TRIANGULAR NUMBERS IN THE NEW TESTAMENT
Author(s): F. H. Colson
Source: The Journal of Theological Studies, Vol. 16, No. 61 (OCTOBER, 1914), pp. 67-76
Published by: Oxford University Press
Stable URL: http://www.jstor.org/stable/23947922
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NOTES
AND
TRIANGULAR
STUDIES
67
IN
NUMBERS
NEW
THE
TESTAMENT.
In
a recent
number
Dr
of the Journal
called
M°Neile
attention
to
an article by Dr van den Bergh van Eysinga in the Zeitschriftfur die
neutestamentlicheneutestamentliche
Wissenschaft,
for the number
of the Beast.
=
666
1 + 2 + 3 ...
was then brought
numbers
(=
n.n. Thence
of 8,
from Philo
in
meaning
Dr M°Neile
...
+
Gnostic
did
point
not however
Dr
van
out
pointed
1 + 2 + 3 ...
that
Evidence
+8.
to shew that these'
further
into
=
‫י‬
wisdom.
here
‫י‬
triangular
Eysinga's
what
triangular,
arithmetical
it will
short
be
we
a
get
in
the
new
The
Gow's
the
most
other
well
to 1 + 2+3
equal
a prima
theory.
found
treatises
seemed
that
being
to me to give
seemed
this
to me
viz.
paper,
is also
Thus
is wisdom'.
mention,
153
coincidence
to say about
and in the
Mathematics,Mathematics,
Theon
=
elsewhere
symbolism
number
This
for looking
what I have
it was
article
36 again
and
was suggested
a new explanation
this
for the words
in
Johannine
17.
36, while
+
at any rate
significant
In
1 + 2 + 3 ... + 1(‫ מ‬were held to have the symbolic
value of
the conclusion
was drawn that 666 had the symbolic
value
which
known
in which
facie ground
materials
for
of Greek
History
of Gerasa
of Nicomachus
and
of Smyrna,
series.
both of which are edited
in the Teubner
‫י‬
The doctrine
to the earlier
of triangular
numbers'
seems to belong
form of Pythagorean
is clearly
of the name
arithmetic.
The meaning
both
with
the
above
writers
mentioned
explained
by
diagrams.
Arrange
your
units
(symbolized
(1)
α
(2)
by a) in parallel
α
=
αα
lines
as follows
α
(3)
a
3
α
a
(4)
=6
a
α
:—
a
α
α
α
and
so on.
The
whole
number
in each
case,
3, 6, 10, &c.,
=
a
α
α
IO
α
α
α
will form
an
equilateral triangle, with the side (technically called the gnomon) equal
to the last number
1
May I explain
of the series.
Thus
for the benefit of those who
algebra, that the sum of 1 + ‫נ‬...+ 3
+ « = «.
number is half the product of two consecutive
fishes
153
have
——and
^
will
be
conceived
of
forgotten their elementary
that therefore
numbers, e.g. 666 —
36.
a triangular
'·
Γ 1
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THE
68
why
any
STUDIES
THEOLOGICAL
triangle with a side of 17.
that may attach
to
an equilateral
as forming
see
OF
JOURNAL
symbolic
is thus
It
value
17
to
easy
also
will
attach
to !53·
As
said
I have
other
kindred
been
have
the
of triangular
theory
two
ideas,
of which
very much
popularized
its later
books
were
appears
from history
about
roo
Gerasa,
from
or
at
least
Nicomachus
whose
became
in the second
the subject
was
the type
Theon
The
bour
He
popular.
and
Arabian
of the arithmetician
by Nicomachus
branch
was
in
In
commentaries.
or calculator.1
into
the
was
Latin
sixth,
Lucian's
He
of
of mathematics.
translated
Boethius
by
dis
Gow,
says
revived
and
time
he
succeeded
by
who was most probably
living and working at a. d. i 40.
in the interests
write avowedly
of philosophy,
of which
arithmetic
as the vestibule.
