The Mathematical Basis of the Arts by Joseph SchillingerReview by: R. C. ArchibaldIsis, Vol. 40, No. 3 (Aug., 1949), pp. 293-295Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/227267 .

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Reviews 293

x960 population. To achieve this two things are necessary. There must be an increase in the yield of the land now cultivated and there must be more cultivated land. The crucial question is, of course, how much of an increase is neces- sary? The answer really penetrates the surface optimism and shows exactly what will have to be accomplished. We will have to increase the present world's cultivated acreage by 66%. To increase the yields we will have to fertilize more heavily. Assuming a plentiful supply of nitrate just where it is needed, the world will have to increase its production of phosphate by 8oo% and of potash by i8oo%. There is also another factor (which is not emphasized). This in- crease in cultivation must occur in the sparsely populated regions, primarily in Africa and South America, where the present inhabitants are relatively well fed. Africa and South America must then develop at an unprecedented rate so that they can feed distant overpopulated regions. The food must also be given to the underfed countries for these countries can not buy it.

John D. Black's essay, The economics of freedom from want, is full of interesting and valuable information. Particularly useful is his emphasis on the fact that increased production alone does not furnish a solution. More food does not decrease hunger if there are at the same time more people to eat it. As he sees it, about 40%'o of the world's population are now under control and live under conditions where lack of food is not a limiting factor in their increase. The real problem is to bring the other 60%o who live in "Malthusian countries" to a stage where they will limit their breeding. These countries as Dr Black points out, will have to solve their own problems, with perhaps some outside aid.

All in all, it seems to the reviewer that "Free- dom from want" is still some distance in the future. Perhaps "Lloyds of London" or even some local bookmaker will give us odds on its arrival by zg6o.

CONWAY ZIRRLE

JOSEPH SCHILLINGER: The Mathematical Basis of the Arts. Part one, Science and Esthetics. Part two, Theory of Regularity and Coordination. Part three, Technology of Art Production. xii + 696 pp. New York: Philosophical Library, 1948. $12.00

Music is the oldest of the arts with a def- inite mathematical basis. The theory started with the Pythagoreans and was ardently and in extraordinary fashion pursued by later Greeks, the most noted among such theorists being Aristoxenus of Tarentum a contemporary and pupil of Aristotle. Euclid of Alexandria, who flourished about 3oo B.C., wrote two works on music which have come down to us. The great Claudius Ptolemy in the second century of our era was also the author of a remarkable

work on music. In all such work, rational fractions corresponded to various tones of nu- merous scales. In connection with such relations Paul Tannery pointed out a sort of logarithmic relationship

I 2 3 I 13\28 -=-X 3, -= (-3 X- 2 34 2\4/9

being immediately interpreted in music by: The octave is composed of a fifth and a fourth; the octave is composed of two fourths and of a major tone. Thus mathematical multiplication is changed into musical addition. There are many more complicated cases of this kind in Ptolemy's work.

From the time of Pythagoreans on to the i8th century the Quadrivium (geometry, arith- metic, music and astronomy) was regarded as basic in a liberal education; and so we find such mathematicians as Boetius, Cardan, Kep- ler, Wallis, Mersenne (who recognized har- monics associated with fundamental tones later important in developments by Rameau, Desargues, Descartes, Christiaan Huygens, d'Alembert, and Euler) all writing works on music, while exact mathematical treatment of vibrating strings was fast developing.

In 1509 Pacioli published a work in which Golden Section (a nineteenth-century term), equivalent to Pacioli's Divine Proportion, was fundamental. This is a proportion already occur- ring in Euclid's Elements. In the title of his work Pacioli refers to it as one for "every Stu- dent of Philosophy, Perspective, Painting, Sculp- ture, Architecture, Music, and other branches." Golden Section was also fundamental in the comparatively recent work of Jay Hambidge in connection with Greek sculpture, architecture, vases, and other objects; for example see his Dynamic Symmetry, the Greek Vase, Yale Uni- versity Press, 1920, and the periodical he edited, also published by the YUP: The Diagonal, a magazine devoted to the explanation of the redis- covered principles of Greek design, their appear- ance in nature and their application to the needs of modern art. In applying scientific experi- mental method to the study of aesthetic objects G. T. Fechner was led to the conclusion (I871) that the rectangle of most pleasing proportions was one in which the adjacent sides are in the ratio of parts of a line divided in golden sec- tion. Such rectangles of text on a printed page are sometimes referred to as printer's oblongs or golden oblongs.

