A Collection of Word Oddities and Trivia, Page 15

Last revision: June 20, 2014

MATHEMATICS

The system for naming large numbers used here is the one shown in MWCD10. Words such as unvigintillion, trigintillion, novemnonagintillion, etc., which are found in extensions of this system, are generally not used here; "and" is not allowed in the name of a number. Number names are limited to those that represent positive integers between 1 and 10 66 . Names involving googol and googolplex are not used.

Most of the information in this mathematics section was contributed by Eric Brahinsky and Paul Wright.

EIGHT is the first whole number, if all whole numbers are arranged in alphabetical order; ZERO would be the last number [Stuart Kidd].

By standard dictionary alphabetizing procedure (i.e., ignoring spaces, punctuation, etc.), the first ten number names alphabetically are:

eight

eight billion

eight billion eight

eight billion eighteen

eight billion eighteen million

eight billion eighteen million eight

eight billion eighteen million eighteen

eight billion eighteen million eighteen thousand

eight billion eighteen million eighteen thousand eight

eight billion eighteen million eighteen thousand eighteen.

The last whole number excluding zero is two vigintillion two undecillion two trillion two thousand two hundred two.

The first ordinal integer name alphabetically is eight billion eighteen million eighteenth and the last, excluding zeroth, is two vigintillion two undecillion two trillion two thousand two hundred twenty-third.

The first perfect square alphabetically is eight billion eighteen million eight hundred forty-four thousand three hundred four (8,018,844,304), which is the square of 89,548. The last is:

two vigintillion two undecillion two hundred two decillion two hundred eleven nonillion six hundred ninety-eight octillion two hundred ninety-seven septillion seventy-nine sextillion twenty-seven quintillion two hundred thirty-two quadrillion six hundred sixty-three trillion two hundred forty-nine billion sixty million one hundred sixty-seven thousand nine

The shortest names for numbers are ONE, TWO, SIX, and TEN.

According to Paul Wright, the longest single-word number name is QUATTUORQUINQUAGINTAQUADRINGENTILLION (37 letters) it is a one with 1,365 zeros. However, he writes, “If you want to go totally long and ridiculous: TREMILLIAMILLIAMILLIATRECENTTRETRIGINMILLIAMILLIATRECENTTRETRIGINMILLIATRECENTDOTRIGINTILLION (94 letters) is a one with 10,000,000,000 zeros after it!”

The longest names for numbers have 758 letters each, and there are 984,770,902,183,611,232,881 (or 3 44 ) such numbers.

The smallest natural number whose name is spelled with:

3 letters is 1 (one);

4 letters is 4 (four);

5 letters is 3 (three);

6 letters is 11 (eleven);

7 letters is 15 (fifteen);

8 letters is 13 (thirteen);

9 letters is 17 (seventeen);

10 letters is 24 (twenty-four);

15 letters is 103 (one hundred three);

20 letters is 124 (one hundred twenty-four);

25 letters is 1104 (one thousand one hundred four);

30 letters is 1117 (one thousand one hundred seventeen);

40 letters is 13,373 (thirteen thousand three hundred seventy-three);

50 letters is 113,373 (one hundred thirteen thousand three hundred seventy-three);

100 letters is 11,373,373,373 (eleven billion three hundred seventy-three million three hundred seventy-three thousand three hundred seventy-three).

Hans Havermann has greatly expanded the above list at chesswanks.com/num/a080777.html. He has also compiled a list of the largest numbers less than 10 66 whose name has a given number (3-758) of letters at chesswanks.com/num/a052196.html.

The smallest natural number whose name uses the five vowels a, e, i, o, u (in any order) is 1005 (one thousand five).

The smallest natural number whose name uses the six vowels a, e, i, o, u, y (in any order) is 1025 (one thousand twenty-five).

The smallest natural number whose name uses the five vowels a, e, i, o, u in order is 1084 (one thousand eighty-four).

The smallest natural number whose name uses the six vowels a, e, i, o, u, y in order would appear to be 1,000,000,000,008,020 (one quadrillion eight thousand twenty).

Eight number names use only one distinct vowel: THREE, SEVEN, TEN, ELEVEN, TWELVE, and SEVENTEEN use only e; SIX uses only i; and TWO uses only o. Numbers whose names use exactly two distinct vowels, however, go as high as six vigintillion six nonillion six octillion six trillion six billion six million six (using i and o)."

