Alhazen's problem
Encyclopedia
Ibn al-Haytham's work on catoptrics
in Book V of the Book of Optics
contains the important Islamic mathematical
problem known as Alhazen's problem.
The problem comprises drawing lines from two points in the plane
of a circle meeting at a point on the circumference
and making equal angles with the normal
at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to canon off the edge of the table and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror were the light will be reflected to the eye of an observer." This leads to an equation of the fourth degree.
This eventually led Ibn al-Haytham to derive the earliest formula for the sum of the fourth power
s, and using an early proof
by mathematical induction
, he developed a method for determining the general formula for the sum of any integral powers. He used his result on sums of integral powers to perform an integration
, in order to find the volume of a paraboloid
. He was thus able to find the integral
s for polynomial
s up to the fourth degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal
and integral calculus
.
Ibn al-Haytham solved the problem using conic section
s and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as Christiaan Huygens, James Gregory,Guillaume de l'Hôpital
, Isaac Barrow
, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry
and derivation by complex number
s. An algebraic solution to the problem was finally found in 1997 by the Oxford
mathematician Peter M. Neumann
.
Catoptrics
Catoptrics deals with the phenomena of reflected light and image-forming optical systems using mirrors. From the Greek κατοπτρικός ....
in Book V of the Book of Optics
Book of Optics
The Book of Optics ; ; Latin: De Aspectibus or Opticae Thesaurus: Alhazeni Arabis; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Muslim scholar Alhazen .-See also:* Science in medieval Islam...
contains the important Islamic mathematical
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
problem known as Alhazen's problem.
The problem comprises drawing lines from two points in the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
of a circle meeting at a point on the circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
and making equal angles with the normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to canon off the edge of the table and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror were the light will be reflected to the eye of an observer." This leads to an equation of the fourth degree.
This eventually led Ibn al-Haytham to derive the earliest formula for the sum of the fourth power
Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying n by itself four times. So:Fourth powers are also formed by multiplying a number by its cube...
s, and using an early proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
by mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
, he developed a method for determining the general formula for the sum of any integral powers. He used his result on sums of integral powers to perform an integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, in order to find the volume of a paraboloid
Paraboloid
In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
. He was thus able to find the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s for polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s up to the fourth degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
and integral calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
.
Ibn al-Haytham solved the problem using conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as Christiaan Huygens, James Gregory,Guillaume de l'Hôpital
Guillaume de l'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞...
, Isaac Barrow
Isaac Barrow
Isaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...
, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
and derivation by complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. An algebraic solution to the problem was finally found in 1997 by the Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...
mathematician Peter M. Neumann
Peter M. Neumann
Peter Michael Neumann OBE is a British mathematician. He is the son of the mathematicians Bernhard Neumann and Hanna Neumann and, after gaining a B.A. from The Queen's College, Oxford in 1963, obtained his D.Phil from Oxford University in 1966...
.