The school sets an example for Sitara; the Sitaras set an example for the Pappus.
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Those currents help keep the whole seed structure afloat by increasing the drag on the falling pappus.
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That happens because of the way air currents interact as they flow among the filaments of the pappus.
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The pappus seemed to be acting like a balloon or a parachute, keeping the seed from succumbing to gravity.
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The group tested their ideas with simplified artificial models, etched silicon discs that were porous like the bristly pappus.
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Dr. Cummins said the filaments make the pappus four times as efficient at staying afloat as a simple flat disc.
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The vortex also contributes to the support of the pappus, because it forms a low-pressure area so that air rises.
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This tuft, called a pappus, is made up of a sparse thicket of filaments, or bristles, that look something like the sprouting hair on the head of the Chinese crested dog.
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Above the pappus, the air flow took the form of something called a separated vortex ring, a kind of swirling eddy that had been considered a theoretical possibility, but was thought to be too unstable to exist in reality.
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A Pappus chain In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD.
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Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem. He also gives his name to the Pappus chain and to the Pappus configuration and Pappus graph arising from his hexagon theorem.
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Soon afterwards, the Romans arrested and imprisoned Rabbi Akiva, and the Romans arrested Pappus ben Judah and imprisoned him next to Rabbi Akiva. Rabbi Akiva asked Pappus who had brought him there. Pappus replied that Rabbi Akiva was happy that he had been seized for occupying himself with the Torah.
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The fruit is a hairless, speckled, four- angled achene about 3 millimeters long. There is usually no pappus, but some achenes have vestigial pappus structures on their flat tops.
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Pappus configuration In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point..
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D is the harmonic conjugate of C with respect to A and B, so that the cross- ratio equals −1. Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII. Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.
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Only little is known of his life. The mathematician Pappus of Alexandria refers to him as Aristaeus the Elder. Pappus gave Aristaeus great credit for a work entitled Five Books concerning Solid Loci which was used by Pappus but has been lost. He may have also authored the book Concerning the Comparison of Five Regular Solids.
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In 1917, a mysterious beggar gifts Lamar Jimmerson the Codex Pappus, a handwritten book allegedly containing the secrets of Atlantis. The beggar initiates Lamar into the Gnomon Society, tells him of their current Master of Gnomons—Pletho Pappus. In 1919, Lamar travels to Malta, in search of Pappus. Instead, he meets Sydney Hen and shares the Codex with him.
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The fruit is a hairy achene with a long pappus.
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The fruit is a hairy achene with a long pappus.
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The fruit is a black achene with a small pappus.
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The fruit is a small ribbed achene without a pappus.
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The pappus is absent or consists of a minute fringe.
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Cuspidia cernua has bristle-like pappus on top of the fruitlets and the fruiting head remains intact, while both Didelta-species have chaffy pappus and the fruiting head breaks into several triangular segments when ripe.
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Female heads have longer phyllaries and multiple layers of pappus hairs.
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Seeds have a pappus which lets them float over long distances.
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The fruit is an achene tipped with a pappus of scales.
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The fruit is a hairy achene with a bushy tawny pappus.
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The fruit is an achene with a whitish pappus of bristles.
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The fruit is an achene with a short scaly brown pappus.
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The fruit is an achene over a centimeter long including the pappus.
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The fruits have 0 to many pappus bristles, fused at the base.
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The fruit is a white achene with a pappus of white bristles.
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The fruit is an achene about long including its pappus of scales.
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The fruit is an achene with pappus of five distinct whitish points.
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The fruit is an achene over a centimeter long, including its pappus.
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Pappus tells us about the conchoid of Nicomedes in his Mathematical collection.
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Pappus of Cirsium arvense The pappus is the modified calyx, the part of an individual floret, that surrounds the base of the corolla tube in flower heads of the plant family Asteraceae. The term is sometimes used in other plant families such as Asclepiadaceae (milkweeds), whose seeds have a similar structure attached, although it is not related to the calyx of the flower. The Asteraceae pappus may be composed of bristles (sometimes feathery), awns, scales, or may be absent. In some species, the pappus is too small to see without magnification.
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The ray floret yields a fruit with a plumelike pappus, and the fruit from a disc floret has a more "bristle-like contorted pappus". ; SpeciesFlann, C (ed) 2009+ Global Compositae Checklist Barnadesia species records. Bolivia Checklist. eFloras.Urtubey, E. (1999).
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The fruit is a black achene a few millimeters long with no pappus.
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The fruit is a hairy, ridged achene with a pappus of long bristles.
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The fruit is an achene with a whitish or brownish pappus of bristles.
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The fruit is an achene about a centimeter long, including its tiny pappus.
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The center is filled with yellow disc florets with purple anthers. The fruits are small achenes of two types. The ray florets yield hairy, curved fruits with no pappus, while the disc florets yield fruits with a long, hairlike pappus.
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The Pappus graph The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges., p. 28. The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra.
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He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived. Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work. The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415).
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The fruit is a hairless achene tipped with a pappus of long, white bristles.
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The achenes are hairy with reddish pappus hairs. It flowers from July until September.
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The fruit is a hairy achene 2–3 mm long tipped with a pappus.
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In species such as Dandelion or Eupatorium, feathery bristles of the pappus function as a "parachute" which enables the seed to be carried by the wind. The name derives from the Ancient Greek word pappos, Latin pappus, meaning "old man", so used for a plant (assumed to be an Erigeron species) having bristles and also for the woolly, hairy seed of certain plants. The pappus of the dandelion plays a vital role in the wind- aided dispersal of its seeds. By creating a separated vortex ring in its wake, the flight of the pappus is stabilized and more lift and drag are produced.
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The whitish fluffy head of a dandelion, commonly blown on by children, is made of the pappus, with tiny seeds attached at the ends, whereby the pappus provides a parachute like structure to help the seed be carried away in the wind. Ligulate floret: A = ovary, B = pappus, C = anthers, D = ligule, E = style with stigmas Disc floret: A = ovary, B = pappus, C = anthers, D = style with stigmas A ray flower is a 3-tipped (3-lobed), strap- shaped, individual flower in the head of some members of the family Asteraceae. Sometimes a ray flower is 2-tipped (2-lobed).
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The fruit is an achene; fruits from the disc florets generally have a white pappus.
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These are topped by a pappus of winged scales that have merged at their base.
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Achenes about long, and not hairy; pappus to long. It grows easily from stem cuttings.
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The fruit is an achene tipped with a spreading cluster of long, white pappus bristles.
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Several generalizations of Steiner chains exist, most notably Soddy's hexlet and Pappus chains.Ogilvy, p. 60.
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The fruit is an achene; fruits on the disc florets often have a long pappus.
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The fruit is an achene; fruits on the disc florets have a pappus of bristles.
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Flora of North America The fruit is an achene with a body only about a millimeter long attached to a soft pappus up to 7 millimeters long. The pappus catches the wind, which disperses the seed. The plant also reproduces vegetatively via its creeping stolons.
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The fruit is an achene; fruits on the disc florets often have a white bristly pappus.
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Fruits are tiny, flattened achenes with a ring of pappus bristles, falling off as a unit.
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The fruit is a grooved achene tipped with a spreading cluster of long, plumelike pappus bristles.
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The fruit is an achene; fruits on the disc florets often have a white bristly pappus.
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This construction is due to Pappus of Alexandria (circa 300 A.D.) and the proof comes from .
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This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon AbCaBc. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points X, Y, and Z, called the Pappus line.
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Hypatia has often been called the first woman to have contributed to mathematics, but Pappus died before the earliest suggested birth date of Hypatia. Therefore, Pandrosion is a likely candidate for an earlier female contributor to mathematics than Hypatia. Pandrosion was also described by Pappus as a teacher of mathematics, and although all of Pandrosion's students that were recorded by Pappus were male, Edward J. Watts suggests that Hypatia may have known of, or even known, Pandrosion.
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The fruit is an achene; the fruits of the disc florets sometimes have a pappus of scales.
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The fruit is an achene which may be well over a centimeter in length including its pappus.
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The fruit is an achene with a hairy body half a centimeter long and a light- colored pappus.
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The fruit is a stout, hairy achene which may be over a long including the long, spiky pappus.
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The fruit is a long, narrow achene which may be 2 centimeters in length including its plumelike pappus.
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Brown knapweed (Centaurea jacea) is different in having pale brown bract appendages, no pappus. Flowers August until September.
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The pappus of the dandelion has been studied and reproduced for a variety of applications. It has the ability to retain about 100 times its weight in water and pappus-inspired mechanisms have been proposed and fabricated which would allow highly efficient and specialized liquid transport. Another application of the pappus is in the use of minute airflow detection around walls which is important for measuring small fluctuations in airflow in neonatal incubators or to measure low velocity airflow in heating and ventilation systems.
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The Pappus configuration, augmented with an additional line (the vertical one in the center of the figure), solves the orchard-planting problem. A variant of the Pappus configuration provides a solution to the orchard-planting problem, the problem of finding sets of points that have the largest possible number of lines through three points. The nine points of the Pappus configuration form only nine three-point lines. However, they can be arranged so that there is another three-point line, making a total of ten.
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The name "Syntrichopappus" derives from a Greek name: "syn" = "joined together", "tricho" = "hair", of the "pappus", which means many bristles fused at the base (however some species have no pappus).Jepson Desert Manual: Syntrichopappus, Dale E. Johnson, 2002 Ed., p 184 The common name "xerasid" derives from Greek, meaning "son of dryness".
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On top of the cypselas there may be two to five stiff scabrous bristles, which are equivalent to sepals (and are called pappus). Also, on top of the cypsela and within the pappus is a yellow, orange or white strap-like corolla which ends in five teeth, together comprising a ligulate floret.
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The fruit is an achene with a brownish pappus. "Mollis" means "soft", referring to the soft hairs on the leaves.
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The fruit is an achene; fruits from the disc florets are coated in white hairs and have a white pappus.
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Aldama is characterized by having a perennial herbaceous habit, a pappus usually of awns and scales, and a multiseriate involucre.
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The fruit is a hairy achene which may be up to a centimeter long, including its pappus of long bristles.
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The fruit is a very narrow achene which may exceed one centimeter in length including its pappus of plumelike bristles.
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N. N. Tzvelev & Andrey Aleksandrovich Fedorov, Flora of Russia, Achenes fusiform, transversely rugose, with five broad, rounded ribs. Pappus white.
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The fruit is up to half a centimeter (0.2 inches) long including its pappus. Blooming occurs in September through November.
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The fruit is an achene; fruits on the disc florets often have a thick pappus of white or brown bristles.
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Fruits: Achenes can vary between and in length, are smooth and bear a pappus of to long with white hairs.
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Mathematicae collectiones, 1660 The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form: the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus: Χωρογραφία οἰκουμενική (Chorographia oikoumenike or Description of the Inhabited World), commentary on the four books of Ptolemy's Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ (The Rivers in Libya), and Ὀνειροκριτικά (The Interpretation of Dreams). Pappus himself mentions another commentary of his own on the Ἀνάλημμα (Analemma) of Diodorus of Alexandria. Pappus also wrote commentaries on Euclid's Elements (of which fragments are preserved in Proclus and the Scholia, while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy's Ἁρμονικά (Harmonika).
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The arbelos is the shaded region (grey). Archimedes first introduced the arbelos in proposition four of his book: The figure is used in propositions four through eight. In propositions five, Archimedes introduces the Archimedes's twin circles, and in proposition eight, he makes use what would be the Pappus chain, formally introduced by Pappus of Alexandria.
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Each contains yellow disc florets and ray florets. The fruit is an achene up to a centimeter long including its pappus.
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The disc florets are yellow with purple anthers. The fruit is a hairy achene with a pappus of many white scales.
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Crepis sancta makes both light seeds with pappus as well as heavier seeds without pappus. In the city the plants were found to make more heavy seeds in comparison to the plants in nonurban areas.Cheptou, P., Carrue, O., Rouifed, S., & Cantarel, A. (2008). Rapid evolution of seed dispersal in an urban environment in the weed Crepis sancta.
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This is one of the equivalent forms of Pappus's (hexagon) theorem. When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the Pappus configuration. The Reye configuration is formed by four quadruply perspective tetrahedra in an analogous way to the Pappus configuration.
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Top view of the flower The plant bears daisylike yellow-centered white or yellow flowers with three-toothed ray florets. The leaves are toothed and generally arrowhead-shaped. Its fruit is a hard achene covered with stiff hairs and having a feathery, plumelike white pappus at one end. Calyx is represented by scales or reduced to pappus.
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The fruit is about half a centimeter (0.2 inches) long including the pappus of bristles at the tip. Blooming occurs in August.
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It contains several hairy yellow disc florets. The fruit is an achene tipped with a large pappus of over 100 long, fine bristles.
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The disc florets at the center are yellow. The fruit is a club-shaped achene about 3 millimeter long with a small pappus.
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A different date is given by a marginal note to a late 10th- century manuscript (a copy of a chronological table by the same Theon), which states, next to an entry on Emperor Diocletian (reigned 284–305), that "at that time wrote Pappus". However, a real date comes from the dating of a solar eclipse mentioned by Pappus himself, when in his commentary on the Almagest he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi in 1068 after Nabonassar". This works out as 18 October 320, and so Pappus must have been writing around 320.
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Pappus of the dandelion which produces a separated vortex ring in order to stabilize flight. There has been research and experiments on the existence of separated vortex rings (SVR) such as those formed in the wake of the pappus of a dandelion. This special type of vortex ring effectively stabilizes the seed as it travels through the air and increases the lift generated by the seed. Compared to a standard vortex ring, which is propelled downstream, the axially symettric SVR remains attached to the pappus for the duration of its flight and uses drag to enhance the travel.
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They are distinguished technically by the fact that the ray florets (when present) are sterile, and by the presence on the disk flowers of a pappus that is of two awn-like scales that are caducous (that is, easily detached and falling at maturity). Some species also have additional shorter scales in the pappus, and one species lacks a pappus entirely. Another technical feature that distinguishes the genus more reliably, but requires a microscope to see, is the presence of a prominent, multicellular appendage at the apex of the style. Further, the florets of a sunflower are arranged in a natural spiral.
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Tropicos, Pappobolus S.F. Blake Pappobolus is distinguished from closely related genera by its combination of shrubby habit and usually caducous pappus. Within the genus there is considerable variation in the pappus, a structure that has traditionally been considered a key to defining genera in Asteraceae, and this led to earlier confusion in defining the genus. Most of the species of the genus were originally described as members of Helianthus, based on having a pappus of two caducous awns. When it was recognized that they were not part of the exclusively North American Helianthus, they were transferred to a genus called Helianthopsis.
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In this section the long hair is often interspersed with short scales. F. bechuanica and F. dentata (section Felicia) also have conspicuous short scales. Pappus with long barbes nearer to the base and shorter teeth nearer to the top occur in F. tenella and the section Longistylus. A glabrous base and teeth higher up the pappus occurs in the sections Dracontium and Neodetris.
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Flowers are produced in summer, as late as September. The fruit is an achene a few millimeters long tipped with a pappus of bristles.
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Flowers from May until July. The achenes are smooth ribbed, beakless, with similar pappus to Tragopogon pratensis. It exudes a milky juice from its stem.
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This is its first printed publication. Book VIII was lost before the scholars of Almamon could take a hand at preserving it. Halley's concoction, based on expectations developed in Book VII, and the lemmas of Pappus, is given in Latin. The commentary of Eutocius, the lemmas of Pappus, and two related treatises by Serenus are included as a guide to the interpretation of the Conics.
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The inflorescence is 2.5–5 cm diameter, pink-purple, with all the florets of similar form (no division into disc and ray florets). The seeds are 5 mm long, with a downy pappus, which assists in wind dispersal. As in other species of Cirsium (but unlike species in the related genus Carduus), the pappus hairs are feathery with fine side hairs.Blamey, M. & Grey-Wilson, C. (1989).
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Blooming occurs in November and December. The fruit is an achene two millimeters long tipped with a tiny pappus of bristles.Chrysopsis floridana. Flora of North America.
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On the top of the cypselae is a whorl of crown-like or free pappus bristles. C. congestum has eight homologous sets of chromosomes (2n=16).
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Each head is less than long and filled with bright pinkish-purple or magenta flowers. The fruit is a tiny achene tipped with a bristly pappus.
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At the tip of the body is a large pappus made up of several long, barbed, bristly scales each up to a centimeter in length themselves.
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The flowers are pollinated by native bees.Jepson Manual Treatment of Agnorhiza reticulata The fruit is an achene a few millimeters long which usually lacks a pappus.
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It contains several hairy yellow to reddish disc florets. The fruit is an achene covered densely in long hairs and tipped with a large pappus of bristles.
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The head is discoid, containing only yellow disc florets, and no ray florets. The fruit is an achene about 7 millimeters long, not counting its white pappus.
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The inflorescence is a solitary flower head with yellow to reddish disc florets. The fruit is a ribbed cypsela about one millimeter long with a large pappus.
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The head contains many white to pink-tinted flowers with large, protruding, darker colored anthers. The fruit is an achene over a centimeter long including its pappus.
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Centaurea solstitialis is an annual herb from the family Asteraceae. During the vegetative stage it forms a rosette of non-spiny leaves (5–20 cm diameter). As the summer approaches, it produces a flowering stem (1 m) which will produce numerous spinous capitula containing numerous (10-50) yellow flowers. Flowers within capitula are pollinated by insects and each capitula will produce a mix of (10-50) pappus and non-pappus seeds.
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Helmut Tiefenthaler, Die Wanderregion Klostertal im Entwicklungsrückblick, Bludenzer Geschichtsblätter 98 (2011)p. 11 In much literature the first ascent is ascribed to David Pappus and companions on 25 July 1610 during his exploration of the borders of the Sonnenberg and Bludenz Counties. Pappus however did not mention this in his detailed account. The "Rote Wand" that he did describe climbing is the 2,105 m high Garsellakopf on the border with Liechtenstein.
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The quadratrix is mentioned in the works of Proclus (412–485), Pappus of Alexandria (3rd and 4th centuries) and Iamblichus (c. 240 – c. 325). Proclus names Hippias as the inventor of a curve called quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappus only mentions how a curve named quadratrix was used by Dinostratus, Nicomedes and others to square the circle.
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The head is lined with hairy purple or purple- tipped phyllaries. Blooming occurs in July and August. The head is an achene with a pappus of bristles.Erigeron lanatus.
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The flower head contains many golden yellow ray florets, the outer ones usually darker in color. The fruit is an achene with a plumelike pappus of white bristles.
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The fruit is up to a centimeter (0.4 inches) long including its long pappus. They are wind- dispersed.Ericameria paniculata, Mojave or punctate rabbitbrush Flora of North America.Chrysothamnus paniculatus.
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If there are ray florets they are less than a millimeter long. The fruit is a hairy achene a few millimeters long, sometimes with a pappus of tiny scales.
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The head contains several yellow disc florets. The fruit is a ribbed achene tipped with a pappus of fringelike scales, the fruit around half a centimeter long in total.
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The tips of the outer phyllaries curve outward. The fruit is a hairy, ribbed achene a few millimeters long with a pappus of bristles and scales on its tip.
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It spreads at the top with several yellow ray florets a few millimeters long and black-tipped disc florets. The fruit is a shiny black achene with no pappus.
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Florets along the edges of the head produce fruits. The fruit body is a millimeter in length with a white pappus about 5 millimeters long.Pluchea indica. Flora of China.
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In members of the Asteraceae the fruit is achene-like, and is called a cypsela (plural cypselae). Although there are two fused carpels, there is only one locule, and only one seed per fruit is formed. It may sometimes be winged or spiny because the pappus, which is derived from calyx tissue often remains on the fruit (for example in dandelion). In some species, however, the pappus falls off (for example in Helianthus).
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Achenes oblong 3 mm, pappus of 1 row of plumose, dense, appressed, caduceus, shiny brown hairs 2- to 3- branched and united at the base into clusters. Fl. V-VIII.
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The fruit is an achene tipped with a cluster of pappus bristles which are not plumelike as are those of the Stephanomeria species with which this plant was once classified.
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Each triangle can be extraverted in three different ways; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the Pappus graph..
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The head is discoid, containing only yellow disc florets, and no ray florets. The fruit is an achene about 5 or 6 millimeters long, not counting its white to brownish pappus.
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The flower head has a center of glandular yellow disc florets and a fringe of yellow ray florets. The fruit is an achene a few millimeters long with a white pappus.
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The flower head has 5 to 30 florets in shades of blue or purple, or occasionally white or yellow. The achene is ribbed and has a pappus of bristles and hairs.
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The head is enclosed in a layer of phyllaries which are glandular and sticky. The fruit is a ribbed achene with a pappus a few millimeters long. Close-up of flowers.
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It is dioecious, with male plants bearing heads of staminate flowers and female plants bearing heads of larger pistillate flowers. The fruit is an achene with a long, soft, barbed pappus.
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The one-seeded indehiscent fruits called cypselas are five-ribbed, with five pappus scales on top. Cypselas from subterranean capitula are larger (4–5 mm long) and heavier, more or less ovate in shape as seen from the side and with erect, about 3 mm long pappus scales, and come in two distinct types, one about 10 mg, glabrous and whitish in color, that shows quick germination, the other about 7⅓ mg, wrinkled, hairy and brown with delayed germination. Besides these two types of subterranean seeds, there are three types of seeds produced by the aerial flowerheads. Cypselas from the aerial flowerheads germinate quickly under favorable conditions, are smaller (about 3 mm long), lighter (about 3⅓ mg), with diverging pappus scales, and come in three distinct types.
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Unable to decipher the corrupted text, he abandoned it. Subsequently, David Gregory (mathematician) restored the Arabic for Henry Aldrich, who gave it again to Halley. Learning Arabic, Halley created De Sectione Rationis and as an added emolument for the reader created a Neo-Latin translation of a version of De Sectione Spatii reconstructed from Pappus Commentary on it. The two Neo-Latin works and Pappus' ancient Greek commentary were bound together in the single volume of 1706.
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The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation. In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin.
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The tip of the ligule is often divided into teeth, each one representing a petal. Some marginal florets may have no petals at all (filiform floret). The calyx of the florets may be absent, but when present is always modified into a pappus of two or more teeth, scales or bristles and this is often involved in the dispersion of the seeds. As with the bracts, the nature of the pappus is an important diagnostic feature.
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The periferous cypselas are about ovate, glabrous, whitish in color and having about 3 mm long pappus scales, subtended by an involucral bract and remain in the flowerhead until it disintegrates. The central cypselas are brown, conical with the narrow end at the base, heavily wrinkled and hairy, with about 1 cm long pappus scales that end in a long bristle. The third type of aerial cypsela is intermediate between periferous and central types, and dark green in color.
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Flora of North America Vol. 21 Page 375 Rayless arnica Arnica discoidea Bentham, Pl. Hartw. 319. 1849. The fruit is an achene about 7 millimeters long, not counting its light-colored pappus.
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The fruit from each floret is a cylindrical achene up to long, not considering the large pappus of up to 50 hairlike white bristles which may be an additional centimeter in length.
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It is enveloped in several pointed phyllaries which are covered in bristly hairs. The fruit is an achene well over a centimeter in length which is tipped with a pappus of bristles.
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Each head contains many yellow disc florets and sometimes one or two ray florets as well. The fruit is an achene a few millimeters long topped with a brown or white pappus.
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The fruit is an achene with a whitish body a few millimeters long. At the tip of the body is a large pappus made up of 5 to 10 long, bristly scales.
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The fruit is a tiny achene with a pappus of bristles.Flora of North America, Erigeron aequifolius H. M. Hall, Univ. Calif. Publ. Bot. 6: 174. 1915. Hall’s fleabane Hall, Harvey Monroe 1915.
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In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles.
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There are no pappus bristles on top of the cypselas. The yellow, strap-like corolla is usually 17–24 mm long, ends in five teeth, and carries some black hairs on the tube.
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Each head contains up to 40 yellow disc florets surrounded by a fringe of up to 20 yellow ray florets. The fruit is an achene up to a centimeter long including its pappus.
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Between the pappus and the achene is a stalk called a beak, which elongates as the fruit matures. The beak breaks off from the achene quite easily, separating the seed from the parachute.
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The fruit is an achene with a brown, hairy to hairless body a few millimeters long. At the tip of the body is a large pappus made up of five long, bristly scales.
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With great pomp and ceremony he was buried in a grave with Karl on 8 June 1601.Benians, p. 709; Johannes Pappus, Angehengtem Programmate publico Rectoris Academiae, Abdankung Genealogia. (undated.) no page number. .
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Each petal may measure over 2 centimeters in length. The flowering period is January to June. The fruit is an achene which may be over a centimeter long, including its pappus of bristles.
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The prominent synantherologist S.F.Blake, however, much earlier named the genus Pappobolus based on species that have a pappus of numerous caducous awns.S. F. Blake. 1916. Pappobolus macranthus. Hooker's Icones Plantarum 31: plate 3057.
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Federico Commandino translated the Collection of Pappus into Latin in 1588. The German classicist and mathematical historian Friedrich Hultsch (1833–1908) published a definitive 3-volume presentation of Commandino's translation with both the Greek and Latin versions (Berlin, 1875–1878). Using Hultsch's work, the Belgian mathematical historian Paul ver Eecke was the first to publish a translation of the Collection into a modern European language; his 2-volume, French translation has the title Pappus d'Alexandrie. La Collection Mathématique (Paris and Bruges, 1933).
