closed. of planet SATURN. Chandrais temporarily closed. Yantra: For theKetu +ve Yantra influence: Forjourneys. ofthe planet +ve influence MOON. of planet KETU. Venus governs the Mercury refined is attributes- the planet romance, of intellect beauty, and communication. It rules traders and businessmen. It rules our intelligence, speech, self- GO TOP GO TOP http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraconfidence, humor, wit, astrology, mathematics, and short Page 6 of 7 Start using this Yantra on any Wednesday Morning. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraStartjourneys. using thisPage 4 ofYantra 7 on any Thursday Morning. Start using this Yantra onStart any Shaniusing Thursday this Yantra Morning. on any WednesdayYantra Morning. : For the Chandra+ve influence Yantra of: Forplanet the SATURN. +ve influence of planet MOON.Online shopping of astro-remedies http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantrahttp://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7 Online Ketushopping Yantra :of ForPage astro-remedies the 2 of +ve7 influence of planet KETU.

Guru Yantra (Brihaspati Yantra): For the planet JUPITER. Navgriha Yantra: For the +ve influence of ALL THE NINE PLANETS. Online shopping of astro-remediesStartOnline using shopping ofthis astro-remedies Yantra on any Wednesday Morning. Jupiter rules religion, philosophy, spirituality, wealth, and is temporarily closed. This yantra is beneficial for all the people and used to balance is temporarily closed. progeny. If favorable, this planet gives name, fame, success, Start using this Yantra on any Thursday Morning. the influences of all the planets. honor, wealth, progeny and good relationship with progeny, and is temporarily closed. is temporarily closed. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantrait brings benefits to whatever planets or house it is associated Page 2 of 7 The Saturn gives its nativesThewith. Saturn integrity, gives its nativeswisdom,The moon integrity, rules spirituality, wisdom, peace ofspirituality, mind, comfort,fame, fame, general well-being, and http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraThe moon rulesPage 3 of 7 peaceGuru Yantra of mind, (BrihaspatiKetu comfort,bestows Yantra) spiritual general: For tendencies, the planet well-being, asceticism, JUPITER. and nonattachment and Online shopping of astro-remedies Ketu bestows spiritual tendencies, asceticism, and nonattachment http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra is temporarilyNavgriha closed. Page 5 of 7 Yantra: For the +ve influence of ALL THE NINE PLANETS. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7 patience, ability to lead,also the authority, fortune of long a person. life,to worldly organizational It makes desires us moody, and ambitions. emotional, Ketu and is also a half-planet that patience, ability to lead, authority, long life, organizational GO TOP GO TOP alsoabilities, the sincerity, GO TOPfortune honesty, of lovea person. ofto justice, worldly It awareness makes desires of right us moody, and ambitions. emotional, Ketu and is also a half-planet that sensitive. The Moon bestows gives wisdom, illumination, powers intuitive of discrimination, nature, spiritual knowledge, GO TOP Online shopping of astro-remedies is temporarily closed. Jupiter rules religion, philosophy, spirituality, wealth, GO TOP and abilities, sincerity, honesty,sensitive.Thisand wrongyantra love because is of The beneficial it justice, rulessensuality, Moon the for inner all awareness taste, theconscience, gives peoplelove and of nonattachment, poetry,and theillumination, of used knowledge right fine to arts, balance of music, Guru the self. intuitivejewelry Yantra It gives and psychic nature,(Brihaspati abilities to its Yantra): For the planet JUPITER. George P. H. Styan16 Yantra & Agrippa-typeprogeny. magic bestows If favorable, wisdom, this planet powers gives name, of fame, discrimination, success, spiritual knowledge, Feed Back | About Us | Contact Us | Terms of Use | Disclaimer | Privacy Policy | Security . Navgriha Yantra: For the +ve influence of ALL THE NINE PLANETS. and asceticism.Online shopping of astro-remedies Copyright © 2004-2010 AstrologyPlus.in- All rights reserved. attractive appearance. natives and makes them masters of the healing arts, natural is temporarily closed. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraAstrology Plus provides energized yantras crystal shri yantrathe sriyantra lockets astroremedies influences rudrakshas bead zodiac gems pandents, other astro remedies of all the planets. Page 5 of 7 and wrong because it rulessensuality, the inner GO TOP taste, conscience,Start honor, love using wealth,thisand of Yantra nonattachment, poetry, progeny the onStarthealing, any knowledge using and Thursday tantric fine this good Yantra healing, Morning. arts, relationship on of healing any music, theTuesday with persons progeny, self.jewelry Morning. affected Itand bygives and evil spirits psychic abilities to its Start using this Yantra on any Thursday Morning. it brings benefits toand whatever ghosts. planets or house it is associated and asceticism. attractiveShukra Yantra: For the +ve influence of planetappearance. VENUS. natives and makes them masters of the healing arts, natural http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Venus governs the refined attributes- romance, beauty, with. Jupiter rules Page religion, 5 of 7 philosophy, spirituality, wealth, and http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantrahttp://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraPage 4 of 7 ShukraThis Yantra yantra: For the Buddha is +ve beneficial influence Yantra: Forof planet the for +ve VENUS. all influence the of people planet MERCURY. and usedPage to 3 of 7balance http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraStart using thisPage 7 of 7 YantraOnline on Startshopping any ofusing astro-remedies KetuThursday rules this no signYantra Morning. of the zodiac on butany it has Tuesday authority over Morning. Pisces. healing, Onlinetantric shopping healing,of astro-remediesOnline healing shopping of astro-remedies persons affected by evil spirits http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra is temporarily closed. Pageprogeny. 