". . .I shdl affirm, that the Acidity of any Liquor does consist in keen Particles of Salts, put in motion; and I hope no Body will offer to dispute whether an aeid has points or no, seeing every ones Experience does demonstrate it, they need but taste an acid to be satisfied of it, for it pricks the Tongue like anything keen, and finely cut; but a demonstrative and convincing Proof that an Acid does consist of pointed Parts is, that not only all acid Salts do Crystallize into Edges, but 511 Dissolutions of different things, csused by aeid Liquors, do assume this figure in their Crystallization: these Cry&ls consist of Points differing both in Length and Bigness from one another, and this diversity must beattributed to the keener or blunter Edges of the different Sorts of Acids. "As for Alkali's, they are known by pouring s n aeid upon tbem, for presently or soon after, there rises a violent Ebullition, whieh remains until the aeid finds no more bodies to rarifie. This effect may make us reasonably conjecture that an Alkali is a terrestrious and solid matter wbosepores are figured after such a manner that the aeid points entering into tbem do strike and divide whatever opposes their motion.. "There me as many different Alkali's, as there are bodies that have different pores, and this is the reason why an aeid will Ferment with one strongly, and with another not at all; for there must he a due proportion between the acid points, and the pores of the Alkali!'
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This section is especially useful for informing the student that many other investigators other than Rutherford, Bohr, and Moseley made significant contributions to our knowledge of atomic structure. The last four chapters deal with the chemical hond. Its electrical nature, the significanceof electron pairs, the geometry of electron groupings, and electron waves, all are discussed. An informative discussion of the Lewis and Bohr controversy over the static vs. the dynamic nature of the atom is given in considerable detail. A lucid discussion on how Lewis was led from the view of s cubic carbon atom to thst of a. tetrahedral atom is given. The arguments that required the introduction of such concepts as the wave nature of material particles, quantum numbers, the uncertainty principle, and orbital and spin angular momentum are clearly treated. Although the text introduces students to chemistry from the historical viewpoint, the treatment is quite close to the current method of treatingmmy topics: coordination compounds, valence band theory, molecular orbital theory, bond geometry, bond order, and isomerism. The book is highly recommended as supplementary reading for serious students specializing in chemistry.
MODDIE D. T.LYMR Howard University Washington, D. C.
The dedication of scientists is embodied in the quotation of Professor T. 0. Bergman (1738-1784) "The tables which we have at present contain only a few substances, and each of these compared only a with afew others. There is no reproach to the authors of them, for the task is laborious and long. Although, therefore, I have been employed opon it with all the diligence I could exert, and as much as my many other engagements would permit, yet I am very far from venturing to assert, that that whieh I offer is perfect, since I know witb certainty, that the slight sketch now proposed will require above 30,000 exact experiments before it can be brought to any degree of perfection. But when I refleoted on the shortness of life, and the instability of health, I resolved to publish my observations, however defective, lest they should perish with my papers, and I shall relate them as briefly ss possible. In itself it is of small consequence by whom science is enriched; whether the truths belonging to it are discovered by me or by another. Meanwhile, if God shall grant me life, health, and the necessary leisure, I will persevere in the task whieh I have begun.'' The book continues with a disoossion of Structure, Three-Dimensional Chemistry, and The Elements and Their Compounds: Pemdlcity. The dualistic theory of Berzelius and his introduction of symbolism is discussed. I t is noteworthy that the symbolism introduced by Berzelius is e s sentially that which is still used today. The next three chapters are concerned with Electrons and Atomic Structure, Electrons and Periodicity, and The Structural Units of Matter. The tetrahedral carbon atom of vm't Hoff and Le Be1 is discussed.
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Journal of Chemical Educofion
Mathematics for Scientists
Thor A . Bak, University of Copenhagen, and Jonas Lichtenberg, Royal Danish College of Education. W. A. Benjamin, Inc., New York, 1966. xiv 487 pp. Figs. and tables. 16 X 23.5 cm. $12.50.
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Mathematical Methods in the Physical Sciences
Mary L. Ems, DePanl University, Chicago. John Wiley and Sons, Inc., 778 pp. New York, 1966. xiw Figs. and tables. 16 X 23.5 em. $11.95.
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Mathematics for Quantum Chemistry
Jay Martin Andemon, Bryn Mawr College, Bryn M a w , Pennsylvania. W. A. Benjamin, Inc., New York, 154 pp. Figures. 14.5 X 1966. x 22 em. Clothbound, $8.50; paperbound, $3.95.
