sexagesimal
The article below
the hyperlinks is about the sexagesimal of
SATOCONOR.COM
J.G. van der Galiën ‘The Dawn Of Science’ 1.1.
(2002)
Full paper
SATOCONOR.COM Journal of History
The Dawn Of Science
A New Hypothesis
about the Foundation of the Sexagesimal Number System in the cradle of
civilisation
By Johan G. van der Galiën
For comments e-mail: johan.van.der.galien@satoconor.com
Version 1.3. September 25, 2005 (version 1.0. from August 13, 2002)
Download PASCAL program FACTOR1.pas
Download PASCAL program SXGSML8.pas
Download PASCAL program CALCSEX2.pas
Download PASCAL program CALCSEX3.pas
Abstract:
Several old
hypotheses, which can be found in the literature and on the Internet, about why
the Sumerians took the sexagesimal sytem as their number base are discussed.
They all fall in to the categories of factorability, astronomy, combination of
earlier number bases or interaction of language with writing. Most of them I
will disprove in this publication.
A new hypothesis is
presented about an decimal system as starting point used in the period before
the early Uruk (> 3500 BC). In this system the Sumerian temple order did
resolving of natural numbers in factors. They discovered that 60 is, among the
integers (n), the number with the most solely consecutive factors under the
breakpoint of factors (√n, see paragraph 2.1.).
In this article the
rediscovery of this fact by my self, based on empirical results of a PASCAL
program, and the mathematical proof of Dean Hickerson is given.
Because of this
unique property of 60 the priest caste, who were in the middle of all
economical activities, which took a lot of calculating, regarded 60 as highly
mystical. In other words holy, which is illustrated by the fact that there
supreme god Anoe was coupled to this number. What might have amplified the
opinion of the priest is the fact that because of this consecutive sequence all
the factors are easy remembered and consequently easy to work with in mental
calculations. That is why, in my opinion, Sumerian economic calculations
shifted from a decimal number system to a base 60 (sexagesimal) system!
1. Introduction
The
deepest root of science is most likely the foundation of the sexagesimal number
system widely used by the Mesopotamians. [The Sumerians where the first people
to develop a calculating system and during developing of their sexagesimal
number base system they must also have discovered the prime numbers for the
first time. They must have regarded these strange and odd (prime) numbers as
being a problem for (mental) division in their sexagesimal system.] Because of
this event we prosper of all the blessings of nowadays science and technology.
The sexagesimal astronomical and mathematical clay tablets of Mesopotamian
science found its way to classical
2. Number Theory and 60
2.1. The
factorisation of integers and specially that of the number 60
One can
resolve all integers in to pairs of so called factors (in this article used as
synonym for divisors). Multiplication of the two factors from a pair yields the
resolved number. In Table 1 this resolving of the first twelve integers is
shown.
|
1= |
2= |
3= |
4= |
5= |
6= |
7= |
8= |
9= |
10= |
11= |
12= |
|
1*1 |
1*2 |
1*3 |
1*4 2*4 |
1*5 |
1*6 2*3 |
1*7 |
1*8 2*4 |
1*9 3*3 |
1*10 2*5 |
1*11 |
1*12 2*6 3*4 |
Table 1: The
resolving in factors of the first twelve integers.
If one
does resolving of factors of integers (n), for instance by means of a computer
program like the PASCAL program FACTOR1 (see Appendix), one only has to check
the integers for possible factors from 1 up to rounded √n. Because the
factors above √n are then already determined. In other words the factor
pairs repeat them self above √n in the reverse order. I call √n the
breakpoint for factors as you can see below:
12=
1*12
2*6
3*4
------Breakpoint
√12 ≈ 3,464 rounded = 3
4*3
6*2
12*1
It can
happen that √n is also a factor of n like in the case of n = 16. √16
= 4 and consequently the corresponding factor pair 4*4 = 16.
Numbers
with only one pair of factors (1*n, n being the number it self.) are called
prime numbers (p). The prime numbers play an important role in fundamental
number theory and for instance in encryption. So the numbers 2, 3, 5, 7 and 11
from Table 1 are prime. The number 1 is not considered prime because of the
fundamental theorem of Number Theory which says that each not prime number
(composite) has a set of unique prime factors. If the number 1 was prime then
one could write infinite sets of prime numbers for a composite because a
composite = p1a*p2b*p3c……
R.E.
Smalley, the 1996 Nobel Laureate in Chemistry who got this prize for the
discovery of the fullerenes (Like the archetypal fullerene C60 also
known as the Buckyball.), said in a lecture: "…60 is the most factorable
of all integers. That’s why the Babylonians used it as the base of their number
system…".1
The number
60 has 12 factors by the way, and they are:
60=
1*60
2*30
3*20
4*15
5*12
6*10
A simple
PASCAL program (See Appendix program: FACTOR1) proofs that there are an
infinite amount of numbers with more than 12 factors, take for example
120=
1*120
2*60
3*40
4*30
5*24
6*20
8*15
10*12
120 has 16
factors! But it is not only the multiples of 60 which have more than 12
factors. Take for example the number 1000:
1000=
1*1000
2*500
4*250
5*200
8*125
10*100
20*50
25*40
1000 also
has 16 factors! As a matter of fact; while the numbers grow, so do the amount
of possible factors. In my Number Theory research I discovered that the numbers
in the 50 millions can have up to 700 factors.2 Take for instance
the number 47.345.760 it has 504 factors!
So 60 is NOT
the number with the most factors! But what does make 60 so special and
different from all the other integers? For this we must look at the fact that
all the factors below the square root of 60 (Ö 60 » 7,746) are consecutive from 1 up to 6. I wrote a PASCAL (See Appendix
program: SXGSML8) to inspect if there are more numbers with solely consecutive
factors under Ö n. I checked all the
integers up to 47.345.760 with this program and I found an infinite sequence
that includes all the prime numbers (p)14, all the so called
"even primes" (2p)15 and the six numbers from Table 2.
