History of Typewriters:  It is believed that 'writing machines', similar to typewriters, were around in the early 1700s, the earliest was probably created by Henry Mill who in 1714 obtained a patent from what sounds to have closely resembled a typewriter.

In 1829 the 'typographer' was patented by William Burt and is sometimes called 'The First Typewriter', although more accurate perhaps is London's Science Museum's description of Burt's work as 'the first writing mechanism whose invention was documented'. 

Burt's version of the typewriter was never commercially produced, no buyer was ever found for the invention which was in fact slower than writing by hand. But one great benefit of Burt's typewriter was the distinction incurred between an 'index typewriter' like this one which had a round dial to select characters, over the 'keyboard typewriter' as it came to be known where multiple keys existed to create character impressions in much the same manner as the typewriters we know today. 

Since the first successful commercial typewriters were introduced in the late 1860s many unusual designs have emerged, some plain and simple, others intricate and stunningly detailed.  One of the simplest and earliest designs had a wheel with letters round the edge which was turned manually until the required letter appeared in front of the paper and was pushed to form an impression.   More complicated typewriters had double keyboards, one for lower case, the other for capitals, and were created in brass and mother of pearl hand painted with glorious gilded leaves and flowers.  These are the kind of unusual models to watch out for at non-specialist auctions and they're almost certain to attract high prices on eBay.

These early models sometimes crop up at specialist typewriter auctions where they invariably fetch a high price.  Not the place to buy in expectation of high resell fees on eBay but worth visiting for research and experience.

Like most collectibles, value depends mainly on rarity, not just age.  For example, one of the earliest serviceable typewriters, the Underwood, created from 1900 to 1932 was made in the millions and can still be found in working condition, consequently they are worth very little.