Nicomachus
was a Pythagorean
a Platonist.
date
and
home
of Nicomachus
(or possibly
birthplace)
interest in our present enquiry.
He came
of some
in
Gerasa
the
is given
date
sistent
floruit
was
to
Testament
men
is a matter
from
Arithmetic,
but
with
of Smyrna,
these
they regard
and Theon
This
of some
but
it appears
later,
the time that the New
the favourite
Apuleius
is early,
mention
written.
it became
by
numbers
shall
about
B.C.,
extremely
century
was
Both
time
I
with
some
about
twenty
Decapolis,
as about
a.d.
100,
by Gow
decades
earlier.
and
Liddell
at a. d. 50.
If so, he is both
of the Apostles.
miles
from
the
but
the
Jordan.
evidence
is
in fact
Scott
a contemporary
himself
probably
and
a
give
near
con
his
neigh
'
bound
are two other
closely
up with the
triangulars'
first
is
the
are
or
numbers.
These
systems.
square
τετράγωνοι
the same as our square
but they are conceived
of as generated
numbers,
in the following
with
unit a;
Start as in the triangulars
one
way.
In
both
writers
The
then
add
a 'gnomon'
of 5, making
'gnomon'
so on.
of
the
making
number
square
3,
the
number
square
9,
then
then
4,
of 7 making
a
16,
and
α
Thus
the
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
α
α
α
α
τετράγωνοι
are
conceived
of as
the
sum
of successive
odd
numbers.
Again,
1
closely
connected
with these
are the ετερομηκεκ.1
ίριθμιΐ! ώϊ Νικόμαχο! δ Γιρασηνό! (Philopalris
I do not know how to translate this word.
Boethius gives ' altera parte longiores
1‫■נל‬
Not ‫ י‬oblong',
No
diagram
which is προμήκη!.
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NOTES
is given,
but it can
cα
α
Thus
by one
consecutive
c1
unit
seen
three
with
two
units
a
a
α
α
α
α
α
α
α
α
α
α
form a rectangle
It is
is of course
of which
therefore
the
double
always
of successive
even
one
add
side
of some
It
is
of two
product
numbers.
form a compact
systems
: then
so on.
α
the sum
that these
69
a
the other.
and
It is also
Start
12, and
will always
than
STUDIES
supplied.
6, making
numbers,
number.
be
add
the ίτίρομήκης
shorter
be
easily
6, then
4, making
AND
triangular
will thus
whole.
(1) τρίγωνοι(sum of successive numbers).
1 + 2 + 3...
n + 1
ιn
»·——·
» =
+
(2) τΐτράγωνοι(sum of successive odd numbers).
‫מ‬/
\1+
1+ 3 + ‫·״‬5 + »" = .(—)
(3) έτερομηκεις(sum of successive even numbers).
η n+ 2
2 + 4 + 6 ...
A passage
these
last
interest
from
η — -‫י‬
2
+
which
the τετράγωνοι
attach
and
τετράγωνοι
is worth
to
then
to
suppose
arithmetic,
some
of
Apocalypse.
great interest
Testament
first few of
4th
5th
6th
7th
8th
9th
roth
16
25
36
49
64
81
100
12
2ο
30
42
56
90
IIO
that
to
an
form
add
the number
the
8th
and
(φιλαλληλία),
that succeed,
those
now
New
have
Testament.
It will not be questioned
But
is there
that
forth
how
they
and
And
that
his
another
(1)
choice
element
first
we
shall
each
thus
naturally
the
numbers
us
writer
is largely
by the
Nicomachean
the
let
the
help
is accomplished
influenced
distinctly
corresponding
its place
in the
8 to make
(‫׳‬κοσμική ■πρόνοια).'
offer is that this
to
it, has
(2)
result
a similar
the Cosmos
so call
the
έτερομηκης
which represents
the
you must add
τετράγωνος,
then proceeds.
them
as set
(ένατενιζοντες)
harmony
orders
I
for
and
:—
3rd
in numbers,
usage.
the
9
if I may
books
shewing
He arranges
lines as follows
of
relations
romance
4
and perfect
generate
nature
that in universal
providence
The
suggestion
the
the
2nd
mutual
which
describes
I
He
8th έτερο μη κης viz. 72.
we contemplate
'When
other
he
as
1st
shews
at their
in parallel
ίτερομήκιις
you must
τετράγωνος,τετράγωνος,
series:
e. g. 64 being
marvel
quoting,
numbers.
these
2
6
έτερομηκειςέτερομηκεις
He
which
in
Nicomachus
two systems
.