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Many teachers are convinced that mathematics has not yet found its proper place in the education of chemists. In his inaugural lecture (1965) at the University of Queensland, "What is Applied Mathematics," Professor A. F. Pillow placed this question in a broader context when he noted that "throughout the world there is a growing concern a t the widening gulf between modern pure mathematics and the sciences. Modern pure mathematics encourages mathematicians to create new fields of activity whieh are exciting and esthetically pleas-
ing in their own right without reference, either general or specific, to scientific or engineering problems. The probability of such mathematics becoming applicable in reasonable time decreases. One cannot help feeling that mathematics, science and engineering form basically a coherent body of intellectual pursuits which are subdivided because of human limitations. If intellectual hedonism is to be avoided, it becomes important to maintain an even development. The crucial step for this purpose is. to recognize applied mathematics as a legitimate self-propagating discipline of scientific activity, side by side with pure mathematics, the sciences, and engineering." The program envisaged by Professor Pillow would include the development of special interdisciplinary undergraduate programs between applied mathematics and chemistry. The University of Toronto has already initiated a program of this kind which deserves to be more widely studied and perhaps copied. The gulf between pure mathematics and chemistry is so wide thst our present practice of simply sending our chemistry students to take courses in the department of mathematics does not, and cannot reasonably he expected to produce acceptable training. Books like those under review probably arise from the recognition by scientists that the student needs help in order to bring his mathematics closer to scientific applications. One can suppose tbat such b o k s are designed for three possible uses: (1) as textbooks in formal courses (2) for independent study (3) as references. The book of Bak and Lichtenberg is the result of a collaboration between a theoretical chemist and a mathematics teacher. I t is based on a one-year course given by the authors for chemists, biochemists, and M.D!s doing basic research. In such a course, the teachers would doubtless have emphasized particular scientific problems related to the textbook material. The book itself does not do this and hence fails to capture the interest of the scientific reader. I t is almost incredible tbat one reaches page 212 of this 487 page book before finding the first mention of a scientific s;pplication of mathematics in either text or problems. Nor are there more than half a doxen further references to scientific problems in the remainder of the book. The rather difficult style would tend to discourage use of this book for reference or independent study. For instance, here is a typical definition: "A tensor (linear vector funct,ion) in N-dimensional space is an operator which in every (usual right angle) coordinate system appears as a square matrix of order N." Thus despite the promising nature of the collaboration, this hook does not provide a. well-lighted bridge between applied mathematics and science. The book by Boas is written by an experienced college teacher of mathematics and physics. Designed for science students with one year of calculus, it is consistently readable witb a nice balance between the mathematics and its applications. Usually the author introduces a topic in physical terms and then brings in
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(Continued on page A 3 6 f )
BOOK REVIEWS the mathematical analysis. This seems to be the correct way to proceed in teachine science students. It allows them to participate in the process of abstraction that leads to making a. mathematical model for the physicd system. This book would also he suitable for independent study. I n a. way it is a sort of "mathematical physics without tears!' A student who had mastered it would be well prepared for a subsequent rigorous course in theoretical physics. There are extensive problems, hut unfortunately they are mostly routine exercises in 2'8 and y's which provide the student no opportunity to apply hi newly learned mathematics to the analysis of physical situations. So far as this author is concerned, "Physical Sciences" in the title appears to be synonymous with "classical physics" and scant mention is made of chemistry, astronomy, geology, or modern physics. The book might have been considerably enlivened by inclusion of more recent physical science drawn from a wider variety of fields. It is doubtful that one would turn to Boas for reference purposes with more comprehensive sources available. Despite its rather grand title, the book by Anderson covers only a few topics. The author states that "this book attempts to lay down a central core of physiod and mathematical background for quantum chemistry in general, but for moleculm spectroscopy in particular." This ambitious program is wisely no1 attempted in the 149 pagesavailable. What we actually get is an excellent 41-page summary of orthogonal functions, which is always clear, concise, and to the point [except for serious misprints in eq. (2-102)l. This is followed by 56 pages on abstract linear algebra. The final section of the hook attempts to discuss the classical merhal~ic.;of vibration, and n,tations in 30 1,nge.i 2nd predirtnldy) f s i l i t u preient the nlawrixi ndrountelv. I'ntii this frrlal chapter ~ndersan.mttin'tainsa.rigid wall of separation between mathematics and science. Does he expect the science student to study all the mathematics first before being shown any of the applications? The important place of this book will be as a valuable supplement to a c o m e in molecular structure or quantum chemistry. These three books left me with the feeling that the task outlined by Professor Pillow has not yet been seriously begun. For example, none of these books makes reference to the existence of giant computers which have already produced a revolutionary expansion in the theoretical treatment of chemical problems. (It is as if a physician wrote a treatise on bacterial diseases without reference to antibiotics.) We are far from havinu solved
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chemistry teaching. The authors of these books deserve our thanks for attacking some aspects of an important problem in a practical way. We may hope that many more books will be written in this field so that we may
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Journal o f Chemical Education
learn through experiment how best to present to university students a coherent understanding of mathematics as applied in the chemical sciences. It seems that we might seriously consider a solution in which we would withdraw our chemistry students from the pure mathematicians and arrange for them to be taught by specialists in applied mathematics, by theoretical chemists, or both. WALTERJ. MOORE Indiana University Bloominglon
Molecular Aspects of Symmetry Robin M . Hochstrasser, University of Pennsylvania, Philadelphia. W. A. Benjamin, Inc., New York, 1966. 355 pp. Figs. and tables. xiii 16 X 23.5 em. $14.75.
