Gears: Volume 3: A Concise History (Springer Series in Solid and Structural Mechanics, 12) 3030401634, 9783030401634

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Springer Series in Solid and Structural Mechanics 12

Vincenzo Vullo

Gears Volume 3: A Concise History

Springer Series in Solid and Structural Mechanics Volume 12

Series Editors Michel Frémond, Rome, Italy Franco Maceri, Department of Civil Engineering and Computer Science, University of Rome “Tor Vergata”, Rome, Italy

The Springer Series in Solid and Structural Mechanics (SSSSM) publishes new developments and advances dealing with any aspect of mechanics of materials and structures, with a high quality. It features original works dealing with mechanical, mathematical, numerical and experimental analysis of structures and structural materials, both taken in the broadest sense. The series covers multi-scale, multi-field and multiple-media problems, including static and dynamic interaction. It also illustrates advanced and innovative applications to structural problems from science and engineering, including aerospace, civil, materials, mechanical engineering and living materials and structures. Within the scope of the series are monographs, lectures notes, references, textbooks and selected contributions from specialized conferences and workshops.

More information about this series at http://www.springer.com/series/10616

Vincenzo Vullo

Gears Volume 3: A Concise History

123

Vincenzo Vullo University of Rome “Tor Vergata” Rome, Italy

ISSN 2195-3511 ISSN 2195-352X (electronic) Springer Series in Solid and Structural Mechanics ISBN 978-3-030-40163-4 ISBN 978-3-030-40164-1 (eBook) https://doi.org/10.1007/978-3-030-40164-1 © Springer Nature Switzerland AG 2020, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Aphorism

Historia est testis temporum, lux veritatis, vita memoriae, magistra vitae, nuntia vetustatis. (History is the witness of the times, the light of truth, the life of memory, the teacher of life, the messenger of antiquity) Marcus Tullius Cicero, De Oratore, 2.36

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To my wife Maria Giovanna, my sons Luca and Alberto, my nephew Nicolò and my students

Preface

Gears and gear drives are among the oldest artificial mechanisms invented by man. Rudimentary geared mechanisms, to be considered as the distant predecessors of the current gears, were in fact designed and built after the introduction of the potter’s wheel, and therefore almost certainly in times that blend with the mists of prehistory. Their evolution, from their introduction to today, has not been constant over time. It has been characterized by periods of acquisition of empirical knowledge, through the accumulation of field experiences, and scientific knowledge, through scientific theories and scientific methods. The set of acquired knowledge has led to the current gears, which today play an irreplaceable role in power mechanical transmissions with constant transmission ratio as well as in traditional, semi-automatic and automatic geared transmissions for motor vehicles, mechanical and electric industry, wind turbine generators, aerospace industry, especially the helicopter industry, etc. This is due to their extreme versatility and reliability of use. Gears and gear drives in fact allow to solve any practical problem within speed and power ranges that effectively meet the various needs that may arise, even in the most advanced current technology areas, and in those reasonably foreseeable for the future. In the first two volumes of this monothematic treatise on gears, entitled Gears, we discussed the main aspects concerning their design, namely in Vol. 1, the geometric and kinematic design of the various types of gears most commonly used in practical applications, also considering the problems concerning their cutting processes; in Vol. 2, various strengths’ problems of the gears and their load carrying capacity under actual working conditions, also providing the theoretical basis for a better understanding of the calculation relationships of the ISO standards. In order to provide a more complete picture of gears, including their evolution over time, we considered it useful and appropriate to also write a brief history of the same gears, which is the subject of this third volume. As the title of this book indicates, this is a brief history of gears. The historical information provided in this volume is in fact reduced to the essential one, necessary to provide the curious reader with news on the main stages of gear development and related knowledge, from the beginning of the history of the ix

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technological man to the present day. Therefore, this volume does not claim to provide a complete picture of the historical development of the gears, for which a specific encyclopedia would be necessary. It is intended only to provide an exhaustive picture of the most significant milestones that determined and continue to determine the current state of knowledge about gears. The first archeological evidences and the first historical documents concerning the beginning of the path of knowledge on gears are certainly to be traced back in the mists of ancient times. The first scientific contributions on gears are to be ascribed to the first two centuries of Hellenism. Further scientific knowledge was acquired, after the long medieval night, starting from the late sixteenth century, with an exponential progression in the twentieth century and at the beginning of this third millennium. Nevertheless, in the current state of knowledge, which is remarkable, the path of knowledge concerning gears cannot be considered concluded. Large dark areas of knowledge still exist on the gears, to be considered as real black holes. The knowledge still to be acquired concerns not only those specific to the numerous disciplines involved in gear design (geometry and kinematics; static and dynamic loads, including those due to impact; friction and efficiency; dynamic response and noise emission; static and fatigue tooth root strength; contact stresses and surface fatigue durability; nucleation of fractures of any kind and their propagation until breakage; full film, mixed and boundary lubrication; scuffing and wear; materials and heat treatments; new materials; cutting processes and other manufacturing processes; etc.), but also those arising from their mutual interactions. Other important challenges are those related to the new fields of application (see, e.g., those related to the helicopter and aerospace industry and wind power generators), which require the introduction and design of new types of gear drives that are not reflected in the current technological landscape. Another important challenge is the one concerning the formulation of a unified scientific theory of gears, able to simultaneously consider the geometric–kinematic aspects and those of strength and load carrying capacity. The scholars of vaunted credit, who claim to know everything and to be able to speak or write about any subject, and who boast or attribute themselves knowledge in any field, consider the gears as a synonym of obsolescence, a symbol of the past or, when they are benevolent, a nineteenth-century old stuff. This way of thinking of those we have benevolently referred to as scholars of vaunted credit (no one can therefore accuse this author of not being kindly gentlemen) implies at least the ignorance of the historical evidence that the theory of the gears, considered as mechanisms, is much older, having it characterized the birth of science, in the first Hellenism (see Chap. 3). But the unjustified conviction of these so-called scholars hides a far more serious ignorance. In fact, they prove not to know that the most significant contributions for the calculation of the load carrying capacity of the gears were brought gradually in the entire twentieth century (with the exception of Lewis, 1892—see references in Chap. 3) and that in this beginning of the third millennium, new and equally significant contributions of high scientific value appeared and continue to appear at the horizon (see, e.g., Chaps. 10 and 11 of Vol. 2).

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On the contrary, the true scholars and experts of gear power transmissions are well aware that the gears were yesterday, continue to be today and for a long time yet will continue to be an ongoing scientific and technological challenge. Even today, surely it is worth investing significant financial resources, in terms of manpower, tools and means, in R&S activities on this important area, as all the technologically advanced countries continue to do. This depends on the fact that the gears are a very complex multidisciplinary field, as few in mechanical engineering, and knowledge still be acquired are numerous. We can affirm, without fear of being denied, that gears are the result of ancient knowledge in continuous updating. For the development of new knowledge and new scientific theories concerning any subject, it is necessary to know everything that has been done in the past on the same subject, because who has no past has neither present nor future, because he has no memory. The memory of what has been done by those who preceded us is the basis on which to build; and when the basic school is lacking, the product of any study or research is poor because it is built on sand. There is no better praise of the heritage of the past than the following famous sentence written at the turn of the eleventh and twelfth centuries by Bernard of Chartres … “nos esse quasi nanos gigantium humeris insidentes, ut possimus plure eis et remotiora videre” i.e., “we are like dwarfs sitting on the shoulders of giants, so we can see more things than they could and far away” …. This volume intends first of all to highlight the most relevant contributions on gears, provided by scientists, researchers and scholars of the distant and recent past. The contributions described are those of which there is a historical documentation, consisting of direct or indirect writings (the indirect writings are here to be understood as those we have inherited from contemporary authors, or immediately following, with respect to the authors who were the architects of the new knowledge attributed to them), archeological evidences, epigraphs, reliefs on monuments, engravings, etc. The review of some reckless statements on the basis of these mandatory points, considered essential here, has had many surprises. Some of these surprises concern the ancient world, while other errors of historical perspective refer to times much closer to us. We leave the reader the pleasure of discovering more in the text of this brief history. However, we cannot fail to highlight some emblematic examples. As for the ancient world, the following two examples are to be mentioned: • The claim that the first literary description of a gear was made by Aristotle is a myth to be demolished. The great Greek philosopher, founder of the Peripatetic School, never spoke of gears, and the aforementioned statement is doubly false: In fact, on the one hand, it is now established that the writing on which it is based is a pseudo-Aristotle and, from the another, this pseudo-Aristotle, who lived in a historical period in which the gears were already known, and therefore called by their name, does not speak of gears, but only of circles. • Some argue that the very ingenious Chinese south-pointing chariot dates back to the twenty-seventh century B.C., while others postpone the dating of this chariot placing it between the eleventh and eighth centuries B.C., during the Western

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Zhou Dynasty. Both the first and the second dating are artificial to create another myth that has to be debunked, because the first historical documentation dates back to the third century A.D. As for errors of historical perspective in more recent times, it is sufficient to mention the following three examples: • The friction laws, which are the basis of the evaluation of the load carrying capacity of the gears against abrasive wear, are attributed to various authors, except to the one who first formulated and wrote them in his notebooks, namely Leonardo da Vinci. • The totality of textbooks on gears gives, as a reference of Tredgold’s approximation, books written by Tredgold where this approximation does not appear. It is indeed formulated in Buchanan's textbook of 1823 edited by Tredgold, with his notes and additional articles where the approximation that bears his name is described. • No one brings back the Lewis’ parabola, which constitutes one of the milestones of the gear strength calculation, to Galileo, who was instead the first to formulate the theory of the constant strength cantilever beam subject to bending. Unfortunately, all the aforementioned myths and false news, as well as other myths and other false news described in the text, have been taken for good and have passed from one book to another, even the more recent ones, without any filter. Undoubtedly these factual circumstances, which do not correspond to the existing and available evidence, have constituted a great stimulus to the writing of this brief history, which wants to be a small contribution to restore historical truths that are indisputable until proven otherwise. A no less important stimulating factor in writing this brief history is the fact that the formulation of a historical judgment on a statement (or fact) cloaked in myth, legend or false truth leads to actualizing the same statement (or fact). Therefore, a statement concerning a remote or very remote fact, examined through historical judgment, becomes alive actuality, because the vibrations of that remote or very remote fact are propagated to the present. The acquisition of new knowledge on gears was neither continuous nor uniform over time. Rather brief periods of intense activity, which have produced significant theoretical works as well as corresponding practical achievements of great scientific and technological value, have alternated with long periods of stagnation, in which only acquisitions of empirical knowledge are to be reported, deriving from successive refinements due to the slow accumulation of experiences handed down from generation to generation. In accordance with what has been proposed by Russo (2015; see references in Chap. 1), from the point of view of the development of knowledge concerning natural phenomena and related innovations, four historical periods can be identified starting from the appearance of technological man to date. The first and third of these periods are characterized by a lack of scientific sensitivity and mentality, for which they are called pre-scientific ages; the second and fourth of these periods are instead characterized by a scientific mentality and method, for which they are called

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scientific ages. To emphasize these four historical periods, which have followed and alternated gradually over time, a special chapter is dedicated to each of them. The four chapters dedicated to these historical periods are preceded by a brief general introduction chapter. Therefore, this brief history of gears consists of the following five chapters: Chapter 1—A brief general introduction is reported, in which, however, a surprising curiosity is also highlighted: the very recent discovery of a unidirectional biological gear in a planthopper species insect, widespread in many European gardens. However, it does not invalidate the general consideration of gears as artificial mechanisms. Chapter 2—The main evolutionary stages of the gears are described, in the long period of the first pre-scientific age, which goes from the dawn of the technological man to the birth of ancient science. Some news wrapped in the legend are examined in light of historical documents, and two myths, those of the ancient world mentioned above, are unmasked. Chapter 3—The development of gears in the first scientific age, which goes from the beginning of Hellenism to the diaspora of the scientists of the Museum of Alexandria, is described. This development, which sees the gears set up almost in today's forms, is analyzed by framing it within the revolution in the way of thinking about natural phenomena that characterizes the golden age of Hellenism, in which science in the modern sense of the term is born and the scientific method is stated. The main contributions of scientists, researchers and scholars of the main centers of Hellenism are described, with particular attention to those of the greatest scientist of the ancient world, Archimedes. The great innovations that make use of geared mechanisms are described in more detail, focusing attention on the qualifying theoretical contributions that underlie the same mechanisms, which for the first time appear in the history of technology. Chapter 4—The stagnation in the development of gears in the second pre-scientific age, which includes the long period between the diaspora of the scientists of the Museum of Alexandria and the Renaissance, is described. It is shown that the so-called rebirths before the Renaissance (four in all) are actually pseudo-rebirths, given the absence of noteworthy contributions, because the scientific method had been irretrievably lost. As far as gears are concerned, the return to the methods of the first pre-scientific age is demonstrated. However, the small and not very significant steps forward, resulting from the refinement of field experiences, are highlighted. In this context, the appreciable empirical contributions in the field of clocks are described, of which we are indebted to good craftsmen-clockmakers. A particular attention is dedicated to the Renaissance, in which the pre-scientific man has difficulty in dying and the scientific man has difficulty in being born. The most significant contributions of the most representative figures of the Renaissance, concerning gears, are analyzed and described. Particular attention is given to the figure par excellence of the Renaissance, Leonardo da Vinci, an immense genius for vastness of horizons and depth of thought. His uncommon capacity for innovation in the field of gears is brought to light, and his numerous and incomparable contributions are described in detail,

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including those concerning pure and applied science, which make Leonardo a figure that transcends his time and projects it into the next age of scientific awakening. Chapter 5—The initially slow and then increasingly rapid awakening of the second scientific age, which goes from Galileo to the present day, is described. Attention is first focused on the first two centuries of scientific activity, from Galileo to Lagrange, because in this time the mechanics establishes itself as queen of the sciences. However, it is emphasized that, in these two centuries, stagnation continues to prevail in the field of gears, which continue to look to the past again and do not immediately benefit from new scientific achievements. Subsequently, attention is focused on the slow detachment from pure empiricism and the equally slow penetration of science into the field of gears. The main scientific contributions are described, which initially concern only the geometric–kinematic aspects. It is also highlighted that, gradually, these aspects begin to be seen from the perspective of the new cutting processes that are gradually conceived and developed. Still later, the attention is focused on the mechanics of solids and on material strength theories that, even in this case, first gradually and then with forced charge, enter the field of gears. In this regard, this brief history only describes the contributions of the pioneers, who have begun new lines of research on gears, starting from Lewis to the present day. Therefore, only the milestones on the state of knowledge concerning the load carrying capacity of the gears in its most diverse aspects are described. Those who have helped to widen the horizon of specific lines of research are not mentioned, even if in many of their works theoretical analysis and experimental research worthy of the highest appreciation are identifiable. Not even the numerous and excellent works published in the last twenty years of the past century and in this beginning of the third millennium are mentioned: They concern pitting, micropitting, tooth root fatigue bending strength, tooth fatigue fracture, tooth interior fatigue fracture, spalling, scuffing, etc. This is because these works, rather than being history, are recent news or, to use a journalistic term, they are chronicle in progress. However, to give a concrete demonstration of the fact that gears are ancient science in continuous updating, in the final part of the chapter, brief considerations are presented on the historical aspects of three particular types of damage, which are related to the load carrying capacity of the gears in terms of abrasive wear, micropitting and tooth flank breakage. The chapter closes with epigraphic news on the main monographs and textbooks on gears from 1900 to the present, with a due exception to Olivier’s 1842 treatise, which is probably the first monograph that was written and published on the gears. Rome, Italy

Vincenzo Vullo

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The First Pre-scientific Age: From the Down of Technological Man to the Bird of Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 The First Scientific Age and Birth of the Science: From the Beginning of Hellenism to the Diaspora of Scientists of the Museum of Alexandria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Prodromes of the First Scientific Age and Centers of the Hellenistic Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Golden Age and the Genius of Archimedes . . . . . . . 3.4 Rapid Decline and Death of Science . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...... ......

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23 31 52 55

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4 The Second Pre-scientific Age: From the Diaspora of Alexandrian Scientists to the Renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Loss of the Scientific Method and So-called Early Rebirth of the Imperial Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Subsequent Pseudo-rebirths: From the End of the Imperial Age to the Renaissance . . . . . . . . . . . . . . . . . . 4.2.1 The Second Pseudo-rebirth Between 5th and 6th Century A.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Third Pseudo-rebirth: The So-called Islamic Renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Fourth Pseudo-rebirth: The Rebirth of Western Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Renaissance, Precursor of the Second Scientific Age . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

.. 76 .. 83 . . 128

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5 The Second Scientific Age: From Galileo to Today . . . . . . . . . . . 5.1 The First Two Centuries, from Galileo to Lagrange: Mechanics Establishes Itself as the Queen of the Sciences, but the Gears Still Look to the Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 From Desargues to Today: From Empiricism to the Slow Penetration of Mechanical Science into Gear Design . . . . . . . . 5.3 From Galileo to Today: Entry of Solid Mechanics and Material Strength Theories in Gear Design . . . . . . . . . . . . 5.3.1 Wear Load Carrying Capacity of Gears . . . . . . . . . . . . 5.3.2 Micropitting Load Carrying Capacity of Gears . . . . . . . 5.3.3 Tooth Flank Breakage Load Carrying Capacity . . . . . . 5.4 Main Monographs and Textbooks on the Gears . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . 132 . . 156 . . . . . .

186 207 210 214 218 222

Correction to: The First Scientific Age and Birth of the Science: From the Beginning of Hellenism to the Diaspora of Scientists of the Museum of Alexandria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Name Index of Non-authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Chapter 1

Introduction

Abstract In this chapter, a brief introduction is made, in which the four historical ages into which the entire history of the gears is divided are defined. The fact that the gears are to be considered artificial mechanisms conceived by the technological man, understood as a human already become able to conceive and manufacture artificial mechanisms to be used for his daily needs, is then clarified. Finally, a surprising curiosity is highlighted, concerning a very recent discovery of a unidirectional biological gear in a planthopper species insect, widespread in many European gardens. However, it is shown that this discovery does not invalidate the general consideration of gears as artificial mechanisms.

Today gears play an irreplaceable role in power mechanical transmissions with constant transmission ratio as well as in traditional, semi-automatic and automatic geared transmissions for motor vehicles, mechanical and electric industry, wind turbine generators, aerospace industry, especially the helicopter industry, etc. This is due to their extreme versatility and reliability of use. Gears and gear drives in fact allow to solve any practical problem within speed and power ranges that effectively meet the various needs that may arise, even in the most advanced current technology areas, and in those reasonably foreseeable for the future. To have knowledge of the gears the more extensive and thorough as possible, we here consider useful and appropriate to add to Vol. 1 and Vol. 2, which deal with the various problems of geometric-kinematic calculation and assessment of mechanical strength and load carrying capacity of gears under actual working conditions, this Vol. 3, concerning a their concise history. The information provided here is reduced to the essential ones so that the reader can know the main milestones characterizing the development of gear knowledge, from the beginning of the history of Homo Sapiens Sapiens already become a technological man to this day. This in order to better understand the various steps and related interconnections, which determined and continue to determine the gear current knowledge. In the wake of Linnaeus [7], who first coined the binomial name Homo Sapiens, we used the term Homo Sapiens Sapiens instead of the perhaps more correct AMH (Anatomically Modern Human) to indicate the only non-extinct subspecies of Homo © Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_1

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1 Introduction

Sapiens. Without entering here into an issue that does not concern us, it is sufficient for us to point out that it is not to be excluded that the most distant origins of the technological man, understood in the more general meaning specified below (but not in the narrower meaning used in this textbook), are to be put even further in the past, even making them go back to other species (or subspecies?) of Homo Sapiens now extinct, such as (we limit ourselves here to mention only two species or subspecies of Homo Sapiens, as they are unanimously classified as such by specialists and scholars): • Homo Sapiens Neanderthalensis, species or subspecies of archaic humans who lived within Eurasia from 400,000 until 40,000 years ago [9]; • Homo Sapiens Idàltu, also called Herto Man because of the human fossils found in 1997 in Herto Bouri in a region of Ethiopia, under volcanic layers, who lived in Pleistocene Africa around 160,000 years ago (see [1, 11]). However, this factual evidence is not relevant here, since the introduction of rudimentary mechanisms that can make us think of gears is certainly to be placed in the prehistoric age very close to us, and perhaps even at the down of the historical age. However, before starting this brief history, it is good to clarify that gears and gear drives were considered unanimously as exclusively artificial mechanisms until 2013, when Burrows and Sutton [2], two researchers of the University of Cambridge, discovered that the Issus Coleoptratus, an insect of planthopper species widespread in many European gardens, jumps from one plant to another with the aid of a pair of tiny, one-way gears. The ballistic jumping movements of this insect, in its youthful age, are due to a biological gear-like mechanism in his hind legs. The young specimens of these insects (but not adults) possess in fact a raw of cuticular gear-like teeth around the curved medial surfaces of their hind leg trochanters. The teeth of one trochanter engage with and sequentially move past those on the other trochanter during the preparatory cocking and the propulsive phases of jumping. Measurements made by the aforementioned researchers have shown that these biological gear-like mechanisms, which are not connected all the time, but only for the duration of the jump, are able to ensure that both hind legs move at the same angular velocities to propel the insect body without yaw rotation. Figure 1.1 shows the juvenile form of a common insect Issus Coleoptratus, and its gear-like mechanism for jumping. Of course, this factual biological circumstance transcends the temporal beginning of this brief history of gears, which is certainly to be asked after the introduction of the potter’s wheel. The times of biological evolution are indeed very long. They are not even remotely comparable with those of the prehistory of human technology and corresponding technological man, whose most distant origins can at most be made to coincide with the moment when the first Homo Sapiens Sapiens (or Homo Sapiens, as Homo Sapiens Nearderthalensis or Homo Sapiens Idàltu) picked up a stone from the ground to be used as a tool to crush a nut, an almond or something similar or to better defend himself against external aggressions. We have no claim to go so far in time. Therefore, as technological man we mean here a human being now able to

1 Introduction

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20 µm

Fig. 1.1 a Juvenile form of a common insect Issus Coleoptratus; b its gear-like mechanism for jumping

conceive and manufacture artificial mechanisms to be used for his daily needs, and this happened almost simultaneously or immediately before the introduction of the potter’s wheel. With these limitations placed on the historical framework of reference, even considering that the discovery of Burrows and Sutton was made at the beginning of this third millennium, it is quite probable that the first artificial mechanism conceived by a human of our species that had the appearance of a gear had been devised without the awareness of the existence of these biological gears. For these reasons, in this brief history, gears or their distant predecessors will continue to be considered as artificial mechanisms, even if it is impossible to hide the amazement for an exceptional discovery and the admiration for nature that never ceases to surprise an attentive observer, as the research man must be. This brief history, whose first documents are to be traced back in the mists of ancient times, well shows that the path of knowledge regarding the gears cannot yet be considered concluded. Large dark areas of knowledge, to be considered as true black holes, still exist, as yet the lack of a unified and comprehensive scientific theory of the gears unequivocally demonstrates. Anyway, to develop such a theory, it is necessary to know all that has been done in the past, because who has no past has neither present nor future, because he has no memory. We can affirm, without fear of being denied, that gears are the result of ancient knowledge in continuous updating.

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1 Introduction

Historians of science have now agreed that scientific progress has not had a continuous evolution in the history of humanity, but was characterized by two scientific revolutions: the first, very nearly coinciding with the Hellenism, is called by Russo [10] The Forgotten Revolution; the second is the daughter of the Renaissance, and it is what historians regarded as the only scientific revolution up to the entire mid-20th century, and beyond. The two historical ages, the one between the beginning of the history of humanity and the Hellenistic scientific revolution, and the one between the two scientific revolutions, are considered as pre-scientific ages. Therefore, the history of scientific progress is characterized by the following succession of historical ages: • a first pre-scientific age: from the down of technological man to the birth of science; • a first scientific age, which saw the birth of the science and scientific method: from the beginning of Hellenism to the diaspora of scientists of the Museum of Alexandria; • a second pre-scientific age, where humanity sinks into the darkness of ignorance: from the diaspora of Alexandrian scientists to the Renaissance; • a second scientific age, in which the humanity frees itself of para-science, rediscovers the lost knowledge, and on them builds the modern world: from Galileo to today. This framework, which sees the alternation of pre-scientific and scientific historical ages, brings to mind the historical courses and resorts of Vico’s memory (see [3, 12]). It is necessary that the thinking humanity is on the alert, because it is possible that the history can be repeated. However, the reflections on this risk, the consequences of which would be disastrous, go beyond the subject of this brief history of the gears. We limit our discussion here to the subject that interests us. Based on the aforementioned picture of scientific evolution, we consider it appropriate to focus our attention on all four of these historical ages, separately. In this historical framework, for reasons of space, we merely provide the basic information on the development of the gears in the above four ages. Of course, the development of the gears is framed in the historical context in which it occurred. This is because, without a clear perception of this historical contest, there is a risk of not understanding the real extent of innovations regarding gears and the repercussions of this innovations on the advancement of knowledge related to the gears themselves and to the mechanical devices of which the gears constitute a fundamental characterizing part. The lingering of this historical contest then finds a further justification in the fact that the advancement of scientific and technological knowledge in the panorama of mechanical sciences and, more generally, of the physical sciences and mathematics, and that of specific knowledge concerning gears are often strongly interrelated and interdependent, so that they influence each other. With reference to the second scientific revolution, we just mention scholars and researchers who have opened new frontiers of knowledge on the gears, i.e. the initiators of specific areas of research. We apologize in advance for any unintentional omission. As for the contributions made on the gears in the last quarter of the 20th

1 Introduction

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century and at the beginning of this third millennium, we refer the reader to specialized journals, where periodically papers and works of review on the gears appear, more or less extensive, perhaps focused on particular aspects, but still interesting (see e.g., [4–6, 8]). Before closing this introduction, it should be pointed out that we have tried to frame this brief history of gears in the broader context of a hypothetical history of technology, understood as the application of scientific principles and methods. Unfortunately, the stories of technology that have been written do not provide an organic chronological development of events where the product of intellectual speculation is closely related to its practical applications, for which it has become technology. From a philosophical point of view, many scholars argue that technology is an asymptotic construction over time. However, no historian has ever marked the points of this hypothetical asymptotic curve, in order to highlight the milestones and the tendency to the limit. This observation applies to every aspect of technology and, in particular, to the gears that make up a small portion of it. In the four chapters that follow, dedicated to the aforementioned four ages, we will try to correlate the practical achievements of gears with the ideational effort of those who conceived them, in order to highlight and enhance the fatigue of often anonymous ancient predecessors of actual engineers who sacrificed much of their life to devise new innovations useful for everyday life or for their own defense. We must not forget that our current highly sophisticated gear technology is nothing but the result of the centuries-old elaboration and/or reworking of practical achievements often made by more or less ingenious common men. The history of gears starting from geared archaic mechanisms also wants to be a due recognition of those generations of innumerable ingenious men who preceded us, of whom we often know nothing, except for the few of which have been preserved historical memory. Undoubtedly, the first archaic devices that may resemble rudimentary geared mechanisms were developed in the mists of time that are lost in the foggy desert of prehistory and were the work of anonymous men. Very little information has come down to us on the technological development of the ancient world and even more limited is the information regarding the gears. The sources on which to base a systematic analysis that allows to follow the development of the gears over time are indeed very scarce, for a double reason: in fact, on the one hand, the aristocratic culture of Greeks and Latins originated numerous monumental literary, philosophical, juridical, historical works, while few were the technological works, with the probable exception concerning works on military art; on the other hand, with the fall of the ancient empires, the most prestigious libraries were destroyed and the little that was saved mainly concerned the works falling within the aforementioned aristocratic culture. With some exceptions, which we will highlight from time to time, the development of gears from prehistory to the present is to be considered mainly as the development of an anonymous product. But if it is true that many creators of this development have never emerged from anonymity, it is equally true that this development is characterized by detailed historical events attributable to well-known scientists, inventors and scholars.

