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Interpolation
598
Boo~: REVIEWS.
[J. F. I.
in square form in three rows and three columns, and each one is provided with a stationary pointer at which the angular movement may be read. The axles of the dials which pierce the board may be coupled by gearing arranged at the opposite face of the board. The gearing may be covered and made inaccessible to the operator so that the angular movements of the dials may be read without a knowledge of these connections. The plan of operation contemplated is to tabulate against the motion of a given dial the motions of the others. That is, regarding the motion of the operating dial as the independent variable, to determine the value of the functions from the movements of the others. Many variations may be made including, by the use of epicyclic gearing, functions of more than one variable. W h a t advantage there is in a device of this kind over representation by co/Jrdinated graphs is not apparent. The dial machine can represent only linear functions, w h i l e by the common graphic method functions of any character may be represented. The graph method, moreover, can be practically applied to the analysis or design of complex mechanisms by plotting against a common base, which represents the movement of a driving element, the movements of each component operating part. LucmN E. PICOLET. INTERPOLATION. By J. F. Steffensen, Sc.D., Professor of Actuarial Science at the University of Copenhagen. ix-248 pages, 23 x I5 cm., cloth. Baltimore, The Williams and Wilkins Company, I927. Price, $8. The process of interpolation by the method of differences as applied to the construction of mathematical tables or to statistical data to which the method is applicable has long been developed into a form suitable for practical use, and adequate texts are available which cover the details of these and other applications. The author of this volume points out that despite those developments a non-rigorous point of view is still dominant in existing texts and that formulas and methods developed on the assumption that the function under consideration is a polynomial are also applied to /unctions which are not defined. The work of recent investigators, he further states, has developed a number of formulas with remainder terms in a form suitable to the ready determination of the limits of error. The present text, which is based upon lectures of the author to actuarial students at the University of Copenhagen, is in harmony with these recent developments. The reader is reminded that the work is a text-book and not a reference manual, a fact which becomes apparent upon encountering the subtilities of the operator notation which is employed. The compass of the work is best set forth by an enumeration of the topical headings, which are as follows: Introduction, Displacement-Symbols and Differences, Divided Differences, Interpolation-Formulas, Some Applications, Factorial Coefficients, Numerical Differentiation, Construction of Tables, lhverse Interpolation, Elementary Methods of Summation, Repeated Summation, Laplace's and Gauss's Summation-Formulas, Bernoulli's Polynomials. Euler's Summation-Formula, Lubbock's and Woolhouse's Formulas, Mechanical Quadrature, Numerical Integration of Differential Equations, The Calculus of Symbols, Interpolation with Several Variables, Mechanical Cubature, On Differential Coefficients of Arbitrary Order.
April, I927.]
BOOK R E V I E W S .
599
The critical consideration of the remainder term in the development of the formulas and the reduction to a practical form of this term adapted to numerical application in determining the limits of error of a derived value is a unique and valuable feature and in some cases a real necessity. Some twenty pages at the beginning of the work are devoted to an explanation of the special notations adopted and to the demonstration of theorems to be applied in deriving the formulas. Every feature in the derivation of an applicable formula is minutely examined and examples worked out in detail illustrate the conditions under which each should be used. Attention is also given to errors arising from " r o u n d i n g o f f " tabular values. In this division, as indeed throughout the work, derivations are made in the most general terms. The author places a rather high estimate on the analytical capacity of readers of limited mathematical attainment in saying: " The mathematical equipment required in order to master this book is very small." As such things are viewed here, only a reader well trained in analytical operations, and presumably of considerable mathematical attainment, could profitab.ly undertake a perusal of the work. As to the considerable price, a perfunctory examination discloses costly type-setting of a multiplicity of special symbols on nearly every page which, it should b e mentioned, constitute an unusually creditable piece of mathematical typography. This presentation of the subject of interpolation is deductive, rigorous and exhaustive in character, and probably no other text-book in English combines all of these qualities in the degree attained in this one. LUClEN E. PICOLET. INTRODUCTION TO CONTEMPORARY PHYSICS. By Karl K. Darrow, Ph.D. 453 pages, illustrations, 8vo. New York, D. Van Nostrand Company, I926. Price, $6. Readers of the Bell System Technical Journal have followed with interest the series of articles on " Some Contemporary Physics," by Dr. Karl K. Darrow, member of the Technical Staff. These articles have now been collected, many of them being recast and expanded into book form and published under the title "Introduction to Contemporary Physics." The surprising changes that have taken place in physics in the past quarter of a century make this book of especial value in that it brings together, in consecutive and logical order, the phenomena that have been studied and the theories that have been developed to account for them. The discovery of X-rays and radio-activity demand an explanation, and this led to an attack upon the atom, the indivisible unit of Dalton--and this attack resulted in giving us as many complex structures for the atom as there are elements. The author devotes his first and second chapters to a discussion of the experimental electron and the experimental atom. The " experimental " atom the author defines as the atom found in n a t u r e - - t h e atom having measurable qualities of mass, magnetic moment and charge. These are distinct from the various atom-models that have been suggested from the various theories. A similar distinction is made between the electron-model and the " experimental " electron whicb is discussed in the first chapter.