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There are many disc florets with a yellow corolla of up to long, in the center of each of these are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a triangular appendage. Around the base of the corolla are many yellowish white pappus bristles of two different lengths. The longer pappus bristles are about long, toothed, with a long pointy tip.
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Paul Guldin (original name Habakkuk Guldin; 12 June 1577 (Mels) – 3 November 1643 (Graz)) was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. (This theorem is also known as the Pappus–Guldinus theorem and Pappus's centroid theorem, attributed to Pappus of Alexandria.) Guldin was noted for his association with the German mathematician and astronomer Johannes Kepler. Guldin composed a critique of Cavalieri's method of Indivisibles.
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a narrowly triangular appendage. Around the base of the corolla are numerous yellowish white, toothed, persistent pappus bristles, which are all of the same length, up to about . Very rarely with a few short (0.2 to 0.3 mm long) basal scales represent short pappus.
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The head contains 27 to 38 yellow ray florets and many yellow disc florets. The fruit is a cypsela which can be almost long and is tipped with a pappus of two short awns.
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They are up to a centimeter long or longer. There are few to many whitish or purplish disc florets with purple anthers. The fruit is a ribbed cypsela, sometimes with a pappus of scales.
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It requires wet soil, and is sensitive to hydrological changes. The seed is a sticky achene without the large pappus which would suggest wind dispersal, so it is presumed to be transmitted by animals.
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The fruit is an achene which may be up to a centimeter (0.4 inches) long, including its pappus of scales. There are three varieties of this plant. The rare var. winkleri (Winkler's blanketflowerGaillardia aestivalis.
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The fruit is an achene with a gray or brown, sometimes speckled body a few millimeters long. At the tip of the body is a large pappus made up of five long, bristly scales.
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The fruit is an achene with a brown to nearly black body a few millimeters long. At the tip of the body is a large pappus made up of five long, flat, barbed scales.
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The inflorescence is an open array of flower heads with a fringe of violet ray florets around a center of yellow disc florets. The fruit is a hairy achene with a long white pappus.
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Another method included in the same section, and attributable in the same way indirectly to Pandrosion, is a correct and exact method for constructing the geometric mean, simpler than the method used by Pappus.
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Webb's An Irish Flora Cork University Press The achene is purple black, without bristles at the tip. The pappus is the same as Lactuca serriola. In the northern hemisphere, it flowers from July until September.
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It flowers from July until September in the northern hemisphere. The achenes are grey, tipped with bristles. The pappus is white with equal length hairs. Similar to Mycelis muralis but showing more than 5 florets.
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Enneapogon nigricans, known by the common names blackheads, bottle washers, pappus grass, purpletop grass, and niggerheads,Enneapogon nigricans. USDA PLANTS. is a perennial Australian grass. Distinctive lance-shaped seedheads appear in late spring and summer.
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Plants only emerge from seeds near the surface, however, some seed can germinate (4%) while buried deep (4 cm). The seed carries a pappus of hairs, indicating the use of wind as a dispersal agent.
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The fruit is an achene; achenes from the disc florets may have a pappus of scales.Flora of North America, Calycadenia fremontii A. Gray in W. H. Emory, Rep. U.S. Mex. Bound. 2(1): 100. 1859.
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The fruit is an achene; those developing from the disc florets have a pappus of scales.Flora of North America, Calycadenia pauciflora A. Gray in W. H. Emory, Rep. U.S. Mex. Bound. 2(1): 100. 1859.
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The pits in the common base of the florets (called the receptacle) are each circled by a row of long hairs. The fruits (or cypselas) are 4–5 mm migh and topped by the changed calyx called pappus. On the outer cypselas the pappus takes the shape of trifid scales, the middle branch of which is prolonged as a rigid hair. On the inner cypselas, there is a row of rigid hairs encircling a row of lanceolate scales that change at the tip into a rigid hair.
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Dirk Struik (1953) Lectures in Analytic and Projective Geometry, page 19, Addison-Wesley In Cambridge, England, John J. Milne gave readers the benefit of his reading of Pappus. In 1985, Alexander Jones wrote his thesis at Brown University on the subject. A revised form of his translation and commentary was published by Springer-Verlag the following year. Jones succeeds in showing how Pappus manipulated the complete quadrangle, used the relation of projective harmonic conjugates, and displayed an awareness of cross-ratios of points and lines.
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He also breaks with the Greek tradition of associating powers with geometric referents, with an area, with a volume and so on, and treats them all as possible lengths of line segments. These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass. The bulk of the remainder of this book is occupied by Descartes's solution to "the locus problems of Pappus."Pappus discussed the problems in his commentary on the Conics of Apollonius.
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Pandrosion is credited with developing a method for calculating numerically accurate but approximate solutions to the problem of doubling the cube, or more generally of calculating cube roots. It is a "recursive geometric" solution, but three- dimensional rather than working within the plane. Pappus criticized this work as lacking a proper mathematical proof. Although Pappus does not directly state that the method is Pandrosion's, he includes it in a section of his Collection dedicated to correcting what he perceives as errors in Pandrosion's students.
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These anthers are about long, and have triangular appendages at their tip of about long. The style of the disc florets is about long and the two line-shaped branches in which it splits are long and have a delta-shaped appendage at both tips. Surrounding the base of the corollas of both ray and disc florets are two whorls of white to straw-coloured pappus bristles. The outer whorl consists free barbed bristles of long, alternating with the pappus of the inner whorl.
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The fruit of Vincetoxicum nigrum is a slender, tapered follicle that ranges in color from green through light brown and is tightly packed with seeds, each bearing a fluffy pappus to allow distribution by the wind.
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The name is derived from the Latin word pecten, meaning "comb." It refers to the marginally-bristled leaves or the pappus form. These plants vary in appearance but they usually bear yellow daisy-like flower heads.
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The large clusters of scented flowers appear in spring, mauve to lilac to magenta-blue in color. The clusters have a bursting fireworks appearance. The seed has a fluffy pappus and is easily dispersed by wind.
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The mature plants produce many achene, although most seeds fall within a few metres of the parent plant. This is because the plant grows a very small pappus, which makes wind-borne seed distribution very inefficient.
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The head is lined with sticky, spiny phyllaries and packed with white to lavender flowers. The fruit is an achene with a thick body a few millimeters long and a pappus about 1.5 centimeters in length.
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Involucral bracts are canescent and covered with cobweb-like hairs, each bract ends with a single spine. The fruit is a smooth rotund achene with lateral hilum measuring long and wide surmounted by a white pappus.
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The inflorescence is a cluster of fuzzy flower heads under a centimeter long containing long, protruding white disc florets and no ray florets. The fruit is an achene a few millimeters long with a rough bristly pappus.
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Alexandria, being the center of the Hellenistic world, produced a number of great mathematicians, astronomers, and scientists such as Ctesibius, Pappus, and Diophantus. It also attracted scholars from all over the Mediterranean such as Eratosthenes of Cyrene.
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The inflorescence produces hairy, glandular flower heads filled with yellow disc florets and a fringe of up to 125 thin, flat white to purple-tinged ray florets. The fruit is an achene with a pappus of bristles.
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The inflorescence is a solitary flat-topped woolly flower head containing many yellow disc florets. There occasionally appears a yellow ray floret, but they are usually absent. The fruit is an achene with a pappus of bristles.
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The head is discoid, containing no ray florets but many funnel-shaped disc florets with lobes that resemble ray florets. The disc florets are yellow with brown throats. The fruit is an achene with a whitish pappus.
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It flowers throughout the summer and early fall. The achenes are 2–3 mm long, four-angled, with faint lengthwise stripes. The fluffy pappus bristles are 11–13 mm long. After flowering and setting seed, it dies.
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Pappus mentions other treatises of Apollonius: # Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio") # Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area") # Διωρισμένη τομή, De Sectione Determinata ("Determinate Section") # Ἐπαφαί, De Tactionibus ("Tangencies") # Νεύσεις, De Inclinationibus ("Inclinations") # Τόποι ἐπίπεδοι, De Locis Planis ("Plane Loci"). Each of these was divided into two books, and—with the Data, the Porisms, and Surface-Loci of Euclid and the Conics of Apollonius—were, according to Pappus, included in the body of the ancient analysis. Descriptions follow of the six works mentioned above.
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As in all Asteraceae, the fruit does not open and only one of the compartments contains a single seed. The top of this so-called cypsela is ornamented with a single row of around thirty hair-shaped, long pappus bristles, but only approximately ten bristles in F. bergerana and six to eight in F. annectens. An individual bristle hair consists of many cells and with few to many teeth along its margins. Pappus is particularly long in the section Lignofelicia, such as in F. comptonii, F. heterophylla, F. ovata, F. venusta and F. welwitschii.
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These encircle numerous disc florets, which are partly sterile, partly fertile, and have an up to long yellow corolla. In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a triangular appendage. Pappus is missing at the ray florets, but surrounding the base of the corolla of the disc florets are six to eight, up to 2 mm (0.1 in) long, quickly shunted pappus bristles.
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In his surviving writings, Pappus does not indicate the authors' date whose works he uses or the time (but see below) at which he himself wrote. If no other date information were available, all that could be known would be that he was later than Ptolemy (died c. 168 AD), whom he quotes, and earlier than Proclus (born ), who quotes him. The 10th century Suda states that Pappus was of the same age as Theon of Alexandria, who was active in the reign of Emperor Theodosius I (372–395).
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Since Michel Chasles cited this book of Pappus in his history of geometric methods,Michel Chasles (1837) Aperçu historique sur l'origine et le développement des méthodes en géométrie, especially page 302; see also pages 12, 78, and 518. it has become the object of considerable attention. The preface of Book VII explains the terms analysis and synthesis and the distinction between theorem and problem. Pappus then enumerates works of Euclid, Apollonius, Aristaeus and Eratosthenes, thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation.
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Abulfaraj, in Münter, "Der Jüdische Krieg," p. 18, Altona and Leipsic, 1821 Marcius Turbo pursued him and sentenced to death the brothers Julian and Pappus, who had been key leaders in the rebellion. Lusius Quietus, the conqueror of the Jews of Mesopotamia, was now in command of the Roman army in Judea, and laid siege to Lydda, where the rebel Jews had gathered under the leadership of Julian and Pappus. The distress became so great that the patriarch Rabban Gamaliel II, who was shut up there and died soon afterwards, permitted fasting even on Ḥanukkah.
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The heads contain yellow disc florets, and some species have yellow ray florets. The fruit has a pappus with an outer row of wide, membranous scales and an inner row of longer, narrower scales. ; SpeciesCalostephane. The Plant List.
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The fruit is an achene which may be well over a centimeter in length including its pappus. There are three varieties of this species; var. subsquarrosa is an uncommon type known only from southwestern Montana and northwestern Wyoming.
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The inflorescence is generally a cluster of glandular flower heads with black- tipped yellow disc florets and sometimes one or more tiny greenish or purplish yellow ray florets. The fruit is a flat black achene with no pappus.
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The inflorescence is a solitary sunflower-like flower head or cyme of several heads. The flower head has several yellow ray florets measuring 6 millimeters to over a centimeter long. The fruit is an achene tipped with a pappus.
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It is very similar to the more widespread C. parryi but has leaves that are pubescent on both surfaces, and a distinctive pappus attached to the achenes, long narrow bristles instead of the short scales characteristic of C. parryi.
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The Nature Conservancy. 2012. This plant is distinguished from other members of the tribe Liabeae by its reddish disc florets and its plumelike pappus of bristles.Gutiérrez, D. G. (2010). Inkaliabum, a new Andean genus of Liabeae (Asteraceae) from Peru.
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It yields one large fruit, a cypsela up to 1.6 centimeters long including its barrel-shaped body and its long, spreading pappus of brown or black bristles.Crupina vulgaris. Flora of North America. It can weigh up to 36 milligrams.
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The head is enclosed in a layer of phyllaries which are glandular and sticky. The fruit is a ribbed achene with a pappus or long.Flora of North America, Baccharis plummerae A. Gray, Proc. Amer. Acad. Arts. 15: 48. 1879.
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There is a pappus present that forms a minute crown on the body of the achene. The plant contains terpenes which make it quite aromatic. Many people regard the species to have a pleasant smell.Native Salvias and Artemisias MyMotherLode.
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In contrast to F. venusta, the leaves of F. oleosa overlap, point upwards, and contain translucent resin or oil ducts. The involucral bracts are narrowly lance-shaped and also contain ducts. The pappus consists of bristles with short teeth.
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The disc florets are bright yellow. One plant can produce many stems which mat together due to their spininess and form a small thicket. The fruit is an achene about half a centimeter long with many rigid pappus scales.
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The outer phyllaries may be very long and leaflike. It contains tubular or funnel-shaped disc florets in shades of yellow or red. The fruit is a hairy cypsela with a plumelike pappus made up of tufts of bristles.
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It may be missing from history because it was never in history, Apollonius having died before its completion. Pappus of Alexandria, however, provided lemmas for it, so at the very least some edition of it must once have been in circulation.
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There are also yellow ray florets, but they are so short they may be nearly invisible inside the involucre of phyllaries. The fruit is a hairy achene a few millimeters long which is linear in shape with a pappus of scales.
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The filaments each have a distinct bulbous collar just below the anthers. The pappus of the disc florets is persistent, and the cypselae are narrowly oblong, and have always five wings covered with blunt hairs that become slimy when wet.
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The fruit is a rough-textured, ribbed cypsela with a pappus of many bristles.Garberia heterophylla. Flora of North America. The fruits remain on the plant, and their long, brown pappi make the shrub stand out among other plants during the winter.
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The flower heads are borne in open inflorescences. Each head contains up to about 35 disc florets, usually lavender to dark magenta or pinkish purple, sometimes blue. The fruit is a ribbed, rough-textured cypsela with a pappus of bristles.
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Plants for a Future: Cirsium arvense Bruichladdich distillery on Isle of Islay lists creeping thistle as one of the 22 botanical forages used in their gin, The Botanist. The feathery pappus is also used by the Cherokee to fletch blowgun darts.
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Each head contains many yellowish disc florets and many pistillate florets around the edges. The latter may have minute ray florets. The fruit is an achene a few millimeters long which is tipped with a small pappus of toothed scales.
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The final propositions deal with multiplying together the numerical values of Greek letters in two lines of poetry, producing two huge numbers approximately equal to and .Pappus of Alexandria, trans. into Latin by Friedrich Hultsch. Pappi Alexandrini collections quae supersunt.
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The inflorescence bears one or more flower heads, each up to long. The head is lined with sticky, twisted, spiny phyllaries and contains pink to purple flowers. The fruit is an achene a long topped with a pappus of about centimeters.
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The inflorescence bears flower heads with five bright yellow ray florets, each with three lobes. The center of the head contains six disc florets which are yellow with black anthers. The fruit is a dark brown achene with no pappus.
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Twelve to twenty bright yellow ray florets, tubes long, rays x with four veins. A dull-yellow to brown disc floret, corolla long, all hairless and expanding from the middle. ;Fruits and reproduction: Achenes long, ribbed with no hairs. Pappus long.
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Seeds are ordinarily dispersed intact with the fruiting body, the cypsela. Anemochory (wind dispersal) is common, assisted by a hairy pappus. Epizoochory is another common method, in which the dispersal unit, a single cypsela (e.g. Bidens) or entire capitulum (e.g.
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The phyllaries are coated in knobby yellow resin glands. At the tip of the inflorescence are minute yellowish ray florets each under a millimeter long, and one or two yellow disc florets. The fruit is an achene with no pappus.
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Most of what is known about Hipparchus' text comes from two ancient sources: Ptolemy and Pappus. The work is also mentioned by Theon of Smyrna and others, but their accounts have proven less useful in reconstructing the procedures of Hipparchus.
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The flowering period extends from June through August. The flowers are hermaphrodite and are pollinated by insects (usually bees, wasps and butterflies) (entomogamy). The fruits are achenes long, each with a feathery yellow pappus. Seed dispersal is by wind (anemochory).
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The flower heads are in arrays or clusters. They contain up to 50 long yellow or orange disc florets and no ray florets. The fruit is a veiny cylindrical cypsela with a long pappus of many barbed, white bristles.Cacaliopsis nardosmia.
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They are sometimes unlobed but have toothed margins. The leaves have woolly fibers, especially on the undersides. The flower heads contain long lavender or purplish florets. The fruit is an achene which may exceed 2 centimeters in length including its pappus.
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The arbelos is defined by two circles, CU and CV, which are tangent at the point A and where CU is enclosed by CV. Let the radii of these two circles be denoted as rU and rV, respectively, and let their respective centers be the points U and V. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to CU (the inner circle) and internally tangent to CV (the outer circle). Let the radius, diameter and center point of the nth circle of the Pappus chain be denoted as rn, dn and Pn, respectively.
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The tip of the pappus hair is wider in F. echinata and species in the section Lignofelicia, but mostly ends pointy. In fresh plants, the pappus is mostly white to bone-colored, sometimes yellowish, yellow-brown in F. dentata, although in herbarium specimens stronger colors may develop such as fox-red in F. burkei. Fertilised and fully ripe cypselas may be long, dependend on the species. The color varies from yellow-brown (sections Lignofelicia and Longistylus), red-brown (Anhebecarpaea), dark brown (Dracontium), black (Neodetris), while species of the section Felicia may either be red-brown, dark brown or black.
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The sterile cyselas are about 11 mm long, hairless except for one whorl of pappus hairs of about 5 mm at the tip. The female florets are purplish in color, tube-shaped, densely covered in hard white hairs, with a tube of about 7 mm long and lobes of less than ¼ mm. In the female florets, the stigmas have two lobes, the lobes being exserted inside the corolla tube. The cyselas in the female florets are slender, angular cylindrical, 5–6 mm, set with dense bristles and two whorls of about fifty rough, purple pappus hairs of about mm.
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They are coated in green phyllaries with spines up to a centimeter long. The head contains white, pink, or purplish flowers up to 2 centimeters long. Blooming occurs in June through August. The fruit may be 2 centimeters long including its long pappus.
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Anther bases are sharply sagittate, with oblong tips; the ends of the styles manifest a somewhat swollen node, with a cylindrical superior appendage. The smooth brownish fruits are four to five millimeters in diameter, but distinctly ovoid; many pappus bristles are exhibited.
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It forms a mound just a few centimeters tall. The branches are covered in small, linear leaves. The flower heads each contain 6 or 7 white ray florets and 7 to 9 red disc florets. The fruit has a plumelike white pappus.
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The corollae are typically five-lobed. Style tips may be triangular or round, but typically hair-tufted. The glabrous club-shaped fruits are less than 1.5 millimeters across, black or gray in color, absent a pappus structure. The species is cross-pollinated.
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The fruit is a cylindrical, beaked, ribbed cypsela with a pappus of many white bristles. This genus is closely related to the dandelions of genus Taraxacum. Plants of both genera undergo apomixis, producing fertile seeds via asexual reproduction.Van Dijk, P. J. (2003).
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The heads are 2.5 to 3.5 cm wide. The somewhat rounded head is covered in layers of phyllaries with fringed tips and sometimes spines. The head contains many yellow florets. The fruit, including its pappus, can be well over one centimeter long.
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Involcre is 3 to 5 millimeters long and about 2 millimeters in diameter. Pale to bright yellow ray florets and 4 veined spreading disc florets that turn red brown. ;Fruits: Ribbed achenes 2 millimeters long and without hairs. Pappus 3.5 millimeters long.
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The inflorescence is a flower head with a bell-shaped involucre of woolly-haired phyllaries. There are 12 or 13 yellow ray florets and about 30 disc florets at the center. The fruit is an achene with a pappus of scales.Thymophylla tephroleuca.
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The head is enclosed in a layer of phyllaries and the female flowers yield fruits, each an achene with a white pappus about a centimeter long. The earliest name for the species is Baccharis salicifolia Nutt., coined in 1840.Nuttall, Thomas 1840.
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Inside are many purple, lavender, pink, or white ray florets and a center packed with up to 120 tubular yellow disc florets. The fruit is an achene with a pappus of reddish bristles on top. Corethrogyne filaginifolia; formerly Lessingia filaginifolia var filaginifolia.
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These bisexual florets have obtuse and irregular anther bases. They have pistillate ray florets that can be yellow or white. From these pistillates, they produce achenes, which are indehiscent and angled. The pappus, a modified calyx, is not present or extremely small.
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The inflorescence contains several flower heads, each lined with dark green phyllaries. The head contains many golden yellow disc florets and generally either 8 or 13 yellow ray florets each over a centimeter long. The fruit is an achene tipped with a pappus of bristles.
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The pappus has simple white hairs, the inner longer than the outer. The flower heads are yellow, small with only 4-5 yellow ray florets. wide more or less, on branches 90 degrees to the main stem, in loose panicle.Webb, D.A., Parnell, J. and Doogue.
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The inflorescence is an array of nearly cylindrical flower heads each containing five yellow disc florets but no ray florets. Many flowers are produced after times of stress. Blooming typically occurs in August through October. The fruit is over a centimeter long including its pappus.
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The pappus also has the property of being able to change its morphology in the presence of moisture in various ways that aid germination. The change of shape can adjust the rate of abscission, allowing increased or decreased germination depending on the favorability of conditions.
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The inflorescence bears several large flower heads each up to wide. They are lined with spiny, woolly to cobwebby phyllaries and bear many narrow glandular purple flowers each about long. The fruit is a cylindrical achene long topped with a white pappus in length.
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If taken in the wrong dose it can be very dangerous. The seedlike fruit has a pappus of plumose, white or pale tan bristles. The entire plant has a strong and distinct pine-sage odor when the leaves of mature plants are rubbed or bruised.
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Some species have ray florets in shades of yellow, or white with yellowish bases. Some species lack true ray florets but have flat yellowish disc florets that look like rays. The fruit is a ribbed, glandular cypsela, usually with a pappus on the end.
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The inflorescence is a solitary flower head atop an erect peduncle. The hairy head has several yellow disc florets each around a centimeter long and at the center many yellow disc florets. The fruit is a silky-haired achene tipped with a white pappus.
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The pappus-tipped seeds are dispersed on the wind or on clothing or fur. The plant also reproduces vegetatively by rooting from the nodes on sections of stem. The climbing herbage can become weedy and dense, sometimes covering other vegetation.Moon, M., et al. (1993).
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The leaf blades are sometimes borne on petioles, which may have spines. The flower head contains 2 female disc florets and 2 to 4 male disc florets. The latter are whitish, greenish, or yellowish. The fruit is a rough- edged cypsela with a pappus.
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The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the inversion centered at point A transforms the arbelos circles CU and CV into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle.
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While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth" ().Quoted by Pappus of Alexandria in Synagoge, Book VIII Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.
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The involucral bracts between the individual heads are thin and papery. The pappus consists of a circle of scales around the tip of the cypselas. Flowering usually appears from June to September, rarely extending to December. This species has seven sets of homologue chromosomes (2n=14).
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The involucre is 13 mm length, and 8 mm diameter and has four lines of involucral scale. The achene has about 4 mm length and 1.3 mm diameter and lanceolate shape. There is no pappus. It blooms from June to September; the fruits (achenes) mature in November.
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They are oblong and lobed. On the upper plant the leaves are small, linear in shape, and smooth edged without lobes. Flower heads occur at intervals on the branches and contain 5 to 9 ray florets. The fruit is an achene with a pappus of white bristles.
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The inflorescence is a spike of several flower heads. The heads contain several flowers which are usually purple, but sometimes white. The fruit is an achene tipped with a long pappus of feathery bristles. The plant reproduces sexually by seed and vegetatively by sprouting from its rhizome.
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The heads are numerous terminating the branches. Flowers are pink to purplish, the marginal ones not enlarged. The outer and middle involucral bracts are broad, striate, smooth with broadly rounded tips; the inner bracts are narrower with hairy tips. Pappus present with bristles 6–11 mm long.
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The often nodding flower head contains up to 50 yellow ray florets. The fruit is an achene with a brownish or reddish body a few millimeters long. At the tip of the body is a large pappus made up of 15 to 30 silvery, hairy scales.
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The genus name derives from the Greek: angeion, a vessel or cup, and anthos, flower, and "allud(es) to the cup-like shape of the ring of broad pappus-scales in A. tomentosus."Black, J.M. & Robertson, E.L. (1957) Flora of South Australia Edn. 2, 4: 923.
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Felicia fruticosa subsp. fruticosa reaches a height of maximally , has lance-shaped to inverted lance-shaped leaves of up to long and wide. The heads are on practically leafless, up to long peduncles. The ray florets are violet, rarely white, the short pappus bristles are long.
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In Ptolemy's Almagest, Coma Berenices is not listed as a distinct constellation. However, Ptolemy does attribute several seasonal indications (parapegma) to Conon. Conon was a friend of the mathematician Archimedes whom he probably met in Alexandria. Pappus states that the spiral of Archimedes was discovered by Conon.
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The distinctive fruit is an achene with a brown, hairless body about half a centimeter long. At the tip of the body is a large pappus made up of five long, jointed scales each up to a centimeter in length and lined with bristles and hairs.
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The flower head is solitary atop the stem and at the ends of branches. It has yellow ray florets with 2 to 4 teeth at the tips and tubular yellow disc florets at the center. The fruit is a cypsela with a pappus of scales.Buphthalmum salicifolium.
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These biennial or perennial plants bloom from June to September. Flowers are hermaphroditic. Pollination occurs through insects (entomophily). The seeds fallen to the ground are transported for some meters by the wind due to the pappus (anemochory) are subsequently dispersed by insects, especially by ants (myrmecochory).
