Euler’s mathematics and “Letters to a German Princess”
Mysterious numbers left by Euler
Hiroki Sumida-Takahashi
Department of Mathematics, Graduate School of Science
Hiroshima University

Leonhard Euler, the greatest mathematician of the 18th century, has a large number of numerical errors in his major work “Introduction to Inﬁnite Analysis”(1748) (Japanese translation[4]), as pointed out in the complete collection [3]. [9] introduced the following hypothesis, statingthat it is diﬃcult to attribute these errors to simple calculation errors.
The subtle errors in the large number of numerical values in “Introduction
to Inﬁnite Analysis” are not mere mistakes, but are Euler’s intentional“problems”, and the solutions are written in “Letters to a Princess”.
In this essay, we summarize the reasons for the hypothesis and speculate about the intent of the“problems”.
First, in §1, we present the numerical list and errors, and explain the abnormalities. Next, in§2, we point out phrases and ﬁgures that are considered to be hints for solving the “problems”.
In §3 , we provide proposed solutions to the “problems” and explain how to derive and conﬁrm
them. Finally, in §4, we infer the intent of the “problems” by referring to some of Euler’s claims.
The “problems” are thought to be related to the various academic ﬁelds that Euler explored,
and are assumed to be extremely important for understanding his behavioral principles. At
present, only incomplete solutions to the “problems” have been presented, but we hope that the
day will come when we can ﬁnd widely understood solutions by the challenge of many researchers
who admire Euler.
Furthermore, Euler’s personality as a problemer is widely known, as he left behind many
important sources over vast areas (see [2][5][6]). Our hypothesis emphasizes this once again, anddoes not contradict the conventional understanding on Euler.

§1. Numerical errors in “Introduction to Inﬁnite Analysis”
The numerical errors are only near the last digit. The list below shows the diﬀerences
between the numbers calculated in [7] and those in the “Introduction”. For example, the correct
truncated value for log10 2 with seven decimal places is 0.3010299, but the “Introduction” says
it as 0.3010300 .
Volume 1
Chapter 6, p.77
log10 2
= 0.3010299 +1
1/ log10 2= 3.3219280 −3
Chapter 7
P1
l7
l8
l9

log n
1.9459101490553133051053527 +1112
2.0794415416798359282516963+1
2.1972245773362193827904904+1
1

Chapter 8
P2
01
03
05
07
09
11
13
15
17
19
21
23
25
27
29

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

P3
05
07
09
11
13

Chapter 112(2n+1 − 1)π n |Bn+1 | 4−(n + 1)!
π+0.0018424752035+0.0001975800715+0.0000216977373+0.0000024011369+0.0000002664133

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+0.0000000000000000000000010167−0.0000000000000000000000000028ksk = (−1) 2

ck = (−1) 2

−0
+2
−6
+7
−3
+5
−6
+8
−6
+4
−4
+2
−1
+0
−1

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

2π n |Bn+1 |2n−1 π(n + 1)!
−0.0065510747882−0.0003450292553−0.0000202791060−0.0000012366527−0.0000000764958
1

−1
−1
−128
+1
−1

(

03
05
07
09
11

−

−1
−1

)
1
1
PA ζ(k) 1 − k
PB ζ(k) k
2
2
E = 1.00001704136304482548818 +1998 ε = 0.00097753376477325984896+2
F = 1.00000188584858311957590
ξ = 0.00024420070472492872273+1
H = 1.00000002323715737915670
θ = 0.00001525902225127271503 −1526
λ = 0.00000023841863595259255 −101
L = 1.00000000003186677514044
Q = 1.00000000000000053965957
π = 0.00000000023283064370808−1
W = 1.00000000000000000000913
ϕ = 0.00000000000022737367545−1
ω = 0.00000000000000355271368
−1

2

−0
+4
−4
+8
−4
+4
−5
+2
−6
+3
−5
+3
−3
−1
−2
+2

Chpater 15

∑

1
pk

PC

υ(k) =

02
04
06
08
10
12
20
30

0.452247420041065 +157
0.076993139764246+60.017070086850637+20.004061405366518−30.000993603574437 −804
0.000246026470035−20.000000953961124−10.000000000931326−3

p:primeVolume 2
Chapter
21 (∼ cos log 2)
√
2
log
2
= 0.4257207
+67
√ 10
2
2 √
= 2.665144+422
10= 25.954553 +1317