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In the four chapters that follow we will give voice not only to the brilliant men who, with their scientific contributions and related technologies, determined memorable innovations to be considered milestones in the field of gears, but also to events of uncertain date, known for their disruptive effects in gear technology rather than for their anonymous creators, who never entered history. Thus we will show that innovations and inventions concerning gears, like any human action, possess historical propelling charges variously graded over time, for which, in the face of an innovation/invention that immediately deflagrates and propagates everywhere, a hundred others may languish for centuries, as happened for most of Leonardo da Vinci’s findings.

References 1. Asfaw B, DeGusta D, Gilbert H, Richards GD, Suwa G, Howell FC (2003) Pleistocene Homo sapiens from Middle Awash, Ethiopia. Nature 423(6491):742–747 2. Burrows M, Sutton G (2013) Interacting gears synchronize propulsive leg movements in a jumping insect. Science 341(6115):1254–1256 3. Croce B (1922) La filosofia di Giambattista Vico, 2nd edn. Laterza, Bari 4. Hlebanja J, Hlebanja G (2013) An overview of the development of gears. In: Power transmissions. Mechanisms and machine science. Springer, Berlin, pp 55–81 5. Hyatt G, Piper M, Chaphalkar N, Kleinhenz O, Mori M (2014) A review of new strategies for gear production. Procedia CIRP 14:72–76 6. Korka Z (2007) An overview of mathematical models used in gear dynamics. RJAV IV(1):43–50 7. Linnaeus C (1758) Systema Naturae: Per Regna Tria Naturae Secundum Classes, Ordines, Genera, Species, Cum Characteribus, Differentiis, Synonymis, Locis. Tomus I et Tomus II, Editio Decima, Reformata, Holmiae, Imprensis Laurentii Salvii 8. Özgüven HN, Houser DR (1988) Mathematical models used in gear dynamics—a review. J Sound Vib 121(3):383–411 9. Pääbo S (2014) Neanderthal man. In: Search of lost genomes. Basic Books, New York 10. Russo L (2015) La rivoluzione dimenticata. Il pensiero scientifico greco e la scienza moderna, 9th edn. Giangiacomo Feltrinelli Editore, Milano 11. Stringer C (2016) The origin and evolution of Homo sapiens. Philos Trans R Soc Lond B Biol Sci 371(1698):20150237 12. Vico G (1744) Principi di Scienza Nuova. Tomo I. Stamperia Muziana, Napoli

Chapter 2

The First Pre-scientific Age: From the Down of Technological Man to the Bird of Science

Abstract In this chapter, the long period of the early pre-scientific age is considered, which ranges from the dawn of the technological man to the birth of ancient science to the beginning of Hellenism. The main evolutionary stages of the gears in this historical period are described. Some news wrapped in the legend are examined in the light of the historical documents available and the following two myths are unmasked: the first concerns the reckless claim that Aristotle provided the first literary description of the gears; the second concerns the groundlessness (at least until proven otherwise) of the exaggerated backdating of the very ingenious Chinese south-point chariot, unmotivably dated back to the twenty-seventh century B.C., when instead its first historical documentation dates back to the third century A.D.

The first pre-scientific age begins at the dawn of the history of humanity and bird of civilization, and ends in the second half of the 4th century B.C. As dawn of the history of humanity and bird of civilization, we here mean the one in which the first rudimentary artificial mechanisms and the simplest of the so-called simple machines (the latter in the sense of the six classical simple machines defined a few millennia later by the scholars of the Renaissance) appeared, conceived and manufactured by humans to satisfy their daily needs. We are therefore at the dawn of the technological man defined in the previous chapter. The end of this first pre-scientific age, here placed in the second half of the 4th century B.C., must not be understood in an absolute sense. Important and significant symptoms that something absolutely new was maturing about the way of conceding natural phenomena are clearly traceable starting from the beginning of the 6th century B.C., as we will see in the next chapter, where we will focus our attention on the anticipatory prodromes of the first scientific age. Here we limit ourselves to considering only the historical development of the gears in the contest of man-conceived machines. In this framework, we must recognize that gears in very rudimentary form were made and used from the very beginning of human history and civilization. Actually, with the exception of the potter’s wheel, the first man-conceived machines used these rudimentary gears, characterized by mating parts with wooden pins, having the function of coarse teeth.

© Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_2

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However, the historical origins of the gears cannot be precisely determined. The oldest descriptions suggest gears originate from the ancient Mesopotamian world, and from ancient China and Greece, but almost all of them are vague and unreliable, so much that it is hard to tell where the history begins, and mythology ends. Moreover, these descriptions are so inconsistent and nebulous that it is even difficult to determine whether the described devices incorporated rudimentary gears or not. In the current state of knowledge, a categorical affirmation of this kind in a positive sense, not supported by more reliable documentary evidences, would constitute a more then risky Pindaric flight. Everything indicates that the first rudimentary gear wheel, a distant relative of the current ones, is to be correlated with the problem of water lifting, for water supply from rivers, waterways, lakes, wells or similar water resources. In this regard, the oldest machine that we know is the shaduf or shadoof, which is an irrigation device better known as swape or counterpoise lift [15] or as well pole, well sweep or simply sweep. This irrigation device is documented in Mesopotamia, on reliefs of Accadic monuments of 2500 B.C. [8]. There are also indications that the Minoans in Crete were using the shaduf in the Meso-Minoan period (about 2100–1600 B.C.), while it appeared, almost simultaneously (about 1600 B.C.), in the Pharaonic Egypt, during the 18th Dynasty, and in China (here the shaduf is known as jiég¯ao) in the first year of the Shang Dynasty (about 1600–1046 B.C.), where it would have been invented by Yi Yin (see [24]). Among other things, the shaduf is still used in some of the Near and Far East regions. We can reasonably think that a semblance of gear wheel has been originated when someone had the idea of equipping a wheel with a horizontal axis of coarse radial paddles, able to be actuated by the flowing motion of the water, as well as of lateral buckets for drawing and lifting water from a water source. So, the first rough predecessor of the hydraulic noria (or simply noria, also called Persian wheel, paddle-wheel, Egyptian water wheel and water wheel) was born and, along with it, a new water-lifting technology. It contained, albeit in embryo, the first idea of gear wheel. It is noteworthy that someone argues that this predecessor of the noria had vertical axis. We have not inherited archaeological evidence of these early rough predecessors of the hydraulic noria, perhaps also due to the perishability of the material with which they were made. We can only get an idea of how they were made and used, based on similar devices that were implemented in the various historical periods following their first introduction, almost to the eve of our days, perhaps without substantial changes. Figure 2.1 shows a reconstruction of a hydraulic noria, along with its main parts, complying with the oldest descriptions we received. This new machine still represented a remarkable technological advancement, as it allowed not only to increase considerably the quantity of water raised, but also to automate the same action of water lifting. In fact, this was reduced to a continuous motion of rotation, driven by a running water stream. The subsequent development, consisting in the introduction of the animal power as an energy source, led to the conception of the sakia, which is generally considered as the true water wheel. We are not able to specify when and where it happened, because no written document,

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Fig. 2.1 Reconstruction of a hydraulic noria or paddle-wheel, and its parts

no pictorial image or engraving on stone, or other similar archaeological find has come down to us. Despite the aforementioned missing archaeological evidences, we have no difficulty today in imagining how these devices were made, as they have been used almost to the present day (for example, in India, Egypt and other Middle Eastern countries as well as in some European countries, as in Portugal and Spain, particularly in the Balearic Islands) and perhaps continue to be used in some of these countries. They could be set in motion by human muscle power, exercised with hands and feet, or by animal power (oxen, camels, donkeys, mules, etc.). Figure 2.2 shows a sakia used in recent times, set in motion by a donkey and used as water lifting device. The continuous evolution of the aforementioned first rough predecessor determined the genesis of paddle-driven water-lifting wheels, similar to water wheels, as we now know them. According to Wikander [20], these wheels would be documented in ancient Egypt in the 5th century B.C., that is, a century before the Ptolemaic age began. Still Wikander [20], and even De Miranda [7] argue that water wheels originated from Ptolemaic Egypt, so they would appear to the historical horizon in the late 4th and early 3rd century B.C. According to Oleson [14], it is possible that both the compartmented wheel (i.e., a water wheel with compartmented rim), and the paddle-wheel (i.e., a wheel moved by the power of water or hydraulic noria) were invented together at the end of the 4th century B.C., while the sakia (or saqiya), i.e. the true water wheel, was invented one or two centuries later, again in Egypt.

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Fig. 2.2 Sakia used as water lifting device

The above statements are supported by archaeological findings in Faiyum, Egypt, where the oldest archaeological evidence of water wheel was found, in the form of a sakia, dating back to the 3rd century B.C. Furthermore, a papyrus dating back to the 2nd century B.C. was found, also at Faiyum, which mentions a water wheel used as irrigation device. The same statements are also supported by the following evidences: • A fresco of the 2nd century B.C. found in Egypt, Alexandria, in which a compartmented sakia is painted. • The lost Peri Alexandreias of Callixenus of Rhodes (cited by Athenaeus of Naucratis), which mentions the use of a sakia in Ptolemaic Egypt during the reign of Ptolemy IV Philopator in the late 3rd century B.C. (see [3, 11, 12]). We cannot leave unmentioned the thesis of Colin [5], who argues that the water wheel understood in a wider meaning has Aramaic origin, as in the word noria he found an Aramaic etymology of the Arabic word n¯a’¯ura or n¯a’¯ora (see also [10, 23]). According to this thesis, there is a possibility that the water wheel has appeared during the 7th or 6th century B.C., in Mesopotamia, when Aramaic gradually replaced Accadic as spoken and written language. In fact, notoriously, Aramaic was for several centuries, from 7th century B.C. to 7th century A.D., the lingua franca in the Middle East, and bridged the Accadic and Arabic as dominant language of the region. Despite the fact that Colin does not specify what type of water wheel can be treated, and does not support his thesis with documentary or physical evidence, it is possible that a hydro-powered water wheel, i.e. a water wheel driven by the flowing water of a river, i.e. a hydraulic noria, existed as early as the 7th century B.C. Evidence of this statement comes from the text of an Accadic tablet dated to the neo-Assyrian age, though it should be noted that the interpretation of the cuneiform test does not provide convincing evidence [7, 9]. Another evidence is related to the fact that the word noria derives from the common Semitic noun for river or water flow, which appears in Accadic, Aramaic, Arabic, and Hebrew.

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Fig. 2.3 Tympanum, i.e. compartmented noria or sakia

As for the gears, it is my opinion that norias and sakias a little more elaborate than those mentioned above, and similar to those used until the last century in the Near East, constitute the link between the first pre-scientific age and the first scientific age. As no documentary evidence has come down to us that can support their conception and design based on scientific theories, we believe that this is the right section for their description, and this despite the fact that the earliest evidence belongs to the historical period of the first scientific age. In this description, we do not follow the historical chronology of their appearance, which according to the current evidence is that described above, but the history of actual evidence, of whatever nature they may be. As we mentioned above, the first, oldest documented example of water wheel is the sakia, which is also called Persian wheel or tablia. The Latin tympanum (Fig. 2.3) is a compartmented sakia or a compartmented noria, and takes its name of Greek origin from its resemblance with a drum or tambourine (in this regard, see [24]). As we saw above, the sakia found in Faiyum dates from the 3rd century B.C. [7]. It is a mechanical water-lifting device, consisting of an early gear system with a vertical wheel, to which buckets, jars or scoops are attached, and a horizontal wheel that is set in motion by human and animal power. Both wheels of sakia were made of wood, and can be considered as the paradigm of spur gear wheels having wooden cylindrical pin teeth as drive elements. These pin teeth had axes parallel to the axis of the wheel, so the latter could be considered as a very distant precursor of the face gear wheel of the current face gears. The other above mentioned archaeological evidence of a

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Fig. 2.4 Conjectural representation of a treadwheel

sakia, the one found in Alexandria and consisting of a Hellenistic painting-fresco of a tomb of Ptolemaic Egypt from the 2nd century B.C., also leads to the same conclusions with reference to its configuration: its wooden cylindrical pin teeth had axes parallel to the axis of the gear wheel, also made of wood. The sakia (Fig. 2.2) should not be confused with the noria (Fig. 2.1) which was conceived and developed before the sakia, and represents a kind of duality. In fact, the noria (it is also called water wheel) is a gear device placed on the bank of a river and powered by its flowing water, while the sakia is a gear device powered by animals or men (in this second case it is also called treadwheel) and used to lift the water out of a well or a body of standing water. Figure 2.4 shows a conjectural representation of a treadwheel. Both sakia and noria are constituted by a horizontal gear wheel, which meshes with a vertical gear wheel, but the sakia wheel is a water-lifting device, while the noria wheel is a typical device of water-mills for operating grain or other cereals (windmills were also equipped with rudimentary geared mechanisms similar to those of watermills). It is very likely that some norias of this first pre-scientific age were hybrid, i.e. consisting of water wheels assisted secondarily by animal power; however, in this regard, we have no historical evidence. Animal-powered sakias, water-powered norias, and hybrid noria-sakia devices were in everyday use in the Islamic word throughout the Middle Ages. Figure 2.5 shows an example of an advanced hybrid noria-sakia device, both driven by hydraulic and animal power, used in the 13th century A.D. in the Islamic Middle Ages, and described by Al-Jazary in 1206 (see Al-Jazari, translated by Hill [1]). Other examples similar to sakias and norias, used in ancient Egypt and the Mesopotamian basin of the Fertile Crescent, could be mentioned, but they would add nothing new to the history of the gear development. These examples of water

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Fig. 2.5 Example of an advanced hybrid noria-sakia device, driven by both hydraulic power and animal power, as described by Al-Jazary in 1206

lifting devices in fact have the same characteristics as those described above, and the few variations that can be found in them are so insignificant as not to call our attention. Instead, it is worth focusing our attention on another more than interesting gearing device that, due to the historical location of its first documented realization, dating back to the 3rd century A.D., should be discussed in Chap. 4, concerning the second pre-scientific age. This is the very ingenious South-Pointing Chariot. The discussion of this topic is done in this chapter in the wake of those who exaggeratedly over-anticipate the historical date of introduction of this device, even placing it in

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Fig. 2.6 Conjectural model of south-pointing chariot incorporating a differential gear mechanism, exhibited at the Smithsonian Institution of Washington

the 27th century B.C. This historical location almost at the dawn of the history of human civilization is not supported by any archaeological or documentary evidence. Therefore, until proven otherwise, in my humble opinion that is supported by that of the scholars mentioned above, it is to be considered as a legend. However, let the historians and archaeologists clear the fog on the interesting question inherent in the dating of the device under consideration. Instead we focus our attention on the correlated historical evidences and above all on its characteristics, especially those pertaining to the geared mechanisms that characterized it. However, there is no doubt that the very ingenious South-Pointing Chariot is a more than interesting and documented example of geared device, characterized by wooden cylindrical pin teeth (Fig. 2.6). It was an ancient Chinese two-wheeled vehicle, which was carrying a movable pointer (a doll, monk or any figure with an extended arm) to indicate the south, no matter how the chariot was rotated. This device is a rotation neutralizer and was used as a non-magnetic compass for navigation, but could also have had other practical applications. This ancient mechanism is anything

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but primitive or simplistic. In fact, it configures a complex mechanical system, despite the fact that it is a pin-tooth gear drive. Some believe and write that this south-pointing chariot dated back to the 27th century B.C., but this daring statement has no documented historical foundation. At the present state of historical evidence, it is to be considered a legend (see [13, 18, 25]). The first documented realization of this chariot is the one created by Ma Jun (200–265 A.D.) of Cao Wei during the Three Kingdoms, i.e. about eight hundred years before the first navigational use of a magnetic compass. No ancient southpointing chariot has come down to us, but many ancient Chinese texts talk about them, saying that they were used intermittently until about 1300 A.D. Subsequent Chinese texts of the 6th century A.D. have provided some design and operating details of these chariots, but they are especially made as an effort to create the legend of their long before Ma’s time use, making them even date back to the Wester Zhou Dynasty (1050–771 B.C.). Most likely several types of south-point chariot were designed and manufactured. Perhaps they worked differently, but they were all characterized by a geared mechanism, able to ensure that the pointer was correctly directed. This mechanism did not automatically detect which direction was south. The pointer was manually oriented towards the south at the beginning of a journey, and the mechanism against acting to the steering direction, ensured that the pointer rotated with respect to the body, maintaining its orientation to the south whenever the chariot wheels were steering to change the direction of travel. Some authors argue that some chariot’s mechanism may have had differential gears, for which they could be considered archetypes of the first differential in the world, in the case where the aforementioned legend was disproved by archaeological finds or proven historical documents. Although the fascinating hypothesis that some types of south-pointing chariot would use differential gears has gained wide acceptance, it should be noted that the mechanism can perform its design functions without planetary gear trains. In any case, this device is to be considered, rightly, as a milestone in the history of technology. Various reproductions of south-pointing chariots have been carried out in different parts of the world, almost all built with differential gear trains. One of these is shown in Fig. 2.6, exhibited at the Smithsonian Institution of Washington and then replicated at Ohio State University. Another similar reproduction is that exhibited at the Science Museum in London. Several theories of the synthesis of constant direction pointing chariots, with and without differential gear trains, have been also proposed. One of these, concerning a constant direct point chariot with a differential gear train, is the one made by Bagci [2]. There are no other examples of geared devices worthy of being mentioned in this first pre-scientific age, which ends with Aristotle (384–322 B.C.) in the 4th century B.C. He is, by definition, the coryphaeus of natural philosophy, which tries to deduce quantitative statements on particular physical phenomena directly from general philosophical principles, found and defined through qualitative observation of nature. Given the important role that phenomenological analyses have in the

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direct reading of experience, Wieland [19], like others, has been able to affirm that “Aristotelian physics is a theory of experience” (see also [17]). This Aristotelian way of thinking influenced, in a substantial manner, the evolution of scientific thought, putting a brake on the development of science. Nevertheless, the Western world would be indebted to Aristotle for the most ancient description that has come down to us that, according to some authors, makes us think to gears. In fact, according to these authors, with reference to wheel drives in windlasses, in around 330 B.C., Aristotle would have written that “the direction of rotation is reversed when one gear wheel drives another gear wheel” (see, for example, [16]). The aforementioned statement of these authors was based on an extensive interpretation of the solution of a specific problem (see below), described in the Problemata (i.e., Problems or Mechanical Problems or Mechanica), a text traditionally attributed to Aristotle, as found in the Aristotelian Corpus. This attribution is however very controversial today. Wilson [21] estimated that Problemata was probably composed between 280 and 260 B.C., while Winter [22] suggested that the author was Archytas of Tarentum (428–347 B.C.). Instead, Coxhead [6] states that at present it is only possible to conclude that the author was one of the Peripatetic School. Today, almost everyone recognizes that the Problemata are a pseudo-Aristotle. They are a collection of mechanical problems and their solutions, written in a question and answer format. It is virtually certain that this collection, gradually assembled by the Peripatetic School, reached its final form in some unidentifiable place between the 3rd century B.C. and the 6th century A.D. The work is divided by topic into 36 sections, and the whole contains almost 900 problems. All problems and related phenomena are explained using a particular way of looking at the lever principle. The problem that interest us here is that related to the wonder that an entertainment machine or device must inspire in the viewer, to whom it must show only the stunning effect, hiding its cause. The problem considers three circles (with the generalization to more than three circles) of equal diameter, tangent to each other in a horizontal plane, and with their centers arranged on a straight line. Using the lever principle, the unknown author of the pseudo-Aristotle writes that, when a circle rotates around its axis in a given direction of rotation, the adjacent circle will rotate around its axis in the opposite direction to the previous one, and so will the third circle with respect to the second (and any subsequent circles with respect to the adjacent ones). In this way the craftsman of the machine, moving only one circle without showing it to the viewer, will rotate all the other circles with opposing rotations, which will arouse the desired wonder. The pseudo-Aristotle speaks therefore only of circles. It is possible that these circles indicated friction wheels, while it is risky to assume that they were gear wheels, also because the energy and power involved in handling a gear train are not very compatible with the manual ones of the craftsman, especially in the case in which the principle of operation of the mechanism is applied to the aforementioned generalization to more than three circles, whatever the material with which the corresponding cylinders are made (note that the pseudo-Aristotle says that the small wheels are made of bronze and steel). The aforementioned attribution of the first description of the gears is no longer sustainable, for a twofold reason: in primis, it is now clear that

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Fig. 2.7 Explanation figure from pseudo-Aristotle Problemata: a single rotating circle; b multiple rotating circles

the Problemata are certainly a pseudo-Aristotle; in secundis, the Problemata do not speak absolutely of gears, but simply of circles, almost certainly representative of friction wheels. The above sentence attributed to Aristotle is a Pindaric flight of those who, without any exegesis of the sources, only work with imagination and confuse myth with reality. The above deductions are supported by the original Greek text below, taken from the pseudo-Aristotle Problemata, 848a, 19.37, as well as the related explanation figure (Fig. 2.7). The English translation of the original Greek text is also reported below. Other interesting considerations on this pseudo-Aristotle problem can be found in Bur [4].

Because of the fact that opposed motions simultaneously put the circle in motion i.e. one end of the diameter A moving forwards and the other end B moving backwards (Fig. 2.7a), some have set up a construction so that from one movement, many circles move in opposite directions at the same time, just like they dedicate in temples, having made the little wheels out of bronze and steel. For if circle AB were to touch circle , when the diameter of AB is moved forward, the diameter of  will move backwards, so long as the diameter is moved on itself (i.e. it does not roll forwards, but rather pivots around its center point). Consequently, the circle AB is moved in opposition to  and the same circle again will move the adjacent circle EZ in the opposite direction to itself due to the same principle (Fig. 2.7b). In the same way, if there were more [circles], they would all do this for the same reason. Taking this underlying nature of the circle, then, craftsmen construct a machine concealing the principle so that only the marvel of the mechanical device is visible, while the cause is unknown”.

The aforementioned thesis is also supported by the observation that the mechanism described by the pseudo-Aristotle is compared to those used as temple automata (see translation from the Greek), that is as a device which used a self-animation technology to set in motion the heavy statues within the temple or in the endless processions of

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which we have a memory, to present an inexplicable ϑαυμα (thauma, miracle) to the eyes of viewer. It is not difficult to be convinced that, with the gears presumably not very accurate of that time, set in motion by a crank operated manually by the craftsman, the mechanism would not have had any success even in the case of a single pair of gear wheels. It could be successful only in the case of small automata to arouse wonder, based on the use of a limited number of friction wheels, called circles by the pseudo-Aristotle. The term circle used by the pseudo-Aristotle also deserves careful consideration, in light of the historical context in which the text is placed by the scholars, which goes from Archytas to the late Peripatetic School. The unknown author of the text, to indicate a rotating wheel, does not use the equivalent Greek terms (τρoχ´oς, oδoς, ´ ´ to indicate a gear wheel, already known to the Greek world of that era, but μηχανη) the term κκλoς ´ (circle). This denomination must also be evaluated in the light of the initial proposition of the Problemata, according to which the answer to a lot of marvelous problems set out is easily understood based on the geometry of the circle, defined as “the first principle of all marvels”. Just to make the devil’s advocate, it is also to say that, in support of the unsustainable thesis that the pseudo-Aristotle speaks of gear wheels, Dante Alighieri cannot be called into question; in fact, as we will see in Sect. 4.2.3, he calls circles the gear wheels of the clocks of his time. The imaginative language of a poet, moreover than Dante’s caliber, cannot be questioned in support of a technical language. In this regard, it should be borne in mind that the Greeks, in terms of language precision, were incomparable masters and did not allow exceptions. In any case, the now established fact that Aristotle is not the author of the Problemata, for which they are to be attributed to an unknown author, known today as pseudo-Aristotle, becomes absorbent with respect to any sophistical contrary statement, whose roots are cut to the base.

References 1. Al-Jazari IR (1973) The book of knowledge of ingenious mechanical devices (translated from Arabic and annotated by Hill DR). Kluwer Academic Publishers, Dordrecht 2. Bagci C (1988) The elementary theory for the synthesis of constant direction pointing chariots (or rotation neutralizers). Gear Technol 31–35 3. Bakman D, Rankov NB (2013) Ship and shipsheds: large polyremes. Cambridge University Press, Cambridge 4. Bur T (2016) Mechanical miracles: automata in ancient Greek religion. Master of Philosophy Thesis, Faculty of Arts, University of Sydney 5. Colin GS (1932) La noria morocaine et les machines hydraulique dans le monde arabe. Hesperis 14:22–60 6. Coxhead MA (2012) A close examination of the pseudo-Aristotelian mechanical problems: the homology between mechanics and poetry as techne. Stud Hist Philos Sci Part A 43(2):300–306 7. De Miranda A (2007) Water architecture in the lands of Syria: the water-wheels. L’Erma di Bretschneider 8. Hill DR (1984) A history of engineering in classical and medieval times. Croom Helm & La Salle, London

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9. Johns CHW (1901) Assyrian deeds and documents recording the transfer of property, etc. vol II. Deighton Bell&Co. London, G. Bell&Sons, Cambridge 10. Laufer B (1934) The noria or Persian wheel. In: Jackson AVW (ed) Oriental studies in honor of Cursetji Erachji Pavry. Oxford at the University Press, pp 238–250 11. Morrison JS, Coates JF, Rankov NB (2000) The Athenian trireme: the history and reconstruction of an Ancient Greek Warship. Cambridge University Press, Cambridge 12. Murray WM (2011) The age of titans: the rise and fall of the great Hellenistic navies. Oxford University Press, Oxford 13. Needham J (1986) Science and civilization in China, vol 4, Part 2. Caves Books, Ltd., Taipei 14. Oleson JP (2008) The Oxford handbook of engineering and technology in the classical world. Oxford University Press, Oxford 15. Potts DT (2012) A companion of the archaeology of the Ancient near east. Wiley, New York 16. Radzevich SP (2016) Dudley’s handbook of practical gear design and manufacture, 3rd edn. CRC Press, Taylor & Francis Group, Boca Raton 17. Ruggiu L (2007) Aristotele, Fisica: Saggio introduttivo, traduzione, note e apparati. con testo greco a fronte, Mimesis Edizioni, Milano 18. Santander M (1992) The Chinese South-Seeking Chariot: a simple mechanical device for visualizing curvature and parallel transport. Am J Phys 60(9):782–787 19. Wieland W (1962) Die aristoteliche Physik. Untersuchungen uber die Grundlegung der Maturwissenschaft und die sprachlichen Bedingungen der Prinzipienforrschung bei Aristoteles. Göttingen 20. Wikander Ö (2008) Sources of energy and exploitation of power (Chap. 6). In: Oleson JP (ed) The Oxford handbook of engineering and technology in the classical world. Oxford University Press, Oxford 21. Wilson A (2008) Machines in Greek and Roman technology. In: Oleson JP (ed) Oxford handbook of engineering and technology in the classical world. Oxford, pp 337–368 22. Winter TN (2007) The mechanical problems in the corpus of aristotle. Faculty Publications, Classic and Religious Studies Department, University of Nebraska 23. Worlidge J (1669) Systema Agriculturae, being the Mystery of Husbandry Discovered and layed Open, 1st edn. T. Johnson for Samuel Speed, near the Inner Temple Gate in Fleet Street, London 24. Yannopoulos SI, Lyberatos G, Theodossiou N, Li W, Valipour M, Tamburrino A, Angelakis AN (2015) Evolution of water lifting devices (pumps) over the centuries worldwide. Water 7:5031–5060 25. Yonkxiang L (ed) (2014) A history of Chinese science and technology, vol 3. Springer, Berlin

Chapter 3

The First Scientific Age and Birth of the Science: From the Beginning of Hellenism to the Diaspora of Scientists of the Museum of Alexandria

Abstract In this chapter, the development of gears in the first scientific age (the age from the beginning of Hellenism to the diaspora of the Alexandria Museum scientists) is described. This development is characterized by the conception and construction of gears with characteristics almost analogous to modern gears; it is framed in the historical framework of the revolution of the way of thinking about natural phenomena, which is peculiar to the golden age of Hellenism, in which science in the modern sense of the term was born and the scientific method was affirmed. The main contributions to this development, given by scientists, researchers and scholars of the main cultural centers of Hellenism, are described, with particular attention to those of Archimedes, rightly universally considered the gretest scientist of the ancient world, and one of the greatest scientists of every age and time. The great innovations that made use of geared mechanisms are described in greater detail, focusing attention on the qualifying theoretical contributions that underlie the same mechanisms, which for the first time appeared in the history of technology, intended as the daughter of science.

