was to be said after the beautiful studies of Messrs. Lacassagne and
Marro, and after my own.[47]
[47] See _Nouvelle Revue_; also my _Uomo Delinquente_, 4th ed., 1889.
However, the researches made by Messrs. Severi, Lucchini, and Boselli
on 4,000 new criminals have given results of a high importance and
first of all a proportion eight fold greater than that of the
alienists of the same district (Florence and Lucca). The prevalency
of this practice is enormous; it amounts to 40 in 100 among military
criminals and to 33 in 100 among criminals under age; the women give a
proportion of only 1·6 in 100, but this would be increased to 2 in 100
if we included certain kinds of fly-tattooing (_tatouages mouches_)
resembling beauty spots, which are found even in high life prostitution.
What chiefly astonishes us in these researches, next to the frequency
of the phenomena, is the specific character of the tattooings: their
obscenity, the vaunting of crime, and the strange contrast of evil
passions and the highest sentiments.
M. C..., aged 27 years, convicted at least fifty times for affrays, and
the assaulting and wounding of men and horses, has the history of his
crimes literally written on his skin; and in this respect, let us note
that the infamous De Rosny, who only lately committed suicide in Lyons,
had her body covered with tattooings in the form of erotic figures; one
could read there the list of her lovers and the dates at which she left
them.
F. L..., a carrier, aged 26 years, several times convicted, bears on
his breast a heart pierced by a poniard (the sign of vengeance), and
on his right hand a female singer of a _café chantant_, of whom he was
enamoured. By the side of these tattooings, and others which propriety
forbids us to cite,[48] one sees with surprise the picture of a tomb
with the epitaph: "To my beloved father." Strange contradictions of the
human mind!
[48] See _Atlas de L'Homme Criminel_. 1888. Alcan.
A certain B..., a deserter, has on his chest a St. George and the cross
of the Legion of Honor, and on the right arm a woman, very little
dressed, who drinks with the inscription: "Let us wet the interior a
little."
Q. A..., a laborer, convicted many times for theft, expelled from
France and Switzerland, has on his chest two Swiss gendarmes with the
words "Long live the Republic!" On his right arm he has a heart pierced
through, and at the side the head of a fish—a mackerel, to signify
that he will poniard a bully, his rival.
We have seen on the left arm of another thief, a pot with a lemon tree,
and the initials V. G. (_vengeance_); which in the strange language
of the criminals means: treason, and, afterwards, revenge. He did not
conceal from us the fact that his constant thought was to revenge
himself on the woman who loved him and then abandoned him. His desire
was to cut off her nose. His brother offered to perform the operation
for him, but this he refused, reserving for himself the pleasure of
executing his purpose when he should ultimately be liberated.
One sees, therefore, from these few examples, that there is among
criminals a kind of hieroglyphical writing, but which is not regulated
or fixed. The system is founded on daily happenings and slang, as
would be the case among primitive mankind. Very often, in fact, a key
signifies among thieves the silence of secrecy; and a death's-head (the
bare skull), revenge. Sometimes points are used instead of figures. In
this way one criminal marked himself with 17 points, which means, to
his mind, that he proposes to inflict injury on his enemy seventeen
times, whenever he meets with him.
The criminal tattooers of Naples have the habit of making long
inscriptions on their bodies; but initials are used instead of words.
Many Camorrists of Naples carry a tattooing which represents iron bars,
behind which there is a prisoner and underneath the initials Q. F. Q.
P. M.; which means: "Quando finiranno queste pene? Mai!" (When will
these pains end? Never!) Others bear the epigraph C. G. P. V., etc.,
which means: "Courage, galeriens, pour voler et piler; nous devons tout
mettre à sang et à feu!" (Courage, convicts, to steal and to rob; we
must put all to the sword and fire!) We see here at once that certain
forms of tattooing are employed by criminal federations, and serve as
a sort of rallying-call. In Bavaria and in the South of Germany, the
pickpockets, who are united together in real alliances, recognise each
other by the epigraphic tattooing "T and L," which means _Thal und
Land_ (valley and country); words which they must exchange in a low
voice when they meet each other, in order not to be denounced to the
police. A thief R..., who has on his right arm a design representing
two hands crossed, and the word _union_ (unity) surrounded by a
garland of flowers, told us that this tattooing is extensively adopted
by malefactors in the South of France (Draguignan). According to the
revelations made to us by emerited Camorrists, a lizard or a serpent
denotes the first grade of this dangerous association.
I pass over in silence, and for good reasons, the tattooings spread
over all the remaining parts of the body.
