I've come across this theory before and it has fascinated me. But is it true? There are certainly examples in the past of Democratic states going to war but throughout most of the 20th century it seems that states with functioning democracy's have largely remained peaceful with one another. Does this theory have any justification as we go further back in time?
Also would people have any recommended literature on the topic?
This question turns entirely upon what you decide qualifies as a 'functioning democracy'. If a functioning democracy is, as we might expect today, one in which the goverment of a state is elected by the population, with a secret ballot and suffrage limited only by age and nationality, then of course there are no wars between democracies that precede the C20th, because there are no democracies that precede the C20th.
How much can the franchise be limited before it ceases to be a democracy? How indirect can representation become before you don't count it as democratic? How many functions of the state can be performed by non-elected officials? The answers you get to your question depend on how you first answer these.
Three Possible Contenders from the Long C18th
If you would count a federal republic as a democracy and an elected parliment as a democracy, then you have the series of three Anglo-Dutch wars in the latter half of the C17th. This can be extended to four if you include the events of 1688 as a war (as some do), and more if you count independantly the subsequent military actions against Scotland and Ireland, each of which had it's own independant parliament.
There are further scuffles between those two powers throughout the Eighteenth Century.
You then have the French Revolutionary Wars in which both states were involved, this time against a common enemy in a France which was defined by its radical political system and governed by an elected National Convention.
During the Napoleonic Wars, France at least maintained a parliament throughout the consulship, though it would be a stretch to describe that as democratic in practice. But there were side actions that took place in North America between the UK and the USA.
If you're interested in the earlier part of this period at all, and interested in the relationship between the nature of representative politics and the experience of war, I would recommend J. Miller (2000) After the Civil Wars. It's a political history of Restoration England which seeks to illustrate the way in which the memory and threat of war influenced the development of balanced and representative political institutions.
The way i heard it was that democratic states don't go to war against other democratic states.