Errors in “Introduction to Inﬁnite Analysis”
Hiroki Sumida-Takahashi
Faculty and School of Engineering
The University of Tokushima§1. Errors in “Introduction”
Leonhard Euler, who is celebrated for the 300th anniversary of his birth in 2007,
continues to inﬂuence many researchers with his voluminous works. Among them, “Introduction to Inﬁnite Analysis”(E101, E102), the ﬁrst part of his famous trilogy on analysis
that forms the basis of his works, is an extremely important work.
Trilogy on analysis
the number of volumes written published
Introduction to Inﬁnite Analysis
2
1745
1748
Foundations of Diﬀerential Calculus
1
1748
1755
Foundations of Integral Calculus
3
1763 1768 ∼ 1770
In “Leonhardi Euleri Opera Omnia”[3], it has been pointed out that there are a large
number of numerical calculation errors in this work. There are more than 100 such errors,
but when we examined these errors, we found that many of them were not necessarily
wrong. This is because the complete collection handles the numbers as “rounded oﬀ”,
and if we consider that the numbers are “rounded down”, nearly half of the numbers arecorrect. On the other hand, dozens of new errors that were not recorded in the complete
collection were discovered, leaving about 80 errors that cannot be explained by “roundingoﬀ” or “truncation”. Of course, there is a high possibility that it was a simple calculation
error, so we reconsidered each numerical value and calculation method, but in the end
we were unable to attribute the cause of these errors to a simple calculation error. If
we assume that these errors are not caused by chance, we must consider the following
possibility.
Aren’t these errors intentional?
In this essay, we will introduce this hypothesis. First, we will answer some questions
that most people may have.
Question 1 Would a great mathematician do such a strange thing?
Question 2 Is there any beneﬁt in changing correct values to wrong values?
Question 3 Why hasn’t there been a hypothesis like this until now?
Question 4 Why do we propose such a hypothesis?
Answer 1 Euler was invited by Frederick the Great to establish and manage the Academy
at the time. The year before the publication of “Introduction”, the great musician Johann
Sebastian Bach, also known as the Great Bach, famously posed a riddle to King Frederick
the Great called “The Musical Oﬀering”. One of the dedicated scores cannot be played1

as it is, because the ﬁrst half of notes are reversed. The problem was how to play thesestrange musical scores. Great Bach was already 62 years old at the time, but he had thissense of playfulness.
It is also well known that the great mathematician Newton sent Leibniz a ciphertext
using anagrams regarding the ﬂow rate method in nealy same period. It is not unnatural
to encrypt results that one would like to refrain from making public, but for which one
would like to claim a right of ﬁrst refusal.
There were great predecessors like those mentioned above, so it wouldn’t be so strangethat young Euler between the ages of 38 and 41 gave riddles using his specialty in calculations. Please judge whether it is strange or not after considering the numerical data
and the hypothesis in this essay.
Answer 2 Even if the numerical values that mathematicians and scientists do not use
are slightly wrong, there is no problem practically. Furthermore, it is not necessarily inthe best interest of the reader to write down the correct numerical value. This is because,
for enthusiastic calculators who recalculate, if correct numbers are written, it only serves
as a check. However, if a numerical value is written with some interesting fact includedin the error, the calculator who correctly calculates the numerical value will be rewarded
with the achievement. In other words, most mathematicians and scientists do not haveany practical problems, and there are possibly beneﬁts for calculators who check thevalues. It would not be unnatural for Euler, a great calculator, to try to provide results
to calculators in future generations.
Answer 3 The corrected numbers in Euler’s Complete Works are due to rounding, so
it is not possible to understand the meaning by examining the errors. Furthermore, as
studying errors may lead to criticizing Euler, who was considered a great calculator, sucha task may be reluctant to do for scholars who respect Euler. In any case, there has neverbeen any merit in checking errors.
Answer 4 This is because we believe that this hypothesis will advance our understandingof Euler’s actions and writings. We also assume that Euler’s intentions in including errors
were both grand and delicate. We believe that this hypothesis is worth considering, but
we must leave the ﬁnal judgment to the readers.