The longest whole-number name consisting entirely of alternating consonants and vowels is NINETY-SEVEN. However, if all integers are allowed, NEGATIVE NINETY-SEVEN would qualify.

In writing the names of the natural numbers one, two, three, four, five, six, etc., remarkably the letter A is not used until ONE THOUSAND. The letter A is not used twice until ONE QUADRILLION ONE HUNDRED THOUSAND. [Paul Wright] [Note that, for example, ONE HUNDRED AND ONE includes an A, but for this page the preferred U. S. naming convention which does not use the word AND is used, as explained at the top of the page.]

The letter B is not used until ONE BILLION. No number name contains two B’s.

The complete list of smallest natural numbers whose names contain each respective letter of the alphabet follows:

a - one thousand, b - one billion, c - one octillion, d - one hundred, e - one, f - four, g - eight, h - three, i - five, j - (none), k - (none), l - eleven, m - one million, n - one, o - one, p - one septillion, q - one quadrillion, r - three, s - six, t - two, u - four, v - five, w - two, x - six, y - twenty, z - (none)

The complete list of the smallest natural numbers whose names contain two a’s, two b’s, etc., follows:

a - one quadrillion one thousand, c - one decillion one octillion, d - one hundred, e - three, f - fifteen, g - eighty-eight, h - thirty-three, i - thirty-five, l - eleven thousand eleven, m - one novemdecillion one million, n - nine, o - forty-one, p - one septdecillion one septillion, q - one quintillion one quadrillion, r - thirty-three, s - sixty-six, t - thirteen, u - one hundred four, v - seventy-five, w - twenty-two, x - sixty-six, y - twenty thousand twenty.

The above list was provided by Paul Wright and Eric Brahinsky.

The smallest whole number using all 23 letters that are used in naming numbers is

one octillion one septillion one quadrillion one billion one million two thousand five hundred sixty-eight (1,001,000,000,001,000,001,001,002,568).

The largest number whose name does NOT contain the letter "n" is EIGHTY-EIGHT.

The first Roman numeral alphabetically is C (100); the last is XXXVIII (38). The first odd Roman numeral alphabetically is CCCI (301); the last is XXXVII (37).

The only integer names with no repeated letters are ZERO, ONE, TWO, FOUR, FIVE, SIX, EIGHT, TEN, FORTY, FORTY-SIX, SIXTY, SIXTY-ONE, SIXTY-FOUR, EIGHTY, EIGHTY-FOUR, and FIVE THOUSAND.

The smallest number whose name has all of its letters repeated is THREE TRILLION FOUR HUNDRED FOUR.

The shortest number names with all letters repeated are ONE NONILLION ONE, TWO NONILLION TWO, SIX NONILLION SIX, and TEN NONILLION TEN, each with 15 letters.

TWO is the only number name (positive integer) consisting entirely of letters from the second half of the alphabet. No number name is spelled only with letters from the first half of the alphabet.

No number name except TWO is spelled with letters from any single row of the typewriter keyboard. No number name is typed only with left-hand, or only with right-hand, letters.

The largest number that can be spelled using only the last 18 letters of the alphabet (I through Z) is 66,000,000,000,066,000,066,000,066,000,066. But disallow the letter I, and the largest that can be spelled with just the last 17 (J through Z) drops down to . . . 2.

The following number names use each of the five vowels a, e, i, o, u exactly once: FIVE THOUSAND, SIX THOUSAND TEN, EIGHT THOUSAND, NINE THOUSAND, TEN THOUSAND SIX.

The following number names use each of the six vowels a, e, i, o, u, y exactly once: SIX THOUSAND TWENTY, TEN THOUSAND THIRTY, TEN THOUSAND FIFTY, TEN THOUSAND SIXTY, TWENTY THOUSAND SIX, TWENTY-SIX THOUSAND, THIRTY THOUSAND TEN, FIFTY THOUSAND TEN, SIXTY THOUSAND TEN, EIGHTY THOUSAND, NINETY THOUSAND.

