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Hermeticism

Hermeticism or Hermetism is a philosophical and religious system based on the purported teachings of Hermes Trismegistus (a Hellenistic conflation of the Greek god Hermes and the Egyptian god Thoth). These teachings are contained in the various writings attributed to Hermes (the Hermetica), which were produced over a period spanning many centuries (c. 300 BCE – 1200 CE) and may be very different in content and scope.

One particular form of Hermetic teaching is the religio-philosophical system propounded by a specific subgroup of Hermetic writings known as the 'religio-philosophical' Hermetica, the most famous of which are the Corpus Hermeticum (a collection of seventeen Greek Hermetic treatises written between c. 100 and c. 300 CE) and the Asclepius (a treatise from the same period mainly surviving in a Latin translation). This specific, historical form of Hermetic philosophy is sometimes more restrictively called Hermetism, to distinguish it from the philosophies inspired by the many Hermetic writings of a completely different period and nature.

A more open-ended term is Hermeticism, which may refer to a wide variety of philosophical systems drawing on Hermetic writings, or even merely on subject matter generally associated with Hermes (most notably, alchemy often went by the name of "the Hermetic art" or "the Hermetic philosophy"). The most famous use of the term in this broader sense is in the concept of Renaissance Hermeticism, which refers to the wide array of early modern philosophies inspired by, on the one hand, Marsilio Ficino's (1433–1499) and Lodovico Lazzarelli's (1447–1500) translation of the Corpus Hermeticum, and on the other, by Paracelsus' (1494–1541) introduction of a new medical philosophy drawing upon the 'technical' Hermetica (i.e., astrological, alchemical, and magical Hermetica, such as the Emerald Tablet).

In 1964, Frances A. Yates advanced the thesis that Renaissance Hermeticism, or what she called "the Hermetic tradition", had been a crucial factor in the development of modern science. While Yates's thesis has since been largely rejected, the important role played by the 'Hermetic' science of alchemy in the thought of such figures as Jan Baptist van Helmont (1580–1644), Robert Boyle (1627–1691) or Isaac Newton (1642–1727) has been amply demonstrated.

Throughout its history, Hermeticism was closely associated with the idea of a primeval, divine wisdom, revealed only to the most ancient of sages, such as Hermes Trismegistus. In the Renaissance, this developed into the notion of a prisca theologia or "ancient theology", which asserted that there is a single, true theology which was given by God to some of the first humans, and traces of which may still be found in various ancient systems of thought. Thinkers like Giovanni Pico della Mirandola (1463–1494) supposed that this 'ancient theology' could be reconstructed by studying (what were then considered to be) the most ancient writings still in existence, such as those attributed to Hermes, but also those attributed to, such as Zoroaster, Orpheus, Pythagoras, Plato, the 'Chaldeans', or the Kabbalah. This soon evolved into the idea, first proposed by Agostino Steuco (1497–1548), that one and the same divine truth may be found in the religious and philosophical traditions of different periods and places, all considered as different manifestations of the same universal perennial philosophy. In this perennialist context, the term 'Hermetic' tended to lose even more of its specificity, eventually becoming a mere byword for the purported divine knowledge of the ancient Egyptians, especially as related to alchemy and magic. This generic and pseudo-historical use of the term was greatly popularized by nineteenth- and twentieth-century occultists, despite their occasional use of authentic Hermetic texts and concepts.

Etymology
The term Hermetic is from the medieval Latin hermeticus, which is derived from the name of the Greek god Hermes. In English, it has been attested since the 17th century, as in "Hermetic writers" such as Robert Fludd.

The word Hermetic was used by John Everard in his English translation of The Pymander of Hermes, published in 1650.

Mary Anne Atwood mentioned the use of the word Hermetic by Dufresnoy in 1386.

The synonymous term Hermetical is also attested in the 17th century. Sir Thomas Browne in his Religio Medici of 1643 wrote: "Now besides these particular and divided Spirits, there may be (for ought I know) a universal and common Spirit to the whole world. It was the opinion of Plato, and is yet of the Hermeticall Philosophers." (R. M. Part 1:2)

Hermes Trismegistus supposedly invented the process of making a glass tube airtight (a process in alchemy) using a secret seal. Hence, the term "completely sealed" is implied in "hermetically sealed" and the term "hermetic" is also equivalent to "occult" or hidden.

Late Antiquity
In Late Antiquity, Hermetism emerged in parallel with early Christianity, Gnosticism, Neoplatonism, the Chaldaean Oracles, and late Orphic and Pythagorean literature. These doctrines were "characterized by a resistance to the dominance of either pure rationality or doctrinal faith."

Plutarch's mention of Hermes Trismegistus dates back to the first century CE, and Tertullian, Iamblichus, and Porphyry were all familiar with Hermetic writings.

The texts now known as the Corpus Hermeticum are dated by modern translators and most scholars to the beginning of the second century or earlier. These texts dwell upon the oneness and goodness of God, urge purification of the soul, and expand on the relationship between mind and spirit. Their predominant literary form is the dialogue: Hermes Trismegistus instructs a perplexed disciple upon various teachings of the hidden wisdom.

In fifth-century Macedonia, Joannes Stobaeus or "John of Stobi" compiled a huge Anthology of Greek poetical, rhetorical, historical, and philosophical literature. Among the excerpts of ancient philosophical literature preserved by Stobaeus are also a significant number of discourses and dialogues attributed to Hermes.

Renaissance
After centuries of falling out of favor, Hermeticism was reintroduced to the West when, in 1460, a man named Leonardo di Pistoia brought the Corpus Hermeticum to Pistoia. He was one of many agents sent out by Pistoia's ruler, Cosimo de' Medici, to scour European monasteries for lost ancient writings.

In 1614, Isaac Casaubon, a Swiss philologist, analyzed the Greek Hermetic texts for linguistic style. He concluded that the writings attributed to Hermes Trismegistus were not the work of an ancient Egyptian priest but in fact dated to the second and third centuries CE.

Even in light of Casaubon's linguistic discovery (and typical of many adherents of Hermetic philosophy in Europe during the 16th and 17th centuries), Thomas Browne in his Religio Medici (1643) confidently stated: "The severe schools shall never laugh me out of the philosophy of Hermes, that this visible world is but a portrait of the invisible." (R. M. Part 1:12)

In 1678, flaws in Casaubon's dating were discerned by Ralph Cudworth, who argued that Casaubon's allegation of forgery could only be applied to three of the seventeen treatises contained within the Corpus Hermeticum. Moreover, Cudworth noted Casaubon's failure to acknowledge the codification of these treatises as a late formulation of a pre-existing oral tradition. According to Cudworth, the texts must be viewed as a terminus ad quem and not a terminus a quo. Lost Greek texts, and many of the surviving vulgate books, contained discussions of alchemy clothed in philosophical metaphor.

In 1924, Walter Scott placed the date of the Hermetic texts shortly after 200 CE, but W. Flinders Petrie placed their origin between 200 and 500 BCE.

Modern era
In 1945, Hermetic texts were found near the Egyptian town Nag Hammadi. One of these texts had the form of a conversation between Hermes and Asclepius. A second text (titled On the Ogdoad and Ennead) told of the Hermetic mystery schools. It was written in the Coptic language, the latest and final form in which the Egyptian language was written.

According to Geza Vermes, Hermeticism was a Hellenistic mysticism contemporaneous with the Fourth Gospel, and Hermes Tresmegistos was "the Hellenized reincarnation of the Egyptian deity Thoth, the source of wisdom, who was believed to deify man through knowledge (gnosis)."

Gilles Quispel says "It is now completely certain that there existed before and after the beginning of the Christian era in Alexandria a secret society, akin to a Masonic lodge. The members of this group called themselves 'brethren,' were initiated through a baptism of the Spirit, greeted each other with a sacred kiss, celebrated a sacred meal and read the Hermetic writings as edifying treatises for their spiritual progress." On the other hand, Christian Bull argues that "there is no reason to identify [Alexandria] as the birthplace of a Hermetic lodge as several scholars have done. There is neither internal nor external evidence for such an Alexandrian lodge, a designation that is alien to the ancient world and carries Masonic connotations."

God as 'the All'
In the religio-philosophical Hermetica, the ultimate reality is called by many names, such as God, Lord, Father, Mind (Nous), the Creator, the All, the One, etc. In the Hermetic view, God is both the all (Greek: to pan) and the creator of the all: all created things pre-exist in God and God is the nature of the cosmos (being both the substance from which it proceeds and the governing principle which orders it), yet the things themselves and the cosmos were all created by God. Thus, God ('the All') creates itself, and is both transcendent (as the creator of the cosmos) and immanent (as the created cosmos). These ideas are closely related to the cosmo-theological views of the Stoics.

Prisca theologia
Hermeticists believe in a prisca theologia, the doctrine that a single, true theology exists, that it exists in all religions, and that it was given by God to man in antiquity. To demonstrate the truth of the prisca theologia doctrine, Christians appropriated the Hermetic teachings for their own purposes. By this account, Hermes Trismegistus was (according to the fathers of the Christian church) either a contemporary of Moses or the third in a line of men named Hermes—Enoch, Noah, and the Egyptian priest-king who is known to us as Hermes Trismegistus.

