problems can be attacked either inside or outside of the plant.
If the policy of the industrialist is that all problems are to
be investigated only within the establishment, a research
laboratory must be provided for the plant or for the company.
At present, in the United States, probably not more than one
hundred chemical manufacturing establishments have research
laboratories or employ research chemists, although at least
five companies are spending over $100,000 per year in research.
In Germany, and perhaps also in England, such research
laboratories in connection with chemical industries have been
much more common. The great laboratories of the Badische Anilin
und Soda Fabrik and of the Elberfeld Company are striking
examples of the importance attached to such research work in
Germany, and it would be difficult to adduce any stronger
argument in support of its value than the marvelous
achievements of these great firms.
A frequent difficulty encountered in the employment of
researchers or in the establishment of a research laboratory,
is that many manufacturers have been unable to grasp the
importance of such work, or know how to treat the men in charge
so as to secure the best results. The industrialist may not
even fully understand just what is the cause of his
manufacturing losses or to whom to turn for aid. If he
eventually engages a researcher, he is sometimes likely to
regard him as a sort of master of mysteries who should be able
to accomplish wonders, and, if he can not see definite results
in the course of a few months, is occasionally apt to consider
the investment a bad one and to regard researchers, as a class,
as a useless lot. It has not been unusual for the chemist to be
told to remain in his laboratory, and not to go in or about the
works, and he must also face the natural opposition of workmen
to any innovations, and reckon with the jealousies of foremen
and of various officials.
From the standpoint of the manufacturer, one decided advantage
of the policy of having all problems worked out within the
plant is that the results secured are not divulged, but are
stored away in the laboratory archives and become part of the
assets and working capital of the corporation which has paid
for them; and it is usually not until patent applications are
filed that this knowledge, generally only partially and
imperfectly, becomes publicly known. When it is not deemed
necessary to take out patents, such knowledge is often
permanently buried.
In this matter of the dissemination of knowledge concerning
industrial practice, it must be evident to all that there is
but little cooperation between manufacturers and the
universities. Manufacturers, and especially chemical
manufacturers, have been quite naturally opposed to publishing
any discoveries made in their plants, since "knowledge is
power" in manufacturing as elsewhere, and new knowledge gained
in the laboratories of a company may often very properly be
regarded as among the most valuable assets of the concern. The
universities and the scientific societies, on the other hand,
exist for the diffusion of knowledge, and from their standpoint
the great disadvantage of the above policy is this concealment
of knowledge, for it results in a serious retardation of the
general growth and development of science in its broader
aspects, and renders it much more difficult for the
universities to train men properly for such industries, since
all the text-books and general knowledge available would in all
probability be far behind the actual manufacturing practice.
Fortunately, the policy of industrial secrecy is becoming more
generally regarded in the light of reason, and there is a
growing inclination among manufacturers to disclose the details
of investigations, which, according to tradition, would be
carefully guarded. These manufacturers appreciate the facts
that public interest in chemical achievements is stimulating to
further fruitful research, that helpful suggestions and
information may come from other investigators upon the
publication of any results, and that the exchange of knowledge
prevents many costly repetitions.
INDUSTRIAL FELLOWSHIPS
If the manufacturer elects to refer his problem to the
university or technical school--and because of the facilities
for research to be had in certain institutions, industrialists
are following this plan in constantly increasing numbers--such
reference may take the form of an industrial fellowship and
much has been said and may be said in favor of these
fellowships. They allow the donor to keep secret for three
years the results secured, after which they may be published
with the donor's permission. They also secure to him patent
rights. They give highly specialized training to properly
qualified men, and often secure for them permanent positions
and shares in the profits of their discoveries. It should be
obvious at the outset that a fellowship of this character can
be successful only when there are close confidential relations
obtaining between the manufacturer and the officer in charge of
the research; for no such cooperation can be really effective
unless based upon a thorough mutual familiarity with the
conditions and an abiding faith in the integrity and sincerity
of purpose of each other. It is likely to prove a poor
investment for a manufacturer to seek the aid of an
investigator if he is unwilling to take such expert into his
confidence and to familiarize him with all the local and other
factors which enter into the problem from a manufacturing
standpoint.
THE MELLON INSTITUTE OF INDUSTRIAL RESEARCH[3]
[3] For a detailed description of the Mellon Institute and its
work, see Bacon and Hamor, J. Ind. Eng. Chem., 7 (1915),
326-48.
According to the system of industrial research in operation at
the Mellon Institute of Industrial Research of the University
of Pittsburgh, which is not, in any sense of the word, a
commercial institution, a manufacturer having a problem
requiring solution may become the donor of a fellowship; the
said manufacturer provides the salary of the researcher
selected to conduct the investigation desired, the institute
furnishing such facilities as are necessary for the conduct of
the work.
