[This is a condensed version of an article [5] in Mathematics Today]
A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon’s Theorem states that `The area of a tangent sweep of a curve is equal to the area of its tangent cluster’. We shall illustrate how this theorem can help to solve a range of integration problems.
Continue reading ‘Mamikon’s Visual Calculus and Hamilton’s Hodograph’
.
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definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities.
. Another triangle, with sides 
and 
is stacked upon the first. This process continues, giving hypotenuse lengths 
for all 
. 
. More generally, for an N-gon the ratio is easily shown to be 
. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum 
is the angle through which the disk has rotated. The centre of the disk is at 
.
as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result
. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,
. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:
are the
In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [
, 
, 
using the integral
the amplitude and wrote 
. It can be difficult to understand what motivated his definitions. We will define the elliptic functions 
of a real variable, we often have to find the values of 
for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton’s method or as the Newton-Raphson method. It usually produces highly accurate approximations to the roots of the equation 
.
, which obviously does not converge, but alternates between 
and 
for 
is rotated in 3-space about the x-axis.
and 
were known to mathematicians in India about 400 years before Taylor’s time.
or 
? Of course, it depends on the values of x and y. We might consider a particular case: Is 
or 
?
is chosen randomly with probability 
of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.
converges it is easy to solve numerically for a given 
.
In Part I we described an attempt to fit a logarithmic spiral to the sequence 
. While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large 
, the sequence 
converges for 
. This result was first proved by Euler. For an earlier post on the power tower, click 
, the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of 
.
is beautifully behaved, oscillating regularly along the entire real line 
(it is also well-behaved for complex 
ThatsMaths 
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