What body parts are effected by Stigmata?

"Stigmata" received mixed reviews by critics, who compared it with the previous Arch Enemy album, "Black Earth". Jason Anderson of Allmusic noted that "as often happens with highly anticipated follow-ups, "Stigmata" disappointed some fans with what some called a more restrained sound and lesser material" and criticized the singer Johan Liiva and the addition of the session drummer Peter Wildoer. Nevertheless, Anderson praised Michael Amott stating that "these problems are really minor when balanced against Amott's massive playing and songwriting, so while "Stigmata" is a fraction off from the band's debut form, Arch Enemy still delivers some nice melodic death metal on the sophomore effort." Paul Schwarz of "Chronicles of Chaos" first noted that "Arch Enemy have made an interesting, and slightly unexpected, progression from their "Black Earth" debut of two years ago". He commented that the band "have not drawn their'melody' from pop or even folk" and what characterises the difference between both albums "can be summed up in three words: classic heavy metal." Schwarz finished with a good review writing that "overall "Stigmata" is a great album and one of the best directions Arch Enemy could have chosen to follow after their critical success with "Black Earth"."

Stigmata (Arch Enemy album)

Stigmata is the second studio album by Swedish melodic death metal band Arch Enemy. The album was Arch Enemy's first to see worldwide release, in Europe and North America on Century Media Records, and in Japan again on Toy's Factory records. "Stigmata" features session drummer Peter Wildoer, who had also appeared in Christopher Amott's solo project Armageddon on the 1997 album "Crossing the Rubicon", shortly before the recording of the album. The album was reissued on 25 May 2009, featuring a new layout, packaging, and bonus tracks. Stigmata not only contains a title track, but a track named after their previous album as well. The Sri Lankan metal band Stigmata named themselves after this album.

Critics Paul Schwarz and EvilG mentioned that the sounds resembles to the Carcass album "Heartwork". Schwarz mentions and compares again "Black Earth": "Opener "Beast of Man" kicks in with searing riffing and thundering double bass, which creates the false impression that the new material will be far more like the "Heartwork" era Carcass sound which characterised much of "Black Earth". Arch Enemy have, in fact, gone the opposite way - in terms of heaviness, not quality -, and produced an album which utilises far more melody, especially in its choruses, than "BE" did." EvilG of Metal Rules praises writing that "I think the reason why this CD is so great is that it has re-captured the Carcass "Heartwork" sound and feel while at the same time incorporating more melody and even some power metal sounds."

Personnel credits adapted from "Stigmata" album liner notes.

Stigmata (singular "stigma") is a term used in Christian mysticism to describe the manifestations of bodily wounds, scars and pain in locations corresponding to the crucifixion wounds of Jesus Christ, such as the hands, wrists, and feet. An individual bearing the wounds of stigmata is referred to as a stigmatist or a stigmatic.

Stigmata

The term originates from the line at the end of Saint Paul's Letter to the Galatians where he says, "I bear on my body the marks of Jesus." "Stigmata" is the plural of the Greek word στίγμα "stigma", meaning a mark, tattoo, or brand such as might have been used for identification of an animal or slave.

Stigmata are primarily associated with the Roman Catholic faith. Many reported stigmatics are members of Catholic religious orders.
St. Francis of Assisi was the first recorded stigmatic in Christian history. For over fifty years, St. Padre Pio of Pietrelcina of the Order of Friars Minor Capuchin reported stigmata which were studied by several 20th-century physicians.

A high percentage (perhaps over 80%) of all stigmatics are women. In his "Stigmata: A Medieval Phenomenon in a Modern Age", Ted Harrison suggests that there is no single mechanism whereby the marks of stigmata were produced. What is important is that the marks are recognised by others as of religious significance. Most cases of stigmata have been debunked as trickery. Some cases have also included reportings of a mysterious chalice in visions being given to stigmatics to drink from or the feeling of a sharp sword being driven into ones chest.

Reported cases of stigmata take various forms. Many show some or all of five Holy Wounds that were, according to the Bible, inflicted on Jesus during his crucifixion: wounds in the wrists and feet, from nails; and in the side, from a lance. Some stigmatics display wounds to the forehead similar to those caused by the Crown of Thorns. Stigmata as crown of thorns appearing in the 20th century, e.g. on Marie Rose Ferron, have been repeatedly photographed. Other reported forms include tears of blood or sweating blood, and wounds to the back as from scourging.