The money paid in to found a fellowship is paid over by the
institute in salary to the investigator doing the work. In
every case, this researcher is most carefully selected for the
problem in hand. The institute supplies free laboratory space
and the use of all ordinary chemicals and equipment. The
chemist or engineer who is studying the problem works under the
immediate supervision of men who are thoroughly trained and
experienced in conducting industrial research.
At the present time, the Mellon Institute, which, while an
integral part of the University of Pittsburgh, has its own
endowment, is expending over $150,000 annually for salaries and
maintenance. A manufacturer secures for a small
expenditure--just sufficient to pay the salary of the fellow,
as the man engaged on the investigation is called--all the
benefits of an organization of this size, and many have availed
themselves of the advantages, twenty-eight companies
maintaining fellowships at the present time.
Each fellow has the benefit of the institute's very excellent
apparatus, chemical and library equipment--facilities which are
so essential in modern research; and because of these
opportunities and that of being able to pursue post-graduate
work for higher degrees, it has been demonstrated that a higher
type of researcher can be obtained by the institute for a
certain remuneration than can be generally secured by
manufacturers themselves. There is a scarcity of men gifted
with the genius for research, and it requires much experience
in selecting suitable men and in training them to the desirable
degree of efficiency, after having determined the special
qualities required. Important qualifications in industrial
researchers are keenness, inspiration and confidence; these are
often unconsidered by manufacturers, who in endeavoring to
select, say, a research chemist, are likely to regard every
chemist as a qualified scientific scout.
All researches conducted at the Mellon Institute are surrounded
with the necessary secrecy, and any and all discoveries made by
the fellow during the term of his fellowship become the
property of the donor.
When the Mellon Institute moved into its $350,000 home in
February, 1915, the industrial fellowship system in operation
therein passed out of its experimental stage. During the years
of its development no inherent sign of weakness on the part of
any one of its constituent factors appeared; in fact, the
results of the fellowships have been uniformly successful.
While problems have been presented by companies which, upon
preliminary investigation, have proved to be so difficult as to
be practically impossible of solution, there have been so many
other problems confronting these companies that important ones
were found which lent themselves to solution; and often the
companies did not realize, until after investigations were
started, just what the exact nature of their problems was and
just what improvements and savings could be made in their
manufacturing processes.
Fellowships at the Mellon Institute are constantly increasing
in the amounts subscribed by industrialists for their
maintenance and, as well, in their importance. The renewal,
year after year, of such fellowships, as those on baking,
petroleum and ores, goes to show the confidence which
industrialists have in the Mellon Institute. Again, the large
sums of money which are being spent by companies in bringing
small unit plants to develop the processes which have been
worked out in the laboratory, demonstrate that practical
results are being secured.
Where there have been sympathy and hearty cooperation between
the Mellon Institute and the company concerned, the institute
has been able to push through to a successful conclusion large
scale experiments in the factory of the company, which in the
beginning of the fellowship seemed almost impossible: it may be
said that the results of the fellowships at the Mellon
Institute indicate that a form of service to industry has been
established, the possibilities of which no man can say.
A FEW CLASSIC UNKNOWNS IN MATHEMATICS
BY PROFESSOR G. A. MILLER
UNIVERSITY OF ILLINOIS
KING HIERO is said to have remarked, in view of the marvelous
mechanical devices of Archimedes, that he would henceforth
doubt nothing that had been asserted by Archimedes. This spirit
of unbounded confidence in those who have exhibited unusual
mathematical ability is still extant. Even our large city
papers sometimes speak of a mathematical genius who could solve
every mathematical problem that was proposed to him. The
numerous unexpected and far-reaching results contained in the
elementary mathematical text-books, and the ease with which the
skilful mathematics teachers often cleared away what appeared
to be great difficulties to the students have filled many with
a kind of awe for unusual mathematical ability.
In recent years the unbounded confidence in mathematical
results has been somewhat shaken by a wave of mathematical
skepticism which gained momentum through some of the popular
writings of H. Poincare and Bertrand Russell. As instances of
expressions which might at first tend to diminish such
confidence we may refer to Poincare's contention that
geometrical axioms are conventions guided by experimental facts
and limited by the necessity to avoid all contradictions, and
to Russell's statement that "mathematics may be defined as the
subject in which we never know what we are talking about nor
whether what we are saying is true."
The mathematical skepticism which such statements may awaken is
usually mitigated by reflection, since it soon appears that
philosophical difficulties abound in all domains of knowledge,
and that mathematical results continue to inspire relatively
the highest degrees of confidence. The unknowns in mathematics
to which we aim to direct attention here are not of this
philosophical type but relate to questions of the most simple
nature. It is perhaps unfortunate that in the teaching of
elementary mathematics the unknowns receive so little
attention. In fact, it seems to be customary to direct no
attention whatever to the unsolved mathematical difficulties
until the students begin to specialize in mathematics in the
colleges or universities.
