The Science of the Shulba: A Study in Early Hindu Geometry c25: Errata

The Science of the Śulba: A Study in Early Hindu Geometry c25

Language: English

Errata

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Page 45, line 11, *for* tringle *read* triangle

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,, ,, ,, 17, *for* doidical *read* boidical

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,, 68, ,, 10, · *for* *abcd* *read* *abcD*

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,, 189, ,, 29, *for* account *read* accounts

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SOME OF THE UNIVERSITY PUBLICATIONS ON MATHEMATICS

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*Matrices and Determinoids*. Vol. I (*Readership Lectures delivered at the Calcutta University*), by C. E. Cullis, M.A., Ph.D., D.Sc. Sup. Royal 8vo pp. 442. 1913. English price 24s. net.

Contents :—Chap. I—Introduction of Rectangular Matrices and Determinoids. ” II—Affects of the Elements and Derived Products of a Matrix or Determinoid. ” III—Sequences and the Affects of Derived Sequences. ” IV—Affects of Derived Matrices and Derived Determinoids. ” V—Expansions of a Determinoid. ” VI—Properties of a Product formed by a Chain of Matrix Factors. ” VII—Determinoid of a Product formed by Chain of Matrix Factors. ” VIII—Matrices of Minor Determinoids. ” IX—Rank of a Matrix and Connections between the rows of a Matrix. ” X—Matrix Equations of the First Degree. ” XI—Solution of Any System of Linear Algebraic Equations.

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Prof. Cullis will earn the gratitude of mathematical students for affording them the opportunity of obtaining a right perspective of an important branch of pure mathematics, whose developments so far have appeared in scattered notes and memoirs not always easy of access. Examples are abundant, and, while a large number of them are llustrative, there is a good collection of suggestive exercises indicating the directions in which further original work may be done. —The Journal of Education.

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The chief feature of this book is that it deals with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it just as a determinant is related to a square matrix. The author endeavours to set forth a complete theory of these two subjects, and uses the first volume to give the most fundamental portions of the theory. Two more volumes are promised, the second to give the more advanced portions of the theory, and the third its applications.

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∗ The right of publication of this book is held by the Cambridge University Press (Fetter Lane, London, E. C. 4, on behalf of the Calcutta University and copies of the book may be had of the firm.

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This is new ground and the author has done a splendid piece of work and with the publishers deserves much credit.—*Mathematical Teacher (Syracuse, U. S. A.).*

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Matrices and Determinoids, Vol. II. Sup. Royal 8vo pp. 573. 1918. *English price 42s. net.

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| *Contents :*—Chap. | XII— | Compound Matrices. |
| " | XIII— | Relations between the Elements and Minor Departments of a Matrix. |
| " | XIV— | Some Properties of Square Matrices. |
| " | XV— | Ranks of Matrix Products and Matrix Factors. |
| " | XVI— | Equigradent Transformations of a Matrix whose Elements are Constants. |
| " | XVII— | Some Matrix Equations of the Second Decree. |
| " | XVIII— | The Extravagances of Matrices and of Spacelets in Homogeneous Space. |
| " | XIX— | The Paratomy and Orthotomy of Two Matrices and of Two Spacelets of Homogeneous Space. |

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The outstanding feature of the work, which the author properly emphasises, is the detailed discussion of rectangular, as distinguished from square, matrices. For this reason alone the work ought to give a great stimulus to the subject, and we hope that the publication of the whole treatise will not be long delayed. Until it is finished, it will be difficult, if not impossible, to give a proper appreciation of it, especially as the author introduces so many new symbols and technical terms. One thing, however, is certain; we now have the outlines of a calculus of matrics in which the operations of addition, subtraction, and multiplication are definite.—*Nature.*

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The present volume worthily maintains the traditions of the Cambridge University Press, and is a most valuable addition to the rapidly growing series of volumes for which the Readership at the University of Calcutta is responsible.—*Science Progress.*

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**The right of publication of this book is held by the Cambridge University Press (Fetter Lane, London, E. C. 4) on behalf of the Calcutta University and copies of the book may be had of the firm.*

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Matrices and Determinoids, Vol. III, Part I. Royal 8vo pp. xx + 682. 1926. *English price £3 3s. net. Indian price Rs. 45.

