# Tuesday 9 June 1663

Up and after ordering some things towards my wife’s going into the country, to the office, where I spent the morning upon my measuring rules very pleasantly till noon, and then comes Creed and he and I talked about mathematiques, and he tells me of a way found out by Mr. Jonas Moore which he calls duodecimal arithmetique, which is properly applied to measuring, where all is ordered by inches, which are 12 in a foot, which I have a mind to learn.

So he with me home to dinner and after dinner walk in the garden, and then we met at the office, where Coventry, Sir J. Minnes, and I, and so in the evening, business done, I went home and spent my time till night with my wife.

Presently after my coming home comes Pembleton, whether by appointment or no I know not, or whether by a former promise that he would come once before my wife’s going into the country, but I took no notice of, let them go up and Ashwell with them to dance, which they did, and I staid below in my chamber, but, Lord! how I listened and laid my ear to the door, and how I was troubled when I heard them stand still and not dance. Anon they made an end and had done, and so I suffered him to go away, and spoke not to him, though troubled in my mind, but showed no discontent to my wife, believing that this is the last time I shall be troubled with him.

So my wife and I to walk in the garden, home and to supper and to bed.

## 45 Annotations

## First Reading

## TerryF • Link

duodecimal arithmetique

"The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. The number ten may be written as A, the number eleven as B, and the number twelve as 10." http://en.wikipedia.org/wiki/Duodecimal

"It is obvious...that when a number system based on ten and a measuring system based on twelve inches to the foot collide, 'decimals' cannot be used, and usually it is necessary to reduce the whole problem to inches, with long computations and eventual divisions by 1,728 to obtain cubic feet. We were able to show [shipping] clerks [attempting to calulate and add the volumes of several boxes] that by using duodecimal multiplication with the inches now simply duodecimals of feet (i.e., 2'6? is simply 2;6 - two and six-twelfths feet) - and pointing off three places in the answer, the problem is amazingly simplified....

[ this paper recounted the preceding later than the following ]

"A book by Simon Stevin seems to have mentioned [duodecimals] as early as 1585 [*La Theinde* (The tenth)...in which decimals were introduced in Europe;...Robert Norton published an English translation of La Theinde in London in 1608....titled Disme, The Arts of Tenths or Decimal Arithmetike and it was this translation which inspired Thomas Jefferson to propose a decimal currency for the United States (note that one tenth of a dollar is still called a dime).]"

"My Love Affair with Dozens" by F. Emerson Andrews, *Michigan Quarterly Review*, Volume XI, Number 2, Spring, 1972 http://www.dozenalsociety.org.uk/basicstuff/andre…

The text of "DISME: The Art of Tenths, or, Decimall arithmetike" is online at http://home.wxs.nl/~hopfam/Dime.html

## Bradford • Link

"I staid below in my chamber, but, Lord! how I listened and laid my ear to the door, and how I was troubled when I heard them stand still and not dance."

Whyn'tcha go in and take a few turns yourself, wallflower? That'd make two pair (unless Ashwell is at it again, her triangle I mean). As a character tells Creon in Sophocles' "Antigone," "Natures like yours chiefly torment themselves."

## Pedro • Link

Duodecimal…which are 12 in a foot, which I have a mind to learn

Duodecimal that good old measuring system, length and money. Somehow I don’t think Sam will have as much trouble as we had on February 15th 1971 with the change to decimal currency.

And then them daft Europeans trying to get rid of our mile and our pint!

## dirk • Link

For those interested:

Basic arithmetic in base twelve

http://base12.org/basic_arithmetic.html

## graybo • Link

12 inches in a foot.

And one inch equates to 25.4 milly-meaters, I'll be bound.

OK, now work out what petrol/gasoline costs in USD per US gallon when British pumps charge 0.969 GBP per litre. (Using Google counts as cheating).

## dirk • Link

"to the office, where I spent the morning upon my measuring rules very pleasantly till noon"

There's something charmingly childlike about Sam playing around with his new toy during office hours -- not unlike my first experience with a desktop computer many, many years ago. And then again, who could afford to do the same in a time-managed and strictly controlled modern office environment?

