This series is focused on vocabulary; a few verb paradigms are given, but apart from that, the grammar is presented inductively.
Laas D'Aguen 1859 The Text
An edition in Italian, English, French and Greek also exists, by Peridou, "Dialogoi Synetheis".
Laas D'Aguen published a number of works related to Hellenic studies, including a French-Greek dictionary, first published in 1858 (1874, second edition)
This educated nineteenth century form of Greek was closer to Ancient Greek than the Demotic Greek used in Greece since the 1970's. See the wikipedia article on Καθαρεύουσα
Why study Early Modern Greek? I agree with Stedman, and a number of other academic textbook writers that studying Καθαρεύουσα is a good entry point for Classical Greek - as the textbooks such as this one and Stedman's (which is designed as an introduction to Ancient Greek) deal with more everyday matters, which are easier to assimilate.
Constantinides in his NeoHellenica (1892) has this to say:
"Modern Greek, like many other European languages, has only in comparatively recent times assumed the form of a single fixed and definite language understood by the whole nation, and in this form it differs so little from ancient Greek that were a foreigner to address a Greek in the language of Lucian, he would be readily understood ; in fact many of my pupils, reading with me a passage from a good modern author, have asked me whether it was ancient or modern Greek, and were not a little astonished when they were told that they might regard it as either. It is not too much to say that any one who has a competent knowledge of ancient Greek can learn to speak the modern language in a month, though of course fluency can only be acquired by constant practice."
Structure - the word or phrase it read once in Greek, once in English, and then repeated twice in Greek.
Lesson 18 Greetings
Lesson 19 Parting
Lesson 20 Asking and Thanking
Lesson 21 Affirming and Denying
Lesson 22 Expressions of Surprise
Lesson 23 Probability
Lesson 24 Sorrow
Lesson 25 Blame
Lesson 26 Anger
Lesson 27 Joy
Lesson 28 Consulting and Opinions.
Lesson 29 Eating and Drinking
Lesson 30 The News
Lesson 31 Arriving and Departing
Lesson 32 Naming and Questioning
Lesson 33 Age
Lesson 34 Time
Lesson 35 Weather
Lesson 36 Dialogue One - Greetings
Lesson 37 Dialogue Two - Visiting