Why is 刻 used to mean ''a quarter of an hour''?
刻 means ''to carve'' but my dictionary also says it means ''to set a time limit''.


well, you need a better dictionary 

in 國語辭典, the first explanation of the entry “刻”, as noun is:


  古代計時的名稱，一晝夜共分一百刻。今以十五分鐘為一刻。《周禮．夏官司馬．挈壺氏》：「縣壺」漢．鄭玄．注：「漏之箭晝夜共百刻，冬夏之間有長短焉，大史立成法，有四十八箭。」


http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=W00000004456

some chinese astronomy background:

in the past, time is divided into year, month, day, and hour (“辰” / “時“)

in 商 dynasty a day was divided into 8 periods, and evolved to 10 periods.

then, in 周 dynasty, a day was divided into 12 periods; afterward, in 漢 dynasty, people used 十二辰 (子，丑 . . 亥) to name these periods.

to know time unit shorter than a day, people relied on sundial (日晷) &amp; water clock (漏壺)





the vertical ruler in the bottom bucket is called “箭”, that, marks are craved onto, to indicate 100 equal divisions. together, in one operation (24 hours), the ruler (箭) would flow upward continuously, one just read the mark on the ruler, one can know the approximately which hundredth of a day.

the history is, most dynasties in the past used 100 periods (刻) for 24 hours, which means 1 刻 is 14 minutes 24 seconds.

it’s till 1640s, the first emperor (順治) of 清 dynasty adopted jesuit ferdinand verbiest’s (南懷仁) recommendations; both the definitions &amp; methods of chinese astronomy were changed / improved. one of these “improvements” is, one day (24 hours) is divided into 96 刻.

therefore,

1 day = 12 辰 / 大時 = 24 hours (小時)

1 辰 / 大時 is divided into “初” &amp; “正”

24 hours = 96 刻 —-> 1 刻 = 15 minutes

such definition is valid since 1640s, and certain dynasties in the past.

info added.

for anyone who is interested in this topic, the internet archive has the complete collection of joseph needham’s book “science and civilisation in china”, a download of 12 files.

https://archive.org/details/ScienceAndCivilisationInChina/page/n4/mode/2up

the water clock (aka clepsydra) is in volume 3, page 379-399 of the pdf

https://archive.org/download/ScienceAndCivilisationInChina/Needham_Joseph_Science_and_Civilisation_in_China_Vol_3_Mathematics_and_the_Sciences_of_the_Heavens_and_the_Earth.pdf

not a leisure read, you’re reminded