Maybe because the author of the book was trying to translate the individual characters "所 (소)" and "以 (이)," but if that were the case, where is the "by which" part? In "Du's Handbook of Classical Grammar," it also says 所以 (소이) can be translated as "that by which," adding that the 以 means "by." It also says that "that by which" can translate as "the reason why." In "A New Practical Primer of Literary Chinese," 所以 (소이) is translated as "the means by which" and "the reason why." It seems all three books want to translate the "以" in 所以 (소이) as "by," but I have a different opinion.
My Korean dictionary says 所 (소) can mean "a place," "a thing," or "what." The character that follows 所 seems to usually be a verb in the passive voice, so 所得 (소득) translates as "what (所) was earned (得)," which can translate simply as "income"; 所望 (소망) translates as "what (所) was desired (望)," which can translate simply as "desire"; and 所聞 (소문) translates as "what (所) was heard (聞)," which can translate simply as "a rumor" or "hearsay." These examples suggest that the 以 in 所以 (소이) should also be translated as a verb in the passive voice.
As a verb, 以 (이) can mean "to use," "to take," or "to consider," but I think it can also mean "to reason," which is very close in meaning to the verb "to consider." My Korean dictionary says 以 (이), as a noun, can mean "reason," which translates in Korean as "이유 (理由)" or "까닭." Based on the way Chinese characters often work, if there is a noun meaning, then there is also a similar verb meaning, and vice versa. For example, 問 (문) can mean "a question," but it can also mean "to question" or "to ask." Therefore, since 以 (이) can mean "a reason," we can assume it can also mean "to reason." That would mean that 所以 (소이) has, at least, four possible translations:
- "what (所) is used (以)"
- "what (所) is taken (以),"
- "what (所) is considered (以)"
- "what (所) is reasoned (以)," which can translate as the noun "reason."
One of the example sentences in "Du's Handbook" is as follows:
此吾所以悲也“This is why I am sad.”
The above sentence can literally translate as "This is (此) my (吾) reason (所以) [for] being sad (悲也)."