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Flower heads occur on the spreading branches. Each has up to 13 or 14 ray florets, each with an elongated tube and a whitish ligule up to 1.5 centimeters long. The fruit is an achene tipped with a spreading cluster of long, tan, plumelike pappus bristles.
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Megha Suri (9 May 2009) Pappus say they aren't one, brave chant with a smile. The Times of India. It has also been used in film titles such as Aur Pappu Paas Ho Gaya (2007), Pappu Can't Dance Saala (2011). In Telugu the word refers to lentils.
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The disc florets are 4- to 5-dentate. The receptacle is 2–3 times as high as wide. The pappus may be crown-shaped and short, or lacking. Matricaria species are used as food plants by the larvae of some Lepidoptera species (caterpillars) including lime-speck pug.
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In the middle are seven to thirteen disc florets, each 3¾–5 mm long. The ripe indehiscent and one-seeded fruits (called cypselas) are brown in color and 3½ mm long. There is no hairy or scaly pappus present. The species has thirty chromosomes (2n=30).
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The flower head is about half a centimeter long and is enclosed in narrow, sometimes purple-tinged phyllaries. The flowers are pinkish, purplish, or white. The fruit is a dark-colored, resinous achene about half a centimeter long, including its pappus of white or purplish bristles.
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It is not a cedar, fir, or pine, but a member of the aster family, Asteraceae. It is a leafy evergreen shrub with glandular, resinous foliage. It flowers in yellow flower heads which have only disc florets. The fruits are woody, bristly seeds with a pappus.
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The face is fringed with 3-14 (typically 5) broad, 3-lobed ray florets which are usually white, but sometimes yellow. The center contains yellow disc florets with yellow anthers. The fruit is a hairy achene; fruits on disc florets have a pappus of stiff white hairs.
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For instance, species in the Asteraceae on islands tended to have reduced dispersal capabilities (i.e., larger seed mass and smaller pappus) relative to the same species on the mainland. Reliance upon wind dispersal is common among many weedy or ruderal species. Unusual mechanisms of wind dispersal include tumbleweeds.
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There is one brown seed per fruit capsule, a cylindrical capsule with 5 or 6 ribbed parts measuring 3-7 mm long. The seeds come attached with a pappus-like structure and separate easily from the fruits. Seeds are sprouted during the months of late January to mid-March.
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It contains pink or purple ray florets up to 1.5 centimeters long and many disc florets. The fruit may be nearly a centimeter long including its pappus. There were previously two varieties of this species, but these subtaxa are no longer recognized. This plant may hybridize with Xanthisma grindelioides.
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The flowers are hermaphrodite The outer flowers are ligulate, bright yellow and feminine, while the inner ones are tubular, dark yellow and bisexual. The diameter of the flower varies from . The flowering period extends from May through late September. The fruits are glabrous achenes with hairy appendages (pappus).
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Each head is lined with glandular green or purplish phyllaries. It contains purple ray florets which may be up to 2.2 centimeters long, and yellow or purplish disc florets. Blooming occurs in summer, or as late as October. The fruit is an achene with a pappus of bristles.
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The glandular leaves are triangular with serrated edges. The inflorescence is a dense cluster of fuzzy flower heads containing long, protruding disc florets in shades of white, pink, and blue. There are no ray florets. The fruit is an achene a few millimeters long with a rough, bristly pappus.
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The fruit is a cypsela with a pappus of several plumelike bristles or scales. Carminatia tenuiflora is found in Guatemala, El Salvador, most of Mexico, southern Arizona, southwestern New Mexico, western TexasTurner, B. L. 1997. The Comps of Mexico: A systematic account of the family Asteraceae, vol. 1 – Eupatorieae.
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The head has a center of many yellow disc florets surrounded by up to 25 yellow ray florets. It blooms in the Spring. S&S; Seeds−California Native Seeds: Encelia actoni The fruit is an achene about half a centimeter long, usually lacking a pappus. It reseeds well.
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Each head has a bullet-shaped involucre lined with woolly, purple-tipped phyllaries. The head is discoid, containing no ray florets but many funnel-shaped pink, lavender, or purple disc florets with lobes that resemble ray florets. The fruit is an achene with a whitish pappus of bristles.
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The hanging flower heads contain several yellow disc florets and no ray florets. The fruit is a hairy achene up to a long including its pappus. Most of the parts of the plant are very resinous and have a tarlike or hoplike scent. It has a bitter taste.
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These surround numerous bisexual yellow, later burgundy-washed disc florets of long. The two style branches each have a long triangular appendage. The pappus bristles are numerous, yellowish white in colour, and do not detach. Although they vary in length, they do not occur in two distinct rows. The longer pappus bristles have teeth along their length and are 5–7 mm (0.2–0.28 in) long, the shorter scaly and ½–1 mm (0.02–0.04 in) long. The dry, one-seeded, indehiscent fruits called cypsellae are obovate to elliptic, about 4 mm (0.16 in) long and 1½ mm (0.06 in) wide, evenly silky hairy, with a brownish scaly surface when mature, and a light ochre-coloured marginal ridge.
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The species of the section Neodetris are erect, rarely creeping or hanging annual or perennial herbs or shrubs, mostly with flat, hairy leaves with an etire margin, sometimes with teeth and rarely with hairless or succulent leaves. The flower heads are small to relatively big, with blue or white, sometimes violet ligulate florets and yellow, very rarely reddish brown or dark blue, disc florets encircled by an involucre of two worls of bracts. The pappus that crowns the brown to black, medium size, shortly haired cypselas is mostly dehyscent, and pappus is absent in the ligulate florets. The 31 species are concentrated in the greater Cape area, extending from Lüderitz in Namibia to Durban.
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When development is complete, the mature seeds are attached to white, fluffy "parachutes" which easily detach from the seedhead and glide by wind, dispersing. The seeds are able to cover large distances when dispersed due to the unique morphology of the pappus which works to create a unique type of vortex ring that stays attached to the seed rather than being sent downstream. In addition to the creation of this vortex ring, the pappus can adjust its morphology depending on the moisture in the air. This allows the plume of seeds to close up and reduce the chance to separate from the stem, waiting for optimal conditions that will maximize dispersal and germination.
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The inflorescence produces one or more tiny flower heads which are oblong or shaped like tops on close inspection. Each is a few millimeters wide, enclosed in phyllaries studded with stalked resin glands, and tipped with minute yellowish florets. The fruit is an achene a few millimeters long with no pappus.
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The inflorescence contains several flower heads, each lined with woolly green phyllaries. The head contains many golden yellow disc florets and generally either 8 or 13 narrow yellow ray florets each up to a centimeter long. The fruit is an achene around a centimeter long, including its pappus of bristles.
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A few mathematicians of late Antiquity wrote commentaries on the Almagest, including Pappus of Alexandria as well as Theon of Alexandria and his daughter Hypatia. Ptolemaic astronomy became standard in medieval western European and Islamic astronomy until it was displaced by Maraghan, heliocentric and Tychonic systems by the 16th century.
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The shorter pappus bristles are up to 0.7 mm (0.03 in) long, wide scales. The cypselae are large, up to 4 mm (0.16 in) long and 2 mm (0.08 in) wide, elliptic, smooth to slightly scaly, with marginal ridges, with long silky brown hairy along the edge and at the base.
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The inflorescence bears one to four flower heads lined with thick phyllaries. The head contains about 15 yellow disc florets surrounded by about 13 yellow ray florets each about long. The fruit is an achene with a pappus made up of two awns.Flora of North America, Grindelia fraxinipratensis Reveal & Beatley, 1972.
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The bristles of the pappus are scabrous, barbellate, or plumose. The receptacle (base of the flower head) is often smooth, with a fringed margin, or honey-combed, and resemble daisies. They may be in almost all colors, except blue. There are many capitula and generally flat-topped corymbs or panicles.
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The inflorescence is a single flower head or a cluster of a few heads, each lined with woolly phyllaries. The head contains yellow disc and ray florets. The fruit is an achene which may be over a centimeter long including its long pappus. There are two varieties of this species.
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The inflorescence is a wide array of flower heads. A dioecious species, the male and female plants produce different flower types which are similar in appearance. The flowers and foliage are glandular. Female flowers yield fruits which are ribbed achenes, each with a fuzzy body long and a pappus about long.
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Baccharis vanessae is a sticky, glandular shrub producing dense, branching, erect stems approaching 2 meters in maximum height. The leaves are linear and up to long. This dioecious shrub produces male and female flower heads on different individuals. The fruit is an achene with a pappus up to a centimeter long.
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The inflorescence bears one or more flower heads lined with spreading or curling, pointed phyllaries. The head has a center of many yellow disc florets and a fringe of many lavender to purple ray florets each long. The fruit is a flat achene about 1 cm long including the pappus.
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The flower head contains up to 200 white or yellow ray florets. The fruit is an achene with a brown to nearly black, sometimes speckled body up to a centimeter long. At the tip of the body is a large pappus made up of about five long, bristly, barbed scales.
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The ligulate florets have a bluish purple strap ending in five teeth, 2½–2⅞ cm long, while the tube is darker. The one-seeded indehiscent fruit (called cypsela) is cylindrical with five to ten longitudal ribs, 5–6 mm long, and is crowned with papery pappus scales. The pollen is yellow.
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The fruit is an achene with a gray or brown body a few millimeters long. At the tip of the body is a large pappus made up of 5 to over 20 long, hairy scales, each of which may exceed one centimeter in length. There are four Microseris laciniata subspecies.
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The head is lined with layers of hairless phyllaries. It contains up to 33 coiling white ray florets and many yellow disc florets. The fruit is a cypsela with a pappus of many barbed bristles. This plant is known from dry habitat and moist spots, such as streambanks and seeps.
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Creeping groundsel is easily dispersed by wind-blown seed, stem fragments, and dumped garden waste. Achenes are 3 millimetres to 4 millimetres long, ribbed or grooved with short hairs in the grooves and a tapering cylindrical shape. The parachute- like hairs, the pappus, are 5 millimetres to 7 millimetres long.
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Flowers occur singly or in small clusters along the stiff branches. Each head contains up to 15 or 16 ray florets, each with an elongated tube and a pink ligule 6 or 7 millimeters long. The fruit is an achene tipped with a spreading cluster of long, plumelike pappus bristles.
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The head contains many bright purple flowers. The fruit is an achene a few millimeters long which lacks a pappus. It flowers from July until September, and the seeds ripen from August to October. The Red Star- thistle has been identified as a Priority Species by the UK Biodiversity Action Plan.
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The chromosome number is 2n=38. The flowering season is between May and August (Central Europe). The hairy flowers are composed of yellow disc florets in the center and orange-yellow ray florets at the external part. The achenes have a one-piece rough pappus which opens in dry conditions.
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Commentary by Adin Even-Israel (Steinsaltz), volume 1, page 356. Jerusalem: Koren Publishers, 2012. The Rabbis taught that once the Roman government forbade Jews to study the Torah. Pappus ben Judah found Rabbi Akiva publicly gathering people to study Torah and asked Akiva whether he did not fear the government.
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Each head is lined with hairless, glandular phyllaries. The head is discoid, containing no ray florets and just a few pale purple, pinkish, or nearly white funnel-shaped disc florets with narrow lobes. The fruit is an achene with a whitish pappus of bristles which may be fused into points.
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The flower heads appear singly in leaf axils, each lined with purple-tipped, glandular, woolly phyllaries. The head is discoid, containing no ray florets but a few tubular light lavender to nearly white disc florets with long, narrow lobes. The fruit is an achene with a whitish pappus on top.
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For any nonsingular cubic plane curve in the Euclidean plane, three real inflection points of the curve, and a fourth point on the curve, there is a unique way of completing these four points to form a Pappus configuration in such a way that all nine points lie on the curve..
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The plant has narrow, floating leaves and white ray florets. The fruit is a cypsela with a pappus of coarse bristles and small tufts of hairs that become sticky and slimy when wet. These may be adaptations to zoochory, in which seeds are dispersed by animals, perhaps waterfowl.Nordenstam, B., et al. (2009).
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Members of Steviopsis are perennial herbs that have heads composed entirely of disk flowers, a pappus of capillary bristles, narrow corollas with spreading lobes, and glands on the cypselae (achenes). The base chromosome number is x=10, which distinguishes it in part from the morphologically similar Brickellia. The genus is endemic to Mexico.
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They are deeply veined, coated in woolly hairs, and glandular but not shiny. The inflorescence is a cyme of sunflower-like flower heads borne on a hairy, leafless peduncle. The flower head has several yellow ray florets measuring up to 1.5 centimeters long. The fruit is an achene tipped with a pappus.
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The leaves are no more than a centimeter long. The flower head has many ray florets which are usually white, sometimes purple-tinged. They are 5 to 7 millimeters long. The fruit is an achene about half a centimeter long including the pappus, which is an elongated bristle surrounded by fused scales.
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The flower heads are tiny, fluffy and can be pale dusty pink or whitish. The fruit is an achene about 2 or 3 mm long, borne by a pappus with hairs 3 to 5 mm long, which is distributed by the wind. The plant over-winters as a hemicryptophyte. Eupatorium cannabinum bluete.
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The leathery, pointed leaves are up to 5 by in size. The flower head is turbin-shaped and has several ray florets and disc florets surrounded by 40 to 60 resinous phyllaries. The fruit is a few millimeters long and is tipped with a brown pappus about half a centimeter long.Hazardia orcuttii.
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The acute, barely overlapping phyllaries number six to ten. The ray flowers number 6-13. The rays surround 40-70 disc flowers, each three to four millimeters in diameter. The strigose (hairy) fruit measures 3-4 millimeters (0.12-0.16 inches), and its pappus can vary between 0.3 and 1.0 millimeters (0.012-0.40 inches).
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It is a thin, spindly plant which reaches a meter in height. It starts from a basal rosette of leaves and branches extensively, often forming a weedy thicket. It produces small daisylike flowers with rectangular yellow ray florets. The fruit is an achene about a centimeter long topped with a white pappus.
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The inflorescence is a branching panicle of many yellow flower heads at the top of the stem, sometimes with over 200 small heads. Each head contains about 5-14 yellow ray florets a few millimeters long surrounding 6-20 disc florets. The fruit is an achene tipped with a pappus of bristles.
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The inflorescence bears one or more flower heads lined with pointed, roughly hairy phyllaries. The head has a center of many yellow disc florets and a fringe of 16 to 18 yellow ray florets roughly a centimeter long. The fruit is a woolly achene 2 to 3 millimeters long tipped with a pappus.
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Flower heads occur at intervals along the mostly naked stems, especially near the tips. Each has a cylindrical base covered in hairless phyllaries. It contains 3 to 6 florets, each with an elongated tube and a flat pink ligule. The fruit is an achene tipped with a spreading cluster of plumelike pappus bristles.
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There are smaller, narrower leaves along the lower part of the stem. The inflorescence is 1-3 flower heads lined on the lower outside with hairy phyllaries. The head has 45–90 blue or purple ray florets surrounding many yellow disc florets. The fruit is an achene with a pappus of bristles.
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The inflorescence is an array of clusters of flower heads. Each head is lined with phyllaries that are coated densely with stalked knobby resin glands. It bears yellow, lobe-tipped ray florets a few millimeters long and several black-anthered disc florets. The fruit is a flat, hairless achene with no pappus.
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The flower head opens into a face of up to 10 yellow ray florets. There are no disc florets. The fruit is a narrow achene 7 or 8 millimeters long tipped with a pappus of white hairlike bristles.Flora of North America, Longleaf or tapertip hawksbeard, Crepis acuminata Nuttall, Trans. Amer. Philos. Soc.
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There are three to seven pappus bristles on top of the cypselas in the centre, three to five with bordering the involucre. The yellow to yolk-yellow, strap-like corolla is usually 25–37 mm long, ends in five teeth, and does not carry black hair. Flowers are present from May to July.
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The inflorescence is usually a solitary flower head with a bell-shaped base up to 1.5 centimeters wide. It is lined with green or yellowish phyllaries with white edges. It contains several yellow ray florets and many disc florets. The fruit is an achene at least a centimeter long including its pappus.
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The pappus is initially wrapped tightly around the corolla, and so allowing access of light to the cypselas, but is later spreading, to create better lift for the ripe fruit. The pollen is approximately globular (50×45 μm), has three very wide furrows and has a pattern of very small spines and ridges.
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The genus Hymenonema contains two species, H. graecum, that is known from the Cyclades, and H. laconicum. H. graecum has yellow ligulate florets with yellow anther tubes, which sit on 3½-5 mm long and ¾-1¾ mm thick, sparsely hairy cypselas, encircled by pappus that differs maximally 5 mm in length, and basal leaves with a narrow top segment of usually 3–10 mm. H. laconicum differs in having yolk yellow ligulate florets with a purple spot at the base and dark purple anther tubes, which sit on 4¾-6 mm long and 1½-2 mm wide densely hairy cypselas and the pappus differing 8–15 mm in length, and basal leaves with a wide top segment of 10–35 mm wide.
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The genus Hymenonema contains two species, 'H. laconicum, which is known from the southern Peloponnesos and H. graecum. H. graecum has yellow florets with yellow anther tubes, which sit on 3½-5 mm long and ¾-1¾ mm thick, sparsely hairy cypselas, encircled by pappus that differs maximally 5 mm in length, and basal leaves with a narrow top segment of usually 3–10 mm. H. laconicum differs in having yolk yellow ligulate florets with a purple spot at the base and dark purple anther tubes, which sit on 4¾-6 mm long and 1½-2 mm wide cypselas and the pappus differing 8–15 mm in length, and basal leaves with a broader top segment of 10–35 mm wide.
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A high proportion of the first type of subterranean cypselas germinates quickly, while in the second type germination is spread over time due to inhibition by hormones from the fruit skin. The cypselas in the aerial flower heads form from April to May and neither shows delayed germination, but the three types differ in the way they disperse. Those bordering the involucre have short and scaly pappus and are subtended by a bract, to be released with the flower head when it breaks free from the dead mother plant, resulting in dispersal over a short distance. The cypselas at the centre of the flower head however carry much longer pappus and dislodge much sooner to be carried off by the wind over larger distances.
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Flowering occurs during the summer. Each head is 3 to 5 centimeters wide and long and has an involucre of phyllaries which are purple, curve outward, and taper into hard, toothed spines. The head bears many hairlike pinkish purple flowers. The fruit is an achene with a plumelike pappus up to 2 centimeters long.
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The foliage is made up of woolly leaves divided into many thin, flat, threadlike segments. The inflorescence is a narrow cluster of several flower heads. The fruit is a tiny resinous achene with a pappus of hairs.Flora of North America Vol. 19, 20 and 21 Page 530 Island sagebrush, Artemisia nesiotica P. H. Raven, Aliso.
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The species may be annuals, biennials or perennials. They are well-branched with erect stems, characterized by their numerous white, lavender or pink ray flowers and yellow disc flowers. Some members of this group have no ray flowers. The pappus (= modified calyx, forming a crown) is shorter than in Aster, and consists of bristles.
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The force of the "flick" can kill small ants and flies. Some bees in the genus Leioproctus (L. conospermi, L. pappus and L. tomentosus) feed exclusively on one or two species of Conospermum obtaining both nectar and pollen. Some appear to be camouflaged, having white eyes, milky-coloured wings and bodies covered with white hair.
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Each head is about 13 millimeters long and wrapped in flat, wide, purplish green overlapping phyllaries. At the tip of the head are a number of long white to pink disc florets. The bloom period is August through November. The fruit is a hairy cylindrical achene 3 millimeters long with a pappus of bristles.
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The cubic distance-regular graphs have been completely classified. The 13 distinct cubic distance-regular graphs are K4 (or tetrahedron), K3,3, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.
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The leathery oval leaves are up to 2.5 centimeters long. They are shiny and hairless on the upper surfaces and woolly-haired on the undersides. The inflorescence is a panicle of bell-shaped flower heads containing disc florets. The fruit is an achene up to 8 millimeters long including its pappus of barbed white hairs.
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It flowers from March or April to June. The inflorescence has 5 hardened phyllaries surrounding a head of 5 yellow ray flowers with several yellow disk flowers. The ray flowers have 3 strong lobes, or teeth. The fruit is an achene with a pappus of 30–40 white bristles about long, fused at their base.
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The solitary flower head has 5 to 13 yellow ray florets and up to 90 yellow or purple-tinged disc florets. The fruit is a cypsela which may be over a centimeter long including its pappus of bristles. The plant grows on grasslands and playas. It flowers in summer and fall, especially after rain.
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Flora of North America, Brickellia greenei A. Gray The inflorescences hold widely spaced flower heads, each about long and lined with narrow, pointed phyllaries. Each flower head holds a nearly spherical array of about 60 thready disc florets. The fruit is a hairy cylindrical achene about long with a pappus of bristles.Gray, Asa 1877.
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Each head is lined with rigid, hairy and glandular phyllaries and filled with white, pink, or pale yellow flowers. The flowers around the edges are larger and open-faced, and the ones in the center of the head are smaller and somewhat tubular in shape. The fruit is an achene with a pappus of scales.
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The smallest 4-crossing cubic graph is the Möbius-Kantor graph, with 16 vertices. The smallest 5-crossing cubic graph is the Pappus graph, with 18 vertices. The smallest 6-crossing cubic graph is the Desargues graph, with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known.
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The fruit is a cypsela with a pappus of white bristles.Britton, Nathaniel Lord & Brown, Addison (1913). An Illustrated Flora of the Northern United States, Canada and the British Possessions: From Newfoundland to the Parallel of the Southern Boundary of Virginia, and from the Atlantic Ocean Westward to the 102d Meridian, Volume 3., p. 538.
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This biennial or short-lived perennial plant usually has a stout taproot and/or pubescent stems when young. It has deeply lobed leaves and vibrant pink flowers. The sepals have black tips that look like spots, which is the origin of its common name. The fruit is an achene with a short, stubby pappus.
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The inflorescence is a solitary flower head or a cluster of 2 or 3 heads, each with up to 11 yellow ray florets which may be up to 4.5 centimeters long. The fruit is an achene about a centimeter long, not counting its pappus. The seeds are edible and taste similar to sunflower seeds.
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Trichoptilium incisum sends up stems from a basal rosette of sharply-toothed leaves which are covered in curly hairs and oil glands. Atop each stem is a small rounded bright yellow flower head with only disc florets. Each head is a hemispherical button about a centimeter in diameter. The fruit is bristly with pappus.
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They are known generally as rock daisies. Perityle is a variable genus, with its members sharing few characteristics. They include small herbs to spreading shrubs and most bear yellow or white daisylike flower heads. The fruit is generally a flat seed with thickened margins which may or may not have a pappus or scales.
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The leaves are deeply divided into many variously-shaped lobes which may have toothed edges or smaller lobes. The inflorescence bears flower heads lined with glandular, hairy to woolly phyllaries. They are filled with numerous yellow ray florets but no disc florets. The fruit is an achene up to a centimeter long including its pappus.
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He migrated with his brother to Strasbourg at the conclusion of the war in 1589, and died there in 1593.Michaela Waldburg, Waldburg und Waldburger – Ein Geschlecht steigt auf in den Hochadel des Alten Reiche. 2009, Accessed 15 October 2009 He is buried in the Strasbourg Minister.Johannes Pappus, Angehengtem Programmate publico Rectoris Academiae, Abdankung Genealogia.
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They can be almost any shade of yellow, orange, red, purplish, brown, white, or bicolored. They are sometimes rolled into a funnel shape. There are many tubular disc florets at the center of the head in a similar range of colors, and usually tipped with hairs. The fruit usually has a pappus of scales.Gaillardia.
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The genus is distinguished from other genera in tribe Astereae mainly by the structure of the fruit. These achenes or cypselas are roughly club-shaped but usually incurved and flattened. They often have a membranous rim or wing around the edge that is sometimes wavy or fringed. The pappus is less than one millimeter long in most species.
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The flower head is urn-shaped and covered in phyllaries. The head opens slightly at the top, revealing many yellow disc florets and sometimes one or more tiny yellow ray florets, although these may be absent. The fruit is a long, thin achene coated in ashy gray hairs and tipped with a pappus of long, white bristles.
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All leaves are hairless, base stalkless, ear-shaped, arrow-shaped or semi-stem-clasping. Blue flower-heads are bore on top of the stem and branches arranged in loose corymbose inflorescence, each flower head containing ray flowers but no disc flowers. Achenes light brown, oblanceolate, long 2.3 mm, width of 1 mm. Pappus white, thin, hairy, 3 mm long.
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Rainiera stricta is an herbaceous perennial with both basal and cauline, alternate, petiolate leaves. It has 30-70 discoid heads arranged in a raceme-like or thyrse-like capitulescence. The disk florets are about 5 per head, and have yellow, sometimes purple-tinged corollas. The cypselae (achenes) are glabrous and have a pappus of white or straw-colored bristles.
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They measure up to long. The inflorescence is a flower head lined with green, sometimes purple-speckled, phyllaries and containing many yellow ray florets and no disc florets. The fruit is a cylindrical achene up to long not including the large pappus of up to 30 silvery white bristles which may be an additional in length.
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The inflorescence is a solitary flower head or a few heads clustered together atop the woolly stem. The flower head is enclosed in hairy purple phyllaries and contains 8 or 13 yellow ray florets up to a centimeter long. The fruit is an achene tipped with a pappus half a centimeter in length. Blooming occurs in August through October.
