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The dial machine, an apparatus for the elementary mathematical laboratory
April, I927.]
BOOK REVIEWS.
597
The book contains a large amount of information on modern methods of inorganic analysis and on the theoretical side with elaborate explanation of the physico-chemical principles applicable to analytical problems. HENRY LEFFMANN. MODERN ASTRONOMY: ITS RISE AND PROGRESS. By Hector Macpherson, M.A., Ph.D. vi-I96 pages, illustrated, small 8vo. New York, O x f o r d University Press, American Branch, 1926. Price, $2. This is a popular treatment of the subject but goes 'quite comprehensively into the history of astronomy and the development of the modern information and theories. In examining a book of this kind, one turns with some interest to the discussion of Mars which is a sort of storm centre in modern popular astronomy. It is to be regretted that the author has been so enthusiastic in his presentation of what may be called the " Lowellian" views as to Mars. The popular view concerning the possibility of Mars having intelligent beings on its surface seems to be based very largely, as so many other popular views are, upon a mistranslation. W h e n Schiaparelli saw the lines on the Martian surface he called them in his own language "canali " which means "channels." The word was unfortunately translated " canals," and this has come into the English language, apparently permanently, concerning this ruddy-faced planet. " Channels," of course, suggest natural conditions; " canals" carry with them the idea of intelligent purposeful construction. Thus an important point is gained before the general public in developing the notion that Mars contains beings capable of doing work. The phantasy does not stop here. Changes in the tint of the surface are ascribed to the growth and disappearance of vegetation and even some suggestions are made as to the character of this. It is true that the author in speaking of Schiaparelli's work states that the word he used means " channels," but he uses " canals " in all reference to the markings on Mars. Speculation has gone so far as to suggest that there are beings on that planet who are comparable to ourselves and it may be remembered that during the recent near approach to the earth, an American astronomer made the remarkable request that broadcasting should cease for a limited time in order that any message from Mars might be detected. Imagination has no place in scientific work except possibly as a suggestion for research. certainly not in inferential matters. So far as the general text of the book is concerned, it is very readable and doubtless will afford to the unscientific reader considerable entertainment and interest, but it is greatly to be regretted that the interpretations have been so liberal and there has not been more of close scientific criticism presented. HENRY LEFFMANN. THE DIAL MACHINE, an apparatus for the elementary mathematical laboratory. By T. C. J. Elliott. 93 pages, I8.4 × I2.7 cm., flexible cloth. Peterborough, The Peterborough Press, I926. Price, 4/6. The Dial Machine is a means of illustrating mechanically the idea of functions and variables. The " m a c h i n e " is rather a system than a definite mechanism. A form which is illustrated consists of a square board upon which are mounted nine dials with graduations. These dials are arranged
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.
BOOK REVIEWS.
597
The book contains a large amount of information on modern methods of inorganic analysis and on the theoretical side with elaborate explanation of the physico-chemical principles applicable to analytical problems. HENRY LEFFMANN. MODERN ASTRONOMY: ITS RISE AND PROGRESS. By Hector Macpherson, M.A., Ph.D. vi-I96 pages, illustrated, small 8vo. New York, O x f o r d University Press, American Branch, 1926. Price, $2. This is a popular treatment of the subject but goes 'quite comprehensively into the history of astronomy and the development of the modern information and theories. In examining a book of this kind, one turns with some interest to the discussion of Mars which is a sort of storm centre in modern popular astronomy. It is to be regretted that the author has been so enthusiastic in his presentation of what may be called the " Lowellian" views as to Mars. The popular view concerning the possibility of Mars having intelligent beings on its surface seems to be based very largely, as so many other popular views are, upon a mistranslation. W h e n Schiaparelli saw the lines on the Martian surface he called them in his own language "canali " which means "channels." The word was unfortunately translated " canals," and this has come into the English language, apparently permanently, concerning this ruddy-faced planet. " Channels," of course, suggest natural conditions; " canals" carry with them the idea of intelligent purposeful construction. Thus an important point is gained before the general public in developing the notion that Mars contains beings capable of doing work. The phantasy does not stop here. Changes in the tint of the surface are ascribed to the growth and disappearance of vegetation and even some suggestions are made as to the character of this. It is true that the author in speaking of Schiaparelli's work states that the word he used means " channels," but he uses " canals " in all reference to the markings on Mars. Speculation has gone so far as to suggest that there are beings on that planet who are comparable to ourselves and it may be remembered that during the recent near approach to the earth, an American astronomer made the remarkable request that broadcasting should cease for a limited time in order that any message from Mars might be detected. Imagination has no place in scientific work except possibly as a suggestion for research. certainly not in inferential matters. So far as the general text of the book is concerned, it is very readable and doubtless will afford to the unscientific reader considerable entertainment and interest, but it is greatly to be regretted that the interpretations have been so liberal and there has not been more of close scientific criticism presented. HENRY LEFFMANN. THE DIAL MACHINE, an apparatus for the elementary mathematical laboratory. By T. C. J. Elliott. 93 pages, I8.4 × I2.7 cm., flexible cloth. Peterborough, The Peterborough Press, I926. Price, 4/6. The Dial Machine is a means of illustrating mechanically the idea of functions and variables. The " m a c h i n e " is rather a system than a definite mechanism. A form which is illustrated consists of a square board upon which are mounted nine dials with graduations. These dials are arranged
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.