NINETY-SEVEN is the longest number name with strictly alternating vowels and consonants, unless you count NEGATIVE NINETY-SEVEN. [Charles Turner]

Some words of four or more letters composed solely of letters used in Roman numerals are: DIVIDIVI (a tropical American tree), CIMICIC (a yellow oil), CIMICID (an insect), CIVIC, CIVIL, CIXIID (one of the Cixiidae, a family of insects of order Homoptera), CLIM (dial. past of climb; in W3), DILL, DILLI (a bag or basket also called dilly bag), DIXID (a type of midge, of the family Dixidae), IDIC (pertaining to or consisting of ids), ILICIC, IMID (variant spelling of imide), IMIDIC (related to an imide), IMMI (unit of measure formerly used in Switzerland = 1.5 l), IMMIX, IXIL (Mayan tribe in central Guatemala), LILL, LIVID, MIDI, MILD, MILL, MIMI (heroine of the opera La Boheme), LLIVILLIVI (a place name in Chile), MIMIC, VILI (a brother of Odin in Norse mythology), VILL, VILLI (plural of villus), VIVID.

MIX is the longest English word that is also a valid number in Roman numerals. BACILLICIDIC (destructive to bacilli) and MILLIMILLINARY (correct within a millionth part) have the longest uninterrupted string of Roman numeral letters in a word (10) [Dmitri Borgmann]. ARMADILLIDIIDAE (pill bugs) has a sequence of nine Roman numeral letters.

Among words consisting only of Roman numeral letters, the words with the highest total value are MIMIC (2,102) and IMMIX (2,012).

If U is allowed in Roman numeral words, there are ILLUVIUM, DILUVIUM (a coarse glacial deposit), MILLILUX (1/1000th of a lux), ILLICIUM (a genus of evergreen trees), and CULICID (mosquito family).

In Latin, there are JUDICIUM, DILUCULUM ("dawn" or "daybreak), LIXIVII (genitive of LIXIVIUM), and LILICIDIUM (the slaughter of the lily). Note that in ancient Rome, there were no U and J and these would have been written as V and I.

[Pierre Abbat, Bruce D. Wilner, Dan Tilque, Rudy Wang, Rick Shepherd, Stuart Kidd, Susan Thorpe, Paul Wright, Philip Bennett and Pertti Malo contributed to the Roman numeral section]

FABACEAE (a family of beans in some classifications) is the longest word which can also be a hexadecimal numeral (consisting only of A, B, C, D, E, and F). Other hexadecimal words are ACCEDED, BEDFACE (a word used in the building industry), DEFACED, and EFFACED. DEADBEEF in hex is often used in computers to mark an invalid number or address. DEADBEEF is the name of an attack on a public-key cryptographic protocol, consisting of publishing a key with the same key ID as a previously existing key. In PGP and GPG, the key ID is the last few digits of one of its numbers in hex. To demonstrate this attack, someone created a key ending in 0xdeadbeef, so it is called a dead-beef attack. The hex code "DEADC0DE" (including a zero) also coded as "0xdeadc0de" (0x meaning hex) is sometimes used in programming [Sven Silow, Ted Clarke, Bruce D. Wilner, Pierre Abbat, Mike Geller, Scott McCarter].

FORTY is the only number which has its letters in alphabetical order. FIRST shares this property [Stuart Kidd]. According to Stuart Kidd, ONE is the only number with its letters in reverse alphabetical order.

In German, EINS and ACHT are the only numbers with their letters in alphabetical order [Gerd Baron].

GOOGOL is a 1 followed by 100 zeros. Mathematician Edward Kasner supposedly asked his nephew Milton Sirotta to suggest a name for the number, and he came up with this word, which is now found in many dictionaries. The million, billion, trillion, quadrillion system skips over this number. A googolplex is 1 followed by a googol of zeros.

According to Pascal Kaeser, in these languages you can count until one million (and more), without using the vowel e: Aymara, Chichewa, Haoussa, Ouigour. These languages are officially written with the Latin alphabet.

In German, EINE MILLION is the first number using the vowel "O" and the first using the consonant "M." EINE QUADRILLION is the first number using the consonant "Q." EINE SEXTILLION is the first number using the consonant "X." EINE SEPTILLION is the first number using the consonant "P." EINE OKTILLION is the first number using the consonant "K." "J" and "Y" seem to appear never in German numbers. (Myriade is not used as a certain number.) [Gerd Baron] [EINE QUADRILLION is 10 24 , like in the old British system, or one septillion in modern/American usage; and analogously with the others.]