Many stigmata show recurring bleeding that stops and then starts, at times after receiving Holy Communion, and a significant portion of stigmatics have shown a strong desire to frequently receive Holy Communion. A relatively high percentage of stigmatics also exhibit inedia, claiming to live with minimal (or no) food or water for long periods of time, except for the Holy Eucharist. Some exhibit weight loss, and closer investigation often reveals evidence of fakery.

Some stigmatics claim to feel the pain of wounds with no external marks; these are referred to as "invisible stigmata". Some stigmatics' wounds do not appear to clot, and seem to stay fresh and uninfected. The blood from the wounds is said, in some cases, to have a pleasant, perfumed odor, known as the Odour of Sanctity.

Individuals who have obtained the stigmata are many times described as ecstatics, overwhelmed with emotions upon receiving the stigmata. No case of stigmata is known to have occurred before the thirteenth century.

In his paper "Hospitality and Pain," Christian theologian Ivan Illich states: "Compassion with Christ... is faith so strong and so deeply incarnate that it leads to the individual embodiment of the contemplated pain." His thesis is that stigmata result from exceptional poignancy of religious faith and desire to associate oneself with the suffering Messiah.

St. Francis of Assisi is the first recorded stigmatic in Christian history. In 1224, two years before his death, he embarked on a journey to Mount La Verna for a forty-day fast. One morning near the feast of the Exaltation of the Cross, a six-winged angel reportedly appeared to Francis while he prayed. As the angel approached, Francis could see that the angel was crucified. He was humbled by the sight, and his heart was filled with elation joined by pain and suffering. When the angel departed, Francis was purportedly left with wounds in his hands, feet, and side as if caused by the same lance that pierced Christ’s side. The image of nails immediately appeared in his hands and feet, and the wound in his side often seeped blood. In traditional artistic depictions of the incident, Francis is accompanied by a Franciscan brother.

St. Francis' first biographer, Thomas of Celano, reports the event as follows in his 1230 "First Life of St. Francis":

When the blessed servant of God saw these things he was filled with wonder, but he did not know what the vision meant. He rejoiced greatly in the benign and gracious expression with which he saw himself regarded by the seraph, whose beauty was indescribable; yet he was alarmed by the fact that the seraph was affixed to the cross and was suffering terribly. Thus Francis rose, one might say, sad and happy, joy and grief alternating in him. He wondered anxiously what this vision could mean, and his soul was uneasy as it searched for understanding. And as his understanding sought in vain for an explanation and his heart was filled with perplexity at the great novelty of this vision, the marks of nails began to appear in his hands and feet, just as he had seen them slightly earlier in the crucified man above him.

His wrists and feet seemed to be pierced by nails, with the heads of the nails appearing on his wrists and on the upper sides of his feet, the points appearing on the other side. The marks were round on the palm of each hand but elongated on the other side, and small pieces of flesh jutting out from the rest took on the appearance of the nail-ends, bent and driven back. In the same way the marks of nails were impressed on his feet and projected beyond the rest of the flesh. Moreover, his right side had a large wound as if it had been pierced with a spear, and it often bled so that his tunic and trousers were soaked with his sacred blood.
From the records of St. Francis' physical ailments and symptoms, Dr. Edward Frederick Hartung concluded in 1935 that he knew what health problems plagued St. Francis. Hartung believed that he had an eye ailment known as trachoma and quartan malaria.

Quartan malaria infects the liver, spleen, and stomach, causing the victim intense pain. One complication of quartan malaria occasionally seen around Francis' time is known as purpura, a purple hemorrhage of blood into the skin. According to Hartung "If this were the case of St. Francis, he would have been afflicted by "ecchymoses", an exceedingly large "purpura". The purple spots of blood may have been punctured while in the wilderness and there appear as an open wound like that of Christ."

A later medical hypothesis was proposed in 1987 to explain the wounds, it claimed that St. Francis may have contracted leprosy.

For over fifty years, Padre Pio of Pietrelcina reported stigmata which were studied by several 20th-century physicians, whose independence from the Church is not known.
The observations were reportedly inexplicable and the wounds never became infected. His wounds healed once, but reappeared.