In the _Revista de Antropologia Criminal_, a new publication which
has just appeared in Madrid, Mr. Sallilas has published an excellent
study relative to the tattooing of Spanish criminals. According to
him, this is a frequent custom among murderers. The predominance of
the religious character is there noticeable, but always with the seal
of lewd obscenity, universally observed. I have lately had occasion to
verify up to what point the impulsion which leads criminals to inflict
on themselves this strange operation, is atavistic. One of the most
incorrigible thieves I have met, who has six brothers tattooed like
himself, begged of me, notwithstanding he was half covered with the
most obscene tattooings, to find him a professional tattooer who should
complete what might well be called the carpeting of his skin. "When
the tattooing is very odd and grotesque, and spreads over the whole
body," he said, "it is for us thieves what the black dress coat and
the decorated vest is to society. The more we are tattooed the greater
is our esteem for one another; the more an individual is tattooed, the
more authority has he over his companions. On the other hand, he who
is not much tattooed enjoys no influence whatsoever with us; is not
considered a thorough scoundrel, and has not the estimation of his
fellows." "Very often," another told me, "when we visited prostitutes,
and they saw us covered all over with tattooings, they overwhelmed us
with presents, and gave us money instead of demanding it." If all that
is not atavism, atavism does not exist in science.
Of this characteristic, of course, as of all the other characteristics
of criminals, one may say that it is to be met with among normal
people. But the chief thing here is its proportion, its commonness,
and the exaggerated extent to which it is practised. Among honest,
respectable people its peculiar complexion, its local and obscene
coloring, and the useless, vain, and imprudent display of crime are
wanting.
Again, it will probably be objected that this is not psychology, and
that only through the latter science can we trace out the picture of
the criminal. I could well answer here, that these tattooings are
really psychological phenomena. And I may add that Mr. Ferri, in the
introductory part of his work on Homicides, has given us in addition to
a real statistical psychology, an analysis of all criminal propensities
and of their extent before and after the crime.
Among born criminals, for example, 42 in 100 always deny the crime
with which they are charged, while among occasional criminals, and in
particular among maimers, only 21 in 100 deny all; of the first 1 in
100, and of the second 2 in 100 confess their crime with tears; etc.[49]
[49] _L'Omicidio_, Turin, 1890.
CESARE LOMBROSO.
[Prof. Lombroso has in preparation for this series of
criminological studies, an essay on the physiognomy of the
Anarchists.—ED.]
THE SQUARING OF THE CIRCLE.
AN HISTORICAL SKETCH OF THE PROBLEM FROM THE EARLIEST TIMES TO THE
PRESENT DAY.[50]
[50] From Holtzendorff and Virchow's _Sammlung
gemeinverständlicher wissenschaftlicher Vorträge_, Heft 67.
Hamburg: Verlagsanstalt, etc.
I.
#Universal interest in the problem.#
For two and a half thousand years, both trained and untrained minds
have striven in vain to solve the problem known as the squaring of the
circle. Now that geometers have at last succeeded in giving a rigid
demonstration of the impossibility of solving the problem with ruler
and compasses, it seems fitting and opportune to cast a glance into the
nature and history of this very ancient problem. And this will be found
all the more justifiable in view of the fact that the squaring of the
circle, at least in name, is very widely known outside of the narrow
limits of professional mathematicians.
#The resolution of the French Academy.#
The Proceedings of the French Academy for the year 1775 contain at
page 61 the resolution of the Academy not to examine from that time
on, any so-called solutions of the quadrature of the circle that
might be handed in. The Academy was driven to this determination
by the overwhelming multitude of professed solutions of the famous
problem, which were sent to it every month in the year,—solutions
which of course were an invariable attestation of the ignorance and
self-consciousness of their authors, but which suffered collectively
from a very important error in mathematics: they were _wrong_. Since
that time all professed solutions of the problem received by the
Academy find a sure haven in the waste-basket, and remain unanswered
for all time. The circle-squarer, however, sees in this high-handed
manner of rejection only the envy of the great towards his grand
intellectual discovery. He is determined to meet with recognition, and
appeals therefore to the public. The newspapers must obtain for him
the appreciation that scientific societies have denied. And every year
the old mathematical sea-serpent more than once disports itself in the
columns of our papers, that a Mr. N. N., of P. P., has at last solved
the problem of the quadrature of the circle.
#General ignorance of quadrators.#
But what kind of people are these circle-squarers, when examined by
the light? Almost always they will be found to be imperfectly educated
persons, whose mathematical knowledge does not exceed that of a modern
college freshman. It is seldom that they know accurately what the
requirements of the problem are and what its nature; they never know
the two and a half thousand years' history of the problem; and they
have no idea whatever of the important investigations and results
which have been made with reference to the problem by great and real
mathematicians in every century down to our time.