2
take
had
based
of
the
a
very
on
Old
?
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THE
7<‫נ‬
The
smaller
or another.
numbers
His
4 is. square,
course
10 triangular,
on this.
in
systems
occur
inferred
from
so frequently
to
shew
12.
It
Here
rise
to
we
must
however
addicted
to
dispense
with the use
the
the
one
anything
that
is true
way
of these
can of
12 ίτΐρομηκψ,
but no argument
the first place
they are all Old Testament
to the three
the numbers
belonging
place
and
In
among
early
All
their
we
from these
said
4, 7, 10,
be
are
appearance.1
with a
incompatible
nothing
But when
result.
second
the
STUDIES
THEOLOGICAL
cannot
favourites
be based
numbers,
OF
JOURNAL
naturally
Nicomachean
of thousands
arithmetic.
we
disregard
series,
to represent
can
large
be
so far we have
get a more
the thousands.
numbers
can
nothing
say is, that
of Nicomachean
knowledge
higher
that
numbers,
we can
be
suggestive
No
one,
to
expected
multitudes.
Apart
we have—
(а)(a)The number of the Beast 666 (xiii 18).
(б)(6)1,260 days (xi 3 and xii 6).
(i)
blood
The
pours
out
of the winepress
for 1,600
stades
(xiv
(d)(d)The wall of the New Jerusalem is 144 cubits high (xxi 17).
Of these
but
one
(a)
point,
(de Is.
van
Eysinga
he
which
has
has
shewn
not
that 666
observed,
is the
20).
'
of 36,
triangle'
In
may be mentioned.
was held peculiarly
sacred
et Os. 76) we are told that 36
and that it was called
the great tetractys, because
by the Pythagoreans,
it was the sum of the first four odd and the first four even numbers.
This
Plutarch
is of course
to 1 + 2 + 3 ... + 8."
merely a way of stating that it is equal
'
‫י‬
observed
that elsewhere
the name
of the
great tetractys
=
seems
to be reserved
for 10 which
1 + 2 + 3 + 4.
36, says Plutarch,
was
their
and signified
the Cosmos.
This
last state
greatest
oath,
It should
ment
be
seems
in connexion
with the
Beast.
We
suggestive
evidence
that the honour
to 36 was
specific
given
to 666, but it would naturally
follow from the general
extended
principle
numbers.
When
we take into account
of triangular
the identity of what
it seems
we call the digits in 666,'
or Nico
likely that in Pythagorean
have,
of course,
rather
no
1 It
may help readers to estimate the value of the argument, if I subjoin the
The total proportion of numbers belonging to the three systems is
I and 50—44 per cent
following:
between
50 and 100—18 per cent
100 and 1,000—8 per cent.
1,000 and a,000—4J per cent.
‫ י‬Zeller
suggests that 36 was also sacred as being the sum of the first three cubes
= 1 + 8 + 37.
I do not know whether there is any definite evidence that this was
It also possesses
the rare, if not unique, property of being both
recognized.
square and triangular.
• This of course is not so
But the
striking in the Greek numerical system.
words Ιζακόσιο 1, itfxovra, ?£ would by their similarity strike the imagination.