## JohnT • Link

Mr Coventry for so long has become simple " Coventry ". I have not checked but I think the same has happened to Creed for so long " Mr Creed ". Is it because Coventry has been slightly more intimate in talking of his recent problems and is thereby more of a peer/colleague? Or has his present vulnerability lowered his status ? Use of the surname on its own is a difficult social problem in the English class stucture. As recently as 30 years ago, one used the surname alone to one's equals but the title "mr" to inferiors or superiors. Only a male thing, though . Different rules applied to "miss" or Mrs" and then "ms " came.

## daniel • Link

arithmetic and jealousy

Sam is all in a pickle!

## Australian Susan • Link

Titles

I have just been reading many of Charlotte Bronte's letters and in those she consistently refers to her husband as "Mr Nicholls" except to her closest correspondent, Ellen. She also always refers to her father as "Mr Bronte" and her Aunt (who lived with them after the death of their mother in 1822) always as Aunt Branwell, never Aunt Elizabeth, even when writing to her siblings. This has been taken to indicate estrangement, but i don;t think so. Sam recently referred to one of his Aunts as Aunt James (using her surname), but seems very friendly (if rather patronising) towards her. The rigidities of the English class system reached their final state in high Edwardian times and have been declining ever since: but an example of longevity - my father worked in the Bank of England all his working life (with a sojourn in Holland and Germany in the 1940s)and men there were designated as A. Smith Esq. (for Esquire) if they were bank clerks or above and Mr A. Smith if they were messengers or similar. The use of the surname between peers stems from the usage in the English public (private to Americans) school system. Masters referred to boys by their surnames (using Junior or Tertius for lesser siblings) and the boys picked this up. Sam's usage shows an earlier manifestation of this.One wonders what Sam said to their faces: just the surname? Which is why it has crept into the Diary as usuage developed?

Decimals

I was reminded of the fictional atttempts of Mr Palliser (Trollope novels) to introduce decimal coinage into Victorian Britain: he devised "quints" - a fifth of a penny.

## Australian Susan • Link

Another thought: anyone else remember Tom Lehrer's New Math song with sums in base 8?

## TerryF • Link

"How do I love thee? Let me count the ways."

http://www.amherst.edu/~rjyanco94/literature/eliz…

## TerryF • Link

Tom Lehrer's New Math song's lyrics

http://www.stlyrics.com/songs/t/tomlehrer3903/new…

## Australian Susan • Link

For those who want them - lyrics to New Math

http://www.sing365.com/music/lyric.nsf/SongUnid/E…

and Amazon ref for The Prime Minister

http://www.amazon.com/gp/product/014043349X/ref=p…

## Australian Susan • Link

TerryF beat me to it! Even without Tom L's inimitable voice and tone, the words still make good reading.

## Alan • Link

Please enlighten me. Did they dance to music? Something else? Tks.

## TerryF • Link

Australian Susan, I'll step aside for awhile.

## dirk • Link

Susan, more on Tom Lehrer

-- with actual recordings, playable & downloadable... [scroll down]

http://www.haverford.edu/physics-astro/songs/lehr…

There's even (very relevant today) a "Slide Rule Song".

http://www.haverford.edu/physics-astro/songs/lehr…

## Robert Gertz • Link

And of course for Sam's jealous side there's always "The Masochism Tango", appropriate for a dance lession from Hell.

Though I wonder if Tom ever wrote a song strictly for the jealous.

"What is wrong with me?" Sam sighs, pulling back from the door. "There's at least one other person up there with him. Ah, and Bess just called the maid up as well. Pepys, get hold of yourself, one more night and we are rid of the fellow."

"I don't hear creaking. I think Sam'l has given it up." Bess turns to Pembleton while executing a perfect step-and-turn.

"Well, that's good. I hope the poor man takes himself in hand." Pembleton shakes head. "Anyway, ladies..." he smiles at the group now numbering an eager three. "That was an excellent effort by all."

"Now, who's up for sex?"

## TerryF • Link

"Please enlighten me. Did they dance to music?"

In one day's entry it was said that they couldn't dance because Ashwell wasn't there - presumably to provide musical accompaniment.

