Sacred Geometry and Mathematics

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Sacred Geometry and MathematicsBy V. Wy. Fra. Timothy Edwards VII

Presented at the Regular Convocation of Ontario College March 11 2002 and the secondpart at The Metropolitan Study group June 18 2002. This draft was updated Aug. 30 2005

and is a work in progress. Please send you comments to [email protected]

Introduction

Let me start by saying that my given name is Timothy Alan Edwards; however I refermyself as Timothy Edwards. One because timothy comes from Timo Theos which isGreek for Fearing or Honoring God. Timothy with Edwards because the total of my nameequals

Sum all the letters 20+9+13+15+20+8+25+5+4+23+1+18+4+19 =184

Then sum the sum

Until we get the number 4 a sign of completeness. This is an example of TheosophicalSubtraction.

If I added Alan 1+12+1+14 =28, 28+184 = 212 it would give me 5 sign of unstablepower. I add the month and day my birth 1+12 = 4 and it gives me 9. This is a negativebecause it will return the same number as it started with 1+9 =10 =>1, 2+9=11=>2,3+9=12=>3 I would be always influenced by what day it was. So the stronger numberwill be 4 + 4 =8 this way I can retain who I am on any given day.


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This is an exercise in Numerology. In mathematics it is described as modulo 9 on the sumderived from a sum base 26. Numerology is not the subject of this paper, what governsnumbers is. Here is a list of interpretations of the numbers

1 Genesis EA Neophyte Initiated Birth

2 Exodus MLKTh FC Zelator Passed Baptism

3 Leviticus YSD MM Theoriticus Raised Confirmation

4 Numbers HD MMM Practicus Advanced Communion

5 Deuteronomy NTzCh WM Philosophus Installed Healing

6 Joshua TPhRTh GM Portal Received Anointing

7 Judges GBRH The Constitution Adeptus Minor Exalted Holy Orders

8 Ruth ChSD The Old Charges Adeptus Major Matrimony

9 Samuel BNH The Work Adeptus Exemptus

1

0Kings ChKMH GAOTU Magister

1

1Chronicles KThR GGOTU Magus

1

2 Ezra AN SPh MH Imperator

This is a guide and you should ultimately develop your own interpretation of your ownlist of the numbers Gematria includes the sum of words to show synonyms by theirnumber. Gematria: part of the Literal Qabalah, gematria is a form of Hebrew numerologythat uses the numerical values of the letters of the Hebrew alphabet. Names or words areconverted into numbers in order to find hidden meanings. Words which have the samenumerical value are thought to have an important and significant relationship.

You can derive any number by applying corrections to get the wanted effect. Just as Ichanged my name to become a 4 instead of the 5 which it is naturally. I impose my willto get what I want. I am not limited to fate I just change my position move an object thereand align myself to get the best grid. One thing that you can take from numerology andgematria is the meaning of numbers. What is associated with 1 ,2 ,3 etc. with one I thinkof myself, 2 a relationship, 3 the result of 2 connected forms a line and 3 is the midpoint.And we build from there. What we want is something that is independent of base or itsmodulus. Something deeper that Numerology and Gematria will not provide.


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One which is associated with the monad symbolizes the unity of God.

Two is the duad. A sign of separation that cannot exist without the existence of thetrinary. This is because as you divide the unity there is three reflected. There is sectionone and section 2 and the space that is nether section 1 or 2. the famous separation of

man from God at the Fall.

Three or Trinary represents the manifested world as it is the first polygon created with theminimal sides. The living god has 3 persons as he is manifest and living with us. He is 3as He is creative. He is 3 as He is with us in the beginning he is with us now and he willbe until the end. His unity cannot be deigned as he is the first cause and his trinary cannotbe denied because he is creative.

Four is a symbol of physical perfection. It returns to 1 through theosophical contraction

which combines theosophical addition 1+2+3+4 = 10 and theosophical subtraction 10 1+0 = 1. Theosophical contraction asserts that there are only 3 whole numbers and all

others are derived there from.

Five is a symbol of ambivalence and pensiveness. It is the middle of a series from 1 to 9.

