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 Title: Lecture: The Two Christmas Annunciations
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 were not able to think as clearly as we can now. Geometry and
 our geometry and mathematics. Euclid was the first to give geometry to
 Euclid gave in the way of geometry had already lived in the Mysteries
 geometry was taught in the Mysteryschools to selected individuals who
 content of geometry in an age in which, from the moment of awaking
 geometry with quite another soulconstitution than we have today,
 Title: Lecture: The Alphabet
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 geometry, music and astronomy. In this ascent through the various
 arithmetic and in a similar way with geometry. From geometry one got
 today learns geometry and as he proceeds begins to experience forms as
 grade, and makes geometry sound musically, all this, as far as I know,
 to the sixth part of what was to be learned — from geometry to music.
 Title: Lecture: Goethe and the Evolution of Consciousness
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 partiality for mathematics, for geometry, for mechanics.
 Title: Lecture: The Universe
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 occupy ourselves with arithmetic and geometry, we look back
 into ourselves. We recognize the laws of geometry simply
 Title: Lecture: The Position of Anthroposophy among the Sciences
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 which is usually called geometry and, at least in those parts of it
 outside us. And what we conceive as space, the space we use in geometry
 — Euclidean geometry, at first — is nothing more than an
 space we speak of in geometry, we extend our consciousness
 bring the cubical shape with us from our geometry, knowing that its
 space of geometry (which the child learns to know from inner
 is called modern or synthetic geometry.” You see, when one
 passes from analytic to synthetic geometry — which enables us,
 these ends and obtain the curves. In analytical geometry we do not
 lay hold of the forms, whereas in synthetic geometry we live within
 things inwardly. In this way the student learnt Geometry
 through inward exercises, and this geometry, as something involving
 synthetic geometry that I became clear about the clairvoyant's
 procedure. Naturally, not everyone who has studied synthetic geometry
 Title: Lecture: Anthroposophy and the Visual Arts
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 place, Euclidean geometry — we set out, as you all know, from a
 one advances beyond ordinary perception, which suffices for geometry,
 analytical geometry, as one constructs Euclidean space. One can,
 Title: Lecture: Three Epochs in the Religious Education of Man
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 they read their destiny. External calculations, geometry and
 Title: Lecture: Zarathustra
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 his great predecessors — Geometry from the Egyptians,
 Egyptians from whom he learnt Geometry, nor among the Phoenicians
 Title: Lecture: Hermes
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 mathematical sciences, geometry (which Pythagoras afterwards learnt
 was brought down by Hermes into mathematics and geometry and he
 geometry, landsurveying — in short all they needed for their
 mathematics, geometry, the science of landsurveying and
 Title: Tension Between East and West: Lecture 1: Natural Science
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 we have mathematics, geometry; for the organic, the
 Title: Tension Between East and West: Lecture 2: Psychology
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 as are operations in mathematics or geometry.
 Title: Tension Between East and West: Lecture 3: East and West in History
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 comprehended with ordinary mathematics and geometry: living
 geometry, astronomy and music. What mattered was
 Title: Purpose of the Goetheanum and Anthroposophy
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 formed only in the imagination. Geometry is not a spontaneous
 must first produce geometry within ourselves, and by means of
 geometry we create the forms which will lead us into the
 Title: Karma of Vocation: Lecture VIII
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 instruments, was also familiar with geometry and mathematics,
 occupied with geometry, and at first taught his son music
 Title: Necessity for Spiritual Knowledge: Lecture 2
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 Space, in terms of mathematics, of geometry and of mechanics.
 Title: Necessity and Freedom: Lecture V: The "I" is Found on the Physical Plane in Acts of Will
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 more a matter of mathematics and geometry. Just as today we can
 Title: The Building at Dornach (Bn/GA 289): Lecture II: Bau Lecture II
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 Whereas elsewhere the dynamics of Geometry are put before us in repetition
 Title: Forming of Destiny: Lecture 3: The Subconscious Strata of the SoulLife and the Life of the Spirit After Premature Death
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 instance, in reference to Geometry. I have once before mentioned how
 much we all unconsciously know of Geometry. This is actually a fact.
 For when we learn Geometry we do not learn it from outside things, it
 reality we know everything that can be learnt of Geometry. We know it
 Title: Lecture: Foundations of Esotericism: Lecture XXXI
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 of great initiates who taught geometry, the practice of surveying,
 Title: First Lecture (First Scientific LectureCourse)
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 to outer Nature. I mean Geometry, — all that is known by means
 of pure Geometry. What a cube or an octahedron is, and the relations
 In Arithmetic, in Geometry and in Phoronomy or Kinematics we have the
 Geometry and Kinematics are not yet Natural Sciences in the proper
 scientists proceed. Equipped with arithmetic, geometry and
 truths of arithmetic, geometry and kinematics, — these we
 Title: Second Lecture (First Scientific LectureCourse)
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 contains, as by arithmetic, geometry and kinematics I get a clear
 Title: Tenth Lecture (First Scientific LectureCourse)
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 of arithmetic and geometry — by the arrangement of lines,
 arithmetical thinking. Geometry, you know, was a very ancient
 pure Geometry — was a thing handed down from ancient time.