"As above, so below"
"As above, so below" is a popular modern paraphrase of the second verse of the Emerald Tablet (a compact and cryptic text attributed to Hermes Trismegistus and first attested in a late eight or early ninth century Arabic source), as it appears in its most widely divulged medieval Latin translation:

Quod est superius est sicut quod inferius, et quod inferius est sicut quod est superius.

That which is above is like to that which is below, and that which is below is like to that which is above.

Three parts of the wisdom of the whole universe
"The three parts of the wisdom of the whole universe" is a phrase derived from the Emerald Tablet referring to three disciplines purportedly known to and taught by Hermes Trismegistus.

Alchemy
Alchemy, or the operation of the Sun, is not merely the changing of lead into gold, which is called chrysopoeia. It is an investigation into the spiritual constitution, or life, of matter and material existence through an application of the mysteries of birth, death, and resurrection. The various stages of chemical distillation and fermentation, among other processes, are aspects of these mysteries that, when applied, quicken nature's processes to bring a natural body to perfection. This perfection is the accomplishment of the Great Work (magnum opus).

Astrology
In Hermetic thought, the movements of the planets are believed to have meaning beyond the laws of physics and actually hold metaphorical value as symbols in the mind of the All, or God, which have influence upon the Earth, but do not dictate our actions; wisdom is gained when we know what these influences are and how to deal with them, and this wisdom is astrology, or the operation of the stars. The discovery of astrology is attributed to Zoroaster, who is said to have discovered this part of the wisdom of the whole universe and taught it to man.

Theurgy
Theurgy, or the operation of the gods, is one of the two different types of magic, which are – according to Giovanni Pico della Mirandola's Apology – completely opposite to each other. The first is Goëtia (γοητεια), black magic reliant upon an alliance with evil spirits such as demons. The second is Theurgy, divine magic reliant upon an alliance with divine spirits such as angels, archangels, and gods.

"Theurgy" translates to the "science or art of divine works" and is the practical aspect of the Hermetic art of alchemy. Furthermore, alchemy is seen as the "key" to theurgy, the ultimate goal of which is to become united with higher counterparts, leading to the attainment of divine consciousness.

Reincarnation
Reincarnation is mentioned in Hermetic texts. Hermes Trismegistus asked: "O son, how many bodies have we to pass through, how many bands of demons, through how many series of repetitions and cycles of the stars, before we hasten to the One alone?"

Rebirth
Rebirth appears central to the practice of hermetic philosophy. The process would begin with a candidate separating themselves from the world before they rid themselves of material vices; they are then reborn as someone completely different from who they were before.

Good and evil
Hermes explains in Book 9 of the Corpus Hermeticum that nous (reason and knowledge) brings forth either good or evil, depending upon whether one receives one's perceptions from God or from demons. God brings forth good, but demons bring forth evil. Among the evils brought forth by demons are: "adultery, murder, violence to one's father, sacrilege, ungodliness, strangling, suicide from a cliff and all such other demonic actions".

This provides evidence that Hermeticism includes a sense of morality. The word "good" is used very strictly. It is restricted to references to God. It is only God (in the sense of the nous, not in the sense of the All) who is completely free of evil. Men are prevented from being good because man, having a body, is consumed by his physical nature, and is ignorant of the Supreme Good. Asclepius explains that evil is born from desire which itself is caused by ignorance, the intelligence bestowed by God is what allows some to rid themselves of desire.

A focus upon the material life is said to be the only thing that offends God: "As processions passing in the road cannot achieve anything themselves yet still obstruct others, so these men merely process through the universe, led by the pleasures of the body."

One must create, one must do something positive in one's life, because God is a generative power. Not creating anything leaves a person "sterile" (i.e., unable to accomplish anything).

Cosmogony
A creation story is told by God to Hermes in the first book of the Corpus Hermeticum. It begins when God, by an act of will, creates the primary matter that is to constitute the cosmos. From primary matter God separates the four elements (earth, air, fire, and water). Then God orders the elements into the seven heavens (often held to be the spheres of Mercury, Venus, Mars, Jupiter, Saturn, the Sun, and the Moon, which travel in circles and govern destiny).

"The Word (Logos)" then leaps forth from the materializing four elements, which were unintelligent. Nous then makes the seven heavens spin, and from them spring forth creatures without speech. Earth is then separated from water, and animals (other than man) are brought forth.

The God then created androgynous man, in God's own image, and handed over his creation.

Fall of man
Man carefully observed the creation of nous and received from God man's authority over all creation. Man then rose up above the spheres' paths to better view creation. He then showed the form of the All to Nature. Nature fell in love with the All, and man, seeing his reflection in water, fell in love with Nature and wished to dwell in it. Immediately, man became one with Nature and became a slave to its limitations, such as sex and sleep. In this way, man became speechless (having lost "the Word") and he became "double", being mortal in body yet immortal in spirit, and having authority over all creation yet subject to destiny.

Alternative account of the fall of man
An alternative account of the fall of man, preserved in Isis the Prophetess to Her Son Horus, is as follows: "God, having created the universe, then created the divisions, the worlds, and various gods and goddesses, whom he appointed to certain parts of the universe. He then took a mysterious transparent substance, out of which he created human souls. He appointed the souls to the astral region, which is just above the physical region.

He then assigned the souls to create life on Earth. He handed over some of his creative substance to the souls and commanded them to contribute to his creation. The souls then used the substance to create the various animals and forms of physical life. Soon after, the souls began to overstep their boundaries; they succumbed to pride and desired to be equal to the highest gods.

God was displeased and called upon Hermes to create physical bodies that would imprison the souls as a punishment for them. Hermes created human bodies on earth, and God then told the souls of their punishment. God decreed that suffering would await them in the physical world, but he promised them that, if their actions on Earth were worthy of their divine origin, their condition would improve and they would eventually return to the heavenly world. If it did not improve, he would condemn them to repeated reincarnation upon Earth."

Religious and philosophical texts
Some of the most well-known Hermetic texts are:
 * The Corpus Hermeticum is the most widely known Hermetic text. It has 17 chapters, which contain dialogues between Hermes Trismegistus and a series of other men. The first chapter contains a dialogue between Poimandres and Hermes. Poimandres teaches the secrets of the universe to Hermes. In later chapters, Hermes teaches others, such as his son Tat and Asclepius. It was first translated into Latin by Marsilio Ficino (1433–1499), whose translation set off the Hermetic revival in the Renaissance.
 * The Emerald Tablet is a short work attributed to Hermes Trismegistus which was highly regarded by Islamic and European alchemists as the foundation of their art. The text of the Emerald Tablet first appears in a number of early medieval Arabic sources, the oldest of which dates to the late eighth or early ninth century. It was translated into Latin several times in the twelfth and thirteenth centuries. Among Neo-Hermeticists, "As above, so below" (a popular modern paraphrase of the second verse of the Tablet) has become an often cited catchphrase.
 * The Asclepius (also known as The Perfect Sermon, The Perfect Discourse, or The Perfect Teaching) was written in the second or third century and is a Hermetic work similar in content to the Corpus Hermeticum. It was one of the very few Hermetic works which were available to medieval Latin readers.

Other important original Hermetic texts include Isis the Prophetess to Her Son Horus, which consists of a long dialogue between Isis and Horus on the fall of man and other matters; the Definitions of Hermes Trismegistus to Asclepius; and many fragments, which are chiefly preserved in the anthology of Stobaeus.

There are additional works that, though not as historically significant as the works listed above, have an important place in Neo-Hermeticism:
 * A Suggestive Inquiry into Hermetic Philosophy and Alchemy was written by Mary Anne Atwood and originally published anonymously in 1850. This book was withdrawn from circulation by Atwood but was later reprinted, after her death, by her longtime friend Isabelle de Steiger. Isabelle de Steiger was a member of the Golden Dawn. A Suggestive Inquiry was used for the study of Hermeticism and resulted in several works being published by members of the Golden Dawn:
 * Arthur Edward Waite, a member and later the head of the Golden Dawn, wrote The Hermetic Museum and The Hermetic Museum Restored and Enlarged. He edited The Hermetic and Alchemical Writings of Paracelsus, which was published as a two-volume set. He considered himself to be a Hermeticist and was instrumental in adding the word "Hermetic" to the official title of the Golden Dawn.
 * William Wynn Westcott, a founding member of the Golden Dawn, edited a series of books on Hermeticism titled Collectanea Hermetica. The series was published by the Theosophical Publishing Society.
 * Initiation into Hermetics is the title of the English translation of the first volume of Franz Bardon's three-volume work dealing with self-realization within the Hermetic tradition.
 * The Kybalion is a book anonymously published in 1908 by three people who called themselves the "Three Initiates", and claims to expound upon essential Hermetic principles.

History of scholarship on the Hermetica
After the Renaissance and even within the 20th century, scholars did not study Hermeticism nearly as much as other topics; however, the 1990s saw a renewed interest in Hermetic scholarly works and discussion.

Societies
When Hermeticism was no longer endorsed by the Christian church, it was driven underground, and several Hermetic societies were formed. The western esoteric tradition is now steeped in Hermeticism. The work of such writers as Giovanni Pico della Mirandola, who attempted to reconcile Jewish kabbalah and Christian mysticism, brought Hermeticism into a context more easily understood by Europeans during the time of the Renaissance.