#A cyclometric type.#
Yet great as is the quantum of ignorance that circle-squarers intermix
with their intellectual products, the lavish supply of conceit and
self-consciousness with which they season their performances is still
greater. I have not far to go to furnish a verification of this. A book
printed in Hamburg in the year 1840 lies before me, in which the author
thanks Almighty God at every second page that He has selected him and
no one else to solve the 'problem phenomenal' of mathematics, "so long
sought for, so fervently desired, and attempted by millions." After
the modest author has proclaimed himself the unmasker of Archimedes's
deceit, he says: "It thus has pleased our mother nature to withhold
this mathematical jewel from the eye of human investigation, until she
thought it fitting to reveal truth to simplicity."
This will suffice to show the great self-consciousness of the author.
But it does not suffice to prove his ignorance. He has no conception
of mathematical demonstration; he takes it for granted that things are
so because they seem so to him. Errors of logic, also, are abundantly
found in his book. But apart from this general incorrectness let us see
wherein the real gist of his fallacy consists. It requires considerable
labor to find out what this is from the turgid language and bombastic
style in which the author has buried his conclusions. But it is this.
The author inscribes a square in a circle, circumscribes another about
it, then points out that the inside square is made up of four congruent
triangles, whereas the circumscribed square is made up of eight such
triangles; from which fact, seeing that the circle is larger than the
one square and smaller than the other, he draws the bold conclusion
that the circle is equal in area to six such triangles. It is hardly
conceivable that a rational being could infer that something which is
greater than 4 and less than 8 must necessarily be 6. But with a man
that attempts the squaring of the circle this kind of ratiocination
_is_ possible.
Similarly in the case of all other attempted solutions of the problem,
either logical fallacies or violations of elementary arithmetical or
geometrical truths may be pointed out. Only they are not always of such
a trivial nature as in the book just mentioned.
Let us now inquire whence the inclination arises which leads people to
take up the quadrature of the circle and to attempt to solve it.
#The allurements of the problem.#
Attention must first be called to the antiquity of the problem. A
quadrature was attempted in Egypt 500 years before the exodus of the
Israelites. Among the Greeks the problem never ceased to play a part
that greatly influenced the progress of mathematics. And in the middle
ages also the squaring of the circle sporadically appears as the
philosopher's stone of mathematics. The problem has thus never ceased
to be dealt with and considered. But it is not by the antiquity of the
problem that circle-squarers are enticed, but by the allurement which
everything exerts that is calculated to raise the individual out of the
mass of ordinary humanity, and to bind about his temples the laurel
crown of celebrity. It is ambition that spurred men on in ancient
Greece and still spurs them on in modern times to crack this primeval
mathematical nut. Whether they are competent thereto is a secondary
consideration. They look upon the squaring of the circle as the grand
prize of a lottery that can just as well fall to their lot as to that
of any other. They do not remember that—
"Toil before honor is placed by sagacious decrees of Immortals,"
and that it requires years of continued studies to gain possession of
the mathematical weapons that are indispensably necessary to attack
the problem, but which even in the hands of the most distinguished
mathematical strategists have not sufficed to take the stronghold.
#About the only problem known to the lay world.#
But how is it, we must further ask, that it happens to be the squaring
of the circle and not some other unsolved mathematical problem upon
which the efforts of people are bestowed who have no knowledge of
mathematics yet busy themselves with mathematical questions? The
question is answered by the fact that the squaring of the circle is
about the only mathematical problem that is known to the unprofessional
world,—at least by name. Even among the Greeks the problem was very
widely known outside of mathematical circles. In the eyes of the
Grecian layman, as at present among many of his modern brethren,
occupation with this problem was regarded as the most important and
essential business of mathematicians. In fact they had a special word
to designate this species of activity; namely, τετραγωνίζειν, which
means to busy one's self with the quadrature. In modern times, also,
every educated person, though he be not a mathematician, knows the
problem by name, and knows that it is insolvable, or at least, that
despite the efforts of the most famous mathematicians it has not yet
been solved. For this reason the phrase "to square the circle," is now
used in the sense of attempting the impossible.
#Belief that rewards have been offered.#
But in addition to the antiquity of the problem, and the fact also
that it is known to the lay world, we have yet a third factor to point
out that induces people to take up with it. This is the report that
has been spread abroad for a hundred years now, that the Academies,
the Queen of England, or some other influential person, has offered a
great prize to be given to the one that first solves the problem. As a
matter of fact we find the hope of obtaining this large prize of money
the principal incitement to action with many circle-squarers. And the
author of the book above referred to begs his readers to lend him their
assistance in obtaining the prizes offered.