 19th century, the Geometry itself began to grow uncertain. It
 carry out the proof. Now in the whole of Euclid's Geometry there is
 — the space of ordinary Geometry — the three angles of
 besides the ordinary geometry handed down to us from Euclid other
 Euclidean Geometry which we ourselves think out. Might it not be
 processes outside are governed by quite another geometry, and it is
 Euclidean geometry and all the formulae thereof?
 algebra, geometry and kinematics, then we are doing far more
 remain utterly unconscious of the origin of the geometry you use.
 mathematics, in our geometry, in our ideas of movement. These
 akin to the realm of human Will, in which geometry and arithmetic
 whence come geometry and kinematics — and on the other hand
 Title: Origins of Natural Science: Lecture III
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 geometry, and who uses his analytical formulas and geometrical concepts
 mathematician starts from the concepts of Euclidean geometry, the
 nonEuclidean geometry.)
 So what can we say when we look back at the geometry of earlier
 space layout of analytical geometry, which establishes a point
 example, geometry with the body's orientations to
 mathematics, of geometry, is brought up out of our limb system of
 movement. We would not have geometry if we did not place ourselves
 all such moving about, in all such inner geometry. There is always
 corresponding to the pentagram. In the blood the whole of geometry
 projection of geometry on the blood and its experiences. This was
 Title: Origins of Natural Science: Lecture IV
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 abstracted mathematics and geometry, and therefore he penetrates
 Title: Origins of Natural Science: Lecture VI
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 applied to everything concerned with geometry, number, weight, etc.
 that is related to form, space, geometry or arithmetic, we have to
 we want to discover the nature of geometry and space, if we want to
 Title: Warmth Course: Lecture III
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 point would disappear form his space realm. What kind of a geometry
 geometry. He would be able to speak only of distance and the like, of
 of plane figures and would have a twodimensional geometry. We men
 have at the outset a threedimensional geometry. A being with a
 unidimensional geometry would have no possibility of understanding
 twodimensional geometry would be unable to follow the motion of a
 threedimensional geometry. Now I may just as well do what I am
 extension in one dimension as one does in geometry in the case of a
 the way it is done in geometry. It is always true that when I observe
 Title: Warmth Course: Lecture V
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 impulses, as I have explained we do in geometry, this is because we
 Title: Warmth Course: Lecture VII
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 Geometry really comprehends space forms but can never comprehend the
 Title: Lecture: Younger Generation: Lecture II
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 Grammar, Dialectic, Rhetoric, Arithmetic, Geometry, Astrology, Music
 Title: Lecture: Younger Generation: Lecture IX
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 lead them gradually to what can be known, to arithmetic, geometry,
 rhetoric, one was able later to pour into arithmetic, geometry,
 rhetoric, arithmetic, geometry, astronomy and music, grew out of an
 Title: Lecture: Mission of Michael: Lecture IV: The Culture of the Mysteries and the Michael Impulse.
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 such as mathematics and geometry, which are employed in the endeavor
 through logic and the Ahrimanic wisdom through mathematics, geometry,
 Title: Spiritual Hierarchies: Lecture 1
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 reflection in the physical world. Just as the geometry of Euclid has
 Title: Lecture: The Proclamations to the Magi and the Shepherds
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 possible. Nor could mathematics or geometry be grasped in the way they
 who first gave geometry to the world at large. The
 geometry presented to mankind by Euclid had already been cultivated
 nevertheless a fact that the geometry and arithmetic learnt by
 geometry to children by calling upon the intellect in an age when from
 Title: Rosicrucianism/Initiation: Lecture V: Occult Schools in the 18th and First Half of the 19th Century
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 know the true and authentic World Geometry, to know the way in which
 Title: Lecture: Moonbirth and Sunbirth. Necessity and Freedom. Stages of the Ancient
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 the Geometry, the Mensuration of that time, and so forth. For what did
 space: Mensuration, Geometry, is an earthly science. This in fact was
 Geometry, Architecture and Mensuration. He saw that a real science of
 the true sciences of Earth are Geometry, Architecture, Music and
 knowledge, we can study only Geometry, Mensuration, Music and
 overcome bodily vision, nay, if one has even overcome Geometry and can
 Title: Kingdom of Childhood: Lecture 5
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 teaching of Geometry. You can build up your Geometry lessons to reach
 had to teach Geometry to an elderly lady because she loved it so
 Geometry at all, so I began and made everything lead up to the
 children of eleven or twelve you can quite well take Geometry up to
 Title: Kingdom of Childhood: Questions and Answers
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 and colour. (Illustration). Linedrawing belongs only to Geometry.