5 contracts to 2. 1+2+3+4+5 = 15 1+5 = 6 1+2+3+4+5+6 = 21 2+1 =3 3+2+1

6 The pattern is 5,6,3,6,3 compare 2 the pattern is 2,3,6,3,6,3 the 2nd and 3rd

numbers are mirrors of each other and the 1st ones reduce nicely. Namely 1,4,7,10,13,16 the mirror become the next number through contraction

23: 2 1+2=3

56: 5 1+2+3+4+5= 15 1+5=6

89: 8 1+2+3+4+5+7+8+9 = (8x9)/2= 363+6=9

Six is a symbol of creation as the world was created in 6 days

7 is a symbol of the 7 Asian churches 7 arch angels and 7 seals of revelations. It alsocontracts to 1

8 is the Christ number as it is perfected (4) on 2 levels (4)x2 = 8

9 is a perturbation period as nothing can escape. One cannot call on 9 to make change.

For example 1+9 = 18 1+8=9 this is true for all whole numbers. 9 is the mirror of 8 orof the Christ which is the Anti-8 or the Anti-Christ.

The reason that I have pursued Masonry and Rosicrucianism is through my researches inmathematics. I liked the order that Mathematics brings and I was attracted to thePythagorean ideas. Masonry uses the symbols of the square and compass which whereused by the ancients to prove there theorems. It was hoped that you could gain andunderstanding of god through mathematics using a strait edge and compass alone.


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Rosicrucianism shows that there is alchemy in the elements that the chemicals contain ananswer to what creates life and the soul. The society is organized so that the membersresearch and share their findings.

The Pythagoreans asserted that all is number; the masons refer to the Supreme Being as

the GAOTU and GGOTU. We hope that the Supreme is a Loving and Logical beingworking in our favor for universal good. He created us in a world where there is an orderto things. The laws are intuitive and self evident. If we are obedient and know the rulesthat govern our world then we will have a long life filled with happiness. Geometry andMathematics is the one religion that all men agree. This is why Anderson claimed thePythagoras was the founder of Freemasonry. Not that freemasonry existed at that time butbecause geometry was a great unifier of the people. Anderson was taking the greatpersonalities of the past and retroactively making them members of freemasonry becauseof there association with Architecture and freemasonry. In the work we state thefreemasonry is synonymous with Geometry. The pursuit of mathematics is to enable us totake self evident truths and discover answers to our questions. The answers are not

always self evident and we need to explore there hidden mysteries. When we look at ourworld and translate to the language of mathematics we enter a world of abstraction. Thisworld is the bridge between man and the Supreme Being. It is not the language of Godbecause it is imperfect and contradicts its self. This is not universally understood by mostpeople and I will give a few examples. It is not evident in the physical world because welive in a space-time continuum that quantas energy in packets. However using our willand mathematics we may ascend the ladder of Abstraction each rung leading us closer toGod.

At the same time we must remember to remove any superstitions from our research.There should be no taboos towards what other people have found regardless of their

affiliation. Hold that science will show us what is true and trust that god will bless ourresults. The Hidden Mysteries of Geometry can from time to time disprove somepreconceived notions of truth. Gnostics believe that God is knowable and knowledge willover come all obstacles raze ignorance from despair and establish happiness in the pathsof science.


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Numbers

For some people there is a reassurance that 1 + 1 = 2. This statement seems quitereassuring. Who can argue this? It is based on common since. However this is based onthe assumption that there is a cumulative law which states that a + b = b + a. and theassociative law a + (b + c) = (a + b) + c we are using at least base 3 because 1 + 1 in base2 (binary) = 10 we have 2 operators +, = and 3 operands 1, 1, 2 and you cant prove anassumption. Assumptions are self evident truths and we base our rules to preserve thesetruths however we could create a law where a + b ~= b + a and create a whole systembased on that (Lie Group or Lie Algebra)

The human base is 10; we have 10 fingers so we created a number system based on 10.We use other bases and they have come down from different cultures. A base 12 in our

English reading of the numbers. Ten eleven twelve 3teen 4teen 5teen 6teen 7teen 8teen9teen in French we have a base 20 like quart vent which is 4 score in English. Time usesbase 60 12. Computers ended up using base 2 because it is the simplest base. Base 2 isconsidered the celestial base because it consists of on and off, something and nothing,ying and yang. DNA uses base 4 GCTA; we consist of the combination of these 4elements AFWE