§2. Lists of Errors
The numerical errors exist only near the last digit and are not considered to be atypographical error. In the list below, after writing down the correct numbers, we wrote
down only the incorrect numbers in the “Introduction”. (Calculation is according to [7].)For example, in the list of P0, the correct truncated value with 7 signiﬁcant digits is
E = 4.216965, but the Introduction says E = 4.216964. Also, the correct cutoﬀ value to7 digits after the decimal point for log 10 2 is 0.3010299, but in the “Introduction” it is
written as 0.3010300. Since log10 2 = 0.30102999 · · · , if we round it oﬀ, 0.3010300 is morecorrect. This number is extremely delicate.

2Volume 1
P0
log10 5A = 1.000000
lA = 0.0000000
B = 10.00000
lB = 1.0000000
C = 3.162277
lC = 0.5000000
D = 5.623413
lD = 0.7500000
E = 4.216965 4 lE = 0.6250000
F = 4.869675 4 lF = 0.6875000
G = 5.232991
lG = 0.7187500
H = 5.048065
lH = 0.7031250
I = 4.958068 9 lI = 0.6953125
K = 5.002864 5 lK = 0.6992187
L = 4.980416
lL = 0.6972656
M = 4.991627
lM = 0.6982421
N = 4.997243 2 lN = 0.6987304
O = 5.000053 2 lO = 0.6989746
5
P = 4.998647
lP = 0.6989525
Q = 4.999350
lQ = 0.6989135
R = 4.999701
lR = 0.6989440
S = 4.999877 6 lS = 0.6989593
2
T = 4.999965 3 lT = 0.6989669
8
V = 5.000009 8 lV = 0.6989707
W = 4.999987 4 lW = 0.6989688
7
X = 4.999998 7 lX = 0.6989698
7
Y = 5.000003
lY = 0.6989703
2
Z = 5.000000
lZ = 0.6989700
log10 2 =0.3010299 300
1/ log10 2 =3.3219280 77
P1
l1
l2
l3
l4
l5
l6
l7
l8
l9
l10

√
C = √ABD = √ BCE = √CDF = √DEG = √DF
H =√ F G
I = √F H
K = √ HIL = √IK
M =√ KLN = √KM
O = √KN
P = √N O
Q = √OPR = √ OQ
S = √ORT = √OSV = √OTW =√ T V
X = √W V
Y = √V X
Z = XYlog n0.0000000000000000000000000
0.6931471805599453094172321
1.0986122886681096913952452
1.3862943611198906188344642
1.6094379124341003746007593
1.7917594692280550008124773
1.9459101490553133051053527 4639
2.0794415416798359282516963
42.1972245773362193827904904
52.3025850929940456840179914

3

πk
2k k!
+1.5707963267948966192313216916−0.6459640975062462536557565638+0.0796926262461670451205055494−0.0046817541353186881006854639+0.0001604411847873598218726608−0.0000035988432352120853404585+0.0000000569217292196792681177−0.0000000006688035109811467232+0.0000000000060669357311061956−0.0000000000000437706546731374+0.0000000000000002571422892860−0.0000000000000000012538995405+0.0000000000000000000051564551−0.0000000000000000000000181239+0.0000000000000000000000000550
k−1

01
03
05
07
09
11
13
15
17
19
21
23
25
27
29

P3
01
03
05
07
09
11
13
15
17
19
21
23
25

πk
2k k!
+1.0000000000000000000000000000−1.2337005501361698273543113749+0.2536695079010480136365633663−0.0208634807633529608730516372+0.0009192602748394265802417162−0.0000252020423730606054810530+0.0000004710874778818171503670−0.0000000063866030837918522410+0.0000000000656596311497947236−0.0000000000005294400200734623+0.0000000000000034377391790986−0.0000000000000000183599165215+0.0000000000000000000820675330−0.0000000000000000000003115284+0.0000000000000000000000010167−0.0000000000000000000000000028k

