Introduction

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of

spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Roger Cotes. Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals). Epi half spirals

The shape of spirals in the family depends on the parameters. The curves in

polar coordinates 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 reference point (analogous to t ...

, (''r'', ''θ''), ''r'' > 0 are defined by one of the following five equations: :

\frac = 
\begin
 A \cosh(k\theta + \varepsilon) \\
 A \exp(k\theta + \varepsilon) \\
 A \sinh(k\theta + \varepsilon) \\
 A (k\theta + \varepsilon) \\
 A \cos(k\theta + \varepsilon) \\
\end

''A'' > 0, ''k'' > 0 and ''ε'' are arbitrary

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

constants. ''A'' determines the size, ''k'' determines the shape, and ''ε'' determines the angular position of the spiral. Cotes referred to the different forms as "cases". The equations of the curves above correspond respectively to his 5 cases. The Diagram shows representative examples of the different curves. The centre is marked by ‘O’ and the radius from O to the curve is shown when ''θ'' is zero. The value of ''ε'' is zero unless shown. The first and third forms are

Poinsot's spirals In mathematics, Poinsot's spirals are two spirals represented by the polar equations : r = a\ \operatorname (n\theta) : r = a\ \operatorname (n\theta) where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named afte ...

; the second is the

equiangular spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...

; the fourth is the hyperbolic spiral (more correctly called by its alternative name: the "Reciprocal Spiral" since it has no connection with the hyperbola, or the hyperbolic functions which feature in the

); the fifth is the

epispiral The epispiral is a plane curve with polar equation :\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle inversion of the rose curve. In astronomy the epispiral is related to the equations ...

. For more information about their properties, reference should be made to the individual curves. Cotes Spirals

Classical mechanics

Cotes's spirals appear in

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

, as the family of solutions for the motion of a particle moving under an inverse-cube

central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...

. Consider a central force :

\boldsymbol(\boldsymbol) = -\frac\hat\boldsymbol,

where ''μ'' is the strength of attraction. Consider a particle moving under the influence of the central force, and let ''h'' be its specific angular momentum, then the particle moves along a Cotes's spiral, with the constant ''k'' of the spiral given by :

k = \sqrt

when ''μ'' < ''h''² ( cosine form of the spiral), or :

k = \sqrt

when ''μ'' > ''h''², Poinsot form of the spiral. When ''μ'' = ''h''², the particle follows a hyperbolic spiral. The derivation can be found in the references.

History

In the ''Harmonia Mensurarum'' (1722), Roger Cotes analysed a number of spirals and other curves, such as the ''Lituus''. He described the possible trajectories of a particle in an inverse-cube central force field, which are the Cotes's spirals. The analysis is based on the method in the Principia Book 1, Proposition 42, where the path of a body is determined under an arbitrary central force, initial speed, and direction. Depending on the initial speed and direction he determines that there are 5 different "cases" (excluding the trivial ones, the circle and straight line through the centre). He notes that of the 5, "the first and the last are described by Newton, by means of the quadrature (i.e. integration) of the hyperbola and the ellipse". Case 2 is the equiangular spiral, which is the spiral ''par excellence''. This has great historical significance as in Proposition 9 of the Principia Book 1, Newton proves that if a body moves along an equiangular spiral, under the action of a central force, that force must be as the inverse of the cube of the radius (even before his proof, in Proposition 11, that motion in an ellipse directed to a focus requires an inverse-square force). It has to be admitted that not all the curves conform to the usual definition of a spiral. For example, when the inverse-cube force is centrifugal (directed outwards), so that ''μ'' < 0, the curve does not even rotate once about the centre. This is represented by case 5, the first of the polar equations shown above, with ''k'' > 1 in this case. Samuel Earnshaw in a book published in 1826 used the term “Cotes’ spirals”, so the terminology was in use at that time. Earnshaw clearly describes Cotes's 5 cases and unnecessarily adds a 6th, which is when the force is centrifugal (repulsive). As noted above, Cotes's included this with case 5. The mistaken view that there are only 3 Cotes's spirals appears to have originated with E. T. Whittaker's '' A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'', first published in 1904. Whittaker's "reciprocal spiral" does have a footnote, which refers to Cotes's "Harmonia Mensurarum" and Newton's Proposition 9. However, it is misleading, since the spiral of Proposition 9 is the

, which he does not recognise as a Cotes's spiral at all. Unfortunately, subsequent authors have followed Whittaker's lead without taking the trouble to verify its accuracy.

References

Bibliography

* * Roger Cotes (1722) ''Harmonia Mensuarum'', pp. 31, 98. *

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...

(1687) ''

Philosophiæ Naturalis Principia Mathematica ( English: ''Mathematical Principles of Natural Philosophy'') often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin an ...

'', Book I, §2, Proposition 9, and §8, Proposition 42, Corollary 3, and §9, Proposition 43, Corollary 6 * * *

External links

* {{Spirals Spirals Classical mechanics

Classical mechanics

History

See also

References

Bibliography

External links