5 of 7 If favorable, this planet gives name, fame, success, the influencesis temporarilyof all closed.the planets.is temporarily closed. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraVenus governsand theghosts. Mercury refined is attributes- the planet honor, romance, of intellect wealth, beauty, and communication.Page progeny 2 of 7 It rules and good relationship with progeny, and traders and businessmen. It rules our intelligence, speech, self- GO TOP GO TOP http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraconfidence, humor, it wit, brings astrology, benefits mathematics, to and whatever short planetsPage 6 of or 7 house it is associated http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraShukra Yantra: For theBuddha +vejourneys. influence Yantra: Forof planet the +ve VENUS. influencePage of 4 of 7planet MERCURY. Ketu rules no sign ofwith. the zodiacOnline shoppingbut it has of astro-remedies authority over Pisces. OnlineStart shopping using this Yantraof astro-remedies on any Wednesday Morning. Starthttp://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra using this Yantra on any Thursday Morning. GO TOPOnline shopping of astro-remedies Page 3 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra is temporarily closed. isPage temporarily 2 of 7 closed. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Mercury is the planetis oftemporarily intellect closed. and communication.Page 5It of 7 rules Venus governs the refined Guru attributes- Yantra (Brihaspati romance, Yantra): For the planet beauty, JUPITER. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraNavgriha Yantratraders: For the and+ve influence businessmen. of ALL THE NINE PLANETS. It rules our intelligence, speech, self- Page 2 of 7 Online shopping of astro-remedies is temporarily closed. GO TOP Jupiter rules religion, philosophy, spirituality, GO TOP wealth, and This yantra confidence, is beneficial for all the humor, people and wit,used to balance astrology, mathematics, and short http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantraprogeny. If favorable, this planet gives name, fame, success, Page 6 of 7 the influences of all the planets. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra journeys.honor, wealth, progeny and good relationship with progeny, and Page 4 of 7 it brings benefits to whatever planets or house it is associated Feed Back | About Us | Contact Us | Terms of Use | Disclaimerwith. | Privacy Policy | Security . Copyright © 2004-2010http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra AstrologyPlus.in-Start using All rightsOnline thisOnlinereserved.shopping shoppingYantra of astro-remedies of astro-remedies on any Wednesday Morning. Page 3 of 7 Start using this Yantra on any Thursdayis temporarily Morning. closed. http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantrahttp://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraAstrology Plus provides energized yantras crystal shri yantra sriyantra lockets astroremedies rudrakshasis temporarily bead zodiac closed. gems pandents, other astro Pageremedies 5 of 7 Page 2 of 7 GO TOP http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7 GO TOP Guru Yantra (Brihaspati Yantra): For the planet JUPITER. Navgriha Yantra: For the +ve influence of ALL THE NINE PLANETS. GO TOP StartOnline using shopping this Yantra of on astro-remedies any Thursday Morning. is temporarily closed. Jupiter rules religion, philosophy, spirituality, wealth, and This yantra is beneficial for all the people and used to balance Online shoppingShukra ofprogeny. Yantraastro-remedies: For is thetemporarilyIf +ve favorable, closed.influence of thisplanet planet VENUS. gives name, fame, success, the influences of all the planets. honor, wealth, progeny and good relationship with progeny, and Venus governs the refined attributes- romance, beauty, it brings benefits to whatever planets or house it is associated FeedFeed Back Back | About |Us About | Contact UsUs | Terms| Contact of Use | Disclaimer Us | Terms | Privacy Policyof Use | Security | Disclaimer . | Privacy Policy | Security . Copyrightwith. © 2004-2010 AstrologyPlus.in- All rights Onlinereserved. shopping of astro-remedies Online shopping of astro-remedies http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraCopyright © 2004-2010 AstrologyPlus.in-is temporarily closed. All rights reserved.Page 4 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraAstrology Plus provides energized yantras crystal shriOnline yantra sriyantra shopping lockets astroremedies of rudrakshasastro-remedies bead zodiac gems pandents, other astro remedies is temporarily closed. Page 3 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraAstrology Plus provides energized yantras crystal shri yantra sriyantra lockets astroremediesPage 7 ofrudrakshas 7 bead zodiac gems pandents, other astro remedies http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra is temporarily closed. Page 5 GO of TOP 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 2 of 7 GO TOP Start using this Yantra on any Thursday Morning.