Boo~: REVIEWS.
[J. F. I.
in square form in three rows and three columns, and each one is provided with a stationary pointer at which the angular movement may be read. The axles of the dials which pierce the board may be coupled by gearing arranged at the opposite face of the board. The gearing may be covered and made inaccessible to the operator so that the angular movements of the dials may be read without a knowledge of these connections. The plan of operation contemplated is to tabulate against the motion of a given dial the motions of the others. That is, regarding the motion of the operating dial as the independent variable, to determine the value of the functions from the movements of the others. Many variations may be made including, by the use of epicyclic gearing, functions of more than one variable. W h a t advantage there is in a device of this kind over representation by co/Jrdinated graphs is not apparent. The dial machine can represent only linear functions, w h i l e by the common graphic method functions of any character may be represented. The graph method, moreover, can be practically applied to the analysis or design of complex mechanisms by plotting against a common base, which represents the movement of a driving element, the movements of each component operating part. LucmN E. PICOLET. INTERPOLATION. By J. F. Steffensen, Sc.D., Professor of Actuarial Science at the University of Copenhagen. ix-248 pages, 23 x I5 cm., cloth. Baltimore, The Williams and Wilkins Company, I927. Price, $8. The process of interpolation by the method of differences as applied to the construction of mathematical tables or to statistical data to which the method is applicable has long been developed into a form suitable for practical use, and adequate texts are available which cover the details of these and other applications. The author of this volume points out that despite those developments a non-rigorous point of view is still dominant in existing texts and that formulas and methods developed on the assumption that the function under consideration is a polynomial are also applied to /unctions which are not defined. The work of recent investigators, he further states, has developed a number of formulas with remainder terms in a form suitable to the ready determination of the limits of error. The present text, which is based upon lectures of the author to actuarial students at the University of Copenhagen, is in harmony with these recent developments. The reader is reminded that the work is a text-book and not a reference manual, a fact which becomes apparent upon encountering the subtilities of the operator notation which is employed. The compass of the work is best set forth by an enumeration of the topical headings, which are as follows: Introduction, Displacement-Symbols and Differences, Divided Differences, Interpolation-Formulas, Some Applications, Factorial Coefficients, Numerical Differentiation, Construction of Tables, lhverse Interpolation, Elementary Methods of Summation, Repeated Summation, Laplace's and Gauss's Summation-Formulas, Bernoulli's Polynomials. Euler's Summation-Formula, Lubbock's and Woolhouse's Formulas, Mechanical Quadrature, Numerical Integration of Differential Equations, The Calculus of Symbols, Interpolation with Several Variables, Mechanical Cubature, On Differential Coefficients of Arbitrary Order.
April, I927.]
BOOK R E V I E W S .
599
The critical consideration of the remainder term in the development of the formulas and the reduction to a practical form of this term adapted to numerical application in determining the limits of error of a derived value is a unique and valuable feature and in some cases a real necessity. Some twenty pages at the beginning of the work are devoted to an explanation of the special notations adopted and to the demonstration of theorems to be applied in deriving the formulas. Every feature in the derivation of an applicable formula is minutely examined and examples worked out in detail illustrate the conditions under which each should be used. Attention is also given to errors arising from " r o u n d i n g o f f " tabular values. In this division, as indeed throughout the work, derivations are made in the most general terms. The author places a rather high estimate on the analytical capacity of readers of limited mathematical attainment in saying: " The mathematical equipment required in order to master this book is very small." As such things are viewed here, only a reader well trained in analytical operations, and presumably of considerable mathematical attainment, could profitab.ly undertake a perusal of the work. As to the considerable price, a perfunctory examination discloses costly type-setting of a multiplicity of special symbols on nearly every page which, it should b e mentioned, constitute an unusually creditable piece of mathematical typography. This presentation of the subject of interpolation is deductive, rigorous and exhaustive in character, and probably no other text-book in English combines all of these qualities in the degree attained in this one. LUClEN E. PICOLET. INTRODUCTION TO CONTEMPORARY PHYSICS. By Karl K. Darrow, Ph.D. 453 pages, illustrations, 8vo. New York, D. Van Nostrand Company, I926. Price, $6. Readers of the Bell System Technical Journal have followed with interest the series of articles on " Some Contemporary Physics," by Dr. Karl K. Darrow, member of the Technical Staff. These articles have now been collected, many of them being recast and expanded into book form and published under the title "Introduction to Contemporary Physics." The surprising changes that have taken place in physics in the past quarter of a century make this book of especial value in that it brings together, in consecutive and logical order, the phenomena that have been studied and the theories that have been developed to account for them. The discovery of X-rays and radio-activity demand an explanation, and this led to an attack upon the atom, the indivisible unit of Dalton--and this attack resulted in giving us as many complex structures for the atom as there are elements. The author devotes his first and second chapters to a discussion of the experimental electron and the experimental atom. The " experimental " atom the author defines as the atom found in n a t u r e - - t h e atom having measurable qualities of mass, magnetic moment and charge. These are distinct from the various atom-models that have been suggested from the various theories. A similar distinction is made between the electron-model and the " experimental " electron whicb is discussed in the first chapter.