Old Maths, New Science

In the mid-19th century, the most sought-after feat of engineering and industry was a trans-Atlantic telegraph cable. But  when the first successful cable was laid between Britain and America in 1858, the speed of transmission was only about 0.1 words per minute. This post will be an analysis of Thomson's 1855 paper "On the Theory of the Electric Telegraph", published in the journal of the Royal Society of London. Derivation of the PDE Thomson starts by defining the variables that he uses throughout the work. I shall define the same variables using more modern lettering conventions. Let $C$ be the capacitance of the wire, $R$ the resistance of the wire, $V$ the potential at a point $P$ on the wire at a time $t$, and $I$ the current at the same point in the same instant. The charge $Q$ at $P$, called "quantity of electricity" by Thomson, is given by \[Q = VC = It\] Thus, in an infinitesimal length $dx$ of wire at $P$, we have a charge of $VCdx$ and, i...

A common feature of mathematics across the world is the desire by mathematicians to split the subject into categories. The advantages of this are obvious; if you organise a conference on Algebraic Topology, it is relatively clear what kind of lectures a prospective attendee can expect, for example. But there is one categorisation which I believe needs to stop: the notions of pure and applied mathematics. This is not to say that maths does not have varying degrees of use in the real world - that is of course true. It's not very hard to argue that partial differential equations have had a bigger impact on physics and engineering than commutative algebra. But I feel that the name "applied maths" is a little misleading. It seems to imply that this "applied maths" is in some way more real than so-called "pure maths" which is ridiculous. Mathematics exists entirely inside its own, objectively perfect; it isn't real at all. To make this point, I wi...