In
fact this is the one point about the number which does strike Irenaeus,
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AND
NOTES
machean
666
circles
for the present
leave
would
be
a
the question
STUDIES
71
I
numbers.
king amongst
of the variant 616.
very
will
are usually
in two connexions
which
occur
1,260
days
(6)(6) The
2
and xiii 5),
months
of
to be another
assumed
42
(xi
way
expressing
' a
This
and
time, times, and half a time (xii 14) or 3^ years'.
may be so,
without
some
whether
the seer would
one may doubt
special
though
the fact
of only 360 days.
consist
Anyhow
is ίτψομήκη■: = 35 .36, as also is 42 = 6, 7.
numbers.
both square
They
(r) and (d) 144 and 1,600 are of course
in
of
so far as their
scheme
to
the
Old
Testament
numbers,
belong
But
as they represent
Testament
Old
'roots'
are typical
numbers.
reason
have
remains
the year
made
that
1,260
for the squaring.
reason
there is no apparent
numbers
may be noted.
Apocalyptic
It is not of
is the most striking.
the thousands
144,000
(a)(a) Amongst
is
and
it
attained
a
course
number,
by the
square
properly
speaking
not areas,
lengths
other
Two
ordinary
of multiplication.
process
necessarily.
Still
speculation.
It
it harmonizes
Given
12,000
with the seer's
tribe
per
it follows
of square
love
numbers.
(i)(i)The number of the 241 elders has given rise to a good deal of
belong
are
always
(I
an
Old
Testament
But
the argument,
on the assumption,
not
with the
coupled
28 round
no doubt
that when
it does
that the 24 elders
an exception)
is to say there are always
*7 ο
= —This
can
triangular
only
and
number,
it may be observed
That
28 is of course
And
not
systems.
not think xi 16 is really
do
creatures.
4 living
into
is
to our three
the Throne.
be
brought
a writer mentions
to bring their
an assumption
our minds.
Such
before
unexpressed)
(though
case.
in one familiar
I think, be made
Many
would,
by many readers
in speaking
the
in
the
miracle
of
the
feel
that
Evangelists
5,000,
probably
2 small
and
fishes wish to direct our minds to the
of 5 barley-loaves
two
closely
connected
and
numbers,
juxtaposed
he wishes
sum
mystic
number
Leaving
the
five
belong
out
largest
7.
these
Still
two
numbers
to the three
it is of course
doubtful
of the
systems.
conjectural.
we
numbers,
have
Apocalypse
(exclusive
I find it difficult to resist
the
of
fact
all
that
thousands)
the belief,
that
1 I
present in the seer's mind,
suspect that one at any rate of the associations
the number 34 to the Elders, was the fact that the Greek
when he assigned
That the alphabet had a romance for him, which it has
alphabet has 34 letters.
Which of us could describe the deity as A and Ζ ? The
not for us, is obvious.
on the alphabet in Dionysius Thrax (for centuries the accepted grammar
' There are
24 letters from Alpha to OmegaThe
book) opens with the words,
An interesting illustration of this will be
name στοιχΰα was in itself suggestive.
found in Mr Gaselee's Parerga Coptica i.
1
a It is also in
Pythagorean arithmetic a
perfect' number, as being the sum of
its factors 1 + 2+ 4 + 7 +14 = '8.
chapter
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THE
72
St
John
though
we turn to the
When
the
Lucan
probably
Old
narrative
writings
basis
Testament
In
is influenced
by
careful
of a pretty
I
is which
that
may
the
find in
which
Fourth
the
in
though
Gospel
themselves
if taken
with the facts of
significant,
stand in a category
three numbers
by
call
definite
statistical
numbers,
large
reader
would
naturally
of a collection
counting
These
by chance.
we
books
these
what
as
themselves,
books
phenomena,
become
unconvincing,
the Apocalypse.
numbers
an
on
working
STUDIES
THEOLOGICAL
arithmetic.
Nicomachean
and
OF
JOURNAL
as
regard
of people
or things
the
result
assembled
are
153 fishes (John xxi);
120
(about)
at the election
of Matthias
and
cost (Acts i);
1
276 in St Paul's ship (Acts xxvii).
All
are
three
these
153
120
that
15·16
—~—
=
=1
this number
276
23.24
—-—
=
=
17.18
=
666
=
—7—
+ 2 + 3 ...
like
I have
triangular.
