1.
Neusis
–
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of length in between two given lines, in such a way that the line element, or its extension. That is, one end of the element has to lie on l. A neusis construction might be performed by means of a neusis ruler, in the figure one end of the ruler is marked with a yellow eye with crosshairs, this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance a from the origin, the yellow eye is moved along line l, until the blue eye coincides with line m. The position of the element thus found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, length a is called the diastema. Neuseis have been important because they provide a means to solve geometric problems that are not solvable by means of compass. Examples are the trisection of any angle in three parts, the doubling of the cube, and the construction of a regular heptagon, nonagon. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis, Sir Isaac Newton followed their line of thought, nevertheless, gradually the technique dropped out of use. Modified by the recent finding by Benjamin and Snyder that the regular hendecagon is neusis-constructible, T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. One hundred years after him Euclid too shunned neuseis in his influential textbook. The next attack on the neusis came when, from the fourth century BC, under its influence a hierarchy of three classes of geometrical constructions was developed. 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 methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as a not inconsiderable error. R. Boeker, Neusis, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, the most comprehensive survey, however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics, H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum. MathWorld page Angle Trisection by Paper Folding

2.
Meetkunde
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

3.
Passer (gereedschap)
–
A pair of compasses, also known simply as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as tools to measure distances, Compasses can be used for mathematics, drafting, navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. Prior to computerization, compasses and other tools for manual drafting were often packaged as a bow set with interchangeable parts, today these facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. The handle is usually half a inch long. Users can grip it between their pointer finger and thumb, there are two types of legs in a pair of compasses, the straight or the steady leg and the adjustable one. Each has a purpose, the steady leg serves as the basis or support for the needle point. The screw on your hinge holds the two legs in its position, the hinge can be adjusted depending on desired stiffness, the tighter the screw the better the compass’ performance. The needle point is located on the leg, and serves as the center point of circles that are drawn. The pencil lead draws the circle on a paper or material. This holds the lead or pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass, to use a pair of compasses, place the points on a ruler and open it to the measurement of ½ of the measurement of the circle that is desired. For instance, if one desires to draw a 3 inch circle, next, place the point on the spot that you wish the center of your circle to be, and then rotate the section that has the pencil lead around the point, using the handle. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry, although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles

4.
Liniaal
–
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing, engineering and building to measure distances or to rule straight lines. The ruler is a straightedge which may also contain calibrated lines to measure distances, rulers have long been made from different materials and in a wide range of sizes. Plastics have also used since they were invented, they can be molded with length markings instead of being scribed. Metal is used for more durable rulers for use in the workshop,12 inches or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket, longer rulers, e. g.18 inches are necessary in some cases. Rigid wooden or plastic yardsticks,1 yard long, and meter sticks,1 meter long, are also used, classically, long measuring rods were used for larger projects, now superseded by tape measure or laser rangefinders. Desk rulers are used for three purposes, to measure, to aid in drawing straight lines and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges, a line gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic, units of measurement on a basic line gauge usually include inches, agate, picas, and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, measuring instruments similar in function to rulers are made portable by folding or retracting into a coil when not in use. When extended for use, they are straight, like a ruler, the illustrations on this page show a 2-meter carpenters rule, which folds down to a length of 24 cm to easily fit in a pocket, and a 5-meter-long tape, which retracts into a small housing. A flexible length measuring instrument which is not necessarily straight in use is the tailors fabric tape measure and it is used to measure around a solid body, e. g. a persons waist measurement, as well as linear measurement, e. g. inside leg. It is rolled up when not in use, taking up little space, a contraction rule is made having larger divisions than standard measures to allow for shrinkage of a metal casting. They may also be known as a shrinkage or shrink rule, a ruler software program can be used to measure pixels on a computer screen or mobile phone. These programs are known as screen rulers. In geometry, a ruler without any marks on it may be used only for drawing lines between points. A straightedge is used to help draw accurate graphs and tables. A ruler and compass construction refers to using an unmarked ruler

5.
Constructie met passer en liniaal
–
The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in cases were unable to do so. Gauss showed that some polygons are constructible but that most are not, some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its not drawing a circle, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon

6.
Hoek (meetkunde)
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles

7.
Derdemachtswortel
–
In mathematics, a cube root of a number x is a number such that a3 = x. All real numbers have one real cube root and a pair of complex conjugate cube roots. For example, the cube root of 8, denoted 3√8, is 2, because 23 =8, while the other cube roots of 8 are −1 + √3i. The three cube roots of −27i are 3 i,332 −32 i, the cube root operation is not associative or distributive with addition or subtraction. In some contexts, particularly when the number whose root is to be taken is a real number, one of the cube roots is referred to as the principal cube root. The cube roots of a x are the numbers y which satisfy the equation y 3 = x. For any real number y, there is one real number x such that x3 = y, the cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one, then we can define an inverse function that is also one-to-one. For real numbers, we can define a cube root of all real numbers. If this definition is used, the root of a negative number is a negative number. If x and y are allowed to be complex, then there are three solutions and so x has three cube roots, a real number has one real cube root and two further cube roots which form a complex conjugate pair. This can lead to interesting results. For instance, the roots of the number one are,13 = {1 −12 +32 i −12 −32 i. The last two of these lead to a relationship between all roots of any real or complex number. If a number is one root of any real or complex number. For complex numbers, the cube root is usually defined as the cube root that has the largest real part, or, equivalently. It is related to the value of the natural logarithm by the formula x 13 = exp . This means that in polar coordinates, we are taking the root of the radius