In German, 22 is ZWEIUNDZWANZIG (with three Z's), 20,020 is ZWANZIGTAUSENDZWANZIG (with four Z's), and 20,022 is ZWANZIGTAUSENDZWEIUNDZWANZIG (five Z's). [Paul Wright]

In Finnish/Suomi, 8 is KAHDEKSIKKOKASI (with 5 K's) or KAHDEKSIKKO (with 4 K's). [Paul Wright]

ZENZIZENZIZENZIC (eighth power of a number) has more Z's than any other word in English. The word is in the OED. It was suggested by Robert Recorde, 16th century writer of popular math textbooks, although in his spelling the last letter of the word was k.

In a post in sci.math in 1995, Matthew P. Wiener suggested that the longest word in mathematics that is an accepted standard term is RHOMBICOSIDODECAHEDRON. He listed these other long mathematical words of dubious validity: DODECAHEMIDODECAHEDRON, DODECICOSIDODECAHEDRON, ICOSICOSIDODECAHEDRON, ICOSIDODECADODECAHEDRON, PSEUDODIFFERENTIABILITY, QUASIRHOMBICOSIDODECAHEDRON, and SUPERRENORMALIZABILITY. Some other long words in mathematics are PSEUDORHOMBICUBOCTAHEDRON, RHOMBIDODECADODECAHEDRON, DIRHOMBICOSIDODECAHEDRON, RHOMBITRUNCATED ICOSIDODECAHEDRON, and MEIOCATALECTICIZANT (a word used by James Joseph Sylvester in his Collected Works vol II, page 24, probably coined by him and probably used only by him) [Charles Turner, Jason Douglas, James A. Landau].

LOGARITHM and ALGORITHM are anagrams [Charles Turner].

FOUR has four letters; no other number has this property in English such that the number of letters in the name equals the number. [In fact, according to Brahinsky, each whole number less than 4 is smaller than the number of letters in its name, and each whole number greater than 4 is larger than the number of letters in its name. The smallest ratio of a number to the number of letters in its name is 1 : 3 (for ONE [3 letters]); the largest such ratio is approximately 3.9x10 64 (for NINE HUNDRED VIGINTILLION [23 letters]).

In other languages, words spelled with a number of letters equal to the number itself are:



Afrikaans VIER

Albanian NJIMBËDHIET (11)

Afar KONOY

Basque BEDERATZI (9)

Bislama: TU, TRI

Catalan U

Cebuano UPAT

Chinese (standard Mandarin) I, ER, SAN, SHIH

Czech TŘI

Danish TO, TRE, FIRE

Diola KUGNEN LI SIBAKI (14)

Douala MUTOBA

Dutch VIER

Esperanto DU, TRI, KVAR

Estonian NELI, KUUSTEISTKÜMMEND (16)

Ewondo ZAMGBAL

Finnish VIISI

German VIER

Greek TEΣΣARAKAIΔEKA (14)

Haitian Creole DE, TWA

Hausa UKU, HUDU, BIYAR

Hawaiian UMIKUMAMALUA (12), UMIKUMAMAKOLU (13)

Hungarian NÉGY

Irish (Gaelic) DÓ, TRÍ

Italian TRE

Japanese NI, SAN (see note), KOKÓNOTSU (9)

Khmer PRAIM BEI (8)

Klingon wej

Latvian PEEZI

Lingala ZOMI NA MIBALE' (12)

Lithuanian DU, PENKI, SEPTYNÌ

Malgache ENINA AMBIN'NY FOLO (16)

Manx QUEIG, SHIAGHT

Middle English O

Maori KOTAHI TEKAU MA ONO (16)

Mongolian ZURGAA

Norfuk FOOR

Norwegian TO, TRE, FIRE, ÅTTETAL

Old English þRÌ

Papiamentu SINKU

Polish PIATY, DZIESIÂTKA (10)

Portuguese CINCO

Quechua TAWA, PISQA, QANCHIS

Romanian CINCI

Sango BALE'-OKO NA USE' (12)

Serbo-Croatian TRI

Slovak TRI

Soninke' TANMU-DO-FILLO (12)

Spanish CINCO

Swedish TRE, FYRA, FEMMA

Tagalog APAT, LABING-DALAWA (12; also spelled LABINDALAWA)

Tahitian MAHA, HO'E 'AHURU (10)

Tok Pisin TU, TRI

Turkish DÖRT

Welsh TRI, CHWECH (see note)

Wolof FUKK AK JUROOM BENN (16)

Zulu ISHUMI NESIHLANU (15), AMASHUMI AMABILI NA MBILI (22)

Note - Tahitian. Ho'e 'ahuru - the apostrophes mark glottal stops, and are considered letters, so this is "ten."