P2 sk = (−1) 2

ck = (−1) 2

00
02
04
06
08
10
12
14
16
18
20
22
24
26
28
30

6
88
2
5
0
1
24
0
0
56
3
0
49

2(2n+1 − 1)π n |Bn+1 | 4
1
2π n |Bn+1 |−−(n + 1)!
π2n−1 π(n + 1)!
+0.2975567820597+0.0186886502773
01 −0.2052888894145+0.0018424752035
4 03 −0.0065510747882+0.0001975800715
4 05 −0.0003450292553+0.0000216977373
245 07 −0.0000202791060+0.0000024011369
70 09 −0.0000012366527+0.0000002664133
2 11 −0.0000000764958+0.0000000295864
13 −0.0000000047597+0.0000000032867
15 −0.0000000002969+0.0000000003651
17 −0.0000000000185+0.0000000000405
19 −0.0000000000011+0.0000000000045+0.0000000000005

ζ(k) =

∞
X
1
n=1

nk

=1+

1
1
1
1
1
+ k + k + k + k + ··· .
k
2
3
4
5
6

4

4

9

5
59
64
58
26
65
08
0
0
1
2
27
5
5
6

20
1
· π2
1·2·3 124
1
ζ(6) =· π6
1 · 2 · 3···7 3285ζ(10) =
· π 101 · 2 · 3 · · · 11 3
212
35
ζ(14) =
· π 141 · 2 · 3 · · · 15 1
216
43867 18
ζ(18) =·π
1 · 2 · 3 · · · 1921
220854513 22
ζ(22) =·π
1 · 2 · 3 · · · 23322476977927 26
ζ(26) =·π
1 · 2 · 3 · · · 27
1
ζ(2) =

221· π4
1·2·3·4·5 3
26
3
ζ(8) =· π8
1 · 2 · 3···9 5210691 12
ζ(12) =·π
1 · 2 · 3 · · · 13 105
2143617 16ζ(16) =·π
1 · 2 · 3 · · · 17 152181222277 20
ζ(20) =·π
1 · 2 · 3 · · · 21552221181820455 24
ζ(24) =·π
1 · 2 · 3 · · · 25
273
ζ(4) =(These values are correct.)PA ζ(k) 1 − 21k
PB ζ(k) 21k
A = 1.23370055013616982735431
α = 0.41123351671205660911810
B = 1.01467803160419205454625
β = 0.06764520210694613696975
C = 1.00144707664094212190647
γ = 0.01589598534350701780804
δ = 0.00392217717264822007570
D = 1.00015517902529611930298
8E = 1.00001704136304482548818 50816 ε = 0.00097753376477325984896
F = 1.00000188584858311957590
ξ = 0.00024420070472492872273
4
η = 0.00006103889453949332915
G = 1.00000020924051921150010
H = 1.00000002323715737915670
θ = 0.00001525902225127271503 69977
ι = 0.00000381471182744318008
I = 1.00000000258143755665977
K = 1.00000000028680769745558
κ = 0.00000095367522617534053
λ = 0.00000023841863595259255
154
L = 1.00000000003186677514044
M = 1.00000000000354072294392µ = 0.00000005960464832831555
N = 1.00000000000039341246691
ν = 0.00000001490116141589813
O = 1.00000000000004371244859
ζ = 0.00000000372529031233986
P = 1.00000000000000485693682
o = 0.00000000093132257548284
Q = 1.00000000000000053965957
π = 0.00000000023283064370808
7
R = 1.00000000000000005996217
ρ = 0.00000000005820766091685
σ = 0.00000000001455191522858
S = 1.00000000000000000666246
T = 1.00000000000000000074027
τ = 0.00000000000363797880710
υ = 0.00000000000090949470177
V = 1.00000000000000000008225
ϕ = 0.00000000000022737367545
4
W = 1.00000000000000000000913
X = 1.00000000000000000000101
χ = 0.00000000000005684341886
ψ = 0.00000000000001421085471
ω = 0.00000000000000355271368
75