 shade. Drawing should only be done in Geometry and all that is
 Title: Art of Healing: Lecture I
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 geometry. In fact we comport ourselves with regard to
 mathematics or geometry, only in Anthroposophy we are not
 Title: Spiritual Science and the Art of Healing: Lecture I
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 maybe to the attainment of mathematics or geometry. In fact we
 same way as we do with regard to mathematics or geometry, only
 Title: Education: Lecture II: Principles of Greek Education
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 palæstric. It was known that geometry is inspired in the human
 Title: Education: Lecture IX: Arithmetic, Geometry, History
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 geometry, indeed all mathematics, occupy a unique position in education.
 arithmetic or geometry, or when we lead him on to writing from the basis
 the remarkable thing is that arithmetic and geometry work upon both
 education arithmetic and geometry are really like a chameleon; by
 definite age, arithmetic and geometry must be taught throughout the
 In geometry,
 instead of beginning geometry with triangles and the like, where the
 into arithmetic and geometry. This whole mode of thought is extended
 and attention, especially into our teaching of arithmetic and geometry.
 inner life and soul, just as we did in the case of arithmetic and geometry,
 Title: Human Values in Education: Lecture V
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 descriptive geometry. The pupils sit facing him with their
 geometry. Now coming into another school and looking at its
 name or another: Diploma in Geometry or Mathematics or whatever
 subject, or geometry or something else. The essential substance
 Title: Human Values in Education: Lecture IX
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 Geometry, Astronomy or Astrology, and Music. To take one
 Title: Practical Course/Teachers: Lecture X: Arranging the Lesson up to the Fourteenth Year
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 on to geometry, whereas we have hitherto restricted the
 elements of geometry to drawing. In drawing, of course, we can
 is a square.” But what geometry adds to these, with its
 history by introducing physical concepts and with geometry by
 lesson with geometry.
 Geometry offers you an extraordinarily good opportunity of
 combining the object lesson with geometry itself.
 turn geometry into an object lesson. But there is a certain
 place in a geometry lesson, you can teach the child in seven or
 eight hours at the most all the geometry necessary to introduce
 demonstrate the first rudiments of geometry graphically
 of the aim of geometryteaching. I beg you to notice the use of
 Foreign languages. Geometry.
 Title: Practical Course/Teachers: Lecture XII: How to Connect School with Practical Life
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 relatively good geometry and arithmetic textbooks. They have
 bookshops in Vienna to get older geometry books, because I
 class we took over the Fialkowski geometry books from the boys
 great joy, because it proves that geometry books written in
 quite horrible. The arithmetic and geometry books are very bad.
 the Austrian Benedictines. The mathematics and geometry books
 geometry and mathematics. The Benedictine feeling in general is
 address the people unless he is familiar with geometry and
 child who simply has a geometry lesson after a French lesson.
 Title: Study of Man: Lecture III
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 nerves. Origin of geometry lies in movement. Man keeps earth alive.
 our bone system. How is it, for instance, that man can do geometry?
 The higher animals have no geometry; that can be seen from their way
 animals have a geometry, only we do not notice it.” Now, man can
 form a geometry. But how, for example, does he form the conception of
 These movements to which you give fixed forms in geometry — when
 In evolving geometry he is copying something that he himself does in
 Title: Study of Man: Lecture XIV
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 Thus all the teaching, even what is given in geometry and arithmetic
 even in arithmetic and geometry. That is why I said yesterday (In
 Title: True/False Paths: Lecture Ten: Influences of the ExtraTerrestrial Cosmos Upon the Consciousness of Man
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 with which we calculate, or study geometry, if we penetrate with full
 Title: True/False Paths: Lecture Ten: Influences of the ExtraTerrestrial Cosmos Upon the Consciousness of Man
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 with which we calculate, or study geometry, if we penetrate with full
 Title: Evolution, Earth, Man: Lecture XII
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 that a certain knowledge of geometry is necessary to understand how
 one needs geometry for it. Now the earth is formed in this very way
 out of the universe — formed from knowledge of geometry, in
 geometry in the world, that everything is in accord with geometry.