I have nothing to say about 0. Theres not much to talk about. It is the absents of number,the void, non-existence, the unknowable, the ineffable, these are degrees of nothingness.It has an Arabic origin, as is our number system. It took a while for our predecessors torap their heads around. At first 0 was used to show the positional had no value however

the totality had a value 100 would be expressed a 1-- or C in the roman numerals. Asthe positional system developed the need for 0 arise. It was then a jump to 0 by its self. Inlogic there is a value called NULL its properties are that it has no value and cannot beexpressed as having a position either. It truly has a dimension of -1. For instance

NULL + NULL = NULL

NULL NULL = NULL

NULL + 1 = NULL

NULL*0 = NULL

This too people had trouble with. Try eating a negative apple. Now we are quite aware ofnegative balances in our bank accounts. We derive the number line. If you move to theright you are positive left is negative. Dexter and Sinister in heraldry. -3, 2, -1, 0, 1, 2,3 until we get to the last element n, yet n + 1 still exists. And n and (n+1) or n -1


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The number system that starts with 0 and continues 1, 2, 3 n, n+1 is called theintegers. But if we where to think in terms of numbers representing a distance or pointson a number line then we will need to mark of midpoints. So we get numbers like 0, ,, , 1 . Now we have a numerator n, and a denominator d Q=n/d. Where n and d areintegers. An irrational number is a number that cannot be expressed as ratio of 2 integers.

The numbers that include the Z or N, Q, and ~Q collectively are called the Real Rnumbers. An example is the square root of 2. An algebraic number is a number that canbe expressed as a polynomial X^2 = 2 is an algebraic number conversely pi the ratio between the circumference of a circle and its diameter is not algebraic it is atranscendental number. Then we get into a problem when we try to answer the equationX^2 = -1 for x. ix x is -1 then the answer is (-1) * (-1) = 1and 1x1 = 1, -1x 1 = -1 but xmust be the same. So I was invented the imaginary number. Complex numbers arenoted as z = a + bi where a, b are R Real and I is the square root of -1. Now instead of apoint on a line we have a point on a field. The next is the hyper complex it consists of theR and 3 imaginary H = a +bi +cj +dk where a, b, c, d are Real numbers and I, j, k arelinearly dependent i*i = j*j = k*k = -1, ij=k ik=-j jk=i they are also known as the

Hamiltonians and can be reduced to a matrix with complex components. An example ofan application of hyper complex numbers is the rotation matrix of an electron with a spin.

The number can be thought of as a point or the distance between 2 points so numbershave a magnitude. Magnitude is an example of a scalar. The number can be thought of asa position on the number line position can be thought of as a vector.

Real numbers can be expressed as a list of positional numbers 99 = 9*10^1 + 9*10^0 andgeneralized as number = sum as n approaches infinity e (n) * base^n + the sum as mapproaches negative infinity f (m) * base ^m infinity is not a number and the rational

number must resolve within a finite number of steps. 1=1.0000000 or .999999 it onlybecomes one when a limit is imposed. 1+1 =2 as far as integers are concerned but arational number it approaches 2 but never becomes 2

lim_(m --> oo) sum_(n = 1)^m (9)/(10^n) = 1

0.9999... = 1

Thus x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999...

9x = 9

x = 1.

There are some numbers that have no meaning they are phi or ineffable they are 0/0 1/0and infinitely large and infinitely small.


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The Point

The point is represented by a dot on the page. The Page is a representation of theManifold. The Manifold has the rest of available dimensions and point is described it

terms of those dimensions. In the beginning after the word and numbers there was only apoint Ain Soph (0,0) or Ain Soph Aur (0,0,0): The Origin or first point. That point thenbecame Kether The Crown (1,0,0) Chochmah (0,1,0) Binah (0,0,1)

A number on the number line is an example of a point consisting on one component it isnot moving and has no magnitude

On a plane the point has 2 components its x, y position or (r, theta) where r is the distancefrom the origin and theta is the angle in degrees or radians