A few primarily Hermetic occult orders were founded in the late Middle Ages and early Renaissance. In England, it grew interwoven with the Lollard-Familist traditions.

Hermetic magic underwent a 19th-century revival in Western Europe, where it was practiced by groups such as the Hermetic Order of the Golden Dawn and Ordo Aurum Solis. It was also practiced by individual persons, such as Eliphas Lévi, William Butler Yeats, Arthur Machen, Frederick Hockley, and Kenneth M. Mackenzie.

Many Hermetic, or Hermetically influenced, groups exist today. Most of them are derived from Rosicrucianism, Freemasonry, or the Golden Dawn.

Rosicrucianism
Rosicrucianism is a movement which incorporates the Hermetic philosophy. It dates back to the 17th century. The sources dating the existence of the Rosicrucians to the 17th century are three German pamphlets: the Fama, the Confessio Fraternitatis, and The Chymical Wedding of Christian Rosenkreutz. Some scholars believe these to be hoaxes of the time and say that later Rosicrucian organizations are the first actual appearance of a Rosicrucian society.

The Rosicrucian Order consists of a secret inner body and a public outer body that is under the direction of the inner body. It has a graded system in which members move up in rank and gain access to more knowledge. There is no fee for advancement. Once a member has been deemed able to understand the teaching, he moves on to the next higher grade.

The Fama Fraternitatis states that the Brothers of the Fraternity are to profess no other thing than "to cure the sick, and that gratis".

The Rosicrucian spiritual path incorporates philosophy, kabbalah, and divine magic.

The Order is symbolized by the rose (the soul) and the cross (the body). The unfolding rose represents the human soul acquiring greater consciousness while living in a body on the material plane.

Hermetic Order of the Golden Dawn
Unlike the Societas Rosicruciana in Anglia, the Hermetic Order of the Golden Dawn was open to both sexes and treated them as equals. The Order was a specifically Hermetic society that taught alchemy, kabbalah, and the magic of Hermes, along with the principles of occult science.

The Golden Dawn maintained the tightest of secrecy, which was enforced by severe penalties for those who disclosed its secrets. Overall, the general public was left oblivious of the actions, and even of the existence, of the Order, so few if any secrets were disclosed.

Its secrecy was broken first by Aleister Crowley in 1905 and later by Israel Regardie in 1937. Regardie gave a detailed account of the Order's teachings to the general public.

title: Hermeticism


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Cubic equation

In algebra, a cubic equation in one variable is an equation of the form $$ax^3+bx^2+cx+d=0$$ in which $a$ is not zero.

The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients $a$, $b$, $c$, and $d$ of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:

 * algebraically: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
 * trigonometrically
 * numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.

The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.

History
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians. Babylonian (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task which is now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century.

In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations (Diophantine equations). Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like T. L. Heath, who translated all of Archimedes's works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two conics, but also discussed the conditions where the roots are 0, 1 or 2.

In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form $y = 0$, 23 of them with $x^{3} + px^{2} + qx = N$, and two of them with $p, q ≠ 0$.

In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote: "“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”"

In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: $q = 0$. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Muʿādalāt (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.

In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation $x^{3} + 200x = 20x^{2} + 2000$. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/602 + 42/603 + 33/604 + 4/605 + 40/606), which has a relative error of about 10−9.

In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form $x^{3} + 12x = 6x^{2} + 35$. In fact, all cubic equations can be reduced to this form if one allows $m$ and $n$ to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it. In 1535, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fior, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form $x^{3} + 2x^{2} + 10x = 20$, for which he had worked out a general method. Fior received questions in the form $x^{3} + mx = n$, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Later, Tartaglia was persuaded by Gerolamo Cardano (1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution).

Cardano's promise to Tartaglia said that he would not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano from Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income.

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers.

François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète.

Factorization
If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators. Such an equation $$ax^3+bx^2+cx+d=0,$$ with integer coefficients, is said to be reducible if the polynomial on the left-hand side is the product of polynomials of lower degrees. By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients.

Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form $$qx-p$$ with $q$ and $p$ being coprime integers. The rational root test allows finding $q$ and $p$ by examining a finite number of cases (because $q$ must be a divisor of $a$, and $p$ must be a divisor of $d$).

Thus, one root is $$\textstyle x_1= \frac pq,$$ and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is $$\frac aq x^2+ \frac{bq+ap}{q^2}x+\frac{cq^2+bpq+ap^2}{q^3}$$ (The coefficients seem not to be integers, but must be integers if $x^{3} + mx = n$ is a root.)

Then, the other roots are the roots of this quadratic polynomial and can be found by using the quadratic formula.

Depressed cubic
Cubics of the form $$t^3+pt+q$$ are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple change of variable to that of a depressed cubic.

Let $$ax^3 + bx^2 + cx + d = 0$$ be a cubic equation. The change of variable $$x=t-\frac b{3a}$$ gives a cubic (in $t$) that has no term in $x^{3} + mx^{2} = n$.

After dividing by $a$ one gets the depressed cubic equation $$t^3+pt+q=0,$$ with $$\begin{align} t={}&x+\frac b{3a}\\ p={}&\frac{3ac-b^2}{3a^2} \\ q={}&\frac{2b^3-9abc+27a^2d}{27a^3}. \end{align} $$

The roots $$x_1, x_2, x_3$$ of the original equation are related to the roots $$t_1,t_2,t_3$$ of the depressed equation by the relations $$x_i = t_i-\frac b{3a},$$ for $$i = 1, 2, 3$$.

Discriminant and nature of the roots
The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant.

Discriminant
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.

If $p / q$ are the three roots (not necessarily distinct nor real) of the cubic $$ax^3+bx^2+cx+d,$$ then the discriminant is $$a^4(r_1-r_2)^2(r_1-r_3)^2(r_2-r_3)^2.$$

The discriminant of the depressed cubic $$t^3+pt+q$$ is $$-\left(4\,p^3+27\,q^2\right).$$

The discriminant of the general cubic $$ax^3+bx^2+cx+d$$ is $$18\,abcd - 4\,b^3d + b^2c^2 - 4\,ac^3 - 27\,a^2d^2.$$ It is the product of $$a^4$$ and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as $$\frac{4(b^2-3ac)^3-(2b^3-9abc +27 a^2d)^2}{27a^2}.$$

It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real, the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants.

To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in $t^{2}$, and $a$. The proof then results in the verification of the equality of two polynomials.

Nature of the roots
If the coefficients of a polynomial are real numbers, and its discriminant $$\Delta$$ is not zero, there are two cases:
 * If $$\Delta>0,$$ the cubic has three distinct real roots
 * If $$\Delta<0,$$ the cubic has one real root and two non-real complex conjugate roots.

This can be proved as follows. First, if $r$ is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.

As stated above, if $r1, r2, r3$ are the three roots of the cubic $$ax^3+bx^2+cx+d$$, then the discriminant is $$\Delta=a^4(r_1-r_2)^2(r_1-r_3)^2(r_2-r_3)^2$$

If the three roots are real and distinct, the discriminant is a product of positive reals, that is $$\Delta>0.$$

If only one root, say $r1, r2, r3$, is real, then $r1, r2, r3$ and $r1$ are complex conjugates, which implies that $r2$ is a purely imaginary number, and thus that $r3$ is real and negative. On the other hand, $r2 – r3$ and $(r2 – r3)2$ are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is $$\Delta<0.$$

Multiple root
If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real.

The discriminant of the depressed cubic $$t^3 + p t + q$$ is zero if $$4p^3 + 27q^2 = 0.$$ If $p$ is also zero, then $r1 – r2$, and 0 is a triple root of the cubic. If $$4p^3 + 27q^2 = 0,$$ and $r1 – r3$, then the cubic has a simple root $$t_1 = \frac{3q}{p} $$

and a double root $$t_2 = t_3 = -\frac{3q}{2p}.$$

In other words, $$t^3 + p t + q = \left(t - \frac{3q}{p}\right)\left(t + \frac{3q}{2p}\right)^2.$$

This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas.

By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic $$ax^3 + b x^2 + c x + d$$ is zero, then

 * either, if $$b^2 = 3ac,$$ the cubic has a triple root $$x_1 = x_2 = x_3 = -\frac{b}{3a},$$ and $$ax^3 + bx^2 + cx + d = a\left(x + \frac{b}{3a}\right)^3$$
 * or, if $$b^2 \ne 3ac,$$ the cubic has a double root $$x_2 = x_3 = \frac{9ad - bc}{2(b^2 - 3ac)},$$ and a simple root, $$x_1 = \frac{4abc - 9a^2d - b^3}{a(b^2 - 3ac)}.$$ and thus $$a x^3 + b x^2 + c x + d = a(x - x_1)(x - x_2)^2.$$

Characteristic 2 and 3
The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.

The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its formal derivative. In these characteristics, if the derivative is not a constant, it is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas.

A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root.

Cardano's formula
Gerolamo Cardano is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro and Niccolo Fontana Tartaglia. The formula applies to depressed cubics, but, as shown in, it allows solving all cubic equations.