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In recent yesrs a number of books or parts of books have appeared in which some results of group theory are presented in an elementary way for application to systems of physical and chemical interest. This trend is not fallowed by Professor Hochstrasser; the present volume would be undertaken most profitably Bfter a good introductory course in quantum mechanics. The level of presentation is similar to that found in the books by Tinkham and by Heine, in which group theory and its applications in physics are considered. More than 40 percent of the text is devoted to mathematical background, group thwry itself, including representations, and a discussion of the various point groups. In the remainder of the book numerous applications are discussed, including angular momenta of atoms and molecules, MO states and symmetry, selection rules for electric and magnetic dipole transitions, vibrrttiond and vibronic states, spin-orbit coupling, singlettriplet transitions, spin-vibronic interactions, and the symmetry and spectra of organic molecular crystals. The author adopts a. fundamental viewpoint; he derives theorems, beginning at the basic principles. The style is brief though precise, and it is inevitable that numerous mathematical expressions appear. For these reasons a student would find it very difficultto work systematically through the book by himself; an instructor and course atmosphere would be invaluhble. However, in the sections covering applications numerous detailed examples are given; it should be possible for a student to follow these and obtain useful Unforresults for similar systems. tunately, few problems or exercises are provided. The typography and proofreading seem of high caliber; only rare ermm were noted, and they were minor. This book should be exceedingly useful to those working in electronic spectroscopy; it could do much to broaden the outlook of those in related areas. I believe the price is too high; a volume of this nature should
be owned by the user, ratner than by the institution or library. D. F. EGGERS,JR. Uniwersily of Washinglm SeatU..
The Principles of Chemical Equilibrium
Kenneth Denbigh, University of London. Cambridge, a t the University Press, 494 pp. Figs. 1966. 2nd ed. xxi and tables. 14.5 X 22 cm. Paperbound, $3.95; clothbound, $9.50.
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My high opinion of this book has not changed since its first publication in 1955 (J. CHEM.EDUC.,33 305 [1956]), but I would not c d the present volume a. revision. The only change in the Table of Contents is the addition of two pages to Chapter 1. The rest of the changes are decidedly minor, and represent a tidying up, an updating of the references, m d a use of mare contemporary terminology. But, after all, if you do an excellent jab the first time, there is not much that can be added to a basic treatment of thermcdynamics-the subject having been around for some time now. This is still a fine book, and should be carefully examined before anyone chooses another volume from which to teach classical and statistical thermodynamics.
J. A. CAMPBELL Harvey Mudd College Claremoni, California
Thermoanalytical Methods of Investigation Paul D. Garn, University of Akron, Akron, Ohio. Academic Press, Inc., 606 pp. New York, 1965. xvi Figs. and tables. 16 X 23.5 em. $19.50.
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In the ten-year period preceding the publication of Dr. Garn's book, thermoanalytical methods grew from a few scab tered experiments with rocks and homemade DTA units or quartz spring McBain type balances, to a sophisticated multimillion dollar industry which has placed complex electronic thermal boxes in many laboratories. The growth parallels that of gas chromatography, but unlike that field, there are few useful books to which a new investigator can turn for a comprehensive assessment of the field. Dr. Gam hss attempted to fill part of that void with his extensive treatise. In its 17 chapters, hediscusses thevarious types of apparatus, methods of sample preparation and a ~ e r s t i n econditions for makine differ-
the qualitative and quantitative interpretations of DTA curves. He also reports attempts to extract kinetic dats from such curves. His review of thermogravimetric apparatus, procedures, and associated techniques is similarly detailed. Through(Continued ma page AAS64)
