Chapter XXIV: Part 24

M. A. Binet presents two physiognomical pictures of the same face, one representing disgust or scorn, the other a good-humored and happy smile. The upper parts of both faces are exactly alike and yet the eyes of the former look disdainful while the very same eyes of the latter are full of jest and merriment. This is the fact. M. Binet psychologically interprets the fact as a phenomenon of automatic inhibition. The fact is interesting, but its interpretation seems doubtful.

Charles Dunan discusses the metaphysical aspect of the problem of life.

B. Pérez’s article is a contribution to pathological psychology with special reference to M. Magalhâes’s work on the subject. Pessimism, M. Pérez says, is a disease only if exaggerated, yet he believes that medico-psychological studies which consider the relation between the physical system and morality are very helpful even if carried too far.

M. J.-M. Guardia’s article will have a special interest for Americans. Three men arose in Spain of late, Valentin Almirall, M. L. Mallada, and J.-M. Escudor, who spoke bold and hard words of truth to their country. Cuba is the hen that lays golden eggs for Spain, but the Cubans are treated with great contempt in Spain; and yet the Spaniards are by no means their intellectual superiors, for while Spain is poor in philosophy, Cuba is the only country of Latin America where philosophy has taken root. M. Guardia sketches in the first article the history and philosophy of Don Félix Varélay y Moralès who is the harbinger of the other Spanish-Cuban philosopher, José de la Luz. The second article in the February number treats of the latter (1800-1862) whom Guardia calls the master.

George Mouret with reference to Frontera’s book on Zeno’s argument against motion makes a few remarks concerning the Eleatic sophism about Achilles and the tortoise.

An injury of a thalamus opticus produces in horses and other animals the effect of their making rotatory movements when intending to walk straight on. Forel proved that a similar effect is produced in ants by a lesion of one of their lobes. M. Binet publishes in the present essay his experiments on certain water-beetles, exhibiting diagrams of their normal and abnormal walk. (Paris: Félix Alcan.)

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ZEITSCHRIFT FUR PSYCHOLOGIE UND PHYSIOLOGIE DER SINNESORGANE. Vols. II and III.

CONTENTS: November, 1891. No. 6.

UEBER BRÜCKES THEORIE DES KÖRPERLICHEN SEHENS. By Dr. _C. du
Bois-Reymond_.

MEIN SCHLUSSWORT GEGEN WUNDT. By _C. Stumpf_.

ERWIDERUNG. By _O. Flügel_.

LITTERATURBERICHT.

CONTENTS: December, 1891. No. 1.

VERSUCH, DAS PSYCHOPHYSISCHE GESETZ AUF DIE FARBENUNTERSCHIEDE
TRICHROMATISCHER AUGEN ANZUWENDEN. By _H. v. Helmholtz_.

UNTERSUCHUNGEN ÜBER BINOKULARES SEHEN MIT ANWENDUNG DES
HERINGSCHEN FALLVERSUCHS. By Dr. _Richard Greeff_.

BEMERKUNGEN ZU DEM AUFSATZE VON DR. SOMMER “ZUR PSYCHOLOGIE DER
SPRACHE.” By Prof. _A. Pick_.

LITTERATURBERICHT.

Dr. C. du Bois-Reymond believes that corporeal vision is either produced by one eye running in succession over several places or two eyes viewing two aspects of the object. Mach’s theory of the influence of shade upon the production of the third dimension in vision which affords quite a new and a better explanation of the phenomenon is not mentioned. Stumpf closes his controversy with Wundt with a few remarks in answer to Wundt’s reply (in _Philos. Studien_ VII, pp. 298-327); and Flügel objects to Professor Rehmke’s proposition made in a criticism of Flügel’s book “Die Seelenfrage,” that Herbart’s psychology, being atomism, is at bottom materialism.

Dr. Richard Greeff describes Hering’s apparatus for investigating the cause of binocular vision. Wheatstone believes that the perspective of the two retina pictures produces the effect of corporeality while Brücke declares that it is mainly due to muscle-sensations. Hering sides with Wheatstone, and the experiments as described by Greeff prove that the third dimension is unfailingly perceived whenever the ocular axes diverge, while in other cases the same result is not attained.

Dr. Sommer had presented in a former article the facts of an interesting case of aphasia, (see _The Monist_, Vol. I, No. 4, p. 629) where the patient, his name is Voit, could remember and pronounce words only when writing them. Prof. A. Pick objects to Dr. Sommer’s regarding the case as contrary to our present experience and following two French authorities Ballet and Bernard, adduces cases of Aphasia by right-sided hemiphlegia where patients could read only when they were able to write or represent to themselves the writing motions of their hand. Thus one patient of Charcot could only read print, and not written words “because,” as he said, “it was easier for him to reproduce in his mind the written letter.” This reminds one of the case a deaf-mute who said: “I feel whenever I think of the motions of my fingers although they are perfectly at rest. I see internally an image of my moving fingers.” Professor Pick concludes that the case Voit is a good argument against Max Müller’s proposition of the identity of language and thought. Max Müller however includes in his conception of word any symbol of an idea. The finger motion of a deaf-mute is a word, and the writing motion of Voit is also a word, according to Professor Max Müller’s theory.

Prof. H. v. Helmholtz publishes the tables of his experiments in applying the psycho-physical law upon color differences of trichromatic eyes, and presents the three fundamental colors diagrammatically in an equilateral triangle in the centre of which lies white. A curve winding round this centre shows the relation of the rainbow spectrum in the system of three fundamental colors. The results do not as yet agree with the investigations of A. König and C. Diterici who make similar inquiries with bichromatic eyes. (Leipsic: O. R. Reisland.)

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VIERTELJAHRSSCHRIFT FÜR WISSENSCHAFTLICHE PHILOSOPHIE. Vol. XVI. No. 1.

CONTENTS:

BEITRÄGE ZUR LOGIK. (Erster Artikel.) By _A. Riehl_.

DIE DIMENSIONEN DER WAHRSCHEINLICHKEIT UND DIE EVIDENZ DER
UNGEWISSHEIT. By _Ad. Nitsche_.

UEBER DIE FORTSCHREITENDE ENTWICKLUNG DES MENSCHENGESCHLECHTS.
II. By _F. Rosenberger_.

ERNST PLATNER’S WISSENSCHAFTLICHE STELLUNG ZU KANT IN
ERKENNTNISSTHEORIE UND MORALPHILOSOPHIE. I. By _B. Seligkowitz_.

UEBER SPRACHREFLEX, NATIVISMUS UND ABSICHTLICHE SPRACHBILDUNG.
X. By _A. Marty_.

Prof. A. Riehl begins in this number a series of articles on logic. The first two chapters are (1) concepts and definitions. Riehl distinguishes between a definition and a predicating sentence (_Aussage_), for instance, “Space has three dimensions,” is a mere definition, but “Space is the form of our intuition,” is an _Aussage_. (2) Conceptual sentences and judgments. The former are merely representative and cannot as the latter be said to combine or separate ideas.