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The flower heads are solitary or borne in inflorescences of up to 20. The head is spherical to cylindrical and covered in several layers of spreading or curving spine-tipped phyllaries. It contains long, tubular disc florets in shades of white, pink, or purple. The fruit is a cypsela tipped with a pappus of barbed bristles or scales.
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The evergreen foliage is dark green, glandular, sticky, and very aromatic. New twigs and leaves are somewhat woolly, but older parts are hairless. The narrow inflorescence holds clusters of flower heads lined with rough, shiny, slightly hairy phyllaries and containing yellowish disc florets. The fruit is an achene up to 2 millimeters long, sometimes with a pappus.
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The upper leaves are smaller and have woolly, glandular surfaces. The inflorescence is sparsely flowered in flower heads which open in the evening and close early in the morning. Each small head has five short light yellow ray florets with lobed tips, and six yellow disc florets. The fruit is an achene a few millimeters long with no pappus.
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The flower head is enclosed in five waxy, gland-studded phyllaries. It bears 20 to 30 flowers, which are disc florets. Each flower is white or purplish and has a long, curling style protruding from it. The fruit is a cylindrical achene topped with a pappus of bristles, the whole unit measuring over one centimeter long.
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They bear woolly, cottony heads of flowers. They have narrow strap-shaped untoothed leaves. The flower heads are small, gathered into dense, stalkless clusters. The fruits have a hairy pappus, or modified calyx, the part of an individual disk, ray or ligule floret surrounding the base of the corolla, in flower heads of the plant family Asteraceae.
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The fruit is an achene a few millimeters long with a pappus which may be up to 4 centimeters in length.Flora of North America, Cirsium undulatumUnited States Department of Agriculture Plants Profile Cirsium undulatum has been shown to have its seed production reduced by an exotic weevil Larinus planus which was released to control canada thistle.
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The inflorescence is a panicle of woolly flower heads containing disc florets. The fruit is an achene 1 to 2 millimeters long with a pappus of barbed white hairs up to 5 millimeters long. Volcanic debris on Mount Taranaki has been colonized by this species, which occurs in dense stands up to 100 years old.Clarkson, B. D. (1990).
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The inflorescence is a cyme of up to 16 sunflower-like flower heads with deep yellow ray florets each up to 2.5 centimeters long. At the center are up to 100 disc florets with long yellow corollas and dark brown anthers. The fruit is a brown, winged achene over a centimeter in length including the pappus at the tip.
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Each head has a cylindrical base under a centimeter long and contains 9 to 12 light lavender or pinkish flowers. Each flower is a ray floret with an erect tube and a strap-shaped ligule with a toothed tip. The ligule is just under a centimeter long. The fruit is a cylindrical, ribbed achene with a white pappus.
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The plant produces an inflorescence just a few centimeters to half a meter tall consisting of flower heads which are cylindrical to hemispheric in shape. Each head contains many yellow to orange disc florets and sometimes a few ray florets. The fruit is a long, narrow achene 1 to 2 centimeters in length including its pappus of plumelike bristles.
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The inflorescence is a wide array of several flower heads. Each head contains up to 20 disc florets with bright green tubes and whitish or pinkish corollas and pinkish anthers. The fruit is a ribbed cypsela with a pappus of many white or purple-tipped bristles. The plant occurs on the Gulf Coastal Plain in sandy, swampy habitat.
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There are no bracts on the common base of the florets. The indehiscent one-seeded fruits (called cypselas) are long, covered in velvety hairs, and are adorned by ten longitudinal ribs. The sepals which are changed to barbed long hairs called pappus are white in color. The pollen is tricolpate and has some small spines of less than high.
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The head is lined with spiny, purple-tipped phyllaries which curve outward. The head contains many red, purplish, or rose pink flowers, each up to 4.5 centimeters long. The fruit is an achene with a brown body 6 or 7 millimeters long topped with a pappus which may be 4 centimeters in length. The flower heads attract hummingbirds.
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They are usually opposite but are sometimes arranged alternately. The flower heads are solitary or paired, or occasionally in arrays of several. They are just a few millimeters wide and usually contain 3 or 4 white, pink, or purple funnel-shaped disc florets, sometimes more. The hairy, ribbed cypsela is tipped with a pappus of long scales.
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The herbage is coated in soft hairs. The inflorescence bears many flower heads. Each head has narrow, pointed, hairy phyllaries, a large dense center of many yellow disc florets, and a short fringe of many rectangular yellow ray florets, which are only about 2 millimeters long each. The fruit is an achene tipped with a pappus of bristles.
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The focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope. Extract of page 3.
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The largest heads may be up to 5 centimeters (2 inches) in diameter. They are packed with white or lavender disc florets but no ray florets. The fruit is a flat brown achene with a long pappus which may reach 2 centimeters long. Unlike many other thistles, this species tends not to be a troublesome noxious weed.
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Atop the stems are solitary flower heads which are ligulate, containing layered rings of ray florets with no disc florets. The florets are yellow with toothed tips. The fruit is a cylindrical achene with a pappus of scales. Fruits near the center of the flower head are rough, while those growing along the edges of the head are smooth.
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S. maculatus is an annual of up to 1½ m high, there are more than five leaflike bracts subtending each cluster of flowerhead, and these bracts are pinnately divided. The yellow florets carry some black hairs. The cypselas do not have pappus at their top (but are encased by the paleae). The spined wings along the stems are uninterrupted.
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8: 213. 1874. The inflorescence holds one to several daisylike flower heads, which nod as buds and then pull erect when the face opens. Each head has a center filled with yellow disc florets and usually several yellow ray florets around the edge. The fruit is a cylindrical achene about half a centimeter long with a bristly pappus.
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Cirsium is a genus of perennial and biennial flowering plants in the Asteraceae, one of several genera known commonly as thistles. They are more precisely known as plume thistles. These differ from other thistle genera (Carduus, Silybum and Onopordum) in having feathered hairs to their achenes. The other genera have a pappus of simple unbranched hairs.
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Female plants bearing slightly larger flower heads containing pistillate flowers, and male plants producing staminate flowers. The fruit is an achene up to a centimeter long, most of which is a long pappus attached to a small hard body.Flora of North America Vol. 19, 20 and 21 Page 408 Evergreen or everlasting pussytoes Antennaria suffrutescens GreeneGreene, Edward Lee 1898.
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The larvae feed on Echinops ruthenicus, Centaurea orientalis, Centaurea salonitana and Cirsium sublaniflorum. They bore the flowers of their host plant and mainly feed on young fruit. After overwintering they feed on last year's fruit, but have also been recorded feeding on the pupae of other insects. Pupation takes place in the pappus of the fruit.
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Around the base of the corolla are numerous, brownish white, short-toothed, persistent pappus bristles, which are all of the same length, up to about . The dry, one-seeded, indehiscent fruits called cypsellae are elliptic, about long and 1 mm wide, evenly silky long haired or with the hairs limited to the base and the edge.
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The inflorescence is a solitary bell-shaped, sunflower-like flower head sometimes tucked amongst the uppermost leaves. The head contains about 13 yellow ray florets which may be 2 to 3 centimeters long or more. At the center are yellow disc florets. The fruit is an achene about 7 millimeters long which does not have a pappus.
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The bell-shaped involucre is lined with pointed phyllaries that curl back as the head matures. The head is discoid, with no ray florets but several tubular golden disc florets with raylike lobes. The plant blooms in July through November.Golden Gate National Recreation Area: 2008 Endangered Species Big Year The fruit is an achene with a whitish pappus.
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The inflorescence is generally a single flower head, or sometimes more than one. The head has a bell-shaped base with curving phyllaries which are green to tan. The head contains a few white ray florets and has white disc florets at the center. The fruit is a hairy achene which is roughly a centimeter long including its pappus.
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Wild rye ear with awns Awns on the fruit of an Australian species of grass In botany, an awn is either a hair- or bristle-like appendage on a larger structure, or in the case of the Asteraceae, a stiff needle-like element of the pappus. Awns are characteristic of various plant families, including Geraniaceae and many grasses (Poaceae).
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The plant produces cylindrical flower heads just a few millimeters wide, containing usually 3-4 bright yellow disc flowers. The phyllaries (green bracts surrounding the flower head) are concave. The disc florets have ray-like lobes, but there are no true ray flowers. The fruit is an achene about half a centimeter long including a short pappus.
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The fruit is an achene a few millimeters in length. It is coated in rough hairs and usually has a pappus on the tip.Flora of North America, Solidago multiradiataAiton, 1789. Northern or Rocky Mountain goldenrod , verge d’or à rayons nombreux The plant has been noted to be among the first species to resprout after oil spills in Alaska.
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In Book V, after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus's treatise on this subject), and of the volumes of different solid figures which have all the same superficial area, and, lastly, a comparison of the five regular solids of Plato. Incidentally, Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere. According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "Lesser Astronomical Works" (Μικρὸς Ἀστρονοµούµενος), i.e., works other than the Almagest.
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The treatise which has given rise to this subject is the Porisms of Euclid, the author of the Elements. As much as is known of this lost treatise is due to the Collection of Pappus of Alexandria, who mentions it along with other geometrical treatises, and gives a number of lemmas necessary for understanding it. Pappus states: :The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or problems, and consequently some geometers think that they are really theorems, and others that they are problems, being guided solely by the form of the enunciation. But it is clear from the definitions that the old geometers understood better the difference between the three classes.
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The fruit is a rough- textured, glandular, purple-mottled cypsela that turns gray with age. It has a pappus composed of one outer layer of reddish scales and two inner layers of white bristles.Jean Trudel , Gardens and scenery: summer flowers , Saint- Laurent (Quebec), Livre-Loisirs Ltée, 65 p. It has been in culture in Europe since 1728 at the latest.
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It is the only United Kingdom dandelion type flower with grass like leaves. The flower heads are 5 cm wide. They only open in the morning sunshine, hence the name 'Jack go to bed at noon'. The achenes are rough, long beaked pappus radiating outwards interwoven like a spider's web of fine white side hairs (referred to as a "Blowball").
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Pappus states that he was with the students of Euclid at Alexandria. Euclid was long gone. This stay had been, perhaps, the final stage of Apollonius’ education. Eudemus was perhaps a senior figure in his earlier education at Pergamon; in any case, there is reason to believe that he was or became the head of the Library and Research Center (Museum) of Pergamon.
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Buried seed can remain viable in the soil seed bank for at least seven years and possibly for up to twenty years or more. Yearly seed production and seed dormancy are highly variable depending on environmental conditions. The slender and smooth achenes are about 3 mm long and are brown with gray markings. They are tipped with a pappus of slender bristles.
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De Locis Planis is a collection of propositions relating to loci that are either straight lines or circles. Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat (Oeuvres, i., 1891, pp. 3–51) and F. Schooten (Leiden, 1656) but also, most successfully of all, by R. Simson (Glasgow, 1749).
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The inflorescence contains several flower heads, each lined with hairy green phyllaries. The head contains many golden yellow disc florets and generally either 8 or 13 yellow ray florets each up to a centimeter long. The fruit is an achene with a body about a millimeter long tipped with a pappus of 3 or 4 millimeters. The bloom period is May to July.
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The inflorescence is a cluster of several flower heads, each a bullet-shaped body covered in purple or purple-tinged green phyllaries. The head opens at the tip to bloom with several white to purple tubular disc florets; there are no ray florets. The fruit is an achene tipped with a pappus, the whole unit sometimes exceeding a centimeter in length.
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The seeds are spread to new growing sites by means of the wind. These seeds are small and light, and covered with a pappus of fine hairs. The temperatures dropping at night appears to stimulate germination. It has proteoid, cluster roots which form a mat only a few centimetres thick, which is found just below the surface of the soil.
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S. maculatus is an annual of up to 1.5 m high, there are more than five leaf-like bracts subtending each cluster of flowerhead, and these bracts are pinnately divided. The yellow florets carry some black hairs. The cypselas do not have pappus at their top (but are encased by the paleae). The spined wings along the stems are uninterrupted.
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Cauline leaves are tomentose on the underside and contain spines on the lobe tips. Flower heads are 2-5 per cluster, densely matted with cobwebby hairs at the base of the phyllaries and spiny towards the tips. Corollas are pink to purple, approx. .4-.6 in (1-1.4 cm) long, and the fruits are brown to gold, with a bristly, minutely barbed pappus.
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Porphyry of Tyre (; , Porphýrios; , Furfūriyūs; – ) was a Neoplatonic philosopher born in Tyre during Roman rule. He edited and published The Enneads, the only collection of the work of Plotinus, his teacher. His commentary on Euclid's Elements was used as a source by Pappus of Alexandria. He wrote original works on a wide variety of topics, ranging from music to Homer to vegetarianism.
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Clusters of woolly leaves grow near the spines. The inflorescence bears up to 7 flower heads which are each enveloped in four or five woolly phyllaries. Each head contains up to four or five tubular yellow flowers each around a centimeter long. The fruit is a hairy achene which may be nearly 2 centimeters long, including its pappus of long bristles.
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The narrow, pointed leaves are usually no more than a centimeter long and most occur in clusters along the branches. The inflorescence bears up to seven flower heads which are each enveloped in four woolly phyllaries. Each head contains four yellow cream flowers each around a centimeter long. The fruit is a hairy, ribbed achene with a pappus of bristles.
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They are mostly hairless and a shiny dull green on top and grayish hairy underneath. Flowers are borne in large panicles at the ends of branches and shorter panicles in the leaf axils. The species is dioecious, with flower heads that look like "plump shaving brushes". Male plants have heads with short phyllaries and a single layer of pappus hairs.
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In contrast, Felicia fruticosa subsp. brevipedunculata grows to tall, has slightly larger leaves up to long and wide, that are always distinctly inverted lance-shaped. The heads are practically without stalk and sit directly in the rosette of the short shoots. The ray florets are light violet to white, the short pappus bristles are with , slightly longer than in the subsp. fruticosa.
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7: 287. 1840. The inflorescences at the tip of the slender stem holds clusters of nodding flower heads, each just over a centimeter long and lined with greenish phyllaries with curling tips. The bell-shaped flower head holds a spreading array of 20 to 40 disc florets. The fruit is a hairy cylindrical achene about 4 millimeters long with a pappus of bristles.
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The inflorescences hold solitary flower heads, each about 2.4 centimeters long and lined with woolly gray-green to grayish purple phyllaries. Each flower head holds an array of about 60 red, yellowish, or grayish disc florets. The fruit is a hairy cylindrical achene about a centimeter long with a pappus of bristles.Flora of North America, Brickellia incana A. Gray, Proc. Amer. Acad. Arts.
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The involucral bracts of the secondary flowerheads are merged into a brown, durable, hard cup with a fibery fringe. Each contains only one cypsela of about 8 mm long, 5 mm at its widest, somewhat dorsally compressed, narrow at its base and widest beyond halflength, with the pappus at its tip consisting of a 2 mm high cup, narrowest at its base.
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Using the Dandelin spheres, it can be proved that any conic section is the locus of points for which the distance from a point (focus) is proportional to the distance from the directrix.Brannan, A. et al. Geometry, page 19 (Cambridge University Press, 1999). Ancient Greek mathematicians such as Pappus of Alexandria were aware of this property, but the Dandelin spheres facilitate the proof.
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The flower head has a base made up of three layers of pointed phyllaries coated in gray or silvery hairs. The head has a fringe of many yellow ray florets each up to 5 centimeters (2 inches) long, surrounding many small disc florets of the same color. The fruit is an achene about a centimeter long with a small pappus.
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Each head has a bell-shaped base about half a centimeter (0.2 inches) long which is lined with phyllaries with pointed, darkened tips. The head contains 8-15 yellow ray florets just a few millimeters long, surrounding 20-60 disc florets. The fruit is an achene with a pappus of bristles.Flora of North America, Solidago spithamaea, M. A. Curtis ex A. Gray, 1842.
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Lasthenia maritima is an annual herb with short, decumbent to prostrate stems lined with fleshy lobed or unlobed leaves up to 9 centimeters long. The inflorescence bears flower heads lined with hairy phyllaries and ringed with 7 to 12 gold ray florets each about 3 millimeters long. The fruit is a small, hairy achene often topped with a brownish pappus.
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The leaves are alternate, and some species can be slightly hairy. Extensions from the leaf base down the stem, called wings, can be lacking (Cirsium arvense), conspicuous (Cirsium vulgare), or inconspicuous. They can spread by seed, and also by rhizomes below the surface (Cirsium arvense). The seed has tufts of tiny hair, or pappus, which can carry them far by wind.
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Pandrosion of Alexandria () was a mathematician in fourth-century-AD Alexandria, discussed in the Mathematical Collection of Pappus of Alexandria and known for developing an approximate method for doubling the cube. Although there is disagreement on the subject, Pandrosion is believed by many current scholars to have been female. If so, she would be an earlier female contributor to mathematics than Hypatia.
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The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. According to , the corresponding statement for an external angle bisector was given by Robert Simson who noted that Pappus assumed this result without proof. Heath goes on to say that Augustus De Morgan proposed that the two statements should be combined as follows: : (3 vols.): (vol. 1), (vol.
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The inflorescence produces one or more large sunflower-like flower heads at the top of the hairy stem. The head has narrow, hairy phyllaries at the base. It contains up to 21 yellow ray florets each up to 4.5 centimeters long and many yellow disc florets. The fruit is an achene which may be nearly 2 centimeters long including its pappus.
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They are coated in woolly hairs and resin glands, and the edges are smooth or slightly serrated. The inflorescence is made up of one or more flower heads. The head has lance-shaped phyllaries and has up to 20 yellow ray florets which can be up to 6 centimeters long. The fruit is an achene over a centimeter long tipped with a pappus.
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The head has a base with long, narrow phyllaries which may be over 2 centimeters long. The head contains up to 60 or more lavender, pale blue, or white ray florets which may be over 3 centimeters long. The bloom period is March through June. The fruit is an achene which may be over a centimeter long, including its pappus of bristles.
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Plants in most other areas are mostly gynoecious, reproducing asexually via apomixis. The plant forms mats by spreading stolons and sprouting new stems. The flower heads are lined with an outer layer of phyllaries which are variable in color from white to red, green, or brown. The fruit is an achene with a pappus that helps it disperse on the wind.
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The yellow flowers turn purple-red at the base and are larger than the involucre. It blooms in June and July. The fruit is a white, thick, long and deeply furrowed achene surmounted by a small pappus. S. libanotica leaves are whole, slightly toothed towards the base, oblong or oblong-lanceolate, more or less acute that narrow at the petiole.
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Symphyotrichum chilense is a rhizomatous perennial herb growing to heights between 40 centimeters and one meter. The hairy leaves are narrowly oval-shaped, pointed, and sometimes finely serrated along the edges. The inflorescence holds aster flower heads with centers of yellow disc florets and fringes of many narrow light purple ray florets. The fruit is a rounded, hairy achene with a pappus.
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Soddy's hexlet is a three-dimensional analog of the Steiner chain. The simplest generalization of a Steiner chain is to allow the given circles to touch or intersect one another. In the former case, this corresponds to a Pappus chain, which has an infinite number of circles. Soddy's hexlet is a three-dimensional generalization of a Steiner chain of six circles.
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It also bears one or more tiny, glandular flower heads, each with 1 or 2 disc florets and sometimes 1 or 2 lobed white ray florets. The fruit is an achene; those arising from the disc florets may have a pappus of scales at the tip.Carr, Gerald D. 1975. Brittonia 27(2): 140–141Flora of North America, Calycadenia hooveri G. D. Carr, Brittonia.
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The center of each head contains golden yellow disc florets and a fringe of bright golden ray florets approaching 3 centimeters in maximum length. The fruit is a hairy achene up to a centimeter long, not counting its off-white pappus. Seeds are dispersed on the wind. An individual plant can live twelve years, surviving periodic wildfire by resprouting from its long, slender rhizome afterward.
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Lasthenia minor is an annual herb growing erect to a maximum height near 35 centimeters. The woolly stem may be branched or not and has oppositely-arranged pairs of linear leaves. The flower heads are under a centimeter wide and have hairy phyllaries and golden yellow ray and disc florets. The fruit is an achene up to about two millimeters long with a pappus of scales.
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From the above equation, it can thus be stated: the position of particle from the point of start is proportional to the angle \theta as time elapses. Archimedes described such a spiral in his book On Spirals. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon.Ivor Bulmer-Thomas, "Conon of Samos", Dictionary of Scientific Biography 3:391.
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The inflorescence compound cyme is in diameter, pink-purple, with all the florets of similar form (no division into disc and ray florets). The flowers are usually dioecious, but not invariably so, with some plants bearing hermaphrodite flowers. The seeds are 4–5 mm long, with a feathery pappus which assists in wind dispersal.Flora of Northwest Europe: Cirsium arvense Blamey, M. & Grey-Wilson, C. (1989).
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They are rounded off into a short, 0.5 to 1.7 mm long, but distinctly pronounced rostrum. The inner achenes are up to 18 mm long, their yellowish beaks are 4 to 5 (rarely to 10) mm long. A pappus is missing or it consists only of two to three awn-like, 1 to 3 mm large bristles.Erich Oberdorfer : Plant sociology excursion flora for Germany and adjacent areas .
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Achnophora tatei is commonly known as the Kangaroo Island Daisy. The genus name Achnophora is derived from the Greek word 'achne' which means chaff and 'phoros' meaning bearing. This refers to the clearly visible scales of the receptacle and pappus. Tatei is named after Ralph Tate (1840-1901), a British born botanist and geologist, who was a Professor of Natural Science at the University of Adelaide.
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Two species native to the mountains of Southeast Asia and formerly included here are now separated as the genus Leucomeris in subfamily Wunderlichioideae. These plants produce flower heads containing whitish or yellow disc florets each with five deep lobes. The style has short, smooth branches, and the fruit is a lightly hairy cypsela with a pappus of bristles or scales.González-Medrano, F., et al. (2004).
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He was thus able to function as an independent scholar, and does not appear to have held any academic posts, although he did lecture on the De sphaera of Sacrobosco at the University of Padua in 1559. Barozzi translated many works of the ancients, including Proclus’s edition of Euclid's Elements (published in Venice in 1560), as well as mathematical works by Hero, Pappus of Alexandria, and Archimedes.
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Felicia tenella subsp. cotuloides grows to a height of only and is also higher up richly branching and laxly covered in leaves. The plants are often reddish and densely covered in relatively large glands. The involucre is small, mostly only about across, the style appendages short with , and the pappus is 1 mm long, near the base with strong teeth but narrowed near the tip.
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Below the head is an involucre of glabrous green bracts long with brownish edges. Flowering mostly occurs from late spring to early summer. The seed-like achenes are long and have ten "ribs" along their edges but lack a pappus. Ox-eye daisy is similar to shasta daisy (Leucanthemum × superbum) which has larger flower-heads ( wide) and to stinking chamomile (Anthemis cotula which has smaller heads ( wide).
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The slightly hairy leaves are several centimeters long and generally oval-shaped, sometimes with small teeth and basal lobes. The inflorescence holds one or more flower heads which are knobby clusters of yellow disc florets but no ray florets. The phyllaries surrounding the flower head are particularly sticky. The fruit is an achene up to a centimeter (0.4 inches) long, not including its pappus hairs.
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The flower head is lined with sticky yellow-green phyllaries and contains several yellowish protruding flowers. The fruit is a hairy achene a few millimeters long with a wispy pappus at the tip. The species grows in sagebrush and woodland habitatFlora of North America, Yellow or sticky-leaf rabbitbrush, Chrysothamnus viscidiflorus (Hooker) Nuttall C. viscidiflorus contains an unusual m-hydroxyacetophenone derivative, named viscidone, and chromanone derivatives.
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Ericameria brachylepis is a bushy shrub growing 100–200 cm (40-80 inches) high with branches covered in thready leaves up to 2.5 centimeters (1.0 inch) long. The inflorescence is a cluster of flower heads, each head lined with phyllaries and resin glands. The flower head contains several yellow disc florets and no ray florets. The fruit is a small achene topped with a white pappus.
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It is also suspected to be allelopathic, releasing a toxin from its roots that stunts the growth of nearby plants of other species. Its seed is an achene about a quarter of an inch long, with a small bristly pappus at the tip which makes the wind its primary means of dispersal. The leaves are a pale grayish- green. They are covered in fine short hairs.
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On Sizes and Distances (of the Sun and Moon) (Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], Peri megethon kai apostematon) is a text by the ancient Greek astronomer Hipparchus. It is not extant, but some of its contents have been preserved in the works of Ptolemy and his commentator Pappus of Alexandria. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts.
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S. grandiflorus is an annual or biennial of up to ¾ m high with one, two or three leaflike bracts subtending each cluster of flowerheads and these are spiny dentate. The yellow to orange florets do not have black hairs. The cypselas are topped by three to seven bristles of smooth pappus hairs (and are encased by the paleae). The spined wings along the stems are uninterrupted.
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The linear or lance-shaped leaves are up to 8 centimeters long and their edges are lined with widely spaced teeth. Solitary flower heads occur on erect peduncles. Each head contains up to 10 ray florets, each with an elongated tube and a fringed pink ligule roughly a centimeter long. The fruit is an achene tipped with a spreading cluster of long, plumelike pappus bristles.
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Pappus ben Judah found Rabbi Akiva publicly gathering people to study Torah and asked Akiva whether he did not fear the government. Akiva replied with a parable: A fox was once walking alongside of a river, and he saw fish swimming from one place to another. The fox asked the fish from what they fled. The fish replied that they fled from the nets cast by men.