Cardano's result is that if $$t^3+pt+q=0$$ is a cubic equation such that $p$ and $q$ are real numbers such that $$ \frac{q^2}4+\frac{p^3}{27}$$ is positive (this implies that the discriminant of the equation is negative) then the equation has the real root $$\sqrt[3]{u_1}+\sqrt[3]{u_2},$$ where $$u_1$$ and $$u_2$$ are the two numbers $$-\frac q2 + \sqrt{\frac{q^2}4+\frac{p^3}{27}}$$ and $$-\frac q2 - \sqrt{\frac{q^2}4+\frac{p^3}{27}}.$$

See, below, for several methods for getting this result.

As shown in, the two other roots are non-real complex conjugate numbers, in this case. It was later shown (Cardano did not know complex numbers) that the two other roots are obtained by multiplying one of the cube roots by the primitive cube root of unity $$\varepsilon_1=\frac{-1+i\sqrt 3} 2,$$ and the other cube root by the other primitive cube root of the unity $$\varepsilon_2=\varepsilon_1^2=\frac{-1-i\sqrt 3} 2.$$ That is, the other roots of the equation are $$ \varepsilon_1\sqrt[3]{u_1}+\varepsilon_2 \sqrt[3]{u_2}$$ and $$\varepsilon_2\sqrt[3]{u_1}+\varepsilon_1 \sqrt[3]{u_2}.$$

If $$4p^3+27q^2 < 0,$$ there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called casus irreducibilis, meaning irreducible case in Latin.

In casus irreducibilis, Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols $$\sqrt{{~}^{~}}$$ and $$\sqrt[3]{{~}^{~}}$$ as representing the principal values of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general.

The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be $p = q = 0$. It results that a root of the equation is $$C-\frac p{3C}\quad\text{with}\quad C=\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}4+\frac{p^3}{27}}}. $$ In this formula, the symbols $$\sqrt{{~}^{~}}$$ and $$\sqrt[3]{{~}^{~}}$$ denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is $$\textstyle \frac{-1\pm \sqrt{-3}}2.$$

This formula for the roots is always correct except when $p ≠ 0$, with the proviso that if $b = 0$, the square root is chosen so that $d = 0$. However, Cardano's formula is useless if $$p=0,$$ as the roots are the cube roots of $$-q.$$ Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not irreducible; this includes the case $$4p^3+27q^2=0.$$

This formula is also correct when $p$ and $q$ belong to any field of characteristic other than 2 or 3.

General cubic formula
A cubic formula for the roots of the general cubic equation (with $b/a$) $$ax^3+bx^2+cx+d=0$$ can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for real coefficients, but also for coefficients $1 = –1$ belonging to any field of characteristic other than 2 or 3. If the coefficients are real numbers, the formula covers all complex solutions, not just real ones.

The formula being rather complicated, it is worth splitting it in smaller formulas.

Let $$\begin{align} \Delta_0 &= b^2 - 3ac,\\ \Delta_1 &= 2b^3 - 9abc + 27a^2d. \end{align}$$

(Both $$\Delta_0$$ and $$\Delta_1$$ can be expressed as resultants of the cubic and its derivatives: $$\Delta_1$$ is $b ≠ 0$ times the resultant of the cubic and its second derivative, and $$\Delta_0$$ is $b ≠ 0$ times the resultant of the first and second derivatives of the cubic polynomial.)

Then let $$C = \sqrt[3]{\frac{\Delta_1 \pm \sqrt{\Delta_1^2 - 4 \Delta_0^3}}2},$$ where the symbols $$\sqrt{{~}^{~}}$$ and $$\sqrt[3]{{~}^{~}}$$ are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "$–p / 3$" before the square root is either "$p = q = 0$" or "$p = 0$"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields $C ≠ 0$ (this occurs if $$\Delta_0=0$$), then the other sign must be selected instead. If both choices yield $a ≠ 0$, that is, if $$\Delta_0=\Delta_1=0,$$ a fraction $0⁄0$ occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section). With these conventions, one of the roots is $$x = - \frac{1}{3a}\left(b+C+\frac{\Delta_0}{C}\right)\text{.}$$

The other two roots can be obtained by changing the choice of the cube root in the definition of $C$, or, equivalently by multiplying $C$ by a primitive cube root of unity, that is $a, b, c, d$. In other words, the three roots are $$x_k = - \frac{1}{3a}\left(b+\xi^kC+\frac{\Delta_0}{\xi^kC}\right), \qquad k \in \{0,1,2\} \text{,}$$ where $−1⁄8a$.

As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if $$\Delta_0=\Delta_1=0,$$ the formula gives that the three roots equal $$\frac {-b}{3a},$$ which means that the cubic polynomial can be factored as $$\textstyle a(x+\frac b{3a})^3.$$ A straightforward computation allows verifying that the existence of this factorization is equivalent with $$\Delta_0=\Delta_1=0.$$

Trigonometric solution for three real roots
When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. Galois theory allows proving that when the three roots are real, and none is rational (casus irreducibilis), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines. More precisely, the roots of the depressed cubic $$t^3 + pt + q = 0$$ are $$t_k = 2\,\sqrt{-\frac{p}{3}}\,\cos\left[\,\frac{1}{3} \arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\,\right) - \frac{2\pi k}{3}\,\right] \qquad \text{for } k=0,1,2.$$

This formula is due to François Viète. It is purely real when the equation has three real roots (that is $$4p^3 + 27q^2 < 0$$). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when $−1⁄12a$.

This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in.

The formula can be proved as follows: Starting from the equation $±$, let us set $+$. The idea is to choose $u$ to make the equation coincide with the identity $$4\cos^3\theta - 3\cos\theta - \cos(3\theta) = 0.$$ For this, choose $$u = 2\,\sqrt{-\frac{p}{3}}\,,$$ and divide the equation by $$\frac{u^3}{4}.$$ This gives $$4\cos^3\theta - 3\cos\theta - \frac{3q}{2p}\,\sqrt{ \frac{-3}{p} } = 0.$$ Combining with the above identity, one gets $$\cos(3\theta) = \frac{3q}{2p}\sqrt{\frac{-3}{p}}\,,$$ and the roots are thus $$t_k = 2\,\sqrt{-\frac{p}{3}}\,\cos\left[ \frac{1}{3} \arccos\left( \frac{3q}{2p}\sqrt{\frac{-3}{p}} \right) - \frac{2\pi k}{3} \right] \qquad \text{for } k=0,1,2.$$

Hyperbolic solution for one real root
When there is only one real root (and $–$), this root can be similarly represented using hyperbolic functions, as $$\begin{align} t_0 &= -2\frac{|q|}{q}\sqrt{-\frac{p}{3}}\cosh\left[\frac{1}{3}\operatorname{arcosh}\left(\frac{-3|q|}{2p}\sqrt{\frac{-3}{p}}\right)\right] \qquad \text{if } ~ 4 p^3 + 27 q^2 > 0 ~\text{ and }~ p < 0,\\ t_0 & = -2\sqrt{\frac{p}{3}}\sinh\left[\frac{1}{3}\operatorname{arsinh}\left(\frac{3q}{2p}\sqrt{\frac{3}{p}}\right)\right] \qquad \text{if } ~ p > 0.\end{align}$$ If $C = 0$ and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.

When $C = 0$, the above values of $–1 ± √–3⁄2$ are sometimes called the Chebyshev cube root. More precisely, the values involving cosines and hyperbolic cosines define, when $ξ = –1 + √–3⁄2$, the same analytic function denoted $p = 0$, which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted $t^{3} + pt + q = 0$, when $t = u cos θ$.

Omar Khayyám's solution
For solving the cubic equation $p ≠ 0$ where $p ≠ 0$, Omar Khayyám constructed the parabola $p = ±3$, the circle that has as a diameter the line segment $t_{0}$ on the positive $x$-axis, and a vertical line through the point where the circle and the parabola intersect above the $x$-axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the $x$-axis (see the figure).

A simple modern proof is as follows. Multiplying the equation by $p = −3$ and regrouping the terms gives $$\frac{x^4}{m^2}= x\left(\frac{n}{m^2}-x\right).$$ The left-hand side is the value of $C_{1/3}(q)$ on the parabola. The equation of the circle being $S_{1/3}(q)$, the right hand side is the value of $p = 3$ on the circle.

Solution with angle trisector
A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots.

A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.

Three real roots

Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form ($x$), $m = 2$, as shown above, the solution can be expressed as

$$t_k=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right) \quad \text{for} \quad k=0,1,2 \,.$$

Here $$\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)$$ is an angle in the unit circle; taking $n = 16$ of that angle corresponds to taking a cube root of a complex number; adding $2$ for $x^{3} + m^{2}x = n$ finds the other cube roots; and multiplying the cosines of these resulting angles by $$2\sqrt{-\frac{p}{3}}$$ corrects for scale.

For the non-depressed case ($$) (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining $x$ such that $n > 0$ so $y = x^{2}/m$. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables $A$ and $B$, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the $C$-axis. Consequently, the roots of the equation in $x$ sum to zero.