Note - Welsh. According to Phil Couch, the word for six "may contain six typed characters but it has only 4 letters. Welsh (more properly Cymraeg) has double character letters CH, DD, FF, NG, LL, PH, RH and TH."

Note - Japanese. According to Graan Jenever, if the words are written in Japanese characters in Hiragana, no words meet the rule, although a non-standard way of writing the word for "three" could involve a total of three strokes of the pen. He also writes that if the words are written in Kanji, 1 is the only number to match the rule strictly, although "if you stretch the rule a bit and count the strokes in the character though, one (single horizontal line), two (two horizontal lines) and three (three horizontal lines) will qualify."

Note - Yiddish. The above list previously included the Yiddish FIER and FINEF. However, Alain Gottcheiner writes, "Yiddish words for 4 and 5 are spelled FIR and FINF. In the Polish dialect (Galitsianer), vowels are often diphtonguized (with some complex cases, like VEYE for VER "who ?"), and epenthetic E's abound, giving us FIER and FINEF, but this doesn't appear in writing."

In Lithuanian the accusative case of ASTUONI (eight) is ASTUONIS (8 letters).

There are also Russian and

There are also Hebrew and . (Here the vowels are not counted as letters; they are not shown in most writing.)

There is also the Greek

Ralph Beaman points out that Chinese words are listed in dictionaries by the number of strokes required to form them; thus, strokes are analogous to letters. The Chinese characters for 1, 2, and 3 use one, two and three strokes, respectively; for example, the number 3 resembles the capital letter E with the vertical bar removed. The common Chinese for 15 (using the ideographs for ten and five) has six strokes; but, for legal documents and other uses to prevent fraud, there are more elaborate "long" forms. The long form for 15 is classified as a fifteen-stroke character.

[Dmitri Borgmann, Pascal Kaeser, James E. F. Landau, Pierre Abbat, Bruce D. Wilner, Brendan O'Sullivan-Hale, Fredrik Viklund, Jussi Tolvi, Juozas Rimas, Carlos Andre Branco, Tom Green, Alan Barry, Audubon L. Bakewell IV, Miroslav Sedivy, Jason Flatley, Anthony Jan Cabatic, Barry Shulman, Stuart Kidd, Sam Ettinger]

THE BIBLE

The second longest word in the Bible is CHU-SHAN-RISH-A-THA-IM (17 letters). It is first recorded in Judges 3:8 and it is the name of "the king of Mesopotamia" and means "the Ethiopian of double wickednesses" [Philip Bennett].

There are eight 16-letter words: BA-SHAN-HA-VOTH-JA-IR (Deuteronomy 3:14), COVENANTBREAKERS (Romans 1:31), EVILFAVOUREDNESS (Deuteronomy 17:1), KIB-ROTH-HAT-TA-A-VAH (Numbers 11:34), LOVINGKINDNESSES (Psalm 25:6), SE-LA-HAM-MAH-LE-KOTH (1 Samuel 23:28), UNPROFITABLENESS (Hebrews 7:18), CHE-PHAR-HA-AM-MO-NAI (Joshua 18:24).

The longest unhyphenated words in the Bible are COVENANTBREAKERS (Romans 1:31), EVILFAVOUREDNESS (Deuteronomy 17:1), LOVINGKINDNESSES (Psalm 25:6), UNPROFITABLENESS (Hebrews 7:18) (all 16 letters) [Philip Bennett].

The shortest names in the Bible are AI (Joshua 7:2), AR (Numbers 21:15), ED (Joshua 22:34), ER (Genesis 38:3), IR (1 Chronicles 7:12), NO (Jeremiah 46:25), OG (Numbers 21:33), ON (Numbers 16:11), PE (Psalm 119:129), SO (2Kings 17:4), UR (Genesis 11:28), and UZ (Genesis 10:23) [Philip Bennett].