Shukra Yantra: For the +ve influence of planet VENUS. Start using this Yantra on any Thursday Morning. Venus governs the refined attributes- romance, beauty, GO TOP

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 4 of 7

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Shukra Yantra: For thePage 7 of +ve 7 influence of planet VENUS.

Online shopping of astro-remedies is temporarily closed. Venus governs the refined attributes- romance, beauty,

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 4 of 7 Feed Back | About Us | Contact Us | Terms of Use | Disclaimer | Privacy Policy | Security . http://www.astrologyplus.in/planetyantras.php#Brihaspati%20YantraCopyright © 2004-2010 AstrologyPlus.in- All rights Onlinereserved. shopping of astro-remedies Page 7 of 7 is temporarily closed. Astrology Plus provides energized yantras crystal shri yantra sriyantra lockets astroremedies rudrakshas bead zodiac gems pandents, other astro remedies

GO TOP

Start using this Yantra on any Thursday Morning.

Shukra Yantra: For the +ve influence of planet VENUS.

Venus governs the refined attributes- romance, beauty,

http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 4 of 7 http://www.astrologyplus.in/planetyantras.php#Brihaspati%20Yantra Page 7 of 7 Navagraha Planet Yantras B1-14

Navagraha Planet Yantras arranged as 9 9 composite magic squares ×

George P. H. Styan17 Yantra & Agrippa-type magic Planet Yantras arranged as 9 9 composite fully-magic squares B1-15 ×

Our “Navagraha (nine-planets) Yantra magic square” is deﬁned by the eight composite 9 9 fully-magic matrices ×

Nj = E L1 + Mj E E E ⊗ ⊗ − ⊗

= E L1 + (Mj E) E; j = 1, 2,..., 8, ⊗ − ⊗ where the eight classic 3 3 magic matrices ×

George P. H. Styan18 Yantra & Agrippa-type magic Eight classic 3 3 magic matrices B1-16 ×

6 1 8 2 9 4 8 1 6 = = 7 5 3 = = 7 5 3 = = 3 5 7 M1 L1   , M2 FL1   , M3 L1F   , 2 9 4 6 1 8 4 9 2

6 7 2 8 3 4 2 7 6 M = L0 = 1 5 9 M = FL0 = 1 5 9 M = L0 F = 9 5 1 4 1   , 5 1   , 6 1   , 8 3 4 6 7 2 4 3 8

4 9 2 4 3 8 M = FL F = 3 5 7 M = FL0 F = 9 5 1 7 1   , 8 1   . 8 1 6 2 7 6

George P. H. Styan19 Yantra & Agrippa-type magic The Luoshu magic matrix M7 = FL1F B1-17

The magic square deﬁned by

4 9 2 M7 = FL1F = 3 5 7 8 1 6    

was chosen as “The Table of Saturn in his Compass” by Heinrich Cornelius Agrippa von Nettesheim (1486–1535).

Gerolamo Cardano (1501–1576) named this magic square “Luna” after the moon (rather than Saturn).

In the magic matrix M7 the numbers in the rows, columns, and two main diagonals all add up to same magic sum 15.

George P. H. Styan20 Yantra & Agrippa-type magic The magic matrix M2 = FL1 B1-18

Paracelsus (1493–1541) [born Philippus Aureolus Theophrastus Bombastus von Hohenheim] named the magic square deﬁned by

2 9 4 M2 = FL1 = 7 5 3 , 6 1 8     Talisman de Saturne, ou Samedi (Saturday).