Ad. Nitsche criticises Johannes v. Kries’s idea that the calculus of probabilities is admissible only if the chances are equivalent. Equivalent Chances (_gleiche Spielräume_), he objects, are apparently impossible, yet he admits that upon the degree of a knowledge of the conditions will depend the reliability of the probability.

The Object of B. Seligkowitz’s article is to rescue from oblivion a philosopher who especially as a critic of Kant deserves to be better known than he is, Ernst Platner (1744-1818.)

The tenth and concluding article of A. Marty on the origin of language reviews Paul Regnaud’s work _Origine et philosophie du langage_. (Leipsic: O. R. Reisland.)

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THE AMERICAN JOURNAL OF PSYCHOLOGY. December, 1891. Vol. IV. No. 2.

CONTENTS:

A SKETCH OF THE HISTORY OF PSYCHOLOGY AMONG THE GREEKS. By
_Charles A. Strong_.

STUDIES FROM THE LABORATORY OF EXPERIMENTAL PSYCHOLOGY OF THE
UNIVERSITY OF WISCONSIN. By Prof. _Joseph Jastrow_, Ph. D.

THE SIZE OF SEVERAL CRANIAL NERVES IN MAN AS INDICATED BY THE
AREAS OF THEIR CROSS-SECTIONS. By _Henry H. Donaldson_, Ph. D.

VISUALISATION AS A CHIEF SOURCE OF THE PSYCHOLOGY OF HOBBES,
LOCKE, BERKELEY, AND HUME. By _Alexander Fraser_, B. A.

ANATOMICAL OBSERVATIONS ON THE BRAIN AND SEVERAL SENSE-ORGANS
OF THE BLIND DEAF-MUTE, LAURA DEWEY BRIDGMAN. II. By _Henry H.
Donaldson_, Ph. D.

PSYCHOLOGICAL LITERATURE. I. Nervous System. By Prof. _H. H.
Donaldson_.

A LABORATORY COURSE IN PHYSIOLOGICAL PSYCHOLOGY. II. By _E. C.
Sanford_.

PSYCHIATRY. PSYCHOSES FOLLOWING ACUTE SURGICAL AND MENTAL
AFFECTIONS AND IN MULTIPLE NEURITIS. By _William Noyes_, M. D.

The post mortem examination of Laura Bridgman shows a brain in which the olfactory bulbs and nerves, the optic nerves, the auditory nerves, and possibly the glossopharyngeal, had all been more or less destroyed at their peripheral ends. This destruction caused a degeneration—most marked in the optic nerves—which extended towards the centres and involved them indirectly.... This case represents a maximum loss in these defective senses with a minimum amount of central disturbance, thus offering the very best sort of opportunity for education by way of the surviving senses.... Mental association was for Laura Bridgman limited to various phases of the dermal sensations and the minor and imperfect senses of taste and smell.... The motor centre there had lost some, but not all its associative connections. (Clark University, Worcester, Mass.)

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INTERNATIONAL JOURNAL OF ETHICS. January, 1892. Vol. II. NO. 2.

CONTENTS:

THE ETHICAL ASPECTS OF THE PAPAL ENCYCLICAL. By _Brother
Azarias_.

THE THREE RELIGIONS. By _J. S. Mackenzie_, M. A.

THE ETHICS OF HEGEL. By _Rev. J. Macbride Sterrett_.

A PALM OF PEACE FROM GERMAN SOIL. By _Fanny Hertz_.

AUTHORITY IN THE SPHERE OF CONDUCT AND INTELLECT. By _Prof. H.
Nettleship_, Oxford.

DISCUSSIONS AND REVIEWS.

Brother Azarias paraphrases and praises the ethics of the Papal Encyclical. J. S. Mackenzie starts from Kant’s famous remarks that two things fill our minds with reverence, the starry, heavens above and the moral law within. The worship of these two separately and the worship of them in combination are set forth as the three great religions of the world. Fanny Hertz pleads for the abolishment of war. She quotes largely from Bertha Suttner’s novel, “Die Waffen nieder,” and from Friederich’s letters. Authority, according to Professor Nettleship, is “the power which in the sphere of conduct, in the long run determines our practice and in the sphere of intellect in the long run determines our assent.” There are roughly speaking four kinds of authority: (1) the authority of law, (2) the authority of religious bodies, (3) the authority of society or public opinion and (4) the authority of great men. Where is the seat of authority? “For each individual,” Professor Nettleship maintains, “the absolute guide can, in the long run be no other than his own conscience.” The origin of conscience and the criterion whether the voice of conscience be true or not are not explained. (Philadelphia: _International Journal of Ethics_, 118 S. Twelfth Street.)

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MIND. New Series. No. 1. January, 1892.

CONTENTS:

PREFATORY REMARKS. _The Editor._

THE LOGICAL CALCULUS. (1) General Principles. By _W. E.
Johnson_.

THE IDEA OF VALUE. By _S. Alexander_.

THE CHANGES OF METHOD IN HEGEL’S DIALECTIC. (1) By _J. Ellis
McTaggart_.

THE LAW OF PSYCHOGENESIS. By _Prof. C. Lloyd Morgan_.

DISCUSSIONS: The Feeling-Tone of Desire and Aversion. By _Prof.
H. Sidgwick_. Sur la Distinction entre les Lois ou Axiomes et
les Notions. By _George Mouret_.

CRITICAL NOTICES.

W. E. Johnson says: “As a material machine is an instrument for economising the exertion of force, so a symbolic calculus is an instrument for economising the exertion of intelligence. And, employing the same analogy, the more perfect the calculus, the smaller would be the amount of intelligence applied as compared with the results produced.” He continues:

“But as the exertion of _some_ force is necessary for working the machine, so the exertion of _some_ intelligence is necessary for working the calculus.”

Here we feel inclined to stop our author. That which makes of a certain amount of metal, brass, and wood a machine, is the form in which they are composed, and this form is instrumental in using a certain amount of energy for doing a certain kind of work. Intelligence is not analogous to force but to the form of force. Not intelligence is necessary to run the instrument of intelligence, but some power, some force, some energy, and this power needed for running the instrument of intelligence, as it exists in man, is generally called will. So we are at variance with Mr. W. S. Johnson from the outset. Mr. Johnson from his standpoint considers it “important to examine the kind and degree of intelligence that are demanded in the employment of any symbolic calculus. It will appear that the _logical_ calculus stands in a unique relation to intelligence; for it aims at exhibiting, in a non-intelligent form, those same intelligent principles that are actually required for working it.”

