I was blabbing from memory, and inaccurately substitued the Roman Catholic
Church for feudal lords.  Anyway here is my source, which seems less
confident than the "dies mali" etymology:
 
"Webster's Universal Dictionary of the English Language", 1937:
 
dis'mal, a.; comp. dismaller; superl. dismallest. [M.E. dismal, diesmal, dysmall,
dismal; origin uncertain, perhaps originally signifying a decimal or tithing day,
when tithes were to be collected by the feudal lords, from OFr. disme, a tithe;
L. decimus, a tenth; from decem ten] Having or exhibiting depression,
melancholy, or gloom; lacking cheer; presaging ill; characterized by darkness
or dolefulness; as, a dismal appearance; dismal prospects.
 
----- Original Message -----
From: Marc LeBrun
To: math-fun
Sent: Wednesday, October 15, 2003 2:08 PM
Subject: Re: [math-fun] a dismal arithmetic

>=David Wilson
 > The word "dismal" apparently derives from "decimal" in relation to the
tithing day, or "decimal day",
 > of the Roman Catholic Church, when one tenth of one's yearly income was
given to the Church,
 > a dismal day indeed.

Fascinating!  So I guess the economists aren't the only practitioners of a
"dismal science"!  According to the web sources I looked at, "dismal" is
actually derived from Latin dies + mali, "evil days", and yet:

   "These days of evil omen were known as Dies Aegyptiaci (Du Cange,
Glossarium, s.v.) or Egyptian days, either as having been instituted by
Egyptian astrologers or with reference to the "ten plagues"; so Chaucer, "I
trowe hit was in the dismal, That were the ten woundes of Egipte" (Book of
the Duchesse, I 206). There were two such days in each
month."  --www.1911encyclopedia.org

So there's also an association with 10 plagues.  (Us enlightened moderns of
course call our biweekly tithe "payroll tax withholding"
generally >>10%!)  Crossing these words seems irresistible:

   "However, a number of learners continue to have difficulties with
English words, and continue to make confusions like "dismal numbers" for
"decimal numbers".  Indeed, some may never get past this problem."  --Ryan,
A. and P. Meara, "A diagnostic test for 'vowel blindness' in Arabic
speaking learners of English.", PRYFISGOL CYMRU ABERTAWE [!]

Also, on the Eastern US seaboard "dismal" is nouned as a synonym for
"pocosin", which is not an exotic trigonometric, but rather a swamp or marsh.

Apropos bogs, here's answers to those "Quiz" questions (would dismals make
for fun elementary-grade "enrichment" puzzles?):

Q: What is the unity?
A: 9.  (Multiplying by 1 changes each non-zero digit to 1).

Q: What's the smallest dismal prime?
A: 19.  The dismal primes are the numbers >9 ending in 9.

Q: Notice that p(12) = 22 = p(21); does p(13) = 92 = p(n) for any other n?
A: Yes, p(31)=92; p(n) depends only on the multiset of the digits in n,
independent of order.
(Extra credit: what is p(111)?  Hint: A003465)



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