In the Cartesian plane

When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa ($$-coordinate) of the horizontal intercept of the curve (point R on the figure). Further,  if the complex conjugate roots are written as $[0, n/m^{2}]$, then the real part $$ is the abscissa of the tangency point H of the tangent line to cubic that passes through $t$-intercept R of the cubic (that is the signed length OM, negative on the figure). The imaginary parts $t$ are the square roots of the tangent of the angle between this tangent line and the horizontal axis.

In the complex plane
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.

The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than $x/m^{2}$ then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than $y^{2}$, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is $y^{2} + x(x − n⁄m^{2}) = 0$, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.

Galois group
Given a cubic irreducible polynomial over a field $x$ of characteristic different from 2 and 3, the Galois group over $y$ is the group of the field automorphisms that fix $t$ of the smallest extension of $h$ (splitting field). As these automorphisms must permute the roots of the polynomials, this group is either the group $y^{2}$ of all six permutations of the three roots, or the group $t^{3} + pt + q = 0$ of the three circular permutations.

The discriminant $1⁄3$ of the cubic is the square of $$\sqrt \Delta =a^2(r_1-r_2)(r_1-r_3)(r_2-r_3),$$ where $x$ is the leading coefficient of the cubic, and $−k2\pi⁄3$, $k = 1, 2$ and $x = t − b⁄3a$ are the three roots of the cubic. As $$\sqrt \Delta$$ changes of sign if two roots are exchanged, $$\sqrt \Delta$$ is fixed by the Galois group only if the Galois group is $t = x + b⁄3a$. In other words, the Galois group is $g ± hi$ if and only if the discriminant is the square of an element of $g$.

As most integers are not squares, when working over the field $g = \overline{OM}$ of the rational numbers, the Galois group of most irreducible cubic polynomials is the group $√tan ORH$ with six elements. An example of a Galois group $√slope of line RH$ with three elements is given by $\overline{BE}$, whose discriminant is $\overline{DA}$.

Derivation of the roots
This section regroups several methods for deriving Cardano's formula.

Cardano's method
This method is due to Scipione del Ferro and Tartaglia, but is named after Gerolamo Cardano who first published it in his book Ars Magna (1545).

This method applies to a depressed cubic $g ± hi$. The idea is to introduce two variables $x$ and $$v$$ such that $$u+v=t$$ and to substitute this in the depressed cubic, giving $$u^3 + v^3 + (3uv + p)(u+v)+ q= 0.$$

At this point Cardano imposed the condition $$3uv+p=0.$$ This removes the third term in previous equality, leading to the system of equations $$\begin{align}u^3 + v^3&=-q \\ uv&=-\frac p3.\end{align}$$

Knowing the sum and the product of $\pi⁄3$ and $$v^3,$$ one deduces that they are the two solutions of the quadratic equation $$\begin{align} 0 &= (x - u^3)(x - v^3) \\ &= x^2 - (u^3 + v^3)x + u^3v^3 \\ &= x^2 - (u^3 + v^3)x + (uv)^3 \end{align}$$ so $$x^2 + qx -\frac {p^3}{27}=0.$$ The discriminant of this equation is $$\Delta = q^2 + \frac{4p^3}{27}$$, and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root): $$-\frac q2 \pm \sqrt {\frac {q^2}{4} +\frac {p^3}{27}}.$$ So (without loss of generality in choosing $±h$ or $$v$$): $$u = \sqrt[3]{-\frac q2 + \sqrt {\frac {q^2}{4} +\frac {p^3}{27}}}.$$ $$v = \sqrt[3]{-\frac q2 - \sqrt {\frac {q^2}{4} +\frac {p^3}{27}}}.$$ As $$u+v=t,$$ the sum of the cube roots of these solutions is a root of the equation. That is $$t=\sqrt[3]{-{q\over 2}+\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} +\sqrt[3]{-{q\over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}$$ is a root of the equation; this is Cardano's formula.

This works well when $$4p^3+27q^2 > 0,$$ but, if $$4p^3+27q^2 < 0,$$ the square root appearing in the formula is not real. As a complex number has three cube roots, using Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by Rafael Bombelli in his book L'Algebra (1572). The solution is to use the fact that $$uv=-\frac p3,$$ that is, $$v=\frac{-p}{3u}.$$ This means that only one cube root needs to be computed, and leads to the second formula given in.

The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two primitive cube roots of unity, which are $$\frac {-1\pm \sqrt{-3}}2.$$

Vieta's substitution
Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of, and avoids the problem of computing two different cube roots.

Starting from the depressed cubic $\pi⁄3$, Vieta's substitution is $\pi⁄3$.

The substitution $S_{3}$ transforms the depressed cubic into $$w^3+q-\frac{p^3}{27w^3}=0.$$

Multiplying by $A_{3}$, one gets a quadratic equation in $K$: $$(w^3)^2+q(w^3)-\frac{p^3}{27}=0.$$

Let $$W=-\frac q 2\pm\sqrt{\frac{p^3}{27} + \frac {q^2} 4}$$ be any nonzero root of this quadratic equation. If $Δ$, $r_{1}$ and $r_{2}$ are the three cube roots of $K$, then the roots of the original depressed cubic are $r_{3}$, $A_{3}$, and $A_{3}$. The other  root of the quadratic equation is $$\textstyle -\frac {p^3}{27W}.$$ This implies that changing the sign of the square root exchanges $Q$ and $S_{3}$ for $A_{3}$, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when $p(x) = x^{3} − 3x − 1$, in which case the only root of the depressed cubic is $81 = 9^{2}$.

Lagrange's method
In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees. This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six. Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher; as was later proved by the Abel–Ruffini theorem. Nevertheless, modern methods for solving solvable quintic equations are mainly based on Lagrange's method.

In the case of cubic equations, Lagrange's method gives the same solution as Cardano's. Lagrange's method can be applied directly to the general cubic equation $t^{3} + pt + q = 0$, but the computation is simpler with the depressed cubic equation, $u^{3}$.

Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves. More precisely, let $K$ be a primitive third root of unity, that is a number such that $t^{3} + pt + q = 0$ and $t = w – p⁄3w$ (when working in the space of complex numbers, one has $$\textstyle \xi=\frac{-1\pm i\sqrt 3}2=e^{2i\pi/3},$$ but this complex interpretation is not used here). Denoting $w(t + w) = p⁄3$, $t = p⁄3w – w$ and $t = w – p⁄3w$ the three roots of the cubic equation to be solved, let $$\begin{align} s_0 &= x_0 + x_1 + x_2,\\ s_1 &= x_0 + \xi x_1 + \xi^2 x_2,\\ s_2 &= x_0 + \xi^2 x_1 + \xi x_2, \end{align}$$ be the discrete Fourier transform of the roots. If $w^{3}$, $w_{1}$ and $w_{2}$ are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is, $$\begin{align} x_0 &= \tfrac13(s_0 + s_1 + s_2),\\ x_1 &= \tfrac13(s_0 + \xi^2 s_1 + \xi s_2),\\ x_2 &= \tfrac13(s_0 + \xi s_1 + \xi ^2 s_2). \end{align}$$

By Vieta's formulas, $w_{3}$ is known to be zero in the case of a depressed cubic, and $w_{1} − p⁄3w_{1}$ for the general cubic. So, only $w_{2} − p⁄3w_{2}$ and $w_{3} − p⁄3w_{3}$ need to be computed. They are not symmetric functions of the roots (exchanging $w_{i}$ and $− p⁄3w_{i}$ exchanges also $i = 1, 2, 3$ and $p = q = 0$), but some simple symmetric functions of $0$ and $ax^{3} + bx^{2} + cx + d = 0$ are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the $K$ as roots of a polynomial with known coefficients. This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the $a$ as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher.

In the case of a cubic equation, $$P=s_1s_2,$$ and $$S=s_1^3+s_2^3$$ are such symmetric polynomials (see below). It follows that $$s_1^3$$ and $$s_2^3$$ are the two roots of the quadratic equation $$z^2-Sz+P^3=0.$$ Thus the resolution of the equation may be finished exactly as with Cardano's method, with $$s_1$$ and $$s_2$$ in place of $K$ and $$v.$$

In the case of the depressed cubic, one has $$x_0=\tfrac 13 (s_1+s_2)$$ and $$s_1s_2=-3p,$$ while in Cardano's method we have set $$x_0=u+v$$ and $$uv=\tfrac 13 p.$$ Thus, up to the exchange of $u$ and $$v,$$ we have $$s_1=3u$$ and $$s_2=3v.$$ In other words, in this case, Cardano's method and Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem.

Computation of $u$ and $w$
A straightforward computation using the relations $t^{3} + pt + q = 0$ and $ξ^{3} = 1$ gives $$\begin{align} P&=s_1s_2=x_0^2+x_1^2+x_2^2-(x_0x_1+x_1x_2+x_2x_0),\\ S&=s_1^3+s_2^3=2(x_0^3+x_1^3+x_2^3)-3(x_0^2x_1+x_1^2x_2+x_2^2x_0+x_0x_1^2+x_1x_2^2+x_2x_0^2)+12x_0x_1x_2. \end{align}$$ This shows that $w$ and $w3$ are symmetric functions of the roots. Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving $$\begin{align} P&=e_1^2-3e_2,\\ S&=2e_1^3-9e_1e_2+27e_3, \end{align}$$ with $ξ^{2} + ξ + 1 = 0$, $x_{0}$ and $x_{1}$ in the case of a depressed cubic, and $x_{2}$, $s_{0}$ and $s_{1}$, in the general case.