We abstain here from discussing the details of this highly suggestive article which contains much that is of interest to logicians. The author claims especially with regard to his interpretation of the universal and particular that his results exactly correspond with the interpretation given by Dr. Venn and Mr. Peirce, and worked out by Dr. Keynes.

The Germans distinguish between _Urtheil_ and _Beurtheilung_, the first being judgment in general, the latter a judgment that declares something to possess value from the view of truth, beauty or goodness. In this sense Mr. S. Alexander deals with the idea of value. He states two main principles. (1) That value is “the efficiency of a conscious agent to promote the efficiency of society” and this, the author says, was maintained indirectly in opposition to the view that value was determined by pleasure. (2) That value is itself no something separable from other mental facts by a wide gulf, but was itself a fact of a purely natural order. “Sollen” is one kind of “Sein.”

Mr. J. Ellis McTaggart in discussing the changes of method in Hegel’s Dialectic arrives at a conclusion which according to the author must be admitted to be quite un-Hegelian. Hegel apparently regarded the procession of the categories with its advance through oppositions and reconciliations as presenting absolute truth. From this the author dissents, “for,” he says: “the true process of thought is one in which each category springs out of the one before it, and not by contradicting it, but as the expression of its deepest nature, while it, in its turn, is seen to have its deepest reality in again passing on to the one after it. There is no contradiction no opposition, and consequently no reconciliation. There is only development, the rendering explicit what was implicit, the growth of the seed to the plant. In the actual course of the dialectic this is never attained. It is an ideal which is never quite realised, and from the nature of the case never can be quite realised. In the dialectic there is always opposition, and therefore always reconciliation. We do not go straight onward, but more or less from side to side. It seems inevitable, therefore, to conclude that the dialectic does not completely and perfectly express the nature of thought.”

Prof. C. Lloyd Morgan starting from the proposition that “the business of consciousness is the control of action” shows that “we identify ourselves rather with the action of our control centres than with our lower animal instincts. Through experience we learn, and habits being formed by individual repetition become innate.” Professor Morgan reviews use-inheritance natural selection, sexual selection, the law of beauty, and conduct and verification with regard to psychogenesis. “Our nature,” he says, “is intellectual, æsthetic, moral, and sensitive”:

“The false is rejected as incongruous to our nature as intellectual; the ugly is avoided as incongruous to our nature as æsthetic; the wrong is shunned as incongruous to our nature as moral; so is the painful, so far as possible, avoided as incongruous to our nature as sensitive.... The guidance of pleasure and pain is of great importance—so great that some are found to argue that in moral matters we are influenced solely by considerations of happiness.... Only by extending the meaning of the words pleasure and pain so as to be coextensive with what I have here termed congruous and incongruous can it be said that our actions and our thoughts are determined by pleasure and pain.” (London: Williams & Norgate.)

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THE PHILOSOPHICAL REVIEW. Vol. I, No. 1. January, 1892.

CONTENTS of No. 1.

PREFATORY NOTE.

THE CRITICAL PHILOSOPHY AND IDEALISM. By Prof. _John Watson_.

PSYCHOLOGY AS SO-CALLED “NATURAL SCIENCE.” By Prof. _George T.
Ladd_.

ON SOME PSYCHOLOGICAL ASPECTS OF THE CHINESE MUSICAL SYSTEM. By
_Benj. Ives Gilman_.

REVIEWS OF BOOKS AND SUMMARIES OF ARTICLES.

CONTENTS of No. 2:

PSYCHOLOGY, EPISTEMOLOGY, AND METAPHYSICS. By Prof. _Andrew
Seth_.

A PLEA FOR PSYCHOLOGY AS A “NATURAL SCIENCE.” By Prof. _William
James_.

ON SOME PSYCHOLOGICAL ASPECTS OF THE CHINESE MUSICAL SYSTEM.
II. By _Benj. Ives Gilman_.

DISCUSSIONS: Dr. Münsterberg’s Theory of Mind and Body and its
Consequences. By _Charles A. Strong_.

REVIEWS OF BOOKS AND SUMMARIES OF ARTICLES.

This is a new magazine which will be an additional proof that the philosophical interest in America is by no means so poor as the inhabitants of the old world generally suppose it to be. The character of the journal, it is to be expected, will be in harmony with the publications of its scholarly editor, Prof. J. G. Schurmann, whose position is clearly set forth in a little volume of his “Belief in God,” in which he conceives God in three ways (1) as the cause or ground of the world (2) as the realising purpose of the world, and (3) as the father of spirits.

Professor Watson reviews in an elaborate article Edward Caird’s work “The Critical Philosophy of Emanuel Kant.” “The philosophy of Kant,” says Watson, “was accepted at first by submissive disciples, but it had afterwards to submit to a severe process of criticism which culminated in the Absolute Idealism of Hegel. The synthesis of Kant, as based upon an untenable opposition of the phenomenal and the real, was weighed and found wanting.... We must be grateful to any one who helps us, not merely to see Kant, but to see beyond him. This is the task which Professor Caird, in his exhaustive work on the Critical Philosophy, has set himself to perform,” and adds Watson, “he has done it in a way that leaves nothing to be desired.”

Professor Ladd criticises Professor James’s Psychology as so-called natural science.

“What we wish to have in the name of cerebral psychology, is a description, in terms of a comprehensible theory of molecular physics; and, also, a statement of the formulæ which define the relations between the molecular changes and the ‘corresponding’ orders of mental phenomena. But this is precisely what Professor James avoids doing, even to the extent which so-called ‘nerve-physiology’ makes possible. And, nothing worthy of the name ‘science’ _is_ possible for any one in this branch of cerebral psycho-physics.”

Professor James replies to the criticism in the second number of _The Philosophical Review_. He says:

“Psychology is to-day hardly more than what physics was before Galileo, what chemistry was before Lavoisier. It is a mass of phenomenal description, gossip, and myth, including, however, real material enough to justify one in the hope that its study may become worthy of the name of natural science at no very distant day. I wished, by treating Psychology _like_ a natural science, to help her to become one.”

Professor Ladd is a transcendentalist and Professor James has great expectations of the work inaugurated by the Society for Psychical Research.

Theoretically they stand much nearer than practically, as well indicated by Professor James’s remark:

“In Professor Ladd’s own book on ‘Physiological Psychology,’ that ‘real being, proceeding to unfold powers that are _sui generis_, according to laws of its own,’ for whose recognition he contends, plays no organic part in the work, and has proved a mere stumbling block to his biological reviewers.”

He adds in a foot-note:

“I mean that such a being is quite barren of particular consequences. Its character is only known by its reactions on the signals which the nervous system gives, and these must be gathered by observation after the fact. If only it were subject to successive reincarnations, as the theosophists say it is, so that we might guess what sort of a body it would unite with next, or what sort of persons it had helped to constitute previously, those would be great points gained. But even those gains are denied us; and the real being is, for practical purposes, an entire superfluity, which a _practical_ psychology can perfectly well do without.”