8.
Rechthoekige driehoek
–
A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a triangle is the basis for trigonometry. The side opposite the angle is called the hypotenuse. The sides adjacent to the angle are called legs. Side a may be identified as the adjacent to angle B and opposed to angle A, while side b is the side adjacent to angle A. If the lengths of all three sides of a triangle are integers, the triangle is said to be a Pythagorean triangle. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the then the other is height. As a formula the area T is T =12 a b where a and b are the legs of the triangle and this formula only applies to right triangles. From this, The altitude to the hypotenuse is the mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse, in equations, f 2 = d e, b 2 = c e, a 2 = c d where a, b, c, d, e, f are as shown in the diagram. Moreover, the altitude to the hypotenuse is related to the legs of the triangle by 1 a 2 +1 b 2 =1 f 2. For solutions of this equation in integer values of a, b, f, the altitude from either leg coincides with the other leg. Since these intersect at the vertex, the right triangles orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that, In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs. This can be stated in equation form as a 2 + b 2 = c 2 where c is the length of the hypotenuse, Pythagorean triples are integer values of a, b, c satisfying this equation. The radius of the incircle of a triangle with legs a and b. The radius of the circumcircle is half the length of the hypotenuse, thus the sum of the circumradius and the inradius is half the sum of the legs, R + r = a + b 2

9.
Kwadratuur van de cirkel
–
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the area as a given circle by using only a finite number of steps with compass. It may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square. It had been known for decades before then that the construction would be impossible if π were transcendental. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a number of steps. The expression squaring the circle is used as a metaphor for trying to do the impossible. The term quadrature of the circle is used to mean the same thing as squaring the circle. Methods to approximate the area of a circle with a square were known already to Babylonian mathematicians. Indian mathematicians also found a method, though less accurate. Archimedes showed that the value of pi lay between 3 + 1/7 and 3 + 10/71, see Numerical approximations of π for more on the history. The first known Greek to be associated with the problem was Anaxagoras, Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, the problem was even mentioned in Aristophaness play The Birds. It is believed that Oenopides was the first Greek who required a plane solution, james Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of pi and it was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility. The Victorian-age mathematician, logician and author, Charles Lutwidge Dodgson also expressed interest in debunking illogical circle-squaring theories, in one of his diary entries for 1855, Dodgson listed books he hoped to write including one called Plain Facts for Circle-Squarers. The value my friend selected for Pi was 3.2, more than a score of letters were interchanged before I became sadly convinced that I had no chance. A ridiculing of circle-squaring appears in Augustus de Morgans A Budget of Paradoxes published posthumously by his widow in 1872, originally published as a series of articles in the Athenæum, he was revising them for publication at the time of his death. Circle squaring was very popular in the century, but hardly anyone indulges in it today

10.
Verdubbeling van de kubus
–
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a cube whose volume is double that of the first, using only the tools of a compass. As with the problems of squaring the circle and trisecting the angle. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made futile attempts at solving what they saw as an obstinate. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837, in algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 =2, in other words, x = 3√2. This is because a cube of side length 1 has a volume of 13 =1, the impossibility of doubling the cube is therefore equivalent to the statement that 3√2 is not a constructible number. This implies that the degree of the extension generated by a constructible point must be a power of 2. The field extension generated by 3√2, however, is of degree 3 and we begin with the unit line segment defined by points and in the plane. We are required to construct a line segment defined by two separated by a distance of 3√2. Any newly defined point either arises as the result of the intersection of two circles, as the intersection of a circle and a line, or as the intersection of two lines. Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates, therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1. By Gausss Lemma, p is irreducible over ℚ, and is thus a minimal polynomial over ℚ for 3√2. The field extension ℚ, ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√2 is not the coordinate of a point, and thus a line segment of 3√2 cannot be constructed. The problem owes its name to a story concerning the citizens of Delos, the oracle responded that they must double the size of the altar to Apollo, which was a regular cube. This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus as still unsolved, however another version of the story says that all three found solutions but they were too abstract to be of practical value. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that a r = r s = s 2 a. In turn, this means that r = a ⋅23 But Pierre Wantzel proved in 1837 that the root of 2 is not constructible