|
1= |
8= |
12= |
18= |
24= |
60= |
|
1*1 |
1*8 2*4 |
1*12 2*6 3*4 |
1*18 2*9 3*6 |
1*24 2*12 3*8 4*6 |
1*60 2*30 3*20 4*15 5*12 6*10 |
Table 2: The
besides p and 2p additional six numbers with their factors, of the sequence of
numbers with solely consecutive factors under Ön. (ID
number A066522 On-line Encyclopedia of Integer Sequences16)
This
sequence is registrated as ID number A066522 at the On-line Encyclopedia of
Integer Sequences.16 The first 65 terms of this sequence are 1, 2,
3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 22, 23, 24, 26, 29, 31, 34,
37, 38, 41, 43, 46, 47, 53, 58, 59, 60, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86,
89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139,
142, 146, 149, 151, 157. What immediately becomes clear from the results of the
program SXGSML8 is the fact that there are no numbers under 47.345.760 which
have more of these consecutive factors than 60. It is even worse; the only
numbers, which pass the SXGSML8 criteria above 60, seems to be the primes and
the "even primes"! These facts are all empirical! But Dean Hickerson17
has worked out the mathematical proof that only the primes (p), the "even
primes" (2p), 1, 8, 12, 18, 24 and 60 have solely consecutive factors
under Ö n. (2p has only two pairs of factors they are
1*2p and 2*p.) He argues as follows:17
Theorema 2.1.: If n is a positive
integer such that the divisors of n which are <= Ön are
consecutive integers, then n is either p or 2p with p prime, or n is one of the
numbers 1, 8, 12, 18, 24, or 60.
Note: It's
easy to get lost in the details of the proof, so I'll start by explaining how
it works in a special case. Suppose, for example, that the divisors of n that
are <= Ön are the numbers 1, 2, ..., 10. Since n is
divisible by both 2 and 7, it's divisible by 14. So 14 must be > Ö n, which implies n<196. But n is divisible by 8, 9, and 10, so it's
divisible by their least common multiple, lcm(8,9,10)=360. So 360 <= n <
196, which is impossible. Most of the proof consists of showing that we get a
similar contradiction if 10 is replaced by any number k>=3. Proof: Let k be
the largest divisor of n which is <= Ö n. Thus
the divisors of n which are <= Ö n are exactly the
integers 1, 2, ..., k, and the divisors of n which are >= Ö n are just n, n/2, ..., n/k.
If k=1,
then n has only the divisors 1 and n, so either n=1 or n is prime.
If k=2,
then n has only the divisors 1, 2, n/2, and n. If n/2 is not prime, then all of
its proper divisors are either 1 or 2, so it must equal 4. Thus either n=8 or n
has the form 2p.
So suppose
k >= 3. Among the numbers k+1, k+2, k+3, k+4, exactly one is congruent to 2
(mod 4); call this number m. Thus m/2 is an odd number; I claim that it's less
than or equal to k. For if not, then k+1 <= m/2 <= (k+4)/2, so k <= 2.
Now we
know that both 2 and m/2 are among the numbers 1, 2, ..., k, so they are both
divisors of n. Since they are relatively prime, their product, m, must also
divide n. Since m > k, m can't be one of the divisors of n that are <= Ö n. Hence Ö n < m <= k+4,
so n < (k2+4) .
Next,
since n is divisible by all positive integers up to k, it's divisible by the
largest 3 of them, k, k-1, and k-2. So n is divisible by lcm(k,k-1,k-2).
By
well-known facts about greatest common divisors and least common multiples, we
have:
lcm(k-1,k-2)
= ((k-1)(k-2))/gcd(k-1,k-2) = (k-1)(k-2),
gcd(k,
(k-1)(k-2)) = gcd(k, k(k-3) + 2) = gcd(k,2),
and
lcm(k,k-1,k-2)
= lcm(k, lcm(k-1,k-2)) = lcm(k, (k-1)(k-2))
=
(k(k-1)(k-2))/gcd(k,(k-1)(k-2)) = (k(k-1)(k-2))/gcd(k,2) = k(k-1)(k-2) or
(k(k-1)(k-2)/2
So
n >=
lcm(k,k-1,k-2) >= (k(k-1)(k-2)/2
Combining
this with the inequality n < (k+4)2 gives
k(k-1)(k-2) <= 2n < 2(k+4)2.
But this
is false for all k >= 8, so we must have k <= 7 and n < (7+4)^2 = 121.
So to find all numbers n which have the desired property, we need only check
the numbers 1, 2, ..., 120, and we find only the solutions listed in the
theorem.
The
conclusion of my empirical results and the mathematical proof by Dean Hickerson17
is that:
The number
60 has the most only consecutive factors under Ö n of all
integers.
This is
what makes the number 60 different and unique among the integers. And to come
back at the words of Smalley1, it most likely is the reason why the
Babylonians used 60 as a base for their number system as I will make clear in
paragraph 4.3. (Dean Hickerson is skeptical about this conclusion.) Maybe this
and the whole consecutive sequence actually gives a clue to what Smalley owes
to the highest factorability of 60. As he puts it:1
"For
reasons that so far seem obscure but probably are connected somehow to its high
factorability, sixty is also the maximum finite number of ways you can rotate
an object around a central point in 3 dimensional space so that when you finish
rotating it looks exactly the same as before. Such an object has the symmetry
of the icosahedron, the highest finite point group, which has 60 proper
rotational symmetry elements."
But that
is stuff for another article. Lets first return to the main subject of this
article in the next paragraphs.
2.2. The
different number systems:
A real
number can be represented in several ways depending on the so-called number
system that is used. The base of these number systems is sometimes called the
radix or scale (lets called it b for base). Then using Arabic symbols (0-9) the
digits of all real numbers are described by 0, 1, … b-1.4 The number
Decimal is
the number system that we use the most in every day live. Telephone numbers,
the prizes in the supermarket and the calculations we learn at school are all
examples of the use of the decimal number system. But most calculations are
done in the binary number system since this is the way that all computers
perform their tasks, although the output of a calculation by a computer program
is mostly in decimal numbers. Sometimes a programmer or an operator asks for
octal or hexadecimal numbers from the computer because these numbers give good
insight on what is going on a the binary level, which can be a bit confusing by
it self.