## in Aqua scripto • Link

Sam was way ahead of his time; it was used to divide a page into 12 parts in his day down at the print shop. duodecimal was not recorded until 1712

from a dictionary:

duo, duae, duo: num 2

duodeciens , -es adv twelve times

duodecim num twelve:

duodecimus adj Twelfth :

duodeni, -irum adj. twelve each

OED no mention until 1700's

[f. L. duodecim-us twelfth, f. duodecim twelve: see -AL1. Cf. F. duodécimal (1801 Hauy).]

A. adj. Relating to twelfth parts or to the number twelve; proceeding by twelves.

1727 J. JORDAINE (title) Duodecimal Arithmetick and Mensuration improved

B. n. pl. duodecimals, a method of multiplying together quantities denoting lengths given in feet, inches, twelfths of an inch, etc., without reducing them to one denomination; also called cross-multiplication.

The method is essentially that of long multiplication, but in the duodecimal scale instead of the decimal. The successive terms of the result denote square feet, twelfths of a square foot, square inches, etc.

1714 S. CUNN (title) A new and complete Treatise of the Doctrine of Fractions..with an Epitome of Duodecimals.

L. duodecim twelve, an initial element in some 19th-c. technical terms:

duode cimlobate a.divided into twelve lobes (Syd. Soc. Lex. 1883).

duode.. cimfid a. [L. -fidus cleft], divided into twelve parts or segments (Webster, 1828);

[L. (in) duodecim o in a twelfth (sc. of a sheet), abl. of duodecimus twelfth.]

1. The size of a book, or of the page of a book, in which each leaf is one-twelfth of a whole sheet: usually abbreviated 12mo.

1658 PHILLIPS s.v., A book is said to be in Duodecimo, when it is of twelve leaves in a sheet.

1688 Catalogus Librorum..per Benj. Walford 137 English Miscellanies in Octavo and Duodecimo.

## Jesse • Link

"all is ordered by inches, which are 12 in a foot"

One advantage is that quite a few fractions of a foot (e.g. 1/12, 1/6, 1/4, 1/3, 1/2) are whole numbers, i.e. inches.

## in Aqua scripto • Link

12ths. the Romans divided money; Bronze /copper into 1/2;1/3;1/4;1/6 and 1/12 [uncia, one ounce] 12 unciae = 1 Libra

Silver coins it be asses 10 [later 16] 5 asses and 2 1/2 asses

Weights Libra [Pound] 12 Uncia to the Pound.

Measures: Pes [ foot], 12 uncia to the foot, 1 Passus [footstep pace] = 5 pedes [5 feet] 125 Passus = 1 stadium; 8 stadia = mille passus

Mille [1000] Passus was one Roman Mile (1620 Yards).

[Fingers be first V two be hands X]

Capacity 4 Cochlearia to 1 cyathus ; 12 cyathus to 1 sextaruius;

6 to 1 then 8 to 1 ; 20 to 1

Sexterius = 1/2 litre [48 Cochlearia]

For those that want to join the modern with ancient, use the late of the Poets;

metres be latin verse using a feet; quick quick long to make a feet .[anapaest]

## Paul Chapin • Link

Murray nods?

I'm startled to learn that the OED missed Sam's use of the word 'duodecimal'. I checked and confirmed that IAS is quite right, the first citation is from 1714 (with 'duodecimo', referring to book size, cited from 1658).

## Pedro • Link

Duodecimal

Maybe duodecimal is the word that the translator has chosen from the shorthand?

## A. Hamilton • Link

Sam's song for this evening's misery

Pauvre Sam, pauvre misere (w/apologies to Georges Brassens, who wasn't singing about jealousy)

## TerryF • Link

Duodecimal

The L&M transciption is "Duodecimall arithmetique" -

Pehaps contra “My Love Affair with Dozens” by F. Emerson Andrews, Simon Stevin did not even "seem" to have mentioned "duodecimals" in the Robert Norton 1608 English translation of "La Theinde", “DISME: The Art of Tenths, or, Decimall arithmetike”.

## Gerry • Link

Susan is quite correct about surnames and schools. To this day I call my closest friends, male or female, by their surname. This surprised a lot of folks here in the US when I moved some thirty years ago but most of them put it down to an English eccentricity.

Oz sue, at my school brothers were labelled major, minor and minimus. We never had four in my time...I'm not even sure what the next word is,miniscule?

## tel • Link

Duodecimal arithmatique.