PC υ(k) =
02
04
06
08
10
12
14
16
18
20
22
24
26
28
30
32
34
36
√

X 1
pk
p:prime0.452247420041065 222
0.076993139764246
52
0.017070086850637
90.004061405366518
50.000993603574437 3633
0.000246026470035
30.000061244396725
0.000015282026219
0.000003817278702
0.000000953961124
30.000000238450446
0.000000059608184
0.000000014901555
0.000000003725333
0.000000000931326
30.000000000232830
0.000000000058207
0.000000000014551
Volume 22= 1.41421356 1.4142356
√

log
2 2
√ 10
2 √2
10 2

= 0.4257207= 2.665144= 25.954553

74
86
5870

cos log 2= 0.76923890136397 5408
log π
= 1.1447298858494001741434273
37
sin 1= 0.84147098480789 514
cos 1= 0.54030230586813 4341
s + cot s+0.63661977−0.17200818−0.09062597−0.05892836−0.04258548

7
6
4
3

§3. Summary of hypothesis
The contents of hypothesis [7] can be summarized as follows.
6Euler intentionally included errors in the numerical values in “Introduction
to Inﬁnite Analysis” and posed the following problem.

Volume 1 7 Lists
P0
log10 5
P1
Log. values
P2
Sine, Cosine
P3Tan, Cotan
PA
Zeta values A
PB
Zeta values B
PC
Zeta values C
Volume 2 Trans. numbers

Solution 1
All problems
Prime decomp., Decimal sys.
Euler’s formula
Unicursal prob., Euler char.
First irregular prime 37
Second 59, (67,101,103,131)
Third 67, (149,157)
Check for solutions

Solution 2
All Solutions
Alphabet psalms 37, 111, 112
Score for the psalm
Period of psalms (2/1, 1/2)
Mercury
Venus, (Earth, Mars, Comet, Jupiter)
Earth, (Saturn, Comets)
Connection to P0

Below, we will explain the abnormal error of P2 , which is the main problem in theﬁrst half . The following points can be immediately gleaned from the numbers.
1. There are errors in 28 out of 31 numbers.
2. The accuracy is mysterious, with 28 digits after the decimal point.
3. The ratios of errors are rapidly increasing.
4. The errors are within the last digit.
5. The absolute value of only one data is greater than the positive value.
It would be diﬃcult to ﬁnd other numerical data that meet these conditions. However,what is even more unusual is that these errors can be interpreted as follows.
The values of P2 are the coeﬃcient of the Maclaurin expansion of sin x (sine) andcos x (cosine). Strings have been the main element that creates the sound of musicalinstruments. This reminds us of a musical score, and we associate the error 0123456789· · ·with CDEFGABcde· · · . Furthermore, the coeﬃcients are arranged alternately in the orderof their power exponents (that is, Euler’s formula). Euler was a devout Christian, andthe number of errors and precision of 28 digits suggests the hymn’s 8686 meter (common
metre double). Since the number “555” appears in several lists, we include this as threebeats, and obtain the score in the next page.
6. This score is extremely skillfully composed.
This is what seems to be the most unusual point. The ﬁrst, second, and last scalein the common metre double are “C”, that is, the error is 0. The intro “CC GE” starts
from the ﬁrst C and becomes a consonant chord (just intonation) of 1:1 (1:2), 2:3, 4:5,
with unison (octave), perfect ﬁfth , and major third. Note that Euler’s temperament was
just intonation composed of prime numbers 2, 3, and 5 . As for the three-tone chords,
the odd-numbered chords of the seven main chords appear in order: “CEG”, “EGB”, “
GBd”, and ﬁnally “Bdf”, making the piece a song that can be called a model for chords.

7

Additionally, the overall numbers have been nicely adjusted as shown below.
 ′xn = (decimal part of nth coeﬃcient given by Euler ) × 1028
xn = (decimal part of nth coeﬃcient in positive values) × 1028
en = x′n − xn (error)
30
X
30
X

|en | = 111,

n=0

30
X

|en |

en = −3 ⇒ n=0
30
X
n=0

= 37.
en

n=0

Detailed analysis by Professor T.
30
X|en | =

8X

|e2n | +

n=0

n=0

=

15
X

|e2n−1 | +n=1|e2n | +

n=0
30
X

8
X

15
X

30
X

|en | = 37 + 37 + 37

n=17|e2n−1 |= 55 + 56 = 111.

n=1

x′n =

15
X

x′2n +

=

30
X
n=0

x′2n−1 = 1 + (−1)

n=1

n=0

n=0

15
X

xn +

30
X

en

= 3 + (−3) = 0.

n=0In this way, 37 appears as the solution for P1 and PA, and the total number of beats is
59, which is the solution for PB.