|
Base |
Name |
Systems
notation of the decimal number 60 |
Calculation
of the number system notation in to the decimal notation (in other words
decimal 60=) |
|
2 |
Binary |
111100 |
1*25+1*24+1*23+1*22+0*21+0*20 |
|
8 |
Octal |
74 |
7*81+4*80 |
|
10 |
Decimal |
60 |
6*101+0*100 |
|
16 |
Hexadecimal |
|
3*161+12*160 |
|
60 |
Sexagesimal |
1,0 or 00:01:00 or 0o1’0" or Y |
1*601+0*600 |
Table 3: Representation
of the decimal number
You can
describe hexadecimal numbers in two ways, as can be seen in Table 1. The first
way (method A) comes from the fact that one can count after the 9 up to the
decimal symbol 15 with letters from the alphabet. Counting this way from the
decimal 1 to
At this
point I want to introduce a new aspect of number systems: How to represent
fractions? By convention placing a comma (in method A) or a semicolon (;) (in
method B) between the integer and fractional part and using negative powers of
the base to multiply the counter with. For example, the hexadecimal number
C57,3A = 12,5,7;3,10 equals 12*162+5*161+7*160+3*16-1+10*16-2
≈
This
brings us, last but not least, to the main subject of this publication: the
sexagesimal number system! All the principles stated above also apply to the
sexagesimal system. As far as we know it is one of the oldest number system
used by mankind, as I will discuss later on. It, partly, survived until today
in every day live. For instance we still divide 1 hour in 60 minutes and 1
minute in 60 seconds, although there go 24 hours in 1 day and 365 days in a
year, and not 60 hours in 1 day and 60 days in a year as a true sexagesimal
time system would require. But this is of course more determined by
astronomical phenomena than by the number system used to describe time. The
digital clock representation of time 23:06:51 actually means that 23*602+6*601+51*600
= 83211 seconds elapsed since midnight yesterday. It also, partly, survived in
what is called the North American system of angular measurement.7
The unit of this system is called a degree (o) which is divided in
60 equal parts, also!, called minutes (‘) which are subdivided into? Yes, 60
seconds ("). But this is as far as the sexagesimal pattern goes, because a
circle in this system is exactly 360o and not 60o. Also
the fractions of a second are represented in the decimal way! So there is no
(’’’) subdivision of a second. I will get back later why a circle is probably
defined as 360o in this "sexagesimal" system.
It were
the Sumerians, about 5000 years ago, as we soon will discuss in depth, which
used the system first and there representation of the decimal number 60, is a
so called cuneiform carving in a clay tablet, which looks like the symbol: Y.
Because all of this there are 4 system notations for the sexagesimal in Table
1.
We saw in
Table 1 how one can calculate numbers from one base system to the decimal
system. But how is the method to calculate from one base system to all other
possible base systems? My method is to use the decimal systems as an
intermediate base. Thus by calculating the base(a) number to a decimal number and
calculating this by using so called successive divisions5,6 to the
base(b) number. For example calculating the hexadecimal number AC to the
ternary system (base = 3) one can proceed like this:
·
Step 1: Calculating AC to the decimal system: 10*161+12*160
= 172
·
Step 2: Calculating 172 to the ternary using
successive divisions (div, whole divisions by 3 for base 3) and modulo
operations (mod, remainder after a whole division by 3 for base 3):
172 div 3 = 57
172 mod 3 = 1
57 div 3 = 19
57 mod 3 = 0
19 div 3 = 6
19 mod 3 = 1
6 div 3 = 2
6 mod 3 = 0
2 div 3 = 0
2 mod 3 = 2
You proceed with the successive divisions until
you encounter a zero for the div operation!
·
Step 3: You arrange the results of the mod operation
in reverse order. So hexadecimal AC = decimal 172 = ternary 20101.
·
Step 4 check the results ternary 20101 = 2*34+0*33+1*32+0*31+1*30
= decimal 172 = hexadecimal AC
One can
find also another method with uses the binary systems as intermediate base, but
the principles are the same.6 I also discovered the interesting
mathematical theory behind the method of successive divisions on the internet.6
If I do
not mention a number system specifically the numbers are from the decimal
system!
3. The Sexagesimal of
3.1. The
situation before the sexagesimal place notation:
First I
will give a timetable (Table 4) of the Mesopotamian history to put the
development of the sexagesimal system in perspective.
|
Notes |
Year BC |
Period |
Writing/Number System Phase |
Historical Developments |
|
|
>3500 |
|
Decimal metrological systems |
|
|
|
3500 |
|
|
|
|
|
3400 |
Early Uruk |
Mystical status of 60 |
|
|
|
3300 |
|
Numerical tablets and clay bullae / Sexagesimal metrological systems |
Beginning of
large scale settlement of |
|
|
3200 |
Late Uruk |
|
First Urban Centres |
|
Beginning of Proto-Cuneiform |
3100 |
|
Archaic texts from Uruk: Writing Phase Uruk IV |
Age of early civilisations |
|
|
3000 |
Jemdet Nasr |
Writing Phase Uruk III |
|
|
Beginning of Proto-Elamite |
2900 |
Early Dynastic I |
Archaic texts
from |
|
|
Beginning of Hieroglyphics |
2800 |
|
|
Formation of large irrigation networks |
|
|
2700 |
Early Dynastic II |
|
|
|
|
2600 |
|
Texts from Fara |
|
|
|
2500 |
|
|
|
|
|
2400 |
|
Old Sumerian texts / Cuneiform sexagesimal place notation |
Rival city states |
|
|
2300 |
Dynasty of |
Old Akkadian texts |
First regional states |
|
|
2200 |
Gudea of |
|
|
|
|
2100 |
|
Neo-Sumerian texts |
Centralised state of the 3rd Dynasty of Ur |
|
|
2000 |
|
|
|
|
|
1900 |
Old Assyrian |
Old Assyrian texts |
|
|
|
1800 |
Old Babylonian |
Old Babylonian texts |
|
|
|
1700 |
|
|
Hammurapi of |
|
|
1600 |
|
|
|
|
|
1500 |
|
|
|
|
|
1400 |
|
|
|
|
|
1300 |
Middle Assyrian |
|
|
|
|
1200 |
Kassite |
|
Kassite rule |
|
|
1100 |
|
|
|
|
|
1000 |
|
|
|
|
|
900 |
|
|
|
|
|
800 |
Neo-Assyrian |
First Aramaic texts |
Assyrian empire |
|
|
700 |
|
|
Babylonian empire (Nebuchadnezzar II) |
|
|
600 |
Neo-Babylonian |
|
Occupation of |
|
|
500 |
Achaemenid |
|
|
|
|
400 |
Seleucid |
Revival of cuneiform under the Seleucids |
|
Table 4: Timetable
of the Mesopotamian history, this is based on work of Nissen et al.18
See also paragraph 4.3. for an explanation of the Number System entries.