One of the earliest opportunities to impress on the student the
fact that mathematical knowledge is very limited in certain
directions presents itself in connection with the study of
prime numbers. Among the small prime numbers there appear many
which differ only by 2. For instance, 3 and 5, 5 and 7, 11 and
13, 17 and 19, 29 and 31, constitute such pairs of prime
numbers. The question arises whether there is a limit to such
pairs of primes, or whether beyond each such pair of prime
numbers there must exist another such pair.
This question can be understood by all and might at first
appear to be easy to answer, yet no one has succeeded up to the
present time in finding which of the two possible answers is
correct. It is interesting to note that in 1911 E. Poincare
transmitted a note written by M. Merlin to the Paris Academy of
Sciences in which a theorem was announced from which its author
deduced that there actually is an infinite number of such prime
number pairs, but this result has not been accepted because no
definite proof of the theorem in question was produced.
Another unanswered question which can be understood by all is
whether every even number is the sum of two prime numbers. It
is very easy to verify that each one of the small even numbers
is the sum of a pair of prime numbers, if we include unity
among the prime numbers; and, in 1742, C. Goldbach expressed
the theorem, without proof, that every possible even number is
actually the sum of at least one pair of prime numbers. Hence
this theorem is known as Goldbach's theorem, but no one has as
yet succeeded in either proving or disproving it.
Although the proof or the disproof of such theorems may not
appear to be of great consequence, yet the interdependence of
mathematical theorems is most marvelous, and the mathematical
investigator is attracted by such difficulties of long
standing. These particular difficulties are mentioned here
mainly because they seem to be among the simplest illustrations
of the fact that mathematics is teeming with classic unknowns
as well as with knowns. By classic unknowns we mean here those
things which are not yet known to any one, but which have been
objects of study on the part of mathematicians for some time.
As our elementary mathematical text-books usually confine
themselves to an exposition of what has been fully established,
and hence is known, the average educated man is led to believe
too frequently that modern mathematical investigations relate
entirely to things which lie far beyond his training.
It seems very unfortunate that there should be, on the part of
educated people, a feeling of total isolation from the
investigations in any important field of knowledge. The modern
mathematical investigator seems to be in special danger of
isolation, and this may be unavoidable in many cases, but it
can be materially lessened by directing attention to some of
the unsolved mathematical problems which can be most easily
understood. Moreover, these unsolved problems should have an
educational value since they serve to exhibit boundaries of
modern scientific achievements, and hence they throw some light
on the extent of these achievements in certain directions.
Both of the given instances of unanswered classic questions
relate to prime numbers. As an instance of one which does not
relate to prime numbers we may refer to the question whether
there exists an odd perfect number. A perfect number is a
natural number which is equal to the sum of its aliquot parts.
Thus 6 is perfect because it is equal to 1 + 2 + 3, and 28 is
perfect because it is equal to 1 + 2 + 4 + 7 + 14. Euclid
stated a formula which gives all the even perfect numbers, but
no one has ever succeeded in proving either the existence or
the non-existence of an odd perfect number. A considerable
number of properties of odd perfect numbers are known in case
such numbers exist.
In fact, a very noted professor in Berlin University developed
a series of properties of odd perfect numbers in his lectures
on the theory of numbers, and then followed these developments
with the statement that it is not known whether any such
numbers exist. This raises the interesting philosophical
question whether one can know things about what is not known to
exist; but the main interest from our present point of view
relates to the fact that the meaning of odd perfect number is
so very elementary that all can easily grasp it, and yet no one
has ever succeeded in proving either the existence or the
non-existence of such numbers.
It would not be difficult to increase greatly the number of the
given illustrations of unsolved questions relating directly to
the natural numbers. In fact, the well-known greater Fermat
theorem is a question of this type, which does not appear more
important intrinsically than many others but has received
unusual attention in recent years on account of a very large
prize offered for its solution. In view of the fact that those
who have become interested in this theorem often experience
difficulty in finding the desired information in any English
publication, we proceed to give some details about this theorem
and the offered prize. The following is a free translation of a
part of the announcement made in regard to this prize by the
Konigliche Gesellschaft der Wissenschaften, Gottingen, Germany:
On the basis of the bequest left to us by the deceased Dr. Paul
Wolskehl, of Darmstadt, a prize of 100,000 mk., in words, one
hundred thousand marks, is hereby offered to the one who will
first succeed to produce a proof of the great Fermat theorem.
