Alhazen was a polymath; the author of 90 works on topics as diverse as optics, vision, number theory, geometry, theology, astronomy, poetry, healing, and metaphysics. Many of his works, including the highly influential Book of Optics, were authored while he feigned madness to escape the wrath of Cairo’s caliph.
Alhazen explained how the images formed in cameras are upside down, solved ‘Alhazen’s problem’ concerning the reflection of light from curved surfaces, and discovered a general method to find the sum of any integral power. He used this general method to discover the sum of fourth power positive integers and hence find the volume of a paraboloid.
The Life of Alhazen
Abu Ali al-Hasan ibn al-Haytham was born into a Muslim family in about 965 AD in the city of Basra in the Buyid emirate, now in Iraq. Just as Mikolaj Kopernik is known by his Latinized name Nicolaus Copernicus in the west, al-Hasan’s name is often Latinized to Alhazen. Although a number of later artistic representations of him exist, we do not know what he really looked like.
Alhazen became a government minister (vizier) in Basra and spent his spare time working on mathematics, writing a treatise on the ancient (and impossible) problem of squaring the circle.
Alhazen believed in God. He weighed the differences between the religious sects around him in Basra, concluding that he favored the absolute, provable truths of mathematics and the philosophy of Aristotle to the conflicting claims made by the sects.
Lifetimes of Selected Scientists and Philosophers
In about 1000 AD, Alhazen left Basra for Cairo to work for its caliph, Hakim, who was a supporter of the sciences. The economies of Cairo and Egypt were dominated by the River Nile – agriculture was wholly dependent on its annual floods, but these had become increasingly unreliable. Alhazen believed he could make them reliable again by regulating the Nile’s flow with a dam. The caliph was excited by Alhazen’s plans and put him in charge of the project.
Unfortunately, Alhazen could not fulfill his promises, which terrified him, because the caliph had a fearsome reputation for punishing failure brutally. Alhazen feigned insanity and hid in a mosque.
It seems Alhazen did some of his greatest work while taking refuge, including writing his famous Book of Optics.
When the caliph died in 1021, Alhazen reemerged to enjoy about 19 years of freedom until his own death in Cairo in about 1040.
Note: A different account from the physician Ibn Abi Usaibia of Damascus says Alhazen feigned insanity in order to be released from his work in Basra and go to Cairo.
The Science of Alhazen
Alhazen personally acknowledged authorship of 90 books, of which 55 still exist. He immersed himself in the works of Ancient Greek scholars such as Euclid, Apollonius of Perga, Archimedes, and Ptolemy, seeking to use their writings as a foundation to build on.
He made important contributions in the fields of optics and mathematics.
The Book of Optics
Do Our Eyes Emit Rays?
Alhazen was aware of two schools of thought from Ancient Greece about light and vision:
(A) Plato, Euclid, Galen, and Ptolemy promoted the idea that our eyes emit rays which land on objects allowing us to see them. They realized light is not really emitted by our eyes, it comes from luminous sources such as the sun. They used the eye ray theory to explain an number of puzzling problems – for example, you don’t see a small object on the ground even though you are looking at the general area it is in until rays from your eyes actually land directly on it.
(B) Aristotle said our eyes do not emit rays. In the sixth century John Philoponus said our eyes receive rays of light. He said air allows colors to pass through it without becoming colored itself. He supported this with his observation that a stained-glass window casts colors on floors and walls but does not color the air.
In his great work Book of Optics, Alhazen correctly identified that our eyes do not emit rays. He argued that light affects the eye – for example we can damage our eyes by looking directly at the sun – but our eyes do not affect light. Moreover, he said that if we look at a bright object, an afterimage remains with us after we close our eyes. Again this suggests that our eyes have been affected by light.
Otherwise, Alhazen supported most aspects of Galen’s incorrect assessment of how the eye works, such as the lens is the receptive organ of sight.
The Camera Obscura
Camera is a Latin word meaning an arched or vaulted room, while obscura means dark.
In ancient times, different cultures discovered that a tiny hole in an external wall of a dark room allows images of the outdoors to form upside down inside the room, as shown below. The effect can also be seen in a pinhole camera, consisting of a dark box with a small hole in it.
Alhazen carried out experiments with pinhole cameras and candles and explained correctly how the image is formed by rays of light traveling in straight lines.
Four centuries would pass before Leonardo da Vinci suggested that the eye is actually a camera obscura, a fact proven by Johannes Kepler after about another century.
As we have seen, Alhazen was fascinated by light and vision.
This led him to make an intriguing mathematical discovery that suggested a link between algebra and geometry. The link was later reinforced by Omar Khayyam and fully developed by Descartes and Fermat.
An Ancient Greek book entitled Catoptrics compiled probably between the times of Euclid and Theon considered the behavior of reflected light and established the law of light reflection. A work by Hero of Alexandria made the assumption that light rays always take the shortest path between two points.
Alhazen considered an observer and a mirror shaped like the inside of a circle. He pictured a ray of light arriving at the mirror from a light source. He asked the question: at what point on the mirror must a light ray arrive so that it is reflected into the eye of the observer? He sought to solve the problem for the light source and observer in any positions. This question became known as Alhazen’s Problem, and is often called Alhazen’s Billiard Problem.
Alhazen’s Billiard Problem
The behavior of a billiard ball is helpful in understanding Alhazen’s problem.
By means of long, complicated geometrical arguments and proofs, Alhazen solved the problem by considering a circle’s intersection with a hyperbola.
The Sum of Fourth Powers
Alhazen discovered the formula for the sum of fourth powers when he took on the challenge of calculating the volume of a paraboloid. This is the 3D-shape you get by rotating a parabola around a flat base.
Alhazen approached the problem in the way Eudoxus or Archimedes would have, by the method of exhaustion, summing slices of the shape. Archimedes had used this technique brilliantly to find the volume of a sphere.
Alhazen applied the method of exhaustion to the paraboloid and found he needed the formula for the sum of fourth powers to calculate the answer. The formula for the sum of second powers had been discovered by Archimedes, and third powers by the great Indian mathematician Aryabhata. There was, however, no formula for the sum of fourth powers. Alhazen realized he would have to discover this for himself.
And discover it he did.
In fact, the method Alhazen developed to discover the formula was valid for any power, so he could have found the sum of the fifth power, sixth power, seventh power, etc.
Sums of Powers
Alhazen used his method to find sums of powers shown below. He wrote them in words, but modern notation is easier to follow:
Sum of first powers:
Sum of second powers:
Sum of third powers:
Sum of fourth powers:
It is appropriate to end with Alhazen’s own eloquent description of his personal view of what being a scientist means:
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