3.1 Generality As we have already said in both Prefaces of Vol. 1 and Vol. 2, according to the most widespread thinking that is shared by historians of science and scientists, a science, to be such, must have at least the following essential features (see also [39]): • All that a science claims should not involve real objects, but specific theoretical entities. • The theory on which a science is based must have a strictly deductive structure, i.e. it must be characterized by a few fundamental statements (called axioms or postulates or principles), and by a unified method, universally accepted, to deduce from them an unlimited number of consequential properties. • Its applications to real objects must be based on rules of correspondence between the theoretical entities and real objects. The correction of this chapter can be found under https://doi.org/10.1007/978-3-030-40164-1_6 © Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_3

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Historians of science now agree that science, understood as a set of scientific theories, that satisfy the three requirements mentioned above, was born with the Hellenism. Historians of human affairs agree on the date of birth of Hellenism, made to coincide with the date on which Alexander the Great (356–323 B.C.) died. They are thus placed in the wake of Droysen, who also introduced the term Hellenism [11, 12], indicating the historical period that affected the geographical area conquered by Alexander the Great and administrated, at his death, by those who were the dyads, he living (see [5, 11]). This geographical area had Alexandria as the main center of conservation and dissemination of culture. The same historians are not as agreed on the date that marks the end of Hellenism. Most historians hold the end of Hellenism in 31 B.C. (this is the year in which the Battle of Actium took place), when the whole Mediterranean was unified under the domain of Rome. However, there are also many historians who put the end of Hellenism beyond this date. Historians who dramatically expand the duration of Hellenism identify the end of this historical age with that of ancient science, putting it at 415 A.D., when the mathematician Hypatia [she was born between 350 and 370 A.D. and was the daughter of the mathematician Theon of Alexandria (about 335–405 A.D.)] was lynched in Alexandria by a crowd of fanatical Christians, for religious reasons [8]. These two dates are quite questionable, because the transitions from one age to the next typically occur gradually, without such a clean break. The start date of Hellenism, for the truth a little macabre, could be moved a little earlier, as a determining factor in the Cultural Revolution related to it is certainly to be identified by the exceptional opening of horizons due to the conquests of Alexander the Great. The civilization of classical Greece thus opened not only to the Mesopotamian and Egyptian civilizations, but also perhaps to that of China, mediated by the Indian civilization. The possible back dating of the birth of Hellenism still has a marginal importance. Instead, the fracture that arises in the way of thinking between teacher and pupil, is very important. This is the fracture between Aristotle, tutor of Alexander and philosophical thought giant, but anchored in natural philosophy, and Alexander, which instead has an open mind to the world, and communicates this way of thinking to all those around him, who will be his heirs. It is to keep in mind that, despite the unparalleled achievements of classical Greece, Mesopotamian and Egyptian civilizations were certainly superior to that of Greece from the technological point of view. This was a direct result of the slow, but thousands of years of accumulation of empirical knowledge gained from these civilizations, and handed down from generation to generation. Only a methodological leap could determine the recovery of the technological gap between these civilizations and Greece, and the Greeks were able to do this in a commendable way, thus giving birth to the science. The first scientific age certainly starts with the beginning of Hellenism. Instead, the end of the first scientific age cannot be made to coincide with the end of Hellenism, because not all the Hellenism is characterized by a scientific mentality aimed at investigating natural phenomena with a scientific method. Indeed, the first scientific age is consumed and dies in the second half of the 2nd century B.C., when the conquest of all Greek polis by Rome is completed (this happened with the conquest

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of Corinth in 146 B.C., the same year of the destruction of Carthage), and the scientific activities in Alexandria cease dramatically. This latest historical fact was due to the diaspora of the Museum’s scientists, who in the two years between 145 and 144 B.C. ran away from Alexandria, to escape the vicious persecution of the Greek ruling class by Ptolemy VIII Kakergetes, also known as Physcon-potbelly, Ptolemy the Younger, and Ptolemy VIII Euergetes II Tryphon (170–116 B.C.), whose causes are completely unknown to us [15]. The 3rd century B.C. is certainly to be considered the golden age in which the science exploded. The main center where science was born and claimed it was almost certainly Alexandria, thanks to far-sighted policies of Ptolemy I Soter (323–283 B.C.) and his successor, Ptolemy II Philadelphus (283–246 B.C.). However, we must not forget the independent contributions of other Greek polis, among which Syracuse, Rhodes, Samos and Marseille must be specifically mentioned.

3.2 Prodromes of the First Scientific Age and Centers of the Hellenistic Science It should be kept in mind that the Hellenist science and the scientific method that forms the basis of the scientific theories developed at that time were not born from nothing. They are routed in the socio-economic-cultural context of entire Mediterranean Greek world, where a great interest in mathematics and clear symptoms of a new way of conceiving natural phenomena can be found, starting from the end of the 7th century B.C. Without following a chronological order, we believe it is useful to make a brief overview of those that were the prodromes that allowed the birth of Hellenistic Science and also to highlight the interdependencies of the centers of excellence were science developed with the previous scientific-cultural circles of the ancient Greek world. Between the end of the 7th century B.C. and the beginning of the 6th century B.C., a true cultural revolution takes place in the Greek world. Until then the Greeks explained the origins and nature of our world as well as the natural phenomena through myths that called heroes and/or anthropomorphic gods into question. The reversal of the way of thinking is to be ascribed to the Ionian school, of which Thales of Miletus (about 624–546 B.C.) is unanimously recognized as the founder. Miletus was the reference point of this school, which counts as its members, in addition to the leader Thales, Anaximander (about 610–546 B.C.), Anaximenes of Miletus (about 585–528 B.C.), Heraclitus of Ephesus (about 535–475 B.C.), Anaxagoras of Clazomenae (about 510–428 B.C.) and Archelaus of Miletus or more probably of Athens, lived between 5th and 4th century B.C. This last member of the Ionian school was pupil of Anaxagoras and, according to a lost paper of Ion of Chios (about 490–420 B.C.) quoted by Diogenes Laërtius, teacher of Socrates (469–399 B.C.).

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It is of no interest here to summarize the contributions of the aforementioned ancient coryphaeus-initiators of Greek thought, called physiologoi by Aristotle, who identified in them those we argued in nature. They are also known as cosmologists as well as physicalists who tried to explain the nature of matter. Moreover, the individual contributions of these members of the Ionian school are not always well defined, also because only fragmentary news of their works have come down to us, moreover filtered by later thinkers who, drawing water to their mill, sought to find in them the support for their own theses. Just think of the anecdote, taken up by Plato (429–347 B.C.) in his Theaetetus, according to which Thales, to contemplate the stars, does not see the rood and falls into a ditch, arousing the hilarity of his slave. Other not less hilarious anecdotes can be mentioned, especially for Heraclitus, but this is not the place to do it. We are interested here to point out that, although in a different way from each other, the members of the Ionian school were the first in the Western tradition to free themselves from the myth and to use abstract reasoning that, although very far from scientific thought, laid the foundation for its future birth. Before proceeding further in the description of the beginning of the first scientific age, it is however necessary to premise a summary picture of the Hellenistic centers where science was born and developed. Among these centers, Alexandria is certainty to be considered the international pole of attraction for the science at that time [4]. In Alexandria, Euclid (about mid-4th century–about 270 B.C.) worked and taught in the late 4th and early 3rd century B.C. He is the father of geometry and is to be considered the most important mathematician of ancient history, and one of the most important and recognized mathematicians in every time and place. Always in Alexandria, Ctesibius (285–222 B.C.), i.e. the founder of pneumatics as well as of the Alexandrian mechanical school, worked in the first half of the 3rd century B.C. In the same city, Philo of Byzantium (280–220 B.C.) continued the Ctesibius work in the second half of the same century. Also, in Alexandria, at the same time of Ctesibius, Aristarchus of Samos (about 310–230 B.C.), i.e. the founder of the heliocentric theory, and Herophilus of Chalcedon (335–280 B.C.), i.e. the fonder of scientific anatomy and physiology, carried out their activities. The Syracusan Archimedes (287–212 B.C.), almost certainly studied at Alexandria during his youth, and was in constant contact by letter with the Alexandrian contemporary scientists, such as Conon of Samos (about 280–220 B.C.) and Eratosthenes of Cyrene (about 276–194 B.C.). The mathematician Conon was a friend of Archimedes; he worked on the conic sections, as Apollonius of Perga (about 240–190 B.C.) tell us, and, as court astronomer of Ptolemy III Euergetes (246–222 B.C.), discovered the constellation Coma Berenice (Berenice’s hair), by the same Conon so called after Ptolemy’s wife Berenice II. Eratosthenes, to whom we owe the first real measure of land size, directed the Library of Alexandria in the second half of the 3rd century B.C. Still in Alexandria, Apollonius of Perga (to him we owe the theory of conics) worked in the late 3rd century and the beginning of the 2nd century B.C.; etc. Alexandria was the undisputed center of conservation and dissemination of culture, as it was the most equipped center for copying manuscripts,

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their widespread distribution (of course, for a money consideration), and their conservation in its famous Library. Pergamon also performed the same task, albeit to a lesser extent. Archimedes, the greatest genius of antiquity, was born and worked in Syracuse, where his killing by a soldier of Marcellus abruptly interrupted his activity, in 212 B.C., during the sacking of the city. The astronomer Hipparchus of Nicaea (about 190–120 B.C.), to be considered as the maximum 2nd century B.C. scientist, was active in Rhodes; he was the forerunner of the modern dynamics and theory of gravitation. However, it must be borne in mind that, if Alexandria was the cultural pole par excellence of Hellenism, the cities that gave birth to the greatest mathematicians and scientists of the time were also and above all others. Thales of Miletus (about 624–546 B.C.), Apollonius of Perga, Conon of Samos, Archimedes of Syracuse, Hipparchus of Nicaea, etc., testify to what extent the study of scientific disciplines was alive throughout the geographic area where Hellenism was active. In this regard, it deserves to dwell on Syracuse and Sicily (and, more generally, the Magna Graecia), which are traditionally included among the areas of influence of Hellenistic thought. A more detailed investigation, however, leads us to recognize that this geographical area, with Syracuse at the head, had intense and fruitful contacts with Alexandria, to be understood as a mutual exchange relationship, without any cultural subjection. The following quotations taken from the writings of Archimedes on this subject are emblematic (see also [39]): • In the dedication to Eratosthenes of the Method (see below), Archimedes makes a clear distinction: he alone is a mathematician, while Eratosthenes knows how to be only if he applies (see Sect. 3.3). There is therefore no admiration for the Alexandria librarian. Archimedes considers him an amateur who is applying himself, hoping that he will progress, without however expressing any form of superiority, but only a clear distinction between roles and methods. • The Problema Bovinum (Problem of the oxen), also this one dedicated to Eratosthenes in the form of an epigram, constitutes a true gauntlet of challenge launched to the same Eratosthenes and to all Alexandrian entourage, for its resolution. • Archimedes expresses his esteem, albeit very measured, for Dositheus of Pelusium, whose life and activity we know very little, including dates of birth and death; we know, however, that he was a pupil and studied under Conon and that after his teacher’s death replaced him as director of the mathematical school at Alexandria. After the death of Conon, Archimedes dedicated the following treatises to Dositheus: De sphera et cylindro (On the Sphere and Cylinder); De conoidibus et spheroidibus (On the Conoids and Spheroids); De lineis spiralibus (On Spirals); Quadratura Parabolae (On the Quadrature of the Parabola). • Archimedes expresses his esteem, even his admiration, only for Conon. This admiration is highlighted in the treatise On the Quadrature of the Parabola, which is dedicated to Dositheus. The existence of a Syracusan scientific school, headed by Archimedes, is almost certain. The traces of a probable Syracusan scientific circle, and perhaps also a

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Sicilian scientific circle, are contained in the first lines of the On Spirals treatise. In these lines, Archimedes apologizes with Dositheus of the delay with which he sends the treatise, thus justifying: “I wanted to first submit my inquiry to those who deal with mathematical things”. The sincere tone of justification suggests the non-great esteem of Archimedes towards Dositheus and downgrades the role of Alexandrian scientific school, showing an implicit recognition of greater value to other cultural circles, which we can only imagine today. The probable Syracusan scientific circle is also foreshadowed in the first lines of the Arenarius, where Archimedes reports that the discussion on large numbers had taken place [13]. Someone then asked themselves whether it is legitimate to identify in Archimedes an heir and a significant exponent of a school of thought and investigation independent of the other cultural centers mentioned above, that is an epigone of a distinct naturalistic school in Italy, which could have influenced a possible Syracusan school, and therefore the same Archimedes. From the scientific-cultural point of view, it is certain that, at least since the 5th century B.C., Syracuse was Sicily and Sicily was Syracuse. Favored by its strategic position in the heart of the Mediterranean, the Polis founded by Arkhias between 758 and 733 B.C. had become, together with Rome, almost contemporary by birth, the main city of the European area. The other important Sicilian cities (Naxos, Messana, Gelas, Katane, Selinus, Tauromenion, Megara Hyblaia, Leontinoi, etc., and the same Akragas, from which Hieron I came from) could not compete with Syracuse that, under its first tyrant Hieron I (he was a tyrant of Syracuse from 478 to 467 B.C., after being a tyrant of Gelas from 485 to 478 B.C.) increased its well-being and its power to such an extent that Thucydides declared that “Syracuse was not less than Athens”. To the expansionist aims, Hieron I joined a real patronage, so that Simonides, Pindar, Bacchylides, Aeschylus and Epicharmus were at home in Syracuse and active at the tyrant’s court, as well as the philosopher Xenophanes. Subsequently, at the time of Dionysius II the Young man (367–344 B.C.), Plato went to Syracuse three times in the illusion of translating into reality his illusory and idealistic system of government hypothesized in his The Republic. The imperialistic politics of Syracuse began with the foundation of new colonies (e.g., Akrai, Kasmenai, Kamarina and Morgantina, respectively founded in 664, 643, 598, and about 560 B.C., and Akrillai and Heloros, both founded in 7th century B.C.) and the downsizing of the role of Carthage in Sicily, defeated by the Battle of Himera (480 B.C.). This politics was continued by sea, first with the colonization of the Dodecanese islands, and subsequently with the naval Battle of Cumae (476 B.C.), which ended the Etruscan claims on the southern sea and in the south of the Italian peninsula. To counter the Carthaginian claims on Sicily, Syracuse brought war on African soil, and placed cornerstones in Hudrous (Hydruntum-Otranto) and Brentesion (Brundisium-Brindisi), while its influence extended to the Etruscan city of Atria (or Hatria-Adria), which Dionysius I the Elder (it was tyrant of Syracuse from 405 to 367 B.C.) colonized, turning it in an emporium. Between the end of the 4th century B.C. and the beginning of the 3rd century B.C., under the reign of Agathocles (he was a tyrant of Syracuse from 317 to 304 B.C. and self-styled King of Sicily from 304 to 289 B.C., the date of his death), Syracuse played a significant

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role in the international politics of the time, intervening against the Macedons, and carrying out expeditions in Libya (291 B.C.), to close the wheat rout from Sardinia to Carthaginians. In the Syracusan polis, not only political life was fervent, but also artistic life (just think of Epicharmus, Theocritus, Diodorus Siculus, etc.) and the scientific one, which interests us. The cultural vivacity was such that despite the depredations suffered in the looting of the 212 B.C. and the subsequent robberies perpetrated by Gaius Verres, still in the 1st century B.C. Marcus Tullius Cicero (106–43 B.C.) called Syracuse “urbem maximam Graecarum pulcherrimam omnium” (i.e., “the most beautiful and important of the Greek cities”). The cultural life of Syracuse, however, ended dramatically in 212 B.C., with the Roman conquest, even if the residual effervescence of the city died in a long and sad agony, as the monumental universal history of Diodorus Siculus (about 90–27 B.C.) shows. About what interests us here, the historical information assigns to Syracuse an important role in science and technology, to be understood as the art of designing and making machines, and this role was not less than the one mentioned above, concerning politics, history, and arts. Clear evidences of the mechanical techniques related to war operations or other human activities can be identified in Pappus of Alexandria (about 290–350 A.D.) and Diodorus Siculus. We are indebted to Pappus (see his Collectio, III and VIII) for a broad-spectrum description of the mechanical techniques used in the ancient Greek world. Among these techniques, Pappus distinguishes: • mechanical techniques most necessary for daily life, which include those used by the manufacturers of tools that serve to alleviate the human fatigue (e.g., winches and similar), and those used by the manufacturers of war devices (e.g., the catapults to hurl darts, stone projectiles or similar objects); • mechanical techniques used by machine manufacturers in the strict sense of the word, which are those necessary to design and build everyday mechanical devices, such as, for example, water lifting machines from sources of considerable depth; • mechanical techniques used by manufacturers of mirabilia (wonderful things), which exploit different physical phenomena and are designed to delight those who see them in operation. Diodorus Siculus, in fragments of his history (Bibliotheca Historica, XIV, Chap. 50, fragment) describes the mechanical fervor in Syracuse under the impulse of Dionysius, aimed at countering Carthage, and reminds us that in Syracuse the catapult was designed, which was used against Carthaginian ships that threatened the port. It is certain that most of the mechanical techniques and related devices, mechanical groups and machines described by Pappus as well as the auxiliary mechanism of the main mechanism of catapult described by Diodorus Siculus made use of geared mechanisms much more refined and advanced then those described in the previous chapter. This certainty is supported by the archaeological evidence described below and in the following sections.

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By virtue of the aforementioned relations of Syracuse with the other Greek cities of the Mediterranean, and particularly of Magna Graecia, it is legitimate to think that the Syracusan school was an integral part of an autonomous school of thought typical of southern Italy, which faced the main schools of the time, but at the same time distinguished itself from them. The current revaluation of Greek science has focused its attention on Hellenism, overshadowing the scientific fervor in Magna Graecia, also because the few testimonies of this fervor are limited to fragmentary news related to the natural philosophy of the so-called pre-Socratics and Pythagoreans. The first antipope of history, Saint Hippolytus of Rome (170–235 A.D.), reports that Pythagoras of Samos (about 570–495 B.C.) founded his school in Croton, in 530 B.C., where he had taken refuge from Samos to escape the tyranny of Polycrates. Samos certainly had a consolidated and developed tradition of mathematics and mechanics, otherwise the Tunnel of Eupalinus, described by the Greek historian Herodotus, designed by Eupalinus of Megara on behalf of Polycrates and completed between 550 and 530 B.C. would have been impossible to excavate and implement, without the adequate knowledge of trigonometry and availability of suitable detection instrumentation that it presupposes. The same Saint Hippolytus makes us the portrait of a Pythagorean, the Syracusan Ecphantus (about 500–450 B.C.), who claimed the impossibility of certain knowledge of things and was a supporter of the heliocentric theory. Diogenes Laërtius (3rd century A.D.), in his Vitae Philosophorum, mentions two other Pythagoreans, Philolaüs of Croton (about 470–390 B.C.) and Hicetas of Syracuse (about 400–335 B.C.). Philolaüs of Croton (or Tarentum, or Metapontum) was the successor of Pythagoras and, among other things, is credited with originating the theory that the Earth was not the center of the Universe. Hicetas of Syracuse, like his fellows Ecphantus the Pythagorean and Academic Heraclides Ponticus (about 390–310 B.C.) believed that the daily movement of permanent stars was caused by the rotation of the Earth around its axis. Marcus Tullius Cicero also in his treatise Academica accredits Hicetas as an ancient naturalistic philosopher who also argued that the Earth was moving. The cultural framework of Magna Graecia of the fifth and fourth centuries B.C. must necessarily include Archytas of Tarentum (428–347 B.C.), also a Pythagorean one. He was a pupil of Philolaüs (about 470–390 B.C.), and teacher of mathematics of Eudoxus of Cnidus (408–355 B.C.). Archytas was a uncommon polyvalent mind, able to influence students of great stature, like Eudoxus, who studied geometry with him (and medicine in Sicily with Philistion of Locri also referred to the Sicilian). During his life, Archytas became interested in what he could, ranging from civil to military life, from science to music, however always excelling in every field. Diogenes Laërtius has given us an interesting testimony of Archytas’ versatility, stating that he would be the initiator of the combination of mathematics and mechanics that, with Archimedes, would have found the highest synthesis, with the technological repercussions that stunned the ancient world and still they amaze us. Aulus Gellius (about 125–180 A.D.) tell us that Archytas designed and built the first artificial self-propelled flying device, a bird-shaped model propelled by a jet of aeriform substance (probably steam), which he called The Pigeon. Marcus Vitruvius Pollio (about 80–15 B.C.) includes Archytas in a list of twelve authors of works of

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mechanics, but all of the Archytas’ works have been lost. However, independently of other contributions concerning, for example, acoustics and, above all, the solution of doubling the cube problem with the introduction of the curve that bears his name (the Archytas curve), Archytas can be considered as the founder of Applied Mechanics. Regarding the history of the gear, here it should be remembered that some scholars hypothesize that Archytas made use of worm gears, although very coarse. For example, Manna [26] overshadows the possibility that Archytas used this type of gear, whose definitive statement as well as its geometric-mathematical characterization is undoubtedly ascribed to Archimedes (see next section), for use in rudimentary winches for lifting heavy loads, such as the one shown in Fig. 3.1, in place of the rackpinion gear pair. This last type of gear at that time was also used as a mechanism for adjusting the position of the wick of the first oil lamps, as shown in Fig. 3.2. Finally, it should be remembered that Winter [45] has suggested that the pseudo-Aristotelian Mechanical Problems (see the previous chapter) is an important mechanical work of Archytas, not lost after all, but misattributed. Moreover, we must not forget the Eleatic school, which was founded in 5th century B.C. by Parmenides of Elea (510–450 B.C.) and had, as chief exponents, the same Parmenides and, in succession, Zeno of Elea (490–430 B.C.) and Melissus of Samos (about early 5th century–430 B.C.). It was Parmenides who, first, affirmed the rotation of the Earth and erased the sphere of the permanent stars, while Zeno first employed the reductio ad absurdum, with which he attempted to destroy the Fig. 3.1 Facsimile of worm gear supposedly used by Archytas

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Fig. 3.2 Facsimile of a mechanism for adjusting the position of the wick of the first oil lamps

arguments and theories of the early physicalist philosophers by showing that their premises led to contradictions (Zeno’s paradoxes). Moreover, this type of reductio ad absurdum is often used by Archimedes when, before describing the conclusions on a thesis proposed by him, he absurdly sweeps away any hypothetical contrary demonstration, which would produce aberrant results. Today science historians unanimously recognize the fact that the Eleatic school has marked a considerable step in the advancement of science. Both schools, the Pythagorean and the Eleatic ones, although different from each other, had in common the aim of searching for the cause of things outside the world of the senses [24]. For both schools, it is no longer necessary to save the phenomena, in order to combine observation with thought (as the Platonic and Aristotelian dogmatic schools did), but rather to explain the phenomena, to make the observations coherent with the theses formulated. In Magna Graecia, in Sicily and particularly in Syracuse, a new way of observing things was made and progressively established, free from preconceived ideas or even worse theological ones, which left the mind free to be amazed and to construct theories. This way of observing reality is not so far from the following aphorism of Quintus Horatius Flaccus, in Epistula II ad Maximum Lollium, Liber Prior (I), 40–41: “sapere aude, incipe”, i.e. “take courage to acquire wisdom, begin to do this immediately” (see [6]). This aphorism was assumed by the Enlightenment as its manifesto. Perhaps this reference to the Enlightenment in relation to the aforementioned cultural fervor is reckless, but it constitutes a necessary premise without which we would run the risk of diminishing the greatness of Archimedes, who was absolute, unique and unmistakable genius. Lastly, at the down of the first scientific age, it is worth noting the figure of Pytheas of Massalia (about 350–285 B.C.), which can be considered the most famous citizen

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of the Greek Marseille. He was a mathematician, geographer, astronomer, navigator and explorer, but he was also a designer and builder of scientific instruments that can be traced back to the gnomon, which allowed him to establish almost exactly the latitude of Massalia. He was also the first scientist to observe that the tides were connected to the phases of the moon. Between 330 and 320 B.C., Pytheas organized a ship expedition to North Atlantic, circumnavigating England and visiting, in addition to England, also Iceland, Shetland, Thule and Norway beyond the Arctic Circle, where he was the first scientist to describe solidified see, drift ice and the midnight sun. He described his travels in a work not survived, of which, however, excerpts remain, quoted or paraphrased by later authors, as the Greek geographer Strabo in his Geographica, Pliny the Elder in his Naturalis Historia, and Diodorus Siculus in his Bibliotheca Historica.