The so
called cuneiform (from the Latin: cuneus, a wedge shape) carvings in clay
tablets used for numbers or language, were both created in the same period as
we will see later, about 4500 years ago by the Sumerians of Mesopotamia. It is
the oldest number system and written language known to mankind. The main theory
how this cuneiform based manuscript form evolved is well documented on the
internet.8,9 I will here give only a short summary of this material
in the next alinea.
The
emanation of the cuneiform script happened within a region that is nowadays
known as
Not the
need to save a story or poetry was the drive for recording data in the ancient
cultures like the Sumerians, but the need to do bookkeeping of economic
activities probably was! The first stage of bookkeeping was by means of tokens
from stone and later from clay. These tokens represented for instance one jug
of wine or one sheep. Remember that the Sumerians not only developed the first
written records but also the concept of money in the form of sea shell rings
and coins.12,13 This simultaneous development of money and written
records is synergistic, and an underestimated important part of Mesopotamian
archaeological and anthropological research. But lets say that the early
Sumerian culture (4000-3000 BC) had the need to do bookkeeping for commercial
purposes without the concept of money. They bartered for instance 3 jugs of
wine for 1 sheep and recorded this transaction by putting 3 wine jugs tokens
and 1 sheep token in a container (bullae), probably made of clay. A collection
of all these containers was their archive of a past transaction or a negotiated
contract for in the future. So, in the early stage there were no different
signs for quantity and item, they just repeated the sign of the item to the
quantity. The problem with these clay containers was that you only knew the
content of tokens when you break them, destroying its actual purpose of
archiving. So they started to carve the clay envelope with the number of
different tokens and with the number of the different signs for what the tokens
represent. They developed in time a de facto standard of signs (= tokens
= commercial items). A major break through! The second stage begun around 3000
BC because of that economic activities started to grow and become more and more
complicated, and one can say also by the introduction of the concept of money because
the market got to complicated for bartering!
In the
meanwhile Sumerians started to replace the clay envelops + tokens by clay
tablets and they developed signs to represent numbers because it became prone
to errors to write for instance 100 sheep symbols! On the clay tablet there was
in this case the sign for sheep and their notation of the decimal number 100.
This break through is called a metrological numeration system. In these early
days the whole system was not as advanced as ours by which the 3 can mean 3
sheep, 3 jars of oil or 3 jugs of wine.
They had
different metrological numeration’s systems for all possible measures. There
was a metrological system for discrete objects, length, area, volumes of
different kind of liquids, time and many other measures. Most of these
metrological systems where already sexagesimal!18 (See Fig. 3)
So the
base 60 of the sexagesimal metrological measuring systems, and finally of the
sexagesimal place notation, must have evolved just before this era.
To
maintain all of these metrological measure notations was troublesome and the
result was one (cuneiform) notation of language and numeration in the
sexagesimal system.
3.2. The
sexagesimal place notation:
In around
2500 BC the cuneiform sexagesimal number system was well established. The
representation was based on two symbols: one wedge for ‘1’ (in the text I will
use Y for this symbol) and two wedges forming a corner for ‘10’ (in the text I
will use < for this symbol). In the source code of CALSEX2 and 3 (see APPENDIX)
the principles are shown how to represent the Sumerian numbers 1 – 59. This is
in simplified cuneiform notation because the Mesopotamian did grouping of the Y
symbols.
What
catches the eye of the sexagesimal place notation is the fact that there is no
zero and that consequently the signs for 1 and 60 are the same (my notation, an
Y). This seems confusing but the Mesopotamians could determine the actual value
of an Y symbol from the context. Take for instance the cuneiform notation Y Y
<<YY the first Y means 1*60*60, the second Y means 1*60 and <<YY
means 22. Using the notation and calculation method of paragraph 2.4. Y Y
<<YY means 1,1,22 an this is decimal: 1*602+1*601+22*600
= 3682. With the program CALSEX2 (see APPENDIX) one can calculate how large
numbers would be carved in clay tablets by the Mesopotamians. These archaic
people did not invent a special symbol for the number zero yet. They often
marked a zero column with an empty space:
So 1*602+0*601+1*600
= 3601 would then be marked as Y Y, 1*601+1*600 = 61
would be marked as Y Y and 2*600 = 2 finally would be marked as YY.
But how
did Mesopotamians note fractions and numbers with an integer and fraction part?
Answer: In the same way as they noted integers! And one must see the
distinction, just like as with the lack of a zero mentioned above, through the
context of the tablet:
So Y Y
<<YY can mean the integer 1*602+1*601+22*600
= 3682 (as I mentioned earlier), the integer/fraction numbers 1*601+1*600+22*60-1
» 61.367 and 1*600+1*60-1+22*60-2
» 1.0228, or the fraction 1*60-1+
1*60-2 +22*60-3 » 0.01705.
3.3. The
calculations in the sexagesimal place notation:
And now
how the Sumerians did their basic calculations. I will omit the cuneiform
notation in this discussion and will use the Arabian numbers. The addition and
subtraction operations go analogous as in our decimal system:
Example
addition: 59,6,19;4,56 + 3,8;6,5,7 = 59,9,17;11,1,7
Example
subtraction: 59,6,19;4,56 – 3,8;6,5,7 = 59,3,9;58,50,53
It can
easily be done mentally if you remember that every time you add above 60 or
subtract below 0 you have to shift one number to the left in case of adding one
to that number. In case of subtracting you have to subtract one from that
number.
Multiplication
and division are far more complicated in the sexagesimal. If you need to
remember 10 multiplication tables (10*10 = 100 facts, which you know from
elementary school) in the decimal system, you need to remember 60 (60*60 = 3600
facts) of them in the sexagesimal system!
Because of
this problem with mental calculations the Sumerians had developed a workaround
for division and multiplication. They used the following formula for
multiplication:
(a) xy =
[(x+y)2-(x-y)2]/4
To do a
quick multiplication they used tables of squares. So they calculated x+y and
x-y mentally, looked up the corresponding squares for the answers in a table
and finally calculated the multiplication by using the above formula. These
squares tables were most likely derived by a tedious addition process where the
number of additions were carefully counted, e.g. in the case of 62
by 6+6+6+6+6+6 = 36. It is obscure how the Sumerians derived this formula (a);
more research is certainly needed on this topic.