In I634 Albert Girard discussed the sequence of numbers o, I, I, 2, 3, 5, 8, 13, 21, . . . from

I I 2 3 5 8 13 which a series of fractions -, -, -, -, 5,-, -3

I 2 3 5 8 13 21

. . .are approximations to i(V/5- -) the ratio of segments in golden section. This sequence of integers first arose 400 years earlier, however, in a rabbit multiplication problem occurring in a work of Fibonacci, and hence it is called a Fibonacci series, the literature of which is now

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294 Reviews

very extensive. Many writers (1830-1904) dis- cussed this series in connection with leaf arrange- ments or phyllotaxis. On the last-named date was published A. H. Church, On the Relation of Phyllotaxis to Mechanical Laws. In such leaf arrangement, discussion of the theories of Church and T. A. Cook (i. The Curves of Life, London, 1914, 2. Spirals in Nature and Art, London, 1903) evolved from observations of arrangements in logarithmic spirals of florets of sunflowers, pine cones, and other growths should be read in connection with D. W. Thomp- son's criticisms in his remarkable and delightful On Growth and Form, Cambridge, 1917; second greatly enlarged ed., 1942. The first definite suggestion connecting the logarithmic spirals with the septa of the Nautillus seems to have been by Sir John Leslie in his Geometrical Analysis and Geometry of Curve Lines, Edin- burgh, 182I. This was followed in I838 by one of the classics of natural history literature from the pen of Canon Moseley.

It is nearly 28 years since Dr Sarton pub- lished an interesting article on "The principles of symmetry and its applications to science and art" (Isis, 4, 32-38), which was primarily a review of an excellent work by F. M. Jaeger, then a professor of inorganic and physical chem- istry at the University of Groningen. This work was entitled Lectures on the Principle of Symmetry and Its Applications in all Natural Sciences, second improved and enlarged edi- tion, Amsterdam, I920. Dr Sarton discusses briefly also the works of Cook, Hambidge, and Thompson.

And finally among works of real mathemati- cal content, we may note a work by the greatest mathematician that America has produced, G. D. Birkhoff, Aesthetic Measure, Harvard University Press, I933. Chapters are devoted to Ornaments and Tilings, Vases, Diatonic Chords and Harmony, Melody, Musical Quality in Poetry, and Earlier Aesthetic Theories. The preface concludes "The true function of the concept of aesthetic measure is to provide sys- tematic means of analysis in simple formal aesthetic domains. There is a vast difference between the discovery of a diamond and its appraisal; still more, between the creation of a work of art and an analysis of the formal factors which enter into it."

Having thus noted some items among the vast number of previous attempts to analyze eIements of beauty and symmetry, we turn to the posthumous work of Joseph Schillinger (I895-I943) completed before his death and representing, we are told, "25 years of his dis- coveries and research." The author was a Rus- sian and held various positions in his native coun-try before coming (about 1930) to America where he was naturalized in I936. The sketch of him in Thompson's Intern. Cycl. of Music and Musicians, 4th ed. I946, states that he was a member of the faculty of Teachers' College,

Columbia University, and lists his works (192I-

3I) for the theatre, orchestra, and piano. The author tells us in his preface:

This work does not pretend to transform the reader into a proficient artist. Its goal is to disclose the mechanism of creatorship as it manifests itself in nature and in the arts. This system, which in a sense is itself a product of creation, i.e., a work of art, opens new vistas long awaiting exploration.

Then later, in the first chapter, the author continues:

Creation directly from principles and not through the imitation of appearances, is the real way to free- dom for an artist. Orginality is the product of knowledge, not guesswork. Scientific method in the arts provides an inconceivable number of ideas, technical ease, perfection, and, ultimately, a feeling of real freedom, satisfaction and accomplishment.

My life-long study, research, and accomplishments as a creative artist have been devoted to a search for facts pertaining to the arts. As a result of this work I have succeeded in evolving a scientific theory of the arts. The entire system emphasizes three main branches:

"i. The Semantics of Esthetic Expression 2. The Theory of Regularity and Coordination 3. The Technology of Art Production."

The Semantics of Esthetic Expression deals with the relationship between form and sensation, and with the associated potential of form, thus estab- lishing the meaning of esthetic perception. The Theory of Regtularity and Coordination discloses the basic principles of creatorship. The Technology of Art Production embraces all details pertaining to the analysis and synthesis of works of art in indi- vidual and combined media.

That the author succeeds in presenting any "scientific theory of the arts" is nowhere appar- ent to the reviewer. A professor in Teachers' College, Columbia University writes,

Because the laws which he formulates are mathe- matically fundamental, Schillinger's method is ap- plicable not only in the analysis of existing works of art and of musical compositions, but offers a definite and workable procedure for architects, painters, composers, and designers in the industrial fields.

To those who know no mathematics the so- called "mathematical basis" may seem impres- sive. Before indicating what kind of mathe- matics is used we may note what scientific writings of previous authors are quoted. In a paragraph with various erroneous statements there is a reference (on p. 32) to Pacioli and Hambidge; the names of Fibonacci, Hambidge, and Church occur incidentally on page 89. There are similar types of reference to D. C. Miller, The Science of Musical Sounds, I926

(p. 64), to R. W. Wood, Physical Optics (p. 8o), to Helmholtz, Sensations of Tone (p. 57), to Seashore tests in pitch discrimination (p. 58), and to Einstein's General and Special Theory of Relativity (p. 56, 73, 365). Apart from the author's own publications that is all; although

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Reviews 295

there are numerous cases of mention of indi- viduals. Let us now indicate the mathematics of the "mathematical basis" exhibited.

