Karl Alois Kneller.

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that a number of atoms by falling together of their own
accord could make a crystal, a sprig of moss, a microbe,
a living animal? People thought that given millions of
years, these might come to pass, but they could not think
that a million of millions of millions of years could give
them unaided a beautiful world like ours. They had a
spiritual influence, and in science a knowledge that there
was that influence, in the world around them. He admired
the healthy, breezy atmosphere of free thought in Professor
Henslow's Lecture. Let no one, he urged, be afraid of
true freedom. They could be free in their thought, in their
criticisms, and with freedom of thought they were bound
to come to the conclusion that science was not antagonistic
to religion, but a help for religion."

In a letter to the "Times" two days later Lord Kelvin
made certain corrections touching upon the differences
between the possible origin of crystals and that of living
organisms.



38 I. A FUNDAMENTAL LAW : THE CONSERVATION OF ENERGY'.

"I desired to point out that while 'fortuitous concourse
of atoms' is not an inappropriate description of the forma-
tion of a crystal, it is utterly absurd in respect to the coming
into existence, or the growth, or the continuation of the
molecular combinations presented in the bodies of living
things. Here scientific thought is compelled to accept the
idea of creative power. Forty years ago I asked Liebig,
walking somewhere in the country, if he believed that the
grass and flowers which we saw around us grew by mere
chemical forces. He answered, 'No, no more than I believe
that a book of botany describing them could grow by mere
chemical forces'. Every action of human free will is a
miracle to physical and chemical and mathematical science."

In printing, in the "Nineteenth Century" for June 1903,
his own version of his speech, Lord Kelvin concludes with
the exhortation :

"Do not be afraid of being free-thinkers! If you think
strongly enough, you will be forced by science to the be-
lief in God, which is the foundation of all religion. You
will find science not antagonistic but helpful to religion." *

Lord Kelvin's utterances brought down on him a
crowd of objections; most of them however were com-
posed in the heat of anger and are beside the point
in question 2 .

As Haeckel appeals to Helmholtz(f 1894), a pas-
sage from a letter of the latter may fitly be quoted.

"We Mathematical Scientists", writes Helmholtz to his
father, "are disciplined to the most anxious precision in the



1 Reprinted from the Nineteenth Century in The American Catholic
Quarterly Review XXVIII, Philadelphia 1903, 603. Cf. Stimmen aus
Maria-Laach LXV 487.

2 The objections and answers are printed in the Weekly Edition
of the Times from the 8 th to the 15 th May (Supp. in). Cf. A Review
ib. 15 th May p. 313. "The importance of Lord Kelvin's opinion",
concludes the latter article, "is rather enhanced than diminished by
the hostile irrelevancy which some of his critics have displayed."



HELMHOLTZ. 39

establishment of facts and inferences, and compress our
wandering conjectures into very short and meagre hypotheses,
by the aid of which we seek to forecast a goal as yet un-
explored, so that we are, perhaps, too much afraid of that
bolder employment of scientific data which in other circum-
stances may be justifiable.

Your letter seems to me to intimate a certain suspicion
that I am a subscriber to the paltry tirades of Vogt and
Moleschott. Not in the least. I must also enter an energetic
protest against your treating these two gentlemen as re-
presentatives of scientific research. Neither has up to the
present shown by original and specialised investigation that
he has acquired that respect for facts and that prudence
in reaching conclusions which men learn in the school of
physical research. A prudent investigator knows very well
that the fact of his having penetrated a little way into the
intricate process of nature gives him no more right, not a
scintilla more, than any other man to pronounce dogmatic-
ally on the nature of the soul. To my mind, too, you are
not right in designating the majority of prudent scientists
as enemies of Philosophy. Indifferent the greater part un-
doubtedly are, a state of things for which the blame rests
on the extravagant speculations of Hegel and Schelling,
two writers who have, I grant you, been taken to represent
all philosophy. . . ." J

Thus wrote Helmholtz on March 4 th 1857, ten years
after the publication of his essay "On the Conservation
of Force".

On religious questions Helmholtz never made any
open declaration. His biographer describes him as "in
mind and conviction a religious man, in the noblest
sense of the word, although not in the orthodox sense
of membership of a Church" 2 , and says that the famous

1 L. Konigsberger, Hermann v. Helmholtz I, Braunschweig
1902, 291 f.

2 Ib. II (1903) 75. E. Dennert (Die Religion der Natur-
forscher, Berlin 1901, 34) says of Von Helmholtz: "Ich erfuhr aber



40 I. A FUNDAMENTAL LAW. THE CONSERVATION OF ENERGY.

scientist always regarded the philosophical views of his
father Ferdinand Helmholtz "not only with pious inter-
est, but also with the highest appreciation of their
scientific value, and, as appears from later utterances,
with hearty agreement" 1 . Ferdinand Helmholtz was a
follower of the younger Fichte, and a convinced theist
and spiritualist 2 .