3.3 The Golden Age and the Genius of Archimedes The first great figure to be considered is certainly Euclid, who was active in Alexandria during the reign of Ptolemy I Soter. He is the scientist with whom the golden age of the first scientific age begins. Euclid therefore represents for the history of Hellenistic science what Homer represents for the history of Greek literature. According to Proclus Lycaeus Diadocus, also called Proclus of Athens (412–485 A.D.), Euclid supposedly belonged to Plato’s persuasion and collected his Elements based on prior work of Eudoxus of Cnidus (408/406–337 A.C.) and of several pupils of Plato (427/426 or 424/423–348/347 B.C.), particularly Theaetetus (see [2]) and Philip of Opus (Diogenes Laërtius, Vitae Philosophorum, iii. 37), which should not be much older than him. In this regard, it is worth mentioning the unconditional admiration of Plato for geometry (of course transmitted to his pupils), up to the point to write at the entrance of his Academy the famous warning: “’Aγεωμšτρητoς μηδε`ις ε„σ´ιτω”, i.e. “let none but geometers enter here”. Euclid ‘s Elements are to be considered one of the most influential writings in the history of Geometry and Mathematics, which knew a number of editions that was second only to that of the Bible, and served as the main textbook for teaching such disciplines since the time of its publication until the late 19th or early 20th century (Fig. 3.3). Reflecting on this factual circumstance, the historian of mathematics and science, Sarton [40], defined the Elements as a “monument which, due to its symmetry, its intrinsic beauty and clarity is as marvelous as the Parthenon, but incomparably more enduring”. To us it remains only to add that one of the fundamental pillars of gear science, the one concerning the geometrical aspects, could not have made great progress without the Euclid’s Elements. Instead, the first great figure of the engineering tradition of Alexandria is the inventor and mathematician Ctesibius (285–222 B.C.), two years younger than Archimedes (see [9, 19, 23]). Very little we know of his life, while his inventions are well known. He is universally recognized as the father of pneumatics, and is credited with a considerable number of inventions, including a water suction pump, a water organ called

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Fig. 3.3 Frontispiece of the first English translation of the Euclid’s elements

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hydraulis, a more accurate water clock than previous ones, a hydraulic hoist, pneumatic and bronze spring catapults, several automatic devices designed for entrainment, called automata, etc. The inventive capacity of Ctesibius was very early, as evidenced by the invention of a counterweight-adjustable mirror, which he made when was a boy working as a barber in his father’s shop. Unfortunately, all the Ctesibius’ writings have been lost, even those that presumably described the theoretical fundamentals of pneumatics. Fortunately, however, we know some of the inventions of Ctesibius thanks to mentions made by many later inventors, such as Philo of Byzantium, Marcus Vitruvius Pollio, Athenaeus of Naucratis, Pliny the Elder and Hero of Alexandria. The main news of the Ctesibius’ works we have received, however, are those of Athenaeus and, above all, of Vitruvius. Here we limit ourselves to considering the news given by Vitruvius (about 80–15 B.C.), since Drachmann [9] convincingly argued that we should accord our preference to the Latin scholar since he, unlike Athenaeus, had direct access to Ctesibius’ works. In his monumental treatise in 10 Books, entitled De Architectura, Vitruvius shows to be only interested in the Ctesibius’ discussions on pumps, organs and water clocks, but also gives us clear evidence that Ctesibius wrote works concerning the entertainment devices and their technologies. In fact, to justify his selective use of Ctesibius’ works, Vitruvius tell us that he limited himself to focusing his attention on what he himself considered useful, leaving aside everything else. In this regard, he writes: … “reliqua, quae non sunt ad necessitatem sed ad deliciarum voluptatem, qui cupidiores erunt eius subtilitatis, ex ipsius Ctesibii commentariis poterunt invenire” (i.e., … “the rest, which is not suited to our needs, but to the pleasures of entertainment, can be found in the commentaries of Ctesibius himself by those who are interested in such refinements”). In Books 9 and 10 of his De Architectura, Vitruvius gives us a detailed description of the Ctesibius’ pumps and, as far as we are interested here, of the geared mechanisms by which a water clock could activate automatic responses in terms of variation of the length of the hour according to the season. By focusing his attention in more detail on the teeth of these geared mechanisms that characterized the Ctesibius’ devices, Vitruvius writes that … “these teeth, acting on each other, induce a measured revolution and movement. Other racks and other drums, similarly toothed and subjected to the same motion, give rise to various kinds of motions, by which figures are moved, cones revolved, pebbles or eggs fall, trumpets sound and other incidental effects take place”. Here we would like to dwell briefly on the use of two geared mechanisms invented and developed by Ctesibius, characterized respectively by a pinion-rack gear pair and an ordinary gear train consisting of two pairs of cylindrical spur gears. Apart from other features that are not of interest here, the first of the two geared mechanisms is the one used in the Ctesibius’ accurate water clock, which was characterized by a complex and imaginative way of denoting the time. Apart from the hydraulic device for automatic adjustment of the current hour length (see below), it is worth noting the completely new introduction or use of a pinion-rack geared mechanism. The end part of the rod connected to the hydraulic piston was in fact configured in the shape of a rack that operated a pinion to which the hour hand was

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rigidly connected. This hand then marked the hours on a graduated scale marked on a ring in the shape of a circular crown. Figure 3.4 shows a possible reconstruction made in the first half of the nineteenth century of the Ctesibius’ water clock or clepsydra (water thief). As regard the type of gear used, it shows important improvements with respect to previous periods, among which the use of metallic materials is to be highlighted. They allowed a considerable reduction in overall dimensions. The second geared mechanism is the one used in the Ctesibius’ fancy clepsydra, also used as an automaton or self-adjusting device. Figure 3.5 shows a possible reconstruction of this device also made in the first half of the nineteenth century. One the two ends of the rod of a hydraulic piston supports a figure pointing at the current hour of the day; the same figure rises when the water enters, operating the piston. Spillover water operates an ordinary gear train, consisting of two pairs of cylindrical spur gears, which drives and controls the rotation of a cylinder rigidly connected to the output gear wheel, so that the hour lengths are automatically changed to match the date of current day. It should be remembered that the Greeks and Romans divided the day from sunrise to sunset in twelve hours. They need to adjust the hour lengths according to the season, as summer days are longer than winter days; therefore, summer hours were longer than winter hours. In this case, the improvements in the geared mechanisms compared to the past are also very significant. Vitruvius then does not exempt himself from expressing his admiration when he describes Ctesibius’ inventions concerning more specifically moving-figures automata activated by compressed air or hydraulic devices. In this regard, he writes that these automata were activated “by means of water pressure and compression of the air” and as examples he mentions “blackbirds singing by means of waterworks, automated dancing dolls, figures that drink and move, and other amazing devices to the eye and ear”. It is likely that even these automata, designed and built to amaze, used refined geared mechanisms such as those mentioned above. Here we want to call attention to two of the many automata almost certainly designed and built by Ctesibius, both characterized by a pinion-rack geared mechanism and used in the famous Grand Procession in Alexandria sponsored by Ptolemy II Philadelphus. The first of these automata was included in a large self-automatic cart that carried the statue of Nysa, impressively imposing itself and dispensing libations as it passed throughout the streets of Alexandria. The movement of the statue inarguably relied on the wheels of the cart as its source of energy. This method of transforming the circular motion of the wheels into a linear motion of the statue was amazing for that time in that, to automatically move the same statue, it used a completely new technology, different from the technology used in Demetrius Phalerum’s snail decades earlier, and still used in the Panathenaic ship cart centuries later. The second automaton was characterized by a cam mechanism associated with the above described pinion-rack geared mechanism. Ctesibius’ invention consisted of a cam-operated statue of a mysterious deity that figured preeminently in the aforementioned Grand Procession, where it carried out a continuous performance, entertaining the festival crowd by standing up and setting up. The excitement caused by this very simple application of the rack-pinion gear pair was certainly due to the fact

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Fig. 3.4 A possible reconstructions of the Ctesibius’ accurate water clock

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Fig. 3.5 A possible reconstructions of the Ctesibius’ fancy clepsydra

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that this gearing mechanism was a recent invention, whose possibilities of practical applications were to be explored. It should be noted that we have here attributed to Ctesibius the first of the two aforementioned automata on the basis of the Fraser conjecture [15], considered by other scholars to be very attractive. Among them, White [43, 44] also adds that the application of the pinion-rack geared mechanism was then particularly exciting because the toothed gear wheels (of course, in the almost modern sense of the term) were a recent invention. This attribution is also supported by the fact that Ctesibius was almost certainly part of the circle of the court collaborators of Ptolemy II Philadelphus. As we learn from Athenaeus, the second automaton is certainly the work of Ctesibius. In Alexandria, Philo of Byzantium (about 280–220 B.C.), also known as Philo Mechanicus, continued Ctesibius’ activities. We know little of Philo, as most of his large work Mechanike Syntaxis has been lost. Only the military sections of this work, entitled Belopoeica and Poliorcetica, and a few other fragments of the sections Isagoge and Automatopoeica, have come down to us in the Greek original text, while we have inherited very little of the remainder. In addition, this part is derived or from a Latin translation of an Arabic version (see, for example, the section concerning Pneumatics), also made very bad, or from a derivative form incorporated in the works of Vitruvius and Arabic authors. In other terms, the largest part of what he wrote is rather lost. In any case, now we have the certainty of being indebted to Philo for the first description of a water mill, which incorporated gear wheels. In fact, according to recent research, a section of Philo’s Pneumatics that so far has been regarded as a later Arabic interpolation, includes the first description of a water mill in history (see [21, 42]). In another section of his Pneumetics, Philo describes an escapement mechanism, the earliest known, as a part of a washstand. It is not the case here to describe this mechanism (in this regard, see [20]). However, the remarkable comment by Philo on the construction of this mechanism should be mentioned: “its construction is similar to that of clocks”. This comment indicates unequivocally that such escapement mechanisms were already integrated in the ancient water clocks. We are also indebted to Philo for the oldest known application of a chain drive in a repeating crossbow as well as a detailed description of the Polybolos, i.e. the ancient repeating ballista reputedly invented by Dionysius of Alexandria (see [38]). We know very little about this last inventor, who was almost a contemporary of Ctesibius; he was a brilliant Greek engineer in the 3rd century B.C., and worked in the arsenal at Rhodes. The Polybolos he invented was a catapult that like a wooden machine gun could fire repeatedly without a need of reload. This new and completely innovative feature was because, contrary to an ordinary ballista, it had a bolt automatic charging device as well as a magazine capable of hold several dozens of bolts. Apart from other features described by Philo of Byzantium, it interests us here that the Dionysius’ Polybolos had, at both sides, two pairs of sprockets that were connected with two wooden flat chains. These chains were connected to a windlass, which by winding back and forth, automatically fired the machine’s arrows until its magazine was empty. Philo thus accredits Dionysius of

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Fig. 3.6 Reconstruction of a polybolos as described by Philo of Byzantium

Alexandria to be the inventor of the first sprockets as well as of the first sprocket chains. Even Vitruvius confirms these inventions. Figure 3.6 shows a polybolos with pentagonal gear wheels linked by chain drives powered by a windlass, according to the description of Philo. Philo also was the first to describe a gimbal. It was applied to an eight-side ink pot that could be turned in any way, driven by gears, without the ink being poured. This was done by suspending the inkwell at a central plate, after assembling this on a series of concentric metal rings, which were stationary regardless of how the pot could rotate. Other scientists, scholars and technicians continued in Alexandria (and in circles connected with Alexandria) the activities undertaken by Ctesibius and Philo of Byzantium, realizing devices characterized by geared mechanisms very similar to those of our days, more or less useful for daily needs, war machines or entertainment. Drachmann [10] notes in this regard that the first historical documentation of gears similar to the current ones is precisely to be traced back to the first half of the 3rd century B.C., that is, almost to the beginning of the Hellenistic age. It was in this age that kinematic pairs, consisting of two gear wheels meshing with one other were developed, which offered to engineers the ability to solve various problems related to new

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practical applications. In fact, the technical requirements became more complicated, linked as they were not only to the need to transfer motion between axes variously positioned (parallel or orthogonal), but also the need to having machines with gear pairs with a high transmission ratio, and thus characterized by a high reduction of the force to be applied for their appropriate operation. Almost simultaneously, different and new kinds of machines were developed, such as machines for water-lifting, water mills, windmills, military equipment and machinery, gadgets for entrainment and automata, water clocks, measuring instruments as diopters, planetariums, armillary spheres, odometers, etc. Unfortunately, very few archaeological evidences of this intense activity have come down to us, but as we will see below evidences are so striking that they do not cease to amaze us. We must now turn our attention to the very important contributions to Hellenistic science given by Syracuse, where Archimedes, certainly the greatest scientist of antiquity and one of the greatest scientists of all time, lived and worked. As it is well known, he is universally recognized as the father of Mechanics, in the high meaning of the word, that is understood and conceived as the science of machines. Archimedes is the founder of the Statics, as we understand it today. He also is the inventor of a quantitative theory of machines, such as winches and speed reducing gears. This theory does foreshadow, so to speak, a reduced form of Dynamics, thus having a more limited meaning than what we give to it today; it enabled to carry out the study of machines, through their reduction to a succession of balanced static conditions, in each of which the forces are taken almost in balance at each instant of time. Archimedes was an eminent encyclopedic scientific mind, having dealt with science in all fields, from optics to mechanics, from pneumatics to hydrodynamics, while remaining always faithful to the geometric-mathematical world. He transcended the Aristotelian dogmatisms, the Platonic idealisms and every schematism of his time, and never disdained to experiment his ideas, always translating them into the most technologically advanced realizations of his historical period. Archimedes was a very great scientist, in the most modern meaning of the term. His scientific approach is in fact that of discovering and finding, through careful observation of natural phenomena. This emerges clearly from the first pages of his treatise On the sphere and cylinder, where he writes: “although the properties pre-existed according to nature in these figures (sphere and cylinder), even though many illustrious geometers have succeeded before Eudoxus, it happens that they were neglected by everyone, not recognized as such by anyone”. The scientist therefore has the task of discovering the properties inherent to the figures in question, of putting them out, making them and participating in the scientific community and civil society. Archimedes had clearly in mind the close link between geometry and mechanics, conceived as two scientific theories, so much so that in the study of simple machines he devised, he draws from geometry not only the general form of the deductive procedure, but also special technical results. Moreover, surprisingly Archimedes used the laws of mechanics to discover theorems of geometry. With reference to this procedure, it is to be considered emblematic the premise that Archimedes himself writes in the introduction of his treatise The Method of

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Mechanical Theorems. This treatise, shortly named The Mechanical Method or simply The Method, is included in the so-called Archimedes Palimpsest, which holds also other six treatises of Archimedes, namely On the Equilibrium of Planes, On Spirals, Measurement of a Circle, On the Sphere and Cylinder, On Floating Bodies and Stomachion, some of which were already known (see [28–33]). In this premise, directed to Eratosthenes (at the time librarian of the Library of Alexandria), Archimedes writes:

So, I send you the proofs of the theorems I wrote in this book. Knowing that you are a diligent and excellent teacher of philosophy, able to evaluate in mathematical matters the observation that present itself, I decided to write and expose you, in my book, the properties of a method by which you will be able to identify the procedure for investigating mathematical theories through mechanical entities. I am also convinced that this method is not less in demonstrating the same theorems. In fact, even to me some concepts first appeared in mechanical evidence, and only later I proved them by geometric-mathematical way, because the observation made in this way is without geometric proof. However, once some knowledge of the things sought has been acquired in this way, it is then easier to provide proof of it with a rigorous procedure.

The mechanical observations are explained by Archimedes with geometric procedures, but the principles presented in his The Mechanical Method do not mark the primacy of speculative mathematics on the mechanical investigation and on the observation and study of the natural phenomena, or vice versa. Theory and experimentation interpenetrate, and his geometry is both abstract and concrete, because “certain experiences and reasoned demonstrations” always follow the exposition of the theory presented. This way of thinking and working constitutes a drastic hiatus compared to that of the greatest naturalist philosophers and geometers, and to mark origin and descendance, he cities only Euclid (4th–3rd century B.C.) and Eudoxus (390–337 B.C.), and only Democritus (460–370 B.C.) among those who were also philosophers. This same way of thinking and working makes science and the world of scientific research perform a fundamental step that sets the basis of scientific thought, setting its pillars in the manifestation of the phenomenon, observation, experimentation, discovery, and demonstration. Platonic and Aristotelian walls are broken, and scientific activity is definitively freed from archaic conceptions. Therefore, to discover mathematical results, Archimedes uses, in a first time, the mechanical method. We now call it physical method, but at the time of Archimedes physics had not yet established itself as an autonomous science, in the modern meaning of the term. In a second time, Archimedes uses the geometric method to demonstrate propositions already heuristically identified as possible, with the mechanical method (see [1]).

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In addition, for his demonstrations on rigorous scientific bases, Archimedes uses the Method of Exhaustion (this term was introduced only in modern times), which he himself invented and described in his treatise The Quadrature of the Parabola. With this method, Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. In addition, it is to be noted that, for a long time the method of exhaustion has been deemed, wrongly, a precursor of the modern methods of passage to the limit. However, it differs in nothing by them (only the name is different), so Archimedes is the father of the concept of limit, which constitutes the embryo of the modern infinitesimal calculus. Finally, based on the above two methods, Archimedes founded a scientific theory of mechanics, which he described above all in his lost treatise Mechanical Elements that he himself mentions in the treatise On Floating Bodies-Book 1 (see [14]), but also in his treatise On the Equilibrium of Planes. On these rigorous theoretical bases, some of which are exhibited in the other four treatises in the Archimedes Palimpsest, it is not surprising that, just in the time of Archimedes, and perhaps thanks to Archimedes, gear wheels like those of today were introduced for the first time. In fact, as we have already said above, the first documentation of gear wheels similar to the current ones is dated to the first half of the 3rd century B.C. (see [10]). These gears offered to engineers the opportunity of developing new practical applications, and new and unusual technologies, and were the results of new scientific theories as well as of new knowledge about materials, both daughters of Hellenism. Mutatis mutandis, the gears that we use today are the direct descendants of the Hellenistic gearing mechanisms, retrieved through the Arabic and Renaissance studies of ancient works; in other words, they are cultural products that we have inherited from the Hellenistic civilization. The association of the name of Archimedes to the most refined knowledge and the most advanced technologies conceived by Hellenism in the field of gears is not a flight of fancy, but it is founded on a solid foundation, as recent documentary and archaeological findings show. Here, to describe the documents (some of these are well known since antiquity) as well as the archaeological findings, we follow a mixed order, because our goal is to highlight the fundamental contribution of Hellenism to scientific and technological knowledge regarding the gears. At the end of the previous chapter, concerning the first pre-scientific age, we have dismantled the myth that, in the Western World, the first most ancient description of gears is that of the pseudo-Aristotle, and we have described the reasons for this dismantling. However, we have abundant historical documentation that unequivocally demonstrates that, between the end of the 4th century and the beginning of the 3rd century B.C., various Greek scholars used gears in water wheels, clocks, and other mechanisms and machines. However, the first literary documented example of a geared device, made of metal, is possibly formed by the “Archimedes’ Planetarium”, which was one of his admired accomplishments in antiquity. In fact, Marcus Tullius Cicero (106–43 B.C.) in De Republica, I, XIV, 21, 22 (see [27]) informs us that, after the sacking of Syracuse (212 B.C.), the consul Marcus Claudius Marcellus brought to Rome two astronomical mechanisms, designed and built by Archimedes, as a prey of war. The first mechanism reproduced, on a sphere,

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the sky. The second, the planetarium, predicted the apparent motion of the sun, moon, and five planets known at the time (Mercury, Venus, Mars, Jupiter, and Saturn), and was therefore equivalent to a modern armillary sphere. Then Marcellus kept for himself the planetarium as his personal loot from Syracuse, and donated the other mechanism to the Temple of Virtue in Rome. In addition, Cicero describes the amazing impression of Gaius Sulpicius Gallus, who was lucky enough to observe the unique astronomical mechanism, and underlines how the genius of Archimedes was able to generate the motions of the planets, enclosed in a sphere and having very different orbits, by a single rotation. Gaius Sulpicius Gallus indeed examined the Archimedes’ planetarium in the house of his colleague of consulate Marcus Claudius Marcellus Jr., who had inherited it from the grandfather with the same name (the looter of Syracuse). The same Gallus described in his lost work his amazement because the motion of all heavenly bodies under consideration depended on only one conversio. Cicero then repeats the same concepts and the same admiration for Archimedes in two other of his famous work, i.e. Tusculanae Disputationes, I, XXV, 63 and De Natura Deorum, II, XXXIV–XXXV, 88. Archimedes’ planetarium is also cited by: Publius Ovidius Naso (43 B.C.–17 or 18 A.D.), in Fasti, VI, 263–283; Lucius Caecilius Firmianus Lactantius (about 240–320 A.D.), in Divinae Institutiones, II, 5, 18; Pappus of Alexandria (about 290–350 A.D), in Collectio, VIII, 1026, 2–4 (Pappus informs us that Archimedes described the planetarium in a manuscript now lost, entitled Spheropea, i.e. On Sphere-Making); Claudius Claudianus (about 370404 A.D.) in Carmina Minora, where the details are added according to which the mechanism was enclosed in a stellar sphere, made of glass; Martianus Mineus Felix Capella (about 360–428 A.D.) in De nuptiis Philologiae et Mercurii, VI. It should however be remembered that today we do not have a finding on which to base a reliable reconstruction of the real configuration of the gearing mechanism that animated the Archimedes’ Planetarium. Figure 3.7 can give a rough idea of this mechanism, which was able to simulate, inside a star-studded glass sphere, the apparent motion of the stars around the earth, the annual and monthly motions of the moon and sun along the ecliptic as well as the sun and moon eclipses at proper time intervals. A recent reconstruction of the Archimedes’ planetarium was made by Wright, former curator of the Science Museum of London, based on two previous experiences that he himself had treasured up in the reconstruction of the Antikythera Mechanism, subsequent to those made in the past [47]. It consists of a copper globe, set in a wooden box, able to reproduce, through a gearing mechanism made of 24 metal gears, the apparent motion of the sun, moon and five planets known in antiquity. The model reconstructed by Wright, which in 2015 was exhibited at the Antikenmuseum of Basel in Switzerland, respects the aforementioned descriptions, starting from those of Cicero, which categorically state that the Archimedes’ planetarium was a threedimensional gearing mechanism, i.e. it had a spherical shape, unlike Antikythera Mechanism that was a two-dimensional mechanism. Wright himself defines the latter as a “shoebox”, to effectively express its intrinsic two-dimensionality.

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Fig. 3.7 A rough reconstruction of the gearing mechanism of Archimedes’ planetarium

The interest on the Archimedes’ planetarium was rekindled in 1900, when Greek sponge divers discovered off the coast of the Greek island of Antikythera, in a Roman shipwreck dating back to the (80–50) B.C., the archaeological finding called Antikythera Mechanism or Antikythera Machine. The true significance of this mechanism was not recognized until the search of the physicist and historian of science Derek John de Solla Price, from the mid-1950s and culminated in his classical work Gears from the Greeks [37]. This complex geared mechanism turned out to be an extraordinary astronomical calculator. It is the only planetarium come down to us, and consists of at least thirty gears, made of bronze with low content of tin, and with wedge-shaped teeth. It almost certainly incorporated a planetary gear train of equally extraordinary conception, which had been forgotten until its rediscovery in western technology, some 1600 years later (see Sect. 5.4). It should be noted that the triangular shape of the teeth profiles of this mechanism is not very refined, and lays for a not advanced manufacturing technology, in the face of very sophisticated scientific knowledge relating to astronomy and planetary gear trains. Using the most sophisticated analysis techniques then available and with a multiyear work, culminating with the aforementioned publication of 1974, Price de Solla was able to provide a plausible reconstruction of the schematic gear diagram of the Antikythera Mechanism, which is shown in Fig. 3.8c. Subsequent studies and analyses made by various other scholars have led to refine the schematic gear diagram

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Fig. 3.8 a Main fragment of the Antikythera mechanism; b the same fragment after cleaning; c schematic gear diagram of the same mechanism according to Price de Solla; d a possible reconstruction of the mechanism

proposed by Price de Solla, but without distorting it (see, for example [16, 17, 22, 36, 46]). However, all the studies on the subject confirmed that the Antikythera Mechanism is a two-dimensional complex gear drive, that is, a planar gear train that uses planetary gear trains also two-dimensional. Figure 3.8a shows the main fragment of the 82 fragments of the Antikythera Mechanism that survive in the National Archeological Museum of Athens. Figure 3.8b shows the same fragment after cleaning, while Fig. 3.8d shows the perspective view of a possible reconstruction of the Antikythera Mechanism.

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The gear wheels of the Antikythera Mechanism are not the oldest metal gears, which we have inherited from the ancient world. In fact, in 2006, in an excavation for the resettlement of the Civic Market Square of Olbia (during the 2nd century B.C. this site was open see, in front of the ancient port), a fragment of a gear wheel was found. The Soprintendenza per i Beni Archeologici (Superintendence for Archaeological Heritage) of Sardinia dated this finding and all the digging related to it as included from the late 3rd century and the beginning of the 2nd century B.C. This gear wheel, subsequently reviewed with the most advanced techniques, surprisingly showed a manufacturing technology as well as more advanced knowledge than the gear wheels of the Antikythera Mechanism, dating back some two centuries later. It is in fact made of brass and its teeth have curved profiles, very similar to those of the current gears. Pastore [35] has identified this gear wheel as belonging to Archimedes’ planetarium. In his opinion, in fact, a profile like that of the teeth of this gear wheel requires a futuristic mathematical knowledge that, at that time, could only be the product of the mind of an absolute genius; and Archimedes was a divine intelligence, as Cicero recognizes when he writes: “Nam cum Archimedes lunae solis quinque errantium motus in spheram inligavit, …, ne in sphera quidem eosdem motus Archimedes sine divino ingenio potuisset imitari”, i.e. “in fact Archimedes, when he reproduced in a sphere the moviments of the sun, moon and five planets, …, even in the sphere Archimedes could not rebuild the same moviments without a divine intelligence” (Marcus Tullius Cicero, Tusculanae Disputationes, I, 63). According to Pastore [35], it is plausible that the Archimedes’ planetarium, passed from the conqueror of Syracuse to his descendant, the consul Marcus Claudius Marcellus Jr. (namesake of his grandfather, the conqueror of Syracuse) was lost in 166 B.C., in Olbia, during the shipwreck just outside its ancient port, where the ship of the same consul Marcellus, headed for a mission in Numidia, had made a probable supply port. Frankly, the hypothesis that the fragment of gear wheel found in Olbia belonged to Archimedes’ planetarium, although suggestive, seems to the writer author at least a little risky. In fact, the objective references or historical documents in support of this hypothesis are missing almost completely. In any case, the discovery of this archaeological finding, definitely dating from the end of the 3rd and beginning of the 2nd centuries B.C., technological advanced and based on sophisticated mathematical knowledge, involves a thorough review of what until now was known to us about knowledge and scientific achievements of the ancient world. However, let archaeologists and other scholars to determine with reasonable certainty the relationships between the Archimedes’ planetarium and the half-gear wheel found in Olbia. We are interested here underline the configuration of the Archimedes’ planetarium, which aroused the admiration and amazement of Sulpicius Gallus, when he had the good fortune to examine it. In fact, in order to coordinate the relative motions of the sun, moon, and five planets then known, with a single conversio as Cicero tell us, the mechanism had to be necessarily characterized by a planetary gearing (moreover, three-dimensional, that is spherical), as also the discovery of the Antikythera Mechanism has confirmed. Furthermore, Russo [39] notes that this mechanism of unique rotation governing the motions of heavenly bodies involved, among

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them so different, is not compatible with a mechanical model depicting a Ptolemaic algorithm. A fortiori, we must infer that the heliocentric theory of Aristarchus of Samos (320–230 B.C.) was well known to Archimedes, who has taken it as the design basis of his spherical planetarium. Archimedes was undoubtedly the inventor of worm gears. This is attested by Athenaeus of Naucratis (170–223 A.D.) who, about the launch of the powerful Syracusia ship, reports that Archimedes “setting up the helix, took the ship out to see”. As helix Athenaeus intents a worm gear, which could also be interconnected with other gear wheels, to configure a speed reducing drive system with a high ratio of speed reduction. Athenaeus than adds “Archimedes was the first to whom this invention is due”. The same Syracusia, designed by Archimedes and built under his direction by Archias of Corinth and Phileas of Tauromenion (the first of this two architects, supervisors and shipbuilders should not be confused with the homonymous quasi mythological Corinthian citizen, founder of Syracuse), received the astonishment of the ancient world. It was a mighty ship, of dimensions never seen before (someone went so far as to assume that it was 110 m long), donated by Hieron II, tyrant of Syracuse, to King Ptolemy III Euergetes I of Egypt, also because there was not a port in the whole of Sicily able to receive it. The sources of this information are the same Athenaeus, in his Deipnosophistai, and Proclus Lycius Diadocus (412–485 A.D.), in his Commentari in Euclidem. From Athenaeus we also learn that Moschus (perhaps to be identified with Moschionus, a naval technician who lived in the 3rd century B.C., assigned to the construction of the Syracusia) attributes to Archimedes the development of the theory of hoists and winches. Moschus or Moschionus (see [3]), in the first book of his Mechanikà, wrote as follows: … “after long meetings to define the method of launching the ship, only Archimedes the mechanicus, with the help of a few men, was able to push the ship into the sea. He succeeded thanks to the construction of a winch (note: other authors have translated the corresponding Greek word with helix), through which he reduced a huge ship to the sea”. This winch supposedly included various geared mechanisms, including worm gears. It is probable that Demetrius Poliorcetes (337–283 B.C.) conceived and built a winch for a lifting crane a few decades before Archimedes. However, the Syracusan scientist was certainly the first to conceive and build a lifting and towing machine, specifically designed to perform a certain function under a given load to be lifted and towed. Archimedes is therefore the first engineerdesigner of the history to design and build a machine based on a data-sheet. With him a decisive turning point is made: the passage from the Greek technè, that is from the skill of a craftsman who untangles him between art and science, to technology that instead involves the application of physical-mathematical formulae to scientifically solve a technical problem (see also [41]). We too are amazed by the high engineering and design value of Syracusia, but the scientific contributions necessary to develop worm gears, which are typically curved toothed gears, amaze us a lot more. These contributions are based on the study of properties of the spiral and, in this regard, Pappus of Alexandria (Collectio, Vol. I,