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Ursinia anthemoides or solar fire is an annual, herbaceous flowering plant of the genus Ursinia,Ursinia anthemoides, Annie's Annuals and Perennials, 2012, access date 23-03-2012 native to South Africa. It has yellow or orange daisy- like inflorescences. Fruits have both pappus and hairs, making the seeds easily dispersed by wind. It was first described by Carl Linnaeus in the 1759 10th edition of Systema Naturae.
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Around the base of the corolla are many white, shortly toothed, deciduous pappus bristles. The brown, dry, one-seeded, indehiscent fruits called cypsellae are 3 mm (0.12 in) long and 1 mm (0.6 in) wide, have a light brown marginal ridge, and the more or less smooth surface carries short hairs. Felicia rosulata is a diploid having nine sets of homologue chromosomes (2n=18).
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This decision was designed to unite Protestants across Germany behind them. On this occasion, Johannes Pappus, president of the Compagnie des Pasteurs (Kirchenconvent), delivered a sermon on the "ministry, character and election of a bishop", but only the Protestants were present to hear it. After the election, the Catholic canons had fled to Saverne and there elected Charles bishop in opposition to Johann Georg on 9 June.
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The genus name is spelled Brachycome by some authors. Henri Cassini published the name Brachyscome in 1816, forming it from the classical Greek brachys ("short") and kome ("hair"), a reference to the very short pappus bristles. Because the combining form of brachys in Greek compound words is brachy-, Cassini later corrected the spelling to Brachycome. Australian taxonomists still debate whether Cassini's corrected spelling is admissible under the rules of botanical nomenclature.
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The blade is oval to spade-shaped and may be several centimeters long. The inflorescence contains one or more hairy, glandular, daisylike flower heads, each with a center of yellowish disc florets and a fringe of yellow ray florets which approach 3 centimeters in maximum length. The fruit is a cylindrical achene about 7 millimeters long which is covered in stiff hairs and has a white pappus at one end.
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The originals of these printings are rare and expensive. For modern editions in modern languages see the references. # A presentation of the first four books of Conics in Greek by Fredericus Commandinus with his own translation into Latin and the commentaries of Pappus of Alexandria, Eutocius of Ascalon and Serenus of Antinouplis. # Translation by Barrow from ancient Greek to Neo-Latin of the first four books of Conics.
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This is a small annual herb producing a glandular stem up to about 35 centimeters tall. The leaves are linear to lance-shaped, with the lower ones lobed and up to about 7 centimeters in length. The daisylike flower heads contain toothed yellow ray florets and yellow disc florets with yellow anthers. The fruit is an achene; fruits on the disc florets have a long white pappus of plumelike bristles.
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Hymenonema laconicum is a species of herbaceous perennial plant plant in the Asteraceae family. It is small to average height, with a rosette of greyish pinnately segmented leaves, and little branching solid stems carrying one to three heads of orange or yolk yellow ray-flowers, with a purple anther tube, and scaly pappus. The species is an endemic of the central and south-eastern Peloponnesos, and flowers in May and June.
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An arrangement of nine points (related to the Pappus configuration) forming ten 3-point lines. In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard problem. There are also investigations into how many k-point lines there can be.
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The hairy, leathery leaves are oval, up to about long, and usually lined with spiny teeth. The plant produces several flower heads each roughly a centimeter (0.4 inches) wide when open. The flower head is lined with roughly hairy, glandular phyllaries and contains disc florets surrounded with a fringe of tiny yellow ray florets. The fruit is a hairy white achene topped with a pappus of many white or brown bristles.
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The inflorescence contains one or more daisy-like flower heads lined in glandular phyllaries. Each has a center of yellow disc florets and several yellow ray florets up to 3 centimeters long. The fruit is an achene with a white pappus. The plant was first described in 1832 by German- Russian botanist Gustav Heinrich von Bongard, based on material collected near Sitka, now in Alaska (then called Russian America).
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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).
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Pentachaeta is a genus of the family Asteraceae; the entire genus is endemic to California. Of the six species members, at least one, Pentachaeta bellidiflora, is classified as an endangered species. The etymology of the genus name derives from Greek: Penta = five + chaeta = bristle, referring to the pappus scales of P. aurea. It was combined in Chaetopappa, but later work led to the genus being recognized as definitely separate.
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Usually nearly all florets set seed, the involucre will open outward when ripe, and the wind will carry away the easily detaching cypselas, thanks to the hairy pappus. The semi-parasitic plant Thesium namaquense, also called poison bush, may infect at least two of the species, F. filifolia and F. muricata, and when these Felicia's are browsed repeatedly by the same animal, this may cause poisoning, particularly in sheep.
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Receptacle somewhat pitted, width up to 4.5 mm. Pappus finely barbellate, white, 9 mm long. Corolla color violet plum, tubular, 5.5-7.5 mm long, 0.1 mm wide, lobes linear to broadly oblong, 1–1.5 mm x 0.5 mm, apex obtuse. Stamens 7 mm long, exserted about 3e4 mm from corolla, somewhat transparent, 0.1 mm long, apical appendage linear, <0.1 mm long, apex acute; pollen grains round, surface spiny, yellow.
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An example of urban evolution in plants was found in Crepis sancta. This plant makes seeds with pappus that can travel with the wind, for seed dispersal. In urban environments green patches are very rare and are also often very small and far apart. Due to this, the chances of the seeds landing on asphalt or stone and not being able to sprout are way higher than in open fields.
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In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution. Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria (ca. 325 AD) as "a not inconsiderable error".
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Leaves have a whitish vein along their margin. S. grandiflorus is an annual or biennial of up to 0.75 m high with one, two or three leaf-like bracts subtending each cluster of flowerheads and these are spiny dentate. The yellow to orange florets do not have black hairs. The cypselas are topped by three to seven bristles of smooth pappus hairs (and are encased by the paleae).
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Micropus californicus is a plant in the daisy family which is known by the common name slender cottonweed. Its flowerheads resemble very small cotton balls, often rounded with cottony white hairs forming the pappus of each seed. The individual plants are known more informally as Q-tips due to their resemblance to the cotton swabs. It is found mostly in California, and into Oregon and northern Baja California.
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Sturm's ongoing conflict with Marbach was adjudicated in Sturm's favor in 1575. But the 1577 Lutheran Formula of Concord reopened the conflict; the theologian Johannes Pappus agitated for its official imposition in Strassburg, supported in the ensuing pamphlet war by the Swabians Andreas Osiander and Jakob Andrea; Sturm opposed them vigorously and vituperatively. Sturm was relieved of his position in 1581 and retired to Northeim. He died in Strassburg in 1589.
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The inflorescence is a single flower head or an array of a few or many heads. The head is hemispherical to bell-shaped and generally no more than a centimeter wide. The head has a center of many golden disc florets and a fringe of 8 to 12 white ray florets each just a few millimeters long. The fruit is an achene, usually with a pappus at the tip.
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The fruit is an achene long including its long pappus of bristles. The shrub is wildfire-resistant, resprouting vigorously and increasing in herbage and seed production in seasons following a fire.US Forest Service Fire Ecology Fire suppression efforts decrease the abundance of the shrub and frequent burns increase it. The shrub is toxic to sheep, causing photosensitivity, bad wool quality, abortion, and death due to the presence of furanoeremophilanes.
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The leaves are narrow, curving, and hooklike, hardening into sharp spines up to 2.5 centimeters long. The inflorescence bears one or two flower heads which are each enveloped in four to six woolly phyllaries. Each head contains up to 8 tubular yellow flowers up to a centimeter long. The fruit is a densely hairy achene which may be nearly 2 centimeters long, including its pappus of long bristles.
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The common base of the florets (or receptacle) is pitted, and carries scales near its margin. The ligulate florets are yellow and have five triangular teeth at their tip. The fruits (or cypselas) are five-angled and carry few or many rigid, appressed hairs. The cypselas are topped by the changed calyx called pappus, which consists of rigid hairs and scales in 2-3 rows, or sometimes only of scales.
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The flower heads that develop underneath the leaves do not open and are self-pollinated. Each floret is fully enclosed in its involucral bracts, and the corolla shows very little development. The cypselas are relatively large and flattened, blackish in color, with ample hairs, and remain below the soil surface after the plant has died. Any pappus consists of somewhat scale-like bristles, hardly developed or is entirely absent.
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At the base of each floret are numerous pappus bristles of two lengths, the shorter ones white, scaly, persistent and about long. When mature, the dry, one-seeded, indehiscent fruits called cypselae are dark brown with a lighter margin, long and wide, narrowly obovate in outline, with a scaly epidermis, and loosely evenly silky hairy. Felicia fruticosa is a diploid having nine sets of homologue chromosomes (2n=18).
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Cavea is a low perennial herbaceous plant that is assigned to the daisy family. Cavea tanguensis is currently the only species assigned to this genus. It has a basal rosette of entire, slightly leathery leaves, and stems of 5–25 cm high, topped by bowl-shaped flower heads with many slender florets with long pappus and purplish corollas. The vernacular name in Chinese is 葶菊 (ting ju).
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Rafinesquia californica is a species of flowering plant in the daisy family known by the common names California chicory and California plumeseed. It looks like a weedy daisy, bearing heads of elegant white-petaled flowers. The ligules of the flowers are often striped with lavender or pink on the undersides, a feature most noticeable when the heads are closed. Each fruit has a pappus of stiff white or light brown hairs.
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The top of the stem bears a multibranched inflorescence with many flower heads. Each head is just over a centimeter (0.4 inches) wide and has many whitish to light blue ray florets but no disc florets. The fruit is a mottled achene about half a centimeter (0.2 inches) long with a brownish pappus. Lactuca biennis was described botanically in 1794, with the name Sonchus biennis, then transferred to Lactuca in 1940.
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The flower head is surrounded by bracts (sometimes mistakenly called sepals) in two series. The inner bracts are erect until the seeds mature, then flex downward to allow the seeds to disperse. The outer bracts are often reflexed downward, but remain appressed in plants of the sections Palustria and Spectabilia. Some species drop the "parachute" from the achenes; the hair-like parachutes are called pappus, and they are modified sepals.
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The inflorescences are loose with 4 to 30 daisy-like flower heads, white with a yellow center, up to in diameter. In most subspecies, the ligules of the ray florets are about 8 mm long, pure white, female and form fertile achenes, which are triangular to horn-like winged. The achenes of the yellow tubular central flowers are sterile and one-winged. The pappus is always irregularly crown-shaped.
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Some wind-dispersed seeds, such as those of the dandelion, can adjust their morphology in order to increase or decrease the rate of germination. There are also strong evolutionary constraints on this dispersal mechanism. For instance, Cody and Overton (1996) found that species in the Asteraceae on islands tended to have reduced dispersal capabilities (i.e., larger seed mass and smaller pappus) relative to the same species on the mainland.
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This is a perennial herb forming a basal patch of woolly grayish oval-shaped leaves a few centimeters long and many slender erect stems up to 40 centimeters tall. It is dioecious, with male and female plants producing different types of flowers. Both flower types are clustered in many flower heads with whitish phyllaries. The female plants produce fruits which are achenes with a soft pappus a few millimeters long.
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The elliptically shaped fruits are 5 angled achenes which are black-brown in color and 3–4 mm long. Pappus is white and about 5 mm long. This species flowers and fruits in July through November in China. Eupatorium fortunei, in Vietnam Museum of Ethnology In Chinese, the name is Pei-lan (佩蘭), and in ancient literatures, it is also referred to as Lan-tsao (蘭草).
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Lasthenia platycarpha is an annual herb producing an erect stem approaching 30 centimeters in maximum height. The oppositely arranged leaves are up to 6 centimeters long, sometimes lobed, and sometimes coated in hairs. The inflorescence bears small flower heads with centers of gold disc florets and 6 to 13 golden yellow ray florets each just under a centimeter long. The fruit is a hairy achene with a pappus at the top.
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Pappus regretted that he had been seized for occupying himself with idle things. When the Romans took Rabbi Akiva out for execution, it was the hour for the recital of the Shema, and while they combed his flesh with iron combs, he was accepting upon himself the kingship of Heaven by reciting ("Here, O Israel . . . ," Shema Yisrael . . .). His disciples asked him whether his devotion extended even to that point.
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Similarly it is a "coincidence" if ab = ba, of any other law of algebra holds. Fortunately, we can show that the required coincidences actually occur, because they are implied by certain geometric coincidences, namely the Pappus and Desargues theorems. If one interprets von Staudt’s work as a construction of the real numbers, then it is incomplete. One of the required properties is that a bounded sequence has a cluster point.
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A few mathematicians of Late Antiquity wrote commentaries on the Almagest, including Pappus of Alexandria as well as Theon of Alexandria and his daughter Hypatia. Ptolemaic astronomy became standard in medieval western European and Islamic astronomy until it was displaced by Maraghan, heliocentric and Tychonic systems by the 16th century. However, recently discovered manuscripts reveal that Greek astrologers of Antiquity continued using pre-Ptolemaic methods for their calculations (Aaboe, 2001).
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There are many disc florets at the center. The fruit is an achene with a pappus of scales; the ray and disc florets produce fruits that differ in appearance. The flowers are pollinated by several species of bees, including Ceratina nanula, Synhalonia fulvitarsis, Dioxys pomonae, Stelis pavonina, and many species of Osmia. The plant was discovered in 1966 near Last Chance Creek south of the Fremont Junction in Utah.
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The inflorescence is made up of one or more flower heads which may be tucked amongst the uppermost leaves. The head has lance-shaped leaflike phyllaries which may be up to 5 centimeters long. There are 5 to 8 yellow ray florets which measure about 1.5 centimeters in length, and many yellow disc florets in the center. The fruit is an achene about a centimeter long tipped with a short pappus.
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The leaves are lance-shaped to oblong with smooth, toothed, or spiny edges. The inflorescence is a solitary flower head with up to 40 or more lavender or pale blue ray florets, each of which may measure over 3 centimeters in length. Flowering may begin as early as late fall or winter. The fruit is an achene which may be over a centimeter long, including its pappus of bristles.
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The plant is of average height (20–70 cm). It usually has 1-3 yellow inflorescences (heads) per stem, each on a peduncle of 10–30 cm. The narrow yellow ligulate florets with five short teeth at the tip sometimes have a whitish base, the tube being hairy. The fruits (or cypselas) are 4–5 mm long and stiffly hairy, and are topped by 12–14 mm long pappus scales.
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These surround many bisexual disc florets with a yellow corolla up to long. In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a triangular appendage. Surrounding the base of the corolla are many, white, serrated pappus bristles of about long.
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Surrounding the base of the corolla are many, yellowish white, serrated, more or less deciduous pappus bristles, all about equal in length at . The eventually yellowish brown to reddish, dry, one-seeded, indehiscent fruits called cypsellae are elliptic in outline, about long and wide, with a ridge along the margin. The cypselae of the ray florets are hairless, those of the disc florets short-haired. Flowering occurs from September to October.
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Like other Senecios, the 10-30 papilla occur stigmatically into pericarp; each usually with four-pored pollen, the grains in polar view 30-35 micrometers when fully expanded. :Seeds The achenes can be 2.5 to 3.5 millimeters (0.1 to 0.15 in) long, are straight and shallowly grooved; with hairless smooth ribs while the grooves are covered with hairs. The silky white, umbrella-like pappus readily detaches from the fruit when ripe.
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Jones makes these comments about Chasles, Pappus and Euclid:Alexander Jones (1986) Book 7 of the Collection, part 1: introduction, text, translation , part 2: commentary, index, figures , Springer-Verlag :Chasles's contribution to our comprehension of the Porisms tends to be obscured by the inherent unreasonableness of his claim to have restored substantially the contents of Euclid's book on the basis of the meagre data of Pappus and Proclus...one still turns to Chasles for the first appreciation of the interest in the Porisms from the point of view of modern geometry. Above all, he was the first to notice the recurrence of cross-ratios and harmonic ratios in the lemmas, and because these concepts suffuse most of his restoration, it is probable that many of his inventions coincide with some of Euclid's, even if we cannot now tell which they are. Chasles's name is one of the 72 names inscribed on the Eiffel Tower.
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The pappus of the inner whorl consists of feathery bristles of long, either or not barbed and fused in a ring at their base. In both ray and disc florets, the brown, one-seeded, indehiscent, dry fruits (or cypselae) are cylinder- to spindle- shaped in outline, rarely inverted egg-shaped, long and wide. The cypselae have four to five, mostly dark brown, distinct ribs along their lengths. The cypselae are covered in shiny yellow glands.
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The fruit is a rough-textured, pyramidal or prism-shaped cypsela up to a centimeter long including its pappus of many barbed white bristles. This plant is abundant in parts of its range, particularly in Mexico, sometimes becoming weedy. It flowers year-round, especially in spring, and it may be in full flower at the end of the dry season. It is admired for its yellow flower heads and is cultivated as an ornamental plant.
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Under a particular inversion centered on A, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula hn = n dn and the fact that the original points of tangency lie on a common circle. The height hn of the center of the nth circle above the base diameter ACB equals n times dn.Ogilvy, pp. 54-55.
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Each disc floret has two lips, the outer of which is long, flat, and usually bright pink, and easily mistaken for a ligule. The fruit is a glandular achene a few millimeters long which has a pappus of bristles up to a centimeter in length.Flora of North America Vol. 19, 20 and 21 Page 74 Acourtia microcephala de Candolle in A. P. de Candolle and A. L. P. P. de Candolle, Prodr.
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The style in both ray- and disc florets forks, and at the tip of both style branches is a triangular appendage. Surrounding the base of the corolla are white, serrated, deciduous pappus bristles of about long. The eventually yellowish brown, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg-shaped, about long and wide, with a prominent ridge along the margin, with some scales on its surface and unique, short, club-shaped hairs.
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The leaves are borne on spiny petioles and are toothed or lobed, the lowest leaves at the base of the stem reaching 50 centimeters in length. The inflorescence is made up of clustered flower heads which are lined with spiny phyllaries. The head is filled with white or purple flowers up to 2.5 centimeters long. The fruit is an achene a few millimeters long topped with a pappus up to about 2 centimeters in length.
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Surrounding the base of the corolla are many white, deciduous pappus bristles of about long, that are strongly serrated near the base and weakly near the top. The eventually black, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg- shaped, about long and wide, with a prominent, light-coloured ridge along the margin, and with some scattered hairs along its surface. It is a diploid with nine homologous pairs of chromosomes (2n=18).
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The daisylike flower heads are up to 2 centimeters (0.8 inches) wide and have long, hairy, lance-shaped green phyllaries. The center of the head is filled thickly with long yellow disc florets and the circumference is lined with 10–23 yellow ray florets.Flora of North America, Hulsea brevifolia A. Gray, 1867. Shortleaf alpinegold The fruit is an achene 6 to 8 centimeters (2.4-3.2 inches) long bearing a pappus which may be red-tinged.
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The range of Tetradymia argyraea is primarily east of the Cascade Range and Sierra Nevada of British Columbia to California. It extends eastward to southwest Montana, Wyoming, western Colorado and northwest New Mexico, where it grows in sagebrush scrub, woodlands, forest, scrubby open plains, and other habitat. It occupies a large range of elevations from near sea level to but favors the range of . The fruit is a achene with a bristly pappus long.
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It is lined with lobed oval leaves each a few centimeters long and coated in woolly fibers. The inflorescence produces one or more flower heads containing many glandular or bristly yellowish disc florets surrounded 6 to 8 yellow ray florets each up to a centimeter long. The fruit is an achene tipped with a pappus of approximately 8 scales.Flora of North America, Eriophyllum jepsonii Greene, 1891. Jepson’s woolly sunflower Greene, Edward Lee 1891.
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The inflorescence bears one to many flower heads, both at the ends of the stem branches and in the leaf axils. The flower head reaches about 3 centimeters long by 4 wide and is lined with cobwebby, bristly, spine-tipped phyllaries. The flower head is packed with white or pink flowers about 2 centimeters long. The fruit is a brown achene a few millimeters long topped with a pappus one to two centimeters in length.
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The small, toothed, oval-shaped leaves are up to 1.2 centimeters long. The inflorescences at the end of stem branches contain clusters of flower heads, each about a centimeter long and lined with greenish, purplish, or yellowish phyllaries. At the tip of the head are 8 to 12 tubular disc florets.Flora of North America, Brickellia desertorum Coville The fruit is a hairy cylindrical achene 2 or 3 millimeters long with a pappus of bristles.
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Chytri was at an early date an episcopal see. Lequien's list of the bishops of the see (II, 1069) is very incomplete, only eight being recorded: the first is St. Pappus, who suffered martyrdom under Licinius, Maximinus or Constantius Chlorus; the most famous is St. Demetrian, 885-912 (?). The Greek, i.e. Orthodox, see of similar title was suppressed in 1222 by Cardinal Pelagius, the papal legate, while the islands was a Latin crusader kingdom.
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Leaves located higher on the stem lack petioles and may clasp the stem at their bases. The inflorescence is a cluster of several flower heads lined with phyllaries which may be over a centimeter long and are hairy to hairless in texture. Each head contains many yellow disc florets and a fringe of several yellow ray florets. The fruit is an achene which may be over a centimeter long including its pappus.
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The systematic position of Hartwrightia floridana. Bulletin of the Torrey Botanical Club 20: 287–288.Biota of North America Program 2013 county distribution map The species is sometimes referred to by the common name Florida hartwrightia. Although superficially similar to some species in Eupatorium, it can be distinguished by having a basal rosette of leaves, flowers of a different shape, and the fruit which lacks the parachute-like pappus found in Eupatorium.
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The narrow linear leaves are up to 5 centimeters long but only a few millimeters wide and may be very hairy. The inflorescence is a solitary flower head, with up to 22 heads per plant. The flower head bears many yellow, brownish, or whitish ray florets 3 to 12 millimeters long, and has a center of many five-lobed yellow to reddish disc florets. The fruit is an achene tipped with a pappus of bristles.
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It is pentaploid (having five sets of chromosomes), and produces seeds asexually like many other Taraxacum species. After pollination the flower closes, later opening as the familiar spherical seedhead or "clock", as in other dandelions. The seedhead consists of many single-seeded fruits or achenes, each attached to a pappus of fine hairs that acts as a parachute to enable wind-dispersal of the seeds, sometimes over long distances. The seeds remain dormant till autumn.
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Sinauer Associates, Inc., Massachusetts. The classic examples of these dispersal mechanisms, in the temperate northern hemisphere, include dandelions, which have a feathery pappus attached to their seeds and can be dispersed long distances, and maples, which have winged seeds (samaras) and flutter to the ground. An important constraint on wind dispersal is the need for abundant seed production to maximize the likelihood of a seed landing in a site suitable for germination.
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Altervista Flora Italiana, Arnica, Arnica montana L. includes photos and European distribution maps Several species, such as Arnica montana and A. chamissonis, contain helenalin, a sesquiterpene lactone that is a major ingredient in anti-inflammatory preparations (used mostly for bruises). Arnica species are used as food plants by the larvae of some Lepidoptera species including Bucculatrix arnicella. Arnica was previously classified in the tribe Senecioneae because it has a flower or pappus of fine bristles.
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They have spiny bracts and ovate sepals which are mucronate but not spiny. They are followed by fruits known as cypselae which are ovoid and slightly flattened. The calyx persists as pappus or may fall off in rings. John Wilkes's Encyclopaedia Londinensis (volume III, 1810) makes note of the "remarkable" features of the plant including the leaves' spots of white (pictured), which it reports are found in three other species of "Egyptian thistle".
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Its ray florets are white with purple markings, while the center of the headis packed with white disc florets with purple anthers. The fruit is a dark achene which often bears a pappus of a few stiff, light colored bristles, resembling human eyelashes (hence the common name of the plant). Blepharipappus scaber grows in forests at elevations of 300–2200 meters (1000–7300 feet).Flora of North America, Blepharipappus Hooker, Fl. Bor.-Amer.
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The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property.
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Around the base of the corolla are many white pappus bristles with teeth, that are easily discarded. The dry, one-seeded, indehiscent fruits called cypsellae are large, 3 mm (0.12 in) long and 1 mm (0.6 in) wide, inverted egg-shaped, yellowish brown to brown, with a strong marginal ridge, with thick, -1 mm (0.02–0.04 in) with short, robust hairs. Felicia namaquana is a diploid having five sets of homologue chromosomes (2n=10).
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Proclus gives a much shorter proof attributed to Pappus of Alexandria. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.
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The Pappus is very big with barbellate bristles. Goldenrod and visiting Cerceris wasp The many goldenrod species can be difficult to distinguish, due to their similar bright, golden-yellow flower heads that bloom in late summer. Propagation is by wind-disseminated seeds or by spreading underground rhizomes which can form colonies of vegetative clones of a single plant. They are mostly short-day plants and bloom in late summer and early fall.
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The inflorescence is a series of dense clusters of flower heads surrounded by long, narrow bracts covered in obvious bulbous glands. The sticky, glandular flower head has a center of several disc florets surrounded by a few white, yellow, or red ray florets. Each ray floret has three lobes at the tip, the middle lobe being shortest. The fruit is an achene; those developing from the disc florets have a pappus of scales.