Mention may be made here also of the Physicist Kronig
(t 1879) "who has won for himself an honoured name in
the field of research by his investigations into the Theory
of Gases. Clausius cites him as his predecessor in this
province, and corroborates his conclusions". He published
in 1874 a book on "The existence of God and the hap-
piness of man." He denies that through "blind chance,
however long a period of time we assume, the atoms could
by their own forces come together and form living cells. . . .
Without the guidance of a purposive intelligence, organisms
could never have come into existence" 3 . It is interesting
to note e. g. what he has to say from the standpoint of
the mathematical Theory of Probability, against those who
find no difficulty in the hypothesis that in the course of
endless time and after endless attempts the most complex for-
mations may have come fortuitously into existence: "If every
year for a million years, a million men were born, each of
whom lived to the age of ten thousand years and every
minute of his life made 20 throws with 30 dice, the ma-
thematical probability is that amongst all these throws one
of thirty aces would not have even once occurred."



aus bester Quelle, dass er Gottesdienst. ja sogar das Abendmahl bin
und wieder besuchte."

1 Konigsberger, Hermann v. Helmholtz I 333.

2 J. R(einke?) writes about an utterance of Von Helmholtz con-
cerning the petty view that we can clear up the fundamental questions
of life by scientific investigation. Deutsche Rundschau LXXXI, Berlin
1894, 131.

3 Cf. J. Reinke, Die Welt als Tat 3 , Berlin 1903, 1214.



II. MATHEMATICS. 4!

We here conclude our survey of the great masters
to whom we owe the largest and most fruitful scientific
conception of modern times. And although certain people
are good enough to appeal to their authority in proof
of the 'monistic' character of the 'Law of Substance',
to whom are we to appeal for the deepest interpretation
of their great discoveries, to the discoverers themselves,
or to those who profess to be their disciples?

II. MATHEMATICS.

Mathematics is not itself a branch of Physical Science,
but in many instances it is by means of Mathematics
that our knowledge of nature is first raised to the status
of genuine science. Astronomy and Physics are scien-
tific precisely in so far as they deal with number and
measure, in so far as they enter into alliance with Mathe-
matics, and absorb its spirit and method.

In a work like this, then, which is concerned with
the scientific conception of life, and with the essential
spirit of Natural Science, some attention must be ac-
corded to number and measure. We must not fail to
examine the attitude of the leaders of Mathematics
towards Christianity and religious belief in general, and
to enquire whether the temper of Mathematics is recon-
cilable with that of religion.

To show how little we need evade the question we
appeal to the most eminent authorities on the subject,
the most recent writers on the History of Mathematics.
"Like most other Mathematicians", says M. Cantor 1 ,
"Euler was deeply religious without any trace of bigotry.



Allg. deutsche Biographic .VI 427.



42 II. MATHEMATICS.

He was in the habit of conducting the devotions of
his family, and one of his few polemical works was a
"Defence of Revelation against the Objections of the
Freethinkers" the publication of which in Berlin in 1747
in the immediate neighbourhood of the Court of Fre-
derick the Great, indicates a moral courage which lifted
him far above the assaults of mere scoffers."

Leonhard Euler (born at Basle in 1707, died at
St. Petersburg in 1783) does not come within the
period with which this book is concerned. But even in
the nineteenth century the praise which Cantor bestows
on "most great mathematicians" is justified by many
illustrious examples. At the beginning of this century
the leaders of this department were Gauss in Germany,
and Cauchy in France. In the latter half of the cen-
tury two of the highest places were occupied by Her-
mite in France, and Riemann in Germany. If we are
able to show that the relations to Christianity of these
four were the reverse of hostile, we shall have refuted
those writers who condemn religious belief as a kind
of mysticism which cannot co-exist with the habit of
exact thought inculcated by mathematics.