Andrew Seth, the well-known coryphæus of philosophy and psychology at Edinburgh, presses the importance of distinguishing the different standpoints of psychology, epistemology, and metaphysics. Locke, Berkeley, Hume and other English as well as Continental thinkers “speak sometimes from one point of view, sometimes from the other without being aware that the two points of view are different.”

“Psychology, assuming the existence of a subject or medium of consciousness, seeks to explain, mainly by the help of association or processes practically similar, how out of the come-and-go of conscious states, there are evolved such subjective facts as perceptions, the belief in an independent real world, and the idea of the Ego or subject himself.... Metaphysics has to do with the ultimate nature of the reality which reveals itself alike in the consciousness which knows and the world which is known.... The epistemological thing-in-itself to be identified with the metaphysical essence.... The problem of knowledge and the Real, is the question which Epistemology has to face.” (Boston, New York, Chicago: Ginn & Co.)

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VOPROSUI FILOSOFII I PSICHOLOGII.[67] Vol. III. No. 11. January, 1892.

CONTENTS:

POSITIVE PHILOSOPHY AND THE UNITY OF SCIENCE. Part V.
Sociology. By _B. Tchitcherin_.

COUNT GIACOMO LEOPARDI AND HIS PESSIMISM. Part IV. Continued
from No. 10 of this review. (Conclusion.) By _V. Stein_.

AN HISTORICAL SKETCH OF THE CONCEPTIONS OF NATURE.
(Conclusion.) By _M. Menzhir_.

J. V. KIRYEBSKII AND THE ORIGIN OF MUSCOVITE SLAVOPHILISM.
Public lecture delivered November 20, 1891, for the benefit of
the rural districts suffering from the bad harvests. By _Paul
Vinogradoff_.

FOUILLÉ AND THE METAPHYSICS OF THE FUTURE. Part III. General
estimate of Fouillé’s views. Continued from No. 10 of this
review. (Conclusion) By _Aleksei Vnedenskii_.

TELEPATHY. To be concluded in the next number. By _M.
Petrovo-Solovo_. [This is a review of the publications of and
the work done by the Society for Psychical Research in England.]

SPECIAL PART: (1) Wundt’s System of Philosophy. By _K.
Ventzel_. (2) Hegel’s Ontology. By _N. P. Hilyaroff-Platonoff_.
New Researches on Plato. By _A. Kozloff_.

CRITICISM AND BIBLIOGRAPHY. Review of Russian and Foreign
Periodicals. Book Reviews. Bibliographical Index of recent
Philosophical works. Answer to an anonymous letter received
by N. Strachoff on the subject of his article: “Opinions
concerning L. N. Tolstoï.” By _N. Strachoff_. Transactions of
the Moscow Psychological Society. (Moscow, 1892.)

FOOTNOTES:

[67] _Questions of Philosophy and Psychology._

VOL. II. JULY, 1892. NO. 4.

THE MONIST.

OUR MONISM.

THE PRINCIPLES OF A CONSISTENT, UNITARY WORLD-VIEW.

The question, What are the essential features of Monism? was brought home to me when I read in the last number of _The Monist_ the critical remarks made with reference to the new edition of my “Anthropogeny.” I shall here endeavor briefly to draw up the outlines of my conception of the world in a manner which will indicate the most characteristic features of my views. Thus both the agreements with and the divergences from the position editorially upheld in _The Monist_ will plainly appear.

As is the case with the majority of philosophical differences, so also in the present instance I find that the divergences which exhibit themselves in our respective unitary conceptions of the world are in part only apparent and in part occasioned by the divergent significances of our fundamental ideas. But this will, perhaps, be made clearer by the following methodically arranged eight theses.

I. MONISM.

Like all general concepts of fundamental scope, that of monism also is liable to different definitions and divergent modifications,—the natural result of individual differences of subjective conception. In the determinate sense in which monism is at present employed by the majority of philosophers and physical inquirers, the sense which I believe I was the first to establish in 1866 in my “General Morphology” (Vol. I, p. 105), it denotes a unitary or _natural_ conception of the world, in opposition to a _supernatural_ or mystical one, that is, in opposition to _dualism_. For us, accordingly, there exists (in the sense of Goethe) _no_ opposition whatsoever between nature and mind, between World and God. Mental existences, “spirits,” outside nature, or in opposition to nature, do not exist. What are commonly termed the “mental sciences,”—for example, philology, history, and philosophy,—are in reality simply a part of _physical philosophy_, of _Natur-philosophie_. The latter discipline embraces, in our opinion, the entire body of human knowledge; it is based upon _empiricism_, on the experiences, the observations, and the experiments of physical inquiry; but it does not become _philosophy_ until it has brought together and united its empiric products, abstracted general laws from its isolated experiential facts, and _synthetised_ the isolated results which _analysis_ has empirically ascertained.

II. MECHANICALISM.

Since an early date, this important fundamental concept has frequently been used in three different and divergent senses, namely:

_A._ In its widest sense, as synonymous with _monism_; wherein mechanical causes (_causæ efficientes_), in the sense of Kant, are assumed as the sole effective causes and are placed in opposition to the teleological causes (_causæ finales_) in the sense of dualism. “Mechanical conception of the world” is in this sense synonymous with “monistic conception of the world.”

_B._ In its more restricted sense, as a universal _motion_-principle of physics, so that, for example, the postulated ether-vibrations of optics, of electricity, and so forth, as well as the grosser material oscillations of acoustics, heat, and so forth, are designated as mechanical processes subject to definite laws. “Mechanical natural philosophy,” in this sense, is identical with _physics_.

_C._ In its narrowest sense, as that _branch_ of physics which deals with the grosser and visible processes of _motion_; as gravitation, locomotion, and the phoronomy of organisms. Mechanics, in this the most restricted sense, is viewed as opposed to optics, acoustics, etc., as the usages of the schools indicate.

Since, now, the phrases “mechanical laws” and “mechanical explanation,” at the present day even, are frequently understood in these three distinct senses, no end of misunderstandings arise. Such misunderstandings may be best avoided, perhaps, by retaining the notion of mechanics in its narrowest (_C_) sense, and by substituting _physics_ for the next narrower sense (_B_) and _monism_ for its most extended sense (_A_).

III. PSYCHISM.

In exactly the same way as the idea of mechanicalism, so also that of psychism is employed in a three-fold divergent sense. As in the former case _motion_, so here _feeling_ is conceived, now as a universal world-principle, now simply as a vital activity of all organisms, now simply as the particular mental activity of man.