George P. H. Styan21 Yantra & Agrippa-type magic Eight composite 9 9 magic matrices B1-19 ×

L6 L1 L8 L2 L9 L4 L8 L1 L6 N1 = L7 L5 L3 , N2 = L7 L5 L3 , N3 = L3 L5 L7 , L L L L L L L L L  2 9 4  6 1 8  4 9 2      

L6 L7 L2 L8 L3 L4 L2 L7 L6 N4 = L1 L5 L9 , N5 = L1 L5 L9 , N6 = L9 L5 L1 , L L L L L L L L L  8 3 4  6 7 2  4 3 8      

L4 L9 L2 L4 L3 L8 N7 = L3 L5 L7 , N8 = L9 L5 L1 . L L L L L L  8 1 6  2 7 6    

George P. H. Styan22 Yantra & Agrippa-type magic The composite 9 9 magic matrix N1 B1-20 ×

11 6 13 6 1 8 13 8 15 12 10 8 7 5 3 14 12 10  7 14 9 2 9 4 9 16 11  L6 L1 L8  12 7 14 10 5 12 8 3 10    N1 = L7 L5 L3 =  13 11 9 11 9 7 9 7 5  ,      8 15 10 6 13 8 4 11 6  L2 L9 L4      7 2 9 14 9 16 9 4 11         8 6 4 15 13 11 10 8 6     3 10 5 10 17 12 5 12 7   

George P. H. Styan23 Yantra & Agrippa-type magic F-associated magic matrices B1-21

The n n magic matrix M with magic sum m is F-associated whenever × M + FMF = 2mE¯,

where E¯ is the n n matrix with every entry equal to 1/n × and F = Fn is the n n “ﬂip matrix” × 0 0 0 1 ··· 0 0 1 0  ···  F = . ···············    0 1 0 0   ···   1 0 0 0   ···   

The magic matrices N1, N2,..., N8 all have magic sum 81 and rank 5, and all are F-associated, since F9 = F3 F3 and M1, M2,..., M8 are all F-associated. ⊗

George P. H. Styan24 Yantra & Agrippa-type magic An F-associated magic square B1-22

In an n n F-associated magic matrix with magic sum m × the sums of pairs of entries diametrically equidistant from the centre are all equal to 2m/n.

An F-associated magic square is often called (just) “associated” (with no qualiﬁer to the word “associated”) or “regular” or “symmetrical”.

George P. H. Styan25 Yantra & Agrippa-type magic Magic-square planetary talismans (MSPTs) B1-23

Magic-square planetary talismans (MSPTs) were created from 1531–1653 by

Heinrich Cornelius Agrippa von Nettesheim (1486–1535),

Gerolamo Cardano (1501–1576),

Paracelsus (1493–1541) [born Philippus Aureolus Theophrastus Bombastus von Hohenheim] ,

Athanasius Kircher (1602–1680).

for the seven planets: Saturn, Jupiter, Mars, Sun, Venus, Mercury, and the Moon.

George P. H. Styan26 Yantra & Agrippa-type magic Agrippa, Cardano, Paracelsus, Kircher B1-24

(from left to right) Agrippa [Wikipedia]; Cardano: Altai Republic 2011 (External Airmail); Paracelsus: Switzerland 1993 (Scott 928); Kircher: Malta 2002 (Scott 1094).

George P. H. Styan27 Yantra & Agrippa-type magic Agrippa, Cardano, Paracelsus, Kircher B1-25

Agrippa was a German magician, occult writer, theologian, astrologer, and alchemist. His magic-square planetary talismans (MSPTs) were ﬁrst published in 1531 in his De occulta philosophia libri tres,

Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler. His MSPTs were ﬁrst published in 1539 in his Practica arithmetice et mensurandi singularis.

Paracelsus was a Swiss Renaissance physician, botanist, alchemist, astrologer, and general occultist. His MSPTs were ﬁrst published in 1567 in Archidoxa Magica.

Kircher was a German Jesuit scholar who published in the ﬁelds of oriental studies, geology, and medicine. His MSPTs were ﬁrst published in 1653 in Oedipi Aegyptiaci, Tomi Secundi.

George P. H. Styan28 Yantra & Agrippa-type magic Two 3 3 magic-square planetary talismans (MSPTs) B1-26 ×

(left panel) by Agrippa for Saturn; (right panel) by Cardano for the Moon.

George P. H. Styan29 Yantra & Agrippa-type magic Luoshu B1-27

The Agrippa–Cardano magic square

4 9 2 L = 3 5 7 8 1 6   is known as Luoshu or Lo Shu (Luo River Writing). Legend has it that the very ﬁrst magic square was discovered in China about 4000 years ago on the back of a turtle in the River Luo, a tributary of the Yellow River in China.

George P. H. Styan30 Yantra & Agrippa-type magic Two 5 5 magic-square planetary talismans (MSPTs) B1-28 ×

(left panel) by Agrippa for Mars: (right panel) by Cardano for Venus.

George P. H. Styan31 Yantra & Agrippa-type magic Two 7 7 magic-square planetary talismans (MSPTs) B1-29 ×

(left panel32) by Agrippa for Venus; (right panel) by Cardano for Mars.