+
is doubly
χ + 2 + 3 ...
noted
already
1 + 2 + 3...
and
15,
,
15
presumably
+
17.
again
=1
at Pente
that
+ 2 ... +
5,
so
triangular.
+
23.
All three are connected
with some
event or revelation.
supernatural
For the 276 of Acts xxvii is not a mere official statistic.
When all hope of
of the Lord
had stood
and
safety had been
lost, the Angel
by Paul
assured
him that God had granted
him all that sailed
with him.
This
assurance
the
was
twice
to the crew, and was verified
conveyed
by Paul
by
to
all
human
It is in connexion
with
contrary
probability.
'
facts that St Luke
now we were in all in the ship 276 souls'.
says
that the triangular
character
of the 120 and 153
may be noted
sequel
these
It
were
known
at Pentecost
to the fathers.
'rising
Jerome
gradually
and
the
number
which
So,
too,
tractate
153
of 15 degrees
in his
Augustine
is the
'triangle'
(Ep.
by
are
on
53)2
remarks
increments
mystically
John xxi
that the number
from
contained
shews
by
1 to 15 makes
in the Psalter
addition
that
of 17.3
1 It will be remembered that Β
'
But
(supported by the Sahidic) has about 76
here no doubt πλοίφ 00T had been corrupted to ττλο'ιψus of.
8
'Qui centum viginti ab uno usque ad quindecim paulatim et per increments
surgentes, quindecim
graduum numerum efficiunt, qui in Psalterio mystice con
tinentur.'
8 ‫י‬
ab uno usque ad seipsum computatis omnibus
Qui numerus (i. e. septemdecim)
crescens ad centum quinquaginta
tres pervenit.'
I owe this reference to Mr W.
Montgomery.
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AND
NOTES
If we
are
distances,
life, and
the like.
here,
number
of money,
suggested
these
70,
data,
two
books
and
40
notes
ages,
or
round
numbers
on
-
added together gives the triangular number 91 (=
the
be
may
numbers.1
The
up.
to
as
in human
or periods
small
(1) the
all of
days,
various
possibilities
of Anna,
that
juxtaposed
find
we
the
crop
records,
age
73
(2)
either
are
or two odd
that
above,
in
12,
numbers;
of these
one
the
amongst
7,
chronological
Most
however,
numbers
disciples
Testament
Old
typical
sums
Even
of the
groupings
which
other
the
examine
familiar
STUDIES
highest
principle
consciously
Again, in
-)·
χ ι the
ment,
variant
72 is *προμήκης
(=8.9).
very well supported
the argu
and only strengthen
of course,
merely possibilities
which
in so far as they weaken
the counter-argument,
may be
drawn
from the frequency
Luke
These
are,
which
of numbers,
do
to the three
not belong
systems.
The
all
I hardly
think
in these
seven
(120,
144,
xoo
and
books,
only
and
classes
may
exception
a
vast
amount
coincidence
can
be
found,
numbers
can
well
a
coincidence.
Such
not
exception
a principle
to
!53=17,
276=23,
seen
Temple.
can
but
indeed
Augustine
the
unless
Still,
for the
Fourth
be
found
themselves
to
666.
the
what
that
explains
are
Jerome
17
we
I
to
odds
some
prefer
a
and
to
brings
an
15
the
15
in
addition
of
where
principle
for such
Revelation,
as
it
regard
'degrees1
10,
require
point
principle
and
17
symbolic
of
make
some
daresay
general
shewn
Acts,
about
to these
it will
it, but
The
in symbolism.
such
treatment,
any
to
as
between
or
belonging
predilection
still
a hundred
103,
to 1.