Chapter 22 (sin 1 ∼)cos log 2= 0.76923890136397 −989
sin 1
= 0.84147098480789 −275
cos 1
= 0.54030230586813 −2472

s + cot s−0.17200818−0.09062597−0.05892836−0.04258548

−1
−1
−2
−5

In particular, the rate of errors in the lists of sine and cosine is high. We point out thefollowing abnormalities in [9].
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.
6. If we make the errors correspond to musical scales, we can obtainan extremely skillfully composed score.
Naturally, if we can explain the reason of errors by a simple algorithm, we do not need toconsider hypothesis such as Euler’s intentions. However, more than three years have passed
since we presented this issue, and no such explanation has been obtained.
Even more troubling, a list on the next page that is almost identical to the list in P2 waspublished in E128 (written in 1739, published in 1750), and the errors in this list must also
be considered. Most of the numbers are the same, but the last digit of the sine coeﬃcient haschanged from 825014006 to 735005005. None of the numbers that changed have been corrected,
and the number of errors remains at 28.
In the end, it must be said that it is probably impossible to systematically explain the errors
in these two lists from a single algorithm.

3

Maclaurin expansion of the sine of E128

§2. Hints in “Introduction to Inﬁnite Analysis”, “Letters to a Princess”, and“Beautiful Relationships”
Against the hypothesis, there should be a reasonable objection that it is diﬃcult to express
concrete intentions using abstract numbers. This is because numbers can represent a variety ofobjects, allowing for a variety of interpretations. Therefore, if they are fair riddles, there should
be hints that direct the interpretations of the numbers.
In the following, we list main ﬁgures and phrases which we consider to be hints. Due tospace limitations, please see [7] and [8] for details. Furthermore, we would like to apologize tothe problemer and potential solvers for the unscrupulous act of specifying hints on problems.

“Introduction to Inﬁnite Analysis” (written in 1745, published in 1748)
Diagram at the beginning of Volume 1

4

Fig.1

Fig.2

Fig.3

Fig.4
5

Examples from Volume 1, Chapter 6
Example B log10 5 (the numbers on the right are the correct values)
A = 1.000000
B = 10.00000
C = 3.162277
D = 5.623413
E = 4.216964
F = 4.869674
G = 5.232991
H = 5.048065
I = 4.958069
K = 5.002865
L = 4.980416
M = 4.991627
N = 4.997242
O = 5.000052
P = 4.998647
Q = 4.999350
R = 4.999701
S = 4.999876
T = 4.999963
V = 5.000008
W = 4.999984
X = 4.999997
Y = 5.000003
Z = 5.000000

5
5

8
4

3
3

7
5
9
7
8

lA = 0.0000000
lB = 1.0000000
lC = 0.5000000
lD = 0.7500000
lE = 0.6250000
lF = 0.6875000
lG = 0.7187500
lH = 0.7031250
lI = 0.6953125
lK = 0.6992187
lL = 0.6972656
lM = 0.6982421
lN = 0.6987304
lO = 0.6989745
lP = 0.6989525
lQ = 0.6989135
lR = 0.6989440
lS = 0.6989592
lT = 0.6989668
lV = 0.6989707
lW = 0.6989687
lX = 0.6989697
lY = 0.6989702
lZ = 0.6989700

6

3
9
8
8
3

√
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 = XY

7

Example C1 Find the value of 2 12 .
Example C2 When the population of a certain area increases by 1/30 every year, assumingthat 100,000 people initially lived there, ﬁnd the population after 100 years.
Example C3 What rate does the population increase each year, assuming that the number ofpeople was 6 after the ﬂood, and that 200 years later the population reaches 1,000,000?
Example D1 How many years will it take for the population to increase 10 times, assumingthat the population increases by 1/100?
Volume 2 Chapter 22 Problems (Problema) and Solutions (Solutio)

Fig.5
6

Fig.6“Letters to a Princess”(written in 1760-62, published in 1768-70)
First letter“It is all the work of the Almighty that is vast and boundless”.
4th letter

