Structure and Mathematics of the Principal Calendars of the Western World

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This monograph studies the history/structure and mathematics of calendars used in those parts of the world west of the Indian subcontinent. The book begins with a brief account of the origins of calendars, the sparse information being supported by references to biblical and other ancient sources. After this, there is an explanation of the astronomical basis of time measurement. The remainder of the book includes information on the structures of a number of ancient calendars, while those used in more modern times are explained in detail, and precise instructions, including worked examples, are given for the conversion of dates to the Gregorian calendar and for the calculation of Christian, Jewish and Muslim holy days.


“Though the subject of time reckoning has existed from time immemorial, this book displays its originality in several ways: it discusses a variety of ancient and later calendars from all over the world except the Far East; it compares and contrasts the different methods invented and employed by the various civilisations to calculate time; and it devises mathematical formulae to enable the reader to move swiftly between one calendar and another ... the more one studies in depth the mathematical formulae for the calculation of various calendars, the more one is impressed by the ingenuity of our ancestors, who not only discovered the cycle of time, but also invented the calendars to calculate it.” – (from the Preface) Avihai Shivtiel, Visiting Senior Research Fellow, University of Cambridge

“One of the strengths of this work is the sheer range of calendars that Dr. Kapel discusses. Although we are all too likely to take the calendar for granted, Dr. Kapel rightly reminds us that calendars are neither mathematically simple nor manifestly transparent ... This work is for readers with some mathematical knowledge who wish to enter this fascinating area and to see how a wide variety of different calendars were constructed. Without becoming bogged down in the complexities, Dr. Kapel gives a clear and concise outline of the principles on which each of these calendars were constructed.” – Professor Geoffrey Cantor, The University of Leeds

“Dr. Martin Kapel is a teacher with an interest in mathematics, and with many years of dedicated experience in teaching at the highest level. This book is the fruit of his lifelong interest in the field of calendars ... The complexity of the material has inspired the author to anticipate many of the difficulties of the student of the subject and to provide practical help for the necessary calculations in a clear and direct way ... This book is thus not only an aid to the mechanics of different calendars, but also a contribution to the understanding of the peoples of the western world whose calendars are so competently described.” – Nina L. Collins, Lecturer in Modern Hebrew, The University of Leeds

“The book deals primarily with the mathematics of calendars. It describes the calculations needed to convert dates from one system to another in sufficient detail to enable a disciplined reader armed only with arithmetic to perform them. The book’s strengths are these descriptions and the many worked examples … The author has added some historical information to his largely mathematical book.” – J.L. Heilbron, Professor of History and Vice Chancellor, Emeritus, University of California, Berkeley

Table of Contents

Preface by Dr. Avihai Shivtiel
1. The Origins of Calendars
2. The Astronomical Basis of Time Measurement
3. The Muslim Calendar
4. The Development of Solar Calendars
5. The Gregorian Church Calendar
6. The Jewish Calendar
7. Some Obsolete and Other Calendars
8. Computer Programmes for Calendars
Appendix 1: The Equation of Time
Appendix 2: The Gregorian Perpetual Calendar
Appendix 3: The Beginning of the First Julian Period
Appendix 4: The Pronunciation of Jewish Months and Holy Days
Appendix 5: The Trilateral Description of the Jewish Year
Appendix 6: The First Magnitude Planets and Stars
Appendix 7: The Names of the Sidroth
Appendix 8: The Iranian Calendar
Appendix 9: The Baha’i Calendar
Appendix 10: The Sikh Calendar

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