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Mairia petiolata has leaves with particularly coarse and irregular teeth along the margin, a distinct pattern of netted veins on the underside, and slender leaf stalks. It also has relatively graceful and elegant flower heads at the top of long, slender, mostly branched stalks. The involucral bracts have dark red margins, the disc florets are longer than the pappus and the dark red styles extend far beyond the mouth of the florets.
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At night it is able to reduce its metabolism to save energy. The nest is a hanging pocket made of pappus, dry leaves and spiders webs with green moss placed on the outside and lined with natural cotton-like fibres, it is suspended from a bush at about 3m above the ground or even from cables in abandoned buildings. 1-2 eggs are laid in July in Cameroon, December in the Democratic Republic of Congo and February in Gabon.
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The central disc florets of the flower head tend to be more red-violet, with the outer ray florets being yellow. In one variety, almost the entire flower is red, with only the barest tips of the petals touched with yellow. It blooms practically year-round in some areas, but more typically in summer to early fall. The fruit is an achene, almost pyramidal, hairy, and prolonged by a pappus 5 to 8 mm in length.
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20–21 Page 776 合冠鼠麴草属 he guan shu qu cao shu GamochaetaWeddell, Chlor. Andina. 1: 151. 1856. Plants of this genus have "relatively small heads in spiciform (spike-like) arrays, concave post-fruiting receptacles, truncate collecting appendages of style branches in bisexual florets, relatively small cypselae (fruits) with minute, mucilage-producing papilliform hairs on the faces, and pappus bristles basally connate (joined) in smooth rings and released as single units." ; SpeciesGamochaeta.
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There may be shorter leaves on the lower part of the stem and there are few or none on the upper part. The inflorescence is a wide open array of many flower heads, each up to about a centimeter wide. The flower head is lined with hairy, often glandular phyllaries and filled with many yellow ray florets and no disc florets. The fruit is a small, dark cylindrical achene topped with a pappus of brown bristles.
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Title page of Pappus's Mathematicae Collectiones, translated into Latin by Federico Commandino (1589). Pappus of Alexandria (; ; AD) was one of the last great Greek mathematicians of antiquity, known for his Synagoge (Συναγωγή) or Collection (), and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than what can be found in his own writings: that he had a son named Hermodorus, and was a teacher in Alexandria.Pierre Dedron, J. Itard (1959) Mathematics And Mathematicians, Vol.
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The longest leaves near the base of the plant may be 70 centimeters (28 inches) long. The inflorescence is a cluster of several flower heads each up to 3 centimeters long by 3 wide. The head is lined with spiny phyllaries and filled with pale pink or occasionally white flowers. The fruit is an achene with a flat, dark brown body about 5 millimeters long and topped with a pappus which may be 2 centimeters in length.
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Surrounding the base of the corolla are many white, deciduous pappus bristles of about 2 mm (0.1 in) long, that are clearly serrated. The eventually black, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg-shaped, about 1 mm (0.06 in) long and mm (0.02 in) wide, without a ridge along the margin, and set with spoon-shaped hairs on its surface. The species is diploid with a base chromosome number of 9 (2n=18).
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It is adaptable to many conditions and is sometimes used to control erosion. It is a perennial with stout, woolly stems and aromatic, violin-shaped, heavily lobed leaves. The flower heads have many creamy-white to pink or bronze ray florets with lavender to reddish undersides and centers filled with purple disc florets. The fruit is a hard achene with a tuft of plumelike hairs on one end and an array of pappus scales on the other.
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Encelia californica is a bushy, sprawling shrub reaching between 50–150 cm (20-60 inches) in height. It has many thin branches covered in widely spaced green leaves which are a rounded diamond shape. The solitary flower heads are daisylike, with 15 to 25 bright yellow ray florets 1 to 3 centimeters long around a center of protruding yellowish to purplish brown disc florets. The fruit is an achene 5 to 7 millimeters long, with no pappus.
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Microseris lanceolata has the form of a tufted rosette of toothed lanceolate leaves. The flower appears in Spring, which is a yellow head of florets, similar to flatweed (Hypochaeris radicata) or dandelion (Taraxacum). The flower stalk is pendulous before flowering, becoming erect for flowering to attract pollinators and again with the ripening of the seed head. The seed heads ripen to a cluster of fluffy, tan achenes, each having a crown of fine extensions called a pappus.
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These are individual pollen grains of the dandelion - Taraxacum officinale. Segment of pappus fiber showing barbs The species of Taraxacum are tap-rooted, perennial, herbaceous plants, native to temperate areas of the Northern Hemisphere. The genus contains many species, which usually (or in the case of triploids, obligately) reproduce by apomixis, resulting in many local populations and endemism. In the British Isles alone, 234 microspecies are recognised in nine loosely defined sections, of which 40 are "probably endemic".
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Microseris walteri has the form of a tufted rosette of toothed lanceolate leaves. The flower appears in Spring, which is a yellow head of florets, similar to flatweed (Hypochaeris radicata) or dandelion (Taraxacum). The flower stalk is pendulous before flowering, becoming erect for flowering to attract pollinators and again with the ripening of the seed head. The seed heads ripen to a cluster of fluffy, tan achenes, each having a crown of fine extensions called a pappus.
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It is a small shrub growing up to about 25 centimeters (10 inches) tall with branches lined with short, narrow, hairless to glandular, woolly leaves. The inflorescence is a cluster of flower heads at the tips of stem branches. Each head is lined with sticky, glandular phyllaries and contains as many as 20 yellowish disc florets and sometimes a few yellow ray florets but sometimes none. The fruit is an achene topped with a brownish pappus.
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The mostly lance-shaped leaves are lobed and long near the base of the plant, approaching 30 centimeters (12 inches) in length, and smaller and sometimes unlobed farther up the stem. The inflorescence is an open array of many flower heads, each with pointed phyllaries with thick midribs and thinner, hair-lined edges. Each flower head has 5 to 8 golden yellow ray florets but no disc florets. The fruit is a narrow, ribbed achene with a whitish pappus.
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The fruit is an achene with a pappus of varying lengths, but generally shorter than those of other Liatris. A 2005 study expanded the species description of Liatris helleri to include plants with certain similarities. Populations of plants growing in West Virginia and Virginia previously included in another species (Liatris turgida) might now be included within the circumscription of L. helleri. In that case it would be less rare and no longer a true North Carolina endemic.
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The Atellan Farce is highly significant in the study of the Character because it contained the first true stock characters. The Atellan Farce employed four fool types. In addition to Maccus, Bucco, the glutton, Pappus, the naïve old man (the fool victim), and Dossennus, the cunning hunchback (the trickster). A fifth type, in the form of the additional character Manducus, the chattering jawed pimp, also may have appeared in the Atellan Farce, possibly out of an adaptation of Dossennus.
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The inflorescence is a long array of several flower heads, with some occurring in the upper leaf axils as well. Each head has a cylindrical base 1 to 2 centimeters long which is lined with layers of glandular phyllaries. The head contains 10 to 15 ray florets, each with an elongated tube and a pink ligule which may be up to 2 centimeters long. The fruit is an achene tipped with a spreading cluster of long, plumelike pappus bristles.
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The inflorescence bears one or more flower heads lined with several layers of pointed, curling or curving phyllaries. The head has a center of many yellow disc florets and a fringe of blue or purple ray florets each 1 to 2 centimeters long. The fruit is an achene around 3 millimeters in length tipped with a pappus of long hairs.Flora of North America, Dieteria canescens A number of insects can often be found in the flowers.
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These are about 12 mm long and may be 1 – 3 mm across, linear in shape, having a margin rolled toward the reverse, and are fragrant when crushed. Older leaves may lose the woolly covering and become smooth and green. The yellow florets are supported by white bracts at the flowerhead. The fruit produced are achenes, 1.5 – 2 mm long, the pappus are bristles twice this length; this assists in the dispersal of the seeds by wind.
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The narrow, widely spaced leaves are linear or lance-shaped and smooth-edged or slightly serrated, and measure 1.5 to 6.5 centimeters in length but only a few millimeters in width. They are glandular and hairless or with few hairs. The inflorescence is an elongated cluster of many flower heads containing twenty to thirty or more male or female flowers. The fruit is a hairy achene tipped with a plumelike white pappus about 7 millimeters long.
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The taxonomic status of Solidago ptarmicoidei created an extensive debate due to frequency hybridization of S. ptarmicoidei with members of the Ptarmicoidei section of Solidago. It was asserted that S. ptarmicoides should be united with Solidago rather than the genus Aster due to external morphological features such as similar pappus length as well as the same chromosome base (x=9). Information about chromosome number is still a crucial part of current understanding and phylogenies of Soldiago.
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It was only during the work of Panero on the genus that the congeneric nature of plants of the two pappus types was recognized, and this has subsequently been supported by both molecular phylogenetic studies Schilling, E. E. and R. K. Jansen. 1989. Restriction fragment analysis of[chloroplast DNA and the stystemics of Viguiera and related genera (Asteraceae: Heliantheae) American Journal of Botany 76: 1771-1780. and phytochemical analyses.Spring, O., J. L. Panero, and E. E. Schilling.
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The leaves are linear in shape and up to 5 centimeters (2 inches) long, sometimes longest toward the middle of the stem. The inflorescence bears one or more flower heads at separate nodes, surrounded by short bracts tipped with resin glands. The glandular and hairy flower heads have a center of several disc florets as well as whitish, triple-lobed ray florets. The fruit is an achene; those arising from the disc florets have a pappus of scales.
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Ancient writers refer to other works of Apollonius that are no longer extant: # Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola # Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus) # A comparison of the dodecahedron and the icosahedron inscribed in the same sphere # Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements # Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of than those of Archimedes, who calculated as the upper limit and as the lower limit # an arithmetical work (see Pappus) on a system both for expressing large numbers in language more everyday than that of Archimedes' The Sand Reckoner and for multiplying these large numbers # a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm. on Eucl. x., preserved in Arabic and published by Woepke, 1856).
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Inula helenium Inula oculus-christi Ploughman's-spikenard (Inula conyzae) Inula is a large genus of about 90 species of flowering plants in the family Asteraceae, native to Europe, Asia and Africa. They may be annuals, herbaceous perennials or subshrubs that vary greatly in size, from small species a few centimeters tall to enormous perennials over tall. They carry yellow daisy- like composite flowerheads often with narrow ray-florets. Some common characteristics include pappus with bristles, flat capitulum, and lack of chaff.
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A configuration in the plane is denoted by (), where is the number of points, the number of lines, the number of lines per point, and the number of points per line. These numbers necessarily satisfy the equation :p\gamma = \ell\pi\, as this product is the number of point-line incidences (flags). Configurations having the same symbol, say (), need not be isomorphic as incidence structures. For instance, there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations.
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Previously this was done at daytime by measuring the shadow cast by a gnomon, by recording the length of the longest day of the year or with the portable instrument known as a scaphe. Equatorial ring of Hipparchus's time. Ptolemy mentions (Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra, to measure the apparent diameter of the Sun and Moon. Pappus of Alexandria described it (in his commentary on the Almagest of that chapter), as did Proclus (Hypotyposis IV).
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The spined wings along the stems are uninterrupted. S. hispanicus is an annual, biennial or perennial of up to 1.75 m high and it also has one, two or three spiny dentate leaf-like bracts subtending each cluster of flowerheads and the yellow, orange or white florets also lack black hairs. The cypselas however are topped by two to five bristles of scabrous pappus hairs (and are encased by the paleae). In this species the spined wings along the stems are interrupted.
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Hymenonema is a genus of flowering plants in the dandelion family endemic to Greece. On each of the single or few stems, the species have one to three flowerheads consisting of yellow or yolk yellow ligulate florets, scaly pappus, greyish, pinnately segmented leaves in a basal rosette, and few smaller leaves on the 20–70 cm high stems. It contains two species: Hymenonema graecum, that is known from the Cyclades, and Hymenonema laconicum, which occurs in the central and south-eastern Peloponnesos.
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The inflorescence is a usually solitary sunflower- like flower head with a base up to 6 centimeters wide lined with several ray florets, each of which are 2 to 6 centimeters long. The yellow ray florets extend outwards and then become reflexed, pointing back along the stem. The disc florets filling the button-shaped to conical to cylindrical center of the head are greenish yellow. The fruits are achenes each about half a centimeter long tipped with a pappus of scales.
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The approximately ten to thirteen, light violet to white, female ray florets are hairless and encircle many yellow bisexual disc florets, in the lower part loosely set with delicate multi-cellular glandular hairs. In the center of the florets, a style with narrow triangular extensions grows through the five anthers that have merged into a tube. The yellowish pappus bristles are persistent, and come in two different lengths. The longer numerous, powerful, toothed, widened at the top, to 6 mm (0.24 in) long.
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Sphaericorum libri tres Although very little is known about Menelaus's life, it is supposed that he lived in Rome, where he probably moved after having spent his youth in Alexandria. He was called Menelaus of Alexandria by both Pappus of Alexandria and Proclus, and a conversation of his with Lucius, held in Rome, is recorded by Plutarch. Ptolemy (2nd century CE) also mentions, in his work Almagest (VII.3), two astronomical observations made by Menelaus in Rome in January of the year 98.
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The heads contain 4 to 8 yellow ray florets tipped with three lobes and up to 12 tubular yellow disc florets tipped with five curled lobes. The fruit is a ribbed achene up to a centimeter long including its pappus of many narrow scales. This species only occurs in the Andes of central Argentina, where it grows in rocky mountainous habitat at elevations between 2000 and 2600 meters. Associated flora includes Nassauvia axillaris, Berberis buxifolia, and species of Ephedra and Adesmia.
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They are borne on petioles with winged, spiny margins, some spines exceeding a centimeter in length. The inflorescence produces one or more flower heads, each up to 3 centimeters long by 5 wide, wispy with cobwebby fibers, and lined with very spiny phyllaries. The flower head is packed with dark purplish-pink flowers up to about 2.5 centimeters in length. The fruit is an achene with a dark brown body about half a centimeter long and a pappus about 1.5 centimeters long.
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On 1 January 1590, James III gave city rights to the market town of Emmendingen. During this period of deep religious division, the Margrave closely watched the three Christian camps: the Catholics, Lutherans and Calvinists. In 1589 and 1590, he commissioned two colloquia, the first in Baden-Baden, the second in Emmendingen, between Lutheran theologicians from Württemberg and Catholics. In Emmendingen, the Lutheran side was led by Johannes Pappus from Strasbourg, and the Catholic side by James's court chaplain Johannes Zehender.
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The achenes have a dandelion-like groups of prickly hairs called a pappus which help spread its seeds by the wind.BMP: TANSY RAGWORT (Senecio jacobaea) by The WeedWise ProgramRagwort - Control and removal advice by The Donkey Sanctuary The number of seeds produced may be as large as 75,000 to 120,000, although in its native range in Eurasia very few of these would grow into new plants and research has shown that most seeds do not travel a great distance from the parent plant.
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The inflorescence bears several flower heads, each with a fringe of up to 13 red-veined white ray florets just under a centimeter long. The center of each head is filled with protruding tubular disc florets with large dark anthers. The fruit is a hairy, club-shaped achene which may or may not have a pappus at the tip.Flora of North America, Blepharizonia plumosa (Kellogg) Greene Blepharizonia is sometimes treated as a monotypic genus, and sometimes as a genus with two species.
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There are 4-21 flowers arranged in a loose raceme on the upper part of the stem, with rounded pinkish purple flower heads on stems. Each flower head has 30-100 five-lobed, tubular flowers surrounded by spoon-shaped bracts (phyllaries) with translucent, jagged, and often purple edges that fold inward. Each flower has a long, thread-like, divided style protruding from the center. The fruits (cypselae) are long, each with a ring (pappus) of barbed hairs at the top.
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Robert Simson was the first to throw real light upon the subject. He first succeeded in explaining the only three propositions which Pappus indicates with any completeness. This explanation was published in the Philosophical Transactions in 1723. Later he investigated the subject of porisms generally in a work entitled De porismatibus traclatus; quo doctrinam porisrnatum satis explicatam, et in posterum ab oblivion tutam fore sperat auctor, and published after his death in a volume, Roberti Simson opera quaedam reliqua (Glasgow, 1776).
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The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.Encyclopedia of ancient Greece By Wilson, Nigel Guy p. 77 (2006) The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the Catoptrica.
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Leaves have a whitish vein along their margin. S. hispanicus is an annual, biennial or perennial of up to 1¾ m high and it also has one, two or three spiny dentate leaflike bracts subtending each cluster of flowerheads and the yellow, orange or white florets also lack black hairs. The cypselas however are topped by two to five bristles of scabrous pappus hairs (and are encased by the paleae). In this species the spined wings along the stems are interrupted.
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Plants in Calenduleae vary from herbs to shrubs and usually exhibit showy flower heads. The defining characteristics separating members of this tribe from others within the family are a dimorphism of the cypselae and the fact that each cypsela lacks a pappus.[2] Calenduleae is named for its most economically important genus, Calendula, known in homeopathic remedies and as a common ornamental. Other genera from Calenduleae produce ornamentals as well, including Osteospermum and Dimorphotheca (see Asteraceae for a more general description).
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The lanceolate shaped phyllaries are imbricate, with 3-seriate edges, the outer phyllaries are very short, only 1–2 mm wide, and green or purple-tinged. The florets are white, red-purple, or pink, with corolla about 5 mm wide and the corolla is covered with yellow glands. The fruits are black-brown, 5-angled, hairless achenes, that are elliptic in shape and about 3.5 mm long, and are covered with yellow glands. The 5 mm long pappus are white.
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The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 Coxeter 2003, p. 2 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the concept of the "point at infinity".Coxeter 2003, p.
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At the tip of both style branches are slightly elongated triangular appendages. Around the base of the corolla are numerous, white, clearly toothed and persistent pappus bristles of about 4 mm (0.18 in) long. At the base of the florets sit elliptic, short- haired, dry, one-seeded, indehiscent fruits called cypsellae of about long and wide, with a smooth seed-skin, possibly all of which are sterile. Felicia wrightii is a diploid having nine sets of homologue chromosomes (2n=18).
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When the disc floret opens, the style grows to about long, hoovering up the pollen on its shaft. The upper 1–1 mm (0.04–0.06 in) of the style is split into two line- to ellipse-shaped branches with deltoid appendages at its tip of wide and long. Surrounding the base of the corolla of both ray and disc florets are two whorls of white to straw-coloured pappus bristles. The outer whorl consist of free barbed bristles of long that alternate with the inner whorl.
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James Henry Weaver (10 June 1883 in Madison County, Ohio – 7 April 1942 in Franklin County, Ohio) was an American mathematician. Weaver received B.A. in 1908 from Otterbein College and M.A. in 1911 from Ohio State University. He was a teaching assistant at Ohio State University from 1910 to 1912. He entered the mathematics doctoral program at the University of Pennsylvania in 1912 and graduated there in 1916 with advisor Maurice Babb and thesis Some Extensions of the Work of Pappus and Steiner on Tangent Circles.
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Senecio madagascariensis Poiret International Environmental Weed Foundation (IEWF) The flowerhead, which is part of an unfirm corymb, is made up of disc florets and ray florets, and is small, yellow and daisy-like, from 1-2cm in diameter. The plant flowers between late autumn and early spring in its native area. The fruit is an achene that is 1.5-2.5mm long and is brown- coloured, with a pappus that is 4-6.5mm long.Auld BA, Meld RW (1992) 'Weeds an illustrated botanical guide to the weeds of Australia.
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The slewing drive is a modernized take on the worm drive mechanism that dates back many centuries and was widely used during the Renaissance Era. Pappus of Alexandria (3rd century AD), a Greek mathematician is credited for an early version of the endless screw, which would later evolve into the worm drive. This mechanism was also used by Leonardo da Vinci as a component in many of his designs for machines. It can also be found in the notebooks of Francesco di Giorgio of Siena.
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The heads contain yellow disc florets and generally either 5 or 8 ray florets each about a centimeter long. Senecio spartioides' silvery white pappus hairs, that carry the tiny, brown seeds aloft, attract as much attention as the golden yellow flowers. The hairs are common on many members of the Sunflower Family, most famously on Dandelions, Taraxacum officinale. "Oides" is a form of the Greek "oid", which means "similar to" and thus "spartioides" means "similar to spart(ium)", a genus of Fabaceae (Pea Family).
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The fruit below the corolla (called cypsela) is 4½–5 mm long, 1½–3 mm wide, narrowed at the tip, where it carries a collar, with a shaggy appearance due to white hairs. Like in all Asteraceae the calyx has changed and is called pappus. In this case it consists of two circles of rigid, white-yellowish, somewhat papery bristles, that carry small teeth at regular distances along their length, mostly 15–18 mm long, but some of the outer bristles only 2½ mm long.
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Here, Pappus observed that a regular dodecahedron and a regular icosahedron could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle. This observation has been generalized to higher- dimensional dual polytopes. # An addition by a later writer on another solution to the first problem of the book. Of Book IV the title and preface have been lost.
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The shorter pappus bristles are up to 1 mm long. Below the base of each corolla is a red-brown, dry, one-seeded, indehiscent fruit called cypsela that is about long and wide, inverted egg-shape in outline, with a slight ridge along the edge. The surface has few scales and long silky hairs, often with a narrow hairless zone bordering the edge. Heads without ray florets also occur from time to time in related species of the section Lignofelicia, in particular F. whitehillensis, F. filifolia subsp.
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Wu GuanYing said "I grew up in the countryside and was once a cow herder, so I know that the cow is one of the animals closest to human beings. Cows are well-known for their friendliness and their affinity to the humans who care for them." The colours of Fu Niu Lele come from traditional Chinese New Year drawings and toys. In the Official Paralympic mascot video, it becomes apparent that the mark on Fu Niu Lele's forehead is made of a dandelion seed pappus.
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A rosette may produce several flowering stems at a time. The flower heads are 2–5 cm (1" to 2") in diameter and consist entirely of ray florets. The flower heads mature into spherical seed heads sometimes called blowballs or clocks (in both British and American English)"blowball" entry, Collins Dictionary"Blowball", InfoPlease Dictionary"Clock", American Heritage Dictionary containing many single-seeded fruits called achenes. Each achene is attached to a pappus of fine hair-like material which enables wind-aided dispersal over long distances.
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Like in all asterids, the anthers are fused into a tube, though which the style grows, picking up the pollen that is discarded at the inside of the tube. The anthers have spurs at their base and appendages at their tip. The styles have hairless shafts and mostly hairless branches (sometimes with papillae at the outside). The one-seeded indehiscent fruits (called cypselas) are cylindrical and carry a pappus of many white, straw-colored or bright orange or pink bristles (which may be feather-like).
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The plant produces an inflorescence generally 25 centimeters to half a meter tall consisting of a solitary flower head or an array of up to three heads. The head is bell-shaped, sometimes widely so. It contains many orange to red-orange disc florets each about a centimeter long, and a fringe of several orange or reddish ray florets each up to 2 centimeters in length. The fruit is a long, narrow achene which may be 2 centimeters in length including its pappus of plumelike bristles.
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It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that is a transcendental number. The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.
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Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus: This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled . Modern research in this area has been focused on the Antikythera mechanism, another device built BC that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing.
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The fruit is an achene with a brown, gray, blue, or purple body tipped with a pappus of five long, spreading scales, the whole unit measuring 1 or 2 centimeters. This species is suspected to be a hybrid between Microseris douglasii and Uropappus lindleyi which may have evolved independently, possibly three times.Flora of North America The plant comes in a wide variety of appearances with variations in florescence properties and size, as well as height, ranging on a scale between both of the suspected parent taxa.
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The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares). Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation. The Hagia Sophia was designed by mathematicians Anthemius of Tralles and Isidore of Miletus. Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD).
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Surrounding the base of the corolla are about ten, quickly discarded, white, protruding bristly pappus bristles of about 1–2 mm (0.06–0.10 in) long. The relatively large, eventually yellowish to reddish brown, dry, one-seeded, indehiscent fruits called cypsellae are about long and wide, inverted egg-shaped, the surface and the edge of the otherwise hairless marginal ridges are covered with strong, up to 1 mm (0.04 in) long hairs, while the seedskin is covered in scales. The cypselae of the innermost disc florets are hairless.
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The inner whorl consists of feathery bristles of long, with or without barbs and fused at their base in a ring. The pappus extends beyond the disc floret tubes. The eventually brown, dry, one-seeded, indehiscent fruits called cypsellae are cylindrical to spindle-shaped, rarely egg-shaped in outline, about 3–4 mm (0.14–0.18 in) long and wide, with four or five prominently raised, mostly dark brown ribs. The cypselae of both ray and disc florets are covered with many shiny yellowish glands.
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Ratios are not equal in this sense; but they may be the same. :KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB). The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular :(J, G; D, B) = (J, Z; H, E). It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X. :Pappus- collection-7-136 Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.
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Normalis is a perfectly good Latin word meaning "measured with a norma," or square. Halley uses it to translate Pappus' eutheia, "right-placed," which has a more general sense of directionally right. For "the perpendicular to," the mathematical Greeks used "the normal of," where the object of "of" could be any figure, usually a straight line. What Fried is saying is that there was no standard use of normal to mean normal of a curve, nor did Apollonius introduce one, although in several isolated cases he did describe one.
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Slaughter argued that Pappotherium should have been a basal form close to the metatherian-eutherian divergence point; this mammal likely was an arboreal insectivore. Etymologically speaking, the name Pappotherium is a compound of the Latin words pappus (from ancient Greek πάππος, páppos, “grandfather”) and therium (from ancient Greek θηρίον, thēríon, “beast”, a common suffix among extinct mammals), with the full meaning of “mammal- grandfather”. The second part of the unique species' name, pattersoni, was instead chosen in honor of the American paleontologist Bryan Patterson. More recently, it has been recovered as a possible deltatheroidean.