Division
was another problem in the sexagesimal related to the need to remember 60
multiplication tables (or 60*60 = 3600 facts) to do it mentally. These archaic
people solved it by using the formula:
(b) x/y =
x*(1/y)
A division
is turned into a multiplication with a reciprocal according to this formula.
The Sumerians had also large reciprocal tables on clay tablets. I suppose that
the multiplication of an integer and a reciprocal from these tables was also
done by formula (a). But this appears to bring in the problem of squaring
numbers with a integer part and fraction part with formula (a)! The problem is
that they could impossibly have squares tables for all of these occurring
numbers. In fact there are an infinite amount of these numbers. I postulate the
following hypothesis about this:
"For
the Mesopotamians this was not a problem. They handled reciprocals and numbers
with an integer and fraction part just like integers in division and
multiplication, and interpreted the results according to the context of the
starting numbers. In other words they mentally placed the semicolon to separate
the integer part from the fraction part of the answer. Lets look at a specific
case to illustrate this hypothesis. If a Mesopotamian wanted to divide 1,40 by
25 he would proceed as follows:
1. Search for
the reciprocal of
2. He would
add 1,40 to 2,24 (omitting the semicolon and regard both numbers as integers!)
1,40 + 2,24 =
3. He would
then subtract 1,40 from 2,24 (also regard both numbers as integers) 2,24 – 1,40
=
4. As the
next step he would look up 4,4 and
5. He would
then subtract the squares 16,32,16 - 32,16 = 16,0,0 in accordance of formula
(a)
6. He then
would only have to divide this number by 4 mentally (So he only has to know one
multiplication table, the one for 4, to do multiplication’s and divisions)
16,0,0/4 = 4,0,0 in accordance of formula (a)
7. Finally he
would place the "forgotten" semicolon mentally back on its place,
that is 2 comma positions to the left. So the answer is sexagesimal 4;0,0 = 4.
We can check this number by doing the same calculation in the decimal system.
Sexagesimal 1,40 = 1*601+40*600 = 100 decimal and
sexagesimal 25 is also 25 decimal. So the decimal answer is 100/25 = 4 which
also is the same number in the sexagesimal system as we saw in our answer
above!"
A
question, which is still open, is: How do the prime numbers, more in general
the resolving in factors of integers, look like in the sexagesimal system? In
broader sense: Do they have the same decimal value in all number systems? It
seems to be so that these arithmetical properties of integers are independent
of the base of the number system. So the factors of integers in the sexagesimal
have the same decimal value as in the other base system. Thus if this is so for
factorability then this also true for the prime numbers!
I have
written two PASCAL programs (See Appendix programs: CALSEX2 and CALSEX3) which
can be used to test the argumentation’s of this paragraph.
4. Hypotheses About the Foundation of the Sexagesimal Number System.
4.1.
Hypotheses with can be found in the literature:
To start
this chapter I will evaluate the main hypothesis about why some ancient
civilisations adopted other number systems than base 60. There is little doubt
that ancient civilisations in
Now I will
say something about the foundation of the sexagesimal number system by the
Sumerians, and evaluate the information about this topic, which can be found in
the literature. This must be viewed in the light that nothing is known about
the exact circumstances of this invention as Nissen et al correctly
stated.18 So all the hypotheses about these subject are a shot in
the dark! They wait for more scientific evidence from future excavations to be
supported or falsified.
The oldest
hypothesis is probably the one from Theon of Alexandria (fourth century AD). In
his commentary on the first book of the mathematics of Ptolemy he writes as
here below where the original Greek text is given. (Which lacks of some of the
necessary diactritic symbols. The complete Greek text can be found in the
references.19) Along with the English translation:
“…
ενχρηστότερον
δε πάυτωυ των
αριθμων ειναι
τον ξ- δια το των
αλλων άπάντων
των δυναμένων
πλείονα μέρη
εχειν ελάττονα
οντα ενμεταχειριστότερον
ειναι”
"… 60
is among all the numbers the most convenient, because being the smallest among
all those which have the most divisors, it is the easiest to handle."
George
Gheverghese Joseph seems to have found another passage of Theon about this
topic because he writes:20
"Different
explanations have been offered for the origins of the sexagesimal system,
which, unlike base 10, or even base 20, has no obviously anatomical basis.
Theon of Alexandria, in the fourth century AD, pointed to the computational
convenience of using the base 60. Since 60 is exactly divisible by 2, 3, 4, 5,
6, 10, 15, 20 and 30 [the author omits 12], it becomes possible to represent a
number of common fractions by integers, thus simplifying calculations… Indeed,
while base 10 maybe more ‘natural’, since we have ten fingers, it is
computationally more inefficient than base 60, or even base 12. However, this
explanation for the use of base 60 is unconvincing because of its "hindsight"
character. It is highly unlikely that such considerations where taken in to
account when the base was chosen. A second explanation emphasises the relation
that exists between base 60 and numbers that occur in important astronomical
quantities.
Joseph goes
on to say that either 30, the number of days in a lunar month, or 360, the
Babylonians’ consequent estimate of the number of days in the year, was used as
the numerical base before the advantages of calculation in base 60 were
reconized.21"
The hypothesis,
which comes directly from Theon (the Greek text), has been falsified by me in
paragraph 2.1. 60 is NOT among the numbers with the most factors,
since there are numbers with an infinite amount of factors possible!
The
Dutchman Simon Stevin has an similar meaning about the sexagesimal and in his Disme
(1585 AD), in which he deals with astronomical computations, he expresses
his opinion as follows (given is here the Renaissance French text together with
an translation in to Modern English):19,35
"Aians
anciens Astronomes parti le circle en 360 degrez, ils voioient que les
computations Astronomiques d’ícelles, avec leurs partitions, estoient trop
labourieuses pourtant ils ont parti chasque degré en certaines parties, &
les mesmes autrefois en autant, &c. à fin de pouuoir par ainsi tousiours
operer par nombres entiers, en choisissans la soixantiesme progression, parce
que 60 est nombre measurable par plusieurs mesures entieres, à sçauoir 1, 2, 3,
4, 5, 6, 10, 12, 15, 20, 30."
"Asians
Astronomers divided the circle in 360 degrees, so they can do there
Astronomical computations with this partitioning, unmature and laborious
nevertheless they kept with this burden of dividing the degree in these certain
parts (I guess what is mend here is the sexagesimal subdivision of the angular
degree.) and this is also true for the division of the circle, et cetera. To
end this discussion I bring forward plainly that the number 60 has (I guess
that is meant here: many.) many divisors: 1,2, 3, 4, 5, 6, 10, 12, 15, 20, 30."