_A._ In its widest sense: _Panpsychism_. All matter is ensouled, because all natural bodies known to us possess determinate chemical properties, that is to say react uniformly and by law when subjected to the determinate chemical (i. e. molecular-mechanical) influences of other bodies: _chemical affinity_. Simplest example: sulphur and quicksilver rubbed together form cinnabar, a new body of entirely different properties. This is possible only on the supposition that the molecules (or atoms) of the two elements if brought within the proper distance, mutually _feel_ each other, by attraction move towards each other; on the decomposition of a simple chemical compound the contrary takes place: repulsion. (Empedocles’s doctrine of the “love and hatred of atoms.”)

_B._ In its more restricted sense: _Biopsychism_. The _organisms_ alone are regarded as “ensouled,” because here the chemical processes are more complicated and more striking (producing motions in cyclically repeated succession) than in the case of the so-called “dead matter” of the inorganic bodies. In particular does organic “irritability” appear here as a higher form of the physical reaction called “_Auslösung_” [the setting free, disengagement], and “soul-activity” (reflexes) again as a higher form of irritability. However, all the phenomena of organic life ultimately admit of being reduced to “mechanical” (or “physico-chemical”) processes that differ from the processes of the inorganic world only in point of degree or quantitatively, not qualitatively. (“General Morphology,” I, Chap. V; VII, pp. 109-238. “Natural Creation,” VIII, First Edition, Lecture XV.)

_C._ In its narrowest sense: _Zoopsychism_. Irritability, or universal organic soul-activity, such as is the attribute of all organisms, (identical with “life,”) reaches a higher stage through abstraction, through the formation of _ideas_. _Feeling_ and _will_ become more distinctly separated. This real soul-life, which is the attribute only of the higher animals, passes through a long succession of different stages of development, the most perfect of which is the soul of man. The so-called “freedom of the will” is apparent only, as each single volitional action is determined by a chain of precedent actions which ultimately rest either upon _heredity_ (propagation) or upon _adaptation_ (nutrition). As these last are (“mechanically”) reducible to molecular motions, the same also holds true of the former.

IV. THEISM.

The idea of god that alone appears to be logically compatible with monism, is pantheism (or “cosmotheism”) in the sense of Goethe and Spinoza. God according to this view is identical with the sum-total of the force of the universe, which is inseparable from the sum-total of the matter of the universe. In opposition to this view stands _anthropotheism_. This is the outcome of dualism, which places God as a personal being in opposition to the “world” created by him, and consequently is always forced in its reasonings to resort to anthropomorphic expedients.

V. MATERIALISM.

The most important differences of form in which this much misunderstood and variously interpreted movement of philosophy has presented itself, may be classed as follows:

_A._ In its most extended sense: as synonymous with _monism_ (or with mechanicalism). All the phenomena of the world are founded upon material processes, upon _motions_ (mechanicalism) or upon _feelings_ (psychism), both of which, as fundamental qualities, are inseparable from matter. Immaterial forces or immaterial “spirits” (minds) are unknown to us. As Goethe once said, “Mind can never exist and act without matter, matter never without mind.”

_B._ In its more restricted sense: originally matter alone exists and creates _secondarily_ force (or “mind”). The fallacy of this view lies in its regarding the two things “matter and force” as disjoint and separate. According to our view the two are inseparably connected,—united in each atom from the very first.

VI. SPIRITUALISM.

This phase also of the world-conception has been the subject of the same misunderstandings and perverted conceptions as its apparent opposite, materialism.

_A._ In its most extended sense, spiritualism is susceptible of identification with _psychism_—consequently also with monism. For _feeling_ (pleasure and pain) is just as much a thoroughly universal and fundamental property of matter (of each atom!) as is _motion_ (attraction and repulsion). Every single “spirit” is inseparably united with some “matter.”

_B._ In its more restricted sense: originally force alone exists and creates _secondarily_ matter. This view, which is very old and very widely spread (“creation of the world”), is just as false and as one-sided as its contrary (5 _B_).

VII. IMMORTALISM.

The “belief in immortality” is scientifically (_critically_) tenable only as a _general_ proposition, and is in this case identical with the most universal law of physics, the _conservation of energy_ (coincidently, of course, the conservation of matter). On the other hand, the widely disseminated _dogmatic_ belief in a _personal_ immortality, a belief supported by the mass of the ecclesiastical religions, and of utmost importance as the consciously or unconsciously assumed _base_-axiom of a great number of philosophical systems, is, _scientifically_, absolutely untenable. The “human soul” (i. e. the sum-total of the individual life-activity: feeling, motion,—will,—and idea) is simply a transient developmentary phenomenon—a very highly developed “vertebrate-soul.”

VIII. COSMISM.

The determinate, and, as I believe, logical, form of the conception of the world, the principles of which I have advocated for thirty years, and whose most important aspects have been briefly outlined in the preceding paragraphs, may also be designated _cosmism_, to the extent that it proceeds from the fundamental idea that _cosmogeny_ or the “world-process,” as world-_development_, is, within certain limits, (within the limits namely of a reduction to the basic notions: matter and its two inseparable fundamental qualities motion and feeling,) a _knowable_ natural process. Cosmism is opposed, thus, to _agnosticism_.

* * * * *

One highly important principle of my monism seems to me to be, that I regard _all_ matter as _ensouled_, that is to say as endowed with _feeling_ (pleasure and pain) and with _motion_, or, better, with the power of motion. As elementary (atomistic) attraction and repulsion these powers are asserted in every simplest chemical process, and on them is based also every other phenomenon, consequently also the highest-developed soul-activity of man. For the comprehension of this _graduated_ psychical development of matter perhaps my three stages will be useful: III _A._ (Panpsychism), III _B._ (Biopsychism), III _C._ (Zoopsychism). So too consciousness, as the highest psychical action and the one most difficult to be explained, is in my views imply a higher stage of brain-activity, based upon the association, the abstraction, and centralisation of groups of ideas. Perhaps I have expressed myself poorly in these expositions, as I am little accustomed to dealing with philosophical axioms abstractly, and am too exclusively engaged in the concrete activity of my own special department. I cherish the hope, however, of being able within two or three years to devote more of my time to purely philosophical labors; when my work with the Challenger material, which has now absorbed twelve years of unremitting toil, is ended, my special zoological activity will have been completed; and I shall then find the opportunity of contributing more frequently to your highly valued magazines _The Monist_ and _The Open Court_.

ERNST HAECKEL.

THE MAGIC SQUARE.

I.

INTRODUCTORY.

Among the philosophies of modern times there is no other which emphasises so much the importance of form and formal thought as the monism of _The Monist_. An expression thereof is found in the following passages:

“The order that prevails among the facts of reality is due
to the laws of form. Upon the order of the world depends its
cognisability.

“... The laws of form are no less eternal than are matter and
energy and ‘Verily I say unto you, till heaven and earth pass,
one jot or one tittle shall in no wise pass from the law!’

“The laws of form and their origin have been a puzzle to all
philosophers. ‘Ay, there’s the rub!’ The difficulties of Hume’s
problem of causation, of Kant’s _a priori_, of Plato’s ideas,
of Mill’s method of deduction, etc., etc., all arise from a
one-sided view of form and the laws of form and formal thought.”