 Title: Metaporphoses/Soul One: Lecture 8: Buddha and Christ
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 Spiritual Science than the fact that the elements of geometry are found in
 Euclid. Just as it would be absurd to accuse a geometry teacher of practising
 true, genuine Buddhism. Just as a teacher of geometry is not required to
 Title: Metaporphoses/Soul Two: Lecture 6: Positive and Negative Man
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 those with a knowledge of geometry may enter here”?
 geometry and mathematics cannot be accepted on the authority of another
 person. We have to work through geometry by our own inner efforts and can
 system of ideas such as geometry, he will learn to respect the creative
 Title: Nine Lectures on Bees: Lecture IV
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 you are not all of you good geometricians. Geometry is not exactly familiar
 Title: Anthroposophy Science: Lecture I
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 geometry. There we are not confronted by an outer world, we
 good grounding in the knowledge of geometry and mathematics.
 geometry. This is based an a seemingly paradoxical but deeply
 Title: Anthroposophy Science: Lecture V
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 how ordinary analytic geometry relates to socalled synthetic
 or projective geometry. I would like to say a few words on
 this subject. In analytic geometry we discuss some equation
 analytic geometry, we perform operations which we
 we consider projective geometry, we arrive at a very
 evident, for example, when you see how projective geometry
 can do it; but we cannot really. However, projective geometry
 which is not possible for analytic geometry. With projective
 geometry it is really possible. When we move out of mere
 analytic geometry into projective geometry, we get a sense of
 rounding, which analytic geometry describes only externally.
 thinking — of which analytic geometry is the prime
 projective geometry, we do not actually have imagination yet,
 Title: Anthroposophy Science: Lecture VII
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 receive a training in geometry and mathematics. In a similar
 Title: Ways/Architecture: Lecture III: The New Conception of Architecture
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 circle. Here I must ask those who have studied geometry
 all know a little of geometry though you may have forgotten
 all the wisest of geometricians, only of course the geometry
 geometry, instead of using their wonted methods, could apply
 Those who have studied geometry will know that the
 Title: Excursus/Mark: III: Excursus: Lecture I
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 nonphysical. Geometry, for example, was first learnt so that it was
 Title: Temple Legend: Lecture 8: The Essence and Task of Freemasonry from the Point of View of Spiritual Science  2
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 the first man, who had an extraordinary knowledge of geometry at the time
 Title: Temple Legend: Lecture 20: The Royal Art in a New Form
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 the techniques of weighing and calculating, of geometry and
 geometry and mechanics. And as you continue your contemplation of the
 geometry and mechanics; so it is the case that in future man will
 Title: Fruits/Anthroposophy: Lecture 6: From Imaginative Knowledge to Inspirational Knowledge
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 is working with geometry, when it enters in a living way into everything
 working with geometry and mathematics. The work which has to be done
 Title: Fruits/Anthroposophy: Lecture 8: The Social Question
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 used in analytical geometry, the cross formed by ordinate and abscissa,
 Title: Where/How/Spirit: Lecture IV: Bible and Wisdom I
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 usual school geometry, the Euclidean geometry, was written down
 historically. However, who learns geometry at school today, is
 today an expert of geometry is. He can find geometry out of
 Title: Child's Changing Consciousness: Lecture I
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 child will be introduced to many different subjects, such as geometry.
 done in geometry is merely an abstraction. Human beings do
 unconsciously, “body geometry” as an activity akin
 Title: Child's Changing Consciousness: Lecture VI
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 introduce an element of grace. In geometry or physics
 occasions, teachers, in their physics and geometry lessons,
 Title: Cosmic Prehistory: Lecture I: The Threefoldness of Space and the Unity of Time
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 standing at right angles to one another, or even all that geometry has
 angles to one another like the three dimensions of geometry.
 Title: Cosmic Prehistory: Lecture III: Romanism and Freemasonry
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 They must first learn them according to the laws of geometry, how they
 see these forms of the mineral kingdom which may be expressed in geometry
 Title: Answers to Big Questions: Lecture I: The Nature of Spiritual Science and Its Significance for the Present
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 Indeed, the mathematical field, the field of geometry seems to
 geometry could be shaken anyhow? Nevertheless, it is typical
 mathematics and geometry and could invent a geometry that
 case: in our Euclidean geometry, one can draw only one parallel
 Title: Answers to Big Questions: Lecture VII: How Does One Attain Knowledge of the Spiritual World?