39th letter

59th letter7

“Beautiful Relations”(written in 1749, published in 1768)§3. Suggested solutions to “problems”
First, we would like to note that the parts that are interpreted as problems, hints, andsolutions have little impact on the main theory of each work. In other words, even if theapproximations, examples, and diagrams are slightly wrong or strange, they probably will notcause any real harm to anyone.
Hints for “Introduction to Inﬁnite”
Fig.1 shows “Algebra (algebraic calculations)”, “Analysis (sine and diﬀerential calculations)”, “Geometry (determined trees and protractors)”, and “Entry to inﬁnitesimal analysis”
is depicted. This is clearly stated in the introduction as a goal of “Introduction to Inﬁnite
Analysis”.
In Fig.2, “music (harp, trumpet)” and “geometry (protractor, compass, globe)” are depicted.
Sound and light have been used since ancient times as a means of communication between distantplaces. It is thought to be a drawing related to Athena, the goddess of wisdom, art, crafts, andstrategy in Greek mythology, and was used in Bousquet’s publications.
In Fig.3, “anatomy (birds, mammals, humans)”, “geometry (compass, diagram)”, and “astronomy (telescope)” are depicted. A person can be seen writing, but the content is unknown.
Anatomy seems a little abrupt here because the relationship with mathematics is not clear. On
the other hand, in the Pythagorean school of ancient Greece, the etymology of the word Mathematics, “mathematica” (meaning subjects to be learned), was the four branches of “arithmetic,
music, geometry, and astronomy”, so there is no sense of discord in their appearance.
In Fig.4, P stands for “pigeon and olive”, S stands for “a small animal like a ﬂy or bee”, andQ stands for “snake”. P reminds us of the dove that brought the olives to Noah’s ark in Genesis
in the Old Testament, and Q reminds us of Moses transforming into a serpent in the book of
Exodus. Because of this association, it would be natural to think of S’s small animal as a “bee”.
This is because when God gave Moses a serpent, he searched for the problem of where the dovebrought the olives, and in the end he found Jerusalem, the promised land ﬂowing with milk andhoney” was the solution . Recognizing that this type of problem is found in “Introduction toInﬁnite Analysis” will make it easier to understand Euler’s “problems”.
If we read the last number in the list of numbers on the left in Example B , reading it up anddown alternately, it becomes (73-37,44,1)-(59,48,5)-(67,36,2)-(157,2), if we read the last number
from the bottom of the number list on the right, it will be 20-777-028-555-7614. Although it isa meaningless mistake at this point, it is presumed that all solutions will be checked in the end.
7
2 12 in Example C1 is a famous number as the frequency ratio of a perfect ﬁfth in equal
temperament in “music”. The ﬂood in Example C3 is Noah’s ﬂood in the book of Genesis in the
Old Testament, and the six people are presumably Noah’s sons and their wives. This assumption
is not diﬃcult if we realize that the letter pictures of P, S, and Q are quotations from the Old
Testament. In the calculation of example C2 , the number 31/30 = 1.033333 · · · appears, andin the calculation of example D1, the number 101/100 = 1.01 appears.
Fig. 5 shows two of the diagrams for problems in Chapter 22 of Volume 2 where D is not
written for some reason. There are a total of 9 problems to ﬁnd angles, and QEI and QEF
appear at the end of the solution, but please note that the famous QED does not appear. The
number of vertices of these two graphs is 5 . Then, while the dual graph in Fig.112 remains
unchanged, the dual graph in Fig.115 changes to the seven bridges of Königsberg .
As shown in Fig.6, solutions S1 and S6 approximate approximately 42 degrees, S2 approximates 54 degrees, S3 and S7 approximate 67 degrees, S4 approximates 132 degrees, S5 approximates 149 degrees, and S8 approximates 84 degrees, and the ﬁrst solution for S9 is written as8

90 degrees - 90 degrees. Here, note that 90 degrees= 1.57079 · · · .
Draft solutions of “problems” in “Introduction to Inﬁnite Analysis”
P1: The ﬁrst two capital letters in the sentence that caused P1’s error are “PS”. Since“music” has already appeared in example C1, and “Old Testament” has appeared in example
C3, Psalms comes to mind ([7] Chapters 4 and 10 ). Psalm 111, verse 2 , says, “Great is thework of God; everyone who loves it seeks it.” Euler was a Protestant and an explorer of theworld.
P2: Since psalms are songs, there should be a musical scores. By correlating the error in
P2 to the scale, a song with 8686 common meters is composed, and the “music” shown in the
opening diagram appears. (For sheet music, see [8] Appendix C or [9] . The basic beat is
42+6+6=54 in “Hymn” (The Board of Publications The United Church of Christ in Japan) 10,
11 )P3: Using the “geometry” shown in the opening diagram as a hint, the seven bridges of theKönigsberg come to mind from the positions of the seven errors in P3 ([7] Chapter 6 ). Notethat 128 = 27 .
PA, PB, PC: When the large errors in PA and PC are factorized, the smallest irregularprime number 37 and the third 67 appear from PA:1998 = 54 × 37, PC:804 = 12 × 67. An
irregular prime number is a prime number that appears in the numerator of the rational part ofthe zeta value. The combined errors of PA and PB are 472 = 8 × 59, and the second irregularprime number 59 also appears. (For the reason for addition, see Part 2 , Chapters 5 and 11 of[8] ) Below are the irregular prime numbers and exponents up to the 9th.
(37, 32) (59, 44) (67, 58) (101, 68) (103, 24) (131, 22) (149, 130) (157, 62) (157, 110) (233, 84)