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IV) informs us that the “plane spiral theorem proposed by Conon of Samos (280–220 B.C.) was admirably demonstrated by Archimedes”. The design of the Syracusia was also the test field for another application of the Archimedes screw, which was also developed purportedly in order to remove the bilge water that accumulated at the bottom of its bull. The machine specifically built to perform this type of drainage was a device with revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from low-lying body of water into irrigation canals. The Archimedes screw is still used today to pump liquids or granular solids, such as cereals, coal, stone, sand, etc. The Archimedes screw described by Vitruvius (Fig. 3.9) may have had an improvement on the screw pump that was used to irrigate the Hanging Gardens of Babylon (see [7]). Figure 3.10 shows an Archimedes screw in working position, with relative operating scheme. At least two other realizations of Archimedes, containing gears, are still to be remembered. The first realization is represented by the odometer, which consisted of a cart with a geared mechanism, which made dropping a small metal sphere inside a container, after a specific distance. The total distance was determined by counting the number of balls collected in the container. Figure 3.11 shows a fantasy reconstruction of the Archimedes odometer, with gears perhaps less refined than those of the real odometer designed and built by the great Syracusan. The second realization consists of a clock, which shared with the odometer the geared mechanism as well as the principle of operation. It in fact, at each hour, released a small metal sphere which, falling inside a container, produced a sound marking the start of a new hour (Fig. 3.12). Other machines designed by Archimedes, such as winches for lifting of heavy objects, and war machines built for the defense of Syracuse, were certainly characterized by gears, but specific information on these machines does not come down to us. Finally, it should be remembered the Archimedes’ screw, which intentionally was developed in order to remove the bilge water, subsequently was also used for the water lifting. It does fall within the gear area in that, together with the treatise “On Spirals” which has come down to us, it is the basis of the screw gears and spiral bevel gears. It is certainly astonishing the clarity of thought of Archimedes, to such an extent that Kasner (see [18]) , had to say that, if “we had be able to preserve” it “from the Renaissance to today, we would not have encountered the many abstruse problems and paradoxical ones connected with the infinite”. Even more significant is Leibniz who, in reference to those who burned incense on the altar of progress without scientific bases, thus expressed himself: “those who are able to understand Archimedes are not very enthusiastic about modern discoveries”. Apart from reiterating that the absolute genius of Archimedes was universally recognized in every historical age (see also [34]), there is nothing to add to the above considerations. At the end of this section, we must necessarily mention the contribution that the metalworking technologies gave to the development of those concerning the construction of the gears. The well-established existence of very advanced metal gears in 3rd century B.C., highlighted above, finds justification in the equally advanced

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Fig. 3.9 Archimedes screw as described by Vitruvius

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Fig. 3.10 Archimedes screw in working position and correlated operating scheme

knowledge on metal working, including their welding processes. In this regard, it is worth mentioning the contribution made by Rhodes, which is to be considered the most advanced Hellenistic center on this subject. As a demonstration of the high level reached by Rhodes in metalworking technologies, just remember its famous Colossus, one of the Seven Wonders of the Ancient World (Fig. 3.13). It was designed and erected, in 280 B.C., by Chares of Lyndos, the Lysippus’ favorite pupil (it is to be noted that the sculptor Lysippus had already constructed a 22-m high bronze statue of Zeus at Tarentum), to celebrate Rhodes victory over the ruler of Cyprus, Antigonos I Monophthalmos, whose son Demetrius I of Macedon unsuccessfully besieged Rhodes in 305 B.C. As we learn from Pliny the Elder, the Colossus of Rhodes, which was about 33 m high, was composed of bronze elements, fused separately, and then welded together with the cavity welding method, and had an internal bearing structure, composed by iron elements as well welded together. The load-bearing elements of the big legs reached the weight of about 800 kN and, according to Philo of Byzantium, they seemed to be joined together by the “hammering of cyclopean forces” [25]. For the construction of the Colossus of Rhodes, much of the bronze and iron from the various weapons Demetrius’ army left behind was re-melted or forged again. These Hellenistic metalworking technologies, which were rooted in the classic Greek world (just remember the colossal chryselephantine Statue of Zeus at

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Fig. 3.11 A reconstructive design of the Archimedes odometer with the gear used

Olympia, made by Phidias, but also the colossal bronze statue of Athena Promachos of Phidias himself) and which also produced the statue of Hercules of the aforementioned famous bronze smith Lysippus and, subsequently (1st century A.D.) the Nero’s statue of Zenodorus, 45 m high) as well as the countless well-known masterpieces of art, were certainly used to produce artifacts useful for everyday life and scientific instruments, which were inherited from the Arabic civilization. The Archimedes Planetarium and Antikythera Mechanism are a clear demonstration of this statement. To be honest, the aforementioned metalworking technologies have more ancient root than the classic Greece. They are in fact to be attributed to Rhoecus of Samos (6th century B.C.), an architect and statuary who was the head of a family of Samian artists, such as Theodorus, Teleches and perhaps others whose name we do not know. Rhoecus was the first architect of the Doric Order great temple Heraion of Samos, which Theodorus completed, and is credited by Pausanias for having invented, together with the members of his family who succeed him, the art of casting statues of bronze and iron. In addition, Rhoecus and Theodorus are credited, by Pausanias, of the invention of ore smelting as well as of the craft of casting, and, by Herodotus, of having improved the process of mixing copper and tin to form bronze as well as being the first to use it in casting. Theodorus is also credited by Herodotus for discovering

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Fig. 3.12 A reconstructive design of the Archimedes clock, which shows the gear used and the way of working

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Fig. 3.13 Fantasy reconstruction of the Colossus of Rhodes

the art of fusing iron and using it to cast statues and, by Pliny the Elder, for having invented a water level, a carpenter’s square, a lock and key, and a turning lathe.

3.4 Rapid Decline and Death of Science After the conquest of Corinth and the aforementioned diaspora of the Alexandrian scientists, two centuries of obscurantism and ignorance began, because the rough Roman conquerors had neither the culture nor the ability to understand and assimilate the scientific and technological achievements of the golden age of the first Hellenism. It should be remembered that the Romans of the mid-2nd century B.C. were essentially barbarians who, from a cultural point of view, were very different from those of the classical period of almost two centuries later, so much that Quintus Horatius Flaccus, in Epistula I, Liber Alter (II), 156–157 (see also [6]), openly acknowledged the fact that “Graecia capta ferum captorem cepit et artis intulit agresti Latio” i.e. “the conquered Greece conquered the fierce conqueror and introduced arts in rural Latium”. Meanwhile, given the conditions that had arisen, the innovative drive of the first Hellenism went fading more and more, and the works written by the masters were replaced with poor epitomes illustrating the results obtained by them, but they left in shadow the principles and scientific bases on which they were founded, as they were no longer comprehensible. Too late Rome took awareness of the greatness of Greek civilization. When the Romans began to appreciate it at the beginning of the empire (in 30 B.C., which is the date specified as the end of Hellenism by most historians),

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only these poor epitomes had survived, while already memory had been lost of the knowledge acquired during the first Hellenism. Scientists, scholars and highly qualified technologists who characterized and made the golden age of the Hellenistic world great, gradually became less and less numerous, until they disappeared completely after the diaspora of the Museum’s scientists of Alexandria that we mentioned in Sect. 3.1. The ability to do science and to develop new knowledge faded dramatically after the middle of the 2nd century B.C., so much so that already in the 1st century A.D. the scientific method was permanently and irrevocably lost. Hero of Alexandria (about 10–70 A.D.) represents a clear demonstration of this fact occurred in the 1st century A.D. Much of Hero’s original writings were lost, but some of them were preserved in Arabic manuscripts. As our knowledge of the Hellenistic science increase and are thorough, we are increasingly convinced that Hero was a compiler lacking in originality, especially when he leaves the description of the technological aspects of the practical applications of which he speaks, and ventures in the discussion of the theoretical questions. Russo [39] clearly shows that, contrary to what was believed until some time ago, Hero does not belong to a scientific school remained alive by Ctesibius time, but draws his knowledge from reading the works written in the 3rd and 2nd centuries B.C., for the most misunderstood in their scientifically higher and noble content, given the interruption of Alexandrian scientific traditions occurred in the period between 146 and 144 B.C. The use made by Hero of fluid mechanics, pneumatics and mechanical technology is based on the scientific theories of mechanics, hydrostatics and pneumatics, all three dating back to the 3rd and 2nd centuries B.C. The basic elements used by Hero, such as gear wheels, precision screws, and valves, as well as the extraordinary technology, evidenced by war machines and shipbuilding, date from the same centuries. Personal contributions of Hero can be limited, at best, to some of the described practical applications, but not to the scientific knowledge and basic technology he used. To give a concrete idea of the historical perspective errors made by historians about Hero, just to note that the Arabic mathematician al-B¯ır¯un¯ı (973–1052 A.D.) attributed to Archimedes the so-called Hero’s formula for calculating the area of a triangle (see [39]). Furthermore, the deciphering of cuneiform texts from various clay tablets excavated from modern Iraq has unequivocally shown that the algebraic techniques, of which Hero was considered the inventor, were already long time been used in Mesopotamia. Very similar conclusions can be drawn about the practical applications described by Hero. The following examples give a clear demonstration: • The throwing weapons described by Hero, based on Ctesibius, had already criticized by Philo of Byzantium. • Hero’s criticisms to the work of Philo of Byzantium regarding the implementation of automata are aimed at self-exaltation of his alleged innovation contribution concerning the number of simultaneous movements of each automaton, but add

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nothing to the wealth of knowledge and technologies developed in this regard in 3rd century B.C. The already identified sources of some parts of his Pneumatics undoubtedly demonstrate that he drew on the works of Ctesibius and Philo of Byzantium for devices based on pneumatics, and on the writings of Strato of Lampsocus (about 335–269 B.C.) in relation to the hydraulic organ. Hero’s diopter is almost certainly to be traced back to Hipparchus, like Ptolemy (about 100–170 A.D.), Strabo (about 64 B.C.–24 A.D.) and Pliny the Elder (23–79 A.D.) make it clear to us. In his Mechanics, for the realization of screws, Hero only describes simple methods to realize large screws and nuts made of wood, while he not known how to get on the construction of precision metal screws. In the same Mechanics, as a part of a theoretical discussion, Hero describes speed reducing gears, but when he has to transfer the motion from one gear wheel to another of his automata, he never uses gears, but only friction devices, as the refined technology developed in the 3rd and 2nd centuries B.C. had already been lost.

Other examples may be cited, but they would add nothing about the conclusion that follow. Essentially, the works of Hero provide valuable evidence, but late and incomplete, on the level of scientific and technological knowledge of early Hellenism, as a kind of distant echo of greatness now irretrievable lost. Reasonable certainty of this statement is supported by the fact that Hero’s writings do not suspect the existence of complicated gear systems, such as the differential, which is documented by the Antikythera mechanism, which is oldest of Hero of about a century and a half. Objectively, in the light of archaeological finds, including the gradually discovered and decrypted documents, the Hero’s devices, which are characterized by a systematic contamination of demonstration and entertainment aspects, should be taken as by-products of Hellenistic technology, originally developed for other purposes. However, perhaps we should own the playful nature of these devices that they have managed to survive and develop in the new conditions related to the imperial age. For the same reason, we must be grateful to Hero to have left a window ajar, which enabled us to gain an insight, albeit partial, on the Hellenistic world. Hero was basically a compiler, to which only the distant echo of the lost Hellenistic science came. It should not be forgotten that on Hero a debate has existed from some centuries, which gave rise to the so-called Heronian question. This question assumes that, under the name of Hero, a collection of scientific writings by various authors is to be understood (see [24]). The presumption is based on the following facts: the biographical news on Hero, provided by historians and commentators, are often contradictory; the diffusion in antiquity of the name Hero can be traced back to no less than twenty different personalities of the scientific world, more or less relevant; the name Hero in Egyptian has a similar meaning to the qualification of engineer. It should be noted that this section, almost entirely dedicated to Hero of Alexandria, should have found a more logical place in the following chapter. This is because, as we have seen above, from a substantial point of view, the scientific method and

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the related theories are completely lacking in the work of Hero. Even the historical period in which Hero lived and worked (1st century A.D.) follows the almost two centuries the one reserved for this chapter, so it belongs to the next age, here called second pre-scientific age. We have considered here to place the questions concerning Hero in this chapter, as we wanted to privilege his belonging to the circle of Alexandria, which was the indisputable pole of aggregation of the pioneers that triggered the process of birth of science.

References 1. Acerbi F, Fontanari C, Guardini M (2013) Archimede, Metodo, nel Laboratorio del Genio. Bollati Boringhieri, Torino 2. Burnyeat MF (1978) The philosophical sense of Theaetetus’ mathematics. Chic: Univ Chic Press J Behalf Hist Sci Soc 69(4):489–513 3. Cambiano G (1996) Alle origini della meccanica: Archimede e Archita, Arachnion, no 2.1 4. Canfora L (2001) Storia della Letteratura Greca. Gius. Laterza & Figli, Bari 5. Canfora L (2007) Johann Gustav Droysen, Histoire de L’Hellénisme. Anabases, Traditions et réceptions de l’Antiquité 5:277–280 6. Colamarino T, Bo D (1983) Le Opere di Quinto Orazio Flacco. Classici Latini UTET, 2nd edn. Unione Tipografico-Editrice Torinese, Torino (revised and updated reprint) 7. Dalley S, Oleson JP (2003) Sennacherib, Archimedes, and the water screw: the context of invention in the ancient world. Technol Cult 44(1) 8. Donovan S (2008) Hypatia: mathematician, inventor, and philosopher. Compass Point Books, Minneapolis 9. Drachmann AG (1948) Ktesibios, Philon, and Heron: a study in ancient pneumatics. Munksgaard, København 10. Drachmann AG (1963) The mechanical technology of Greek and Roman antiquity, a study of the literary sources. Munksgaard/University of Wisconsin Press, København/Madison 11. Droysen JB (1836) Geschicte des Hellenismus, vol 1. Perthes F, Hamburg 12. Droysen JB (1843) Geschicte des Hellenismus, vol 2. Perthes F, Hamburg 13. Fleck HF (2016) Archimede, L’Arenario, with Greek text, Quaderni di Scienze Umane e Filosofia Naturale 1(1) 14. Fleck HF (2016) Archimedous Ochouménon-α’ (On floating bodies-Book I), with Greek text, Quaderni di Scienze Umane e Filosofia Naturale, vol 2(3), Pubblicazione Elettronica Aperiodica 15. Fraser PM (1972) Ptolemaic Alexandria, vols 1, 2, and 3. Clarendon, Oxford 16. Freeth T (2002) The Antikythera mechanism: challenging the classical research. Mediterr Archaeol Archaeom 2(1):21–35 17. Freeth T (2012) Building the cosmos in the Antikythera mechanism. Proceeding of Science 18. Kasner E, Newman JR (2001) Mathematics and the imagination. Dover Publication Inc, New York 19. Landels JG (1978) Engineering in the ancient world. University of California Press, Berkeley 20. Lewis M (2000) Theoretical hydraulics, automata, and water clocks. In: Örijan W (ed) Handbook of ancient water technology, technology and change in history, vol 2, Leiden, pp 343–369 (356f) 21. Lewis MJT (1997) Millstone and Hammer: the origins of water power. University of Hull Press, pp 1–73 22. Lin J-L, Yan H-S (2016) Decoding the mechanisms of antikythera astronomical device. Springer, Berlin 23. Lloyd GER (1973) Greek science after Aristotle. Norton, New York

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24. Loria G (1914) Le scienze esatte nell’antica Grecia. Ulrico Hoepli, Milano 25. Manna F (1980) Storia della Saldatura, vol 1. Edizioni Scientifiche Italiane SPA, Napoli 26. Manna F (1998) Uomini e Macchine, vol I, vol II, prima parte, and vol II, seconda parte, self-published in Casalnuovo di Napoli, and printed by Offselit, Cava dei Tirreni 27. Nenci F (ed) (2008) Cicerone: La Repubblica, with Latin text, Collana: Classici Greci e Latini, Milano: BUR Biblioteca Universale Rizzoli 28. Netz R, Noel W, Tchernetska N, Wilson N (2011) The Archimedes palimpsest, vol I, Catalogue and commentary. Cambridge University Press, Cambridge 29. Netz R, Noel W, Tchernetska N, Wilson N (2011) The Archimedes palimpsest, vol II, Images and transcriptions. Cambridge University Press, Cambridge 30. Netz R (2004) The works of Archimedes, vol 1. In: The books on the sphere and the cylinder. Cambridge University Press, Cambridge 31. Netz R (2017) The works of Archimedes, translation and commentary, vol 2 on spirals. Cambridge University Press, Cambridge 32. Netz R, Noel W (2007) The Archimedes Codex: how a Medieval Prayer book is revealing the true genius of antiquity’s greatest scientist. Da Capo Press, Perseus Books Group, Philadelphia 33. Netz R, Noel W (2008) Il codice perduto di Archimede. La storia di un libro ritrovatoe dei suoi segreti matematici. BUR Biblioteca Universale Rizzoli, Milano 34. Paipetis SA, Ceccarelli M (eds) (2010) The genius of Archimedes—23 centuries of influence on mathematics, science and engineering. In: Proceedings of an international conference held at Syracuse, Italy, 9–10 June 2010, Springer Science+Business Media, B.V. 35. Pastore G (2013) Il Planetario di Archimede Ritrovato, and The Recovered Archimedes planetarium. Pastore ed., Rome 36. Pennestrí E, Valentini PP, Petti F (2002) Kinematic analysis of the Antikythera gear mechanism by means of graph theory. In: Proceedings of DETC02, 2002 ASME design engineering technical conferences, 29 Sept–2 Oct, Montreal, Canada 37. Price de Solla DJ (1974) Gears from the Greeks. The Antikythera Mechanism—a calendar computer from ca. 80 B.C. Trans Am Phil Soc 64, Part 7 (and reprint in 1975, Science History Publications, New York) 38. Ramsey S (2016) Tools of war: history of weapons in ancient time. Alpha Editions-Publishing, Vij Publishing Group, Delhi 39. Russo L (2015) La rivoluzione dimenticata. Il pensiero scientifico greco e la scienza moderna, 9th edn. Giangiacomo Feltrinelli Editore, Milano 40. Sarton GAL (1959) A history of science, vol 2: hellenistic science and culture in the last three century B.C. Harvard University Press, Cambridge 41. Vittorio A (2010) Archimede Siracusano: invenzioni e contributi tecnologici. Morrone Editore, Siracusa 42. Wilson A (2002) Machines, power and the ancient economy. J Roman Stud 92:1–32 43. White KD (1984) Greek and Roman technology. Cornell University Press, New York 44. White KD (1993) The base mechanic arts?: some thoughts on the contribution of science (pure and applied) to the culture of the hellenistic age. In: Green P (ed) Hellenistic history culture. University of California Press, Los Angeles, pp 211–237 45. Winter TN (2007) The mechanical problems in the corpus of aristotle. Faculty Publications, Classic and Religious Studies Department, University of Nebraska 46. Wright MT (2005) The Antikythera Mechanism: a New Gearing Scheme. Bull Sci Instrum Soc 85:2–7 47. Wright MT (2018) Archimedes, astronomy, and the planetarium. In: Rorrers C (ed) Archimedes in the 21st century, Proceedings of a world conference at the courant institute of mathematical sciences. Springer Nature Switzerland AG, Cham, pp 125–141

Chapter 4

The Second Pre-scientific Age: From the Diaspora of Alexandrian Scientists to the Renaissance

Abstract The stagnation in the development of gears in the second pre-scientific age, which includes the long period between the diaspora of the scientists of the Museum of Alexandria and the Renaissance, is described. It is shown that the socalled rebirths before the Renaissance (four in all) are actually pseudo-rebirths, given the absence of noteworthy contributions, because the scientific method had been irretrievably lost. As far as gears are concerned, the return to the methods of the first pre-scientific age is demonstrated. However, the small and not very significant steps forward, resulting from the refinement of field experiences, are highlighted. In this framework, the appreciable empirical contributions in the field of clocks, due to good craftsmen-clockmakers, are described. A particular attention is dedicated to the Renaissance, in which the pre-scientific man has difficulty in dying and the scientific man has difficulty in being born. The most significant contributions of the most representative figures of the Renaissance, concerning gears, are analyzed and described. Particular attention is given to the figure par excellence of the Renaissance, Leonardo da Vinci, an immense genius for vastness of horizons and depth of thought. His uncommon capacity for innovation in the field of gears is brought to light, and his numerous and incomparable contributions are described in detail, including those concerning pure and applied science, which make Leonardo a figure that transcends his time and projects it into the next age of scientific awakening.

4.1 Loss of the Scientific Method and So-called Early Rebirth of the Imperial Age The scientific contributions of the Greek-Hellenistic world were indeed excellent, but they were not inherited either by the Roman civilization, nor by Islam, and in the Middle Ages oblivion descended on them, so much so that today we speak of the Hellenistic scientific revolution as the forgotten scientific revolution [54]. The Romans made extensive use of wooden gears in their gristmills. Their gears were substantially similar to those of noria that, unlike of sakia, uses the waterpower. Also, the Arabians, who excelled in scientific instruments, particularly in the astronomical ones, did not give substantial contributions concerning the gears, limiting itself to use © Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_4

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sakia and noria as those described in Chap. 2. In the Middle Ages the water power mills were still using wooden gears, whose teeth were engaging pins. During the dark night of the Middle Ages, knowledge contributions of the GreekHellenistic world, and especially those of Archimedes, remained forgotten for centuries. The medieval man, committed in vain superstitions and prejudices, had no time to think about science. So, Archimedes and the knowledge of his time were buried under the blanket of ignorance for many centuries. We must await the Renaissance and ingenious men, like Leonardo da Vinci and Galileo Galilei, who for many of their researches were inspired specifically to the discoveries and inventions of Archimedes and Hellenistic scientists. In this regard, it should not be forgotten that the modernity of Archimedes’ method of scientific investigation, emerging from the reading of the writings of the great Syracusan scientist, aroused the amazement of Galilei, expressed through Salviati on the third day of his Dialogo sopra i due massimi sistemi del mondo tolemaico e copernicano, i.e. Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican [22]. The age in which Archimedes lived was an age of great inventions and amazing discoveries. In an extremely dynamic civil society, and so to speak modern, like Hellenism, sciences made great progresses. Every field of knowledge was studied, and sought everything from astronomy to mathematics, geometry, philosophy, technology, etc., and Archimedes was certainly the most representative man of his times. In fact, with his The Mechanical Method (see Sect. 3.3), he proved, for the first time, that the data on which the technology is based must be provide by science. Nevertheless, all the Hellenistic communities (Syracuse first, and gradually all the others, and last Corinth) were subjugate to the power of Rome. Scientific communities and cultural circles quickly disappeared, and scientific progress came to an abrupt halt. The conquest, manu militari, without the perception of what was irreparably destroyed, was a loss for the humanity of incalculable gravity. The conquerors expropriated the vanquished everything, but were unable to capitalize on their knowledge, and did not have the ability to assimilate and to develop the scientific method born with Hellenism. In other words, they did not continue on the path of progress and research, drawn by Greek scholars. As we already said elsewhere, the first scientific age does not have the same lifetime of Hellenism. The decline of science and technology began in the period between 146 and 144 B.C., but it was very fast, so much so that as early as the 1st century B.C. almost all the knowledge acquired in the golden age of Hellenistic science had been irreparably lost. In the 1st and 2nd century A.D., the Pax Romana allowed a partial revival of scientific research, but the results were poor. About it, just bring to our attention the figure of Hero (see previous chapter), and that of Ptolemy (about 100–175 A.D.), who had never news of the heliocentric theory of Aristarchus of Samos, because the memory of that theory had already been lost. In this regard, the figure of Diophantus of Alexandria, who lived in uncertain times, between the 1st and 4th century A.D., is then emblematic. He was considered, until very recently, as the ultimate scientist of a certain value of the enlarged Hellenistic age. His contributions have been drastically reduced, because the deciphering of cuneiform tablets showed that the methods he described are not original, being a

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long time in use in Mesopotamia. Also, the argument used by Tannery [58, 59] to date Diophantus on 3rd A.D. is inconsistent, as Knorr [29] has shown (see also [54]). In the historical framework that was determined from the diaspora of the Museum’s scientists of Alexandria, it is not possible to define exactly when the exchange of baton between the 1st scientific age and the 2nd pre-scientific age actually occurred. We already said that it is not possible to separate with a sharp cut two historical ages, and this is even truer for two ages characterized one of scientific culture, and the other by pre-scientific culture. Hellenistic civilization and, with it, the Hellenistic science, went into crisis in the second half of the 2nd century B.C. The ability to conceive and elaborate scientific theories declined rapidly, so much as that, between the end of the 1st century B.C. and the beginning of the first century A.D., that is about two centuries the decline began, much of the knowledge acquired in the period between the end of the 4th century and the beginning of the second half of the 2nd century B.C., were already irretrievably lost. The causes of this decline were manifold. According to Préaux [46] and many others, the development of science was impeded by the excessive authority of Aristotele. However, this cause seems evanescent, since science, which was born also with the Aristotle’s contribution, developed without regard to any kind of authority. This is demonstrated by the various positions taken from time to time, on specific issues, by Archimedes, Ctesibius, Herophilus, Aristarchus, and Theophrastus, this last collaborator and then successor of Aristotle in the Peripatetic School [54]. Probably the most serious obstacle to scientific activities, which resulted in the decline of science, was caused by the long period of wars between Rome and the Hellenistic States. The Romans in the 3rd and 2nd centuries B.C. were a rude people, very different from the refined culture of the Augustan age, in which Vergilius (Publius Vergilius Maro) and Horatius lived. Several generations were necessary to ensure that this refined culture could be developed, and this was possible thanks to the continuous contact of the Romans with the Hellenistic civilization. This contact was also mediated through the Greeks deported as slaves. The Roman aristocrats bought educated and acculturated Greeks as slaves, and used them as readers, pedagogues or copyists. In addition, the contact above was mediated through works of art, books and volumes, looted and transported to Rome. Hellenistic libraries were included in the spoils of war, and were used to decorate the villas of the winners. From the middle of the 2nd century B.C., most of the Hellenistic centers, which had given birth to the science, disappeared one after another. Rhodes was the last Hellenistic center that fell. In fact, Rhodes tried to survive, but the Romans progressively reduced its role; finally, it was sacked and looted in 43 B.C. The interest of the Romans to the Hellenistic civilization began to late, when the wealth of previously acquired scientific knowledge had already been almost completely lost, and the only surviving works in this regard were mere epitomes, describing practical applications of the developed theories, most of which aimed at the construction of entrainment devices. The Romans were essentially a practical people, and therefore they were not very interested in the scientific method. In this regard, two factually circumstances are emblematic: the first Latin translations of