The wounds were examined by Luigi Romanelli, chief physician of the City Hospital of Barletta, for about one year. The physician Angelo Maria Merla noted that the wounds were not tubercular in origin but could not make an official diagnosis without further tests. The surgeon Giorgio Festa, a private practitioner, also examined them in 1920 and 1925. Professor Giuseppe Bastianelli, physician to Pope Benedict XV, examined the wounds, but no report of his examinations was made. Pathologist Amico Bignami of the University of Rome also observed the wounds describing them as shallow. Festa who had originally agreed with Bignami later described the wounds as superficial when covered with a scab.

Giorgio Festa noted that "at the edges of the lesions, the skin is perfectly normal and does not show any sign of edema, of penetration, or of redness, even when examined with a good magnifying glass". Dr. Alberto Caserta took X-rays of the hands in 1954 and found no abnormality in the bone structure. Dr. Giuseppe Sala who worked as a physician for Pio between 1956-1968 commented that tests revealed his blood had no signs of abnormality.

There were both religious and non-religious critics who accused Padre Pio of faking his stigmata, saying he used carbolic acid to make the wounds. The historian Sergio Luzzatto recounted that in 1919, according to one document in the Vatican's archive, Pio had requested carbolic acid from a pharmacist. She said it was for sterilization.

Amico Bignami in a report wrote that the wounds were caused by "neurotic necrosis". He suggested they had been inflicted unconsciously by suggestion and artificially maintained by iodine that Pio had used as a disinfectant.

In 1922, physician Agostino Gemelli wrote that Pio was a hysteric and his stigmata were self-induced, not of supernatural origin. Gemelli also speculated that his wounds were kept open with carbolic acid.

Throughout his life, Pio had hidden his wounds by wearing fingerless gloves. At death there were no wounds, only "unblemished skin".

Many stigmatics have been exposed as utilizing trickery. Magdalena de la Cruz for example confessed before she died that her stigmata was deliberate deception.

An early neurologist Désiré-Magloire Bourneville published works which stated that saints claiming to produce miracles or stigmata, and those claiming to be possessed were actually suffering from epilepsy or hysteria. Some modern research has indicated stigmata are of hysterical origin or linked to dissociative identity disorder.

There is a link between dietary constriction by self-starvation, dissociative mental states and self-mutilation, in the context of a religious belief. Anorexia nervosa cases often display self-mutilation similar to stigmata as part of a ritualistic, obsessive–compulsive disorder. A relationship between starvation and self-mutilation has been reported amongst prisoners of war and during famines.

The psychologist Leonard Zusne in his book "Anomalistic Psychology: A Study of Magical Thinking" (1989) has written:

Cases of stigmatism fall into two categories: self-inflicted wounds, which may be either cases of fraud or of unconscious self-infliction, and those that are caused by emotional states... Self-induced (through autosuggestion) itching and subsequent scratching of which the individual is unaware is likely to occur in suggestible persons if the stimulus is a mental or actual picture of the Crucifixion used during meditation and if the main motive is to receive the stigmata. The motive behind that may be unconscious conflict and a desire to escape from an intolerable situation into invalidism where one's needs are taken care of. It then becomes a case of hysterical conversion reaction. Many cases of stigmatism can be explained as fraud or unconsciously self-inflicted wounds.

In his "Stigmata: A Medieval Phenomenon in a Modern Age", Ted Harrison suggests that there is no single mechanism whereby the marks of stigmata were produced. Harrison found no evidence from a study of contemporary cases that the marks were supernatural in origin. He concluded, however, that marks of natural origin need not be hoaxes. Some stigmatics marked themselves in attempt to suffer with Christ as a form of piety. Others marked themselves accidentally and their marks were noted as stigmata by witnesses. Often marks of human origin produced profound and genuine religious responses.

Harrison also noted that the male-to-female ratio of stigmatics, which for many centuries had been of the order of 7 to 1, had changed over the last 100 years to a ratio of 5:4. Appearance of stigmata frequently coincided with times when issues of authority loomed large in the Church. What was significant about stigmatics was not that they were predominantly men, but that they were non-ordained. Having stigmata gave them direct access to the body of Christ without requiring the permission of the Church through the Eucharist. Only in the last century have priests been stigmatized.