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Appendices.

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Chapters on Algebra (being the First Three Chapters of Matrices and Determinoids, Vol. III), by C. E. Cullis, M.A., Ph.D., D.Sc. Sup. Royal 8vo pp. 191. 1920. Rs. 11-4.

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This Volume deals with rational integral functions of several scalar variables as also with functional matrices.

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Functions of Two Variables, by A. R. Forsyth, F.R.S. Sup. Royal 8vo pp. 300. 1914. Rs. 11-4.

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*\* The right of publication of this book is held by the Cambridge University Press (Fetter Lane, London, E. C. 4) on behalf of the Calcutta University and copies of the book may be had of the firm.*

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The author's purpose is to deal with a selection of principles. and generalities that belong to the initial stages of the theory of functions of two complex variables. The consideration of relations between independent variables and dependent variables has been made more complete with illustrations in this publication.

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Analytical Geometry of Hyper-spaces, Part I (*Premchand Roychand Studentship thesis, 1914*), by Surendramohan Gangopadhyay, D.Sc. Demy 8vo pp. 93. 1918. Re. 1-14.

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Do., II. Demy 8vo pp. 121. 1922. Rs. 3-12.

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It deals with certain interesting problems in *n*-dimensional Geometry, the method adopted being one of deduction from first principles. The second part contains certain interesting results in the Geometry of Hyper-spaces, which is now recognised as an indispensable part of the science with extensive applications in mathematical Physics. In the treatment of subject-matter, the easiest possible methods have been adopted, so that the discussions can be followed by an ordinary student of Mathematics without a knowledge of Higher Mathematics.

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Theory of Higher Plane Curves, Vol. I, by Surendramohan Gangopadhyay, D.Sc. (*Third Edition, thoroughly revised and enlarged.*) Demy 8vo pp. 396 + xxi. 1931. Rs. 6-8.

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The work is designed to meet the Syllabus prescribed by the University for the Master's Degee and is intended as an introductory course suitable for students of Higher Geometry. The present volume which is a thoroughly revised and enlarged edition of the earlier includes new materials together with recent researches which will not only be of use to the students for the Master's course but will also encourage independent thinking in students of higher studies engaged in research work.

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Theory of Higher Plane Curves, Vol. II, by Surendramohan Gangopadhyay, D.Sc. (*Second Edition, thoroughly revised and enlarged.*) Demy. 8vo pp. 408. 1926. Rs. 4-8.

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This volume deals with the application of the theory in studying properties of cubic and quartic curves.

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This volume is an endeavour to give as complete an account of the properties of cubic and quartic curves as could be compressed within the limits of a single volume of moderate size, confining the discussion to the prominent characteristics of these curves. The subject has been presented in clear and concise form to students commencing a systematic study of the higher curves, indicating references to original sources as far as practicable. It is very useful to students of higher plane curves.

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Parametric Co-efficient (*Griffith Memorial Prize*, 1910), by Prof. Syamadas Mukhopadhyay, M.A., Ph.D. Demy 8vo pp. 31. Rs. 3-0.

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Collected Geometrical Papers, by Prof. Syamadas Mukhopadhyay, M.A., Ph.D. Crown 4to pp. viii + 158. Rs. 4-0.

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Part II. Crown 4to pp. vi + 137. Rs. 3-8.

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Professor J. Hadamard Paris : “My interest in your new methods in the geometry of a plane arc, which I had expressed in 1909 in an (anonymous) note in the revue generale des sciences, has far from diminished since that time.

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Precisely at my seminaire or colloquium of the College de France, we have reviewed such subjects and all my auditors and colleagues have been keenly interested in your way of researches which we all consider as one of the most important roads open to Mathematical Science.”

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Professor F. Engel, Geissen : “I am surprized over the beautiful new calculations on the right-angled triangles and three-right-angled quadrilaterals (in hyperbolic geometry)..... Your analogies in the Gaussian Pentagramma Mirificum are highly remarkable.”

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Professor W. Blaschke, Hamburg : “I am much obliged to you for your kind sending of your beautiful geometrical work. When, as I hope, a new edition of my Differential Geometry comes out. I shall not forget to mention that you were the first to give the beautiful theorems on the numbers of Cyclic and Sextactic points on an oval.”