As I put
forward in paragraph 2.1. there are an infinite number of integers with more
factors. Thus why didn’t they chose for a base with even more factors?
In his Opera
Mathematica, which was published in 1693 AD, J. Wallis finds the reason,
for the use of the sexagesimal number system by the ancient world, in the
specific arithmetic’s of the number 60. Here below the relevant part of the
book in the original Latin text is given, with a sentence for sentence English
translation:19,36
"Cur
autem voluerint Antiqui fractiones omnes ad unam aliquam denominationem (ut
Sexagesimarum) reducere, causa est manifesta;
Why did
these Antique people of all possibility’s chose for this number system of
measure (in other words the sexagesimal), the cause is detected;
nempe ut
molestiam evitarent fractiones vararium denominationum addendi, subducendi,
aliasve computandi;
truly
resolving factors is difficult in this number system, like adding, like
subtracting, in other words calculating
quae certe
magna erat, dum numeros majusculos commode tractare non potuerint.
although
it (
Cur autem
Sexagesimas prae aliis denominationes elegerint, causa est, Quoniam, si 12
aliumve minutulum numerum pro communi denominatore sumpsissent, pluribus opus
foret subdivisionibus, quam sumpto numero 60.
Why is the
sexagesimal chosen before other number systems, the reason might lie in the
fact, that the former is accepted before other common number systems, in its
divisors, of the accepted number 60.
Eo autem
multo majorem haud commode tractare potuerunt;
Here we
are unable to understand the literature of our ancestors exactly;
cum etiam
in hoc satis sit difficultatis.
so we are
going to have a difficult venture.
Ex
non-majoribus autem, hic caeteris potoir videtur, quoniam plures admittit
divisores;
Out of all
what is seen seems to support the divisor theory;
nimirum
sex primores numeros 1, 2, 3, 4, 5, 6, totidemque his respondentes 10, 12, 15,
20, 30, 60, adeoque omnino duodecim :
without
doubt six of them are the first numbers 1, 2, 3, 4, 5, 6, and as many of them
are the correspondents 12, 15, 20, 30, 60, precisely entirely twelve;
cum nullus
sit numeros eo minor, qui tot admittit;
it is the
smallest number which has so many divisors;
nec qui
plures admittit ullus qui non hujus saltem duplus sit, (puta 120).
the next
number, which has more, is twice as great (that is 120).
Quod de
nullo iterim dici poterit, donec ad 360 ventum est, quem fecerunt numerum
Graduum in integra Peripheria.
No one can
deny that the wind directions are divided into 360 degrees.
Atque hanc
divisionem sexagesimalem (in minuta primara, secunda, tertia, caeteraque)
praesertim in partibus Arcuum, Angulorum, Temporum, Motuumque coelestium;
And that
the division is sexagesimal (in minutes, seconds etc.) -the rest is difficult
to translate, but it seems to mean that for instance angular and time
measurements (among others?) are sexagesimal quantities-;
retinuerent
Arabes (Grecos imitati) et nos post illos etiamnum.
This came
to us through the Arabs (imitating the Greek) and stayed with us until even
now."
Stevin
comes close to the consecutive sequence hypothesis I will postulate in
paragraph 4.3. and I talked about 2.1. He recognises also that although
calculating in such a big number base is difficult, 60 is chosen because of its
many factors.
The
Venetian Formaleoni has written a book in Italian from the Enlightenment era,
published in 1789 AD, called Dei fonti degli errori nella cosmografia e
geographia degli Antichi (English translation of the tittle: About the
sources of errors in cosmography and geography from the antiquity.). In this
work he more or less postulates an astronomical basis for the sexagesimal
system. Here below the original text of relevant passages is given together
with an English translation:19,35
"Divisione
La
divisione sessagesimale fu dunque da principio suggerita dalla natura stessa, e
dalle prime osservazione."
"The
division of the circle in sexagesimal parts; was born: suggested by nature:
through the length of the primitive year. This led to a preference in the
Antiquity for sexagesimal numbers. The solar ecliptic was divided in parts of
sixty, these parts every day a time in sixty segments, the day in sixty hours,
The hour in sixty minutes Therefore the periods of sixty, hundred and eighty,
six hundred, or multiplication’s of six hundred, of three thousand six hundred,
were introduced by ancient chronology, of which traces are found can with the
Chaldeans, the Tartars, the Egyptians, the Indians, and the Chinese. But who
can we asked to determine why antique observatories of the Sky had a preference
for the sexagesimal division? This issue leads us to a search of great
importance. We try to elucidate a point, of which nobody, up to now, has shed
some light. -A little further he continues about this subject-:
And what
is now demonstrated, that the ancient year was to short. It is evident that
they observed the race around the Sky of the Sun in exactly three hundred sixty
days, in order to speak with the language of the truth, the earth did not
employ in their eyes more than three hundred sixty revolutions, every day one
covering the year, on its axis in its turn around of the Sun.
The
sexagesimal division was therefore, from principle, suggested from the same
nature, and the first observations."
Formaleoni
comes to this truth from testimonies of ancient authors. And he comes to the
above conclusion, which has been taken up again and again in different forms.
But why did the Sumerians not divide the circle in to 8 parts of 45, or another
whole division of 360? In other words it does not explain why specific 60 was
chosen!
Moritz
Cantor modifies and deepens the hypothesis of Formaleoni in his first edition
(1880 AD) of Vorlesungen ueber Gesichte der Mathematik (I) page 83. Here
below only the English translation of the original text is given:19
"The
year, which was given the round number of 360 days, gave rise to the circle of
360 degrees, and the division of the circle into six parts, suggested by the
fact that the chord of the inscribed hexagon is equal to the radius, gave rise
to the number 60, the basis of the system."
Athough
this is a sound hypothesis, it emanates from special properties of the number
60. Cantor abandoned it later in favour of the hypothesis of Kewitsch which I
will discuss later one.
Lehmann-Haupt,
in Zeitschrift fur Assyrologie XIV (1899 AD) page 364, postulates that
(only the English translation of the original German text is given):19
"The
sexagesimal comes from the fact that in one Babylonian hour (= two of our
hours) the sun moves 60 apparent sun diameters (60 Babylonian minutes = 120 of
our minutes)."