A point can be thought of as moving in the space and its position is determined over time

the position is a function (f (t), g (t), t)

The point has no dimensions but it can be positioned on a manifold that has multipledimensions

(x, y, z) or (r, theta, phi) on 3 space

(x, y, z, t) in 4 space

(F (t), g (t), h (t), t) in space-time the points position is dependant on time

an example of a space-time function would be

F(t) = t2+9

G(t) = t3+1

H(t) = 2t -5

t = 2

so the position would be

(13, 9 ,-1)

or in infinite space

Sum1 to n En*Gn= (e1, e2, e3 en)*(g1,g2,g3 ... gn)N-Space

The point seems intuitive and real to our minds yet there are no examples in our world.


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The point lives in our minds only. No matter how small we make our point it will containsome mass. The glandules of the pencil markings or the volume of ink illustrate that whatappears to have one or two dimensions from a distance have more when viewed closely.Things that are not physical yet exist are called platonic. This is named after Plato thatasserted that there exists a respective object of every thing that exists in our minds and

that this object is perfect and the same for every one. The point called the origin requiresno coordinates because it is especially perfect.

The universe started with this perfect point. Before the big bang there was the singularity,it contained all the mass of the universe yet there was no volume. We must alsounderstand that there was no manifold at the time of creation so we should think of thispoint without the aid of the paper (The Manifold). We know of god as a single being.After creation he emanates such that he is thought of having three parts Father, Son, andthe Holy Spirit. This corresponds to the origin with its components x, y, z being 0, 0, 0.We are all relative to the origin so the origin is our father 0, the origin starts with its selfso it is the son (0, 0) and it is ever present and connected to us through relativity (0, 0, 0).

The point becomes self aware and knows its own potential and decides to form theuniverse in a big bang becoming E= (e1, e2, e3 en). Now we enter the worlds of

dimensions. The Origin is the first principle, that single point that is perfect and absolute,a spiritual attainment which contains the highest virtue: a communion with God.

The affect of the Point on our minds gives us a feeling of our own smallness compared tothe size of the universe. This can be peaceful when we look at a star. It is a large sunmillions of miles away, a giant compared to us and yet it appears as a single point in thenight sky. This point contains much information:

1. The position (x ,y ,z) the distance from the earth. the time that it existed based on

the distance divided by the speed of light. t=d/c2. The speed. the fact that it is moving toward or away from the earth.3. The colour. this contains information about it chemical makeup using a

spectrometer and the speed that it is moving toward us by its blue or red shift.4. The Brightness. this tells us its size relative to the distance.5. The Mass. this is based on its brightness, size and chemical makeup. this tells us

its gravitational pull on surrounding planets and stars. m1xm2/d2

6. The Osculation. this tells us if has any planets in its system.7. The Intelligence. using radio telescopes and listening if there is broadcasts being

sent. it seems like noise because THEY have better encryption algorithms.8. The Name. We name each star in the sky and the constellation that it belongs to.

Focus on the point. It contains a universe of information yet it is the simplest of all thegeometric objects. The Origin is the perfect point and the universe is equal distancearound it. The Origin is not relative like the current state of the universe but absolute. Wecannot observe the origin using the physical senses (Malkuth) however we can use ourabstract reasoning to bridge the void. By our appreciation of the Beauty in mathematics(Tipheroth) we cross the bridge and understand the crowning glory of the Lord (Kether).


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In our contemplation of our creator this leads to a curiosity on what was before creationand this is the point, the origin, Ain Soph.

In the Beginning there was the Manifold and it was without a Point and its dimension was-1 (void). Then the Lord said let there be point and the point and the manifold where one.

This point was the origin and the lord saw that it was good (absolute and perfect). andfrom this all things where made and Geometry was born.


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The Line

The first dimension forms the line or line segment. It consists of the shortest pathbetween two points. It has at least one point in its middle. The Line is a collection of

points. At the moment of creation we come to a new point it started at the origin thenmoved to the first position (1, 0) . This pure vector is the first principal. All furtheractions stem from this vector line. It is where we are going and where we came from.

