Quadratrix
Encyclopedia
In mathematics
, a quadratrix (from the Latin
word quadrator, squarer) is a curve having ordinate
s which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus
and E. W. Tschirnhausen
, which are both related to the circle.
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 affected a mechanical solution of squaring the circle. Pappus
, in his Collections, treats its history, and gives two methods by which it can be generated.
Another construction is as follows. ABC is a quadrant
in which the line AB and the arc AC 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 to BC and through the corresponding points on the radius AB. The locus of these intersections is the quadratrix.
The cartesian equation to the curve is y = x cot 2a' which shows that the curve is symmetrical about the axis of y, and that it consists of a central portion flanked by infinite branches. The asymptotes are x = 2na, n being an integer. The intercept on the axis of y is 2a/7r; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected. The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.
The quadratrix of Tschirnhausen 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 AB, and the lines drawn parallel to BC through the points of division of AB, are points on the quadratrix. The cartesian equation is y=a cos 2a. The curve is periodic, and cuts the axis of x at the points x= (2n - I)a, n being an integer; the maximum values of y are =a. Its properties are similar to those of the quadratrix of Dinostratus.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a quadratrix (from the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
word quadrator, squarer) is a curve having ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...
s which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus
Dinostratus
Dinostratus was a 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 :...
and E. W. Tschirnhausen
Ehrenfried Walther von Tschirnhaus
Ehrenfried Walther von Tschirnhaus was a German mathematician, physicist, physician, and philosopher...
, which are both related to the circle.
The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
geometers, and is mentioned by Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
, who ascribes the invention of the curve to a contemporary of Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, discussed the curve, and showed how it affected a mechanical solution of squaring the circle. Pappus
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
, in his Collections, treats its history, and gives two methods by which it can be generated.
- Let a helixHelixA helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
be drawn on a right circular cylinderCylinder (geometry)A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. 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 spiralArchimedean spiralThe Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...
is intersected by a right circular coneConical surfaceIn geometry, a conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex...
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.
Another construction is as follows. ABC is a quadrant
Circular sector
A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, r the radius of the circle, and L is the arc length of the...
in which the line AB and the arc AC 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 to BC and through the corresponding points on the radius AB. The locus of these intersections is the quadratrix.
The cartesian equation to the curve is y = x cot 2a' which shows that the curve is symmetrical about the axis of y, and that it consists of a central portion flanked by infinite branches. The asymptotes are x = 2na, n being an integer. The intercept on the axis of y is 2a/7r; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected. The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.
The quadratrix of Tschirnhausen 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 AB, and the lines drawn parallel to BC through the points of division of AB, are points on the quadratrix. The cartesian equation is y=a cos 2a. The curve is periodic, and cuts the axis of x at the points x= (2n - I)a, n being an integer; the maximum values of y are =a. Its properties are similar to those of the quadratrix of Dinostratus.
External links
- Quadratrix of Hippias at the MacTutor archive.
- Hippias' Quadratrix at Convergence