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Prof. Blaschke has quoted S. Mukhopadhyay in the third edition (1930) of the first volume of his classical work on Differential Geometry.

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Professor A. R. Forsyth, London : “The first part of your Collected Geometrical Papers is an attractive record of fine mathematical attainment : and I am glad to learn, not only of the manifest advances you have made in our science, but also of the stimulus your work has afforded to other investigators. I can offer you no better wish and suggest no prouder aim than continual success in your Researches.”

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*Prof. T. Hayashi, Japan:* "Your Collected Geometrical Papers, Part I, is very important to the progress of geometry and is to be highly appreciated by geometricians in the world. We regret indeed that most of your valuable papers have remained unknown till now.

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But the collected papers should be welcome to raise your position in geometrical research."

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*Professor F. Cajori, California:* "I congratulate you upon your success in research. If ever I have the time and opportunity to revise my History of Mathematics I shall have occasion to refer to your interesting work."

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*Professor P. Montel, Paris:* "Many thanks for sending me your beautiful work "Collected Geometrical Papers." I am acquainted with many of the memoirs inserted therein and I am happy in having them in a handy united form. I am thus in a position to judge anew of the simplicity and rigour of your methods which have led you to elegant results."

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*Prof. L. Godeau, Liege:* "A first reading of your papers has roused my keen interest. I intend making an exposition of these questions early to my students of Geometrie Superieure, an exposition to which I reckon to join that of works of M. Juel."

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*Prof. T. Levi-Civita, Rome:* "I have received the valuable, very ingenious papers you have had the kindness to send me. I have no special knowledge of the subjects you have treated in so deep and interesting manner. But also a general reader of mathematical papers is able to appreciate the results of your investigations and the penetrating methods you have employed."

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Vector Calculus (*Griffith Memorial Prize*, 1917), by Durgaprasanna Bhattacharyya, M.A. Demy 8vo pp. 91. Rs. 3-0.

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An attempt has been successfully made in this book by the author to place the foundation of vector-analysis on a basis independent of any reference to Cartesian co-ordinates and to establish the main theorems of that analysis directly from first principles as also to develop the differential and integral calculus of vectors from a new point of view.

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Solutions of Differential Equations (*Premchand Roychand Studentship thesis*, 1896), by Jnansaran Chakravarti, M.A. Demy 8vo pp. 54. Rs. 3-12.

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The subject of the book is an enquiry into the nature of solutions of differential equations, chiefly with reference to their geometrical interpretation, and the investigation of the connection that exists between the complete primitive and singular solution.

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Reciprocal Polars of Conic Sections (Premchand Roychand Studentship thesis, 1900), by Krishnaprasad De, M.A. Demy 8vo pp. 66. Rs. 3.

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An Introduction to the Theory of Elliptic Functions and Higher Transcendentals, by Ganesh Prasad, M.A., D.Sc., Hardinge Professor of Higher Mathematics, Calcutta University. Royal 8vo pp. 110. 1928. Rs. 3-12.

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Theory of Fourier Series, by Ganesh Prasad, M.A., D.Sc. Royal 8vo pp. 152. 1928. Rs. 5-4.

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From a letter to the Registrar from Professor Henri Lebesgue of the Paris University, Member of the Institute of France (translated into English) :

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"PARIS,

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The 19th October, 1928.

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SIR,

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I have the honour to acknowledge the receipt of 'Six Lectures on recent Researches in the Theory of Fourier Series,' by M. Prof. Ganesh Prasad.

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I have pleasure in finding in that work a simple and clear exposition of the actual state of advance of certain of the most important problems concerning trigonometrical series. The documentation is true and complete : it is only once that I have had occasion to find anything in which the erudition of the author appears to be in default : M. Kolomogoroff, pursuing the studies indicated on p. 53, has obtained an example of a function of summable square of which the Fourier Series diverges everywhere.

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For justifying the enunciation which he gives, M. Ganesh Prasad utilises the original demonstration of the first author : then he gives a historical note, very interesting by the side of the old demonstration. M. Prasad gives always, whenever possible, as simple a proof as the question under consideration would allow. Many of these proofs are due to M. Prasad himself, for example, that which M. Prasad gives on pages 60-61 for a criterion for the summability (C 1) which I enunciated at another time.

