Big numbers

[David]

In old English, numbers less than one million seem to be quite straightforward, despite the odd variation. However I have only come across two numbers bigger than one million and neither makes any sense to me. It appears there is no word for a million in old English. I would have tried using þūsend þūsenda (a thousand thousand) or þūsend sīðon þūsend (a thousand times a thousand) but that seems to be too easy for the Anglo-Saxons.


In his Enchiridion Byrthfērth has 16 243 200 and 197 760 960. For 16 243 200 he writes

Ān hund þūsenda and hundtēontiġ siðon sȳxtiġ þūsenda.

For 197 760 960 he writes

Ān hund þūsend and hundtēontiġ siðon tȳn þūsend and nigon siðon hundtēontiġ þūsend and þūsend siðon þūsend and nigon hund þūsenda and ān and þrittiġ þūsend and sȳxtiġ.

Obviously you can get different values depending on the order you do the operation but can anyone tell me how they get the given values.

[brian farrell]

For the number "one million"; some texts appear to use the genitive plural 'þusenda', such that 'an þusend þusenda' represents "one thousand of thousands" and 'an þusend syx hund þusenda' would therefore represent "one thousand six hundred of thousands" i.e. 1,600,000.

As for Byrtferth's equations:-
The original Latin for the first one (atomos in a 30-day month) is:-
     A               B            C         D             E       F
centies et sexagies centum millia et CCXLIII CC (I'm using underline here to represent the Latin over-scored numerals)
((A +B) * C * D) + E + F
= (160 * 100 * 1000) + (243 * 1000) + (200)
= 16,000,000 + 243,000 + 200
= 16,243,200 (which is a correct representation of 541,440 * 30 atomos)

Byrtferth's OE interpretation of the above appears to be totally out of sync.
'an hund þusend and hundteonig siðon syxtig þusenda'
         A         D                     C                      B         D
and appears to ignore "E" & "F"

As for the second equation, Byrtferth's OE interpretation of the original Latin is even more confusing and I've lost several "grey cells" over it. The equation is further complicated in that the original Latin doesn't appear to add up correctly either.
I hope that I haven't made any mistakes; OE & Latin maths is very complex compared to the modern use of brackets and Arabic numerals (and my knowledge of Latin is very basic).

 I will try to tackle the second one again later.

[David]

Brian, that was fantastic. So in the first case should Byrthfērth had written something like

 Ān hund and sȳxtiġ siðon hundtēontiġ siðon þūsend and twā hund þūsenda and þrēo and fēowertiġ þūsenda and twā hund.

Was Byrthfērth’s method of getting millions normal or was he just being perverse with the straightforward way being to say sȳtīene þūsend þūsenda?

Can you tell us where we can find old English mathematics.

[brian farrell]

My apologies if "Latin and OE maths" confused you. I was simply using the term as a comparison against the modern mathematical style & techniques of using brackets and Arabic numbers. If there are any OE maths text books, maybe someone else could advise.

In that first example; 'an hund þusenda and hundteontig siðon syxtig þusenda'; I think that Byrtferth was trying to say something like
"a hundred of thousands (of one hundred) plus a hundred of sixty thousands", or words to that effect.
= (100 * 1,000 * 100) + (100 * 60 * 1,000)
= 10,000,000 + 6,000,000 = 16,000,000

but that's not exactly the way that it actually comes across; and of course the ending is missing i.e. et CCXLIII CC { + 243,200}

I suppose that it is possible that Byrtferth's style of representing numbers was influenced by Latin, rather than the straightforward representation that you suggested. Or was it because he wasn't a mathematician? i.e. did the numbers actually mean something to him or was he just a translator?

I'm still trying to get my head around that second example !

[David]

My best attempt to write 197 760 960 in a form like Byrthfērth that gives the right value is

Ān hund þūsend and seofon hundtēotiġ siðon tȳn siðon þūsend and nigon hundtēontiġ siðon hundtēontiġ þūsend and seofon hund þūsend and nigon hund and þūsend and ān siðon sȳxtiġ.

