There are (at least) two logically consistent ways to interpret a period (full stop) that appears at the end of a string of words in a list entry: (1) as signifying the end of a complete sentence; (2) as signifying the end of a free-standing entry in the list, whether the entry consists of a complete sentence (or multiple complete sentences) or not.
The usual style in U.S. and British publishing today is to include periods at the end of complete-sentence entries in a numbered, bulleted, or unnumbered vertical or display (that is, non-run-in) list, but not to include any end punctuation for sentence-fragment entries. In other words, the preference is to hew to interpretation (1) above, rather than to interpretation (2).
Here is the relevant guideline in The Chicago Manual of Style, sixteenth edition (2010):
6.124 Vertical lists—punctuation and format. A vertical list is best introduced by a complete grammatical sentence, followed by a colon (but see 6.125 [discussing the option of punctuating a vertical list as a single complete sentence]). Items carry no closing punctuation unless they consist of complete sentences.
And here is the relevant guideline from The Oxford Guide to Style (2002):
9.1.4 Displayed Lists ... The punctuation for a displayed list is treated precisely the same as if the items within it had no number or letter separating them. List items that are complete sentences start with capitals and end in full points, regardless of the length of the sentence or number of subheadings. Sentence fragments do not, and are usually (but not exclusively) lowercase.
But the preference endorsed by Chicago and Oxford is a matter of style decision—and it would not be unreasonable to emphasize instead the importance of clearly demarcating the end of a list entry rather than simply distinguishing between complete-sentence entries and sentence-fragment entries.
The value of end punctuation for all entries in a list may be stronger in a system of list formatting that calls for positioning turnover text from the first line of an entry somewhere beneath the right-hand side of the first line of the entry, rather than positioning it consistently beneath the leftmost word of the entry at a standard indent beneath that word. Because we are accustomed to seeing lists with entries whose turnover words align left along a consistent indented margin, we may take for granted how messy the alternative formatting style can be.
In the table excerpt that appears in the original question above, two entries in the rightmost column contain turnovers:
A Zenzizenzizenzizenzike, or Square of Squares Squaredly Squared.
and
A Square of Squares of Squared Cubes, or a Zenzizenzizenzicube.
As it happens, the typesetter handled these two turnover entries inconsistently. In the first instance, the typesetter added an open parenthesis mark before the turnover word (Squared) and right-aligned the word with the right-hand margin, immediately beneath the rightmost word (Squaredly) of the first line of the entry—a line that happened to run full measure. In the second instance, the typesetter indented the turnover word (Zenzizenzizenzicube) seven letter spaces to the right of the leftmost word of the first line of the entry with no open-parenthesis marker and no attempt to shift the turnover line to right-align either with the rightmost word (a) of the first line or with the right-hand margin. In that instance, the first line of the entry ended far short of full measure.
It is easy to see how such a haphazard and ragged approach to laying out multiple line entries in a list could make it difficult for readers to identify where one entry ended and another began, or (potentially) to figure out what entry a particular word was intended to belong with. This problem is exacerbated in the posted example by the very frequent use of initial capitalization for adjectives and nouns throughout the table.
Suppose for example, that the entry in the line beneath "A Zenzizenzizenzizenzike, or Square of Squares Squaredly Squared" had said not "A Fifth Sursolide" but "A Fifth Sursolide, or Squared Square of of Squares Squaredly Squared." In that case, following the pattern used elsewhere in the rightmost column of the table, we might expect to see the typesetter to offer the following entries for the four lines affected by turnovers in consecutive lines (the baseline dashes in the fourth line below are there simply to push the turnover from the third line to what I imagine its typeset position would be; EL&U's text program doesn't give effect to multiple consecutive letter space clicks):
A Zenzizenzizenzizenzike, or Square of Squares Squaredly
A Fifth Sursolide, or Squared Square of Squares (Squared.
A Zenzicubicube, or Square of Cubick (Squaredly Squared.
A Sixth Sursolide. _ _ _ _ _ _ (Cubes.
In such a situation, interpreting the intended end of each table entry would be almost impossible without the inclusion of open parentheses to indicate where the turnovers begin and of periods to signify where each entry ends. I suspect that thinking along these lines encouraged typesetters to add periods to the end of each entry in a list, whether the entry was a complete sentence or a fragment. And of course, once readers are accustomed to a typesetter's formatting choices—assuming that the typesetter enforces them consistently—the choices recede into the background and become relevant only when ambiguity might otherwise result.
As a final point, I note that it is very common for editors today to receive manuscripts in which the author has added periods to the ends of entries in a list consisting of moderately long phrase-like entries that are nonetheless sentence fragments. For example:
In this report, we have three objectives:
To give readers a clear idea of the dangers of ignoring climate change.
To provide practical advice on how to identify the most climate-unfriendly processes that a company uses in its regular operations.
To suggest low-cost alterations to those processes that will enable the company to move toward a net-zero operating model.
Chicago (which the consultancy I work for follows on most matters of style) explicitly opposes including periods at the end of each of these list entries, so I would take them out as a matter of course. But many authors (and some publishers) instinctively view length of entry (rather than complete sentence versus fragment) as the primary basis for including or omitting end punctuation from entries in a numbered or unnumbered list. This, too, can provide a logical rationale for consistently adding periods to or omitting them from the ends of list entries.