The line segment starts at an origin and ends at a terminator (a1, b1), (a2, b2) its length isSquare root ((a1-a2) 2 + (b1-b2) 2 ) or in Euclidean Geometry it is defined as c2=a2+b2

(e c/k +e -c/k) = (e a/k + e -a/k ) (e b/k + e -b/k ) Lobachevskian Geometry

ds = adx2 + 2bdxdy + gdy2 Riemannian geometry

Because the line is considered the shortest path the path is not always created by a strait

edge. if the manifold is a sphere then the shortest path would be along the greater circle.

In the Calculus of variations the shortest path is least amount of time it takes to get

between 2 points.

Two lines are considered parallel if they do not intersect in Euclidean Geometry.

the line has only one dimension. Its length when we draw a line it looks like it has onedimension from a distance. But if we look closer we see the width of the line and onfurther investigation we see the height of the ink or pencil graphite. It has been found

that we live in a universe with up to 11 dimensions but we perceive only 3 or 4 it is thisprinciple where our perception of the line at greater vantage point where the dimensionsreduce to one in the pencil.

Two line are considered orthogonal if the are independent for one another

The matrix is used to prove orthogonally or if the are parallel. If the determinate doesn't =0 then the lines are orthogonal. 1 means that the matrix is a rotation matrix and 0 if thelines are parallel.

There are some special classes of lines. Line Segments, Rays, Bars, Vectors:

Line segments show have a magnitude or metric. It has a definite starting point and a endpoint. The length of the line always has a positive value.

A Ray represents the power or velocity. It has a definite starting point and direction butnot a end point. The end point could be considered to be at infinity. Light is an exampleof a Ray.


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A type of line called the Bar pivots at one end and stays at a fixed length. the Compass isan example of a bar.

Vectors are line segments that have a direction. the length is not the physical descriptionbut the speed and direction that a point (the center) is moving. Vectors belong to a larger

group called Tensors. Scalars (Numbers), Vectors, Matrix, finite dimensional Tensors areall members of Tensors

here are some examples

A = 2, ScalarAn = (1, 2, 3), VectorAnm =

1 2 3

4 2 7-1 4 5

Matrix (a spread sheet is an example of a Matrix)

Alnm =

1 1 2

3 4 2

-1 4 4

+

1 5 3

6 2 7

3 7 8

+

8 1 67 9 2

4 6 1

is an example of a tensor of dimension 3x3x3

the value of element A322 = 9


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The Harris code is an example of 3x3x3 tensor. This is also known as the AIQ BKRCipher and uses the Hebrew alphabet along with the finals to form the 27 elements. On a

Hebrew tensorH the H123 element would be Mf ; Final Mem ; ShLG 333 RKB 222 QIA 111MfSV - 666 K fNH - 555 ThMD - 444TzfTzT - 999 PfPCh - 888 NfEiZ - 777

Or in Samarian

#lg rkb qy)

.sw L.Nh tmd

.cT .px .(z

Or in Hebrew

The line joins the points together. It is a symbol of communications as in the Telephoneline, Lines of Communication. In the Work we talk of the unwavering line between Godand Man depicted in the V. O. T. S. L. The lines of the cross that show our dogmatic (therelationship between God and Man) and moral (From man and man or the Golden rule)lines of behavior.

To do on this paper ...


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The Plane

The product of two lines forms a plane

TriangleHeight and baseThe area of a triangle is Height * Base. The base being the bottom where the trianglerests and the height being the highest point of the triangle.

Types isosceles, right, equilateral, acute, obtuseIsosceles triangle is where 2 sides and 2 angles are equalRight Triangle where one angle is at 90Equilateral where all sides and angles (60) are equalacute where no angle is greater that 90Obtuse where one angle is greater that 90

The three altitudes of a triangle intersect at a common point H called orthocenter of thetriangleIf the triangle is an acute triangle, the orthocenter is located inside the triangle.If the triangle is an obtuse triangle, the orthocenter is located outside the triangle;If the triangle is a right triangle, the orthocenter is same as the right angle.Altitudes, Centroid G, Orthocenter H, The circumcircle and the circumcenter O. TheEuler line OGHof the triangle

180 on a sphere it is greater than 180 on a hyperboid it is less than 180

Pythagorean theorem and Euclids 47 problemProves the sum of the square of the hypotenuse is equal to the sum of the squares on theopposite sides. The is the jewel of the past master to show that he can instruct thebrethren in masonry because he has proven for himself the truth of geometry and therules in which it is govern. Proof is a personal thing before you can prove something youmust prove it to you self and before it will be accepted by the public a noted professionalmust look at your work and accept it. Then the public must be presented with the problemand understand it in a pubication.

descriptive verses projective geometryThis is the point where we have non Euclidian geometry an equally valid geometry that isas rigorous as the Euclidian that contradicts some of the rules of Euclidian geometry.