8

§4. Reasons and refutation methods
The hypothesis helps to explain many strange statements in Euler’s work. In otherwords, the solutions of the problems were shown through a strange description.
Numerical values and phrases in the examples and problems of “Introduction”
7
Chapter 6 provides example problems such as “Find the value of 2 12 ” and “Assumingthat the number of humans increased from 6 people after the ﬂood ...”. We think it isappropriate to interpret the former as the perfect ﬁfth of equal temperament that appears
in music, and the latter as the ﬂood recorded in the book of Genesis in the Old Testament.
Hymns are associated with music and the Bible. It is very likely that the hymns of theCalvinist are found in the Psalms.
The Alphabet of “Introduction”
Euler entered the Faculty of Theology in order to become a pastor. Since Latin, Greek,and Hebrew were required subjects, he had a wealth of knowledge of ancient languages.
The list P0 includes 24 Latin letters from the 14th to 16th centuries, excluding J and U.
The list B includes all 24 Greek letters. Then, the 22 alphabets in the list A , in which Y
and Z are strangely omitted, brings to mind the ancient Hebrew letters. These, together
with the psalms mentioned above, brings to mind the alphabetic psalms. Furthermore,
the numbers 26 (zeta values are listed up to 26) and 18 in the list C can be similarlyinterpreted. That is, 26 is the number of Latin letters from the 18th century (letters Ato Z), and 18 is the number of Latin letters (letters A to T, excluding G and J ) that
correspond in order to the Paleo-Hebrew alphabet. The correspondence between the listof zeta values and the above-mentioned writing system is in perfect harmony.
Symbols of Sun and moon in paper E352
The following symbols are used in the paper entitled “On the Beautiful Relationship
Between Power Sums and Power Sums of Reciprocals”, written in 1749 and published in
1768.

These symbols are used to clearly show that the zeta function is compared to celestialbodies such as the sun and the moon. These analogies were so puzzling to us that webegan to think that Euler might have posed a problem. It is presumed that behind thisparable is the idea that “all things are numbers”. Furthermore, Euler observed an annularsolar eclipse on July 25, 1748, the year before he wrote the paper, and he might compare
this eclipse to a functional equation of the zeta function. Moreover, Euler had arrived in
Berlin from Russia on July 25, 1741, exactly seven years before that date, which shows
his enthusiasm for this eclipse. In fact, he wrote two papers related to this solar eclipse.
Zeta values up to 34 in paper E352

9

In the above paper, the zeta values up to 34 are written to “indicate as much as wehave calculated so far”. It is rare for a mathematician to calculate this much by hand, but
it would not be strange if he were interested in irregular prime numbers. To determine
whether p is an irregular prime number , check the numerator of the zeta values up top − 3. Therefore, in order to determine the smallest irregular prime number 37 , it issuﬃcient to examine the zeta values up to 34 = 37 − 3, and it is natural to ﬁnd thevalues up to 34. Note that if we only want to determine whether a prime number is anirregular prime number, we can use calculations modulo that prime number, so irregularprime numbers with large exponents are considered to be investigated using this type of
calculation.
Structure of the book E343 “Letters to a German Princess”
The strongest evidence lies in this famous book. If we pose a problem, it is natural
to write down the solution. This work was written between 1760 and 1762 and publishedin the same year as the paper E352, which included the symbols for the sun and moon.
We believe that the solutions to the problems were presented in order in the ﬁrst of thethree volumes. The following is a summary of the contents of Volume 1, with parts thatare assumed to correspond to the solutions shown in bold.
Expansion (P1), speed, sound, music (P2), air, atmospheric pressure, airgun, light, luminescence, transmission of light, luminous body, color, refraction, refraction of diﬀerent colors, color of the sky, plane mirror, unevenmirror, Focusing mirror (P3), Focus, Mystery of the eye, Gravity, Shapeof the Earth, Gravitation of the moon, Universal gravitation, Mutual gravitation between celestial bodies, Solar system (PABC), Small changes due
to mutual gravitation, Rising and ebbing tides, Universal gravitation explanation, properties of objects, inertia, change, monads, properties of forces, forcesof other kinds.
The solutions to the ﬁrst half of the problems are explained in [8]. In the second halfof the problems, since E352 compared the zeta function to the sun and the moon, it would
be natural to solution how other celestial bodies in the solar system can be compared to
the zeta function. We infer that the solution is the irregular prime number of the solution
to the zeta value problem PABC . The basis for this is the diagram of the solar system in“Letters to the Princess” on the next page. If we compare it with the diagram of irregularprime numbers below, we will notice that the directions of the planets and the irregularprime numbers almost correspond in order. Furthermore, from letters around the 103rdin ‘ ‘Letters to the Princess”, we can also associate 103 with one of many comets.
We believe that it would be diﬃcult to ﬁnd another diagram similar to Euler’s solar
system. Also, we realized that this book was the solution after we had solved the problems
in the “Introduction”, and we did not derive the solution based on the contents of thebook.
Approximate values repeatedly displayed in “Introduction”
The approximation of log10 2 = 0.3010300 was repeatedly displayed because of thesolution to PABC above: “7 irregular primes = 3 inner planets of the solar system + 1
10