In the history of Astronomy the passage from the
1 8 th to the iQ th century is marked by an event of the
highest importance, at once a great discovery and a
great opportunity. On New Year's Night 1801, at
Palermo, Piazzi discovered the first of what we now
know to be a number of small planets between Mars
and Jupiter. But before the observer had ascertain-
ed the positions of the planet necessary, by the me-
thods of computation then practised, to determine ac-
curately its path, it approached so near the sun that it
was lost from sight in the sun's radiance. Thus the



LEONHARD EULER. KARL FREDERICK GAUSS. 43

planet had no sooner been discovered than it was lost
again, for there was no hope of locating it unless
astronomers knew in what quarter of the heavens to
search for it, and this could not be known without a
knowledge of its path. At this juncture there came to
the aid of Astronomy a young mathematician of only
four and twenty : he brought a completely new method
by which, from the meagre observations of Piazzi, the
path of the vanished planet could be deduced. In the
spot indicated by him Ceres was re-discovered by Olbers
on Jan. I st 1802.

The youthful Scientist, who thus leaped into a
European reputation, was Karl Frederick Gauss, one
of the foremost mathematicians of all time. It sounds
incredible, but it is a well established fact that when
a child of three in the workshop of his father, who was
a simple artisan, he was able to detect any mistake
made in the reckoning of accounts. When at nine years
of age he was sent to school, the teacher of Arithmetic
set as the first exercise to the class the addition of a
long row of figures, each successive line of which ex-
ceeded the preceding one by a constant sum. If the
first line of such a row be added to the last, the second
to the second-last, and so on, the sum will always be
the same; and there is consequently no need to worry
oneself with a tedious addition, since the whole can be
reduced to a simple exercise in multiplication. Young
Gauss perceived this relation at the first glance, and
while his classfellows added and added and in the end
succeeded for the most part in adding wrongly, he wrote
the answer on his slate and waited patiently for the
termination of the affair. . . . This performance, and
similar tokens of extraordinary ability decided his future



44 II. MATHEMATICS.

career. The schoolmaster pressed Gauss's father to let
his son continue his studies, and in the end managed
to persuade him. He had the little spinning-wheel, on
which Karl worked every evening, broken up for fire-
wood, and the future mathematician entered on his career.

The later development of this great genius was in
keeping with his first promise. His dissertation for the
Doctorate on the fundamental theorem of the Theory
of Equations, presented in his twenty-second year, was in
itself a scientific feat. In the first part he showed that
this theorem had not hitherto been really demonstrated,
in the second he himself presented an unimpeachable
demonstration. Two years later, appeared his epoch-
making "Arithmetical Researches", and the discovery
already mentioned. These were followed by a long
series of other works of equal brilliancy.

In his student-days at Gottingen, Gauss was on terms
of particular intimacy with a young Hungarian, Wolf-
gang Bolyai, who was tutor to nobleman, and in that
capacity had come to the German University town.
Bolyai was also a student of Mathematics. The letters
which passed between the friends after their separation
reveal amongst other things the attitude which Gauss
maintained towards the higher problems of life. Thus,
e. g. he concludes a letter to Bolyai from Brunswick,
dated Dec. 3 rd 1802:

"Now good-bye, my dear fellow. May the dream which
we call life be for you a happy dream, a foretaste of that
true life which we shall inherit in our real home, when the
awakened spirit shall labour no longer under the grievous
bondage of the flesh, the fetters of space, the whips of earth-
ly pain, and the sting of our paltry needs and desires. Let
us carry our burdens to the end, stoutly and uncomplainingly,
never losing sight of that higher goal. Glad then shall we



KARL FREDERICK GAUSS. 45

be to lay down our weary lives, and to see the dropping of
the curtain." 1

When Bolyai announces the birth of a son, Gauss
ends his letter of congratulation (June 20*'' 1803) with
the words 2 :

"You hold now in your hand the first links of a chain
which stretches into eternity, of an everlasting life. A weighty
and earnest charge, but a glorious one. May your son one
day bless you as the first author of his happiness!"

When Councilor Eschenburg lost his wife in December
1798, Gauss wrote to Bolyai (Jan. 9 th 1799):

"It is beyond doubt that the happiness which love can
bestow on its chosen souls is the highest that can fall to
mortal's lot. But when I imagine myself in the place of
the man who, after twenty happy years, now in one moment
loses his all, I am moved almost to say that he is the
wretchedest of mortals, and that it is better never to have
known such happy days. So it is on this miserable earth:
'the purest joy finds its grave in the abyss of time'. What
are we without the hope of a better future? Let us keep
a free heart as long as we can, and place our happiness in
ourselves." 3

In the early period of their separation the letters
which passed between Gauss and Bolyai were naturally
numerous. They had arranged a fixed hour at which
to think of one another:

"Your letter", writes Gauss, in 1798, "reached me
on the evening of the last day of last month, just when



1 Briefwechsel zwischen Karl Friedrich Gauss und Wolfgang Bolyai.
Herausgeg. von Fr. Schmidt und Paul S tack el , Leipzig 1899, 47.