Dr. Wolfskehl remarks in his will that Fermat had maintained
that the equation
x <superscript Greek 1> + y <superscript Greek 1> =
z <superscript Greek 1>
could not be satisfied by integers whenever <Greek l> is an odd
prime number. This Fermat theorem is to be proved either
generally in the sense of Fermat, or, in supplementing the
investigations by Kummer, published in Crelle's Journal, volume
40, it is to be proved for all values of <Grrek 1> for which it
is actually true. For further literature consult Hibert's
report on the theory of algebraic number realms, published in
volume 4 of the Jahreshericht der Deutschen
Mathernatiker-Vereinigung, and volume 1 of the Encyklopadie der
mathematischen Wissenschaften.
The prize is offered under the following more particular
conditions.
The Konigliche Gesellschaft der Wissenschaften in Gottingen
decides independently on the question to whom the prize shall
be awarded. Manuscripts intended to compete for the prize will
not be received, but, in awarding the prize only such
mathematical papers will be considered as have appeared either
in the regular periodicals or have been published in the form
of monographs or books which were for sale in the book-stores.
The Gesellschaft leaves it to the option of the author of such
a paper to send to it about five printed copies.
Among the additional stipulations it may be of interest to note
that the prize will not be awarded before at least two years
have elapsed since the first publication of the paper which is
adjudged as worthy of the prize. In the meantime the
mathematicians of various countries are invited to express
their opinion as regards the correctness of this paper. The
secretary of the Gesellschaft will write to the person to whom
the prize is awarded and will also publish in various places
the fact that the award has been made. If the prize has not
been awarded before September 13, 2007, no further applications
will be considered.
While this prize is open to the people of all countries it has
become especially well known in Germany, and hundreds of
Germans from a very noted university professor of mathematics
to engineers, pastors, teachers, students, bankers, officers,
etc., have published supposed proofs. These publications are
frequently very brief, covering only a few pages, and usually
they disclose the fact that the author had no idea in regard to
the real nature of the problem or the meaning of a mathematical
proof. In a few cases the authors were fully aware of the
requirements but were misled by errors in their work. Although
the prize was formally announced more than seven years ago no
paper has as yet been adjudged as fulfilling the conditions.
It may be of interest to note in this connection that a
mathematical proof implies a marshalling of mathematical
results, or accepted assumptions, in such a manner that the
thing to be proved is a NECESSARY consequence. The
non-mathematician is often inclined to think that if he makes
statements which can not be successfully refuted he has carried
his point. In mathematics such statements have no real
significance in an attempted proof. Unknowns must be labeled as
such and must retain these labels until they become knowns in
view of the conditions which they can be proved to satisfy. The
pure mathematician accepts only necessary conclusions with the
exception that basal postulates have to be assumed by common
agreement.
The mathematical subject in which the student usually has to
contend most frequently with unknowns at the beginning of his
studies is the history of mathematics. The ancient Greeks had
already attempted to trace the development of every known
concept, but the work along this line appears still in its
infancy. Even the development of our common numerals is
surrounded with many perplexing questions, as may be seen by
consulting the little volume entitled "The Hindu-Arabic
Numerals," by D. E. Smith and L. C. Karpinski.
The few mathematical unknowns explicitly noted above may
suffice to illustrate the fact that the path of the
mathematical student often leads around difficulties which are
left behind. Sometimes the later developments have enabled the
mathematicians to overcome some of these difficulties which had
stood in the way for more than a thousand years. This was done,
for instance, by Gauss when he found a necessary and sufficient
condition that a regular polygon of a prime number of sides can
be constructed by elementary methods. It was also done by
Hermite, Lindemann and others by proving that epsilon and rho
are transcendental numbers. While such obstructions are thus
being gradually removed some of the most ancient ones still
remain, and new ones are rising rapidly in view of modern
developments along the lines of least resistance.
These obstructions have different effects on different people.
Some fix their attention almost wholly on them and are thus
impressed by the lack of progress in mathematics, while others
overlook them almost entirely and fix their attention on the
routes into new fields which avoid these difficulties. A
correct view of mathematics seems to be the one which looks at
both, receiving inspiration from the real advances but not
forgetting the desirability of making the developments as
continuous as possible. At any rate the average educated man
ought to know that there is no mathematician who is able to
solve all the mathematical questions which could be proposed
even by those having only slight attainments along this line.
THE ABORIGINAL ROCK-STENCILLINGS OF NEW SOUTH WALES
BY DR. CHAS. B. DAVENPORT
COLD SPRING HARBOR, N. Y.
IN a number of places in eastern Australia curious aboriginal
markings are found on the faces of the sandstone cliffs. A good
idea of them is given by the photographs. These came from
Wolgan Gap near Wallerang in the Blue Mountain region of New
South Wales. They are found on overhanging rocks that have
served as shelters or camping places for the aborigines and
which doubtless have protected their works of art.