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M. Prasad presents his researches elegant and interesting, by which he has carried further the classical work of du Bois-Reymond."

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From the review by Professor L. Bieberbach of the Berlin University in the Jahresbericht der deutschen Mathematiker-Vereinigung (translated into English) : "The work gives a comprehensive account of the results on the convergence and summability of Fourier Series, things about which the author has also earned merit."

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Six Lectures on the Mean Value Theorem of the Differential Calculus, by Ganesh Prasad, M.A.. D.Sc., Hardinge Professor of Higher Methematics, Calcutta University. Royal 8vo pp. 108 + viii. 1931. Rs. 3.

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*From a letter to the Registrar from Professor E. R. Hedrick of the University of California, Los Angeles, and President of the American Mathematical Society :*

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“October 28, 1931.

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DEAR SIR,

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I am writing to thank you and to express my appreciation of the book itself and of your kindness in sending it to me. The scholarly work of Professor Prasad is known to mathematicians throughout the world and I feel sure that the present volume will add greatly to his reputation as an eminent mathematician.”

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*From a letter to the Registrar from Professor A. Pringsheim of the University of Munich (translated into English) :*

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“MUNICH.

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10th December, 1931.

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VERY HONOURED MR. MUKHERJEE,

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For the sending of the beautiful book of Prof. Prasad on the mean-value theorem of the Differential Calculus, which has interested me vividly, I express to you my sincerest thanks.”

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Khandakhadyakam, edited by Pandit Babua Misra, Jyotishacharyya. Demy 8vo pp. 217. 1925. Rs. 2.

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The book is an astronomical work by the great Scholar Brahmagupta. It contains the commentary called Vāsanā-Bhāṣya by Amarāja. This is the only available work which describes one of the two systems of astronomy as taught by Āryabhaṭa I (born 476 A.D.), generally known as Ardharātrika system and is different from the Audayika System as taught in his Ārya-bhaṭiyam. It was widely read by Arab Scholars and was known by the name of Alarkand. Hence it is a very important work on the History of Hindu Astronomy.

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The *fraternities* of the Middle Ages, which were so numerous and so varied in their objects, were not all of the same origin. Some were religious, some social, some charitable, and some industrial. The religious guilds, or *gilds*, as they were often called, were associations of laymen for the purpose of performing religious duties, such as the maintenance of a chaplain, the celebration of masses, or the support of a hospital. The social guilds were formed for the purpose of mutual aid and good fellowship, and often included in their objects the relief of the poor, the sick, and the aged. The charitable guilds were established for the purpose of supporting hospitals, almshouses, and other institutions of mercy. The industrial guilds, or *craft guilds*, were associations of men engaged in the same trade or industry, and were formed for the purpose of regulating the trade, protecting the interests of the members, and maintaining the quality of their work.

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The craft guilds, which became so powerful in the later Middle Ages, were the successors of the earlier *collegia* of the Roman Empire. The *collegia* were associations of artisans and tradesmen, which were recognized by the state and had a legal existence. When the Roman Empire fell, these associations were largely destroyed, but the idea of association for mutual protection and the regulation of trade survived. In the early Middle Ages, the *gilds* began to reappear, and by the twelfth century they were firmly established in many of the cities of Europe.

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The craft guilds were organized on a democratic basis, and every member had a voice in the management of the guild's affairs. The officers were elected by the members, and the rules and regulations were made by the guild as a whole. The guilds had the power to fix prices, to regulate the hours of labor, and to determine the conditions of apprenticeship. They also had the power to punish members who violated the rules, and to expel those who were guilty of serious offenses. The guilds were thus a powerful force in the economic life of the Middle Ages, and they played an important part in the development of the modern industrial system.

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The decline of the craft guilds began in the sixteenth century, when the rise of the factory system and the growth of international trade made it difficult for the small, local guilds to compete with the larger and more efficient industrial organizations. The guilds were also weakened by the growth of the power of the state, which began to regulate trade and industry directly, and by the rise of the capitalist class, which sought to break the power of the guilds and to establish a system of free competition. By the end of the eighteenth century, the craft guilds had largely disappeared, but their influence lived on in the trade unions and other labor organizations of the modern world.