However I cannot understand how anyone can give the number in such a convoluted and ambiguous form. How is the reader supposed to know the order of operations which is
(100 000 + 700*10)*1000 + 900*100 000 + 700 000 + 900 + (1000 + 1)*60

[brian farrell]

You are right that ideally the "absolute number" should be better represented in a logical format, but it would seem that Byrtferth was looking at numbers primarily through the eyes of a translator. You are also correct in that a number, or numerical expression, can be ambiguous, particularly for large numbers, but even small numbers could potentially lead to confusion, especially when amplified by the Germanic style of counting.

Is 'an and þrittig þusend' 31,000 or 30,001 ? Byrtferth uses the expression for the former. I suppose the occasional use elsewhere of expressions in the style of 'an þusend and þrittig þusend' were used in an attempt to clarify such numbers.

I have not read Byrtferth, but the impression I have is that he was a translator and collector of data, rather than a scientist/mathematician. My argument being that if someone were to translate (say) Hawkin into French, it doesn't make that person a physicist (and even as a good translator, mistakes are inevitable if the person is "out-of-his depth" on the subject matter).

By-the-by; the number of hours in a year is given in Latin as IX DCCCLXVI (9,766), but the actual number should be 8,766. Byrtferth states 'nigon þusend tida and seofen hund tida and syx and syxtig' (i.e. he translates it, without checking it)

[brian farrell]

I don't think that I can take the analysis of that second example any further. So here goes :-

Atomos in a 365 1/4- day year (i.e. 197,760,960)
    A         B         C          D         E              F         G      H                I           J
decies novies centies centum millia et quatuor mille millia et DCCCCXXXI LX

((A + B) * C * D * E) + (F * G * H) + I + J
= ((10 + 9) * 100 * 100 * 1000) + (4 * 1000 * 1000) + ((500 +400 + 30 + 1) * 1000) + 60
= 190,000,000 + 4,000,000 + 931,000 + 60
= 194,931,060

Byrtferth's equivalent is :-
        D         E                   C                    A        E                B                    C              E           {F?} G                H
an hund þusend and hunfteontig siðon tyn þusend and nigon siðon hundteontig þusend and þusend siðon þusend
                                         I                                                     J
 and nigon hund þusenda and an and þrittig þusend and syxtig

NOTES
a) The Latin number is inaccurate, representing 194,931,060 (not 197,760,960), which is approx., but not exactly, 360 days
b) Byrtferth's number appears to be an attempted and misunderstood translation rather than an "absolute number"
c) It misses out the "4" of four million, that appears in the Latin
d) It is not necessarily a correct translation, even taking (c) into account (I think that he misunderstood the beginning)

The Latin number-suffix "-ies" represents "{number}times ... " , However, because the first number "10" is followed by a smaller number "9", the pair together actually means "19 times". The alternative option being better represented by "novagiens" (ninety times).

The Latin expression therefore begins "19 times 100 times", but Byrtferth seems to have it as "10 times the later expression plus 9 times the later expression plus 100 times ... etc"

For the Latin I used a transcript of "Liber de Computo" Book 10; Rabani Mauri [AD820], who I think was one possible source for Byrtferth

Now! It's time to repair the brain cells with a quick pint!

[David]

Is 'an and þrittig þusend' 31,000 or 30,001 ?

As I said the numbers below a million seem to be straightforward. So if you were expecting a number ān and þrittiġ þūsend would be xxxi . If you were expecting a calculation it might be i and xxx bēoð xxxi. Here I am also using underline for overline.
This is like the modern English one hundred and three thousand. If you are expecting a number it would  be 103 000 but as a calculation it might be 100 + 3000 = 3100.

I would suggest that verbal calculation can only be simple cases and, for clarity, I would change the order i.e. “þrittiġ þūsend and ān” and “three thousand and one hundred”. For written calculations I would use the symbols. For some reason Byrhtfērth tends to avoid the symbols.

For numbers up to a million the old English order seems to be hundreds and units and tens thousand and hundreds and units and tens. For numbers greater than a million I would suggest hundreds and units and tens million etc.

Incidentally I have been misspelling the man’s name. It is Byrhtfērth.