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The top of the stem is occupied by an inflorescence of several flower heads, their hemispheric bases up to 2.5 centimeters (one inch) wide and lined with many small, green phyllaries with curving tips. Each flower head may have up to 30 narrow, pointed yellow ray florets between 1 and 2 centimeters (0.4-0.8 inches) long, surrounding a center of yellow disc florets. The fruit is a brown achene about a centimeter (0.4 inches) long including its long pappus of bristles.Flora of North America, Grindelia ciliata (Nuttall) Sprengel, 1826.
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The three species belonging to the section Anhebecarpaea are erect shrublets with overlapping, initially hairy leaves, large heads with hairless purplish ligulate florets and yellow disc florets, set at the tip of the stems in umbel-like inflorescences. The pappus bristles are all equal in length, firm, indehiscent and are set with many teeth, and the cypselas are yellow- to red-brown, those belonging to the ligulate florets hairless, those of the disc florets with short bristles. The three species are endemic to coastal areas of the Western and Eastern Cape.
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Species assigned to the section Dracontium are erect perennial herbs, with large leaves in a rosette at ground level and smaller bracts along a not or rarely shyly branched stem that carries one to four large heads with blue ligulate florets and yellow disc florets that are encircled by an involucre consisting of three worls of approximately equal sized bracts. The cypselas are brown, topped with one row of firm indehyscent pappus hairs and covered in short hairs. There are four species that are all restricted to the Drakensberg Mountains.
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Felicia echinataThe section Longistylus consists of erect, branching, annual or biennial plants with oppositely set leaves lower and alternately set leaves nearer the top, both types hairy, lancet-shaped, with entire or toothed margins. Stems carry many, small heads with white or yellow ligulate florets and yellow disc florets, surrounded by an involucre of four worls of bracts. The small cypselas are crowned by pappus that consists of dehyscent, almost feathery bristles. The six species can be found from Namibia to southern Angola, and Zimbabwe through Malawi to Uganda.
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E. argophylla is a perennial herb up to tall, appearing silvery because of many small hairs pressed against the leaves. Leaf blades are up to long, with wings running along the sides of the petioles. Appearing in April and May, the flower heads are yellow, at the ends of long peduncles, each head with as many as 35 ray florets and up to 500 tiny disc florets. The achene is strongly flattened, covered with small hairs, and sometimes with a pappus of 2 awns up to 2 mm long (unlike some of the related species).
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Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Pandrosion, a female mathematician of ancient Greece, found a numerically-accurate approximate solution using planes in three dimensions, but was heavily criticized by Pappus of Alexandria for not providing a proper mathematical proof. Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
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The original treatise by Marinus of Tyre that formed the basis of Ptolemy's Geography has been completely lost. A world map based on Ptolemy was displayed in Augustodunum (Autun, France) in late Roman times. Pappus, writing at Alexandria in the 4th century, produced a commentary on Ptolemy's Geography and used it as the basis of his (now lost) Chorography of the Ecumene. Later imperial writers and mathematicians, however, seem to have restricted themselves to commenting on Ptolemy's text, rather than improving upon it; surviving records actually show decreasing fidelity to real position.
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The pappus consists of many white bristles of about 3 mm (0.12 in) long standing out at an angle and set with protruding teeth from the base to the top. The dry, one-seeded, indehiscent fruits called cypselae are narrowly inverted egg- shaped in outline, about 2½ mm (0.1 in) and 1 mm (0.04 in) wide are dark brown in colour with two light ribs along the edge, are set with broad, very short hairs of 0.1 mm (0.004 in) long, and the surface without further sculpture or few scales.
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When the plant dies down in summer, the cypselas remain encased between the hardened bracts, presumably safe from harvester ants. After the first rain, which usually occurs the next winter, the bracts and pappus on the aerial flowerheads unfold, and the cypselas are dispersed by the wind, while many are gathered by ants. The cypselas in the underground flowerheads however germinate through the dead parts of the flowerhead, and remain protected against the ants. These seeds increase the chance that the plant continues its presence in a location that was favorable in the previous year.
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The first record of the conjecture dates back to 36 BC, from Marcus Terentius Varro, but is often attributed to Pappus of Alexandria (). The conjecture was proven in 1999 by mathematician Thomas C. Hales, who mentions in his work that there is reason to believe that the conjecture may have been present in the minds of mathematicians before Varro. It is also related to the densest circle packing of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area of the plane.
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Surrounding the base of the corolla are many white, deciduous pappus bristles of about 2 mm (0.1 in) long, that are strongly serrated near the base and weakly near the top. The eventually yellowish brown, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg-shaped, about long and wide, with a prominent, ridge along the margin, and scales on its surface. The middle long hairs that also occur scattered along its surface have a forked tip, but the hairs along the edge are not widened near their tips.
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Mature shrub It is a vigorous evergreen shrub growing to tall and wide, with silvery green, hairy leaves and yellow, daisy-like composite flowers in diameter. They bloom from early summer through to autumn and into winter in areas with mild climates. The fruits bear a single seed and are either hairless or covered in myxogenic (slime-producing) hairs, and may also be topped by a pappus of white or brown bristles. The Latin specific epithet pectinatus means “comb-like”, possibly referring to the deeply-divided, fernlike leaves.
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R. Jose the Galilean and Pappus discussed the subject with R. Akiva.Hagigah 14a; Genesis Rabbah 21 The tradition, quoted above, of the four who studied the secret doctrine mentions, besides Akiva, Simeon ben Azzai, Simeon ben Zoma, and Elisha ben Abuyah. The fate of the last-named, who was driven from Judaism by his experience, is said to have given rise to restrictive measures. The study of profane books was forbidden,Sanhedrin 100 and an interdiction of the public discussion of these subjects was issued, only R. Ishmael objecting.
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The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix. # A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
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Although functionally female, five staminodes can be found surrounding the style shaft. The style is long, and the style branches ar its tip are purplish in colour, line–shaped to elliptic, 1–1 mm long, with a blunt tip. The ray florets surround many bisexual disc florets with a yellow corolla up to long, which is mostly shorter than the pappus. The tube-shaped part at the base carries glandular hairs, and the five triangular lobes are recurved at the top, often carry a few hairs and have a resin duct along their margin.
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. The anthers are 1–2 mm long, with small triangular appendages at the top. The style is approximately long with two dark red to purplish, narrowly elliptic branches of long, each with a deltoid appendage of about wide and long. Surrounding the base of the corollas of both ray and disc florets are many, whitish or straw-coloured, pappus bristles in two whorls.
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The bone-colored style grows through the anther tube, collecting the pollen on hairs, and displaying it above the anter tube like in all other Asteraceae. The style branches are only and have rounded tips. At the base of the corolla the one-seeded indehiscent fruit (called cypselas) develops, which is not flattened, has four to five hardly visible nerves, turns brown and has lost any hairs when ripe, and is about ½ cm (0.2 in) high. On the top of the cypselas is a crown of six unequal scales (the pappus) of high.
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a triangular appendage. Around the base of the corolla are many white, toothed, persistent pappus bristles of about long, which become slightly wider towards the top. The eventually dark brown, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg-shaped, about long and wide, the surface slightly scaly, and covered with short hairs.
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For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist algebraic matroids that are not linear;. indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero. However, if a matroid is algebraic over a field F of characteristic zero then it is linear over F(T) for some finite set of transcendentals T over FOxley (1992) p.221 and over the algebraic closure of F.
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In this, his sources were primarily written in Greek and secondarily in Arabic, while his translations were primarily in Latin and secondarily in Italian. He was responsible for the publication of many treatises of Archimedes. He also translated the works of Aristarchus of Samos (On the sizes and distances of the Sun and the Moon), Pappus of Alexandria (Mathematical collection), Hero of Alexandria (Pneumatics), Ptolemy of Alexandria (Planisphere and Analemma), Apollonius of Perga (Conics) and Euclid of Alexandria (Elements). Among his pupils was Guidobaldo del Monte and Bernardino Baldi.
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The flower head is encompassed by between 10 and 18 white ray florets, each with a three-toothed shape; the florets tend to curve downwards around the edges and may occasionally have pistils, although these do not produce fruit. Beneath the flower proper, oval bracts of the plant form an involucre, with soft hairs on each; further bracts are bristled and sit at right angles to the flowers. ;Fruits: The fruits are achenes (with no pappus). They are wrinkled, ribbed with ten ridges, and have small glandular bumps across the surface.
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In 1949 the family went to England for "Ath", who had received a Nuffield Foundation fellowship, to do a doctorate on the evolution of the manuscript tradition of the Greek mathematician Pappus of Alexandria. She became an English coach, gaining a reputation as one of the best English coaches on the North Shore line. Many of her students were Asian and in 1972 she was made an MBE for "assistance to international relations, as tutor and 'mother' to Asian students". From 1978-80 she earned an M.A. in drama at Sydney University.
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When Friedrich Hultsch prepared his 1878 translation of Pappus's Collection from Greek into Latin, the manuscript of the Collection that he used referred to Pandrosion using a feminine form of address. Hultsch decided that this must have been a mistake, and referred to Pandrosion as masculine in his translation; many later scholars have followed suit. However, the 1988 English translation of Pappus by Alexander Raymond Jones "argued convincingly" that the original feminine form was not a mistake, and more recent scholarship has followed Jones in taking the position that Pandrosion was a woman.
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Lepidospartum squamatum is a large shrub often exceeding two meters in height which takes a spreading, rounded form, its branches are coated in woolly fibers and stubby leaves no more than 3 millimeters long. The inflorescence is a single flower head or small cluster of up to 5 heads at the ends of branches. The heads are discoid, bearing many yellow tubular disc florets and no ray florets. The fruit is a narrow achene a few millimeters long with a dull white to light brown pappus on top.
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The fruits have 5–7 ribs on each side and are tipped by two rows of small white hairs. The pappus remains at the top of each fruit as a dispersal structure. Each fruit contains one seed, which can be white, yellow, gray or brown depending on the variety of lettuce. The domestication of lettuce over the centuries has resulted in several changes through selective breeding: delayed bolting, larger seeds, larger leaves and heads, better taste and texture, a lower latex content, and different leaf shapes and colors.
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It was extensively studied in the 19th century.A theorem on the anharmonic ratio of lines appeared in the work of Pappus, but Michel Chasles, who devoted considerable efforts to reconstructing lost works of Euclid, asserted that it had earlier appeared in his book Porisms. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
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The head is discoid, containing no ray florets but many funnel-shaped disc florets with long, narrow lobes. These disc florets are usually yellow, but some plants bear white or pink colouration in certain populations. The corolla is generally yellow in most of the disc florets, one diagnostic separating other white and pink species such as L. nemaclada, but some outer florets may have a corolla that is dark purple, especially in white or pink flowering specimens. The fruit is an achene with a whitish pappus of bristles.
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. Around the base of the corolla are many white pappus bristles of about long. The dark brown, dry, one-seeded, indehiscent fruits called cypsellae are ellips- to inverted egg-shaped, about long and wide, with a marginal ridge, while the surface has some weak scales and is evenly covered in long hairs. Felicia elongata is a diploid having eight sets of homologue chromosomes (2n=16).
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The earliest remaining writings regarding levers date from the 3rd century BC and were provided by Archimedes. "Give me a place to stand, and I shall move the earth with a lever" is a remark attributed to Archimedes, who formally stated the correct mathematical principle of levers (quoted by Pappus of Alexandria). One of the earliest examples of a compound lever is from Han dynasty (202 BC - 220 AD) crossbow trigger mechanisms which featured a triple compound lever. Such a mechanism was placed within the crossbow stock itself.
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a narrowly triangular appendage. Surrounding the base of the corolla are many, yellowish white, shallowly serrated, more or less deciduous pappus bristles, all about equal in length at . The eventually yellowish brown to reddish, dry, one-seeded, indehiscent fruits called cypsellae are oval in outline, about long and wide, with a weak ridge along the margin.
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Fleischmannia species were traditionally included as part of the large Eupatorium, recognized in the broad sense based on achenes with five ribs and a pappus of capillary bristles. Fleischmannia is distinguished morphologically by having the inner surface of the corolla lobes markedly papillate (requires magnification to see), the carpopodia symmetrical and stopper-shaped, and the involucres small and with 3 series of subimbricate bracts. The number of flowers per head is typically 20 or more (up to 50). Many species are herbs and somewhat weedy, and have leaves with long petioles.
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Surrounding the base of the corolla of both ray- and disc florets are two whorls of pappus. The outer whorl consists of few white, free, delicate, barbed bristles of long. The inner whorl consists of more white feathery bristles of up to long, barbed near the base. The eventually bright brown, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg- shaped, about long and wide, with four or five ribs along their lengths, with shiny glands and many, long twin hairs on a further smooth seed skin.
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The tube is cylinder-shaped and has a few to many blunt hairs, which are several rows of cells thick and are topped with glands. The straps are white or cream-coloured, elliptic or obtuse in shape, mostly have four veins and is split in three teeth at the tip. From the mouth of the ray floret tube emerges a style that splits in two outward curling, hairless branches each topped by an obtuse conical appendage. Surrounding the base of the ray floret corolla are many, white, barbed pappus bristles which are quickly shed in the ray florets.
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Diagram used in reconstructing one of Hipparchus's methods of determining the distance to the Moon. This represents the Earth–Moon system during a partial solar eclipse at A (Alexandria) and a total solar eclipse at H (Hellespont). Hipparchus also undertook to find the distances and sizes of the Sun and the Moon. He published his results in a work of two books called Perí megethōn kaí apostēmátōn ("On Sizes and Distances") by Pappus in his commentary on the Almagest V.11; Theon of Smyrna (2nd century) mentions the work with the addition "of the Sun and Moon".
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Each flower head carries about twenty ray florets, with a purplish blue ligula of about long and 3 mm (0.12 in) wide. These encircle numerous disc florets, with a yellow corolla of up to 3 mm long. Encircling the base of the corolla are many white, toothed, deciduous, pappus bristles of 2–3 mm (0.08–0.12 in) long. The eventually black, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg- shaped, about 2½ mm (0.1 in) long and wide, set with fine scales and with up to long hair, and are edged with an mostly brighter coloured ridge around the outline.
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Agoseris is native to North America, South America and the Falkland Islands.Flora of North America Mountain- or false dandelion Agoseris Rafinesque, Fl. Ludov. 58. 1817. In general appearance, Agoseris is reminiscent of dandelions and are sometimes called mountain dandelion or false dandelion. Like dandelions the plants are (mostly) stemless, the leaves forming a basal rosette, contain milky sap, produce several unbranched, stem-like flower stalks (peduncles), each flower stalk bearing a single, erect, liguliferous flower head that contains several florets, and the flower head maturing into a ball-like seed head of beaked achenes, each achene with a pappus of numerous, white bristles.
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Two solutions whose sides pass through A, B, C In geometry, the Cramer–Castillon problem is a problem stated by the Swiss mathematician Gabriel Cramer solved by the Italian mathematician, resident in Berlin, Jean de Castillon in 1776., page 1. The problem consists of (see the image): Given a circle Z and three points A, B, C in the same plane and not on Z, to construct every possible triangle inscribed in Z whose sides (or their elongations) pass through A, B, C respectively. Centuries before, Pappus of Alexandria had solved a special case: when the three points are collinear.
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The approximately seven, deep blue ray florets surrounding the disc have a hairy tube, at the top changing into a spreading blade of about 15 mm (0.6 in) long and 5 mm (0.2 in) wide. The dry, one-seeded, indehiscent fruits (or cypselae) of the ray florets lack pappus. The many blackish blue, rarely brown-red and yellow, disc florets are bisexual and about 5 mm (0.20 in) long. Like in all Asteraceae, the five anthers have merged into a hollow tube through which the style grows when the floret opens, while gathering the pollen on its shaft.
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The one-seeded indehiscent fruits are not embedded in the common base of the florets receptacle, is inverted cone-shaped or oblong, has three or four ribs, is at least 3 mm long, and half as wide, with a smooth surface or with tiny wrinkles and hearless. At the tip is one row of scales (the pappus) of ½–1 mm long, that are free in the outer florets, but merged at their foot in disc florets. These scales are split into twelve to fifteen standing, awl-shaped lobes, with a long, narrow tip, divided in side-lobes and without hair.
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The functionally male florets occur in small groups and have very short individual stems, mostly in the centre of a larger cluster of female florets. The corollas are small, have (three or) four triangular lobes, greenish yellow and contain (three or) four stamens, carry yellowish or purplish anthers that are blunt on both ends and the filament is not extended beyond the anther. The fruit at the base of the male flower is much reduced and void, and pappus may consist of some irregular scales or be entirely abstent. Pollen is globe- shaped and has three sunken furrows (a type called tricolpate).
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The characteristics of Pappus's Collection are that it contains an account, systematically arranged, of the most important results obtained by his predecessors, and, secondly, notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the various books as valuable, for they clearly outlined the contents and the general scope of the subjects to be treated. From these introductions, one can judge Pappus's writing style, which is excellent and even elegant when he is free from the shackles of mathematical formulae and expressions.
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The inflorescence is a cluster of 10 to 50 or more small flower heads, each on a short peduncle. The flower head has a center of hairy, glandular, star-shaped yellow disc florets and a fringe of four to nine yellow ray florets each about 2 millimeters long. The fruit is an achene a few millimeters long with a small pappus at the tip. Like many Channel Islands endemics, this plant was threatened with extinction by the herbivory of the feral goats living on the islands; the goats have since been removed and the plant is recovering.
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From it rises a style that is circular in cross-section, ending in two pointy, line-shaped branches of about 1 mm (0.04 in) long, its outer margins functioning as stigma. There is no pappus. The one-seeded, indehiscent, dry fruit (called cypsela) is flattened oval in shape, 2½–3 mm (0.10–0.12 in) long and about 1½ mm (0.06 in) in diameter, with thickened margins, its surfaces and margins being covered with pale brown hairs pressed against its surface. The fourteen or fifteen deep purple disc florets are bisexual, but those in the center of the head do not develop seed.
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It is topped by a purple style that is circular in cross-section, ending in two pointy, recurved, flattened, line-shaped branches of about 1 mm (0.04 in) long, its outer margins functioning as stigma. In the central florets the style ends in triangular grainy branches. The pappus on each of the cypselas of the tube florets consists of one row of about 25 spreading, short, stiff, hooked bristles of 3–4 mm (0.12–0.16 in) long but feathery near middle, and merged at their base, forming a short, white collar. The cypselas are identical to those of the ray florets.
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Flower of Narcissus showing an outer white corolla with a central yellow corona (paraperigonium) Flower of Passiflora incarnata showing corona of fine appendages between petals and stamens An additional structure in some plants (e.g. Narcissus, Passiflora (passion flower), some Hippeastrum, Liliaceae) is the corona (paraperigonium, paraperigon, or paracorolla), a ring or set of appendages of adaxial tissue arising from the corolla or the outer edge of the stamens. It is often positioned where the corolla lobes arise from the corolla tube. The pappus of Asteraceae, considered to be a modified calyx, is also called a corona if it is shaped like a crown.
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From the 17th to the 19th centuries this subject seems to have had great fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus Albert Girard says in his Traité de trigonometrie (1626) that he hopes to publish a restoration. About the same time Pierre de Fermat wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita (see Œuvres de Fermat, i., Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus.
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Baccharis glutinosa is a rhizomatous perennial herb growing to heights between one and two meters. The lance-shaped leaves are up to about 12 centimeters long and have short winged petioles. The foliage and inflorescences are resinous and sticky. The plants are dioecious, with male plants producing clusters of up to 40 whitish staminate flowers and female plants bearing bunches of up to 150 fluffy whitish pistillate flowers with a hairlike pappus attached to each developing fruit.Flora of North America, Saltmarsh baccharis, Douglas’s falsewillow, Baccharis douglasii de Candolle in A. P. de Candolle and A. L. P. P. de Candolle, Prodr.
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Like many geometers of the time, Nicomedes was engaged in trying to solve the problems of doubling the cube and trisecting the angle, both problems we now understand to be impossible using the tools of classical geometry. In the course of his investigations, Nicomedes created the conchoid of Nicomedes; a discovery that is contained in his famous work entitled On conchoid lines. Nicomedes discovered three distinct types of conchoids, now unknown. Pappus wrote: "Nicomedes trisected any rectilinear angle by means of the conchoidal curves, the construction, order and properties of which he handed down, being himself the discoverer of their peculiar character".
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Pappus theorem: proof If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique. Because of the parallelity in an affine plane one has to distinct two cases: g ot\parallel h and g \parallel h. The key for a simple proof is the possibility for introducing a "suitable" coordinate system: Case 1: The lines g,h intersect at point S=g\cap h. In this case coordinates are introduced, such that \;S=(0,0), \; A=(0,1), \;c=(1,0)\; (see diagram).
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Calflora taxon report, Calycadenia mollis A. Gray, soft calycadenia, soft western rosinweed Calycadenia mollis is an annual herb producing an erect, hairy stem up to 90 centimeters tall. The leaves are linear in shape and up to 8 centimeters long, the longest ones often toward the middle of the stem. The inflorescence bears several long clusters of small, very glandular flower heads, each of which has one or more white, yellow, or red triple-lobed ray florets around a center of disc florets. The fruit is an achene; those arising from the disc florets have a pappus of about eight scales.
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Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions.
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A rich repertoire of geometrical and algebraic methods have been developed to solve Apollonius' problem, which has been called "the most famous of all" geometry problems. The original approach of Apollonius of Perga has been lost, but reconstructions have been offered by François Viète and others, based on the clues in the description by Pappus of Alexandria. The first new solution method was published in 1596 by Adriaan van Roomen, who identified the centers of the solution circles as the intersection points of two hyperbolas. Van Roomen's method was refined in 1687 by Isaac Newton in his Principia, and by John Casey in 1881.
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Viète first solved some simple special cases of Apollonius' problem, such as finding a circle that passes through three given points which has only one solution if the points are distinct; he then built up to solving more complicated special cases, in some cases by shrinking or swelling the given circles. According to the 4th-century report of Pappus, Apollonius' own book on this problem—entitled (', "Tangencies"; Latin: De tactionibus, De contactibus)—followed a similar progressive approach. Hence, Viète's solution is considered to be a plausible reconstruction of Apollonius' solution, although other reconstructions have been published independently by three different authors.Simson R (1734) Mathematical Collection, volume VII, p. 117.
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Late fall flowerheads, with purple sheath around silky white pappus Carl Linnaeus, who first named and described Baccharis halimifolia (1775 portrait by Alexander Roslin) Baccharis halimifolia is a fall-flowering shrub growing to about 12 ft (4 m) high and comparably wide, or occasionally a small tree. Its simple, alternate, thick, egg-shaped to rhombic leaves mostly have coarse teeth, with the uppermost leaves entire. These fall-flowering Baccharis plants are dioecious, with male and female flowers on separate individuals. Their flowers are borne in numerous small, compact heads in large leafy terminal inflorescences, with the snowy-white, cotton-like female flower-heads showy and conspicuous at a distance.
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This may be shown by inverting in a circle centered on the tangent point A. The circle of inversion is chosen to intersect the nth circle perpendicularly, so that the nth circle is transformed into itself. The two arbelos circles, CU and CV, are transformed into parallel lines tangent to and sandwiching the nth circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle C0 and the final circle Cn each contribute ½dn to the height hn, whereas the circles C1-Cn−1 each contribute dn. Adding these contributions together yields the equation hn = n dn.
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In the 2nd and 3rd centuries coins were made in his honour in Bithynia that bear his name and show him with a globe; this supports the tradition that he was born there. Relatively little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. Most of what is known about Hipparchus comes from Strabo's Geography and Pliny's Natural History in the 1st century; Ptolemy's 2nd- century Almagest; and additional references to him in the 4th century by Pappus and Theon of Alexandria in their commentaries on the Almagest.
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The Pappus configuration may be formed from two triangles that are perspective figures to each other in three different ways, analogous to the interpretation of the Reye configuration involving desmic tetrahedra. If the Reye configuration is formed from a cube in three-dimensional space, then there are 12 planes containing four lines each: the six face planes of the cube, and the six planes through pairs of opposite edges of the cube. Intersecting these 12 planes and 16 lines with another plane in general position produces a 163124 configuration, the dual of the Reye configuration. The original Reye configuration and its dual together form a 284284 configuration .
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The anthers produce cream-coloured pollen, are themselves deep blue, and have a shortly triangular appendage at the top. The whitish pappus on each of the cypselas of the disc florets consists of many bristles of about 5 mm (0.20 in) long, with short teeth in the lower quarter but feathery further up, the side branches about 0.3 mm (0.012 in) long. The cypselas are elliptical, about 4½ mm (0.18 in) long and 2 mm (0.08 in) thick, with scaly, thickened ribs along the edge and the surface set with blunt hairs of 0.7 mm (0.028 in). Felicia heterophylla is a diploid having five sets of homologue chromosomes (2n=10).
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The functionally female florets are each stiffly enclosed by a large, cone-shaped green and white bract, and clustered with other one-flowered female flower heads, mostly surrounding groups of male florets. The inconspicuous corolla consists of whitish threads and surrounds the base of a whitish style with long arms which have rounded tips. The one-seeded indehiscent fruit (or cypsela) of the female florets are tiny, ovoid, with rows of stiff hairs on the ribs, and felty overall with long twin hairs, which have thin cell walls. The pappus on top consists of scales ending in a long drawn tip and with a row of hairs along the edges.