Desargues theorem and all triangles can derive a point from each other and when youextend the line the intersections create a strait line.I noticed that the reverse angle where you are the point and the triangle is yourperspective then all triangles can be thought of as the same triangle only differentperspectives. \


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The platonic triangle and Kants universally valid truths Kant, Immanuel (1724-1804)

Plato introduced the Idea that there is an ideal representation of all objects. ImmanuelKant extended that Idea to truth and there is a Universal Truth that is independent from

empirical data. The Mathematician David Hilbert wanted to create a foundation forMathematics where there was a universal set of axioms that are always true which wereintended to lead to a 'proof theory', this was proven false by Kurt Gdel. Gdel's resultswere a landmark in 20th-century mathematics, showing that mathematics is not a finishedobject, as had been believed. It also implies that a computer can never be programmed toanswer all mathematical questions.

Euler's eye to construct use triangle ABC and draw a line between the AltitudesOrthocenter and the perpendicular bisectors Circumcenter. theCentroid, Cantor point and the 9-Point center fall on this line as well. De Longcampspoint is out site but runs parallel to it

In a equilateral triangle all of these points are the same.

Named after Leonard Euler(1707 - 1783) Swiss mathematician and physicist who lost hiseye in 1735 invited by Frederick the Great to be a member of the member of the BerlinAcademy 1741. has a transcendental number named after him and he created the identityi which is the square root of negative one.

Pascal's theorem If (A,B,C,D,E,G) are six points on a conic, then the intersections oflines ACand BE,AD and BG, and CGand DEare collinear.

I will attempt that Pascal's theorem is Collinear to Hermes Trismegistus Emerald tabletor Tabula Smaragdina (here after TS)

TS.1 this is trueTS.2 that which is the intersection is from that which isCollinear and that which is Collinear is from that which isthe intersection working the miracles ofABCDEG. Note:Circle and triangles are examples of conics.TS.3 1=ABCDEGTS.4 intersection lines father is CG and the motherDETS.5 the intersections nurse is ABTS.6 separate Subtle A from Gross B and create lines ACand BETS.11 This (The mathematics) is how the world (ABCDEG) was formed

Desargues perspective theorem. when two triangles are in perspective the meets ofcorresponding sides are collinear.


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I will attempt that all triangles can be derived from a single primitive triangle with theproper transformation matrix so that there is but one triangle only multiple perspectives.From this we should conclude that All religions point to a single truth.

Circle

Squire


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The Manifold

A manifold is any other dimensional space

Man's life in the Manifold

The geometry of the craft lodge is referred as a parallelepiped from North to South fromEast to west from the surface or its center to the heavens. This is a spherical manifoldnorth and south stop and the poles and east meets west it starts from the center of theearth and reaches out to infinity. Our reference would be (Radus, Longitude, Latitude) orPoint1= (R, Phi, Theta) the conversion to Cartesian co ordinance would be (x= R CosTheta, y=R Sin Theta, z= R Sin Phi)

Meter Space

An N-Dimensional space where distance is defined

Topology

A space where there is no Metric. Math without numbers!

Banach Space

An N dimensional inner product space where the distance formula is defined by LP and P>= 1.

Hilbert Space

A N dimensional inner product space where distance is determined by LP and P=2

The non-Manifold interacts with the Manifold

How does god interact with creation?

A defense of the Trinity

Conclusion:

A new Sacred Geometry would need to move into a more general representation for lines.They would be defined in a multi Dimensional inner product space. We must alsorecognize the limitation of mathematics when approaching the divine. What is infinity?Some logical elements like Russell's Paradox and Gdel's incompleteness theorem onproofs in general show that we are limited in our understanding. We have a feeling thatthere should be a universal truth and yet we can't describe it mathematically. It is in therealm of faith.