Considering the great results which engineering and other applied sciences accomplish through the assistance of mathematics, we must confess that the forms of thought are wonderful indeed, and it is not at all astonishing that the primitive thinkers of mankind when the importance of the laws of formal thought in some way or another first dawned on their minds, attributed magic powers to numbers and geometrical figures.

We shall devote the following pages to a brief review of magic squares, the consideration of which has made many a man believe in mysticism. And yet there is no mysticism about them unless we either consider everything mystical, even that twice two is four, or join the sceptic in his exclamation that we can truly not know whether twice two might not be five in other spheres of the universe.

MELANCHOLY OR THE GENIUS OF THE INDUSTRIAL SCIENCE OF MECHANICS]

The author of the short article on “Magic Squares” in the English Cyclopædia (Vol. III, p. 415), presumably Prof. DeMorgan, says:

“Though the question of magic squares be in itself of no use,
yet it belongs to a class of problems which call into action
a beneficial species of investigation. Without laying down
any rules for their construction, we shall content ourselves
with destroying their magic quality, and showing that the
non-existence of such squares would be much more surprising
than their existence.”

This is the point. There obtains a symphonic harmony in mathematics which is the more startling the more obvious and self-evident it appears to him who understands the laws that produce this symphonic harmony.

* * * * *

On the wood-cut named “Melancholia”[68] of the famous Nuremberg painter, Albrecht Dürer, is found among a number of other emblems, which the reader will notice in our reproduction of the cut, the subjoined square. This arrangement of the sixteen natural numbers from 1 to 16 possesses the remarkable property that the same sum 34 will always be obtained whether we add together the four figures of any of the horizontal rows or the four of any vertical row or the four which lie in either of the two diagonals. Such an arrangement of numbers is termed a magic square, and the square which we have reproduced above is _the first magic square which is met with in the Christian Occident_.

+--+--+--+--+
| 1|14|15| 4|
+--+--+--+--+
|12| 7| 6| 9|
+--+--+--+--+
| 8|11|10| 5|
+--+--+--+--+
|13| 2| 3|16|
+--+--+--+--+
]

Like chess and many of the problems founded on the figure of the chess-board, the problem of constructing a magic square also probably traces its origin to Indian soil. From there the problem found its way among the Arabs, and by them it was brought to the Roman Orient. Finally, since Albrecht Dürer’s time, the scholars of Western Europe also have occupied themselves with methods for the construction of squares of this character.

The oldest and the simplest magic square consists of the quadratic arrangement of the nine numbers from 1 to 9 in such a manner that the sum of each horizontal, vertical, or diagonal row, always remains the same, namely 15. This square is the adjoined.

+-+-+-+
|2|7|6|
+-+-+-+
|9|5|1|
+-+-+-+
|4|3|8|
+-+-+-+
]

Here, we will find, 15 always comes out whether we add 2 and 7 and 6, or 9 and 5 and 1, or 4 and 3 and 8, or 2 and 9 and 4, or 7 and 5 and 3, or 6 and 1 and 8, or 2 and 5 and 8, or 6 and 5 and 4.

The question naturally presents itself, whether this condition of the constant equality of the added sum also remains fulfilled when the numbers are assigned different places. It may be easily shown however that 5 necessarily must occupy the middle place, and that the even numbers must stand in the corners. This being so, there are but 7 additional arrangements possible, which differ from the arrangement above given and from one another only in the respect that the rows at the top, at the left, at the bottom, and at the right, exchange places with one another and that in addition a mirror be imagined present with each arrangement. So too from Dürer’s square of 4 times 4 places, by transpositions, a whole set of new correct squares may be formed. A magic square of the 4 times 4 numbers from 1 to 16 is formed in the simplest manner as follows. We inscribe the numbers from 1 to 16 in their natural order in the squares, thus:

+--+--+--+--+
| 1| 2| 3| 4|
+--+--+--+--+
| 5| 6| 7| 8|
+--+--+--+--+
| 9|10|11|12|
+--+--+--+--+
|13|14|15|16|
+--+--+--+--+
]

We then leave the numbers in the four corner-squares, viz. 1, 4, 13, 16, as well also as the numbers in the four middle-squares, viz. 6, 7, 10, 11, in their original places; and in the place of the remaining eight numbers, we write the complements of the same with respect to 17: thus 15 instead of 2, 14 instead of 3, 12 instead of 5, 9 instead of 8, 8 instead of 9, 5 instead of 12, 3 instead of 14, and 2 instead of 15. We obtain thus the magic square

=34 =34
\ /
+--+--+--+--+
| 1|15|14| 4|=34
+--+--+--+--+
|12| 6| 7| 9|=34
+--+--+--+--+
| 8|10|11| 5|=34
+--+--+--+--+
|13| 3| 2|16|=34
+--+--+--+--+
34 34 34 34
]

from which the same sum 34 always results. It is an interesting property of this square that any four numbers which form a rectangle or square about the centre also always give the same sum 34; for example, 1, 4, 13, 16, or 6, 7, 10, 11, or 15, 14, 3, 2, or 12, 9, 5, 8, or 15, 8, 2, 9, or 14, 12, 3, 5. We may easily convince ourselves that this square is obtainable from the square of Dürer by interchanging with one another the two middle vertical rows.

II.

EARLY METHODS FOR THE CONSTRUCTION OF ODD-NUMBERED SQUARES.

Since early times rules have also been known for the construction of magic squares of more than 3 times 3, or 4 times 4 spaces. In the first place, it is easy to calculate the sum which in the case of any given number of cells must result from the addition of each row. We take the determinate number of cells in each side of the square which we have to fill, multiply that number by itself, add 1, again multiply the number thus obtained by the number of the cells in each side, and, finally, divide the product by 2. Thus, with 4 times 4 cells or squares, we get: 4 times 4 are 16, 16 and 1 are 17, and one half of 17 times 4 is 34. Similarly, with 5 times 5 squares, we get: 5 times 5 are 25, and 1 makes 26, and the half of 26 times 5 is 65. Analogously, for 6 times 6 squares the summation 111 is obtained, for 7 times 7 squares 175, for 8 times 8 squares 260, for 9 times 9 squares 369, for 10 times 10 squares 505, and so on. The Hindu rule for the construction of magic squares whose roots are odd, may be enunciated as follows: To start with, write 1 in the centre of the topmost row, then write 2 in the lowest space of the vertical column next adjacent to the right, and then so inscribe the remaining numbers in their natural order in the squares diagonally upwards towards the right, that on reaching the right-hand margin the inscription shall be continued from the left-hand margin in the row just above, and on reaching the upper margin shall be continued from the lower margin in the column next adjacent to the right, noting that whenever we are arrested in our progress by a square already occupied we are to fill out the square next beneath the one we have last filled. In this manner, for example, the last preceding square of 7 times 7 cells is formed, in which the reader is requested to follow the numbers in their natural sequence (Fig. 5).