2 Schmidt u. Stack el ante 54.

3 Ib. 1 6. L u d w. H a n s e 1 m a n n , Karl Friedrich Gauss. Zwolf
Kapitel aus seinem Leben, Leipzig 1878, 92 93.



46 II. MATHEMATICS.

I had settled down to celebrate the anniversary of our
friendship. There I sat in my armchair with a pipe
filled for you, and just in the middle of my dream
that you were sitting opposite in your black jacket
and black cap and talking to me about old times, came
your letter with its questions to show that you were
actually at that moment thinking of me, and that my
dream was no dream." 1

As time went on, the two scientists, naturally enough,
had not time for such poetical exercises of friendship.
Their correspondence became rarer and at last al-
together ceased.

Many years went by and their lives led them very
far apart. The spruce Hungarian, "with his black jacket
and black cap", had crusted into a peevish professor.
He held a post in a college in Siebenburg, and had
become a notable old crank, at feud with his wife
and mother-in-law, and discontented with his son, for
whose insubordination the educational theories of the
pedagogue papa were not perhaps blameless.

"His cynicism", writes in 1849 a visitor of his, "passes
into a sort of noble communism. Thus, for example, he
keeps a servant but lets him do what he likes, for he cleans
his own boots (even prepares the grease for them), makes
his own bed, attends to his own cellar, and as a rule to
his own meals too. . . . His room resembles in dirt and
disorder the tub of Diogenes, with the sole addition of a
heap of books and some blackboards. . . ." 2

But the fame of Gauss's discoveries and researches
had penetrated even to Siebenburg, and Bolyai was



1 Schmidt u. Stackel ante 10.

2 Ib. 132.



KARL FREDERICK GAUSS. 47

moved to write once more to his old comrade. He
wrote, sending news of his own fortunes, and extolling
the happiness of a friend to whose lot had fallen all
that the world could give of glory, all that intellect
could give of knowledge, joy or consolation. This
outpouring of his heart affected Gauss deeply, and in
response he opened his inmost heart to the friend of
his youth. He writes April 2O th 1848:

"Melancholy were my feelings on receiving your letter
of Jan. 1 8. It was as the voice of a ghost crying out of
the long vanished past, or at least it cast my thoughts back
to a time since which so many heavy years have passed
over our heads. It is true that my life has been crowned
with much of what the world thinks enviable. But believe
me, my dear Bolyai, the bitterness of life, or at least of
mine, which runs through it like a strand of red, and be-
comes less and less endurable as I grow older, is not com-
pensated in the hundredth part by the joy of life. I will
freely admit that these burdens, which to me have been so
grievous, would have been lighter to many another; but
our temperament is part of ourselves, given to us by the
Creator with our very existence, and we have very little
power to change it. I find, on the other hand, in this very
consciousness of the vanity of life, which nearly all men
must confess to as they draw near the end, my strongest
assurance of the approach of a more beautiful meta-
morphosis. In this, my dear friend, let us find comfort, and
endeavour to call up calmness to bear life out to the end.
Fortem facit vicina libertas senem, says Seneca." *



1 W. Sartorius v. Waltershausen, Gauss zum Gedachtnis,
Leipzig 1856, 103. Grunert, who quotes this passage in his
Archives of Mathematik und Physik (XXVI, Greifswald 1856, Liter.
Bericht civ) remarks : "Absichtlich haben wir die religiose Seite des
grossten Mathematikers und Naturforschers der neuesten Zeit hier be-
stimmter hervorgehoben und stellen sie gegenuber den namentlich fur
die Jugend leicht so verderblich werden konnenden Ansichten einer



48 II. MATHEMATICS.

"Vanity of vanities, and all is vanity", one is driven
to cry out over such a letter as that. So a Gauss,
too, in the full possession of all that Science can pro-
cure of intellectual joy, honour and distinction comes,
at the end of his days, to declare that there is no peace
of spirit in it all, and that life is a riddle and a torment
unless it be completed by a happy eternity. And with
Gauss these are not mere passing moods but convictions
which dwelt with him all his life, and formed the basis
of his moral nature.

