On Spirals

Last updated

On Spirals (Greek : Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. [1] Although Archimedes did not discover the Archimedean spiral,[ dubious ] he employed it in this book to square the circle and trisect an angle. [2]




Archimedes begins On Spirals with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου) and On Conoids and Spheroids (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results of On Spirals.

Archimedean spiral

The Archimedean spiral with three 360deg turnings on one arm Archimedean spiral.svg
The Archimedean spiral with three 360° turnings on one arm

The Archimedean spiral was first studied by Conon and was later studied by Archimedes in On Spirals. Archimedes was able to find various tangents to the spiral. [1] He defines the spiral as:

If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane. [3]

Trisecting an angle

Example of how Archimedes trisected an angle in On Spirals. Archimedes trisect spiral.svg
Example of how Archimedes trisected an angle in On Spirals.

The construction as to how Archimedes trisected the angle is as follows:

Suppose the angle ABC is to be trisected. Trisect the segment BC and find BD to be one third of BC. Draw a circle with center B and radius BD. Suppose the circle with center B intersects the spiral at point E. Angle ABE is one third angle ABC. [4]

Squaring the circle

The circle and the triangle are equal in area. Archimedean spiral circle squaring triangle.svg
The circle and the triangle are equal in area.

To square the circle, Archimedes gave the following construction:

Let P be the point on the spiral when it has completed one turn. Let the tangent at P cut the line perpendicular to OP at T. OT is the length of the circumference of the circle with radius OP.

Archimedes had already proved as the first proposition of Measurement of a Circle that the area of a circle is equal to a right-angled triangle having the legs' lengths equal to the radius of the circle and the circumference of the circle. So the area of the circle with radius OP is equal to the area of the triangle OPT. [5]

Related Research Articles

Archimedes Greek mathematician, physicist, engineer, inventor, and astronomer

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

Circle Simple curve of Euclidean geometry

A circle is a shape consisting of all points in a plane that are 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 constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Tangent straight line touching a point in a curve

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f ) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

Straightedge and compass construction way of drawing geometric objects using only an idealized compass and straightedge

Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

Archimedean spiral spiral with constant width between its turnings; in polar coordinates, distance from the origin is linearly proportional to angle

The 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 that rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation

Perpendicular Relationship between two lines that meet at a right angle (90 degrees)

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

Angle trisection the construction of an angle equal to one third a given angle

Angle trisection is a classical problem of compass and straightedge constructions 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 and a compass.

The Method of Mechanical Theorems, also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so-called because it relies on the law of the lever, which was first demonstrated by Archimedes, and of the center of mass, which he had found for many special shapes.

Thaless theorem The angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved as part of the 31st proposition, in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, who is said to have offered an ox as a sacrifice of thanksgiving for the discovery, but sometimes it is attributed to Pythagoras.

Concyclic points a set of points that lie on a single circle

In geometry, a set of points are said to be concyclic if they lie on a common circle. All concyclic points are the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.


In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line that contains their diameters.

Tomahawk (geometry) tool for angle trisection

The tomahawk is a tool in geometry for angle trisection, the problem of splitting an angle into three equal parts. The boundaries of its shape include a semicircle and two line segments, arranged in a way that resembles a tomahawk, a Native American axe. The same tool has also been called the shoemaker's knife, but that name is more commonly used in geometry to refer to a different shape, the arbelos.

Area of a circle area enclosed by a circle

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

<i>On the Sizes and Distances</i> (Aristarchus)

On the Sizes and Distances 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.

Indiana Pi Bill 1897 attempt to define by legislation a method of squaring the circle

The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative fiat. Despite its name, the main result claimed by the bill is a method to square the circle, rather than to establish a certain value for the mathematical constant π, the ratio of the circumference of a circle to its diameter. The bill, written by the crank Edward J. Goodwin, does imply various incorrect values of π, such as 3.2. The bill never became law, due to the intervention of Professor C. A. Waldo of Purdue University, who happened to be present in the legislature on the day it went up for a vote.

Archimedes quadruplets

In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles.

<i>Book of Lemmas</i> mathematical treatise attributed to Archimedes

The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions (lemmas) on circles.

Measurement of a Circle or Dimension of the Circle is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work.

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.


  1. 1 2 Weisstein, Eric W. "Archimedes' Spiral". MathWorld .
  2. "Spiral". Encyclopædia Britannica. 2008. Retrieved 2008-07-29.[ permanent dead link ]
  3. Heath, Thomas Little (1921), A History of Greek Mathematics, Boston: Adamant Media Corporation, p. 64, ISBN   0-543-96877-4 , retrieved 2008-08-20
  4. Tokuda, Naoyuki; Chen, Liang (1999-03-18), Trisection Angles (PDF), Utsunomiya University, Utsunomiya, Japan, pp. 5–6, archived from the original (PDF) on 2011-07-22, retrieved 2008-08-20
  5. "History topic: Squaring the circle" . Retrieved 2008-08-20.