=175 =175
\ /
+---+---+---+---+---+---+---+
|30 |39 |48 | 1 |10 |19 |28 |=175
+---+---+---+---+---+---+---+
|38 |47 | 7 | 9 |18 |27 |29 |=175
+---+---+---+---+---+---+---+
|46 | 6 | 8 |17 |26 |35 |37 |=175
+---+---+---+---+---+---+---+
| 5 |14 |16 |25 |34 |36 |45 |=175
+---+---+---+---+---+---+---+
|13 |15 |24 |33 |42 |44 | 4 |=175
+---+---+---+---+---+---+---+
|21 |23 |32 |41 |43 | 3 |12 |=175
+---+---+---+---+---+---+---+
|22 |31 |40 |49 | 2 |11 |20 |=175
+---+---+---+---+---+---+---+
175 175 175 175 175 175 175
]

For the next further advancements of the theory of magic squares and of the methods for their construction we are indebted to the Byzantian Greek, Moschopulus, who lived in the fourteenth century; also, after Albrecht Dürer who lived about the year 1500, to the celebrated arithmetician Adam Riese, and to the mathematician Michael Stifel, which two last lived about 1550. In the seventeenth century Bachet de Méziriac, and Athanasius Kircher employed themselves on magic squares. About 1700, finally, the French mathematicians De la Hire and Sauveur made considerable contributions to the theory. In recent times mathematicians have concerned themselves much less about magic squares, as they have indeed about mathematical recreations generally. But quite recently the Brunswick mathematician Scheffler has put forth his own and other’s studies on this subject in an elegant form.

| 7|
| |
| 6| |14|
| | | |
| 5| |13| |21|
+==+==+==+==+==+==+==+
| 4| |12| |20| |28|
---|--+--+--+--+--+--+--|---
3| |11| |19| |27| |35
------|--+--+--+--+--+--+--|------
2 |10| |18| |26| |34| 42
--------|--+--+--+--+--+--+--|---------
1 9| |17| |25| |33| |41 49
--------|--+--+--+--+--+--+--|---------
8 |16| |24| |32| |40| 48
------|--+--+--+--+--+--+--|------
15| |23| |31| |39| |47
---|--+--+--+--+--+--+--|---
|22| |30| |38| |46|
+==+==+==+==+==+==+==+
|29| |37| |45|
| | | |
|36| |44|
| |
|43|
]

The best known of the various methods of constructing magic squares of an odd number of cells is the following. First write the numbers in diagonal succession as in the preceding diagram (Fig. 6). After 25 cells of the square of 49 cells which we have to fill out, have thus been occupied, transfer the six figures found outside each side of the square, without changing their configuration, into the empty cells of the side directly opposite. By this method, which we owe to Bachet de Méziriac, we obtain the following magic square of the numbers from 1 to 49:

+--+--+--+--+--+--+--+
| 4|29|12|37|20|45|28|
+--+--+--+--+--+--+--+
|35|11|36|19|44|27| 3|
+--+--+--+--+--+--+--+
|10|42|18|43|26| 2|34|
+--+--+--+--+--+--+--+
|41|17|49|25| 1|33| 9|
+--+--+--+--+--+--+--+
|16|48|24| 7|32| 8|40|
+--+--+--+--+--+--+--+
|47|23| 6|31|14|39|15|
+--+--+--+--+--+--+--+
|22| 5|30|13|38|21|46|
+--+--+--+--+--+--+--+
]

III.

MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED SQUARES.

The reader will justly ask whether there do not exist other correct magic squares which are constructed after a different method from that just given, and whether there do not exist modes of construction which will lead to all the imaginable and possible magic squares of a definite number of cells. A general mode of construction of this character was first given for odd-numbered squares by De la Hire, and recently perfected by Professor Scheffler.

To acquaint ourselves with this general method, let us select as our example a square of 5. First we form two auxiliary squares. In the first we write the numbers from 1 to 5 five times; and in the second, five times, the following multiples of five, viz.: 0, 5, 10, 15, 20. It is clear now that by adding each of the numbers of the series from 1 to 5 with each of the numbers 0, 5, 10, 15, 20, we shall get all the 25 numerals from 1 to 25. All that additionally remains to be done therefore, is, so to inscribe the numbers that by the addition of the two numbers in any two corresponding cells each combination shall come out once and only once; and further that in each horizontal, vertical, and diagonal row in each auxiliary square each number shall once appear. Then the required sum of 65 must necessarily result in every case, because the numbers from 1 to 5 added together make 15, and the numbers 0, 5, 10, 15, 20 make 50.

We effect the required method of inscription by imagining the numbers 1, 2, 3, 4, 5 (or 0, 5, 10, 15, 20) arranged in cyclical succession, that is 1 immediately following upon 5, and, starting from any number whatsoever, by skipping each time either none or one or two or three etc. figures. Cycles are thus obtained of the first, the second, the third etc. orders; for example 3 4 5 1 2 is a cycle of the first order, 2 4 1 3 5 is a cycle of the second order, 1 5 4 3 2 is a cycle of the fourth order, etc. The only thing then to be looked out for in the two auxiliary squares is, that the same “cycle” order be horizontally preserved in all the rows, that the same also happens for the vertical rows, but that the cycle order in the horizontal and vertical rows is different. Finally we have only additionally to take care that to the same numbers of the one auxiliary square not like numbers but _different_ numbers correspond in the other auxiliary square, that is lie in similarly situated cells. The following auxiliary squares are, for example, thus possible:

+--+--+--+--+--+
|3 |4 |5 |1 |2 |
+--+--+--+--+--+
|5 |1 |2 |3 |4 |
+--+--+--+--+--+
|2 |3 |4 |5 |1 |
+--+--+--+--+--+
|4 |5 |1 |2 |3 |
+--+--+--+--+--+
|1 |2 |3 |4 |5 |
+--+--+--+--+--+
]

and

+--+--+--+--+--+
| 0|10|20| 5|15|
+--+--+--+--+--+
| 5|15| 0|10|20|
+--+--+--+--+--+
|10|20| 5|15| 0|
+--+--+--+--+--+
|15| 0|10|20| 5|
+--+--+--+--+--+
|20| 5|15| 0|10|
+--+--+--+--+--+
]

Adding in pairs the numbers which occupy similarly situated cells, we obtain the following correct magic square:

+--+--+--+--+--+
| 3|14|25| 6|17|
+--+--+--+--+--+
|10|16| 2|13|24|
+--+--+--+--+--+
|12|23| 9|20| 1|
+--+--+--+--+--+
|19| 5|11|22| 8|
+--+--+--+--+--+
|21| 7|18| 4|15|
+--+--+--+--+--+
]

