I have read in various books,One example from a trip to the library: the book "Introduction to Qur'anic Script" (Syet Barakat Ahmad, 1984) says:
that, originally, the Arabic alphabet lacked those dots, and that they
were introduced later.
"The pre-Islamic script could be read in various ways; it had neither short vowels nor dots to identify consonants which shared identical letter outlines. Abu'l-Aswad al-Du'ali (died 69[AH]/688[AD]), said to be the founder of Arabic grammar, invented a system of providing fathah, dammah, and kasra to indicate some vowel sounds. A few years later his students, Nasr ibn 'Asim (89/707 or 90/708) and Yahya ibn Ya'mar (889/707), devised a system of placing dots above and below the letter outline to differentiate between various consonant sounds."
This paragraph is followed by a reproduction of a couple lines of completely unvoweled script, otherwise pretty similar to current Arabic script, with the caption "Early cursive script in a letter sent by the Prophet to the ruler of al-Hasa during the early seventh century."
Bruce