#The problem among mathematicians.#
Although the opinion is widely current in the unprofessional world,
that professional mathematicians are still busied with the solution of
the problem, this is by no means the case. On the contrary, for some
two hundred years, the endeavors of many considerable mathematicians
have been solely directed towards demonstrating with exactness that
the problem is insolvable. It is, as a rule,—and naturally,—more
difficult to prove that something is impossible than to prove that
it is possible. And thus it has happened, that up to within a few
years ago, despite the employment of the most varied and the most
comprehensive methods of modern mathematics, no one succeeded in
supplying the wished-for demonstration of the problem's impossibility.
At last, Professor Lindemann, of Königsberg, in June, 1882, succeeded
in furnishing a demonstration,—and the first demonstration,—that it
is impossible by the exclusive employment of ruler and compasses to
construct a square that is mathematically exactly equal in area to a
given circle. The demonstration, naturally, was not effected with the
help of the old elementary methods; for if it were, it would surely
have been accomplished centuries ago; but methods were requisite
that were first furnished by the theory of definite integrals and
departments of higher algebra developed in the last decades; in other
words it required the direct and indirect preparatory labor of many
centuries to make finally possible a demonstration of the insolvability
of this historic problem.
Of course, this demonstration will have no more effect than the
resolution of the Paris Academy of 1775, in causing the fecund race of
circle-squarers to vanish from the face of the earth. In the future
as in the past, there will be people who know nothing, and will not
want to know anything of this demonstration, and who believe that they
cannot help but succeed in a matter in which others have failed, and
that just they have been appointed by Providence to solve the famous
puzzle. But unfortunately the ineradicable passion of wanting to solve
the quadrature of the circle has also its serious side. Circle-squarers
are not always so self-contented as the author of the book we
have mentioned. They often see or at least divine the insuperable
difficulties that tower up before them, and the conflict between their
aspirations and their performances, the consciousness that they want to
solve the problem but are unable to solve it, darkens their soul and,
lost to the world, they become interesting subjects for the science of
psychiatry.
II.
#Nature of the problem. Numerical rectification.#
If we have a circle before us, it is easy for us to determine the
length of its radius or of its diameter, which must be double that
of the radius; and the question next arises to find the number that
represents how many times larger its circumference, that is the length
of the circular line, is than its radius or its diameter. From the fact
that all circles have the same shape it follows that this proportion
will always be the same for both large and small circles. Now, since
the time of Archimedes, all civilised nations that have cultivated
mathematics, have called the number that denotes how many times larger
than the diameter the circumference of a circle is, π,—the Greek
initial letter of the word periphery. To compute π, therefore, means to
calculate how many times larger the circumference of a circle is than
its diameter. This calculation is called "the numerical rectification
of the circle."
#The numerical quadrature.#
Next to the calculation of the circumference, the calculation of the
superficial contents of a circle by means of its radius or diameter is
perhaps most important; that is, the computation of how much area that
part of a plane which lies within a circle measures. This calculation
is called the "numerical quadrature." It depends, however, upon the
problem of numerical rectification; that is, upon the calculation of
the magnitude of π. For it is demonstrated in elementary geometry,
that the area of a circle is equal to the area of a triangle produced
by drawing in the circle a radius, erecting at the extremity of the
same a tangent,—that is, in this case, a perpendicular,—cutting off
upon the latter the length of the circumference, measuring from the
extremity, and joining the point thus obtained with the centre of the
circle. But it follows from this that the area of a circle is as many
times larger than the square upon its radius as the number π amounts to.
#Constructive rectification and quadrature.#
The numerical rectification and numerical quadrature of the circle
based upon the computation of the number π, are to be clearly
distinguished from problems that require a straight line equal in
length to the circumference of a circle, or a square equal in area
to a circle, to be _constructively_ produced out of its radius or
its diameter; problems which might properly be called "constructive
rectification" or "constructive quadrature." Approximately, of
course, by employing an approximate value for π these problems are
easily solvable. But to solve a problem of construction, in geometry,
means to solve it with mathematical exactitude. If the value π were
exactly equal to the ratio of two whole numbers to one another, the
constructive rectification would present no difficulties. For example,
suppose the circumference of a circle were exactly 3-1/7 times greater
than its diameter; then the diameter could be divided into seven equal
parts, which could be easily done by the principles of planimetry with
ruler and compasses; then we would produce to the amount of such a
part a straight line exactly three times larger than the diameter, and
should thus obtain a straight line exactly equal to the circumference
of the circle. But as a matter of fact, and as has actually been
demonstrated, there do not exist two whole numbers, be they ever so
great, that exactly represent by their proportion to one another the
number π. Consequently, a rectification of the kind just described does
not attain the object desired.