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parts of the Elements of Euclide date back to the 6th century A.D.; the first complete translation of the Euclide’s work appears to have been the one made in 1120 A.D., by Adelard of Bath (1080–1152 A.D.), who translated from the Arabic version attributed to the work of al-Hajj¯aj [16]. However, during the imperial age, a resumption of the studies took place, which led to a partial recovery of ancient knowledge, so some scholars have spoken of early rebirth or first rebirth with reference to this historical period. In reality, it is not a true rebirth, but a pseudo-rebirth, as well as pseudo-rebirths are those described in the next section. Here all these pseudo-rebirths are also called so-called rebirths. All the studies carried out during the so-called first (or early) rebirth did not generate new knowledge and new scientific theories. The same scientific method and the related theoretical concepts were even rejected, as Aelius (or Claudius) Galenus of Pergamon (about 125–210/216 A.D.) and Lucius Mestrius Plutarchus of Chaeronea (about 46–120 A.D.) made implicitly and, respectively, explicitly. Sextus Empiricus (about 160–210 A.D.) then went further and, on a philosophical level, theorized the unsustainability of a scientific age. In this age, the most learned of Romans, when they come in contact with the applied science Hellenistic works, clearly show to be only interested in the results of the theoretical models. On the contrary, the logical links between model and results are systematically eliminated, or they are replaced with links, often arbitrary, but more easily understood in relation to the desired result, and in any case such as to arose the wonder of the reader. Evident but emblematic examples of this erroneous and inadmissible way of conceiving the science, which obviously relate to a pre-scientific culture entirely foreign to science, are as follows (for brevity, here we limit ourselves only to three examples, but an entire book would not suffice to describe them exhaustively): • Varro (Marcus Terentius Varro, 116–27 B.C., was a scholar of Caesar’s age, and generally regarded as the greatest of the Roman scholars), in his treatise De Re Rustica, considers the writings of Theophrastus on agriculture as philosophical books without utility. Theophrastus in fact exposes theories that Varro cannot understand, because he does not have the necessary culture to understand them. Therefore, Varro delivers these theories to the field of philosophy, which is the only theory known by him. • In the age of the Julio-Claudian dynasty, Lucius Annaeus Seneca, (4 B.C.–65 A.D.) in his Naturales Questiones, intrigues his reader with at least two pleasant issues, which are the results of his researches. The first regards the wine freeze when it is struck by lightning. After three days, it turns into a fluid, with arcane property to kill or made mad those who drink it. The second concerns the scientific treatment of catoptrics. In this regard, Seneca leaves open the question of whether there are simulacra of mirrored objects behind the mirror, adding then the highlight, where he describes the obscene use of cosmetic mirrors during sexual intercourse that allows him to conclude successfully his discussion of the mirrors deprecating the depraved use of the same mirrors. • In the age of Flavian emperors, Gaius Plinius Secundus, known as Pliny the Elder (23–79 A.D.), describing the life of bees in his Naturalis Historia, motivates the

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hexagonal shape of honeycomb cells. He says, “each of the six legs of the bee did a side”, so showing that he had not been able to understand the sources from which he takes the subject, i.e. Aristomachus of Soli (3rd century B.C.), and Philiscus of Thasos (4th century B.C.). These sources have not received up to us, but we know their subject, which has come from Pappus of Alexandria, also called Pappus Alexandrinus (about 290–350 A.D.). This author, in a beautiful page of his work entitled Synagoge (or in Latin Mathematicae Collectiones) notes that the bees, in the construction of the hexagonal cells of the honeycombs, solve an optimization problem, using the minimum amount of wax, with the same content of honey. From the aforementioned examples, as well as from the whole so to speak Latin scientific literature, we infer a fortiori that, with the Roman civilization the scientific method had been already lost, which is why the world from a scientific age relegated to a pre-scientific age. Of course, here we look good from infringing other great merits of Rome about the civilization of humanity, but this is not the place to discuss these merits. The total insensitivity of Rome towards science and technology, understood as the daughter of science, is emblematically summarized in a Seneca’s sentence where, referring to the glass sheets of windows and tubing systems to heat the houses, he expresses himself: “vilissimorum mancipiorum ista commenta sunt: sapientia altius sedet nec manus edocet; animorum magistra est”, i.e. “this is the invention of inferior individuals, slave: wisdom sits on a higher throne and does not teach the hands; it is the teacher of the minds”. This way of thinking coincided with that of classical, pre-scientific Greece. The distance from Archimedes is abysmal. The great Syracusan, that is the finest and deepest intelligence of the ancient world, never disdained to get his hands dirty, nor did he consider the manual dexterity unworthy of the aristocratic thinking. He followed his passion and his interest in machines, ennobling the mechanics at the level of science (just remember that in his The Mechanical Method, he points out that the geometric theorems were first found through mechanics, and then demonstrated with geometry), with a way of thinking that came from afar, that is, from the Ionian, Pythagorean and the Eleatic schools. It was precisely his non common ability to translate his scientific ideas into real mechanical systems, useful to the civil society of his city, that is, his manual dexterity, which enabled him to become the scientific advisor to King Hieron II, who was Tyrant of Syracuse from 270 to 215 B.C. After the republican age and that of the first empire, the erasure of the historical memory of the knowledge acquired by the Hellenistic science experienced other events that were sadly significant. Among these, the following events should be mentioned: the siege of Gaius Aurelius Valerius Diocletianus Augustus (Diocletian, 244–311 A.D., and Roman Emperor from 284 to 205 A.D.) at Alexandria in 297 A.D., the Christian destructive fury that followed the edit Cunctos populos of Thessalonica in 380 A.D., the Islamic conquests of the lands of Egypt, in the 7th and 8th centuries A.D., the Venetian sack of Constantinople in 1204 A.D., during the Fourth Crusade, and finally the Islamic conquest of Constantinople in 1453 A.D.

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Here we limit ourselves to narrow our focus on the gears, and in this respect the Latin author who at greater length makes some indirect mention of the subject, albeit in the framework of other topics, is Vitruvius, in his monumental work De Architectura, in the Augustan age. First, we must give to Vitruvius the merit of having introduced Greek words, without any alteration; in fact, contrary to what Pliny the Elder made later, Vitruvius gave up the idea of translating the Greek words into Latin, to avoid the danger of distorting their meaning. Vitruvius described many different machines, used in various practical applications. They all derive from Hellenistic sources (Archimedes, Ctesibius, Philo of Byzantium, etc.), which Vitruvius specifically cites, but they have not come down to us. Another Vitruvius’ merit is therefore that of giving us news of these sources. Leaving aside the description of the construction works and their building methods, which do not interest here, it must be noted that Vitruvius describes machines for engineering structures, such as hoists, cranes, and pulleys, as well as war machines, such as ballistae, catapults, and siege catapults. Vitruvius also describes, with a certain amount of detail, the following machines: • Dewatering devices that use various tympanum configurations (see also Chap. 2), one of which is shown in Fig. 4.1. • Raising water machines and dewatering machines to irrigate fields and drain mines, based on Archimedes’ screws; in this regard, Vitruvius does not directly mention the great Syracuse scientist, but suggests that the source is Archimedes (Fig. 3.9). • Other lifting machines, including endless chains of buckets, and reverse overshot water wheels. • Water clocks (Fig. 4.2) and sundials. • Surveying and measuring instruments, such as diopters, chorobates (i.e., water levels), and odometers. • Aqueducts and inverted siphons. • A steam engine, the aeolipile, conceived as an experiment to demonstrate the nature of atmospheric air motions. • The so-called Vitruvius’ water mill (Fig. 4.3), which is a vertical water mill more efficient than the horizontal water mill, consisting of a water wheel with horizontal axis, which transmits the motion to a milling wheel with a vertical axis by means of a special geared pair. • Other machines that here for brevity are not described. However, all these machines have no novelty worthy of being detected with respect to the Hellenistic machines, to which they can be traced back, also with reference to the most insignificant construction details. Vitruvius was an architect interested in the most diverse aspects of architecture: he is in fact called architect by Sextus Julius Frontinus (about 43–103 A.D.), in his work De Aquaeductu Urbis Romae (On aqueducts of Rome), written in the late 1st century A.D.. The architecture, in Roman times, had a much larger horizon than the current one, in that it included not only the architecture as we know it today, but also construction engineering and management, civil engineering, chemical and material engineering, mechanical engineering, military engineering, and urban planning. He was both an architect and

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Fig. 4.1 Dewatering wheel from a finding found in the Rio Tinto mines in Spain

an engineer, brought to focus on the practical aspects of a problem, rather than the theoretical ones. In his writings, he describes personal experience of design, but it is based on Hellenistic sources. From these sources, he borrows and considers only the practical application aspects, while he clearly leaves aside those related to scientific knowledge upon which they are based. It is noteworthy that tympanum in Roman times had a more widespread diffusion, being used not only for dewatering devices and water lifting devices, but also to operate, by means of the power supplied by slaves, cranes and lifting devices,

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Fig. 4.2 Water clock imagine from a manuscript Italian translation of the De Architectura, IX, made in 1567

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Fig. 4.3 Reconstruction of a water mill according to the Vitruvius’ description

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replacing the more traditional pin-teeth wheels. Figures 4.4 and 4.5 show two applications of the tympanum as a lifting device. The first application (Fig. 4.4) consists of the bas-relief of the Haterii’s Mausoleum at the Vatican Museums, built for himself and his family on the Via Labicana in Rome by Quintus Haterius Thychicus, who

Fig. 4.4 Tympanum used as a lifting device from a bas-relief of the Haterii’s Mausoleum

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Fig. 4.5 Tympanum used as a lifting device: drawing from a bas-relief of the Roman amphitheater of Capua

was a redemptor, i.e. a constructor of public works under Titus Flavius Domitianus (Domitian, 51–96 A.D., and Roman emperor from 81 to 96 A.D.). The second application (Fig. 4.5) consists of a drawing taken from another bas-relief of the Roman amphitheater of Capua, also from the 1st century A.D. The massive use of slaves as a source of energy was a huge brake on the development of new technologies. In this regard, it is emblematic how Gaius Suetonius Tranquillus (about 69–122 A.D.) refers to us about a craftsman who submitted to Titus Flavius Vespanianus (Vespasian, 9–79 A.D., and Roman emperor from 69 to 79 A.D.), at the time when he was rebuilding the Capitolium, which had been set on fire during the recent civil wars, a carriage capable of transporting the columns on the top of the hill with very little expense. The emperor rewarded the good inventor, provided however that he did not speak of the thing, to “not deprive the working people of the work from which it drew food”. Only when the manpower began to run low, the Romans began to devise more efficient machines, however always based on practical empiricism. To demonstrate this tendency, just to recall the torcular oliarium of the late imperial age (Fig. 4.6), described by Rutilius Taurus Aemilius Palladius (4th century A.D.) in his Opus agriculturae (or De re rustica), where he highlights the advantages of this machine (as well as other machines for agricultural use), in terms of manpower savings. However, the gears described by Vitruvius (as well as those described by Hero and other authors until the late Roman Empire) do not show any substantially new features compared to those of the gears of the first Hellenism. Moreover, from the point of view of manufacturing technologies used, these gears are a definite regression in relation to their Hellenistic progenitors. In this regard, the conclusion drawn by Price de Solla (see [48] ) is very paradigmatic: he argues that before the discovery

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Fig. 4.6 Roman torcular oliarium of the 4th century A.D.

of the Antikythera Mechanism, based on the writings of Vitruvius and Hero, we had erroneously underestimated the ancient technology of the gears. The Antikythera Mechanism has open our eyes, helping to change our unjustified ideas about the Hellenistic civilization, and definitely refute the clichés on the contempt of the Greeks for the technology, and on the unbridgeable gap between theory and experimental and applied science, that the institution of slavery would have caused. Byzantines and Arabs have felt us some Hellenistic works, especially those of the imperial age, which were methodologically inferior to those of the early Hellenism, but more usable for practical applications. Another reason may be found perhaps in the fact that the best works had been irretrievably lost, since the scientific culture had died out long ago, and there remained no trace of the scientific method. For example, the Hero’s treatise on mirrors has been preserved, while the one theoretically more relevant written by Archimedes on the same subject (The Catoptrica), and confirmed by several testimonies (e.g. Theon of Alexandria, 335–405 A.D.), has been lost.

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And yet, for the imperial age, some historians have introduced the bombastic and pretentious name of early rebirth of the science. In light of what we have highlighted regarding Hero in Sect. 3.4 as well as the aforementioned considerations, this early rebirth of science is to be referred as a co-called early rebirth. It was in fact a failed attempt to recover the Hellenistic science. Indeed, remarkable results worthy of mention were not achieved, also because the scientific mentality and the scientific method, once they are lost, are not easily invented when a scientific school no longer exists. The same considerations are valid unchanged for all the successive so-called rebirths of which we will speak in the following section.

4.2 The Subsequent Pseudo-rebirths: From the End of the Imperial Age to the Renaissance In the long dark night of the Middle Ages already begun, it is possible to identify some historical periods characterized by a marked awakening of interest in the Hellenistic science, which have kept alive the memory, with special reference to some ingenious artifacts, generated by that science. These periods of awaking are not be considered as true rebirths, but rather as pseudo-rebirths, as the scientific method of the first scientific age was never recovered. The first period was that of the imperial age that we have described in the previous section.

4.2.1 The Second Pseudo-rebirth Between 5th and 6th Century A.D. The second so-called rebirth occurred between the end of the 5th century A.D. and the 6th century A.D., but the level of scientific originality of the main authors of this awakening is to be considered very poor. Among these authors, Simplicius of Cilicia (about 490–560 A.D.), Eutocius of Ascalon (about 480–540 A.D.), Ioannes Philoponus (about 490–570 A.D.), Anthemius of Tralles (474–534 A.D.) and Isidorus of Miletus (442–537 A.D.) are to be mentioned. Their individual contributions are limited not only numerically, but also and above all in-depth of thought, as their scientific writings show. In fact, these writings are mainly commentaries on the works of scientists and scholars of earlier age, especially from the Hellenistic age. However, they often have the merit of having handed down to us an ancient cultural heritage that would otherwise have been irretrievably lost. Simplicius was one of the last members of the Neoplatonist school but, in his commentaries on Aristotle’s Physica Auscultatio and de Caelo, a contribution regarding a more in-depth knowledge of ancient astronomical systems can be found. Moreover, especially when he argues with Philoponus, his contemporary, he leaves us some trace where he shows a certain sensitivity to the observation of natural phenomena.

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Philoponus, who was Byzantine Alexandrian philologist, Aristotelian commentator and Christian theologian, had the merit of breaking with the AristotelianNeoplatonic tradition, questioning methodology and arriving at empiricism in the natural sciences. In his commentary on Aristotle’s Physics, he rejects the Aristotelian dynamics and is the first to propose a Theory of impetus, according to which an object moves and continues to move because of an energy imparted in it by the mover and ceases the movement when that energy is exhausted. This statement is to be considered as the first step towards the concept of inertia of modern physics. He is also credited with having carried out experimental investigations in support of his theses and having used teaching methods of reasoning not dissimilar to those of modern science. Finally, his theological work On the Creation of World, where he applies the theory of impetus to the motion of planets, is recognized in the history of science as the first attempt at a unified theory of dynamics (see [32, 33]). Eutocius was a mathematician who wrote commentaries on several treatises of Archimedes as well as on the Conics of Apollonius. Not all of these commentaries have survived, but what has come down to us is very valuable for the news it provides. For example, we should mention that the main news we have of the Archimedes’ solution of a cubic by intersecting conics is from Eutocius. In this respect, Boyer [9] expresses himself as follows: “It is to Eutocius that we owe the Archimedean solution of a cubic through intersecting conics, referred to The Sphere and Cylinder but not otherwise extant except through the commentary of Eutocius”. He also describes the various solutions that were given to the famous Problem of Delos, namely the problem of the duplication of the cube. However, from the writings we received, it does not appear that Eutocius did any original work. Eutocius dedicated his commentary on Apollonius’ conics to Anthemius of Tralles who, together with Isidorus of Miletus, was the architect of the wonderful and incomparable Hagia Sophia patriarchal basilica in Constantinople. Anthemius was also a skilled mathematician. To him the following contributions are attributed (see [9]): the description of the construction of an ellipse using a string; an essay On the Burning Mirrors, which describes the focal properties of the parabola. Also, Isidorus, colleague and successor of Anthemius in the construction of Hagia Sophia, was a skilled mathematician and perhaps we owe him the well-known construction of the parabola by means of a square and a string as well as the apocryphal book XV of Euclid’s Elements. It should also be pointed out that we are largely indebted to the activities of Eutocius, Anthemius and Isidorus for the conservation of the original Greek text of the works of Archimedes and of the first four books of Apollonius. Anthemius was also able to invent and develop entertainment devices like those of Hero of Alexandria, using mirrors and steam. The two contraptions that Anthemius would have built to annoy his neighbor Zeno are famous in this regard: the first simulated a miniature earthquake, engineered sending steam through leather tubes fixed between the joists and flooring of Zeno’s parlor while he was entertaining friends; the second simulated lightning and thunder as well as a flash of intolerable light into Zeno’s eyes from a slightly hollowed mirror. In addition to his familiarity with steam, some authors credited Anthemius with knowledge of gunpowder or other

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explosive compounds. Some other authors also claim that he wrote about mechanical and hydraulic subjects. However, we are not far from the historical truth when we think that, in their role of designers and builders of Hagia Sophia, Anthemius and Isidorus were able to combine their geometrical knowledge with the mechanical ones of practical interest. These were the necessary skills required for the aforementioned role, and those who had these skills from the time of Hero onwards were called mechanikoi [37]. For the construction of their impressive masterpiece they certainly used lifting machines at least as those described by Vitruvius (similar machines were certainly used for the construction of the equally impressive and wonderful Roman monuments of the classical and imperial age), which were characterized by geared mechanisms. We cannot say anything specific about this subject, but they did not anything new compared to the Hellenistic sources from which they derived, including their geared mechanisms.

4.2.2 The Third Pseudo-rebirth: The So-called Islamic Renaissance The third period of awakening of interest in science is the so-called “Islamic Renaissance”, which began in the 8th century A.D., developed especially in the 9th century A.D. [2] and, progressively decaying, continued until the beginning of the 13th century A.D. It stimulated, for sympathy, a parallel interest in science by Byzantium. In this period, the translations into Arabic of the survived Hellenistic scientific treatises were carried out. Arabs translated the “Elements” of Euclide. The earliest Arabic version was produced by al-Hajj¯aj ibn Yüssuf ibn Matar (786–833 A.D.) sometime about the beginning of the third-ninth century; subsequently the same translator revised his earlier work, using principles that are not explained, thus leaving two versions [16]. The Byzantines instead, on the initiative of Leo the Mathematician (about 790–869 A.D.), Byzantine philosopher and logician associated with the Macedonian Renaissance, prepared a code of the writings of Archimedes. The manuscript of this code, now deceased, come to the Norman Court of Sicily in the 12th century. Then it belonged to the “stupor mundi” (the Emperor Frederick II of Sweden) and, after the battle of Benevento in 1266, ended up in the Vatican Library, where still existed in the 15th century when it was copied in France and Italy, but it disappeared in the 16th century. In their work, the Islamic scientists devoted primarily to the exegesis of the scientific works of the imperial age, and particularly the works of Claudius Ptolomaeus (Ptolemy, 100–170 A.D.) and Galenus of Pergamon (Galen 130–210 A.D.), who were considered the highest authorities respectively in the astronomical and medical fields. We are indebted to Islamic scientists not only the preservation of some Hellenistic works, but also some significant scientific contributions. Just remember, in this regard, the one of Ibn Sahl (940–1000 A.D.), who systematically applied to

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optics the theory of conics, considering not only parabolic and elliptical mirrors, but also plane-convex and bi-convex lenses, bounded by hyperboloids [50]. Today we cannot judge how original the results obtained by Ibn Sahl can be, since Archimedes’ Catoptrica has been lost. According to Lucius Apuleius Madaurensis (124–170 A.D.), Archimedes’ Catoptrica was a voluminous writing that dealt with the magnification obtained with curved mirrors as well as the burning-glasses and the rainbow, while Olimpiodorus the Younger (6th century A.D.) and Theon of Alexandria (335–405 A.D.) inform us respectively that in this work the phenomenon of reflection and that of refraction were analyzed. Russo [54], based on the historical documentation available today, argues that the laws of reflection, deduced from the principle of reversibility of the optical path, were perfectly formulated. The writings of Archimedes had already been translated into Arabic by Th¯abit ibn Qurra (836–901 A.D.), i.e. about one century before Ibn Sahl. The judgment on the originality of the contributions of Ibn Sahl is to be suspended, because it cannot be formulated. However, to get back on topic, we want to emphasize the fact that, in the Islamic world, the interest regarding the Hellenistic science (it was necessarily limited to the issues described in the surviving works) goes hand in hand with the development of appropriate technologies, related to different industries, such as shipbuilding, metallurgy, textile and paper industries. In other words, the close relationship between science and technology was already clear to Islamic thinkers, long before it became a key concept in the modern age. With reference to gears, here we must mention the Ban¯u M¯us¯a brothers (Muhammad Ban¯u M¯us¯a, Ahmad Ban¯u M¯us¯a and al-Hasan Ban¯u M¯us¯a), three 9th-century Persian scholars who lived and worked in Baghdad and are known for their Book of Ingenious Devices on automata and mechanical devices, and al-Jazari (1136–1206 A.D.), who lived and worked in the late 12th century and early 13th century. Based on manuscripts, one of which in the Vatican (see [25, 26]), we know that the Ban¯u M¯us¯a brothers developed 100 mechanical devices. These include 7 fountains, 4 selftrimming lamps (Fig. 4.7 shows one of these, from On Mechanical Devices, written by Ahmad), a gas-mask for approaching polluted wells, an automatic musical instrument and a mechanical grab for excavating in the beds of streams, while the remaining devices consist of trick vessels for dispensing liquids. Not all these devices, including those using gears, show substantial changes than those described by Philo of Byzantium and Hero. In fact, apart from some more complexity in terms of layout, they are all similar in design and operation. The Ban¯u M¯us¯a brothers are also known for their Book on the Measurement of Plane and Spherical Figures, an important work on geometry that was frequently quoted by both Islamic and European mathematicians. Al-Jazari is the paradigm of the Arab Muslim technical culture. He is best known for writing, in 1206, the work entitled The Book of Knowledge of Ingenious Mechanical Devices which, due to its style, Mayr [36] defined a modern do-it-yourself book. The theoretical contributions are completely absent in his works, as the same al-Jazari admits (see [4, 5, 60]). In addition, the originality is almost completely missing. Nevertheless, al-Jazari is important for the detailed description of 100 mechanical devices, many of which use gears. He was, so to speak, an artisan-engineer, more

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Fig. 4.7 One of the four self-trimming lamps designed by Ahmad Ban¯u M¯us¯a

interested in the craftsmanship necessary to construct the devices, which were usually obtained by trial and error rather than by theoretical calculations. The Al-Jazari’s devices include: 5 water-raising machines; water mills; water wheels with cams on their axes to operate automata; a sakia with a crankshaft in a crank-driven sakia chain pump; automated moving peacocks and automatic gates, both driven by hydropower; automatic doors for the most elaborate water clocks; a waitress humanoid automaton, that could serve water, tee or drinks; a hand washing

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automaton, incorporating a flush mechanism; a female humanoid automaton, standing by a basin filled with water; a peacock fountain with automated servants; fountain and musical automata, characterized by an innovative use of hydraulic switching; water clocks and candle clocks; an elephant clock and a castle clock; a monumental water-powered clocks; a weight-driven water clock; a double-action reciprocating piston suction pump; escapement mechanisms; segmental gears; etc. Some of these devices are clearly inspired by earlier devices. For example, the monumental water clock is based on that of a Pseudo-Archimedes; most of automata are inspired by those of Hero, which in turn are based to Ctesibius; the fountains are based on those of Ban¯u M¯us¯a brothers; musical automata are inspired by those of AlHusayn; a candle clock is based on that of Al-Shaghani; a water-raising device driven by hydropower emulates a similar hydropower device introduced by the Chinese many centuries before; etc. However, apart from this lack of originality, we must be grateful to Al-Jazari for some ingenious solutions introduced in his devices, some of which seem new and innovative, at least until proven otherwise. Among these, we highlight the following: • An intricate gear device based on sakia, powered in part by the pull of on ox walking on the roof of an upper-level reservoir, but also by water falling into the spoon-shaped pallets of a water-wheel, placed in a lower-level reservoir (Fig. 2.5). • A sakia characterized not only by a gear system, but also, and for the first time in the history of technology, by a crankshaft (it is to be noted that the earliest evidence of a crank and connecting rod mechanism is the one of Hierapolis sawmill in the Roman Empire, which dates from the 3rd century A.D.). • Segmental gears, i.e. machine elements consisting of a sector of a cylindrical gear, able to transmit or receive reciprocating motion to or from a cogwheel, these also appeared for the first time in the history of technology; • Water clocks, water-raising machines and automata incorporating gear systems as well as cam and camshaft mechanisms, the latter also used for the first time. Figure 4.8 shows a twenty-four-tooth ratchet gear that Al-Jazari believes can be adopted as a mechanism for counting the solar hours and half-hours in water clocks, as well as for other devices. The same figure shows the schematic operating diagram of the same mechanism, which is controlled by the continuous emptying and filling of a tipping bucket and that, in turns, drives a quadrant with presumable timekeeping index, by means of two pulleys (one of these is rigidly connected to the ratchet gear) on which a string is wound [4, 5]. Despite these significant aspects of inventive innovation, which are the result of an appreciable practical ingenuity of the author, mechanical devices of al-Jazari do not present original theoretical contributions compared to those already known from previous times. This is a characteristic common to all the scholars of the Islamic Renaissance. In fact, in the Islamic engineering concept of this age, most of the mathematical relationships that underlie the physical phenomena had not been identified, and the engineers drew on large fund of practical experience, proceeding with trial and error methods, without making calculations.

