Joined-Word Style in American Handwriting and Penmanship: Examining Documents Outside the Telegraph Office
As stated in Wikipedia, scriptio continua, (Latin for "continuous script"), also known as scriptura continua or scripta continua, is a style of writing without spaces, or other marks between the words or sentences. The form also lacks punctuation, diacriticals, or distinguished letter case. In the West, the oldest Greek and Latin inscriptions used word dividers to separate words in sentences; however, Classical Greek and late Classical Latin both employed scriptio continua as the norm."
The way in which words are connected, or not, has a long history, as made clear by the definition of scriptio continua, as seen above.
The shorter writing history in America has some similar traits.
The incidence of words joined together in one long pen stroke is not entirely unusual. In American handwriting, this was done by early penmen to show their virtuosity, and that has persisted for hundreds of years.
Somewhat less common is to observe this joined-word handwriting in others. For telegraph operators who were expected to write at great speed, it became a habit of many to keep the pen on the paper without lifting it unless required, and so words were joined. There is an inherent visual component too, that seems obvious, in which the lines that connect the words are like the electrified wires that run from pole to pole; whether this was ever in the minds of the writers or of the public is not known at this time.
Is it possible that telegraph operators carried the style of writing with them, outside of their jobs and into their public lives? Or was such writing not of operators, but others, and if so, how can that be discerned?
Here is a problematic piece that follows.  First is the issue that its author is unclear. While an A. W. Agassiz does exist in this time period, it is in fact Louis Agassiz who is in charge of the program at Harvard, from which this letter seems to come. [This needs to be cleared up, as a point of research, but in some ways is less important to the central argument here.] Also, the letter seems to suggest that this is in fact the writing of a person who is taking dictation. It would certainly make sense to hire a former telegraph operator for dictation, and if that is here the case, then perhaps this is telegraph handwriting from an operator who is now employed by Harvard. (All readers of this with a better theory are encouraged to send in their own theories.)
David S. Bispham’s writing is a vertical and readable round hand with lines connecting the words, and he worked for the Lehigh Valley Railroad, according to Wikipedia .  While a weak linkage because he is not identified as a telegraph operator, it is enough to put forth the argument that his writing style is “telegraph hand” or “railway hand,” as it is also called. Here is a sample of his writing in a letter to a friend in 1893. 
Did any knowledge or popularity of the telegraph hand add to the attraction of such a style for the general public? I have read no material which discusses this or makes this connection, so I will accept it to be a possibility without any basis in fact.
So what of the those, who appear not necessarily to be former or current telegraph operators, who write in such a fashion?
This is a puzzling question for which I have no answer. It may be that the writing or telegraph operators was publicized and became popular, or that it is some trend for which there is no evidence that has yet been found. However, there is in my study a grouping of such writing that occurs at the end of the nineteenth and the beginning of the twentieth century.
Here is a letter from one librarian to another in 1893.  The words are joined, in some cases. It is notable that it is from John Cotton Dana who is himself an ardent supporter of library handwriting who publishes it in a book,  to Melvil Dewey, who is the first person to have published the letterforms in 1887.  Knowing these facts, it is hard in some respects to make sense of the style, yet I suppose we must assume that John Cotton Dana wrote this way as his person script? How did he arrive at it? There is mention that he worked as a civil engineer before he was a librarian.  Could it be that he learned this script as a civil engineer, and that they too used this script, or was he also trained as a telegraph operator, though it is not reported? More research is required. But for now, his writing stands as a puzzle, or a piece of it.
This example is a “page from the class notes of Eliot Harlow Robinson taken from lectures on trusts given at the Harvard Law School by James Barr Ames, 1907-1908.”   While two short biographies have been found, they do not indicate that the writer worked as a telegraph operator or for a railroad.  See also pages 269 and 270. 
Here is a 1908 letter from Basil L Gildersleeve, a philologist, to Donald Gay Baker, also a scholar.  Born in Charleston, South Carolina in 1831, he did fight during the Civil War, but the details I have seen about his life do not indicate that he was a telegraph operator. Also, he entered the war at a time when his writing style was likely already fixed. Also, while his writing meets two criteria of being vertical in angle and with joins between words, it is not a round hand, and its illegibility suggests his writing could never have been meant to be read by others on a telegram, if it were in this style.
Charles Burton Gulick, another language scholar, also wrote this letter to Donald Gay Baker. This letter, while undated, likely falls into roughly the same time period as the first. This too is in a joined-word style. This writing is very much of the time, and while its angle would not disqualify it from being like telegraph hand, it is at best a rounded Spencerian or similar style script, and its sloppiness also disqualifies it from being the work of anyone who might have ever been a telegraph operator.
How then do we consider these items? Was joined-word script a style for law students and scholars studying Greek manuscripts in the first decade of the twentieth century? Likely not. Was it a period affectation, or some style used in this time? Perhaps.
This topic is still under research, and while it would be very convenient to avoid brining up these samples relative to telegraph handwriting, or railway handwriting, the existence of such writing must be acknowledged. It is far easier to make the unknown public and to discuss it rather than let it continue to muddy the waters of any research into the writing of telegraph operators.