It has been suggested that rare cases of non self-induced stigmata may be explained by painful bruising syndrome.

Skeptical investigator Joe Nickell who has investigated recent cases of stigmata such as Katya Rivas, commented that they are indistinguishable from hoaxing.

In 2002, a psychoanalytic study of stigmatic Therese Neumann has suggested that her stigmata resulted from post-traumatic stress symptoms expressed in unconscious self-mutilation through abnormal autosuggestibility.

Among the Warao of the Orinoco Delta, a contemplator of tutelary spirits may mystically induce the development of "...(imagined) openings in the palms of his hands." That these tutelary spirits are presented by the "itiriti snake" makes for a close analogue with the Seraph who endowed Francis of Assisi with his stigmata.

Buddhist "stigmata" are regularly indicated in Buddhist art.

Some spiritualist mediums have also produced stigmata. During the séances of German medium Maria Vollhardt, it was alleged that bleeding wounds appeared. However Albert Moll, a psychiatrist, considered her phenomena to be fraudulent.


Who is considered the founder of geometry?

Euclid (; – "Eukleídēs", ; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.

Euclid

Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious".

Very few original references to Euclid survive, so little is known about his life. He was likely born c. 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes (c. 287 BC – c. 212 BC) onward, and is usually referred to as "ὁ στοιχειώτης" ("the author of Elements"). The few historical references to Euclid were written centuries after he lived, namely by Proclus c. 450 AD.

A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be fictitious. If he came from Alexandria, he would have known the Serapeum of Alexandria, and the Library of Alexandria, and may have worked there during his time. Euclid's arrival in Alexandria came about ten years after its founding by Alexander the Great, which means he arrived c. 322 BC.

Proclus introduces Euclid only briefly in his "Commentary on the Elements". According to Proclus, Euclid supposedly belonged to Plato's "persuasion" and brought together the "Elements", drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato (particularly Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I (c. 367 BC – 282 BC) because he was mentioned by Archimedes. Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before Archimedes wrote his. Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's "Elements", "Euclid replied there is no royal road to geometry." This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.
Euclid died c. 270 BC, presumably in Alexandria. In the only other key reference to Euclid, Pappus of Alexandria (c. 320 AD) briefly mentioned that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought" c. 247–222 BC.

Because the lack of biographical information is unusual for the period (extensive biographies being available for most significant Greek mathematicians several centuries before and after Euclid), some researchers have proposed that Euclid was not a historical personage, and that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara (à la Bourbaki). However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.

Although many of the results in "Elements" originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

There is no mention of Euclid in the earliest remaining copies of the "Elements", and most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the "Elements" was from Proclus, who briefly in his "Commentary on the Elements" ascribes Euclid as its author.

Although best known for its geometric results, the "Elements" also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the "Elements" was long known simply as "geometry", and was considered to be the only geometry possible. Today, however, that system is often referred to as "Euclidean geometry" to distinguish it from other so-called "non-Euclidean geometries" that mathematicians discovered in the 19th century.

The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is a fragment of the second book of the "Elements" of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 AD.

The classic translation of T. L. Heath, reads:

In addition to the "Elements", at least five works of Euclid have survived to the present day. They follow the same logical structure as "Elements", with definitions and proved propositions.

Other works are credibly attributed to Euclid, but have been lost.

The European Space Agency's (ESA) Euclid spacecraft was named in his honor.

Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.

History of geometry

Geometry (from the ; "geo-" "earth", "-metron" "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See Areas of mathematics and Algebraic geometry.)

The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley (see Harappan Mathematics), and ancient Babylonia (see Babylonian mathematics) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B.C. contained the first statements of the theorem; the Egyptians had a correct formula for the volume of a frustum of a square pyramid;

The ancient Egyptians knew that they could approximate the area of a circle as follows:

Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that  is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.

Ahmes knew of the modern 22/7 as an approximation for, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for for computing his hekat volume found in a cylinder.

Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for of 3.111...

The two problems together indicate a range of values for between 3.11 and 3.16.

Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula:
where "a" and "b" are the base and top side lengths of the truncated pyramid and "h" is the height.

The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if "π" is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used "π" as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did.

The Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars.
Early Indian texts (1st millennium BC) on this topic include the "Satapatha Brahmana" and the "Śulba Sūtras".
According to, the "Śulba Sūtras" contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." The diagonal rope (') of an oblong (rectangle) produces both which the flank ("pārśvamāni") and the horizontal (')  produce separately."
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.
They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."

The "Baudhayana Sulba Sutra", the best-known and oldest of the "Sulba Sutras" (dated to the 8th or 7th century BC) contains examples of simple Pythagorean triples, such as: formula_2, formula_3, formula_4, formula_5, and formula_6 as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."

According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written c. 1850 BC "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BC. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:

would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."

In all, three "Sulba Sutras" were composed. The remaining two, the "Manava Sulba Sutra" composed by Manava (fl. 750-650 BC) and the "Apastamba Sulba Sutra", composed by Apastamba (c. 600 BC), contained results similar to the "Baudhayana Sulba Sutra".

For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the "axiomatic method", still in use today.

Thales (635-543 BC) of Miletus (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to Babylon and Egypt. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato (427-347 BC) is a philosopher that is highly esteemed by the Greeks. There is a story that he had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." However, the story is considered to be untrue. Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid (c. 325-265 BC), of Alexandria, probably a student at the Academy founded by Plato, wrote a treatise in 13 books (chapters), titled "The Elements of Geometry", in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt.

"The Elements" began with definitions of terms, fundamental geometric principles (called "axioms" or "postulates"), and general quantitative principles (called "common notions") from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.Concepts, that are now understood as algebra, were expressed geometrically by Euclid, a method referred to as Greek geometric algebra.

Archimedes (287-212 BC), of Syracuse, Sicily, when it was a Greek city-state, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of "Commentary on the First Book of Euclid", was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The great Library of Alexandria was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.

Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the 4th century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.

In the Bakhshali manuscript, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." Aryabhata's "Aryabhatiya" (499) includes the computation of areas and volumes.

Brahmagupta wrote his astronomical work "" in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles ("i.e." triangles with rational sides and rational areas).

Brahmagupta's formula: The area, "A", of a cyclic quadrilateral with sides of lengths "a", "b", "c", "d", respectively, is given by

where "s", the semiperimeter, given by: formula_8

Brahmagupta's Theorem on rational triangles: A triangle with rational sides formula_9 and rational area is of the form:

for some rational numbers formula_11 and formula_12.

The first definitive work (or at least oldest existent) on geometry in China was the "Mo Jing", the Mohist canon of the early philosopher Mozi (470-390 BC). It was compiled years after his death by his followers around the year 330 BC. Although the "Mo Jing" is the oldest existent book on geometry in China, there is the possibility that even older written material existed. However, due to the infamous Burning of the Books in a political maneuver by the Qin Dynasty ruler Qin Shihuang (r. 221-210 BC), multitudes of written literature created before his time were purged. In addition, the "Mo Jing" presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.

The "Mo Jing" described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid's first and third definitions and Plato's 'beginning of a line', the "Mo Jing" stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus, the "Mo Jing" stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. It stated that two lines of equal length will always finish at the same place, while providing definitions for the "comparison of lengths" and for "parallels", along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided definitions for circumference, diameter, and radius, along with the definition of volume.

The Han Dynasty (202 BC-220 AD) period of China witnessed a new flourishing of mathematics. One of the oldest Chinese mathematical texts to present geometric progressions was the "Suàn shù shū" of 186 BC, during the Western Han era. The mathematician, inventor, and astronomer Zhang Heng (78-139 AD) used geometrical formulas to solve mathematical problems. Although rough estimates for pi (π) were given in the "Zhou Li" (compiled in the 2nd century BC), it was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. Zhang Heng approximated pi as 730/232 (or approx 3.1466), although he used another formula of pi in finding a spherical volume, using the square root of 10 (or approx 3.162) instead. Zu Chongzhi (429-500 AD) improved the accuracy of the approximation of pi to between 3.1415926 and 3.1415927, with ⁄ (密率, Milü, detailed approximation) and ⁄ (约率, Yuelü, rough approximation) being the other notable approximation. In comparison to later works, the formula for pi given by the French mathematician Franciscus Vieta (1540-1603) fell halfway between Zu's approximations.