I have come late in the day to this (literally) but must ask - what system were they using before this? Surely not decimal?

## TerryF • Link

What system was customary? "Surely not decimal?"

tel, likely so. Consider these, courtesy of Dirk in the Background info on Mathematics:

"The first mediaeval Oxford mathematician of note was John of Holywood or John de Sacro Bosco, as he was usually called. He died in 1244. He was educated at Oxford but subsequently moved to Paris and taught there. His best known books are De Algorithmo (or the art of computing in the decimal system)" http://www.maths.ox.ac.uk/about/history/#II

Puzzling equations are found to be calclations of interest due "expressed decimally rather than in £.s.d and would need to be converted in use; the large number of significant figures gives sufficient precision for all ordinary calculations."

http://www.mhs.ox.ac.uk/staff/saj/aide-memoire/

## TerryF • Link

Tel, to be clear, the "Puzzling equations" expressed in decimals referenced above are relevant to A. your Q. since they appear on "a small silver oval medallion (approximately 48 x 37 mm) held in a private collection....The object is revealed as a witness to the virtuoso amateur culture of mathematics in 17th-century England." http://www.mhs.ox.ac.uk/staff/saj/aide-memoire/

## in Aqua scripto • Link

I never remembered any fancy word for adding LSD [never in gested it either]or working out the bakers dozen, we just did it, as long ye remembered thy 12 times tables and a 4 times's one and 20 time's this or that. That be why it be called Reckoning {the third R}.

The majority of calculations only consisted of pennies, hapepences and farthings. Most of us agnostic ones, never had enough to see or touch a pound note or yet alone that fiver [which had to be endorsed on the back].

## Pedro • Link

to see or touch a pound note or yet alone that fiver [which had to be endorsed on the back].

Lady Godiva Serial Numbers in Aqua scripto!

Sometime during 2002…

“One or two members of staff at the Bank of England may have blushed at news that the much-heralded new five-pound note was duff. Although the note had been "rigorously tested" before issue, and the Bank's chief cashier Merlyn Lowther said that the new fiver would be "the most secure five pound note we have ever produced", it transpired that there was one detail the printers didn't get right: the confounded serial numbers! Of all the silly... After all, they simply allow you to verify where and when it was printed. For all the state-of-the-art features of the new note, including a foil hologram, metallic thread and what not, making counterfeits virtually impossible to produce, the serial numbers just seemed to vanish into thin air with a touch of leisurely confrication.”

http://www.centralbanking.co.uk/newsmakers/archiv…

## Australian Susan • Link

Is that John Bosco the same as the Saint of that ilk?

## in Aqua scripto • Link

There be 12 big toes [thumb]to the Paris foot [pied] "...where all is ordered by inches, which are 12 in a foot,..."

No wonder Miss Eliza be at 6's and 7's with Le Maister. He be a saying inches and she be a saying thumbs, [which be twelve to the foot and she says to effect she has only two].

Pouce [not pou = louse] and pied be still in Enlish version of Dictionary.

## GrahamT • Link

Human body parts have been the basis of measurement since biblical times. The cubit is the length of forearm from elbow to finger tips; the yard is nose to finger tips; a hand is four inches (horses for the measuring of); the foot needs no explanation; an inch is the width of a man's thumb. This is recognised in French as Aqua scripto points out. Pouces are still used in French when referring to Anglo-Saxon devices like 3½" Floppy discs.

## tonyt • Link

Is that John Bosco the same as the Saint of that ilk?

As that John Bosco lived 1815-1888 the answer is fairly easy!

## Paul Dyson • Link

Surnames and schools.

At a mixed English grammar school in the mid-sixties, boys were addressed by their surnames, girls by their christian/fore names. This gradually changed until surname address disappeared - quite useful though as a way for teachers to express displeasure. However it's still quite common for boys and young men to call each other by surnames only, even more so to use a nickname, even if it's only the soccer-speak with "-y" suffixed.

## Pedro • Link

Meanwhile in Tangier…

Teviot had arrived on the 11 May and realised that the key to a successful defence of Tangier lay in the construction of fortifications which would allow control of the circle of hills overlooking the town.