Gospel,
will
On
seven
of stating
way
many
all
are
Testament,2
numbers
are
other
There
above
numbers
millions
reduce
account
of the
hardly
of
120
this
that
lend
perhaps
We
have
will
authors
understand
seem
to
taken
which
against
about
of exception
to
becomes
probable.
by the
I
do
be
definite
therefore
somewhere
think
who
readers
Now
the
1,600).
to our three
classes,
belong
coincidence.
sum
1,260,
odds
The
are
call
should
666,
which
ι in 14 or 15.
three
I
276,
153,
I think
indeed
of what
1,600,
them
ask
of
in probabilities.
in the whole of the New
a little
forward,
brought
limits
reasonable
I would
and
do,
before
put
the
within
that they
to let me
wise,
fall
together
I have
the facts, which
is whether
now
question
taken
numbers
with
120 =
and
the
15,
23?
of the
symbolizing
1
and 38 in John ν 5
46 ‫ י״‬John ii 20 (apparently a careful piece of chronology)
are rather marked exceptions.
' The
I have observed, is the
only number answering to this description, which
datum
derived
direct from the
which
as
a
of
iii
Gal.
chronological
17,
430 years
Old Testament may be fairly said not to count.
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THE
74
the law,
to
of
author
THEOLOGICAL
the
7 symbolizing
to
intended
Gospel
to Acts ii at Pentecost
that
fulness.
modern
the
I
Moreover,
symbolism
a
plays
numbers.
doubt
on
the
will
(which
outdoes
the
my
STUDIES
thinks
Eysinga
indicate
critic
add
nations
up
ancient
that
to 17).
It
seems
in
fanci‫׳‬
allegorists
present
the
represented
whether
knowledge,
in the later
of
part
Pythagorean
theory
is nothing
of the sort in Nicomachus
and
large
there
Certainly
Van
Gospel.
the
according
to me
OF
JOURNAL
to the original
seems to
symbolism
Pythagorean
as that of Wordsworth's
view of nature
to Keble's,
'
as shewn in the hymn
There is a book, who runs may read'.
While to
the Maker's
Keble
the sky meant
the Church,
and the
love, the moon
to Wordsworth
of divine
all nature
was a revelation
stars the saints,
Their
Theon.
me
much
with Nicomachus
So
beauty.
the
but
things,
relation
the same
endless
of numbers
properties
I would
different
a sign
were
the New
explain
had
become
to
not
mean
different
in
harmonies
revealed
'
of
Cosmic
Providence'.
proof
and
Testament
numbers
these
did
numbers
the
and
mysteries
phenomena
certain
minds
by
matter
a
that
supposing
of profound
This
took
a different
in different
feeling
probably
shape
it may have been superstitious
in
or at least mystical,
a genuine
others it may have represented
intellectual
Many
curiosity.
to think that in St John the first feeling
readers
was
may be inclined
interest.
minds.
In
predominant,
a threefold
some
in
result.
is in
presumably
chosen.
Secondly,
is
cumstance
a
tendency
St
more
to
Luke
the
second.
when
Firstly,
the Apocalypse,
when
such
likely
to
imagine
that
interest
such
might
numbers
may
be
have
free as it
is largely
imagination
numbers
be
a
an
Such
the
deliberately
the cir
occurred,
actually
be
there
may
Thirdly,
is triangular
when
it is really
from
illustrate
such
a tendency
recorded.
number
less
I can
something
interesting.
a recent experience
of my own.
Forty years ago
match
at Haileybury,
cricket
where
the captain
a great
I remember
a splendid
played
I remember
a friend of mine
of (as we thought)
153 not out.
'
and
although
they were so many yet was not the net broken',
exclaiming
all these years I should
have affirmed most positively
that the score on
this occasion
A few months
was 153.
to look back
ago I had occasion
innings
at the
old
recorded
I can
school
magazines
in the magazine
and
of which
found
I
was
that
it was
at the
time
really
152,
myself
an
and
so
editor.
that the aptness
of the quotation
and the association
from my mind the real facts.
If we apply these principles
to the number
of the Beast
we need
not
only
suppose
obliterated
come
with van Eysinga
to the conclusion
that the old explanation
of
is to be rejected.
On the contrary,
such
a 'gematria'
'gematria'
would
be more acceptable,
if it could
be made
to harmonize
with
a sovereign
number.
