geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a quadratrix () is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

having

ordinate In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph. The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x c ...

s which are a measure of the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

(or quadrature) of another curve. The two most famous curves of this class are those of

Dinostratus Dinostratus ( el, Δεινόστρατος; c. 390 – c. 320 BCE) was a Greece, Greek mathematician and geometer, and the brother of Menaechmus. He is known for using the quadratrix to solve the problem of squaring the circle. Life and work Di ...

and E. W. Tschirnhaus, which are both related to the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

Quadratrix of Dinostratus

The quadratrix of Dinostratus (also called the ''quadratrix of Hippias'') was well known to the

ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...

geometers, and is mentioned by

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...

, who ascribes the invention of the curve to a contemporary of

Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...

, probably

Hippias of Elis Hippias of Elis (; el, Ἱππίας ὁ Ἠλεῖος; late 5th century BC) was a Greek sophist, and a contemporary of Socrates. With an assurance characteristic of the later sophists, he claimed to be regarded as an authority on all subjects, ...

. Dinostratus, a Greek geometer and disciple of

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

, 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 A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...

be drawn on a right circular

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

; a screw surface is then obtained by drawing

lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

from every point of this spiral

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to its axis. The

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

of a section of this surface by a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

containing one of the perpendiculars and inclined to the axis is the quadratrix. # A right cylinder having for its base an

Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...

is intersected by a right circular

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

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. Quadratrix_no_anim

Another construction is as follows. is a quadrant in which the line and the arc are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

to and through the corresponding points on the radius . The

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of these intersections is the quadratrix. Quadratrixfunktion

Letting be the origin of the

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, be the point , units from the origin along the -axis, and be the point , units from the origin along the -axis, the curve itself can be expressed by the equation :

y = x\cot\left(\frac\right).

Because the cotangent

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

is invariant under negation of its argument, and has a simple

pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...

at each multiple of , the quadratrix has

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

across the -axis, and similarly has a pole for each value of of the form , for

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

values of , except at where the pole in the cotangent is canceled by the factor of in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the -axis has ; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...

multiple of , leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge. An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge:

doubling the cube 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 second cube whose volume is double that of the first. As with the related pro ...

and

trisecting an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...

Quadratrix of Tschirnhaus

The quadratrix of TschirnhausSee definition and drawing in the following online source: is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. The mutual intersections of the lines drawn from the points of division of the arc parallel to ''DA'', and the lines drawn parallel to ''AB'' through the points of division of ''DA'', are points on the quadratrix. The Cartesian equation is

y = a \cos\!\big(\tfrac\big)

. The curve is periodic, and cuts the ''x''-axis at the points

x = (2n - 1)a

n

being an integer; the maximum values of

y

are

a

. Its properties are similar to those of the quadratrix of Dinostratus.

Other quadratrices

Other curves that have historically been used to square the circle include: *

* Ozanam's quadratrix *

Cochleoid In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac, the Cartesian equation :(x^2+y^2)\arctan\frac=ay, or the parametric equations :x=\frac, \quad y=\frac. The coch ...

References

External links

Quadratrix of Hippias
{{Webarchive, url=https://web.archive.org/web/20120204120351/http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Quadratrix.html , date=2012-02-04 at the MacTutor archive.
Hippias' Quadratrix
a
Convergence
(MAA periodical) Curves Squaring the circle Area