Applications
Cubic equations arise in various other contexts.

In mathematics

 * Angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.
 * Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.
 * The area of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to $$2\pi/7$$ satisfy cubic equations.
 * Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.
 * The solution of the general quartic equation relies on the solution of its resolvent cubic.
 * The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix.
 * The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation.
 * Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve.
 * Critical points of a quartic function are found by solving a cubic equation (the derivative set equal to zero).
 * Inflection points of a quintic function are the solution of a cubic equation (the second derivative set equal to zero).

In other sciences

 * In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation.
 * In thermodynamics, equations of state (which relate pressure, volume, and temperature of a substances), e.g. the Van der Waals equation of state, are cubic in the volume.
 * Kinematic equations involving linear rates of acceleration are cubic.
 * The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation.
 * The steady state speed of a vehicle moving on a slope with air friction for a given input power is solved by a depressed cubic equation.
 * Kepler's third law of planetary motion is cubic in the semi-major axis.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cubic equation


text:
Martie Maguire

Martha Elenor Maguire (née Erwin, previously Seidel; October 12, 1969) is an American musician who is a founding member of the country band the Chicks and the country bluegrass duo Court Yard Hounds. She won awards in national fiddle championships while still a teenager. Maguire is accomplished on several other instruments, including the mandolin, viola, double bass and guitar. She has written and co-written a number of the band's songs, some of which have become chart-topping hits. She also contributes her skills in vocal harmony and backing vocals, as well as orchestrating string arrangements for the band.

Maguire learned several instruments at a young age, honing her skills with her younger sister, Emily Strayer (born Emily Erwin) and two schoolmates (a brother and sister team, Troy and Sharon Gilchrist) for over five years as a part of a touring bluegrass quartet while in high school. After graduation, the sisters forged an alliance with two other women they had met through the Dallas music scene, Laura Lynch and Robin Lynn Macy, forming a bluegrass and country music band, busking and touring the bluegrass festival circuits for six years. After the departure of Macy, and the replacement of Lynch with singer Natalie Maines, the band widened their musical repertoire and appearance. The result was a trio so commercially successful that it took the country music industry by surprise, with 19 singles hitting the Billboard Country Charts, 2 Diamond Albums, 2 Platinum albums, and 13 Grammy Awards. Maguire subsequently stood by her bandmates as they were engulfed in a 2003 controversy for criticizing George W. Bush, which led to the Chicks being blacklisted by many country radio stations.

Early life
Martha Elenor Erwin, nicknamed Martie, was born October 12, 1969, in York, Pennsylvania.

Encouraged by her parents – educators at private schools – Maguire began playing violin at age five, and by age 12, started to learn to play "fiddle style" after receiving a birthday gift of fiddle lessons. She also was active in her school orchestra.

Her sister Emily shared her love and interest in music, and displayed early talent herself. The two sisters were provided musical instruction on several instruments, and their talent for vocal harmony continued to be nurtured. Although Maguire became famous for her harmony and mastery of the fiddle, she also plays a variety of strings that include the viola, guitar, mandolin, and double bass. Maguire composes songs and arranges the use of stringed instruments for concert and recording performances.

Early career
By 1983, Maguire was touring with her sister Emily and school friends siblings Troy and Sharon Gilchrist. The sisters showed an "almost obsessive" interest in busking at small venues and attending bluegrass festivals. The four students formed the teenage bluegrass group "Blue Night Express", playing together for five years, from 1984 to 1989, while still attending private Greenhill School. "We'd drive down to the west end of Dallas and open our cases, and that was our job," Maguire said of it in a later interview to 60 Minutes II correspondent Dan Rather. "That's how we made money in high school." Also while in high school Maguire performed at Six Flags Over Texas in Arlington, TX, in a strolling duo with Troy Gilchrist for a summer.

In 1987, Maguire was awarded second place for the fiddle in the National fiddle championships held yearly in Winfield, Kansas. Upon graduation from high school, with Emily still in high school, she spent a year attending college at Southwestern University in Georgetown, Texas, from 1988 to 1989. She performed in the school orchestra there and again competed in the national fiddle championships at the Walnut Valley Festival, in Winfield, earning third place that year.

In the 90s she played occasionally with the slo-core band, Bedhead.

The Chicks
In 1989, Maguire and Strayer joined guitarist Robin Lynn Macy and Laura Lynch on double bass in the Dixie Chicks, playing what was at the time predominantly bluegrass music and a mix of country standards. Originally listed as Martha, and then Martie Erwin, Maguire played fiddle, mandolin, viola and harmonized with Strayer on backing vocals.

Maguire had begun writing and recording songs on their debut independent album, Thank Heavens for Dale Evans, and in 2001, one song co-written with Laura Lynch called "Pink Toenails" resurfaced. The song was featured in the film, Don't Say A Word, with Skye McCole Bartusiak performing the song.

When their second album had a more polished sound, Maguire commented, "I hope our fans won't be disappointed [in The Little Ol' Cowgirl]; it's got drums on every track; it's no longer bluegrass, but we have to make a living and you can't do that playing bluegrass."

Two albums later, after Natalie Maines became the lead vocalist of the Chicks, the band's commercial success skyrocketed. Maguire said of their music, "It's very rootsy, but then Natalie comes in with a rock and blues influence. That gave Emily and I [sic] a chance to branch out, because we loved those kinds of music but felt limited by our instruments."

Maguire co-wrote the song, "Cowboy Take Me Away" on the Chicks' album Fly for her sister Emily. It was a tribute to Emily's courtship with country music writer/singer Charlie Robison. Maguire was singled out by BMI in 2000, and awarded Songwriter of the Year for writing and/or co-writing "Cowboy Take Me Away", "Ready To Run" and "You Were Mine". "You Were Mine" was co-written by the sisters about their parents' divorce.

In 1999, the Dixie Chicks performed two songs, recording a cover of the song, "You Can't Hurry Love" as well as Maguire's "Ready to Run" for the Julia Roberts/ Richard Gere film Runaway Bride. In 2003, Maguire was nominated for a Tony Award for co-composing the film score to Urban Cowboy; the musical used her "Cowboy Take Me Away".

After the band achieved massive commercial success with its first two Sony studio albums with Maines, they struggled with their record company over artistic direction on the third. Maguire commented, "I don't think any of us ever trusted Nashville. When you're in that town you know everybody is talking about everybody else. Everybody is wishing for the other guy to fail."

Although Maguire and Strayer often appear quiet and demure compared to their animated bandmate Natalie Maines, the trio have stood united on controversial subjects since they banded together to play in 1995, even when their opinions have had the potential to serve them more harm than good. In 2006, Maguire said, "I'd rather have a smaller following of really cool people who get it, who will grow with us as we grow and are fans for life, than people that have us in their five-disc changer with Reba McEntire and Toby Keith. We don't want those kinds of fans. They limit what you can do."

Court Yard Hounds

With Natalie Maines taking a break from music, Maguire was working on a solo fiddle album; however, it was announced in January 2010 that she and sister Emily Strayer have formed a side project called the Court Yard Hounds, with Strayer on lead vocals. The band made their live debut in March at South by Southwest with an album released in May 2010.

Personal life
Maguire married pharmaceutical representative Ted Seidel on June 17, 1995, and changed her last name to Seidel. She also was stepmother to his son, Carter. However, her marriage did not last, and she and Ted Seidel were divorced in November 1999.

At bandmate Natalie Maines' sister Kim's wedding, Martie met Gareth Maguire, a Roman Catholic teacher and actor from Carnlough, Northern Ireland. The couple became engaged in June 2001 and married on August 10, 2001, in a civil ceremony in Hawaii; as explained at one of their concerts, the Dixie Chicks' song "White Trash Wedding" is based on Martie's relationship with Gareth. Later they had a Catholic "blessing" ceremony in the groom's hometown, Carnlough in County Antrim, Northern Ireland, on March 9, 2002, for all the guests that were not able to attend the first wedding. Maguire said of the ceremony that the Catholic Church would not permit a wedding service due to her prior divorce. Maguire said that there were many "special" people that they had wanted to include, but that they just could not wait to marry. Another reason for the rush was Gareth's need to keep flying back and forth between the United States and Ireland because of visa considerations. After her second marriage, Maguire changed her name to that of her new husband, which is why in her musical career she has been credited by all three of her surnames: Erwin, Seidel, and Maguire.

Maguire and her second husband had three daughters together: fraternal twins Eva Ruth and Kathleen 'Katie' Emilie were born April 27, 2004. Katie was named after Gareth's late sister, Kathleen. Their third daughter, Harper Rosie Maguire was born July 25, 2008. The couple divorced in 2013. In February 2022, it was revealed Maguire is married to Clem Moore.

Maguire has been frank about using in vitro fertilization to conceive their daughters. In an interview in Conceive, she said, "All my paperwork said 'unspecified origin.' We spent three years of active trying before we went to IVF. First I went on Clomid. Then I had some dye tests and found I had a collapsed tube, so I had laparoscopic surgery; the tube wasn't blocked, just spasming." After three attempts at intrauterine insemination, she said, she and her husband didn't think it was worth continuing in that manner, and switched to IVF. In August 2007, Maguire began IVF again, resulting in their third daughter Harper. In 2008, regarding the number of children the Dixie Chicks had produced in the past seven years, (Maines had two; Strayer had four in addition to Maguire's twins and newborn) Maguire told People "We'll have to move over and let the little chicks take over! We've got a new band!"