"The indestructible idea of personal survival after death",
says a biographer, "the steadfast belief which he had in a
Supreme Ruler, a just, eternal, omniscient, omnipotent God,
formed the foundation of his religious life, and in unison
with his matchless scientific achievements formed a perfect
harmony."

He once expressed himself to this effect: 'There is in
this world a joy of the intellect, which finds satisfaction in
science, and a joy of the heart, which manifests itself above
all in the aid men give one another against the troubles
and trials of life. But for the Supreme Being to have
created existences, and stationed them in various spheres
in order to taste these joys for some 80 or 90 years - that
were surely a miserable plan.' . . . 'Whether the soul were
to live for 80 years or for 80 million years, if it were
doomed in the end to perish, such an existence would only
be a respite. In the end it would drop out of being. We
are thus impelled to the conclusion to which so many
things point, although they do not amount to a coercive
scientific proof, that besides this material world there exists
another purely spiritual order of things, with activities as
various, as the present, and that this world of spirit we
shall one day inherit.' This divine wisdom was the food



gewissen Klasse heutiger Naturforscher", that is to say of those, u die
so gern das Gottliche und Geistige in den Staub ziehen und lediglich
an die Materie ketten mochten."



JOHANN FREDERICK PFAFF. 49

and discipline of his soul up to that last silent midnight in
which his eyes closed for ever."

Gauss gives frequent expression to his conviction
of the immortality of the soul. Thus, e. g. he writes
to Olbers 1 :

"I always return to the conviction that the necessary
truth of Geometry cannot be demonstrated, at least by
human reason 'and for* human reason. Perhaps* we shall
in another life attain to a deeper insight into the essence
of space than is possible to us here. Till then, we must
not rank Geometry with Arithmetic, which is of a purely
a priori character, but rather with Mechanics."

Laplace is reported to have, on one occasion, de-
scribed Johann Frederick Pfaff of Berlin (f 1825) as
the greatest mathematician of Germany. When he was
reminded of Gauss, he extricated himself from the dif-
ficulty with the witty remark: "PfafTis the greatest mathe-
matician in Germany, Gauss the greatest in Europe." 3
Certain it is that Laplace held Pfaff in high esteem 4 , and
that the researches of the latter contributed greatly to



1 Karl Friedrich Gauss, Werke VIII, Gottingen 1900, 177: Gauss
an Olbers, Gottingen, 28. April 1817. C. Vogt made a violent
attack (Kohlerglaube und Wissenschaft 4 , Giessen 1856, xxxiv) on
Rud. Wagner, because the latter had maintained (Allg. Ztg. 1855,
1050) that Gauss entertained friendly feelings towards Christianity.
Vogt adduces in proof of the contrary, that Gauss wrote under his
portrait: "Thou Nature art my goddess, to thy laws my services are
bound!" To write a word to refute him would be mere waste
of ink.

2 Doubly underlined by Gauss.

3 Sammlung von Briefen , gewechselt zwischen Joh. Friedr. Pfaff
und Herzog Karl von Witrttemberg, F. Bouterwek, A. v. Humboldt,
A. G. Kastner u. a. Herausg. von Karl Pfaff, Leipzig 1853,
Vorwort vn.

* Ib. 1 60.
Kneller, Christianity. A



50 II. MATHEMATICS.

the advance of his science. This "greatest mathematician
in Germany" stood on very friendly relations with re-
ligion !. In his early letters to a "jacobinically" dis-
posed brother he admonishes the latter: u Be a respecter
of religion", "never disregard the standpoint of religion";
and in his maturer years his friends celebrate his strong
"religious sense" 2 .

Two years after Gauss, there died in Paris another
Mathematician who had rendered to science services no
less worthy, Augustin Louis Cauchy 3 . Born in 1789
at Paris, he was, at the conclusion of a brilliant course
of studies, attached as engineer to the immense opera-
tions by which Napoleon undertook to elevate Cherbourg
to a first-rate naval harbour confronting England. He
was soon compelled by ill-health to resign this position ;
and he thenceforth devoted himself exclusively to
scientific research, and published in rapid succession a
number of brilliant works. That which attracted most
attention was his demonstration of a Theory of Fermat,
at which the most eminent mathematicians, including



1 Ib. Brief vom 4. Jan. 1789, 89 91; cf. 114.

' Ib. Biograph. Einleitung 31.

3 Cf. J. B. B i o t , A.-L. Cauchy. Lettre a M. de Falloux, membre



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