The frontispiece of the book consists of two reproductions of photographs, the one of lily- pads in a pond and trees and hills beyond, while the lower is of mountain peaks. The legend below it is: "Restfulness and restlessness in a landscape. As effected by a sine-curve and a complex curve." The "sine-curve" on p. IO9 consists of a series of approximate semicircles - "Transverse oscillations of a pendulum," illus- trating periodicity. The figure on p. 685 is not quite as bad.

Besides the Fibonacci series mentioned above, and "summation" series, geometric, arithmetic, and "accelerating series" also arise in the dis- cussion; the last-named is illustrated by I, 4,

n 7, ii, i6, 22, . . A "Determinant" - series

n

in the case n= 2, is the series . ..

I 2 I 2 ( ,)-, -, 2, 28, . .2 Prime Number 2 2 2

series, Harmonic or Natural Fraction series, also come up. The terms "factorial," "continuum," are used with wholly non-mathematical mean- ings. There are operations with simple arithmetic, multinomial expansions, permutations (p. I58- I72) and combinations (the table on p. 666 suggests that the simplest facts in this regard were unknown to the author).

Such mathematics is used almost wholly to get imagined corresponding graphs illustrating continuity, periodicity (p. I09-I57), symmetry (p. 2I9-23I), etc. Appendix A (p. 445-64I) "Basic forms of regularity and coordination" consists of nothing but a wholly useless set of graphs corresponding to arithmetic relations. Appendix B, "Relative dimensions" (p. 643- 66i) consists of various 5-place tables, with differences, for (I) 2/nf, n = 2(1)48; (2) 4/n, n = 4(1)48; (3) 6/n, n = 6(i)48; . . .; (23) 46/n, n = 46(1)48. What the use of such tables may be is not clear, since there is a reference for an explanation to a non-existent section of a chapter in Part II. In the section on selective systems there are a number of well-known curves drawn with a geometric chuck. And, finally, there are numerous refer- ences (for example, p. 39, 8i, 277, 664, to the sequence

a,_, a) a. B a

interpreted as equidistant symmetric points on a

a straight line "within the limit of the - ratio." b

When a = 2, b = i, n = I2, we have the se- quence connected with equal temperament of an octave. There is nothing in the book to

make this clear to one who did not know all about it in the beginning. First of all, as in the Paul Tannery consideration mentioned above we have to find x/y such that (x/y)l' = 2.

12 -

Hence x/y = \/2 = I.059463, not I.059263 as given in Smithsonian Physical Tables, 8th ed., I933, p. I92. If the number of vibrations cor- responding to a note C be n, then the number

for C: is nv/2; for D, nV22; and so on. To sum up, the reviewer finds in the volume

nothing of direct mathematical interest that is new, and very little of the vast amount of in- teresting old material in this field. Of course generalities in connection with symmetry, set forth elsewhere, are numerous. But lack of clear articulation of thought and many complexities in other places, repel. Even in the discussion of music, -that general principles of importance have been set forth is by no means clear.

R. C. ARCHIBALD

H. I. MARROU: Histoire de L'Education dans L'Antiquitd. 595 pp. Paris: Editions du Seuil, I948.

The history of education is not replete with stimulating books. As Ralph Waldo Emerson says, "It is ominous, a presumption of crime, that this word Education has so cold, so hopeless a sound.... Education should be as broad as man."

Marrou's History of Ancient Education ful- fills this demand, "to be as broad as man," to a high degree. In the mirror of Greek and Roman education it reflects the beauty and depth of ancient culture as a whole. In addition, Mr Marrou is a master in the art of presentation and has an intimate personal knowledge of the sources and literature pertinent to his subject. Thus we are delighted and informed at the same time, which is a rare experience. Our students now have Jaeger's Paideia in English; it would be another source of enrichment for them if Marrou's work too could be made available in translation.

After all this praise I may permit myself four comments which should be understood more as questions than as critical remarks. First: is it not too narrow to say that Plato "builds his whole system of education on the funda- mental notion of truth, on the conquest of truth by rational thinking?" (p. IOS) Marrou him- self, in his interpretation of Plato's educational philosophy and its influences on posterity, sug- gests that the philosopher was interested in a full and balanced life nourished by all the re- sources which come from the experiences of rhythm, beauty, active virtue, political responsi- bility, and truth, with truth being only one of them. It seems to me important to emphasize that Plato's concept of education is much more embracing than our modern instructors in phi- losophy and pedagogy generally recognize. In addition, why did Marrou, like so many other

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