Our professor taught that Chinese historiographic tradition demands each dynasty to compose the history of the previous one, using the 實錄 or 起居注 provided by said dynasty. Browsing the Wikipedia article for the Twenty-Four Histories I noticed that the 漢書 ends with the fall of the former Han and the next book to be completed does not concern the later Han, but rather the Three Kingdoms, which is then followed by the book of the later Han. Is there a reason for this irregularity, or was my professor simply generalising? She made it sound like a sacrosanct tradition, so I assume one wouldn't break it for trivial reasons! Or is it simply that, while institutionalisation of the Historian's office began in Qin times, it wasn't complete until Tang?
The short answer is, the historical list (i.e. the image in the question) provided with the year-stamp is not really showing the chronology properly. The key point is Chen Shou's Records of the Three Kingdoms, although written in the 3rd century, should be correctly stated as "annotated & compiled in the 5th century" (by Pei Songzhi). So, it should show:
- Ban Gu, Book of Han, 82
- Chen Shou, Record of the Three Kingdoms, 289, and 429 (Pei Songzhi)
- Fan Ye, Book of Later Han, 445
If it was stated this way, it should clearer, that Chinese historiography during the early years did not progress smoothly, with big and inconsistent gaps when the official history was finally adopted.
The longer answer is Chinese historiography during the early years went through substantial changes in approach, philosophy, etc. A few points about the key texts in question:
First, the 2 books on Han dynasty, Book of Han (by Ban Gu, referring to Western Han & Wang Mang) and Books of the Later Han (by Fan Ye, refering to Eastern Han) are actually mostly from Ban Gu. Fan Ye's Book of the Later Han is, in part, based on the work of Ban Gu, i.e. from Dongguan Hanji (東觀漢記).
Second, Ban Gu, the historian for Han period, was imprisoned and died (92 CE) before he could complete his work, i.e., complete his unified Han Dynasty history. His sister, Ban Zhao, helped complete the Book of Han -- the Wikipedia entry shows this.
Finally, to understand the context of Chen Shou's work, Record of the Three Kingdoms, it is better to see it as a very basic work (not really good enough as historiography) and needed Pei Songzhi to help complete it, see: Annotations to Records of the Three Kingdoms.