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AppendixDavid Hilberts 23 Questions

1. Cantor's problem of the cardinal number of the continuum2. The compatibility of the arithmetical axioms3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes4. Problem of the straight line as the shortest distance between two points5. Lie's concept of a continuous group of transformations without the assumption of thedifferentiability of the functions defining the group6. Mathematical treatment of the axioms of physics7. Irrationality and transcendence of certain numbers8. Problems of prime numbers9. Proof of the most general law of reciprocity in any number field10. Determination of the solvability of a diophantine equation

11. Quadratic forms with any algebraic numerical coefficients12. Extension of Kroneker's theorem on abelian fields to any algebraic realm ofrationality13. Impossibility of the solution of the general equation of the 7-th degree by means offunctions of only two arguments14. Proof of the finiteness of certain complete systems of functions15. Rigorous foundation of Schubert's enumerative calculus16. Problem of the topology of algebraic curves and surfaces17. Expression of definite forms by squares18. Building up of space from congruent polyhedra19. Are the solutions of regular problems in the calculus of variations always necessarily

analytic?20. The general problem of boundary values21. Proof of the existence of linear differential equations having a prescribedmonodromic group22. Uniformization of analytic relations by means of automorphic functions23. Further development of the methods of the calculus of variations

The Works of KantUniversal Natural History And Theory On The Heavens (1755)Dreams of A Spirit-Seer(1766)Inaugural Dissertation (1770)The Critique of Pure Reason (1781).Fundamental Principles of the Metaphysic of Morals (1785)Critique of Practical Reason(1788)Crituiqe of Judgement(1790)Metaphysical Foundations of Natural Science (1786)Religion within the Bounds of Reason Alone (1793)Perpetual Peace (1795)Introduction to the Metaphysics of Morals (1797), often translated in two parts: The
http://aleph0.clarku.edu/~djoyce/hilbert/problems.htmlhttp://www.knuten.liu.se/~bjoch509/works/kant/cr_pure_reason.txthttp://www.knuten.liu.se/~bjoch509/works/kant/princ_morals.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_pract_reason.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_pract_reason.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_judgement.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_judgement.txthttp://www.knuten.liu.se/~bjoch509/works/kant/intr_morals.txthttp://aleph0.clarku.edu/~djoyce/hilbert/problems.htmlhttp://www.knuten.liu.se/~bjoch509/works/kant/cr_pure_reason.txthttp://www.knuten.liu.se/~bjoch509/works/kant/princ_morals.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_pract_reason.txthttp://www.knuten.liu.se/~bjoch509/works/kant/cr_judgement.txthttp://www.knuten.liu.se/~bjoch509/works/kant/intr_morals.txt


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Metaphysical Principles of Rightand Metaphysical Principles of Virtue.The Metaphysical Elements of Ethics

The Science of Right

Works of Plato

Euthyphro, Apology, Crito, PhdoCratylus, Thetetus, Sophist, StatesmanParmenides, Philebus, Symposium, PhdrusAlcibiades, 2nd Alcibiades, Hipparchus, Rival LoversTheages, Charmides, Laches, LysisEuthydemus, Protagoras, Gorgias, MenoHippias major, Hippias minor, Ion, MenexenusClitophon, Republic, Timus, CritiasMinos, Laws, Epinomis, Letters
http://www.knuten.liu.se/~bjoch509/works/kant/science_right.txthttp://www.knuten.liu.se/~bjoch509/works/kant/science_right.txt


19/19

Bibilography:

The Elements Euclid

What is number? Richard CourantThe Mathematical Experience: Philip J. Davis, Reuben HershFoundations of Geometry David HilbertMagick: Liber ABA, Vol. 4 Aleister CrowleyThe WorkAnonymousThe Kabbalah Unveiled S L MacGregor MathersUniverse in a Nutshell Prof. Stephan W. HawkingThe Book of the Words Albert PikeThe Canon William SterlingOccult Philosophy HC Agrippa

Sacred Geometry and Mathematics

Documents

Transcript of Sacred Geometry and Mathematics