Irregular primes appear in the numerator of the zeta value with a period of p − 1. If we takethis periodicity into account and plot pairs of irregular primes and exponents clockwise, we get
the diagram above. This diagram reminds us of the solar system, and the zeta value gives riseto the term “astronomy”. The relationship between zeta and the solar system is a unique idea,
but since zeta is represented by the symbols of the sun and moon in “Beautiful Relationships”,
it is easier to make the connection by referring to this essay.
Conﬁrmation of solutions in “Introduction to Inﬁnite Analysis” Volume 2
PC, PB, PA：Conﬁrmation of the musical score in which three irregular prime numbers from67，42, 1317 are expressed. ([8] Appendix C)
P3, P2, P1：Check the sum of errors from “−989, −275, −2472”. Check the solution basedon the error. ([7] Chapter 8 )Graph of P3：Conﬁrmed from the two graphs in Fig. 5 and their dual graph. ([7] Chapter 6)

9Irregular primes：C2≈1.03, D1=1.01，S3≈ S7≈67，S4≈131，S5≈149，S5+S8≈233，S9=
1.57 − 1.57．
S1≈S6≈42∼37, the diﬀrence is +5. S2≈54∼59, the diﬀerence is −5.
Conﬁrmation of solutions in “Letters to a Princess”
P1: Conﬁrmation of the ﬁrst half of PS.111-2 from the last words of the ﬁrst letter. The second
half is the 20th letter.
P2: Check the music score and psalm from the dots in the 4th letter.(42+11+6=59 beats, 11-12,etc.)
P3: Divide the graph on the left where D is not written in the 39th letter vertically into Fig.
112, and divide it horizontally into Fig.115, Conﬁrmation of the seven bridges in Königsberg
from the graph on the right.
PA, PB, PC: You can conﬁrm the diagram on the previous page by solar system diagram in
the 59th letter. If the ratios of the widths of orbits are 16 : 22 : 37 : 67 : 103 : 101 : 131 :
149, the planets and irregular prime numbers almost correspond (see [7] p.183 ). The symbols
of Jupiter and Saturn are 4 and 5, which correspond to 131 and 149, which correspond to S4and S5 in “Introduction”.

10The above suggested solutions can be summarized as follows, using “Letters to the Princess”
as a reference.
7 Lists functions academic areasBlog10 5Arithmetic
log
Theology
P1
P2
sin, cosMusic
P3tan, cot GeometryPAζA
Astronomy
ζB
Astronomy
PBPC
ζC
Astronomy

physics
number
space
sound
light
gravitygravitygravitycontents
all solutions
lyric for a psalm
score for a psalm
graph for a psalm
37
59
67

§4. Intent of “problems”
Euler’s claims are listed below.
· About the fountain of discovery in mathematics (from the introduction of “Introduction”)“Indeed, I would like to state without hesitation that not only are there clearly new things
contained in this book, but also springs have been revealed, and from it there are still many
outstanding things.”· On the further development of mathematics (from “The Sublime Higher Mathematics” addressed to Frederick the Great)“The perceived usefulness of elementary mathematics does not disappear in higher mathematics, but on the contrary, it increases relatively. And mathematics have not yet developed tothe stage where very general practical application demands.
· About the joy of solving riddles ( from “Letters to a Princess” No. 8 )“If you guess the meaning of the mystery and discover that it is perfectly expressed in the
problem of the mystery, you feel the great joy of discovery.”· On the importance of subtle diﬀerences ( from “Letters to a Princess” No. 118 )“Subtle diﬀerences in reports of the same event prove the truth, rather than they weakenit.”· About beings that cannot be reached by arguments (from “Defense against free thinkers”)“When free thinkers try to completely negate the Bible because of contradictions in it (which
are obvious to them), they are acting in a most unfair and irresponsible manner. We must admitthat they have not been able to solve the problems of geometry, the existence or motion of matter,and that no one has denied the truth or reality of these things. ... There are at least equally
great diﬃculties in the doctrine of revelation, which cannot be reached by argument.”
The second volume of “Letters to a Princess” includes many more arguments of Euler.
Judging comprehensively from those assertions and the repeated strange errors and descriptions,
we deduce that Euler’s intentions of the “problems” are as follows.
1. “Introduction” secretly describes the vast world in the subtle mathematics.
2. “Letter” secretly describes the subtle mathematics in the vast world.
3. “Beautiful Relationships” beautifully connects the vast world and the subtle mathematics.

11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, Series 1, Volume 8, 9”, (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] H. Takahashi, “Inﬁnite Eulerian Analysis”, (Gendai Sugaku)[8] H. Takahashi, “The Origin of Inﬁnite Analysis”, (Gendai Sugaku)[9] H. Takahashi, Errors in “Introduction to Inﬁnite Analysis”, (RIMS Kokyuroku 1583)

2010, November 23th (Talk at RIMS in Kyoto on 2010, August 25th).
Translation of an essay in RIMS Kôkyûroku No. 1739.

12