On the 9th of June the Moors attacked with 500 men but William Fiennes beat them off after two hours of hard fighting. Six days later the Moors launched a fresh offensive against the unfinished Pole Fort, but Teviot had been expecting this and was well prepared. He had the ground near the fort littered with “Galtraps” and “Crow’s Feet”, impeding the advance of the enemy cavalry. British hand grenades finished off the Moorish infantry and the assault failed.

( Summary from Childs…The Army of Charles II)

## Second Reading

## Bill • Link

Cross Multiplication, otherwise called duodecimal arithmetic, is an expeditious method of multiplying things of several species, or denominations, by others likewise of different species, &c. e. g. Shillings and pence, by shillings and pence; feet and inches, by feet and inches. This is much used in measuring, &c. and the method is thus: Suppose 5 feet 3 inches to be multiplied by 2 feet 4 inches; say 2 times 5 feet is 10 feet, and 2 times 3 is 6 inches; again, say 4 times 5 is 20 inches, or 1 foot 8 inches; and 4 times 3 is 12 parts, or 1 inch : the whole sum makes 12 feet 3 inches. In the same manner you may manage shillings and pence.

[note from Bill: the ~ is just used to line the numbers up in Phil's editor]

~~Feet ~Inches

~~~~5~~~~3

~~x~2~~~~4

-------------------

~~~10~~~~6

~~~~1~~~~8

~~~~~~~~~1

-------------------

~~~12~~~~3 [i.e. 12 feet, 3 inches]

---A New and Complete Dictionary of Arts and Sciences. 1764.

## Bill • Link

Wow, the mathematical method (algorithm) above is a little deeper than it looks. We have to remember that multiplying feet x feet gives us a modern answer in SQUARE feet, so the 3 inches in the answer above should be interpreted as 3/12 of a square foot. The answer then, in modern terms, is 12 1/4 sq. ft.

(Another way of interpreting the problem is by saying that a surface 5 ft 3 in. by 2 ft 4 in. has the same area as a surface 12 ft 3 in. by 1 ft. That is, 12 1/4 sq. ft)

How would we do this problem? We could convert it to:

5 1/4 x 2 1/3 ( or 5.25 x 4.33) hmmm.

or convert it to just inches:

63x28 and then convert square inches back to square feet. hmmm.

I like the 1764 method better!! And a good reason to prefer metric measurement.

## Bill • Link

Of course, the 1764 duodecimal method is close to the way we would express it in the "dozenel" method (base 12) today, noting TerryF's annotation above and letting A=10. I guess that's why I like it.

~~5.3 (5ft 3in)

x~2.4 (2ft 4in)

--------

~190

~A6

---------

10.30 (10.30 sq.ft or 12 1/4 sq.ft.)

Sam is mastering the advanced practical mathematics of his day and it will help make him successful.

## Sasha Clarkson • Link

The oldest number system still in use today is the Sumerian/Babylonian 'sexagesimal' system, based on 60, which has the advantage of divisibility by 2, 3, 4, 5, 6,10, 12, 15, 20, and 30. Hence many useful fractions can be expressed in terms of exact numbers of 'minutes' (from 'pars minuta prima', meaning "first small part"), and seconds ((from 'pars minuta secunda', meaning "second small part").

This is the basis of both our measurement of time, and of angles. In the days before caclulators trigonometric functions were tabulated in terms of degrees and minutes for everyday use. Astronomers made made regular use of "seconds of arc" to measure apparent distances between heavenly bodies.

https://en.wikipedia.org/wiki/Sexagesimal

If you want to understand the term 'parsec' used by some SF writers, it's explained below :)

https://en.wikipedia.org/wiki/Minute_and_second_o…

## Sasha Clarkson • Link

There's also the Dozenal Society of America, formerly the Duodecimal Society of America.

http://www.dozenal.org/

## Sasha Clarkson • Link

More on the history of the inch (originally three barleycorns) below.

https://en.wikipedia.org/wiki/Inch#History

## Chris Squire UK • Link

Not defined in Sasha Clarkson’s reference:

‘Parsec n. < English ‘a distance corresponding to a parallax of one second’.

A unit of length equal to the distance at which a star would have a heliocentric parallax of one second of arc, approximately equivalent to 3·086 × 10exp13 kilometres (3·26 light years).’ [OED]