Unfortunately, this whole story is a fairy tale, made up by typographers to make themselves feel like they are correct in some absolute way. The account is riddled with historical fabrication. Here are some facts:
There were earlier standards before the single-space standard, and they involved much wider spaces after sentences.
Typewriter practice actually imitated the larger spaces of the time when typewriters first came to be used. They adopted the practice of proportional fonts into monospace fonts, rather than the other way around.
Literally centuries of typesetters and printers believed that a wider space was necessary after a period, particularly in the English-speaking world. It was the standard since at least the time that William Caslon created the first English typeface in the early 1700s (and part of a tradition that went back further), and it was not seriously questioned among English or American typesetters until the 1920s or so.
The standard of one space is maybe 60 years old at the most, with some publishers retaining wider spaces as a house style well into the 1950s and even a few in the 1960s.
As for the ugly white space, the holes after the sentence were said to make it easier to parse sentences. Earlier printers had advice to deal with the situations where the holes became too numerous or looked bad.
The primary reasons for the move to a single uniform space had little to do with a consensus among expert typographers concerning aesthetics. Instead, the move was driven by publishers who wanted cheaper publications, decreasing expertise in the typesetting profession, and new technology that made it difficult (and sometimes impossible) to conform to the earlier wide-spaced standards.
The lies do not just come from random Slate writers or bloggers, but also established typographers, who seem to refuse the clear evidence that they could easily see if they examine the majority of books printed before 1925 or so. Even an authority like Robert Bringhurst is foolish enough not to do his research before claiming that double spacing is a quaint Victorian habit that originated in the dark and inflationary age in typography of the (presumably mid to late) nineteenth century.
Full post (long but worth it):
http://www.heracliteanriver.com/?p=324