32 The entry in position (6,2) should be “39”. George P. H. Styan33 Yantra & Agrippa-type magic Two 9 9 magic-square planetary talismans (MSPTs) B1-30 ×

(left panel) by Agrippa for the Moon: (right panel) by Cardano for Saturn34.

34 In both panels “45” in position (1, 8) should be “54”; in the left panel the “35” in position (2, 2) should be “38”. George P. H. Styan35 Yantra & Agrippa-type magic Generating odd-order Agrippa-type magic matrices B1-31

We may generate odd-order “Agrippa-type” (Agrippa–Cardano) magic matrices using a “magic matrix algorithm”, which we believe to be new and which we now describe.

Our “magic matrix algorithm” generates classic nonsingular n n magic matrices An for any odd number n 3. × ≥

The Matlab algorithm magic(n) by Cleve Moler (1993) also generates classic nonsingular n n magic matrices with n odd but they × (all for n 5?) essentially diﬀer from our Agrippa-type magic matrices An. ≥

George P. H. Styan36 Yantra & Agrippa-type magic One-step forward shift matrix Sn B1-32

We begin the description of our magic matrix algorithm with the n n “one-step forward shift matrix” Sn (n = 2, 3, 4,...), which has×+1 in all n 1 positions on the leading principal super-diagonal and in the bottom-left− corner and so is a circulant. When n = 5 we have

0 1 0 0 0  0 0 1 0 0  S = 0 0 0 1 0 . 5      0 0 0 0 1     1 0 0 0 0     

George P. H. Styan37 Yantra & Agrippa-type magic The n n matrix Tn = Sn S0 B1-33 × − n

We now deﬁne the n n matrix × 0 −1 Tn = Sn S = Sn S ; n = 2, 3, 4,... − n − n with +1 in all n 1 positions on the leading principal super-diagonal and in the bottom-left− corner and with 1 in all n 1 positions on the leading principal sub-diagonal and− in the top-right− corner.

0 −1 We note that Sn = Sn since Sn is orthogonal. When n = 5 we have 0 1 0 0 1 − 1 0 1 0 0  −  T5 = 0 1 0 1 0 .  −     0 0 1 0 1   −   1 0 0 1 0   −   

George P. H. Styan38 Yantra & Agrippa-type magic The n n matrix Un = (Fn nIn)(Sn S0 ) B1-34 × − − n

We now deﬁne the n n matrix with n = 2, 3, 4,... ×

0 Un = FnTn nTn = (Fn nIn)Tn = (Fn nIn)(Sn S ) − − − − n

where Fn is the n n ﬂip matrix. ×

When n = 5 we have

0 1 0 0 1 1 5 0 1 5 − − − 1 0 1 0 0 5 0 6 0 1  −   −  T5 = 0 1 0 1 0 ; U5 = 0 4 0 4 0 .  −   −   0 0 1 0 1   1 0 6 0 5   −   − −   1 0 0 1 0   5 1 0 5 1   −   − −      George P. H. Styan39 Yantra & Agrippa-type magic Generating Agrippa-type magic matrices for odd n 3 B1-35 ≥

Three equivalent formulas for our “magic matrix algorithm”:

2 −1 1 2 An = n(n 1)(Un + E¯n) n(n 3)E¯n (1) − − 2 −

2 −1 2 2(n 1) = n(n 1) Un + − E¯n (2) − n2 + 1

2 + 1 2 = n(n 1)U + n(n + 1)E¯n. (3) − n 2

1 0 The matrix E¯n has every entry equal to , and Un = (Fn nIn)(Sn S ). n − − n

George P. H. Styan40 Yantra & Agrippa-type magic Generating the Agrippa-Cardano magic matrix A5 B1-36

0 1 0 0 0 1 5 0 1 5 − − 0 0 1 0 0 5 0 6 0 1    −  S5 = 0 0 0 1 0 ; U5 = 0 4 0 4 0 ;    −       0 0 0 0 1   1 0 6 0 5     − −   1 0 0 0 0   5 1 0 5 1     − −     

11 24 7 20 3 4 12 25 8 16 1   24 ¯ − A5 = 120 U5 + E5 = 17 5 13 21 9 . 13      10 18 1 14 22     23 6 19 2 15     

George P. H. Styan41 Yantra & Agrippa-type magic Two 4 4 magic-square planetary talismans (MSPTs) B1-37 ×

(left panel) by Agrippa for Jupiter; (right panel) by Cardano for Mercury.

George P. H. Styan42 Yantra & Agrippa-type magic Two 8 8 magic-square planetary talismans (MSPTs) B1-38 ×

(left panel) by Agrippa for Mercury; (right panel) by Cardano for Jupiter.

