Recorder (musical instrument) Dolmetsch in 1957, he first used the bell-key system publicly in 1958. Recorder (musical instrument) The recorder is a woodwind musical instrument in the group known as "internal duct flutes"—flutes with a whistle mouthpiece. A recorder can be distinguished from other duct flutes by the presence of a thumb-hole for the upper hand and seven finger-holes: three for the upper hand and four for the lower. It is the most prominent duct flute in the western classical tradition. Recorders are made in different sizes with names and compasses roughly corresponding to different vocal ranges. The sizes most commonly in use
Proof that e is irrational numbers. Still another proof can be obtained from the fact that Define formula_11 as follows: Then: which implies: for any integer formula_15 Note that formula_16 is always an integer. Assume formula_17 is rational, so, formula_18 where formula_19 are co-prime and formula_20 It's possible to appropriately choose formula_21 so that formula_22 is an integer i.e. formula_23 Hence, for this choice, the difference between formula_22 and formula_16 would be an integer. But from the above inequality, that's impossible.So, formula_17 is irrational. This means that formula_27 is irrational. In 1840, Liouville published a proof of the fact that "e" is irrational followed by