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Apud Weidmannos, 1877, pp. 19–29. Book III contains geometrical problems, plane and solid. It may be divided into five sections: # On the famous problem of finding two mean proportionals between two given lines, which arose from duplicating the cube, reduced by Hippocrates of Chios to the former. Pappus gives several solutions to this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one.
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The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated. # Let a helix be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis.
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Like the other 13 species members of its genus, Eriophyllum latilobum presents generally alternate leaves ranging from entire to nearly compound. The flower heads are grouped in radiate, flat-topped heads, with an hemispheric to nearly conic involucre. Phyllaries are either free, or more or less fused, their receptacle flat, but naked and conic in the center. The ray flowers (the "petals") have yellow ligules entire to lobed. Fruits are 4-angled cylindric achenes in the outer flowers, but are generally club-shaped for the inner flowers; the pappus is somewhat jagged.Mooring, Madroño 38:213–226, (1991) Eriophyllum latilobum occurs as a subshrub between 20 and 50 centimeters in height.
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In his text The Sand Reckoner, the natural philosopher Archimedes gives an upper bound of the number of grains of sand required to fill the entire universe, using a contemporary estimation of its size. This would defy the then-held notion that it is impossible to name a number greater than that of the sand on a beach or on the entire world. In order to do that, he had to devise a new numeral scheme with much greater range. Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad; was 10,000, was 10,0002 = 100,000,000, was 10,0003 = 1012 and so on.
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Young leaves of C. eriophorum can be eaten raw, and young stems can be peeled and eaten raw or cooked, first being soaked in water which removes the bitterness. The flower buds can be used in a similar way to artichokes, but being smaller this is a fiddly business; an edible oil can be extracted from the seeds, and the pappus can be used as tinder for lighting fires. The plant is hardy and can be easily grown in a sunny position in the garden, in a wildflower meadow or in dappled shade in a woodland garden. In the United Kingdom, it flowers between July and September.
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The older geometers regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed, and finally a porism as directed to finding what is proposed ('). Pappus goes on to say that this last definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as "" (to leîpon hypothései topikoû theōrḗmatos), that which falls short of a locus-theorem by a (or in its) hypothesis. Proclus points out that the word porism was used in two senses. One sense is that of "corollary", as a result unsought, as it were, but seen to follow from a theorem.
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Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm; Elements, Prop. VII.2) and the first known proof of the infinitude of primes (Elements, Prop. IX.20).
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He wrote the first discourse on the principles of the sundial, known as Analemma.encyclopedia.com retrieved 15/09/2011 a commentary on this having later been written by Pappus of Alexandria, that is no longer extant. A small number of sentences having survived the centuries and attributed to him are known; these comment on:the differences (for the purpose of defining) between astronomy and natural science (physiologia © 2001–2011 Douglas Harper etymonline.com Retrieved 2011-09-15) the word meanings for cosmos and star, the nature of (the things being) stars and Γαλαξίαςor Kosmas Milt Markatos (2010) hellenes-markatos.gr website (referencing definition of greek) Retrieved 2011-09-15 [Galaxias oxforddictionaries.
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Like other members of the tribe Cichorieae, lettuce inflorescences (also known as flower heads or capitula) are composed of multiple florets, each with a modified calyx called a pappus (which becomes the feathery "parachute" of the fruit), a corolla of five petals fused into a ligule or strap, and the reproductive parts. These include fused anthers that form a tube which surrounds a style and bipartite stigma. As the anthers shed pollen, the style elongates to allow the stigmas, now coated with pollen, to emerge from the tube. The ovaries form compressed, obovate (teardrop-shaped) dry fruits that do not open at maturity, measuring 3 to 4 mm long.
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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.Heath (1956), p. 2. 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).
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In the center of each corolla are five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a narrowly triangular appendage. Around the base of the corolla are numerous, yellowish white, short-toothed, persistent pappus bristles, which are all of the same length, up to about . The dry, one- seeded, indehiscent fruits called cypsellae are dark brown with a yellowish brown ridge along the outline, elliptic, about 3 mm long and 1 mm wide, thinly silky haired but densely hairy on the edge, and the hairs seldomly wear off.
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Cleonides () is the author of a Greek treatise on music theory titled Εἰσαγωγὴ ἁρμονική Eisagōgē harmonikē (Introduction to Harmonics). The date of the treatise, based on internal evidence, can be established only to the broad period between the 3rd century BCE and the 4th century CE; however, treatises titled eisagōgē generally began to appear only in the 1st century BCE, which seems the most likely period for Cleonides' work.Jon Solomon, "Cleonides [Kleoneidēs]", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001). The attribution of the Eisagōgē in some manuscripts to Euclid or Pappus is incompatible with the Aristoxenian approach adopted in the treatise.
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Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones.
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They are slender to broad, up to 15 centimeters long, and usually with 3-5 pairs of lobes along the margins (these sometimes lacking). The peduncle of the inflorescence can be as tall as 45 centimeters but is usually much shorter. The flower head is up to 2 centimeters wide, surrounded by glabrous to hairy phyllaries, and contains yellow ray florets (the outer ones often have a purple strip on the lower surface) but no disc florets. The fruit is an achene between 5-12 millimeters long; the lower part of the achene contains a single seed, while the upper portion of the achene forms a slender beak that possesses a terminal, white pappus.
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In the center of each corolla are five free filaments and five anthers merged into a tube, through which the style grows when the floret opens, hoovering up the pollen on its shaft. At the tip of both style branches is a triangular appendage. Around the base of the corolla is one whorl of vigorous, equally long, white, shortly toothed, but smooth at their base, largely persistent pappus bristles of about long. The eventually black, dry, one-seeded, indehiscent fruits called cypsellae are inverted egg-shaped to elliptic, about long and 1½ mm (0.06 in) wide, have a ridge along the outline, are hairless or rarely have a few bristles near the top, and a further smooth seedskin.
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His original work is lost and his solids come down to us through Pappus. ;China Cubical gaming dice in China have been dated back as early as 600 B.C. By 236 AD, Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations. ;Islamic civilisation After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam). The 9th century scholar Thabit ibn Qurra gave formulae for calculating the volumes of polyhedra such as truncated pyramids.
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A projective plane satisfying Pappus's theorem universally is called a Pappian plane. Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes. There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring and Wedderburn's theorem that finite division rings must be commutative; give a proof that uses only more "elementary" algebraic facts about division rings.
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Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really equivalent (as Chasles makes them) to Euclid's first seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the "lemma to the first Porism" relates intelligibly, being a particular case of it. An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886, ch. viii.). Observing, e.g.
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According to Pappus, given three or four lines in a plane, the problem is to find the locus of a point that moves so that the product of the distances from two of the fixed lines (along specified directions) is proportional to the square of the distance to the third line (in the three line case) or proportional to the product of the distances to the other two lines (in the four line case). In solving these problems and their generalizations, Descartes takes two line segments as unknown and designates them and . Known line segments are designated , , , etc. The germinal idea of a Cartesian coordinate system can be traced back to this work.
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The botanist Christian Gottfried Daniel Nees von Esenbeck established this genus in 1833 because he felt that a plant he examined, now thought to be a cultivated variety of New York aster, was sufficiently distinct from the rest of the genus Aster to warrant its own genus. He emphasized the uniqueness of this plant in having its pappus hairs arranged in a coherent, basal ring. This structure is the basis for the scientific name of this genus, which derives from Ancient Greek (sýmphysis) "growing together" and (thríks; stem trich-) "hair". Unfortunately, this characteristic ring is not generally shared by most New York aster pappi, nor is it characteristic of any other plants included in the modern concept of Symphyotrichum.
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Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8383, the Möbius–Kantor configuration... In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9393, the Pappus configuration.. The Hesse configuration may in turn be augmented by adding four points, one for each triple of non-intersecting lines, and one line containing the four new points, to form a configuration of type 134134, the set of points and lines of the projective plane over the three-element field.
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The exact origins of John Troglita are unclear. He may have been born in Thrace, but his peculiar surname might indicate provenance from Trogilos (Greek: Τρώγιλος) in Macedonia.. According to information provided by the 6th-century historian Procopius of Caesarea and Troglita's panegyrist Flavius Cresconius Corippus, he was the son of a certain Evanthes, and had at least one brother named Pappus. Troglita himself married a "daughter of a king", probably a barbarian chieftain, and had a son, Peter. John Troglita is first mentioned as having participated in the Vandalic War (533–534) under Belisarius, and may be identifiable with another John, who commanded a unit of foederati in the battles of Ad Decimum and Tricamarum.
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The bracts in the inner whorl are long and wide, eventually hairless. Each flower head contains twelve to sixteen pink, functionally female ray florets, with a closed tube at the base of about long set with some glandular hairs, and with a line-shaped strap that radiates out from the head of long, bluntly ending in three lobes, narrower towards the base, and with five to seven veins. From the mouth of the ray floret tubes extends a tube consisting of five infertile staminodes, through which a forked style grows. The ray florets surround many bisexual disc florets with a yellow corolla of about long, which is only slightly longer than the pappus.
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Because the eclipse occurred in the morning, the Moon was not in the meridian, and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 81 Earth radii. In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. This would correspond to a parallax of 7′, which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2′; Tycho Brahe made naked eye observation with an accuracy down to 1′).
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The pale to dark brown, one-seeded, dry, indehiscent fruits (or cypselae) of both ray and disc florets are cylinder-shaped or somewhat flattened, carry apparently two, or four to seven narrow, mostly contrasting ribs, and are covered by shiny, yellow glands and deeply divided, silvery or golden twin hairs on a further unadorned surface. They carry at their tip the modified calyx called pappus that is yellowish white, persistent, and arranged in two whorls. The outer whorl consists of free, up to 3 mm (0.012 in) long, barbed or feathery bristles and an inner whorl of feathery bristles of up to 9 mm (0.36 in) long, that are merged into a ring at the base. It has eighteen homologous sets of chromosomes (2n= 36).
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He was born in the manse at Rothesay, on the Isle of Bute, on 15 January 1717, the son of Rev Dugald Stewart, the local minister, and his wife, Janet Bannantyne. He was educated at Rothesay Grammar School, then entered the University of Glasgow in 1734, where he studied under the philosopher Francis Hutcheson and the mathematician Robert Simson, the latter from whom he studied ancient geometry. A close friendship developed between Simson and Stewart, in part because of their mutual admiration of Pappus of Alexandria, which resulted in many curious communications with respect to the De Locis Planis of Apollonius of Perga and the Porisms of Euclid over the years.John Playfair, Biographical Account of Matthew Stewart, D.D., Transactions of the Royal Society of Edinburgh, Vol.
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According to Frans van Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the combination of these equations in all possible ways, and of new equations thus derived from them leads to the discovery of innumerable new properties of the figure, and here we have "porisms." The discussions, however, between Breton and Vincent, in which C. Housel also joined, did not carry forward the work of restoring Euclid's Porisms, which was left for Chasles. His work (Les Trois livres de porismes d'Euclide, Paris, 1860) makes full use of all the material found in Pappus. But we may doubt its being a successful reproduction of Euclid's actual work.
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In one respect 'Abd-al- Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. [...] Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century.
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Introduced species become invasive when they compete with natives or with crops. Senecio vulgaris is not known to be a strong competitor but it has been known to reduce mint production. There is evidence that it is not a strong invasive and sometimes protective of critically endangered native plants. The approximately long pappus seeds of Senecio vulgaris, each plant capable of producing 25,000 or more seeds (1,700 seeds per plant are more likely) with three generations of the plant per year; seeds that are widely dispersed by the wind, have been identified as a contaminant of cereal and vegetable seeds and a poison to some livestock; there is some inspiration to understand the growth stages and determine some control methods.
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The third type of aerial cypsela has intermediate pappus and may remain in the flower head or be carried off by the wind. The subterranean cypselas spread germination over time, with one type evolved to extend the presence of this annual into the next growing season at a location of proven suitability, while the other type contributes to the soil seed bank, and so hatches against unfavorable years. The aerial cypselas on the other hand spread the progeny into new areas. Hence, Catananche lutea through its fruits shows different survival strategies by having quick and delayed germination, in situ, short distance and long distance seed dispersal, self- and cross-fertilization, as well as having some ripe seeds already early on in the growing season.
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These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms. The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ). :Pappus-collection-7-129 Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then :KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).
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The style of each ray floret is up to long, the upper is split into two line- to ellipse-shaped branches. The ray florets do not contain staminodes. In the center of the head are many bisexual disc florets that are shorter than the longest pappus bristles, and these florets consist of a tube at their base of long, with some irregularly spread glandular hairs, and five upright or recurved triangular lobes of about long, and often have a resin duct along their margin, and sometimes set with long hairs and shiny glands with rounded tips. The five anthers, which are, as is usual in the entire family Asteraceae, fused in a tube, are about 1–2 mm (0.06–0.08 in) long, each with a triangular appendage of about mm (0.02 in) at their tip.
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The taxa in the section Felicia may be woody shrublets or annual herb, nearly always with fully alternately set, thin to succulent, mostly narrow, variably hairy leaves, and many to few heads, always with ligulate florets, that are purple to white or seldomly yellow, and yellow disc florets that may turn reddish when aging, each head encircled by an involucre of three to four worls of overlapping bracts with resin ducts, the outer bracts clearly smaller. Short triangular style braches entirely set with papillae. Cypselas crowned with one row of soft, equal length, more or less dehyscent pappus hairs, and its surface initially with very short hairs, later often without. The twenty eight species and several subspecies occur in southern Africa, but are concentrated in the south-western Cape.
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Lulianos and Paphos (alt. sp. Julianus and Pappus) (2nd-century CE) were two wealthy Jewish brothers who lived in Laodicea in Asia Minor, contemporaries with Rabbi Yehoshua ben Hananiah,Midrash Rabba (Genesis Rabba, P. Toldot § 64:8) and who suffered martyrdom at the hands of the Roman legate. An anecdote about the lives of these two illustrious Grecian-Jewish citizens has come down in the Midrashic literature stating that, during the days of Hadrian, the emperor mulled over the thought of rebuilding Israel's Temple. When the news reached Lulianos and Paphos who were very wealthy, they set-up tables from Acco to Antioch, hoping thereby to allow Jewish pilgrims to exchange their local currency for coins in specie, or else provide other basic needs for the people before proceeding on to Jerusalem.
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Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time. The surface area included between this curve, and its base is found – the first known instance of a curved surface quadrature. The rest of the book treats the trisection of an angle and the solution of more general problems of the same kind through the quadratrix and spiral. In one solution to the former problem is the first recorded use of the property of a conic (a hyperbola) regarding the focus and directrix.
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Aristarchus's 3rd century BCE calculations on the relative sizes of, from left, the Sun, Earth and Moon, from a 10th-century CE Greek copy On the Sizes and Distances (of the Sun and Moon) (Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], Peri megethon kai apostematon) is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BCE. This work calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius. The book was presumably preserved by students of Pappus of Alexandria's course in mathematics, although there is no evidence of this. The editio princeps was published by John Wallis in 1688, using several medieval manuscripts compiled by Sir Henry Savile.
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Piero della Francesca's image of a truncated icosahedron from his book De quinque corporibus regularibus The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509.
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Related Asteraceae genera such as Chrysoma, Euthamia, and Oreochrysum have been included within Solidago at one point or another, but morphological evidence has suggested otherwise. In a study comparing morphological characters of Solidago and related subgroups, the authors consider the subjectivity of classifying a genus, and how to define it within broader tendencies concerning the taxonomy of North American Asteraceae. Little to no differences were observed between Solidago and the subgroups in terms of karyotype. However, external morphological characters such as habit, or the general appearance of the plant and how a suite of traits contribute to its phenotype; pappus size; and the point of freeing of stamen filaments from the corolla tube, are useful classification schemes for Solidago, since they are applied to differentiating between Asteraceae taxa.
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As in all Asteraceae, the anthers are merged over their length to form a tube, through which the style grows, while gathering pollen from the anthers that have opened at the inside of the tube. The cream- colored anthers are pointy at their tip, and carry two, about 3¼ mm long, woolly haired tails at their foot, which are free from the filament. The yellow style is rounded at the tip and is rough on the outside lower than the split and the branches are about ⅓ mm long. The indehiscent, one-seeded fruits (called cypsela) are green when growing and brown when ripe, narrowly inverted cone-shaped, covered in bristles of twin hairs, with on top the calyx that has changed to barbed hairs, plume-like towards the tip, and is called pappus.
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Printzia has tailed anthers and blunt style branches that are only downy at their tips, while Felicia has tail-less anthers and pointy style branches that are downy over the entire length. Garuleum lacks bristles on its cypselas and has finely pinnate leaves, while Felicia has cypselas crowned by one row of hair-like bristles and mostly entire leaves, sometimes with deep cutting teeth. Amellus has pappus consisting of scales and bristles, while Felicia only has bristles. The species of Polyarrhena have a strong likeness to the species of Felicia section Anhebecarpaea, but have bisexual ligulate florets, that are white, and have a pink wash on the outer surface, and male or fully infertile disc florets, while in the section Anhebecarpaea ligulate florets are female and blue-purple in color and the disc florets are bisexual and do set seed.
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Philosophy, surprisingly, was only 6 percent while arts, liberal and mechanical, "magick," etc. was 3 percent. The remaining subjects were as follows: medicine, surgery, and "chymistry," 2 percent; law, 2 percent; voyages and travels, 1 percent; philosophical history, 1 percent, and miscellaneous, 5 percent. Logan's library contained many 17th and 16th century classical works such as a 1615 edition of Archimedes' works, the mathematical treatise of Pappus of Alexandria printed in 1660, an Aratus of Soles from 1672, Elzevir's architecture publication of 1649 from Amsterdam, Johann Vossius' De Quatuar Artibus Popularibus published in 1650, and a 1599 edition of astronomy edited by Barthelemy Pitiscus In one famous episode, Logan was reading a treatise on early astronomy by Johann Fabricius and read that the first printed edition of Greek astronomer Ptolemy's Almagest was printed in Greek in 1538.
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The program has to be gathered from the book itself. At the beginning is the well-known generalization of Euclid I.47 (Pappus's area theorem), then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. This and several other propositions on contact, e.g., cases of circles touching one another and inscribed in the figure made of three semicircles and known as arbelos ("shoemaker's knife") form the first division of the book; Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes (already mentioned in Book I as supplying a method of doubling the cube), and the curve discovered most probably by Hippias of Elis about 420 BC, and known by the name, τετραγωνισμός, or quadratrix.
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The entire head may move tracking the sun, like a "smart" solar panel, which maximizes reflectivity of the whole unit and can thereby attract more pollinators. At the base of the head, and surrounding the flowers before opening, is a bundle of sepal-like bracts or scales called phyllaries, which together form the involucre that protects the individual flowers in the head before opening. The individual heads have the smaller individual flowers arranged on a round or dome-like structure called the receptacle. The flowers mature first at the outside, moving toward the center, with the youngest in the middle. The individual flowers in a head have 5 fused petals (rarely 4), but instead of sepals, have threadlike, hairy, or bristly structures called pappus, which surround the fruit and can stick to animal fur or be lifted by wind, aiding in seed dispersal.
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The Synodicon Vetus is presumed to have been written soon after its last recorded ecumenical council, at the end of the 9th or beginning of the 10th century, between 887 and 920, but most likely at the end of the 9th century.Cyril Mango, The Oxford History of Byzantium It circulated for centuries in handwritten copies, and was first printed in 1601, produced by theologian Johannes Pappus, who worked from a shortened version he obtained from the prolific scribe and bookseller Andreas Darmarios, who was said to have brought an original manuscript out of the Morea. Darmarios had somewhat of a questionable reputation that has led some scholars to blame him for the introduction of various errors and supposed fabrications to the work. It is published complete, both in Greek and Latin, in the Bibliotheca Graeca of Johann Albert Fabricius.
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F. echinata has much in common with the other two species of the section Anhebecarpaea (F. westae and F. nordenstamii), which all have more than two whorls of involucral bracts, white ligulate florets with a purplish wash on the rear, pappus hairs of equal length and the surface of the cypselas of the ligulate florets bold, while the surface of the cypselas of the disc florets is covered in short bristly hairs. F. westae however has narrow lancet-shaped leaves of at most 1½ cm (0.6 in) wide that are inclined upwards and pressed against the stem, while in F. echinata the leaves are narrowly egg-shaped, about 3 cm (1¼ in) wide and curved outward from the stem. F. nordenstamii has long involucral bracts with dense long hairs, while F. echinata has 1 cm long involucral bracts with stiff bristly hairs, later becoming bold.
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Generalization for arbitrary triangles, green area Construction for proof of parallelogram generalization Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by Pappus of Alexandria in 4 ADFor the details of such a construction, see Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, , pp.
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Of the possible means of transport by wind, water, bird or human agency, the authors dismiss immediately scenarios involving dispersal by wind or via bird droppings as wholly implausible: the seeds of Datura metel are not only heavy but also lack any specialised adaptation to wind dispersal such as a wing or a pappus, and the fruits of Datura are not juicy berries which invite consumption by birds [unlike those of, for example, the bird-disseminated Atropa]. The authors settle upon transport by water as by far the most likely mode and, of the two scenarios involving water, human-mediated transportation being the more probable,Renner S., 2004 Plant dispersal across the tropical Atlantic by wind and sea currents; Int. J. Plant Sci. 165 23–33 although not ruling out a scenario whereby the buoyant fruits (and seeds possibly remaining viable after prolonged immersion in salt water) of Datura might have been carried to India by ocean currents.
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The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.. During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227 as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See for more information about the rediscovery of the Archimedean solids during the renaissance.) Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.
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The name for the genus Senecio is probably derived from senex (an old man), in reference to its downy head of seeds; "the flower of this herb hath white hair and when the wind bloweth it away, then it appeareth like a bald-headed man" and like its family, flowers of Senecio vulgaris are succeeded by downy globed heads of seed. The seeds are achene, include a pappus and become sticky when wet. Laboratory tests have suggested maximum seed scattering distances of 4.2 and 4.6 yd (1.9 and 2.9 m) at wind speeds of 6.8 and 10.2 mph (10.9 and 16.4 km/h) respectively (affected by plant height), suggesting that it was more than wind that spread these groundsel seeds throughout the world. The average weight of 1000 seeds is 0.21 gram (2,200,000 seeds per pound) and experienced a 100% germination success before drying and storage and an 87% germination success after drying and 3 years of cool dry storage.
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In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number s such that each two oriented paths of length s can be mapped to each other by exactly one symmetry of the graph. He showed that s is at most 5, and provided examples of graphs with each possible value of s from 1 to 5.. Semi-symmetric cubic graphs include the Gray graph (the smallest semi-symmetric cubic graph), the Ljubljana graph, and the Tutte 12-cage. The Frucht graph is one of the five smallest cubic graphs without any symmetries: it possesses only a single graph automorphism, the identity automorphism..
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The works of Hipparchus were still extant when Pappus wrote his commentary on the Almagest in the 4th century. He fills in some of the details that Ptolemy omits: : Now, Hipparchus made such an examination principally from the sun, and not accurately. For since the moon in the syzygies and near greatest distance appears equal to the sun, and since the size of the diameters of the sun and moon is given (of which a study will be made below), it follows that if the distance of one of the two luminaries is given, the distance of the other is also given, as in Theorem 12, if the distance of the moon is given and the diameters of the sun and moon, the distance of the sun is given. Hipparchus tries by conjecturing the parallax and the distance of the sun to demonstrate the distance of the moon, but with respect to the sun, not only the amount of its parallax, but also whether it shows any parallax at all is altogether doubtful.
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Pappus gave a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case. This porism, expressed in modern language, asserts the following: Given four straight lines of which three turn about the points in which they meet the fourth, if two of the points of intersection of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line. The general enunciation applies to any number of straight lines, say n + 1, of which n can turn about as many points fixed on the (n + 1)th. These n straight lines cut, two and two, in 1/2n(n − 1) points, 1/2n(n − 1) being a triangular number whose side is n − 1\. If, then, they are made to turn about the n fixed points so that any n − 1 of their 1/2n(n − 1) points of intersection, chosen subject to a certain limitation, lie on n − 1 given fixed straight lines, then each of the remaining points of intersection, 1/2n(n − 1)(n − 2) in number, describes a straight line.
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Cicero's De re publica, a 1st-century BC philosophical dialogue, mentions two machines that some modern authors consider as some kind of planetarium or orrery, predicting the movements of the Sun, the Moon, and the five planets known at that time. They were both built by Archimedes and brought to Rome by the Roman general Marcus Claudius Marcellus after the death of Archimedes at the siege of Syracuse in 212 BC. Marcellus had great respect for Archimedes and one of these machines was the only item he kept from the siege (the second was placed in the Temple of Virtue). The device was kept as a family heirloom, and Cicero has Philus (one of the participants in a conversation that Cicero imagined had taken place in a villa belonging to Scipio Aemilianus in the year 129 BC) saying that Gaius Sulpicius Gallus (consul with Marcellus's nephew in 166 BC, and credited by Pliny the Elder as the first Roman to have written a book explaining solar and lunar eclipses) gave both a "learned explanation" and a working demonstration of the device. Pappus of Alexandria stated that Archimedes had written a now lost manuscript on the construction of these devices entitled On Sphere-Making.
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