It will be seen that we are able thus to construct a very large number of magic squares of 5 times 5 spaces by varying in every possible manner the numbers in the two auxiliary squares. Furthermore, the squares thus formed possess the additional peculiarity, that every 5 numbers which fill out two rows that are parallel to a diagonal and lie on different sides of the diagonal also give the constant sum of 65. For example: 3 and 7, 11, 20, 24; or 10, 14 and 18, 22, 1. Altogether then the sum 65 is produced out of 20 rows or pairs of rows. On this peculiarity is dependent the fact that if we imagine an unlimited number of such squares placed by the side of, above, or beneath an initial one, we shall be able to obtain as many quadratic cells as we choose, so arranged that the square composed of any 25 of these cells will form a correct magic square, as the following figure will show:

2|13|24|10|16| 2|13|24|10|16| 2
--+--+--+--+--+--+--+--+--+--+--
9|20| 1|12|23| 9|20| 1|12|23| 9
--+--+--+--+--+==============+--
11|22| 8|19| 5¦11|22| 8|19| 5¦11
--+--+--+--+--¦--+--+--+--+--¦—
18| 4|15|21| 7¦18| 4|15|21| 7¦18
--+--+--+--+--¦--+--+--+--+--¦—
25| 6|17| 3|14¦25| 6|17| 3|14¦25
--+--+--+--+--¦--+--+--+--+--¦—
2|13|24|10|16¦ 2|13|24|10|16¦ 2
--+--+========¦=====+--+--+--¦—
9|20¦ 1|12|23¦ 9|20¦ 1|12|23¦ 9
--+--¦--+--+--+=====¦========+--
11|22¦ 8|19| 5|11|22¦ 8|19| 5|11
--+--¦--+--+--+--+--¦--+--+--+--
18| 4¦15|21| 7|18| 4¦15|21| 7|18
--+--¦--+--+--+--+--¦--+--+--+--
25| 6¦17| 3|14|25| 6¦17| 3|14|25
--+--¦--+--+--+--+--¦--+--+--+--
2|13¦24|10|16| 2|13¦24|10|16| 2
--+--+==============+--+--+--+--
9|20| 1|12|23| 9|20| 1|12|23| 9
--+--+--+--+--+--+--+--+--+--+--
11|22| 8|19| 5|11|22| 8|19| 5|11
]

Every square of every 25 of these numbers, as for example the two dark-bordered ones, possesses the property that the addition of the horizontal, vertical, and diagonal rows gives each the same sum, 65.

As an example of a higher number of cells we will append here a magic square of 11 times 11 spaces formed by the general method of De la Hire from the two auxiliary squares of Figs. 12 and 13. From these two auxiliary squares we obtain by the addition of the two numbers of every two similarly situated cells, the magic square, exhibited in Diagram 14, in which each row gives the same sum 671.

+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11|
+---+---+---+---+---+---+---+---+---+---+---+
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 |
+---+---+---+---+---+---+---+---+---+---+---+
| 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 |
+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 |
+---+---+---+---+---+---+---+---+---+---+---+
| 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
+---+---+---+---+---+---+---+---+---+---+---+
| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10|
+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 |
+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 |
+---+---+---+---+---+---+---+---+---+---+---+
| 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 |
+---+---+---+---+---+---+---+---+---+---+---+
| 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
+---+---+---+---+---+---+---+---+---+---+---+
| 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+---+---+---+---+---+---+---+---+---+---+---+
]

+---+---+---+---+---+---+---+---+---+---+---+
| 0 | 11| 22| 33| 44| 55| 66| 77| 88| 99|110|
+---+---+---+---+---+---+---+---+---+---+---+
| 33| 44| 55| 66| 77| 88| 99|110| 0 | 11| 22|
+---+---+---+---+---+---+---+---+---+---+---+
| 66| 77| 88| 99|110| 0 | 11| 22| 33| 44| 55|
+---+---+---+---+---+---+---+---+---+---+---+
| 99|110| 0 | 11| 22| 33| 44| 55| 66| 77| 88|
+---+---+---+---+---+---+---+---+---+---+---+
| 11| 22| 33| 44| 55| 66| 77| 88| 99|110| 0 |
+---+---+---+---+---+---+---+---+---+---+---+
| 44| 55| 66| 77| 88| 99|110| 0 | 11| 22| 33|
+---+---+---+---+---+---+---+---+---+---+---+
| 77| 88| 99|110| 0 | 11| 22| 33| 44| 55| 66|
+---+---+---+---+---+---+---+---+---+---+---+
|110| 0 | 11| 22| 33| 44| 55| 66| 77| 88| 99|
+---+---+---+---+---+---+---+---+---+---+---+
| 22| 33| 44| 55| 66| 77| 88| 99|110| 0 | 11|
+---+---+---+---+---+---+---+---+---+---+---+
| 55| 66| 77| 88| 99|110| 0 | 11| 22| 33| 44|
+---+---+---+---+---+---+---+---+---+---+---+
| 88| 99|110| 0 | 11| 22| 33| 44| 55| 66| 77|
+---+---+---+---+---+---+---+---+---+---+---+
]

+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 13| 25| 37| 49| 61| 73| 85| 97|109|121|
+---+---+---+---+---+---+---+---+---+---+---+
| 36| 48| 60| 72| 84| 96|108|120| 11| 12| 24|
+---+---+---+---+---+---+---+---+---+---+---+
| 71| 83| 95|107|119| 10| 22| 23| 35| 47| 59|
+---+---+---+---+---+---+---+---+---+---+---+
|106|118| 9 | 21| 33| 34| 46| 58| 70| 82| 94|
+---+---+---+---+---+---+---+---+---+---+---+
| 20| 32| 44| 45| 57| 69| 81| 93|105|117| 8 |
+---+---+---+---+---+---+---+---+---+---+---+
| 55| 56| 68| 80| 92|104|116| 7 | 19| 31| 43|
+---+---+---+---+---+---+---+---+---+---+---+
| 79| 91|103|115| 6 | 18| 30| 42| 54| 66| 67|
+---+---+---+---+---+---+---+---+---+---+---+
|114| 5 | 17| 29| 41| 53| 65| 77| 78| 90|102|
+---+---+---+---+---+---+---+---+---+---+---+
| 28| 40| 52| 64| 76| 88| 89|101|113| 4 | 16|
+---+---+---+---+---+---+---+---+---+---+---+
| 63| 75| 87| 99|100|112| 3 | 15| 27| 39| 51|
+---+---+---+---+---+---+---+---+---+---+---+
| 98|110|111| 2 | 14| 26| 38| 50| 62| 74| 86|
+---+---+---+---+---+---+---+---+---+---+---+
]

IV.

EVEN-NUMBERED SQUARES.