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 Mathematics, geometry teaches the human being the knowledge of
 or geometry becomes the living symbol for the spiritual
 Title: Riddles of the World: Lecture XVI: German Theosophists at the Beginning of the Nineteenth Century
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 geometry. How circles and angles form a group together, how the
 Title: Knowledge of Soul and Spirit: Lecture IV: Initiation
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 today. I can produce evidence. Mathematics, geometry, can
 nobody could be accepted without knowledge of geometry. That
 does not mean that he had to know the entire geometry, but that
 and culture for other fields than geometry is Goethe again. It
 Title: Knowledge of Soul and Spirit: Lecture VIII: The Soul of the Animal in the Light of Spiritual Science
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 something like a kind of geometry in this activity that the
 geometry is even today not so far to be able to copy these
 Title: The Apocalypse: Lecture VI
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 doesn't mean our rather abstract geometry or mathematics, but
 Title: Mystery Trinity: Part 1, Lecture 3
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 dialectic, arithmetic, geometry, music, and astrology gradually
 Title: Freedom/Immortality/Social: Lecture VIII: How Natural Sciences Justify the Supersensible Knowledge
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 geometry that I had, however, no talent to keep mathematical
 Title: Theosophy/Rosicrucian: Lecture I: The New Form of Wisdom
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 geometry, which includes, for instance, the Pythagorean Theorem, is
 geometry are learnt quite independently of how geometry itself
 rudiments of geometry today know about Euclid? Nevertheless it is
 Euclid's geometry that is being taught. Only much later, when the
 the pupil who learns elementary geometry today is concerned with the
 developed in the course of history. Just as the pupil learns geometry
 Title: Theosophy/Rosicrucian: Lecture XII: Evolution of Mankind on the Earth II
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 he found in geometry into his soil, his land; outer nature was
 Title: Turning Points: Lecture 1: Zarathustra
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 of Geometry, from the Phænicians concerning Arithmetic, and
 Title: Turning Points: Lecture 2: Hermes
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 those directions in which such sciences were needed as Geometry
 taught mankind the principles of Agriculture, Geometry and
 script, our Mathematics, Geometry, Surveying, our general conduct
 Title: Turning Points: Lecture 4: Moses
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 rhetoric,logic, arithmetic, geometry and astronomy. Plato and
 Title: Karmic Relationships, VIII: Lecture V
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 Steiner's own geometry teacher. Lord Byron. Voltaire. Swedenborg.
 had a Geometry teacher whom I loved dearly. It was not difficult for
 Geometry. But this teacher was really quite unusual. He had a
 peculiar talent for Geometry that fascinated me, although people who
 wonderful way in which this teacher of mine worked in Geometry, his
 This club foot enabled me to trace the individuality of my Geometry
 contemporaries (for my Geometry teacher lived later than Byron) had,
 again, as I look back upon my Geometry teacher with his modest,
 Title: Karmic Relationships, Volume I: Lecture I
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 classes on modern Synthetic Geometry, and when for the first time it
 was shown me through Geometry itself that a straight line should not
 returns from the opposite side. Geometry expresses it by saying that
 Title: Karmic Relationships, Volume I: Lecture XI
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 treatment of descriptive geometry one could only stand amazed.
 enabled me to consider the problems of Byron and my geometry teacher
 Title: Karmic Relationships, Volume I: Lecture XII
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 of whom I spoke yesterday, Lord Byron and my geometry teacher. (You
 earlier life if I had not found this geometry teacher of mine at his
 side. Byron was a genius. My geometry teacher was not even a genius
 constructs something in geometry, he always knows how to state the
 all. But this geometry teacher, though an excellent geometrician, was
 example, nothing whatever of analytical geometry. He knew nothing of
 the geometry that has to do with calculating and equations; in that
 constructive geometry at the fact that the circle is the locus of the
 geometry it is shown how one sets up such and such an equation and
 circle and to start calling it by the name of our geometry teacher.
 generally realised) that man does not get geometry or mathematics out
 would not have the perception of it. You would have no geometry at
 all if you did not walk and grasp hold of things. Geometry pushes its
 anyone who has a foot such as my geometry teacher had, there resides
 one penetrated more deeply into this geometry teacher of mine, into
 Title: Karmic Relationships, Volume IV: Lecture IX
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 Dialectic, Arithmetic, Geometry, Astronomy and Music. In their combined
 Title: Karmic Relationships, V: Lecture IV
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 contact. I had a geometry teacher
 to you that geometry is one of the subjects to which I owe most because
 of the impulses it quickened in me. This geometry teacher himself
 focused so exclusively upon geometry that in the real sense of the
 geometry — his special subject — underwent changes.