Maguire and Strayer co-wrote a song, "So Hard", about their own personal experiences with infertility and their need to rely on other methods to conceive. They speak out about the difficulties they faced, but also their good fortune; both having options that for many women are financially prohibitive. Mentioning the stigma attached to IVF, Maguire said, "I think we feel a responsibility to break down some barriers. It's much more of a common problem than people realize." A final concern Maguire mentioned was the question of what to do with all the unused frozen embryos."Now that I have children, I see those embryos as possible children. So I have to think about what my options are if there are leftovers again. I could keep them in storage, and maybe they will help my children some day. Or I can try to donate them to stem cell research. I don't think I could give them to another family. I would always worry: what if it's an abusive family? What if they don't get enough love?"

Natalie Maines has noted that she and Maguire learned Transcendental Meditation in the mid-1990s.

Discography

 * Maguire played fiddle on the song, "Somewhere Down The Road" on the eponymous album, Deryl Dodd by Deryl Dodd on Sony.
 * Maguire also played fiddle on The Flatlanders' adaptation of Woody Guthrie's "Sowing on the Mountain" on their 2009 album Hills and Valleys.
 * As part of the Court Yard Hounds, Maguire played fiddle and provided harmony vocals on the song "She Sang Hymns Out Of Tune" on the Cass County album by Don Henley.

title: Martie Maguire


text:
Emily Strayer

Emily Burns Strayer ( Erwin, previously Robison; born August 16, 1972) is an American songwriter, singer, multi-instrumentalist, and a founding member of the country band the Chicks, formerly known as the Dixie Chicks. Strayer plays banjo, dobro, guitar, lap steel, bass, mandolin, accordion, fiddle, piano, and sitar. Initially in her career with the Chicks, she limited her singing to harmony with backing vocals, but within her role in the Court Yard Hounds, she took on the role of lead vocalist.

Early life
Emily Erwin was born in Pittsfield, Massachusetts. Parents Paul Erwin and Barbara Trask moved the family to the suburban town of Addison, Texas, on the northern edge of Dallas, where she was raised with her two older sisters, Julia and Martha. Her parents were both educators and nurtured the growing interest that both Emily and Martha (later nicknamed Martie) shared, and together both sisters became proficient on several instruments while in elementary school.

Emily began playing the violin at age seven, and the banjo at age ten, afterward learning all the string instruments she could find. Years later, Martie joked that Emily was better than she at the fiddle, and because she wanted to keep the fiddle as her instrument, she forced Emily to learn something else. Emily responded by mastering the five string banjo, by reading books to teach herself the chord progressions.

From 1984–1989, Jane Frost (Director of the Patsy Montana Museum and the Walnut Valley Festival in Winfield, Kansas) remembers watching the sisters mature, teaming up with schoolmates Troy and Sharon Gilchrist. The foursome were touring in a teenage bluegrass group they named the Blue Night Express, in part because of the amount of traveling they had to do to reach far away festivals, frequently with a parent or friend of the family towing them back home long after dark at night. They thought it was worth the sacrifices to meet their musical heroes and experienced the friendly camaraderie and tricks of the trade on instruments from others. The sisters were said to have an "almost obsessive" interest in busking at small venues and attending bluegrass festivals. After Martie graduated from secondary school at Greenhill School with Emily still completing her studies, both remained active in the bluegrass scene.

The Chicks

In 1989, after years of attending bluegrass festivals and busking where they could, Emily joined her sister Martie, guitarist Robin Lynn Macy, and upright bass player Laura Lynch. Frost, again, recalls being privy to the discussion that the four women had about the possibility of a successful career as musicians together. Martie felt they could do well. Robin said, ".. It's going to be a 'hot' band," to which Emily responded, "I give it six months, and if we aren't making money by then, I'm out of here!". Strayer was shy, and the youngest member of the group. She had enjoyed playing throughout school, but was at an age where she was already entertaining thoughts of working hard to be accepted by the United States Air Force Academy.

By 1993, the band had evolved into a new direction. Macy left the group for a "purer" bluegrass sound. Lynch, thrust into the position of sole lead singer, was replaced by the sisters in 1995 with singer composer Natalie Maines after the group was unable to garner anything more than local interest. Strayer commented, "We were prepared to pay our dues for as long as it took; we were prepared for longevity. We know that we will always be playing music together, so we wanted to find someone who is just as determined and energetic as we are."

But in later interviews, the sisters revealed what many had thought for a long time: the reason Lynch was replaced was not commitment but a perceived lack of talent. In an article in the April/May edition of Country Music Magazine, Emily stated, "We felt we needed the next calibre of singer."

From there, massive commercial success ensued, with their 1998 and 1999 albums Wide Open Spaces and Fly both achieving diamond record status. Strayer was a key element of the group's look, with her hair dyed blond to match the other two at first, and then allowing it to return to her natural brunette color, and distinguishing herself visually from the other two. Strayer and Maguire's instrumental virtuosity set the Dixie Chicks apart from many other country acts, male and female. Strayer's songwriting has also been a factor in the Chicks' recording career.

Strayer stood by Maines when the controversy over Maines' remarks about U.S. President George W. Bush hit the newswire on the eve before the Iraq War in 2003. She was the only bandmate to realize that - while in the U.K., there was a big anti-war sentiment in the audience - back in the United States, Maines' criticism of President Bush would not be well received. Nevertheless, even when her home was trashed, and plenty of editorials were predicting the end of the Dixie Chicks' successful careers in music, she remained loyal to Maines, as did her sister.

Court Yard Hounds
Strayer spent at least half of her life busking and touring nearly non-stop since high school, first with her older sister, and then with the changing lineups of the Dixie Chicks. After the bandmates celebrated their five Grammy Awards for Taking the Long Way, the trio all had young children and took time out to spend with them, but Strayer was eager to return to writing and touring. When Strayer's marriage came to an end, she began expressing herself by songwriting. Some of her compositions include her feelings about the breakup with her former husband, and expected that soon, the band would resume its former schedule of writing and performing, but Natalie Maines was reluctant to do so. After four years, Strayer became "angry and frustrated", and in sending her new songs to her sister, conveyed her increasing disappointment. With Natalie Maines taking a break from music, Rolling Stone magazine announced in January 2010 that Emily and sister Martie have formed a side project called the Court Yard Hounds, with Strayer as lead vocalist. The band made their live debut in March at South by Southwest with an album released in May 2010. Strayer wrote all but one of the songs on their album.

Personal life
On May 1, 1999, she married country singer Charlie Robison. Their wedding was held at the Cibolo Creek Ranch. Martie Maguire later revealed on a VH1 program that during their courtship, she had written the romantic hit song, "Cowboy Take Me Away" for them. They have three children: Charles Augustus, called "Gus", born November 11, 2002, and twins Julianna Tex and Henry Benjamin born on April 14, 2005. They divorced on August 6, 2008, after nine years of marriage.

She became romantically involved with Martin Strayer while composing the Court Yard Hounds album. With Strayer she has a daughter, Violet Isabel Strayer (September 4, 2012). The couple later married in 2013.

title: Emily Strayer


text:
Johann Georg Walch

Johann Georg Walch (17 June 1693 – 13 January 1775) was a German Lutheran theologian.

Life
He was born in Meiningen, where his father, Georg Walch, was general superintendent. He studied at Leipzig and Jena, amongst his teachers being J. F. Buddeus, whose only daughter he married. He published in 1716 a work, Historia critica Latinae linguae, which soon came into wide use. Two years later he became professor extraordinarius of philosophy at Jena. In 1719, he was appointed professor ordinarius of rhetoric, in 1721 of poetry, and in 1724 professor extraordinarius of theology. In 1728 he became professor ordinarius of theology, and in 1730 professor primarius.

His theological position was that of moderate orthodoxy, greatly influenced by the philosophy and controversies of the Deistic period. His university lectures and published works ranged over the wide fields of church history in its various branches, particularly the literature and the controversies of the church, dogmatics, ethics and pastoral theology.

Of his works the most valuable were Bibliotheca theologica (1757–1765); Bibliotheca patristica (1770); his edition of Luther's works in 24 vols. (1740–1752); Historische und theologische Einleitung in die religiösen Streitigkeiten, welche sonderlich ausser der evangelische-lutherischen Kirche entstanden (5 vols., 1733 ff.); the companion work to this, ''Einleitung in die Religionsstreitigkeiten der evangel.-luth. Kirche (1730–1739), and Philosophisches Lexikon'' (1726, 4th ed. 1775).

Also important is the Historia logicae, published in Leipzig in 1721.

His life, with a complete list of his writings, which amounted to 287, Leben und Charakter des Kirchenraths J. G. Walch, was published anonymously by his son CWF Walch (Jena, 1777). Cf. Wilhelm Gass, Protestantische Dogmatik, iii. p. 205 sq.