This
implies that the Mesopotamians had a way of measuring the apparent sun
diameter. Did they have partly sun blocking lenses of some kind or did they
measure it through the clouds?
The next
hypothesis in line is that from Zimmer, in "Der egentliche Ursprung des
Sexagesimalsystems" Berichte der Philos.-hist. Classe der kgl. Saechs. Ges. Der Wiss. Zu
Leipzig. Session of the 14 Nov. 1901 AD. He comes very close
to Formaleoni and Cantor. He states in German (given is the original test
together with an English translation):19
"In einer
von der Vollzahl 360 (= den 360 Tagen des Rundjahres) ausgegangen 6-Teilung (=
60 Tage)."
"Starting
from the number 360 (= the 360 days of the solar ecliptic) they did a division
by 6 (= 60 days)."
It is
actually the same as the Formaleoni hypothesis.
The
earlier mentioned Kewitsch points to the fact, in Zeitschrift fur
Assyrologie XVII (1904 AD), that neither astronomy nor geometry give the
right answer to the question. According to him the sexagismal was a hybrid
system invented when two influential people came together. One with a decimal
system and one with a base 6 (senary) system, based upon a special mode of
numbering the fingers.19
This is
highly speculative, since how can one base a senary system on (2 * 5 =) 10
fingers?
Loeffler
publishes, in Archiv der Math. Und Physik vol 17 2nd and 3rd Doppelheft (1910
AD) essential the same hypothesis as Theon, Stevin and Wallis, without
referring to them.
He says
that the sexagesimal system has its origin in the schools of the Sumerian
Priests, who recognised the arithmetical properties of 60 as having the first
six numbers as its factors.19 This hypothesis comes close to the one
about the solely consecutive factors under the breakpoint (√60) I will
discuss in paragraph 4.3..
Neugebauer,
in a memoir titled Zur Entstehung des Sexagesimalsystems published in
1927 AD, has a new approach. His hypothesis comes from the metrology18
of the Sumerians (given is the original German text19 together with
an English translation):19
"Aus
dem sexagesimalen Masssystem wird en sexagesimales Zahlensystem."
"A
sexagesimal measuring system becomes a sexagesimal number system."
He is
essentially right but it shifts the problem. A question remains to be answered:
why did the Sumerians initially choose for a sexagesimal measuring system? He
has a very interesting answer by saying that an original decimal number system
interacting with the process of weighing and measuring led to a sexagesimal
number system.26 I will come back about this hypothesis of an
original decimal system as I present my own in paragraph 4.3.
Marvin A.
Powell Jr. has a unique perspective on the matter. In a publication26
from 1972 AD he postulate a hypothesis with has nothing to do with
factorability, astronomy or geometry. His basis of the sexagesimal system comes
from an interaction of language and writing. His hypothesis comes down to the
fact that there is vigesimal (base 20, originated from counting fingers and
toes) aspect in the etymology in the main dialect of the Sumerians and that
there is ternary (base 3, unknown origin) aspect in the etymology of another
Sumerian dialect. This other dialect is found only in a copy from the
Neo-Babylonian period, some 1500 years after the Sumerian language ceased to be
widely spoken. The vigesimal and ternary aspect were supposedly combined (20*3
= 60) to the sexagesimal number system.
One thing
what undermines this hypothesis is the fact that the etymology of the word niš
(= Sumerian for 60) is unknown. It should mean accordingly something like three
times twenty, but it does not!
4.2.
Hypotheses which can be found on the Internet:
And now enough
of the outdated stuff, lets see some hypotheses, about the foundation of the
sexagesimal number system, from contemporary authors, which can be found on the
Internet.
First we
have a familiar hypothesis, which we also know from Theon of Alexandria and
Formaleoni. This Internet site from an unknown author says: "… for the use
of a base sixty counting system is that the number sixty is so rich in factors.
This one number has twelve factors alone."27
Secondly I
found a new astronomical hypothesis which is based on the sixty years cycle of
with the naked eye visible planets Jupiter and Saturn. According to this
hypothesis ancient observatories discovered that once in the sixty years there
is a conjunction of Jupiter and Saturn on the same place of the zodiac. This
gave rise to a mystical status of the number 60 and lead accordingly to the
sexagesimal system.28,29
A weak
point of this hypothesis is that without a doubt one can find many astronomical
phenomena, if one looks for it, which can be expressed by the number 60 or its
multiples. Like the much talked about solar ecliptic of 360 (6*60) days. Also
think of the 60 days cycle in the Chinese calendar which also must have some
kind of astronomical basis.
And
finally I found a site with a kind of review article by O’Connor and Robertson
about our subject.30 Here two new hypotheses along with some already
mentioned are found. I will only discuss here the new ones. The most striking
hypothesis was invented by O’Connor and Robertson them self.
As I
mentioned at the beginning of this paragraph the decimal and vigesimal number
system seems to be chosen for anatomical reasons. O’Connor and Robertson think
that this might also be the case for the sexagesimal, if you use the structure
of the fingers in the right way. It goes as follows: Take for instance the
fingers, except the thumb, of the left hand then you can see three parts
separated by the joints for each finger. This gives us a total of twelve parts
for four fingers. If you count for each part from one to five on the right hand
you can count up to sixty!
A very
creative hypothesis indeed, but why did they not count up to 75 including the
thumb! In other words why chose the Sumerians base 60 and not base 75? One can
also argue; why did they not count up to 300 with the 30 parts of all 10
fingers, and for each part count to 10 on all fingers?
The other
"new" one is similar to the hypothesis of Kewitsch, it suggest a
hybrid number system was formed when one person with a base 5 number system
come together with a person with a base 12 system.
But this
just shifts the problem why were a base 5 and base 12 chosen initially?
4.3. A new
hypothesis about the foundation of the sexagesimal:
This new
hypothesis stands on the fact that there was an original decimal system employed
by the Sumerians before the early Uruk period (>3500 BC, see Table
Do not
make a mistake; most of the hypotheses I talked about in paragraph 4.1. and
4.2., imply, or emerge from the fact that there was original another number
system or systems employed in
This
decimal system allowed basic arithmetic calculations like resolving in factors
of the integers. (In this do not forget; give these ancient people some slack,
they were just as intelligent as we are, they only had less science and
technology to work with.) An indication of how advanced the mathematics were in
this stage is the fact that they already must have found the formula’s xy =
[(x+y)2-(x-y)2]/4 and x/y = x*(1/y). How else could they
do multiplication’s and divisions in their sexagesimal metrological systems.