George P. H. Styan43 Yantra & Agrippa-type magic Generating Agrippa-type magic matrices with n doubly-even B1-39

The matrix algorithm used by Matlab to generate magic matrices Mn for doubly-even n = 4k has been described in some detail by Kirkland & Neumann (1995); see also Moler (1993, 2004).

We ﬁnd that our Agrippa-type magic matrices A4k = FM4k for k = 1, 2, i.e., the Matlab M4k are just our A4k with the rows ﬂipped:

 8 58 59 5 4 62 63 1  49 15 14 52 53 11 10 56     4 14 15 1    41 23 22 44 45 19 18 48   9 7 6 12       32 34 35 29 28 38 39 25  A4 = FM4 =  ; A8 = FM8 =   .  5 11 10 8   40 26 27 37 36 30 31 33    16 2 3 13  17 47 46 20 21 43 42 24   9 55 54 12 13 51 50 16  64 2 3 61 60 6 7 57

George P. H. Styan44 Yantra & Agrippa-type magic Generating Agrippa-type magic matrices with n doubly-even B1-40

Our procedure for generating Agrippa–type magic matrices An for doubly-even n = 4, 8,... uses the (partner) “magic-basis matrix” or “B-matrix” Bn deﬁned by

2 1 2 2 n + 1 An = nBn + FB0 F + (n + 1)E ; Bn = nAn FA0 F E 2 n n2 1 − n − n + 1 −

The 8 8 matrix B8 is “bordered” (really 2-bordered) in that the 4 4 × (8) × “inner-core” matrix B4 say, formed from B8 by removing its top 2 rows, bottom 2 rows, ﬁrst 2 columns and last 2 columns, is also a magic-basis matrix.

George P. H. Styan45 Yantra & Agrippa-type magic Generating Agrippa-type magic matrices with n doubly-even B1-41

8 58 59 5 4 62 63 1 49 15 14 52 53 11 10 56 4 14 15 1  41 23 22 44 45 19 18 48  9 7 6 12 32 34 35 29 28 38 39 25 A4 =  ; A8 = , 5 11 10 8  40 26 27 37 36 30 31 33     16 2 3 13   17 47 46 20 21 43 42 24     9 55 54 12 13 51 50 16       64 2 3 61 60 6 7 57    have the “two-diagonals-in-progression” (2-dip) property, with parameters (n, 1; n 1, n + 1), indicating that the starting numbers are − n and 1, respectively, on the main forwards and backwards diagonals, with corresponding step sizes n 1 and n + 1, −

n, n + (n 1), n + 2(n 1),..., n + (n 1)(n 1); − − − − 1, (n + 1) + 1, 2(n + 1) + 1,..., (n 1)(n + 1) + 1 = n2. −

George P. H. Styan46 Yantra & Agrippa-type magic The 8 8 magic-basis matrix B8 B1-42 ×

−7 7 7 −7 −7 7 7 −7 5 −5 −5 5 5 −5 −5 5 3 −3 −3 3 3 −3 −3 3  −1 1 1 −1 −1 1 1 −1  B8 = − − − − .  1 1 1 1 1 1 1 1   −3 3 3 −3 −3 3 3 −3   −5 5 5 −5 −5 5 5 −5     7 −7 −7 7 7 −7 −7 7    The top row of the 8 8 magic-basis matrix B8 contains precisely × 4 entries equal to 7 and 4 entries equal to 7, and the second row contains − precisely 4 entries equal to 5 and 4 entries equal to 5, and so on. −

We call Bn and partner An “smooth” whenever in the 2-dip magic-basis matrix Bn the numbers in row R add up to 0 and are all equal in absolute value to

2R + n + 1 when R 1 n and 2R n 1 when R 1 n + 1. − ≤ 2 − − ≥ 2

We ﬁnd that A4, A8, B4, B8 are all smooth.

George P. H. Styan47 Yantra & Agrippa-type magic Row-balanced, column-balanced, doubly-balanced B1-43

We will say that the n n magic matrices An and Bn are “row-balanced” whenever × 2 (I + F)(nAn FA0 F) = (n + 1)(n 1)E − n − or equivalently (I + F)Bn = 0 Bn = FBn ⇐⇒ −

We ﬁnd that A4, A8, B4, B8 are all row-balanced.

