Tilia is a genus of between 30 and 45 species of trees in the Northern Hemisphere, in Asia (where the greatest species diversity is found), Europe and eastern North America; it is absent from western North America.
The trees are generally called linden in North America, and lime in Britain. Both names are from the Germanic root "lind". The modern forms in English went from "linde" or "linne" in Anglo Saxon and old Norse, and in Britain the word transformed more recently to the modern British form "lime". In the United States, the modern German name "Linde", from the same root, became more common, partly to avoid confusion with any other uses of the name.
"Tilia" species are large deciduous trees, reaching typically between 20m and 40m tall, some species grow as shrubs. Plants of this genus are found through the north temperate regions.
The Germanic tribes associated the Linden tree with the goddess Freya. Very often, these tribes held their assemblies, called Thing near or under such a tree. Often, trees that were standing alone somewhere were chosen. These assemblies were used to make laws. They usually lasted three days.
The tree produces fragrant and nectar-producing flowers, the medicinal herb lime blossom. They are very important honey plants for beekeepers, producing a very pale but richly flavoured monofloral honey. The flowers are also used for herbal tea, and this infusion is particularly popular in Europe.
Cartesian coordinate system
In mathematics and geometry, the Cartesian coordinate system is a coordinate system used to give the location of points on a plane by using two numbers for each point. It is also called the "rectangular coordinate system". The numbers are usually called the "x coordinate" and the "y coordinate". To find the coordinates of a point, two perpendicular lines, called "axes" (singular: "axis"), are drawn. The point where the axes meet is the coordinate origin, written (0, 0). The x coordinate gives the position of a point measured along the x axis, and the y coordinate gives the position along the y axis. Cartesian coordinates can be used in three dimensions (3D), by adding a third number, the "z coordinate". Each coordinate represents a dimension of space.
Using the Cartesian coordinate system, many shapes, like straight lines and parabolas can be described by using algebraic equations. The results of the equations can be seen by plotting (drawing) a point for each solution to the equations. For example, the x axis is described by the equation formula_1. A circle having radius "r" and centered at the origin can be described with the equation formula_2 (see Figure 2).
The name Cartesian comes from the French mathematician and philosopher René Descartes (Latin: "Cartesius"), who worked to merge algebra and Euclidean geometry. This work was important in creating calculus and cartography.
The idea of this system was developed in 1637 in two works by Descartes. In part two of his "Discourse on Method", Descartes explains plotting the position of a point or object on a surface, using two perpendicular axes as measuring guides. In "La Géométrie", he further explores these ideas.
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed with angles and distance; in the more familiar Cartesian or rectangular coordinate system, such a relationship can only be found through trigonometric formulae.
As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as formula_1) denotes the point's distance from a central point known as the "pole" (equivalent to the "origin" in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by θ or formula_2) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or "polar axis" (which is equivalent to the positive x-axis in the Cartesian coordinate plane).
The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. Hipparchus (190-120 BCE) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.
In "On Spirals," Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's "Origin of Polar Coordinates." Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.
In "Method of Fluxions" (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal "Acta Eruditorum" (1691), Jacob Bernoulli used a system with a point on a line, called the "pole" and "polar axis" respectively. Coordinates were specified by the distance from the pole and the angle from the "polar axis". Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.
The actual term "polar coordinates" has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's "Differential and Integral Calculus". Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.
The polar coordinates "r" and "φ" can be converted to the Cartesian coordinates "x" and "y" by using the trigonometric functions sine and cosine:
The Cartesian coordinates "x" and "y" can be converted to polar coordinates "r" and "φ" with "r" ≥ 0 and "φ" in the interval (−, ] by:
Cylindrical coordinates take the same idea that polar coordinates use, but they extend it further. To get a third dimension, each point also has a "height" above the original coordinate system. Each point is uniquely identified by a distance to the origin, called "r" here, an angle, called "formula_5" ("phi"), and a height above the plane of the coordinate system, called "Z" in the picture.
The same idea as is used by polar coordinates can also be extended in a different way. Instead of using two distances, and one angle only, it is possible to use one distance only, and two angles, called formula_5 and formula_7 ("theta").
Gold (color)
Gold or golden is a yellowish orange color, or orange-tan color that is a bit like the color of the metal gold. The actual color of the metal, used for example in gilding, is called Metallic gold. Gold paint can be made by mixing brown, yellow paint and orange paint.
The web color "gold" (shown at right) is sometimes referred to as golden in order to distinguish it from the element "gold". The use of "gold" as a color term in traditional usage is more often applied to the color "gold (metallic)" (shown below).