"The Nine Chapters on the Mathematical Art", the title of which first appeared by 179 AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu Hui from the Kingdom of Cao Wei. This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three-dimensional shapes, and included the use of the Pythagorean theorem. The book provided illustrated proof for the Pythagorean theorem, contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon, the circle and square, as well as measurements of heights and distances. The editor Liu Hui listed pi as 3.141014 by using a 192 sided polygon, and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary Wang Fan, a mathematician and astronomer from Eastern Wu, would render pi as 3.1555 by using ⁄. Liu Hui also wrote of mathematical surveying to calculate distance measurements of depth, height, width, and surface area. In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. Furthermore, Liu Hui described Cavalieri's principle on volume, as well as Gaussian elimination. From the "Nine Chapters", it listed the following geometrical formulas that were known by the time of the Former Han Dynasty (202 BCE–9 CE).

Areas for theVolumes for theContinuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen Kuo (1031-1095 CE), Yang Hui (1238-1298) who discovered Pascal's Triangle, Xu Guangqi (1562-1633), and many others.

By the beginning of the 9th century, the "Islamic Golden Age" flourished, the establishment of the House of Wisdom in Baghdad marking a separate tradition of science in the medieval Islamic world, building not only Hellenistic but also on Indian sources.

Although the Islamic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy, and were responsible for the development of algebraic geometry.

Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.

Thābit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.

In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof.

Ibrahim ibn Sinan ibn Thabit (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections.

Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example, Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy.

A 2007 paper in the journal "Science" suggested that girih tiles possessed properties consistent with self-similar fractal quasicrystalline tilings such as the Penrose tilings.

The transmission of the Greek Classics to medieval Europe via the Arabic literature of the 9th to 10th century "Islamic Golden Age" began in the 10th century and culminated in the Latin translations of the 12th century.
A copy of Ptolemy's "Almagest" was brought back to Sicily by Henry Aristippus (d. 1162), as a gift from the Emperor to King William I (r. 1154–1166). An anonymous student at Salerno travelled to Sicily and translated the "Almagest" as well as several works by Euclid from Greek to Latin. Although the Sicilians generally translated directly from the Greek, when Greek texts were not available, they would translate from Arabic. Eugenius of Palermo (d. 1202) translated Ptolemy's "Optics" into Latin, drawing on his knowledge of all three languages in the task.
The rigorous deductive methods of geometry found in Euclid's "Elements of Geometry" were relearned, and further development of geometry in the styles of both Euclid (Euclidean geometry) and Khayyam (algebraic geometry) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

Advances in the treatment of perspective were made in Renaissance art of the 14th to 15th century which went beyond what had been achieved in antiquity.
In Renaissance architecture of the "Quattrocento", concepts of architectural order were explored and rules were formulated. A prime example of is the Basilica di San Lorenzo in Florence by Filippo Brunelleschi (1377–1446).

In c. 1413 Filippo Brunelleschi demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror.
Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings, notably Masolino da Panicale and Donatello. Melozzo da Forlì first used the technique of upward foreshortening (in Rome, Loreto, Forlì and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several.

As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli), but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote "De pictura" (1435/1436), a treatise on proper methods of showing distance in painting based on Euclidean geometry. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen's "Optics'.

Piero della Francesca elaborated on Della Pittura in his "De Prospectiva Pingendi" in the 1470s. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective.

Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, was unable to create a consistent structure for the converging lines in paintings, as in London's The Arnolfini Portrait, because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world.
The culmination of these Renaissance traditions finds its ultimate synthesis in the research of the architect, geometer, and optician Girard Desargues on perspective, optics and projective geometry.

The "Vitruvian Man" by Leonardo da Vinci(c. 1490) depicts a man in two superimposed positions with his arms and legs apart and inscribed in a circle and square. The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise "De Architectura".

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Hellenistic geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788–1867).

In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The very old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of non-Euclidean geometry. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity.

It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.

While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms, now known as Hilbert's axioms, were given by David Hilbert in 1894 in his dissertation "Grundlagen der Geometrie" ("Foundations of Geometry"). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

Developments in algebraic geometry included the study of curves and surfaces over finite fields as demonstrated by the works of among others André Weil, Alexander Grothendieck, and Jean-Pierre Serre as well as over the real or complex numbers. Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.