 Title: Karmic Relationships, VII: Lecture Five
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 recounting the story of my life I have mentioned a geometry
 the saying goes, for geometry. Even at the age of nine, a
 geometry book that fell into my hands brought me sheer delight;
 years after, this man was my geometry teacher. He was a most
 the very embodiment of geometry — but of a particular
 kind: descriptive, constructive geometry. In the higher classes
 I was obliged to learn analytical geometry — as it is
 remarkable progress in geometry just because I loved him so
 came into the class and demonstrated geometry in his own
 teacher of geometry, I should certainly never have discovered
 does not experience geometry through his head. You would never
 arms or the legs form angles. In geometry we actually
 limbs teach us geometry. It is only because we have become
 presenting geometry, was clear to me as I studied the
 slightly later time, was the unknown geometry teacher. In that
 came to give his geometry lesson, never once gave me an
 Title: Spiritual Development: Lecture I: The Inner Experience of the Activity of Thinking
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 especially in geometry. As regards the activity of thinking
 that has to be applied in geometry, one need only take a
 Title: Theosophy and Rosicrucianism: Lecture XI: Progressive Development Through the Different Cycles of Culture
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 physical world. An astrology and geometry arose in Egypt
 Title: Theosophy and Rosicrucianism: Lecture XIII: The Rosicrucian Training
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 known to the civilisation of the west, through geometry, the
 Title: Gospel of John (Basle): Lecture I
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 documents. The best expounder of Euclid's Geometry is one who
 knows Geometry, not one who only knows the Greek
 Title: Gospel of John (Basle): Lecture VI
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 applied his knowledge of Geometry and divided the land; he
 Title: Gospel of John: Lecture I: The Doctrine of the Logos
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 Euclid, the old geometrician, first gave us that geometry
 geometry absolutely dependent upon this book of Euclid? I ask
 you, how many pupils today study elementary geometry without
 because geometry originates in a capacity of the human
 spirit. If the pupil has first studied geometry by means of
 as the pupil acquires a knowledge of geometry without knowing
 the parallel of the geometry of Euclid. Will the best
 expounder of geometry be the one who in his own way can make
 understanding previously nothing at all about geometry. On
 philologist, but understood geometry, he would still be able
 philologist would explain the geometry of Euclid. But from
 Title: Gospel of John: Lecture VIII: Human Evolution in its Relation to the Christ Principle
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 and to develop the science of geometry. Maya or illusion was
 Title: Gospel of John: Lecture IX: The Prophetical Documents and the Origin of Christianity
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 geometry, acquiring it by means of their own faculties from
 geometry itself, what do they know of the geometry of Euclid,
 of the original document in which this elementary geometry
 has first learned geometry by means of his own faculties, he
 it the Egyptians became the masters of Geometry? It was
 Title: Gospel of John: Lecture VI: The Atlantean Oracles.
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 And in Egypt we see arising a geometry applied to external things. Such
 Title: Gospel of St. John: Lecture VI
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 mutual relation. In Egypt we see the rise of a science of geometry
 Title: Destinies of Individuals and Nations: Lecture 12: The Group Sculptured for the Building in Dornach
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 be given by showing that calculations based on geometry
 Title: Things Past and Present: Lecture V: Comenius and the Temple of PanSophia
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 external is geometry and mathematics. That person who today
 allows geometry and mathematics to work upon him in a true
 different kind of knowledge comes to him in geometry and
 something within geometry and mathematics which one can
 much wider basis. That which is at the basis of geometry and
 “Geometry as it is brought to expression in the form of
 able to diagram geometry for himself in a clairvoyant way. It
 the forms of geometry. You are within all of that. Just as
 student of the Greek Mysteries etherically spun geometry
 around himself, whole geometry and the other human knowledge
 Title: Things Past and Present: Lecture VIII: Thomas More and His Utopia
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 which we learn in geometry only have validity for the
 Title: Symptom 2 Reality: Lecture VII: Incidental Reflections on the Occasion of the New Edition of 'Goethes Weltanschauung'
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 geometry. And it was they who paved the way for this inner
 Title: Karmic Relationships, VI: Lecture V
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 the intimate impression made by my geometry teacher upon those who
 Title: Karmic Relationships, VI: Lecture VIII
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 domains of geometry, chemistry, physics, music, architecture, and the
 Title: Man: Hieroglyph: Lecture Ten
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 of abstract geometry, man had applied concrete geometry to celestial
 space, the concrete geometry that proceeds from a study of the whole
 Title: Materialism/Anthroposophy: Lecture V
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 heavens, which he describes only by means of geometry,
 Title: Materialism/Anthroposophy: Lecture VIII
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 like geometry, like arithmetic, which, as we know, is mainly
 The geometry
 experience to something abstract, developing geometry out of
 longer has any idea that he produces geometry because
 the course of humanity's evolution, geometry, arithmetic, and
 Title: Materialism/Anthroposophy: Lecture XVII
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 cultivated in their ancient form: geometry, astrology,
 according to the law of geometry; it is subject to the
 geometry, astrology, arithmetic, and music. The Greeks lost
 element, which can be comprehended by means of geometry,
 we note in Greek culture a certain decline of geometry as it
 forming of thought. Arithmetic, geometry, astrology, and
 arithmetic and geometry. In Egypt, people felt universal
 forms the subjects of higher education: geometry, astrology,
 dialectics, geometry, astrology, arithmetic, and music. The
 geometry, astrology, and music had turned into sciences.