(See paragraph 3.3.) Resolving in factors and finding the consecutive sequence
is just a piece of cake in this regard! With this decimal system they firstly discovered
the consecutive sequence. I rediscovered this sequence and talked about it in
paragraph 2.1.
It most
likely has been the priest caste who did these resolving in factors
calculations, since they were in the midst of all economical enterprises. They
must have considered the number 60, because of its arithmetical properties, in
other words most consecutive factors under √n, as highly mystical. Fact
is that the Sumerians coupled their supreme god Anoe to the number 60. They had
a pantheon of about 300 gods, but only the gods who where directly under Anoe
where also given numbers: 45 (Enlil, god of earth and sky), 30 (Enki, god of
the sea) and 15 (Inanna, goddess of love and fertility).31 They
therefore chose 60 as a base for there number system and this resulted in the
sexagesimal metrological systems and sexagesimal place notation.
What also
might have played a role is the fact that because 60 has so much consecutive
factors that all of them are easy to remember and thus easy to work with
mentally in for instance fractions.
5. The Use of Sexagesimal Systems in Other Ancient Cultures
One can
find evidence of the use of sexagesimal systems in other ancient cultures like
of the Hindu,
-o0o-
Notes
& References:
1) Smalley
R.E. ‘From balls to tubes to ropes: New materials from carbon’ Presentation for
the American
http://cnst.rice.edu/aiche96.html
2) Results
not published yet.
3)
Illustration from http://www.molecules.com/csc_pg.shtml
4)
Anonymous, ‘Base (number)’
http://mathworld.wolfram.com/BaseNumber.html
5)
Anonymous, ‘Chapter 3: Hexadecimal and octal notation’
http://www.rz.uni-hohenheim.de/rz/sys/basics/csc102/ch3.html
6)
Anonymous, ‘Conversion between binary, octal, and hexadecimal systems’
http://www.redbrick.dcu.ie/help/reference/CLD/AppendixA.doc2.html
7)
Anonymous, ‘Sexagesimal or north american system’
http://www.tpub.com/engbas/1-33.htm
8) Melville
D.J. ‘Mesopotamian mathematics’ Last modified 15 June 2001
http://it.stlawu.edu/~dmelvill/mesomath/index.html
9)
Anonymous ‘About cuneiform writing’ Cuneiform Writing @ University of
Pennsylvania Museum of Archaeology and Anthropology
http://www.upenn.edu/museum/Games/cuneiform.html
10) CiYuan
L. ‘Traditional chinese astronomical records’
http://www.astro.uni-bonn.de/~pbrosche/iaucomm41/ga2000/as_li.html
11) Anonymous
‘City states (ca 5000 – 2000 BCE)’ The American-Israeli Co-operative Enterprise
(2002)
http://www.us-israel.org/jsource/History/citystates.html
12) Lietar
B. ‘EXCERPT from: The future of money’ The Origins of Money (1997)
http://www.stim.com/Stim-x/10.1/origins/origins.html
13)
Anonymous ‘The world’s first money: Ancient sumerian shell money---Over 5000
years old!’ CollectSource (1998)
http://www.collectsource.com/worlds.htm
14) Its is
a matter of discussion if the prime numbers should be considered as having
consecutive factors under Ön because by definition they only
have one factor in this region, in other words the second factor is above Ön. But
the PASCAL program SXGSML8 (see Appendix) includes them in the sequence
and so do I.
15) The
so-called “even primes” have solely 2 consecutive factors under Ön
namely 1 (1*2p) and 2 (2*p). The other two factors above Ön are p
and 2p. These numbers are registrated as sequence ID number A001747 in the
On-Line Encyclopedia of Integer Sequences:
http://www.research.att.com/~njas/sequences/
15) This
sequence is registrated as ID number A066522 at the On-line Encyclopedia of
Integer Sequences:
http://www.research.att.com/~njas/sequences/
17)
Personal communications, a scientific version of this proof is also given by
reference 16.
18) Nissen
H.J., Damerow P., Englund R.K. ‘Archaic bookkeeping’
19)
Thureau-Dangin F. ‘Sketch of a history of the sexagesimal system’ Osiris VII 95-99 (1936)
20) Joseph
G.G. ‘The crest of the peacock’ page 100, through my reference 21
21)
Aldersey-Williams H. ‘The most beautiful molecule’ Aurum Press (1995) Chapter
4, Reference 11, page 307
22)
Anonymous ‘Ancient India’s Contribution to Mathematics’
http://india.coolatlanta.com/GreatPages/sudheer/maths.html
23)
Anonymous ‘Development of mathematics in ancient china’
http://www.saxakali.com/COLOR_ASP/chinamh1.htm
24)
Anonymous ‘Maya mathematics’
http://www.michielb.nl/maya/math.html
25)
Anonymous ‘Numbering systems’
http://www.cz3.nus.edu.sg/~dcreamer/cz1105/lec/lec2.pdf
26) Powell
Jr. M.A. ‘The Origin of the sexagesimal system: The interaction of language and
writing’ Visible Language VI 5-18 (1972)
27)
Anonymous ‘The Development of sumerian math’
http://members.aol.com/arbuckled/origin.html
28)
Anonymous ‘Basis of the sexagesimal system’
http://www.hssworld.org/hindutva/hindu_calendar/sexagesimal.html
29)
Anonymous ‘Proof of the sexagesimal number system’
http://www.hssworld.org/hindutva/hindu_calendar/proof.html
30)
O’Connor J.J., Robertson E.F. ‘Babylonian numerals’
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html
31) Langen F. ‘7000 Jaar Wereld-geschiedenis: De
oorsprong van onze beschaving; Het mesopotamië van de soemeriërs’ Lecturama
Rotterdam (1977) page 40-
32) See 18,
page 95
33) Johnson
D.W. ‘Exegesis of hindu cosmological time cycles’
http://www.aaronsrod.com/time-cycles/index.html
34)
Anonymous ‘The arabic numeral system’
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html
35)
Translated with the help of the Early Modern English Dictionary Database.
http://www.chass.utoronto.ca/english/emed/emedd.html
36)
Translated with the help of the program Blitz Latin.
http://www.software-partners.co.uk/blitz_latin.htm