Over the last couple of months I have finally got round to something that I have been wanting to do for a few years but have, until now, just never had the time to do – that is making some videos which give an introduction (albeit a rather detailed introduction) to some of my favourite historical works in mathematics and geometry.

One of the works that I have made a series of videos on is Apollonius of Perga – Treatise on Conic Sections. Apollonius was a geometer (sometimes called the Great Geometer) in Ancient Greece possibly around the time of Euclid (indeed he appears to have had some familiarity with Euclid’s work). This is one of my all-time favourite works in classical geometry; Apollonius is not nearly as well-known as Euclid but the influence that Apollonius has had on mathematicians and scientists such as Isaac Newton over the last 2000 years can’t be underestimated. Apollonius’ work is virtually unknown in modern mathematics and geometry but suffice it to say that without Apollonius’ work much of modern geometry may not have existed.

Apollonius’ Treatise is a collection of seven books (originally eight books but one is no longer extant and indeed, books 5, 6 and 7 do not exist in their original Ancient Greek) covering the theory of the conic sections, that is, of circles, ellipses, parabolae and hyperbolae. In a modern-day setting these would usually be dealt with using coordinate geometry; however, coordinate geometry wasn’t a thing at the time of Apollonius and so everything is dealt with in the Treatise using ratio, proportion and a technique, that is all but completely forgotten in modern geometry, called application of areas.

The reason that I wanted to make this series of videos (which covers the first two books of Apollonius) was to give people an idea about how geometry has been done in the past and to show what can be achieved with what would be considered quite primitive techniques nowadays. The techniques may be fairly rudimentary in and of themselves, but the ways in which Apollonius applies those techniques is anything but rudimentary – I would g so far as to say that very few modern mathematicians and geometers would be able to use these techniques with the confidence and dexterity that Apollonius uses them. Of course it would work the other way round as well – probably Apollonius would not be able to use modern-day techniques (possibly from beyond the grave) as well as modern-day geometers.

So why bother reading Apollonius? After all, there is nothing there that from a modern perspective is going to make you into a better geometer. The only real reason to read it is if you are interested in it for whatever reason; if you are interested in the historical development of mathematics and geometry. But even though you wouldn’t really be looking to use these techniques, you can get some insight into the creative mind that produced the work. You can start to see how these things have been visualised in the past. It’s very tempting nowadays to just think of a conic section as just an equation on the page – but in Apollonius, the conic sections are actual geometrical shapes which need to be visualised in order to appreciate their properties.

There are quite a few parts to my series of videos – I have tried to cover as much ground as possible without simply repeating everything in the books. The particular translation that I have used is Thomas Little Heath’s translation which dates from 1896. There are pros and cons to using this version but alas, that would be the case whichever version you used.

Cheerio for now!

I never really used to be interested in history in school – in fact I couldn’t wait to drop the subject. I just couldn’t see the point in thinking about the past when it just didn’t seem relevant to me or anything for that matter. Historians out there will be pleased to hear that I take a very different view of history now and, although I wouldn’t say that I’m a fully fledged historian, I do enjoy reading the odd history book.

My interest in mathematics and history come together when I read mathematics books written by some of the greatest mathematical thinkers in history. It’s amazing to see how problems were solved by the ancient Greeks using the technique of application of areas which is mostly a lost art now and has been replaced by algebraic techniques; or to see how philosophical issues have shaped the development of mathematics such as whether the Axiom of Choice is valid as an axiom.

Well here are my top historical mathematics books with a brief explanation of what’s inside

  • Principles of Mathematics – Bertrand Russell Bertrand Russell is one of my favourite philosophers and mathematicians in history. Not many can match Russell for his depth of knowledge – even today. Don’t be confused by the title – this is not a book on simple mathematics; it’s certainly not for the faint-hearted and will take a good chunk of time to plough through. Most of what is inside the book would now be considered a bit dated but at the time this was ground-breaking stuff.
  • Elements – Euclid Probably one of the most famous mathematics books ever written. I still think it’s incredible that this book was still the standard geometry textbook in most schools up until the 19th Century – about 2000 years after it was written in ancient Greece. Nowadays most textbooks are thrown out after a year to bring in the next lot of ‘updated’ textbooks. If any book shows how timeless mathematics is then this is it.
  • Treatise on Conic Sections – Apollonius of Perga This is another book by an ancient Greek but is nowhere near as well known as Elements. It’s a shame because inside this book are some of the most inventive uses of the technique known as application of areas to prove various properties of the conic sections – circles, ellipses, parabolas and hyperbolas. Again, although the technique has now been replaced with algebraic techniques the solutions are nothing short of beautiful.
  • Contributions to the Founding of the Theory of Transfinite Numbers – Georg Cantor Cantor is the father of the infinite in mathematics. His works took mathematics in a whole new direction – a very controversial direction at the time it would seem. When you read this book it all seems so straightforward to deal with the infinite but the imagination required to come up with some of the arguments and proofs is off the scale.
  • The Continuum – Hermann Weyl This is an unusual book as Weyl decided that he wasn’t happy with the way that mathematics was working at the time – he felt that mathematics had inadvertently created different ‘levels’ and there was a number system on each of these levels that comes about through the logic used and numbers on different levels were being combined when they shouldn’t be. He aims to demolish these levels and create a single number system but his logical system pays a price for this. It sounds like the plot to a novel! Weyl’s philosophy changed several times throughout his life and this gives a bit of insight into his personal philosophy at the time it was written. He later abandoned this work in favour of a diffeent philosophy but then, after a few years, he changed his mind and thought it was a good work after all.

So there you have it – some of my favourite maths books from history. I love reading these historical maths books because it feels like you’re reading the minds of some of the greatest mathematicians, philosophers and scientists to have ever lived. I’m sure there’s loads more of these books for me to read – I’ll be on the lookout for some good ones to read. By the way – some of these books are in the public domain now and you can download some of them for free from the internet. I prefer to buy copies of the actual book but just so you know.