History Of Mathematics Quotes

“In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]”
― Alexey Stakhov, MATHEMATICS OF HARMONY: FROM EUCLID TO CONTEMPORARY MATHEMATICS AND COMPUTER SCIENCE

“The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large.”
― H.E. Huntley, The Divine Proportion: A Study in Mathematical Beauty

“The Great Pyramid was a fractal resonator for the entire Earth. It is designed according to the proportions of the cosmic temple, the natural pattern that blends the two fundamental principles of creation. The pyramid has golden ratio, pi, the base of natural logarithms, the precise length of the year and the dimensions of the Earth built into its geometry. It demonstrates.... As John Michell has pointed out in his wonderful little book, City of Revelation, 'Above all, the Great Pyramid is a monument to the art of 'squaring the circle''.”
― Alison Charlotte Primrose, The Lamb Slain With A Golden Cut: Spiritual Enlightenment and the Golden Mean Revelation

“The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked with the 'key' ... 528, opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.”
― Bonnie Gaunt, Beginnings: The Sacred Design

“We therefore find that the triangles and rectangles herein described, enclose a large majority of the temples and cathedrals of the Greek and Gothic masters, for we have seen that the rectangle of the Egyptian triangle is a perfect generative medium, its ratio of five in width to eight in length 'encouraging impressions of contrast between horizontal and vertical lines' or spaces; and the same practically may be said of the Pythagorean triangle”
― Samuel Colman, Harmonic Proportion and Form in Nature, Art and Architecture

“The Ark of the Covenant is a Golden Rectangle because its rectangular shape is in the proportions of the Golden Ratio.”
― Donald Frazer, Hieroglyphs and Arithmetic of the Ancient Egyptian Scribes: Version 1

“Central to all these interlinked themes was that curious irrational, phi, the Golden Section. Schwaller de Lubicz believed that if ancient Egypt possessed knowledge of ultimate causes, that knowledge would be written into their temples not in explicit texts but in harmony, proportion, myth and symbol.”
― John Anthony West, Serpent in the Sky: The High Wisdom of Ancient Egypt

“Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion), are known as irrational numbers. They lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind.

Today scientists recognize the Comma of Pythagoras, Pi and the Golden Proportion as well as the closely related Fibonacci sequence are universal constants that describe complex patterns in astronomy, music and physics. ...

To the Egyptians these numbers were also the secret harmonies of the cosmos and they incorporated them as rhythms and proportions in the construction of their pyramids and temples.”
― Jonathan Black, Mark Booth

“Since ancient times artists and architects have seen in the golden mean the most aesthetically satisfying geometric ratio.”
― Stephen M. Barr, Modern Physics and Ancient Faith

“Who amongst them realizes that between the Differential Calculus and the dynastic principle of politics in the age of Louis XIV, between the Classical city-state and the Euclidean geometry, between the space perspective of Western oil painting and the conquest of space by railroad, telephone and long range weapon, between contrapuntal music and credit economics, there are deep uniformities?”
― Oswald Spengler, The Decline of the West

“The golden ratio, as well as the Great Pyramid as an expression of it, is an important key to our universe containing the Earth and the Moon. ... The ratio between the Earth and the Moon is in fact the basis for the mathematical concept of 'squaring the circle' ...”
― Willem Witteveen, The Great Pyramid of Giza: A Modern View on Ancient Knowledge

“The golden ratio is a reminder of the relatedness of the created world to the perfection of its source and of its potential future evolution.”
― Robert Lawlor, Sacred Geometry: Philosophy and Practice

“Kepler followed Proclus and believed that 'the main goal of Euclid was to build a geometric theory of the so-called Platonic solids.' Kepler was fascinated by Proclus and often quotes him calling him a 'Pythagorean'. [History of Mathematics]”
― C. Smoryński

“The conclusion that the Egyptians of the Old Kingdom were acquainted with both the Fibonacci series and the Golden Section, says Stecchini, is so startling in relation to current assumptions about the level of Egyptian mathematics that it could hardly have been accepted on the basis of Herodotus' statement alone, or on the fact that the phi [golden] proportion happens to be incorporated in the Great Pyramid.

But the many measurements made by Professor Jean Philippe Lauer, says Stecchini, definitely prove the occurrence of the Golden Section throughout the architecture of the Old Kingdom.... Schwaller de Lubicz also found graphic evidence that the pharonic Egyptians had worked out a direct relation between pi and phi in that pi = phi^2 x 6/5.”
― Peter Tompkins, Secrets of the Great Pyramid: Two Thousand Years of Adventures & Discoveries Surrounding the Mysteries of the Great Pyramid of Cheops

“The impulse to all movement and all form is given by [the golden ratio], since it is the proportion that summarizes in itself the additive and the geometric, or logarithmic, series.”
― Schwaller de Lubicz

“Petrie found nothing that disproved the pyramidologist's assumption that the Great Pyramid had been built according to a master plan. Indeed, he describes the Pyramid's architecture as being filled with extraordinary mathematical harmonies and concordances: those same strange symmetries that had so haunted the pyramidologist.