 Title: Therapeutic Insights: Lecture III
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 Lubatscheffski's geometry, for instance, or Riemann's
 Title: Evolution of Consciousness: Lecture III: InitiationKnowledge  New and Old
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 spiritual world we find that our mathematics and geometry are no
 greater part of our geometry. But in the spiritual world it is the
 Title: Esoteric Studies: Easter: Lecture II: MoonBirth and SunBirth
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 geometry, surveying, etc.; for this second stage comprised everything
 survey: we measure the Earth; and surveying, geometry and the like
 delusive, except in the case of geometry, architecture or surveying;
 geometry, architecture, music and surveying were the sciences that
 — forces employed for ordinary learning — only geometry,
 physical observation has been surmounted — after even geometry
 Title: Easter Festival: Lecture II: MoonBirth and SunBirth
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 architecture, geometry, surveying, etc., of the time. This was
 the earth; both it and geometry are earthly sciences.
 except as it applies to geometry, architecture, and
 however, geometry, architecture, music, and surveying are
 in the body can be used only to study geometry, surveying,
 even geometry have been left behind, when one can
 Title: Festival of Easter: Lecture 2: MoonBirth and SunBirth
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 was Architecture, Geometry, Surveying, etc. For in what did
 of measuring, geometry, or surveying is an earthly science. It
 gained by mere earthly means, unless connected with geometry,
 knowledge; and that geometry, architecture, music, and the
 geometry, surveying, music, and architecture can be studied ...
 physical perception has been overcome, when geometry has
 Title: Occult Movement: Lecture Two: Mediumistic Methods
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 geometry whose lines are drawn by light, and the light
 Title: History of Art: Lecture X: Disputa of Raphael  the School of Athens
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 appear: grammar, rhetoric, dialectic, geometry, arithmetic,
 Title: Background/Mark: Lecture Three: The Tasks of the Fifth PostAtlantean Epoch
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 related to supersensible facts. Geometry, for instance, was
 Title: Esoteric Development: Lecture VII: The Great Initiates
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 highest thoughts of mathematics and geometry, that have an eternal
 Title: Old/New Methods: Lecture Seven: Methods of Initiation, Old and New  2
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 conditions to practise thinking with exactitude except in geometry
 Title: Old/New Methods: Lecture Eight: The Passage of the Human Soul and Spirit through the Physical SenseOrganization
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 earthly shapes only come about in geometry and in whatever other
 Title: Truths and Errors: Lecture IV: Truths of Spiritual Research
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 example, also in geometry.
 why one cannot apply the usual geometry to the things of the
 occupied with modern geometry, as one called it at that time,
 Title: Lecture: The Bible and Wisdom
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 — Euclid's Geometry. Anyone who understands something of Geometry today
 someone who had really studied Geometry today. When such a man comes to
 Title: Lecture: Buddha and Christ
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 the fact that elementary geometry is also to be found in Euclid; and
 geometry of perpetrating ‘Euclidism,’ so is it
 Title: Lecture: The Wisdom Contained in Ancient Documents and in the Gospels
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 later, but even in the case of geometry we cannot go back further
 Title: Eternal Human Soul: Lecture IX: The Supersensible Human Being
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 become engrossed in geometry in such versatile mental pictures
 Title: Education as a Social Problem: Lecture VI: The Inexpressible Name, Spirits of Space and Time, Conquering Egotism
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 knowledge of mathematics and geometry. Everything is
 Title: Karma: Lecture I
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 on me when, in my study of modern synthetic geometry, I was led
 from the other. Geometry expresses it somewhat as follows: The
 Title: Egyptian Myths: Lecture 9
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 arose geometry, mathematics. With the help of this, man could rule the
 Title: Influence of Spiritual Beings Upon Man: Lecture VI
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 the laws of geometry. Matter is no longer an opposing power but is recast
 Title: Influence of Spiritual Beings Upon Man: Lecture VII
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 founded; it was sought through geometry to make the earth suited to
 Title: Universe, Earth and Man: Lecture XI: The progress of man. His conquest of the physical plane in the postAtlantean civilizations. The beginning and upbuilding of the 'I am.' The chosen people.
