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!

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