And further we may expect
that the working
of
a
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AND
NOTES
the gematria
might
in Hebrew
Caesar
van
somewhat
be
is
STUDIES
to get
strained
otherwise
75
we
suitable,
this
result.
need
not
not
does with the fact that it is Hebrew
Eysinga
well worth while to use Hebrew
letters, if thereby
be
a number
If
Nero
as
quarrel
Greek.
It
might
the writer secured
with the properties
a number
of 666—perhaps
we may add
which bore the very appropriate
of
the
term
which
Cosmos—a
meaning
but
to the Pythagorean
would
the ordered
universe,
by
suggest
a Christian
Fourth
would
be
taken
in the sense
It should
Gospel.1
all the more
be
to account
it bears
which
that
added
so
virtues
the
616.
for the variant
in the
often
make
of 666
The
it
facts
seem
Καίσαρ
does
necessary
in
the original
to suggest
reading
strongly that 616 is not indeed
the text, but is the original form of the gematria
which St John reshaped.
to me
And
this is quite
in accordance
satisfy 616.
On the whole,
and
out,
riddle
real
the
the other
of the
or
of what
were
I have
When
which
me
to
these
Burkitt
we
its results
out
at
the
opinion
our
enlarge
mind.
early Christian
It seems
to me a real link
than this.
put it a little higher
the New Testament
writers and the speculation
'
the time.
If science'
seem too lofty a word, it must
worked
any
an
must
the
then
to the
I should
arithmetic
facts
discussed
between
that Greek
pointed
to leave
expressed
that
was
has
Eysinga
was
paper
numbers
interesting
Γαί05
seem
Have
this
appeared
that
van
added,
Professor
Society,
which
fact
it was.
where
value?
Cambridge
Theological
:hat the residual
truth
ideas
which
much
Beast
significance
fact
remarkable
facts
the
with
under
and
be
science
of
remembered
drawbacks
which
we
In the first place,
as we know
it, was
multiplication,
hardly realize.
a difficult process
with Greek
notation.2
In the second
the
place,
Thus
while
Nicomachus
infers
Greeks
at this time had no algebra.
from
form of his diagrams
that the ίτιρομήκΐκ
and
geometrical
of the form r.r+1
no
were respectively
and
r2, he shews
his
of the corresponding
of the τρίγωνοι, where
property
the
τετράγωνοι
knowledge
did
diagram
him.
not help
In
fact the simple
1 + 2 + 3 ···
which
is
have
to
been
triangular
that
learnt
666
now
within
the
beyond
his
numbers
was
with
the
a
'
was
deal
n — n·
of
grasp
ken.
worked
triangle'
great
+
The
2
he
knew
patience
therefore
calculation
and
a
and
fact
seems
schoolboy,
ordinary
by addition,
of 36
of
an
formula
n+1
if St John
which
labour.'
really
could
It
of
must
high
knew
only
be
be
re
1 And in the
1 v. Gow 1. c.
itself (xi 15).
pp. 49-51.
Apocalypse
' This
may perhaps account for the fact that, while the triangularity of 1 so and
153 was observed by Jerome and Augustine, that of 666 is not known apparently by
Irenaeus, or, so far as I know, by any of the Fathers.
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THE
76
that
membered
attention
the
to education,
and
I would
diffidently
suggest
so much
attention
main
and
current
channels
arithmetic
had
in
that the Mystery
is now
given
an
geometry;
and
and
but
throw a great deal of light on the thought
Tpstnmpnt
and earlv Christian
writers.
study
and
gave
great
scientific
and
religions
were
literature,
a closer
which
age
a really
way
even
of education—grammar
and
born
in its own
STUDIES
was not a
philosophy
intellectual
life and
thought
that
of
THEOLOGICAL
was
Church
to which
time-spirit,
OF
JOURNAL
spirit.
the Gnosticism
side-breaths
universal
of the
followed
The
study.
the regular
rhetoric
and
of these
may
of the
terminology
F.
history,
conceivably
Η.
New
C0LS0N.
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