Approximate measures are units of volumetric measurement which are not defined by a government or government-sanctioned organization, or which were previously defined and are now repealed, yet which remain in use.
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Interestingly, it may be that all English-unit derived capacity measurements are derived from one original approximate measurement: the mouthful, consisting of about 1⁄2 ounce, called the ro in ancient Egypt (their smallest recognized unit of capacity). The mouthful was still a unit of liquid measure during Elizabethan times. (The principal Egyptian standards from small to large were the ro, hin, hekat, khar.)
Because of the lack of official definitions, many of these units have will not have a consistent value.
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United States
The vagueness of how these measures have been defined, redefined, and undefined over the years, both through written and oral history is best exemplified by the large number of sources that need to be read and cross-referenced in order to paint even a reasonably accurate picture. So far, the list includes The United States Pharmacopoeia, U.S. FDA, NIST, A Manual of Weights, Measures, and Specific Gravity, State Board Questions and Answers, MediCalc, MacKenzie's Ten Thousand Reciepts, Approximate Practical Equivalents, When is a Cup not a Cup?, Historical Thanksgiving Cookery, Cook's Info, and knitting-and.com., and Modern American Drinks.
Dashes, pinches, and smidgens are all traditionally very small amounts well under a teaspoon, but not more uniformly defined. In the early 2000s some companies began selling measuring spoons that defined a dash as 1⁄8 teaspoon, a pinch as 1⁄16 teaspoon, and a smidgen as 1⁄32 teaspoon. Based on these spoons, there are two smidgens in a pinch and two pinches in a dash. However, the 1954 Angostura “Professional Mixing Guide” states that “a dash” is 1/6th of a teaspoon, or 1/48th of an ounce, and Victor Bergeron (a.k.a. Trader Vic, famous saloonkeeper), said that for bitters it was 1⁄8 teaspoon, but 1⁄4 fl oz for all other liquids.