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Fig. 4.8 Ratchet gear from Al-Jazari

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4.2.3 The Fourth Pseudo-rebirth: The Rebirth of Western Europe The next period of awakening of the technical-scientific culture occurs in the 12th century in Western Europe, and it is the one for which historians used, for the first time, the term “Rebirth”. The first Latin translations of Greek scientific works were made and, at the beginning of the century, Bernardus Carnotensis, i.e. Bernard of Chartres (1070–1130 A.D.), summarized the new sensitivity to the ancient culture with his famous statement … “nos esse quasi nanos gigantium humeris insidentes, ut possimus plure eis et remotiora videre” …, i.e. … “we are like dwarfs sitting on the shoulders of giants, so we can see more things than they could and far away” …. Of course, the giants were the great thinkers, scholars and scientists of the near and distant past (see [53]). In 12th and 13th centuries, the Iberian Peninsula and Sicily, both subtracted to Islam and reconquered, and southern Italy, which remained in touch with Constantinople, constituted important points of contact between the European culture and scientific tradition going back to Hellenism. Even with the help of Arab scholars remained in the reconquered countries, the Arabic translations of Hellenistic works began to be assimilated by the Europeans, who also began studying the manuscripts in Greek arrived in Europe after the sacking of Constantinople in 1204, during the Fourth Crusade. Leaving aside the description of the efforts made in the different European cultural centers, to recover the ancient science, we should mention here two artist-engineers, who may be considered precursors of the Renaissance: Villard de Honnecourt (about 1200 –1250) and Guido da Vigevano or as he himself called Guido da Pavia (about 1280–1349). The first of these artist-engineers is the author of the so-called sketchbook of Villard de Honnecourt, more correctly an album or portfolio, which is probably the most famous document of medieval technology. This sketchbook was discovered in the mid-19th century, and is dated between 1225 and 1235. The album contains about 250 drawings, which also include mechanical devices, lifting devices, a water-driven saw for lumber-cutting machine (Fig. 4.9), war engines such as a trebuchet, a perpetual-motion machine, a number of automata, and many other devices [6]. The device drawn by Villard, more interesting here, is a strange gearing mechanism to turn an angel statue following the position of the sun with its finger. Today it is thought that this nocturnal clock was used not only so that monks could determine the hour of night, but also for calendrical purposes. The device is believed to date from 1237 and is considered the precursor of the first verge-and-foliot mechanism, which is notoriously made up of crown wheel, verge and foliot (see [57]). The question, however, is controversial and currently the consensus it that the drawing of Villard of Honnecourt was not an escapement device (see [1, 6, 56]). Also, the second artist-engineer, in his sketchbook Texaurus regis Francie, draws a number of technological items and ingenious devices, which offer an invaluable insight into the state of technology of the late Middle Ages (see [23, 24]). The

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Fig. 4.9 Water-driven saw for lumber-cutting machine, from a Francesco di Giorgio Martini drawing taken from Villard de Honnecourt

drawings in this sketchbook include floating pontoons, propeller driven boats, mobile assault towers, and a gear driven assault chariot to carry a platoon of soldiers. It is appropriate here to briefly dwell on the sketch of this armored chariot, which Guido da Vigevano designed for an envisaged Crusade, real or imaginary (Fig. 4.10). This chariot was a wind propelled carriage and siege engine, with wheels configured as spoked wheels and driven by a gearing transmission, consisting of a combination of pin gear wheels and lantern pinions, powered by wind blades. It was therefore a mix of two technologies, the overlying part resulting from a windmill and the underlying chariot, used within the limits of that time, and configuring a device that, like the elephants of Hannibal, would have had to knock down enemies, advancing alone. Moreover, it was maneuvered by a steering axle, commanded by a rudder, so many

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Fig. 4.10 Two different drawings of the wind propelled carriage of 1335 by Guido da Vigevano

experts see in this war machine designed by Guido da Vigevano the first forerunner of modern automobiles. Figure 4.10 shows two different drawings of the wind propelled carriage, taken from Guido da Vigevano sketchbook. Several reconstructions have been made of this imaginative carriage such as, for example, the wooden model jointly made by University of Pavia, Polytechnic of Milan and Polytechnic of Turin. Guido da Vigevano introduced an important innovation, i.e. the prefabrication. In fact, he designed his military machines and other machines composed by pieces transportable in battle on horses, to be assembled at their destination through special joints (hinges and butt joints), which he recommended using iron as a material. According to Hall [23], later Renaissance artist-engineers, such as Mariano di Jacopo (better known by the nickname Taccola), Roberto Valturio, Francesco di Giorgio Martini and even Leonardo da Vinci, used or even copied, directly or indirectly, many elements of Guido da Vigevano’s devices. Moon [39] sees this as a confirmation of the theory of technical evolution of the concepts of machine design. However, although Guido da Vigevano is still linked by style and spirit to the Middle Ages, he can rightly be considered as a forerunner of later artist-engineers of the Renaissance, mentioned above. In the 13th century, many different technological inventions spread in Europe. Among them, mechanical clocks, based on speed reducing gear trains and escapement mechanisms, occupy a prominent place. Price de Solla [47, 48], who was undoubtedly the greatest specialist on the subject, pointed out the close analogy between European

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clocks of the 14th century, the above-mentioned Islamic mechanisms, the Hellenistic planetaria such as the Antikythera Mechanism, and a series of Chinese astronomical clocks, dated between the 2nd and 11th centuries A.D. The Price’s clarification is based on the repetition of the same construction details, chief among them the shape of the tooth profiles of gear wheels. Price, however, does not exclude the possibility that some of these inventions may have been a reinvention. As regards the gears, the mechanical clocks are of particular interest, since they are a typical medieval product. Today we are not able to identify who made the first mechanical clock, and when he did it. Identity of view does not exist about the inventor who first built the mechanical clock, using the certainly not perfectly conjugate gear wheels then available. Someone has given this priority to the archdeacon Pacificus of Verona (about 776–844 A.D.), because the epitaph on his tomb records that he built a horologium nocturnum, i.e. a nocturnum clock. However, it seems that it was merely an astronomical observation tube with crosshairs, used to locate the stars, rather than a mechanical clock or water clock [30]. However, this interpretation, although it is supported by illustrations from medieval manuscripts, is not enough to depress the question, because a conclusive supporting documentation is missing completely. However, the following factual circumstances constitute a certainly [34]: • Towards the middle of the 8th century, Pope Paul I sent to Pepin de France a rudimentary horariolum (diminutive of horarium), driven by a weight. • At the beginning of the next century, Caliph Harun el-Rashid gave to Charlemagne a clock that marked the hours. • Pope Sylvester II, the immediate successor of Paul I, is credited with having built a clock which would be the prototype of many other clocks built in Europe after the one thousand years, and moved like that of Paul I. The most widespread opinion today is that they were Christian monks to invent and build the first mechanical clocks, in the late 13th century, moved by the desire to introduce a strict and regulated timing of prayers and church activities in medieval monasteries. Compared to previous water clocks, a mechanical clock is characterized by the escapement mechanism, which swings and ticks in a steady rhythm, and lets the gears move forward in a series of little jumps. The first mechanical clocks had only one set of toothed gears and an escapement device. Through continuous developments, in the four centuries that followed, these mechanical clocks were enriched with other sets of gears, which allowed marking not only the hours, but also the minutes. Usually, the gears of these clocks were made of metal; furthermore, they had wedge shaped teeth. The first escapement was the verge-and-foliot mechanism, while other types of rocker mechanisms, or others yet, were subsequently developed. Of course, the tooth profiles of the main gear wheel, driven by the escapement mechanism, was configured to marry the profiles of the verge’s pallet. The latter had a duty to engage and release the main gear wheel, which was turned by a heavy stone on the end of a cable. The first description of a mechanism similar to the escapement mechanism is however that of Villard de Honnecourt who, in 1237 A.D., applied it to an almost-clock.

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The first writings that speak of mechanical clocks date back to the 11th century, and this implies that they were well known in Europe at that time. At the beginning of the 14th century, they acquired literary dignity through the greatest poet Dante Alighieri (1265–1321), who speaks of them in two metaphors of the third Cantica, Paradiso (Canticle, Paradise) of the La Divina Commedia (Divine Comedy), which was written between 1307 and 1321 [15]. The first of the two metaphors is that made up of verses 139–144 of the Canto X of Paradise, which are to be considered the first known literary reference to a mechanical clock that struck the hours. The verses are as follows: Indi, come orologio che ne chiami ne l’ora che la sposa di Dio surge a mattinar lo sposo perché l’ami,

Then, as a horologe that calleth us What time the Bride of God is rising up With matins to her Spouse that he may love her, che l’una parte e l’altra tira e urge, Wherein one part the other draws and urges, tin tin sonando con sì dolce nota, Ting! Ting! resounding with so sweet a note, che ‘l ben disposto spirto d’amor turge; That swells with love the spirit well disposed; This is one of the first accounts of the existence and use of these clocks, equipped with gears, which worked driven by weights and counterweights or springs. In a typical situation of cenobitic monastic life, marked from dawn to dusk by welldefined hours of prayer and work, Dante describes a kind of alarm-clock-mechanism, in which the geared wheels commanded a device equipped with beating hammers on one or more bells, which make their ticking sound in the pre-dawning silence. In the l’una parte e l’altra tira e urge, tin tin sonando…, is represented poetically the visual image of the movement in opposite directions of the geared wheels of a gear train (where a part of a wheel tira, that is drags, the one which is behind, while the other part of the same wheel urge, that is pushes, the one that is forward to it). Furthermore, the acoustic perception is added to the waking signal, suggestively indicated with the onomatopoeic description tin tin of the repetitive striking of a bell. The second of the two metaphors is that made up of verses 13–15 of the Canto XXIV of Paradise, which are as follows: E come cerchi in tempra d’orïuoli And as the wheels in works of horologes si giran sì, che ‘l primo, a chi pon mente, Revolve so that the first to the beholder quïeto pare, e l’ultimo che voli; Motionless seems, and the last one to fly; These verses describe the geared wheels (cerchi) of a small mechanical clock (orïuolo), but what mattered to Dante here were the very different rotational speeds. In fact, the various wheels of the clock gear train rotate so that the first one, from which the movement originates, seems to stand still, while the last one, which transmits the movement to the indices (or to the mechanism that beats on the bells) seems to fly instead. Even here Dante shows his gigantic stature of immense poet, since the first hemistich of verse 15 could not be slower, nor the second faster. Dante thus

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communicates his feelings to us not only with words, but also with the rhythmic sonority of the verse, very slow in the first hemistich, and much faster in the second hemistich [19]. In the 14th century, in various cities in Europe, monumental mechanical clocks are built, not only to meet the needs of religious institutions, but also to indicate visually and acoustically the time of the population, with clocks installed on public buildings. In Italy, the first clock with gear trains of which we have news was that placed in 1306 on the bell tower of the Church of Sant’Eustorgio (Holy Eustorgius) in Milan. It seems that this clock does not beat the hours, as it was able to do the clock placed in 1328 by Azzone Visconti (1302–1339) on the Tower of the Church of San Gottardo in Corte, the Chapel of the Royal Palace, also in Milan. It also seems that the same type of clock was placed on the Tower of Westminster in London, even in 1288. Other notable examples of clocks of this period were built in Strasbourg (1354), Lund (1380), Salisbury (1386), Rouen (1389), Wells (1390), and Prague (1462). The Salisbury Cathedral clock (1386) is one of the oldest working clocks in the world, and perhaps the oldest. It is noteworthy that most of its parts are still the original ones, but its original verge-and-foliot time-keeping mechanism has been lost; it has been converted into a pendulum mechanism. This happened after 1656, when Christiaan Huygens introduced the pendulum clock, which was characterized by ten or more sets of gears. Pendulum clocks ensured a higher accuracy than previous mechanical clocks. They were a big step forward in technology and manufacture of gears, and found wide use in public buildings. It is noteworthy that Huygens based his patent, obtained in 1657, on the results of studies made by Galileo Galilei on the pendulum, starting from 1602. Galilei himself had the idea of making a pendulum clock in 1637, but only his son, in 1649, partly realized his father’s design, but neither lived to finish it (see also Sect. 5.1). At the dawn of the Renaissance, Richard of Wallingford (1292–1336) and Giovanni Dondi dall’Orologio (1330–1388) made significant contributions to the development of clocks, with innovations also regarding their gear drives. To Richard of Wallingford we owe the earliest Western description on the escapement, which dates back to 1327, the date of publication of his Tractatus Horologii Astronomici (see [1, 41]). This escapement mechanism differed from the verge and foliot mechanism, as it utilized two spur gears mounted on a common axle with their teeth out of phase, and an anchor-shaped pallet rotating between them. The ends of the pallet were alternatively caught and released the projecting teeth on their gears [13]. This mechanism characterized the astronomical clock designed by Richard of Wallingford. This clock however was completed about 20 years after his death by William of Walsham, and was destroyed during Henry VIII’s reformation and dissolution of St Albans Abbey, of which Richard of Wallingford was abbot. Many scholars believe that the clock-like mechanism of Richard of Wallingford was the most complex mechanism existing in the British Isles at the time and one of the most sophisticated anywhere. Based on the literary evidence, surviving in the 20th century, several reconstructions of the Richard of Wallingford’s clock were made, such as those exhibited at the Time Museum in Rockford, Illinois, built by Harvard Horological, at the Wallingford

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Museum built by Eric Watson, and at St Albans Cathedral built in 1988. Richard of Wallingford is also credited for reinventing epicycloid gearing used in his astronomical clock. He also designed and built calculation devices, such as: a Torquetum, which is a medieval astronomical instrument, resulting from the combination of Ptolemy’s astrolabe and the plane astrolabe, designed to make azimuth and zenith measurements and convert them into spherical coordinates, referring to the ecliptic; the Rectangulus, described in his Tractatus Rectanguli, which is an observational instrument that can also be used for coordinate transformations; and an Equatorium, which he called Albion in his Tractatus Albionis and probably used for astronomical calculations such as lunar, solar and planetary longitudes and, unlike most equatoria, could predict ellipses (see [12, 41]). He published other works on trigonometry, celestial coordinates, astrology and other. The only clock-like mechanism documented in the 14th century, having comparable complexity with that of Richard of Wallingford, is certainly the Astrarium of Giovanni Dondi dall’Orologio. We are indebted to Giovanni Dondi dall’Orologio for the earliest powered complex gearing mechanism used for mechanical clock, which is known in Europe. The detailed description of this clockwork is in fact contained in his 1364 treatise Tractatus Astrarii [20, 21]. He invented and built the Astrarium that bears his name, and is to be considered the first clockwork to be made entirely in metal. This Astrarium was an astronomical clock, with internal gearing and elliptical gear wheels, which reproduced the motion of the sun, moon and five planets then known (Mercury, Venus, Mars, Jupiter, and Saturn, as the Antikythera Mechanism). It also pointed to the length of the daylight hours to the latitude of Padua (Dondi was a professor in the University of Padua). Furthermore, the Astrarium, which was a gearing mechanism far ahead of its historical era, measured time. It showed, in addition to the hour, for the first time between the mechanical clocks, even the minutes, in groups of ten. Moreover, the planetary motions foresaw the moon phases as well as the church feasts calculated with a perpetual calendar. The astronomical motions, obtained with a gear train including elliptical gears and a planetary gear train as well as suitable linkages, were highlighted on seven dials. The brass and copper gears were cut by hand, using some dividing instrument that ensured the reduction of pitch error. The Dondi’s original astronomical clock personally built in Padua with a 16-year job has been lost, but two manuscripts of his Tractatus Astrarii in the Biblioteca Capitolare of Padua describe it in detail, so much so it was faithfully reconstructed in 1963, and it is exhibited today in the watch section of the National Museum of Science and Technology Leonardo Da Vinci, in Milan. In addition, it is to be noted that Dondi’s Astrarium has inspired other modern replicas, including some in London’s Science Museum and the Smithsonian Institution. Figure 4.11 shows a perspective detail of the gearing mechanism and the lower section of Astrarium, reconstructed according to the description taken from the Tractatus Astrarii.

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Fig. 4.11 A perspective detail of the gearing mechanism and lower section of Astrarium, reconstructed according to the description taken from the Tractatus Astrarii

4.3 The Renaissance, Precursor of the Second Scientific Age In the historical period in which Dondi dall’Orologio lived and worked, a new age began in Europe, the Renaissance. From the mid-14th century, a stream of Greek writings from Constantinople went to Italy, and from there in Europe, generating the Renaissance per excellence, which is the definitive awakening from long hibernation of the dark medieval night. This flow of Greek writings intensified in the 15th century. Renaissance intellectuals, however, did not have the preparation and cultural background necessary to understand the Hellenistic scientific theories and, therefore, they were attracted by the results of these theories, especially those represented with illustrations, sketches and drawings, more easily understandable. These results included, among other things that do not interest here, gears, gear trains, and all machines that were equipped with geared mechanisms, such as war machines, hydraulic and

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pneumatic machinery, lifting and shipbuilding machinery, automata and entertainment mechanisms, measurement instrumentation, etc., as well as technologies for their construction, including those regarding the fusion of bronze artifacts. Towards the end of the 14th century, Italy began to enjoy, first in Europe, a kind of interpenetration between active life and scientific knowledge. The architecture was the first to benefit from this interpenetration, thanks to the two stars of exceptional splendor, such as Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–1472). The first was an exceptional craftsman, while the second, more prone to speculative aspects (in a certain sense more scientist, even if not in the current meaning of the term) than the first, differed significantly from him, having made every effort to unify the scientific tradition with the practical and concise sense of experience. However, we must focus our attention on Brunelleschi, who, in order to build without supporting reinforcements the stupendous and daring dome of Santa Maria del Fiore in Florence, designed at least three machines, i.e. construction equipment that only the new materials and sources of energy introduced by the Industrial Revolution allowed to imitate over three centuries later. The first of these machines, called Colla Grande, incorporated a system of three shafts, connected to each other and to the driving shaft, driven by a pair of oxen, by means of gear wheels. Many rope winding drums were keyed on the three shafts. The drive system made it possible to realize three different lifting speeds and had as its fundamental component of great technological relevance a reversing gear that today is to be considered as the first realization of this type (obviously, until proven otherwise). This machine also incorporated a differential gear drive. The second machine was a large crane for lifting smaller loads. It was equipped with all the movements that characterize modern cranes, by virtue of two rudders for the control of the vertical motion, one for the positioning of the counterweight, and a further rudder designed for the rotation motion of the crane. Four screw gears then secured the two horizontal displacements, mutually orthogonal, the first at normal speed and the second of a micrometric type. The third machine, designed to build the lantern of the dome, was never completed (only the complex frame was built), due to the death of Brunelleschi. Brunelleschi’s three machines were disassembled when the construction of the dome was finished, and were held in the Piazza di San Giovanni in Florence until 1470, after which their traces disappeared. It will be necessary to wait almost four centuries to meet similar machines with efficiency not much exceeding that of the prototypes of Brunelleschi, which Michelangelo called architect “difficile da imitare ed impossibile da superare”, i.e. “difficult to imitate and impossible to overcome”. As a document from the Bibliotéque Nationale de France testifies, the Colla Grande was set up in Florence at the atelier of Andrea Verrocchio, where Leonardo was able to examine and admire it. Figure 4.12 shows a drawing by Leonardo of Brunnelleschi’s three-speed winch, which highlights the differential gear train for controlling the various movements possessed by this lifting machine. It also shows, on the right side, the reversing gear that uses a (lantern-pinion/pin-teeth-wheel) pair for the transmission of motion between perpendicular axes. The same figure highlights the detail of the reversing gearing mechanism, according to a recent reconstruction

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Fig. 4.12 Drawing by Leonardo of Brunelleschi’s three-speed winch and schematic diagram of the reversing gearing mechanism

of the Colla Grande. It should be noted that the lantern gearing transmission between orthogonal axes was already in use for a long time, as shown in Fig. 4.13, concerning a water lifting device driven by horses. Figure 4.14 shows another detail of the differential gearing mechanism of the Colla Grande as well as a general view of the same lifting machine, based on a drawing by Mariano Taccola (see below). Leonardo da Vinci was certainly the most famous intellectual of the Renaissance, also because with his sublime genius he was able to transcend the culture of his time, which is to be considered as still belonging to a pre-scientific age. Being a recognized universal genius, Leonardo becomes interested in all topics mentioned above, and in

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Fig. 4.13 Lantern gearing transmission between orthogonal axes

those which we have omitted (e.g., anatomical dissections and surgery, perspective and geometry, psychological portraits, musical instruments, etc.), and tried to grapple well as in the study of the Archimedes‘ works, although unsuccessfully, since he lacked, like intellectuals of his time, the scientific method. He was not an isolated genius. Leonard’s roots sink in fact into the cultural fervor that characterized the Renaissance, and many of his technological interests were shared with him, between the second half of the 14th century and the first half of the 15th century, by other technologists of the time, some of which were his precursors. The first technologist to be mentioned is Mariano di Jacopo of Siena (1381–1453), called Il Taccola (The Jackdaw) and better known as Mariano Taccola, who was interested in the works of Philo of Byzantium on pneumatics and military technology. Mariano Taccola is known for his two technological treatises, entitled De ingeneis (On the engines) and De machines (On the machines), where innovative machines and devices are described, with a wealth of drawings that highlight the details (see [45, 55]). These two treatises were largely studied and copied by later Renaissance engineers and artists, almost certainly also by Leonardo himself. Taccola had the great merit of having invented a new way of making technical drawings, with the introduction of what is today called exploded-view, where the parts of an object are shown separately, but with an indication of how they fit together. Figure 4.15 shows two other drawings by Taccola, taken from De ingeneis, both depicting a (lanternpinion/pin-teeth-wheel) pair, the first enslaved to a wind-driven water lifting device, where wind is personalized by the god of winds, Aeolus, and the second enslaved to another water lifting device driven by power supplied by an ox. Moreover, with the help of drawings, Taccola faced major problems of his time, bringing back the mechanical apparatus to combinations of levers, pulleys, winches and gear trains, whose origin dates back to antiquity. On the one hand, Taccola dedicated himself to the description of the machines listed in the ancient technical treatises, received without an iconographic representation, and on the other hand he engaged in the study of military and civil engineering, with the aim of inventing

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Fig. 4.14 Drawing by Mariano Taccola of a simplified version of Brunelleschi’s Colla Grande

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Fig. 4.15 Lantern-pinion/pin-teeth-wheel pair for: a a wind-driven water lifting device; b a water lifting device driven by power supplied by an ox

new devices. He is a kind of crossroads in the history of Renaissance engineering, because his dual training, technical-artistic and humanistic, prefigures the renewal of the professional figure of the engineer who, as an operator often indoctrinated and literally silent, turns into an author of treatises, where the image is conceived as a main vehicle for the communication of technical information. The same technological interest was shared by the anonymous author of the Italian translation of the Pneumatics of Philo of Byzantium, which is probably the first translation of a Hellenistic scientific work in a modern European language. The manuscript of this translation, which is preserved in the British Library, also contains extracts of Vitruvius, the transcription of the Taccola’s work De ingeneis, as well as compilations of various types, including one on incendiary substances [44]. Another technologist to be mentioned is Roberto Valturio da Rimini (1405–1475), who was an engineer, writer and advisor of Sigismondo Pandolfo Malatesta, lord of Rimini. Between 1446 and 1455, several manuscripts of the Valturio’s treatise, entitled De Re Militari (On the Military Arts), saw the light, while its editio princeps is from 1472, that is just seven years after the introduction in Italy of the printed writing. This treatise represents the first book of Engineering, complete with drawings and related captions, that has never been printed in the world. Perhaps exaggerating, this treatise was considered as an encyclopedia of the military arts (see [14, 35, 62]). In reality, it refers to the main classical military writers, such as: Titus Flavius Iosephus (about 37–100 A.D.), Sextus Julius Frontinus (about 40–103 A.D.), Ammianus Marcellinus (about 330–395 A.D.), and Publius Flavius Vegetius Renatus, who lived in the late 4th century and the early 5th century

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A.D. In addition to already known machines, unknown war devices and systems are described, some of which are indeed strange and even absurd, such as the flabellatus chariot, whose four wheels were cylindrical lantern gears driven by pin gears, in turn driven by wind vanes. Figure 4.16 shows, on the overleaf of a page of De re militari, a scythed chariot pulled by horses, and on the recto of the next page, a non-scythed chariot, but “cacciato cum flabella”, i.e. driven by wind energy captured by wind-vanes. About this flabellatus chariot, we can admire the beautiful flight of fantasy, which would have successive imitators (see below), but we must recognize the inadequacy of the technical solution adopted, which would never have allowed the operation of the chariot. Leonardo da Vinci knew both editions, Latin and Italian (the latter published in 1483) of Valturio’s work, and he used it at first to learn a little Latin, and to sketch the letter-question of recruitment-curriculum addressed to Ludovico il Moro, lord of Milan. In this letter Leonardo, making an almost vaunted credit, with drawings taken from Valturio (these drawings are considered the work of Matteo de’ Pasti of Verona, who was an excellent graphic and engraver), but also from Taccola, boasts of being, rather than a painter, a man of sciences and engineer, expert in military matters. Valturio’s work, however, had the merit of giving impetus to Leonardo’s creativity. In this regard, just remember the drawings of Leonardo’s scythed carriages, made after his transfer to Milan. These carriages present frightening rotating sickles protruding from the wheels, with respect to which those of the above mentioned flabellatus chariot of Valturio, from which they derive, seem almost inoffensive blades. Moreover, about what concerns us, Leonardo draws in detail the connection of the gears and the teeth to the axles and wheels, creating graphic works of absolute beauty, which appear strident with the monstrosity of the military vehicle represented. Another technologist still to be remembered is Francesco di Giorgio Martini of Siena (1439–1501), who was a multifaced artist of great importance in the second half of the fifteenth century. He was in fact a painter, sculptor, architect and engineer, but he devoted himself mainly to the architecture and drafting of architectural treatises, including the technologies and machines necessary for their construction. As Mariano Taccola, he was essentially an ingegniarius. This was the term with which, in the Renaissance, a skillful technician and a humanist was indicated who could take, when necessary, the clothes of the inventor and the writer (the term engineer in its present meaning derives from this renaissance way of defining such a professional figure). Because of his multifaced activity, someone has supposed a youthful presence in the environment of Mariano Taccola, who called himself the Archimedes of Siena. Like many of his contemporary craftsmen, Francesco di Giorgio Martini was interested in the study and development of mechanical devices, and carried out this activity in accordance with the still flourishing Vitruvian Tradition (see [14, 51]). His Trattato di Architettura, Civile e Militare, i.e. Treatise on Civil and Military Architecture, is mainly dedicated to civil and military architecture. It contains hundreds of small but perfectly drawn illustrations showing war machines of every kind as well as pumps, mills, cranes, etc. [17, 18]. Furthermore, he describes water wheels fed by pressure pipes, sump and pressing pumps, worm gears, mechanisms with pinion-rack gear pairs, and many other machine elements, which show clearly

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Fig. 4.16 A scythed chariot and the flabellatus chariot

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their derivation from Hellenistic technology, as well as four-wheeled carts equipped with steering mechanism incorporating gear drives (see [38, 40]). Figures 4.17 and 4.18 show a gear transmission for four-wheeled carts with steering mechanism and, respectively, two pages of the aforementioned treatise, where some axonometric drawings of different types of gear drives are represented. Mariano Taccola and Francesco di Giorgio Martini show that they are mainly anchored to Philo of Byzantium and Vitruvius. Numerous drawings of Leonardo, so to speak futuristic (e.g., screw presses, speed reducing gears, threading machines, automatic hammers, wind wheels, siphons, fountains, water levels, devices moved by the rising air, etc.), which so impressed in the past and still continue to excite, are strongly inspired to Hero. On the contrary, other drawings (e.g., transmission chains with flat links, and related rockets, automatic crossbows, etc.) have Philo of Byzantium as their source. In addition, many notes of Leonardo, such as those on optics, burning mirrors, crossbows, hydraulic saws, ball bearings, paddle boats, and more, are clearly based on ancient sources. The creations of the multifaceted genius of Leonardo, that transcend the culture of his time, should not be considered as a fantastic journey into the future. They must be considered rather as an attempt to plunge in the distant past, to learn their innermost secrets, and so make feasible those objects that were unfeasible, because the corresponding technologies, well known to the Hellenistic scholars, had been lost. The works of Philo of Byzantium and Hero still also gave impetus to the discovery of many elements of Hellenistic technology. As regards the metal gears, the need was felt to acquire the technological knowledge, completely missing, to realize all their various types, the shape of which was instead easily be reconstructed by means of the illustrations contained in the manuscripts. The rediscovery of Hellenistic technology was still very slow and gradual, especially as regards metallurgy, whose knowledge was at an extremely low level, yet to Alchemy level. In this regard, just remember that, in the 3rd century B.C., in Rhodes, the casting technologies of bronze artifacts were developed so as to allow to obtain not only large ornamental statues, like the Colossus of Rhodes, but also large pieces of artillery and large elements of shipbuilding. On the contrary, for Leonardo da Vinci the casting of his famous horse for the equestrian monument to Ludovico Sforza in Milan was just a dream in vain chased. Here we leave aside the speculations of Philippus Aureolus Theophrastus von Hohenheim Paracelsus (1493–1541) and other alchemists like him, but we must recognize to them the merit of having called attention to the practical utility of the emerging technical chemistry on the mechanical constructions. Instead here we must mention the two biggest metallurgists of the Renaissance. They were Vannoccio Biringuccio of Siena (1480–1539), and Georgius Agricola of Glauchau (1494–1555), which is the Latinized name of the German naturalist Georg Bauer. Vannoccio Biringuccio, in his office of foundryman master of the Apostolic Chamber and director of papal artillery, during the papacy of Paul III (1534–1549), acquired a huge experience on hot processing of ferrous and non-ferrous metals, including casting techniques. He gathered a considerable documentation on these technologies, which were the subject of his first work entitled Il modo di fondere, spartire

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Fig. 4.17 Gear transmission for four-wheeled carts with steering mechanism

Fig. 4.18 a, b Two pages taken from the Francesco di Giorgio Martini’s treatise

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e congiungere metalli i.e., The way to cast, divide and join metals. This first contribution to the metallurgical knowledge, acquired by experimenting on the field, constituted a valid support for the liberation from the alchemical superstitions. Later Biringuccio decided to expand significantly the range of metallurgical applications, extending well to the problems inherent in the research and exploitation of mineral deposits, for which he worked on writing his monumental work in ten books, entitled Pyrotechnia. This treatise, however, was published posthumously in Venice, in 1540, after his death [8]. Figure 4.19 shows the title page of the fourth edition of Pyrotechnia, published again in Venice in 1558, while Figs. 4.20 and 4.21, each of which consists of eight distinct figures, show melting furnaces and blowing systems and, respectively, equipment and installations for artillery forms, described in the same edition. All the technological topics covered in the ten books of Pyrotechnia are described with a commendable exhibition expertise, deriving from the experience accumulated in a life of work in the field. Moreover, with the same exhibition expertise, Biringuccio also illustrates the chip machining criteria, such us turning the bells and reaming the guns, also focusing on the necessary tools and calibration and checking instrumentation of the latter (Fig. 4.22). Obviously, Biringuccio is anchored to his time, so it would be futile to try, in his Pyrotechnia, the scientific bases of metallurgical and technological-productive processes in the sense attributed to them today. The other great metallurgist, Georgius Agricola, was also a versatile intellectual, so much so that, as a student of the University of Bologna (then the best in Europe), where he gained a doctorate in medicine; he cultivated also philosophical, theological and philological interests, to so much so that he could write and teach Greek, Latin, grammar and linguistics. In his job as a doctor in the famous mining center of Joachimsthal, in Germany, where Agricola took care of the miner’s diseases and the causes that generated them, he undertook the study of the processing cycles and mining technologies of his time. This interest allows to Agricola to develop a series of works in which the main minerals of that mining center were described, for the first time, in terms of physical properties that can be observed, rather than in terms of supposed philosophical prerogatives. During his lifetime, Agricola published several treatises, each having as a subject the arguments of the various disciplines of which he was interested. None of these treatises, however, reached the fame of his latest work, entitled De Re Metallica, Libri XII, published posthumously in 1556, after his death. This treatise, a sort of Summa of technological knowledge of that time, is to be considered a milestone not only in geology, mineralogy, and mining techniques, but also in metallurgy. From this point of view, De Re Metallica, Libri XII, of Agricola goes well with Pyrotechnia of Biringuccio, integrating it about a profitable start of application of the experimental method to metallurgical technologies that interest us here [3]. Figures 4.23, 4.24, 4.25 and 4.26 show four of the numerous woodcuts that Agricola uses to provide annotated diagrams that enrich his textbook and illustrate equipment and processes described therein. These figures have been chosen here to illustrate the various types of gears then used. Once again, the author writer wants to draw attention to the very extensive use of the (lantern-pinion/pin-teeth-wheel) pair as a

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Fig. 4.19 Title page of the 4th edition of Pyrotechnia, published in Venice in 1558

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Fig. 4.20 Melting furnace and blowing systems

gear to transmit motion between orthogonal axes. It was a direct consequence of the difficulty encountered to build efficient and functional bevel gears, so much so that the aforementioned very ancient gear pair (Fig. 4.27 shows a lantern pinion found in Saalburg, a Roman fort located on the main ridge of the Taunus, in Germany, and dating back to 265 A.D.), even after the massive replacement of the ferrous alloys to wood, it was still used until the turn of the 20th century.

