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On the origin of the word 'trice' (in the time sense)

Consideration of the development of mechanical clocks in the middle ages has led me to some interesting etymological asides. I learned, for instance, that a 'moment', as in the phrase "just a moment", has its origin in the obsolete unit of time, the momentum - being one tenth of a punctum which was itself a quarter of an hour. The Moment, in modern terms, is therefore about 90 seconds - and I use the qualifier 'about' deliberately, because in the era in which those units were used the 'hour' varied with the seasons, as did by necessity its sub-units .
Over some period prior to the 15th century, and probably in concert with the development of mechanical clocks and the adoption of "equal hours", the system of division of hours by 60 became the standard. This system evolved into the familiar present-day one where hours are divided into sixty minutes and minutes divided into sixty seconds.
The origin of the names 'minute' and 'second' for those units of time, comes from the Latin description of the act of dividing the hour. Somewhat illogically the word 'minute' comes from the Latin 'pars minuta prima', meaning "first small part". Similarly, but more logically, the 'second' comes from 'pars minuta secunda' meaning "second small part".
But keen Astronomers, calculating the time of lunar eclipses as accurately as possible, didn't stop with the second small part, they kept going with the third 'pars minuta tertia' and even the fourth 'pars minuta quarta'. Following on from the second the third sub-division units were called the tierce. (I don't know what the fourth sub-division units were called)
The Middle English Dictionary at the University of Michigan Press says tierce is an alternative form of terce, teres or tiers meaning (in this context) "Astron. or geom. A sixtieth part of a second of an arc, one thirty-six hundredth of a minute of an arc".
Examination of the origins of 'trice' lead to a great many references to hauling up things with Dutch pulleys and/or possibly tying things up with rope. However in the context of time and with reference to time-dependent phrases such as "gone in a trice", it seems far more likely (to me) that trice is evolved from tierce - a unit of time so small that to most people it would be equivalent to "an instant".
Do I have an etymological leg to stand on, or have I just read too many Wiki pages?

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