comet + 3 outer planets” and the solution to P3. One possible reason is that it represents“seven bridges (3+1+3)”. Please pay attention to the position of the comet in Euler’s
diagram of the solar system on the next page, and the depiction of bridge e in the diagramof the seven bridges.

Euler’s solar system

Diagram of irregular prime numbers

The seven bridges of Königsberg
11

It is assumed that the solutions to P1, P2, and P3 were also given in the “Letters to the
Princess”, as explained in [8]. In this way, we believe that many strange descriptions canbe explained by this hypothesis. Rather than explaining the many strange descriptions
and errors in terms of a “chain of coincidences”, we think it would be easier to explainthem in terms of Euler’s “consistent intention”.
Finally, we will list a method for disproving it. This is because the value of a hypothesis
is determined by the existence of a method to disprove it.
A. Explain the mistake in the numerical value in another way.
B. Indicates that there are other similar numerical data.
Generally, if there is an error in a large number of errors, we can guess the reason
why they are generated. There are 28 errors in the last digit of the 31 numbers in P2.
If they are just calculation errors, we should be able to guess the reason. If it is diﬃcult
to explain the reason, we should present a similar list where the reason for its creation
cannot be explained. But will it be possible to ﬁnd a list of numbers in which the errors
in nearly 30 numbers coincidentally sound like a song?
In addition, B can refute any fake codes in which characters are arbitrarily extracted
from a large number of characters. All we have to do is show that such false codes exist
at a certain rate in any text. However, we believe that it is extremely diﬃcult to ﬁnd anabnormal list like one in “Introduction”.

References[1] Archive staﬀs, “Euler Archive”, (http://www.math.dartmouth.edu/˜euler/)[2] W. Dunham, “Introduction to Euler”, (Springer-Verlag Tokyo, translated by N. Kurokawa,
M. Wakayama, and T. Momotani))[3] L. Euler, “Leonhardi Euleri Opera Omnia”, (Birkhäuser)[4] L. Euler,“Euler’s Inﬁnite Analysis”, ”Euler’s Analytic Geometry”, (Kaimeisha, translated by M. Takase)[5] E.A. Fellmann,“Leonhard Euler 1707-1783: Beitraege zu Leben und Werk”, (Springer
Verlag Tokyo, translated by A. Yamamoto)[6] A. Weil, “Number Theory: An Approach from History”, (Nippon Hyoronsha, translated by T. Adachi and K. Miyake)[7] Hiroki Takahashi, “Inﬁnite Eulerian Analysis” (Gendai Sugaku)[8] Hiroki Takahashi, “The Origin of Euler’s Mathematics”, (Gendai Sugaku), 2007,
September∼

2007, November 23th (Talk at RIMS in Kyoto on 2007, August 23th) .
Translation of an essay in RIMS Kôkyûroku No. 1583.

12