As the Chinese proverb says “The journey of a thousand miles begins with a single step”. The journey that started with Ibn al-Haytham experimenting with his Camera Obscura in the eleventh century was completed eight hundred years later by James Clerk Maxwell and Heinrich Hertz. While Maxwell laid down the mathematical framework that described the behavior of Electromagnetic waves, Hertz conclusively proved the existing of these invisible waves through his experiments. There were several scientists on the way that played a crucial part in development of this Electromagnetic theory such as Gauss, Faraday and Ampere. Then there were others such as Huygens, Fresnel and Young who worked on nature of light, which was not known to be an Electromagnetic wave at that time. Once the theory of Electromagnetic wave propagation was in place there was rapid progress in many fields, particularly in wireless communications (wireless telegraph, radio, radar etc.).
Maxwell’s equations that were proposed in 1861 were initially quite circuitous and were not well accepted. But later on these equations were simplified into the form we now know by Oliver Heaviside. There are still two popular forms of the equations, the integral form and the differential form. We present the integral form of these equations in this article as it is more intuitive and is also easier to represent graphically. The differential form requires understanding of the concepts of divergence and curl and we skip it in this article. The main take away from these equations (presented below) is that a changing Electric field produces a Magnetic field and a changing Magnetic field produces an Electric field. Another important result is that magnetic monopoles do not exist (simply put a magnet, however small, always has a north and south pole).
The dot product with a line segment means that only that component of the field vector is effective that is along the line segment. On the other hand the dot product with a surface means that only that component is considered that is perpendicular to the surface (since the unit vector of a surface is perpendicular to the surface). It means that only those field components are considered that are going perpendicularly in or out of the surface.
For more on history of Maxwell equations visit IEEE Spectrum and for a detailed explanation of the various forms of the Maxwell’s equations visit this page.
In modern Electromagnetic simulation software the differential form is preferred and the algorithm used is called Finite Difference Time Domain (FDTD). However, if the area of interest is quite large (with respect to the wavelength) then the FDTD method becomes prohibitively complex and another method known as Ray-Tracing is used. Please do check out the Ray-Tracing engine that we have developed. Ray-Tracing is becoming increasingly important in RF Planning of Telecom Networks.
Omar Khayyam was a Muslim mathematician and poet of the 11th and 12th centuries (1048-1131). His poetic works known as Rubaiyat of Omar Khayyam were translated from Persian to English and made popular by Edward Fitzgerald in the late nineteenth century. In the field of mathematics his most valuable contribution was the solution he presented to the cubic equations using geometrical methods. Some of this was adapted from earlier works by Greeks but his compilation of the various cases and their solutions was most complete.
Lets assume that the cubic equation also known as the third degree equation (highest power of the unknown variable) is of the form:
Khayyam’s method consisted of constructing a parabola with equation x2=ay and a circle with center (b/2a2,0) and radius b/2a2. Then the x-coordinate of the intersection of the circle and the parabola gives the solution to the cubic equation. The root found by this method is the real and positive root since the length of a line segment cannot be negative or imaginary. These cases (negative and imaginary roots) were not discussed by Khayyam and were worked out much later by other mathematicians. The MATLAB code for this geometrical construction is given below.
% Omar Khayyams Method to Find
% the Roots of a Cubic Equation
% Copyright RAYmaps 2017 (YA)
% Plot the parabola
% Plot the circle
plot(r+r*cos(t), r*sin(t),'r', 'linewidth', 4);
axis([-5 5 -5 5], "square")
title('Khayyams Method to Solve Cubic Equations')
Omar Khayyam’s Method for Solving Cubic Equations
For more on origins of geometrical methods see the following post on Al-Khwarizmi.
For an interactive tool to understand the method of Omar Khayyam visit the following page.
For a proof of validity of Khayyam’s method see the following page on Cornell website or see selected abstract below. Please note slightly different form of the equation where the term a2 has been replaced by a. This is just a constant term and either form works.
Most of intermediate Physics courses present Snell’s law of refraction in one form or another. But a little known mathematician with the name Ibn Sahl (c. 940–1000) found this law about 650 years before Snell (Willebrord Snellius c. 1580–1626). This mathematical expression was lost for centuries until until some scholars recently were able to dig it up from historical records. Even Ibn al-Haytham (author of Book of Optics or Kitab ul Manazir) who came to the fore a few years later did not recognize the brilliance of Ibn Sahl’s simple expression.
Ibn Sahl was aware that Greek’s knew that there was a relationship between the angle of incidence and angle of refraction of a ray traveling from one medium to the other. They thought that ratio of the two angles was a constant i.e. if the angle of incidence was doubled the angle of refraction also doubled. This also meant that the arcs formed by the two angles on a circle centered at the point of incidence were also directly related (in a linear relationship). But Ibn Sahl showed that this was incorrect.
Ibn Sahl showed that it was not the angles but the sine of the angles that were linearly related. We explain it with the help of the figure below. Imagine that a ray of light travels from air to a denser medium (such as water), then the ray bends towards the normal and angle of refraction is smaller than angle of incidence. According to Ibn Sahl the ratio of line segments l1 and l2 as shown in the figure is a constant. This in fact means that the two sines have a constant ratio and this is equal to refractive index of the second medium (n2) with refractive index of air almost equal to 1 (n1).
Ibn Sahl was not aware of the parameter ‘n’ defined as refractive index by later scientists. Also, as is known that for small angles, sine of the angle and angle itself are almost the same, so earlier scientists like Ptolemy might have been tricked into assuming that the angles are directly related. This can be understood by looking at the figure above. If the angle of incidence is continuously reduced, the angle of refraction would also decrease and the lengths of the two line segments in red (l1 and l2) would approach the lengths of the arcs that are formed between the ray and the normal.
Note: Roshdi Rashed found the Ibn Sahl text to have been dispersed in manuscripts in two different libraries, one in Tehran, and the other in Damascus. He reassembled the surviving portions, translated and published them as “Geometry and dioptric in the tenth century: Ibn Sahl, al-Quhi and Ibn al-Haytham”.