We will say that the magic matrices An and Bn are “column-balanced” whenever

Bn(I F) = 0 Bn = BnF (nAn FA0 F)(I F) = 0. − ⇐⇒ ⇐⇒ − n −

We will say that the n n magic matrices An and Bn with doubly-even n 4 are × ≥ “doubly-balanced” whenever they are both row- and column-balanced, and then

Bn = BnF = FBn = FBnF. − −

We ﬁnd that A4, A8, B4, B8 are all doubly-balanced. George P. H. Styan48 Yantra & Agrippa-type magic Magic-basis vectors B1-44

From the special sign pattern in each of the rows B8 together with B8 being smooth and doubly-balanced, we ﬁnd the following procedure for generating n n doubly-balanced Agrippa–Matlab magic-basis matrices for × doubly-even n = 4, 8, 12,...

Let n = 2k and so k = n/2,

Ik 1 0 2 − 0 An = 2 nBn Bn + (n + 1)En ; Bn = Dk bk bk Ik Fk , − Fk ! where the k k diagonal matrix ×

Dk = diag(n 1, n 3, n 5,..., 5, 3, 1); n = 2k. − − −

George P. H. Styan49 Yantra & Agrippa-type magic Magic-basis vectors B1-45

Then the k 1 “magic-basis vector” bk = × (1) b = +1 1 1 +1 +1 1 1 +1 +1 1 0 ; n = 2k, k − − · · · − − − generates the Agrippa–Cardano matrices A4 and A8.

Drury (2011) created an algorithm for generating classic 2-dip smooth row-balanced An with magic-basis vector

(2) b = +1 +1 +1 +1 +1 1 1 1 1 1 0 ; n = 2k, k · · · − − − − − n n the sign change being between the entries in positions 4 and 4 + 1.

From an algorithm given by Cavendish (1894) we ﬁnd that choosing bk =

(3) b = +1 1 +1 1 +1 1 +1 1 +1 1 0 ; n = 2k, k − − · · · − − − also generates classic 2-dip smooth row-balanced An for doubly-even n = 4, 8,... George P. H. Styan50 Yantra & Agrippa-type magic Magic-basis vectors B1-46

(1) (2) (3) When n = 4 our three choices bk , bk , bk of the magic-basis vector coincide and we have

4 14 15 1  9 7 6 12  b2 = +1 1 0 ; A4 = . −  5 11 10 8       16 2 3 13     

We believe that the Agrippa matrix A4 here is the only 4 4 classic 2-dip smooth × doubly-balanced rank-3 fully-magic matrix.

George P. H. Styan51 Yantra & Agrippa-type magic (1) (2) (3) Three choices: Agrippa–Matlab b4 , Drury b4 , Cavendish b4 B1-47

b(1) = +1 −1 −1 +1 0 ; b(2) = +1 +1 −1 − 1 0 ; 4 4

A(1) = A(2) = 8 8

 8 58 59 5 4 62 63 1   8 7 59 60 61 62 2 1   49 15 14 52 53 11 10 56   16 15 51 52 53 54 10 9       41 23 22 44 45 19 18 48   41 42 22 21 20 19 47 48       32 34 35 29 28 38 39 25   33 34 30 29 28 27 39 40    ;   .  40 26 27 37 36 30 31 33   25 26 38 37 36 35 31 32       17 47 46 20 21 43 42 24   17 18 46 45 44 43 23 24   9 55 54 12 13 51 50 16   56 55 11 12 13 14 50 49  64 2 3 61 60 6 7 57 64 63 3 4 5 6 58 57

 8 58 6 60 61 3 63 1   49 15 51 13 12 54 10 56     24 42 22 44 45 19 47 17   33 31 35 29 28 38 26 40  b(3) = +1 −1 +1 −1 0 ; A(3) =   4 8   .  25 39 27 37 36 30 34 32     48 18 46 20 21 43 23 41   9 55 11 53 52 14 50 16  64 2 62 4 5 59 7 57

George P. H. Styan52 Yantra & Agrippa-type magic Agrippa–Fermat magic matrices An for singly-even n 6 B1-48 ≥

Our objective here is to obtain a sequence of classic fully-magic matrices A6, A10,..., starting with, and having many of the properties of the (original) Agrippa–Cardano magic matrix

6 32 3 34 35 1 7 11 27 28 8 30   19 14 16 15 23 24 A6 =   .  18 20 22 21 17 13     25 29 10 9 26 12     36 5 33 4 2 31     

This matrix A6 is the only n n Agrippa magic matrix with n singly-even. × ¯ We now seek n n fully-magic matrices An and B-bar partners Bn/2,n, × for singly-even n = 4k + 2 10 which, like A6, ≥ satisfy our 7 desiderata in Table 6.1 below.

George P. H. Styan53 Yantra & Agrippa-type magic Pierre de Fermat (1601/7–1665) & Père Marin Mersenne (1588–1648) B1-49

We will use a 14 14 “bordered” magic square due to Pierre de Fermat (1601/1607–1665)× and so we will call our A6, A10, A14,..., “Agrippa–Fermat magic matrices”.