Petrie not only noted, for example, that the proportions of the reconstructed pyramid approximated to pi - which others have since elaborated to include those twin delights of Renaissance and pyramidological mathematicians, the Golden Section and the Fibonacci Series ...”
― John Romer, The Great Pyramid: Ancient Egypt Revisited

“While twentieth-century physicists were not able to identify any convincing mathematical constants underlying the fine structure, partly because such thinking has normally not been encouraged, a revolutionary suggestion was recently made by the Czech physicist Raji Heyrovska, who deduced that the fine structure constant, ...really is defined by the [golden] ratio ....”
― Carl Johan Calleman, The Purposeful Universe: How Quantum Theory and Mayan Cosmology Explain the Origin and Evolution of Life

“In his ... 'Geometrical peculiarities of the Pyramids', Ballard shows the relationship between the equal area theory and the golden number. After checking Herodotus' statement via dimensions Ballard concludes: 'I have therefore the authority of Herodotus to support the theory which I shall subsequently set forth, that this pyramid was the exponent of lines divided in mean and extreme ratio.”
― Roger Herz-Fischler, The Shape of the Great Pyramid

“In short, the idea dawns that the one universal principle which possibly ... between force and structure, the embodiment of the Principle of Least Action and the (unknown) force, which in mathematics is known as the attractor which pulls ... in the direction of the most optimal and relatively stable self-organized criticality, could very well be the Golden Ratio dynamic. the universal principle which as the balance between finiteness and infinity, stability and flexibility underlies self-similar fractal forms emerging at the 'edge of chaos' indeed seems to be the Golden Ratio Spiral.”
― Marja de Vries, Whole Elephant Revealed: Insights into the Existence and Operation of Universal Laws and the Golden Ratio

“Thousands of years ago the ancients had an advanced mathematical understanding of universe that is revealed in many sources. There is a consistent link to knowledge of the golden mean, but the way in which the ancients were able to formulate and use this information speaks of a technical grasp of the subject that exceeds what we know about it in the present day.”
― Alison Charlotte Primrose, The Lamb Slain With A Golden Cut: Spiritual Enlightenment and the Golden Mean Revelation

“The Golden Ratio defines the squaring of a circle. Stated in mathematical terms, this says: Given a square of known perimeter, create a circle of equal circumference. According to some, in ancient Egypt, this mathematical mystery was encoded in the measurements of the Great Pyramid of Giza.”
― Marja de Vries, Whole Elephant Revealed: Insights into the Existence and Operation of Universal Laws and the Golden Ratio

“...men are capable of perceiving the Pyramid in an astonishing number of ways. Some have thought the Pyramid was an astronomic and astrological observatory. Some have thought it functioned as the equivalent of a theodolite for surveyors in ancient times... Some think it performed as a giant sundial... Some think it records the mathematics and science of a civilization which vanished... Some think it is a huge water pump. Others have thought it was filled with fabulous treasures... One early investigator came away convinced it was the remains of a huge volcano. Another thought the pyramids were Joseph's granaries. Some thought they were heathen idols which should be destroyed. Some believe the Pyramid captures powerful cosmic energies... Some think it is a tomb. Some think it is a Bible in stone with prophecies built into the scheme of its internal passages... Some think it was a mammoth public works project which consolidated the position of the pharaoh and the unity of the nation. Some think it was built by beings from outer space. Some say it was a temple of initiation. Some hold that it was an instrument of science. Some believe it is an altar of Guild built through direct Divine Revelation. And today, judging by the uses to which it has been put, some apparently think it is an outhouse.”
― William Fix, Pyramid Odyssey

“The Pythagoreans... were fascinated by certain specific ratios, ...The Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the Babylonians by Pythagoras himself after having been taken prisoner in Egypt. Ratios lay at the heart of the Pythagorean theory of music.”
― Graham Flegg, Numbers: Their History and Meaning

“The golden section was discovered by the Egyptians, and has been used in art and architecture, most commonly, during the classical ages of Egypt and Greece.”
― Steven L. Griffing, The Golden Section:An Ancient Egyptian and Grecian Proportion

“Schwaller de Lubicz identifies the Golden Mean as "the fundamental scission," or division of one into two, that creates three things - the original whole and two parts, one in golden proportion to the whole and the other in golden proportion to that.”
― Richard Heath , Matrix of Creation: Sacred Geometry in the Realm of the Planets

“The Golden Mean was considered a fundamental constant by the Egyptians and the fundamental division of the whole into two parts.”
― Richard Heath, Sacred Number and the Origins of Civilization: The Unfolding of History Through the Mystery of Number

“In a way, the Phi Triangle of the Golden Mean could be compared to the path of light sent forth from the great All-Seeing Eye of God in the beginning, and which paved the way for the creation of the Universe.”
― William Eisen, The English Cabalah Volume 2, The Mysteries of Phi

“The Great Pyramid, that monument to spirituality that the Agashan Teachers hold in such high esteem, is built according to the principles of Pi and Phi.”
― William Eisen, The English Cabalah Volume 2, The Mysteries of Phi

“[Euclid's] second postulate, for example, says that any line segment may be extended indefinitely. That is difficult for even the most querulous to argue with. The fifth, on the other hand, states that if two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side (labelled a and b in the diagram below) is less than two right angles (i.e. 180°), then the two lines inevitably must intersect each other on that side if extended far enough. If, on the other hand, a and b do add up to 180°, the two lines never meet so are said to be parallel.
To mathematicians, that looks less like a postulate and more like a theorem in need of proving.”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann

“Russell’s paradox threatened to deal a far more serious blow to set theory than earlier ideological objections. The problem was this: consider a set of objects – all possible types of cheesecake, say. This set may include any number of different cheesecakes (New York cheesecake, German Käsekuchen, lemon ricotta, etc.) but, because a set is not literally a cheesecake, the set of all cheesecakes is not a member of itself. The set of all things that are not cheesecakes, on the other hand, is a member of itself.
But what, Russell wondered, about the set of all sets that are not members of themselves. If this is not a member of itself, then, by definition, it should be (because its members do not include itself). Conversely, if it is a member of itself, then it should not be (because it does). This was Russell’s paradox in a nutshell. His analysis of the paradox revealed it to be similar in form to several others, including the liar’s paradox (‘this statement is a lie’). ‘It seemed unworthy of a grown man to spend time on such trivialities,’ he complained, desperate for a solution, ‘but what was I to do?”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann