Historical:
It is believed that the Chinese scholar Yang Hsiung (also Yang Xiong,
53 B.C. - 18 A.D.) gave his T'ai Hsüan Ching (太玄經 Mystery) to the I Ching (Chou I, or Changes) which he held in high esteem but was in danger of being forgotten, in order to give the latter new impetus. The already at this time very old classic, the I Ching was dated by R.Alan Kunst 1983 in his dissertation
“The Original Yijing”, p.4, to around 800 BC.
Michael Nylan wrote in her translation "THE ELEMENTAL CHANGES", The Ancient Chinese Companion to The I Ching, SUNY, New York, 1994. p.8.: "By the first century B.C. the I Ching consisted of a set of sixty-four texts, each associated with a six line graph ( or "hexagram"), in which component lines could be either solid or broken (signifying yang if solid, yin if broken). Under each hexagram, there are six assigned texts, each of which corresponds to one line of the graphic symbol (hence, the "Line Texts"). The core text of the Mystery, like that of its prototype, the Changes, presents a series of linear complexes.
For the hexagram of the Changes, however, the Mystery substitutes a four-line "tetragram" whose component parts read from top to bottom (i.e., in the opposite order from the Changes). Also in contrast to the Changes, where the lines are categorized as either yin (broken) or yang (unbroken), the divination procedure prescribed in Yang's instructions involves three possibilities for each line of the graph: (1) an unbroken line (correlated with Heaven), (2) a line broken once (representing Earth), or (3) a line broken twice (symbolizing Man as one of the triadic realms, living between Heaven and Earth)".
Besides the tetragrams of Tai Hsüan Ching are made up of an upper and a lower bigram (element from two lines), instead of the corresponding two trigrams (elements from three lines) for the I Ching. From the four lines of T'ai Hsüan Ching, each with 3 possibilities result in 3 ^ 4 = 81 (instead of 64 for the I Ching) possible tetragrams, to which 9 appraisals (altogether 729) were then added.
In addition to these appraisals, you will receive two numbers as the answer, a) that of the trigram and b) that of the appraisal, so that you can read the full version at Michael Nylan's correct translation. (Remember that the appraisals are read from top to bottom!) I have added the original texts in Chinese characters to the answer pages, with the title (always the first character) of the tetragrams and the three-part main text at the top and the relevant of the nine possible appraisals at the bottom.
However, the most notable difference between these two systems, which are similar in principle, is that the order of T’ai Hsüan Ching follows an elegant, strictly mathematical logic and thus supports a coherent philosophy, while for the order of the I Ching unfortunately no plausible explanation has been found until today.
I would also like to mention Derek Walter's committed work, through which I was able to familiarize myself with the Mystery more than 30 years ago. In his book "The Alternative I Ching", Wellingborough, 1987, he addresses the mathematical beauty inherent in T’ai Hsüan Ching. If you would like to know more about this, we recommend ‘Mathematics of the Tai Xuan Jing’ by Frank ‘Tony’ Smith. You can find it at: https://vixra.org/pdf/1412.0175v1.pdf