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 and directed to the earth; geometry was studied so that the earth
 Title: Astronomy Course: Lecture I
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 it; one ought really to bring Arithmetic or Geometry into all
 geometry, and see what countless ideas of every kind are
 extend the domain — widen out the concept of geometry.
 that can be understood today of nonEuclidean geometry. I
 in nonEuclidean geometry, then we should be in the realm of
 Title: Astronomy Course: Lecture IV
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 realm of Geometry. It is therefore an ideal of certain
 resembling Geometry.
 figures derived from Geometry. So, in Geometry too, we should
 analysis is not yet of much use, Geometry makes its presence
 Title: Astronomy Course: Lecture V
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 geometry take us up to a certain point, whence we can go no
 of the phenomena in algebra, analysis and geometry.
 geometry line, or that the Moon's movement can be so
 Title: Astronomy Course: Lecture VII
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 you will have to admit; when we do analytical geometry we
 of analytical geometry in this rigid way? Then too the whole
 Title: Astronomy Course: Lecture IX
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 Mathematics, where — as in Synthetic Geometry —
 Title: Astronomy Course: Lecture XI
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 mathematics and geometry as such, by thinking of the
 Title: Astronomy Course: Lecture XIV
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 “seen”.) All our geometry in this case, all of
 Title: Astronomy Course: Lecture XV
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 Geometry, that is to say, within a rigid threedimensional
 startingpoint at least, through pure Geometry — yet a
 Geometry of a higher kind. Thence we may gain an idea of
 Geometry that simply reckons with ordinary rigid space and
 Title: Astronomy Course: Lecture XVI
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 able thoroughly to illumina with a truth of pre Geometry we
 Analytical Geometry, with all its manifold results, in
 relation to Synthetic Geometry — to the real inner
 experience of Projective Geometry. True, this will only give
 Title: Young Doctors Course: Easter Course: Lecture I
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 reality, with geometry and arithmetic. But if we want to go
 Title: Young Doctors Course: Easter Course: Lecture V
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 physical world, because the geometry that comes to expression
 Title: Origin and Destination of Humanity: Lecture XI: Origin and Goal of the Human Being
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 of geometry and algebra. There are thoughts which do not flow towards
 Title: Origin and Destination of Humanity: Lecture XVI: The Great Initiates
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 of mathematics and geometry to which the human being dedicates himself
 Title: Apocalypse of John: Introductory Lecture
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 history versed in mathematics, namely, the geometry of
 been taught mathematics and geometry will value and
 geometry of Euclid. And then he tests them as the modern
 geometrician tests the geometry of Euclid; he can prize and
 ask ourselves who is better able to judge the geometry of
 geometry, or one who already understands geometry and is
 mere philologist, one who knows nothing of geometry, how many
 explanations of Euclid's geometry by a
 just as information with regard to the science of geometry
 Title: Apocalypse of John: Lecture III
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 discovery of Geometry, when man learnt to divide the earth
 Geometry. The laws within Maya were investigated. In the
 the stars and invent geometry. This was called “exploration
 Title: Esoteric Lesson: Berlin, 1903 or 1904
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 the laws of space without referring to Euclid's geometry book. But
 Title: Esoteric Lesson: Berlin, December 1904
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 the laws of space without referring to Euclid's geometry book. But
 Title: Social Basis For Primary and Secondary Education: Lecture I
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 age it is possible from the beginning of geometry — the straight
 Title: Search for the New Isis: Lecture I: A Christmas Lecture
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 it has become cold and gray mathematics and geometry. Today we see
 the abstract forms that are taught in schools as geometry and
 Title: Search for the New Isis: Lecture III: The Magi and the Shepherds: The New Isis
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 of colour, changed into the grey, colourless geometry and mechanics
 into mathematics, into geometry, into the science of mechanics. But
 from mathematics, phoronomy and geometry, this means the finding of
 Title: Man as a Being: Lecture 1
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 Now let us consider mathematical truths, the truths of geometry. A
 never say that geometry, for instance, has anything of a subjective
 Title: Cosmic Christianity: Lecture V
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 Steiner's own geometry teacher. Lord Byron. Voltaire. Swedenborg.
 had a Geometry teacher whom I loved dearly. It was not difficult for
 Geometry. But this teacher was really quite unusual. He had a
 peculiar talent for Geometry that fascinated me, although people who
 wonderful way in which this teacher of mine worked in Geometry, his
 This club foot enabled me to trace the individuality of my Geometry
 contemporaries (for my Geometry teacher lived later than Byron) had,
 again, as I look back upon my Geometry teacher with his modest,
 Title: History of the Middle Ages: Lecture IV: Arabic Influence in Europe
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 of physics, geometry and technical science. Everything here was
 Title: History of the Middle Ages: Lecture VI: Culture of the Middle Ages
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 Title: Impulse for Renewal: Lecture I: Anthroposophy and Natural Science
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 geometry which proves that the three angles of a triangle
 Title: Impulse for Renewal: Lecture II: The Human and the Animal Organisation
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 within, which corresponds to the geometry in outer life.
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