We also use a 22 22 magic square specially constructed for us à la Fermat× by Christian Boyer in Paris.

Fermat’s 14 14 magic square was given in a letter by Fermat to Père Marin Mersenne× (1588–1648). The original copy of this letter has been lost but a hand-written copy by Félix Vicq d’Azyr (1746–1794) is in the Bibliothèque nationale de France in Paris.

George P. H. Styan54 Yantra & Agrippa-type magic Hand-written copy by Félix Vicq d’Azyr (1746–1794) B1-49x

From Copies de lettres et opuscules de Fermat, Descartes, Galilée, etc., Fol. 1, Copies de lettres de Fermat à Roberval, au P. Mersenne, à Frenicle, etc., by Félix Vicq d’Azyr (1746–1794), MS #10556, Département de manuscrits, Bibliothéque nationale de France, Paris. Via Christian Boyer.

George P. H. Styan55 Yantra & Agrippa-type magic Pierre de Fermat (1601/7–1665) and the Mersenne prime 213466917 1 B1-50 −

(left panel) Pierre de Fermat (1601/1607–1665) and Fermat’s Last Theorem: France 2001; (right panel) The Mersenne prime 213466917 − 1: Liechtenstein 2004, Scott 1297.

George P. H. Styan56 Yantra & Agrippa-type magic Agrippa–Fermat magic matrices An for singly-even n 6 B1-51 ≥

Table 6.1: Desiderata for our Agrippa–Fermat magic matrices An; n = 4k + 2 6. ≥

n(n2+1) 1 2 classic, with entries 1, 2,... n in some order, magic sum 2 2 2-dip with parameters (n, 1; n 1, n + 1) − ¯ 3 nonsingular, with partner B-bar matrix Bn/2,n of full row rank n/2

4 row-balanced, with Bn = FBn − 5 smooth, with all entries of B¯ equal to 1 n/2,n ± 6 bordered, and with n−3 3 7 characteristic polynomial: det(λIn Bn) = λ (λ + 120). −

George P. H. Styan57 Yantra & Agrippa-type magic Factorizing the magic-basis matrix B1-52

When n = 6 we have

5 5 5 5 5 5 − − − 3 3 3 3 3 3  − − −  2 37 1 1 1 1 1 1 − − − B6 = 6A6 FA60 F E =   35 − − 7  1 1 1 1 1 1   − − −   3 3 3 3 3 3   − − −   5 5 5 5 5 5     − − − 

5 0 0 1 1 1 1 1 1 I3 − − − = 0 3 0 1 1 1 1 1 1 .   − − −  F3! − 0 0 1 1 1 1 1 1 1   − − − 

George P. H. Styan58 Yantra & Agrippa-type magic “Reduced magic-basis matrix” or “B-bar matrix” B1-53

The magic-basis matrix B6 is row-balanced in that its last 3 rows ﬂipped coincide with its ﬁrst three rows with a sign change, i.e.,

B6 = FB6 − which we write as

B6 = J6,3D3B¯ 3,6 where 5 0 0 I3 −1 1 −1 1 1 −1 J6,3 = ; D3 = 0 3 0 ; B¯ 3,6 = −1 −1 1 1 −1 1   F3! 1 −1 −1 −1 1 1 ! − 0 0 1  

¯ We call the 3 6 matrix B3,6 a “reduced magic-basis matrix” or “B-bar matrix”. ×

George P. H. Styan59 Yantra & Agrippa-type magic Magic properties GPHS: 12/16/11, 1

¯ The full 13 26 matrix is B13,26 B1-54 ×

¯ Our B-bar matrices Bn/2,n for n = 6, 10, 14, 18, 22, 26 are shown below.

¯ The full 13 26 matrix is B13,26; the B-bar matrix formed by × ¯ removing the (outer) brown border is the 11 22 matrix B11,22. And so on. × ¯ And then removing the green border gives the 3 6 matrix B3,6 for A6. ×

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1

2 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 2

3 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 3

4 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 4

5 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 5

6 -1 1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 6

7 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 7

8 -1 1 -1 1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 8

9 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 9

10 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 10

11 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 11

12 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 12

13 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 13

George P. H. Styan60 Yantra & Agrippa-type magic

Magic properties GPHS: 12/16/11, 1

¯ The full 13 26 matrix is B13,26 B1-55 ×

¯ ¯ The B-bar matrices B5,10 (green border) and B7,14 (yellow border) are based on the 14 14 Fermat magic square. × ¯ ¯ The B-bar matrices B9,18 (blue border) and B11,22 (pink border) are based on the 22 22 Fermat–Boyer magic square. ×