His sons, Johann Ernst Immanuel and Christian Wilhelm Franz Walch were also noted theologians. Son Karl Friedrich Walch became a law professor.

Johann Georg Walch died in Jena on 13 January 1775.

title: Johann Georg Walch


text:
Johann Ernst Immanuel Walch

Johann Ernst Immanuel Walch (29 August 1725 – 1 December 1778) was a German theologian, linguist, and naturalist from Jena.

Life
The son of the theologian Johann Georg Walch, he studied Semitic languages at the University of Jena, and also natural science and mathematics. In 1749 he published Einleitung in die Harmonie der Evangelien, and in 1750 was appointed professor extraordinarius of theology. Five years later he became professor ordinarius of logic and metaphysics; in 1759 he exchanged this for a professorship of rhetoric and poetry.

Amongst his theological works were Dissertationes in Acta Apostolorum (1756–1761); Antiquitales symbolicae (1772); and after his death appeared Observationes in Matthaeum ex Graecis inscriptionibus (1779). He also edited a periodical Der Naturforscher (1774–1778), and during the years 1749-1756 took an active part in editing the Zeitungen von gelehrten Sachen.

"'Most of Walch’s interests and publications were on topics in the humanities. These included early Christian church history; New Testament exegesis and commentary; Latin and Greek linguistics, literature, and inscriptions; Roman history and antiquities; Celtic religion; and the history of medicine.'"

In the later part of his career Walch found interest in the natural sciences, particularly fossils and geology. One significant result was the beautifully illustrated, four-volume Die Naturgeschichte der Versteinerungen ("The Natural History of Petrifactions"), released from 1768 to 1773, and also published in French and Dutch editions. A comprehensive chapter on trilobites in 1771 contains the first use of the word "trilobite" and predates other equally comprehensive treatments by 50 years.

"'By the end of his career, he had completed over 80 publications (books, chapters, and articles) on various topics in the humanities and about 50 publications in natural history. He became ill in Summer 1778 with the onset of hypochondriac (abdominal) seizures... his death on December 1, 1778, [was] from intestinal infections.'"

Walch died in Jena. His brother Christian Wilhelm Franz Walch was also theologian.

title: Johann Ernst Immanuel Walch


text:
STS-9

STS-9 (also referred to Spacelab 1) was the ninth NASA Space Shuttle mission and the sixth mission of the Space Shuttle Columbia. Launched on 28 November 1983, the ten-day mission carried the first Spacelab laboratory module into orbit.

STS-9 was also the last time the original STS numbering system was used until STS-26, which was designated in the aftermath of the 1986 Challenger disaster of STS-51-L. Under the new system, STS-9 would have been designated as STS-41-A. STS-9's originally planned successor, STS-10, was canceled due to payload issues; it was instead followed by STS-41-B. After this mission, Columbia was taken out of service for renovations and did not fly again until STS-61-C in early January 1986.

STS-9 sent the first non-U.S. citizen into space on the Shuttle, Ulf Merbold, becoming the first ESA and first West German citizen to go into space.

Crew

 * Red Team [[Image:Solid red.svg|8px|Member of Red Team]]
 * Blue Team [[Image:Solid blue.svg|8px|Member of Blue Team]]

Support crew

 * John E. Blaha (entry CAPCOM)
 * Franklin R. Chang-Diaz
 * Mary L. Cleave
 * Anna L. Fisher
 * William F. Fisher
 * Guy S. Gardner (ascent CAPCOM)
 * Chuck Lewis (Marshall CAPCOM)
 * Bryan D. O'Connor
 * Wubbo Ockels

Mission background
STS-9's six-member crew, the largest of any human space mission at the time, included John W. Young, commander, on his second shuttle flight; Brewster H. Shaw, pilot; Owen K. Garriott and Robert A. Parker, both mission specialists; and Byron K. Lichtenberg and Ulf Merbold, payload specialists – the first two non-NASA astronauts to fly on the Space Shuttle. Merbold, a citizen of West Germany, was the first foreign citizen to participate in a Space Shuttle flight. Lichtenberg was a researcher at the Massachusetts Institute of Technology (MIT). Prior to STS-9, the scientist-astronaut Garriott had spent 56 days in orbit in 1973 aboard Skylab. Commanding the mission was veteran astronaut John W. Young, making his sixth and final flight over an 18-year career that saw him fly twice each in Project Gemini, Apollo, and the Space Shuttle, which included two journeys to the Moon and making him the most experienced space traveler to date. Young, who also commanded Columbia on its maiden voyage STS-1, was the first person to fly the same space vehicle into orbit more than once. STS-9 marked the only time that two pre-Shuttle era astronaut veterans (Garriott and Young) would fly on the same Space Shuttle mission. STS-9 was also the first Space Shuttle mission to have more than one veteran astronaut.

The mission was devoted entirely to Spacelab 1, a joint NASA/European Space Agency (ESA) program designed to demonstrate the ability to conduct advanced scientific research in space. Both the mission specialists and payload specialists worked in the Spacelab module and coordinated their efforts with scientists at the Marshall Space Flight Center (MSFC) Payload Operations Control Center (POCC), which was then located at the Johnson Space Center (JSC) in Texas. Funding for Spacelab 1 was provided by the ESA.

Shuttle processing
After Columbia's return from STS-5 in November 1982, it received several modifications and changes in preparation for STS-9. Most of these changes were intended to support the Spacelab module and crew, such as the addition of a tunnel connecting the Spacelab to the orbiter's airlock, and additional provisions for the mission's six crew members, such as a galley and sleeping bunks. Columbia also received the more powerful Space Shuttle Main Engines introduced with Challenger, which were rated for 104% maximum thrust; its original main engines were later refurbished for use with Atlantis, which was still under construction at the time. Also added to the shuttle were higher capacity fuel cells and a Ku-band antenna for use with the Tracking and Data Relay Satellite (TDRS).

The mission's original launch date of 29 October 1983 was scrubbed due to concerns with the exhaust nozzle on the right solid rocket booster (SRB). For the first time in the history of the shuttle program, the shuttle stack was rolled back to the Vehicle Assembly Building (VAB), where it was destacked and the orbiter returned to the Orbiter Processing Facility (OPF), while the suspect booster underwent repairs. The shuttle was restacked and returned to the launch pad on 8 November 1983.

Mission insignia
The mission's main payload, Spacelab 1, is depicted in the payload bay of the Columbia. The nine stars and the path of the orbiter indicate the flight's numerical designation, STS-9.

Mission summary

STS-9 launched successfully from Kennedy Space Center at 11:00:00 a.m. EST on 28 November 1983. The shuttle's crew was divided into two teams, each working 12-hour shifts for the duration of the mission. Young, Parker and Merbold formed the Red Team, while Shaw, Garriott and Lichtenberg made up the Blue Team. Usually, Young and Shaw were assigned to the flight deck, while the mission and payload specialists worked inside the Spacelab.

Over the course of the mission, 72 scientific experiments were carried out, spanning the fields of atmospheric and plasma physics, astronomy, solar physics, material sciences, technology, astrobiology and Earth observations. The Spacelab effort went so well that the mission was extended an additional day to 10 days, making it the longest-duration shuttle flight at that time. In addition, Garriott made the first ham radio transmissions by an amateur radio operator in space during the flight. This led to many further space flights incorporating amateur radio as an educational and back-up communications tool.

The Spacelab 1 mission was highly successful, proving the feasibility of the concept of carrying out complex experiments in space using non-NASA persons trained as payload specialists in collaboration with a POCC. Moreover, the TDRS-1 satellite, now fully operational, was able to relay significant amounts of data through its ground terminal to the POCC.

During orbiter orientation, four hours before re-entry, one of the flight control computers crashed when the Reaction Control System (RCS) thrusters were fired. A few minutes later, a second crashed in a similar fashion, but was successfully rebooted. Young delayed the landing, letting the orbiter drift. He later testified: "Had we then activated the Backup Flight Software, loss of vehicle and crew would have resulted". Post-flight analysis revealed the GPCs (General Purpose Computers) failed when the RCS thruster motion knocked a piece of solder loose and shorted out the CPU board. A GPC running BFS may or may not have the same soldering defect as the rest of the GPCs. Switching the vehicle to the BFS from normal flight control can happen relatively instantaneously, and that particular GPC running the BFS could also be affected by the same failure due to the soldering defect. If such a failure occurred, switching the vehicle back to normal flight control software on multiple GPCs from a single GPC running BFS takes a lot longer, in essence leaving the vehicle without any control at all during the change.

Columbia landed on Runway 17 at Edwards Air Force Base on 8 December 1983, at 03:47:24 p.m. PST, having completed 167 orbits and travelled 6.9 e6km over the course of its mission. Right before landing, two of the orbiter's three auxiliary power units (APUs) caught fire due to a hydrazine leak, but the orbiter nonetheless landed successfully. Columbia was ferried back to KSC on 15 December 1983. The leak was later discovered after it had burned itself out and caused major damage to the compartment. By this time, Discovery had been delivered just three weeks before the launch of STS-9. This allowed NASA to take Columbia out of service for an extensive renovation and upgrade program to bring it up to date with Challenger as well as Discovery and later on Atlantis, which would be delivered in 1985. As a result, Columbia would not fly at all during 1984–1985.

title: STS-9