ARTAXERXES and EZRA are both mentioned in Ezra 7:21, ensuring that that verse contains all the letters of the alphabet except J.

Galatians 1:14 is missing the letter K and is the shortest verse with 25 of the 26 letters: "And profited in the Jews' religion above many my equals in mine own nation, being more exceedingly zealous of the traditions of my fathers."

The complete list of such verses:

Missing Q: Daniel 4:37, I Kings 1:9, Joshua 7:24, Haggai 1:1, Ezekiel 28:13, II Chronicles 36:10, I Chronicles 12:40

Missing X: II Kings 16:15, I Chronicles 4:10

Missing J: Ezra 7:21

Missing K: Galatians 1:14



Excluding the Apocrypha, there is only one proper name in the King James Bible beginning with Q: QUARTUS. QUINTUS MEMMIUS is in the Apocrypha. More modern translations have QUIRINIUS, which is rendered CYRENIUS in the KJV (Luke 2:2).

There are no proper names in the King James Bible beginning with W.

Excluding the Apocrypha, there are no proper names in the King James Bible beginning with X. XANTHICUS (an alternate name for the month Nisan) is in the Apocrypha.

There are no proper names in the King James Bible beginning with Y.

Today's English Version (commonly called the Good News Bible) has WAHEB (Num. 21:14), XERXES (Esther 1:1), and YIRON (Joshua 19:38). YAHWEH appears in several contemporary translations. QOHELETH (the Hebrew name translated as Ecclesiastes or "the preacher" in the KJV) appears in the Jerusalem Bible and is the only name in scripture with a Q not followed by a U. The WATER GATE (Neh. 3:26) appears in lower case in the KJV but is a proper name in contemporary translations. John F. Underwood, who contributed this paragraph, says the Good News Bible has proper names for every letter of the alphabet.

Philippi may be the longest Biblical word with all long letters [Byron Davidson].

JOSHUA JUDGES RUTH is a sentence formed by the names of three consecutive books in the Old Testament. It is also the title of an album by Lyle Lovett released in 1992. [Charles Turner]

The shortest verse in the King James version is John 11:35: "Jesus wept." The shortest verse in the New International Version is Job 3:2: "He said,". The shortest verse in the Old Testament is I Chr. 1:25: "Eber, Peleg, Reu."

The following table shows the most frequently-occurring words in the 1611 King James Bible, according to Philip Bennett:

THE 64,041 AND 51,714 OF 34,755 TO 13,643 THAT 12,916 IN 12,674 HE 10,432 UNTO 9,002 I 8,853 HIS 8,478 A 8,284 LORD 7,836 THEY 7,377 BE 7,012 IS 6,992 HIM 6,667 NOT 6,597 IT 6,131 WITH 6,015 ALL 5,621

[For the above statistics and other Bible statistics he compiled, Bennett included the introductory titles which occur in most of the Psalms and the closing remarks which are found in 13 New Testament books. He did not include titles of each book. Hyphenated words are counted as one word.]

Michael Hutching, using the software program Godspeed, found that the word GOD appears exactly 4444 times in the King James version.

(Hans Havermann reported a concordance-based word count shows GOD occurring 4447 times. Philip Bennett resolved the discrepancy when he discovered that in the software search, GOD-WARD, which occurs three times in the KJV, is indexed as a separate word (three times), whereas in the concordance search, GOD-WARD is broken down into two separate words: GOD and WARD. These counts do not include occurrences of gods (244), ungodly (27), godly (15), God's (26) Godliness (15), goddess (5), ungodliness (4), Godhead (3), and Gudgodah (2). The 4447 number includes the word in the title in Psalm 90, "A Prayer of Moses the man of God"; the title does not appear in the Project Gutenberg version.)

LORD occurs 7,836 times in the Bible. In addition, LORD'S occurs 134 times, LORDS 42 times, LORDSHIP occurs twice, and LORDLY occurs once [Philip Bennett].

The most commonly occurring name in the Bible is DAVID, which occurs 1,085 times. David's occurs 54 times [Philip Bennett].

The second most commonly occurring name is JESUS, which occurs 973 times. There are an additional 10 occurrences of JESUS'; there is one occurrence of BARJESUS.

The third most commonly occurring name is MOSES, which occurs 829 times. Moses' occurs 19 times [Philip Bennett].

[Much of the material for this page was contributed by Philip Bennett.]