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Fig. 4.21 Equipment and installations for artillery forms

Leonardo da Vinci (1452–1519) could not have the technological knowledge, although limited, acquired mainly thanks to the contributions of Biringuccio and Agricola. Yet no more than this giant of human thought, which was many things together, honored and valorized the technique, revealing engineer in the broadest and deepest meaning of the term. It is true that Leonardo cannot be considered a modern scientist according to the meaning that we now give to this term. It is equally true that the condition of homo sanza lettere (i.e., man without letters, as he called himself repeatedly, in his notes, but to be understood only in the sense that he did not know either Greek or Latin), which prevented him from direct access to the translations of the Hellenistic works and other writings available at his time, gave

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Fig. 4.22 Deep-hole boring of guns and tools used

him a strong charge of originality. In fact, he was able to make up to this limit, making use of the experience and reason to solve various problems that overlapped in his fertile mind. For this reason, Leonardo is to be considered as the founder of the modern experimental method and, therefore, from this point of view, he is undoubtedly one of the creators of modern science. However, there is another reason to consider Leonardo

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Fig. 4.23 Cylindrical lantern gearing of a Roman derivative tympanum, used as lifting and dewatering device of mines

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Fig. 4.24 Four orthogonal lantern gear pairs of an ancient derivation water wheel

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Fig. 4.25 An orthogonal lantern gear pair driving a material lifting drum

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Fig. 4.26 Gear train consisting of two cylindrical lantern gear pairs, driving a water lifting device, and gear wheels and other mechanical elements undergoing maintenance

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Fig. 4.27 Lantern pinion found in Saalburg in Germany and dating back to 265 A.D.

belonging not only to the Renaissance, which is still to be considered as a prescientific age, but also to the current scientific age. In fact, the discovery in 1965 of Madrid Codex has revealed a Leonardo previously unknown, who is interested in the theory of machines and machine design, in addition to clever inventions. There is an ultimate reason, which makes Leonardo a scientist, and is perhaps the most important. In fact, according to tests made by him on sliding friction and rolling friction in order to reduce the passive resistances arising between machine parts in relative motion, even in the presence of lubricant, he first formulated the fundamental laws of friction, for which he is also to be considered the father of tribology. Furthermore, he then widened the discussion to the laws of motion of fluids inside pipes and ducts, and sensed the distinction between the wall friction and the internal friction between liquid and liquid, this last able to trigger viscous perturbations. White [63] is therefore not entirely wrong when he defines Leonardo as the first scientist of the modern age. Leonardo had a critical and universal genius, that led him to try his hand in various fields of knowledge (geology, cosmography, botany, topography, cartography, mechanical engineering, architecture, civil engineering, military engineering, perspective, ballistics, hydraulics, aerostatics, magnetism, optics and other branches of physics, medical arts, anatomy and study of the human body, etc.). He was, however, never satisfied with only one solution, and kept a thousand times to ask themselves the same problem, modifying it, complicating it, and answering objections of imaginary contradictory until the mental exhaustion. Since Leonardo had the priceless gift from nature to design and synthesize everything he saw, he never ceased to wonder about the reason, because he had an insatiable curiosity. This is not a Renaissance characteristic, but it is typical of a scientific age. He pinned everything in his notebooks, which presumably should have constituted the basis and documentation kit to be used later, to be translated into treatises after their reprocessing. Nevertheless, he never had the time to plan this work, since he was engaged relentlessly in the frantic search for new things to be made and tested. We

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posterity therefore not have treatises of Leonardo, who however left us a wealth of codices, manuscripts and drawings, some of which are perhaps yet to be discovered and deciphered. The various codices, manuscripts and drawings now known show that Leonardo drew and designed a number of machines, many of which include various types of gears, and several mechanisms. Limiting ourselves to only mention a few, they include (see also [42, 52, 64]): • Lifting equipment, such as winches, cranes and mechanical jacks. Figures 4.28, 4.29 and 4.30 show different gear drives for these types of lifting equipment. • Buckets for underwater works, fans and helicoidal impellers. • Excavators and dredges for port works and channels. • Printing presses with automatic device for introducing the sheet. • Roasting-jacks operated by weights or by hot air flow. • Clockwork mechanisms for automata that ring the hours, verge clock escapements and other escapement mechanisms for clocks. Figure 4.31 shows one of Leonardo’s most beautiful mechanical drawings, which fascinates with its three-dimensional effects deriving from a clever use of chiaroscuro that emphasizes contrasts. It represents the well-known “molla tenperata” (compensated or controlled spring), i.e. a spiral device for clocks that ensures a constant spring pre-load, compensating for the decreasing force due to the driving spring unwinding, and thus transmitting a constant energy to the “roca” (conical lantern-pinion) that rises rotating along the toothed spiral. • Mechanisms for transforming a continuous rotation motion into an intermittent motion. Figure 4.32 shows a cylindrical worm gear driving a slider crank mechanism. • Measuring instruments, such as odometers, pedometers, compasses with gimbal inclinometers and scales. • Friction wheels, flywheels and cam actuated levers. • A speed gear consisting of a lantern-pinion that can be meshed with three spur gear wheels of different diameters. • A small car driven by pedal or spring system, a triumphal chariot with transmission system to all four wheels and a self-propelled chariot driven by a spring system. • Machines for making screws and a wire drawing machine. • Threading lathes, with continuous rotational motion or reciprocating motion, and a lathe operated by pedals, and flywheel to ensure the rotational motion continuity. • Machines for making files and rasps, coupled with a lathe-drilling system. • Rolling mills, water and wind mills, water wheels and hydraulic pumps. • A rear axle with differential. • Air and steam pulsometers. • Yarn twisting machines. • A helical motion propulsion system, that is the famous “vite aerea” (aerial screw), which is the ancestor of the helicopter. • Parachute models. • Screw devices to pluck up the window grills.

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Fig. 4.28 Exploded view of a gear drive for a weight lifting machine: Codex Atlanticus, Ambrosian Library in Milan, c. 30v

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Fig. 4.29 Various gear drives for a water lifting devices: Codex Atlanticus, Ambrosian Library in Milan, c. 1069r

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Fig. 4.30 Cylindrical worm gear driving a screw jack with axial ball bearing: Codex Madrid I, Folio 26r

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Fig. 4.31 “Molla tenperata” (compensated or controlled spring) with “roca” (conical lanternpinion): Codex Madrid I, Folio 45r

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Fig. 4.32 Cylindrical worm gear driving a slider crank mechanism: Codex Madrid I, Folio 28v

• War machines such as multiple catapults, ballistae, small and large caliber mortars, guns, mortars with burst of grenades, crossbows, muskets, tanks, testudos having steering and driving all four wheels, scythed chariots pulled by horses, etc. Figures 4.33 and 4.34 show the toroidal worm gear pair of a pre-tensioning device of a giant crossbow and, respectively, the gearing mechanism interlocked to two beautiful, but horrible, scythed chariots that compared with those designed by Mariano Taccola, make the latter look little more than toys for children. • Automatic ignition devices for guns. • Other machine tools, such as drilling machines, perforating machine for pipes, punching presses, lapping machines, sawing machines, cutting machines with sliding saddle etc. • Gimbal mechanisms, double helix reversing mechanisms, intermittent mechanisms, ratchet and pawl mechanisms, single and double slider-crank mechanisms. It should be noted that the machine elements and mechanisms conceived by Leonardo are particularly important for the mechanical engineering in general, and among them, a prominent place is the one of the gears. Leonardo uses a wide variety of gears, such as cylindrical spur and helical gears, bevel gears, pinion-rack gears (Fig. 4.35), worm gears, sprocket wheels, ratchet gears, pin-teeth reversing mechanisms (Fig. 4.36), double helix reversing mechanisms (Fig. 4.37), stepping gear devices, transmission gear trains, planetary gear trains (Fig. 4.38), crawler track transmissions, speed gearboxes. Other machine elements and mechanisms designed by Leonardo include various types of chain, bearing and pushing pins, band brakes, spherical joints, friction wheels, cams, couplings (see [28, 40, 43]).

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Fig. 4.33 Toroidal worm gear pair of a pre-tensioning device of a giant crossbow, Codex Atlanticus, Ambrosian Library in Milan, 149r-b

Leonardo uses not only the types of gears known previously, but he also offers new types of gears. In fact, the following types can be considered as a novelty: • The bevel gear pair, where the bevel gear wheel is a crown wheel, described in Codex Atlanticus, Folio 397r, and Codex Madrid I, Folio 96r (Fig. 4.39). • The worm gear pair, where the worm has toroidal shape (i.e., the globoidal worm), described in Codex Atlanticus, Folio 10r-a and Folio 372r-b, and Codex Madrid I, Folio 17v (Fig. 4.40). • Cylindrical helical gears, described in Windsor’s fragment no. 12722. • A double worm gear reversing mechanism, described in Codex Madrid I, Folio 15r (Fig. 4.37).

Fig. 4.34 Two terrible scythed chariots and related rotation gearing mechanisms of the scythes: Royal Library in Turin, n. 15583r

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Fig. 4.35 Pinion-rack gear pair driven by two cylindrical worm gears: Codex Madrid I, Folio 35v

Fig. 4.36 Pin-teeth reversing mechanism: Codex Madrid I, Folio 17r

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Fig. 4.37 Double helix reversing mechanism: Codex Madrid I, Folio 15r

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Fig. 4.38 Planetary gear train: Codex Madrid I, Folio 13v

Furthermore, as Uccelli [61] and Ceccarelli [11] highlight, Leonardo knows the kinematics of the helicoidal motion, rather he is the first to describe it (Arundel Codex, in folio 140v). In addition, Leonardo clearly shows to have in mind the importance of the tooth profile shape in gear transmissions. This is confirmed by a drawing of a cylindrical spur gear of the Codex Madrid I, in folio 5r, which shows the curvilinear shape of the tooth profile (Fig. 4.41). It should also be noted the great versatility of Leonardo, who often uses the gears in a new and innovative way to solve high engineering content problems, not only for civil applications, but also and especially for military applications. Just to mention here, as an example, the lathe that he called “oval”, which admirably exploited the movement duality, which is a well-known principle of theoretical kinematics. In relation to military applications, it is to remember that every great architect or artist of that time could not only think about designing works of peace, but also had to tray a fortiori the art of war, at that time considered as a real Ars Magna.

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Fig. 4.39 Bevel gear pair: Codex Madrid I, Folio 96r

Among the great Renaissance architect-engineers who dealt with military art, Giovanni Battista della Valle (about 1470–1550) is certainly to be remembered. Little do we know of this remarkable figure of the Renaissance, even if historians emphasize his doctrine and cognition of the letters. However, we know with certainty that he was a captain and that perhaps, in 1516, he was in the service of Francesco Maria Della Rovere, Duke of Urbino, when the fortress of San Leo besieged for three months by the Pope’s troops. Also, in memory of this experience, he gave us a very important work which, in the opinion of many historians, is worthy of appearing well alongside the most famous treatises of the Renaissance (see [10, 27]). It seems that the editio princeps of this work is the one that appeared in Naples in 1521, with the title “Vallo. Libro continente appertinentie à Capitanij, retenere et fortificare una Città con bastioni, con nuovi artificij de fuoco aggionti, come nella Tabella appare, et de diverse sorte polvere, et de espugnare una Città con ponti, scale, argani, trombe, trenciere, artigliarie, cave, dare avisamenti senza messo allo amico, fare ordinanze, battaglioni, et ponti de disfida con lo pingere, opera molto utile con la esperientia del arte militare” (in brief form: Vallo. Book with necessary information for Military Captains: Holding and Fortifying a City). For the wide-ranging discussion of all aspects of the military art of that time, the work had an exceptional fortune, as evidenced by the numerous editions that followed one another throughout the sixteenth century and again in the seventeenth century, with translation into French, German and Spanish. By focusing attention on the topic of interest here, it should be remembered that the work provides detailed technical descriptions on the construction of:

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Fig. 4.40 Toroidal worm gear: Codex Madrid I, Folio 17v

Fig. 4.41 Parallel cylindrical spur gear with curved profile teeth: Codex Madrid I, Folio 5r

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• orlogi (i.e. orologi, clocks), which allowed the changing of the guard at regular intervals; • various types of war machines to attach besieged cities, to build emergency bridges, to extract water from the subsoil, to open passes across the mountains, etc. All the aforementioned devices and machines are characterized by geared mechanisms, which use technologies that are not dissimilar to those described in this and in the previous sections, without substantial changes compared to the past. The period immediately following Leonardo’s death was influenced by Agostino Ramelli (about 1531–1600) and Jacques Besson (about 1540–1576). The works of both these two artist-engineers are characterized by beautiful machine lithographs in isometric views, which highlight the particularities of detail much more than do the sketches of Leonardo’s manuscripts and the works of his predecessors. In the works of both, it is possible to find reminiscences and even almost copying drawings by Leonardo and also by Francesco di Giorgio Martini. However, it is not possible to define the authorship of the common drawings only on the basis of their simple comparison. In 1588 Ramelli published Le diverse et artificiose macchine del Capitano Agostino Ramelli (i.e., The various and ingenious machines of Captain Agostino Ramelli). This book contains a large variety of water-rising machines and some military machines, all of which highlight a great use of linkages and gearing mechanisms, where worm gear and lantern gear pairs are widely used, as Figs. 4.42 and 4.43 show. It is worth noting the introduction of a rotary pump for the first time, a variation of which would become one of the sources for the Wankel rotary engine, invented by Felix Heinrich Wankel (1902–1988). He also invented a book-wheel, a vertically revolving bookstand containing epicycloid gearing with two levels of planetary gears to maintain proper orientation of the books (Fig. 4.44). Some critics have highlighted the fact that many of the machines designed by Ramelli are too complicated, and that the friction between the gears as well as friction in the bearings could have implied the impossibility of their operation. However, even if some of his machines, much more than being intended for effective use, should be considered the result of his inventive imagination, it is still appreciated Ramelli’s effort to explore the design space, as shown by the drawings of toroidal water pumps [49]. In 1571 or 1572, Besson published his Theatrum Instrumentorum (i.e., in French, Théâtre des Instruments), which differed greatly from previous works on engineering and technology, such as those previously described by Valturio, Biringuccio and Agricola. In fact, unlike the works of these last authors, who limited themselves to describing the new inventions or inventions of the past, without much detail, Besson’s work was a collection of his own inventions, with detailed illustrations of each engraved-on copper plates by Jacques Andruet du Cerceau to his specifications. The Besson’s work, also in its posthumous edition of 1578, contains drawing instruments, log-cutting machines, mills, presses (Fig. 4.45), pumps, lathes, pile drivers (Fig. 4.46), lifting devices and construction equipment. Besson makes extensive use of worm gears and, for one of his pile drivers, uses the double helix reversing mechanism designed by Leonardo. Some drawings suggest important improvements to

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Fig. 4.42 Lantern gearing for wheat mill driven by a horse

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Fig. 4.43 Lantern and worm gearing for two heavy lifting and dragging machines, both driven by cranks

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Fig. 4.44 Planetary gear trains for revolving bookstand for readers who are ill or tormented by gout

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Fig. 4.45 Worm gears in a three-screw press

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Fig. 4.46 Double helix reversing mechanism according to Leonardo, used by Besson for one of his pile drivers

lathes and waterwheels, as well as to a machine for cutting logs into lumber, also by virtue of a more rational use of gearing [7]. As we saw above, the military art made extensive use of geared mechanisms. As further evidence of this factual circumstance, it is appropriate to also mention the Lorraine Jean Errard de Bar-le-Duc (1554–1610), a mathematician and military engineer who introduced the concepts of Italian fortifications to France [31]. In 1584 and 1600 he published his two most important works, respectively entitled Premier livre des instruments mathématiques and La fortification démonstrée et réduit en art. As for military art, he is indebted to the work of his predecessor, Captain Giovanni Battista della Valle, as well as to a similar work almost simultaneously written by Albrect Dürer (1471–1528), just one year younger than the Italian captain. Figures 4.47 and 4.48, taken from Errard’s works, respectively show: a possible reconstruction of the counterweighted winch designed by Archimedes to operate the iron hand sinking a Roman ship during the siege of Syracuse, which is moved by a hand powered worm gear pair; a wind powered lantern gearing, which transfers energy to a grain mill.

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Fig. 4.47 Counterweighted winch moved by a hand powered worm gear pair

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Fig. 4.48 Grain mill moved by a wind powered lantern gearing

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In all the aforementioned works, from Captain Giovanni Battista della Valle to Errard, particular hints of originality are not detectable. They clearly have a common source, the Valturio’s De Re Militari. The fact that subsequent authors drew heavily on the works of previous authors, however, without mentioning them, is not surprising. At that time, the use of ideas from other authors in their work was a completely usual occurrence. The considerations to be made regarding the gears are even more dramatic, because the chain of previous inheritance must be extended even further back to Vitruvius. In fact, in the seventeen centuries between Vitruvius and Errand, gear technology and gear knowledge remained substantially crystallized over time, without advances worthy of being appreciated and mentioned. This is not the case to mention the many other scholars of the Leonardo’s era, which would still extremely be deserving of attention. In this regard, we refer the reader to the massive and meticulous work of Manna [34], in three volumes. The star of Leonardo, and his unparalleled artistic and technical-scientific stature are so bright and high to obscure and make any other name comparatively a dwarf. He lived in a great historical epoch that, with the discovery of perspective associated with the ability to carefully observe nature, made even scientific the art. Nobody ever, like Leonardo, was able to meet many needs in a manner so accomplished and admirable, distinguishing, as he said, “the case solvable with only the light of reason that, if properly understood, does need the experience” and “the case which requires that the experience is first used, and then the reason, which serves to demonstrate the reason for which the experience is forced to act in that way”. It is undeniable that this way of thinking of Leonardo is the foundation upon which science is based. Nevertheless, some still says that this giant of the human genius lacked the scientific method, in the modern meaning of the term. Even if that was the case, we believe that Leonardo, with his intelligence almost above human, has bypassed admirably this supposed limit, so much so that he was able to promote new scientific theories, as the laws on friction, which must be ascribed to him, the intuitions on the impact theory between elastic bodies and those on the mechanics of solids and fluids, the foresights on the principles of dynamics, and finally to devise new and innovative mechanisms. Also Moon [39] acknowledges this latest Leonardo capacity, stating the reasons on the basis of a thorough critical analysis which compares the mechanisms designed by the most ingenious and eldest son of the Renaissance, and the mechanisms of Reuleaux, which is the father of theory of mechanisms. In a balcony of the Castle of Certaldo, the writing author had the good fortune to live a great emotion, sitting where Leonardo was sitting, and where on the dusk of an unspecified day he fixed the watchful gaze to his native hills, and subsequently translated his suggestions in the magnificent panorama that is the background of the world’s most famous painting, the Monna Lisa-La Gioconda. This same author like to imagine Leonardo, standing upright, with one foot on the edge of the second pre-scientific age, and the other one on the edge of the second scientific age, that is the age of science in which we are leaving. There is no doubt that, for his sublime ability to drill, with his intelligence, the thick walls of his historical period, and look to the future with daring and unattainable flights, which are true Pindaric flights, Leonardo is certainly to be considered a thinker

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of our age. Like all very great talents who have worked in different historical periods, even Leonardo has gone beyond the boundaries of his age, and he is recognized universal genius of all times and all spaces. In any case, at least, he is to be considered as the trade union between two historical periods, where in the first the old man struggling to become extinct, while in the second the new man lingers to be born. It is impossible here to mention all the countless followers of Leonardo, who were able to design machines, also very clever, that used gears or gear trains. These machines were destined mainly to the war employment, but interesting machines were designed, useful for everyday life or as mirabilia (wonders), to delight those who owned them or saw them work. An exception, however, must be made for a brief mention to Vittorio Zonca of Padua (1568–1602), as he, with his work published posthumously in 1607 and intitled “Nuovo Teatro di Machine et Edificii”, i.e. “New Theater of Machines and Buildings”, offers us perhaps the most reliable industrial panorama of the manufacturing machines of his time. Furthermore, he is the first author who focuses his attention on anti-friction metals, when he finds that every metal is consumed more than bronze when it is rubbed against steel. Zonca is finally associated with Ramelli, due to the widespread use of worm gears. Figure 4.49 shows one of these gear pair applied to a winch driven by a crank [65].

Fig. 4.49 Worm gear pair applied to a winch driven by a crank

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Finally, here we must necessarily make a nod to the development of portable clocks, which were part of this last category of machines, and which developed thanks to the introduction, in the 15th century, of the driving system consisting of a spiral spring and related escapement mechanism, designed to offset the decreasing reaction force of the spring. It is noteworthy that the development of the gears received a strong boost from the design of portable clocks, as well as from new technologies designed to produce them. The first driving mechanism for portable clocks was the famous Pyramid of Leonardo (Fig. 4.50), consisting of a helically grooved drum (it was a kind of spiral bevel gear wheel, ante litteram), thanks to which the connecting rope exerted a greater pulling force on the pyramid shaft, while the spring tended to unwind and its preload decreased. This type of driving system led the boom in clock making, and marked the beginning of Feinmechanik, which later became, for a few centuries, almost exclusively the preserve of the Swiss, whose founder may be considered the horologiorum mathematicus Jobst Bürgi, Latinized into Iustus Byrgius (1552–1632).

Fig. 4.50 Pyramid of Leonardo used as driving mechanism for portable clocks

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References 1. Addomine M (2009) Richard of Wallingford ed il suo Tractatus Horologii Astronomici. La Voce di Hora 27:5–20 2. Agius D, Hitchcock R (eds) (1994) The Arab Influence in Medieval Europe: Folia Scholastica Mediterranea. Ithaca Press 3. Agricola G (1950) De Re Metallica, translated from the first Latin edition of 1556 by Herbert Clark Hoover and Lou Henry Hoover. Dover Publication Inc., New York 4. Al-Jazari I-R (1973) The book of knowledge of ingenious mechanical devices (translated from Arabic and annotated by Hill DR). Kluwer Academic Publishers, Dordrecht 5. Al-Jazari I-R (1979) A compendium on the theory and practice of the mechanical arts (Arabic text edited by al-Hassan AY et al). Institute of the History of Arabic Science, Aleppo 6. Barnes CF Jr (2009) The portfolio of Villard de Honnecourt (Paris, Bibliothèque National de France, MS Fr 19093): a new critical edition and color facsimile. Farnham Burlington, Ashgate Publishing Ltd., Aldershot, United Kingdom 7. Besson J (2001) Il theatrum instrumentorum et machinarum di J. Besson (translated in Italian by M. Sonnino). Edizioni dell’Elefante, Roma 8. Biringuccio V (1990) De la Pirotechnia. The Pirotechnia of Vannoccio Biringuccio, the Classic Sixteenth-Century Treatise on Metals and Metallurgy (translated by Cyril Stanley Smith and Martha Teach Gnudi). Dover Publications Inc., New York 9. Boyer CB (1991) A history of mathematics, 2nd edn. Wiley, New York 10. Brugh P (2019) Gunpowder, masculinity and warfare in German texts 1400–1700. University of Rochester Press, Rochester 11. Ceccarelli M (2000) Preliminary study to screw theory in XVIIIth century. In: Proceedings of a symposium commemorating the legacy, works, and life of Sir Robert Stawell Ball upon the 100th Anniversary of A Treatise on the Theory of Screws, University of Cambridge, Trinity College, 9–11 July 12. Clark JG (2004) A Monastic Renaissance at St. Albans: Thomas Walsingham and His Circle c. 1350–1440, Oxford Historical Monographs. Clarendon Press, Oxford 13. Crosher WP (2014) A gear chronology, significant events and dates affecting gear development. Book Publishers Xlibris LLC, Bloomington, Indiana, USA 14. Cuneo PF (ed) (2002) Artful armies, beautiful battles: art and warfare in early modern Europe. Brill, Leiden 15. Alighieri Dante (2012) The Divine Comedy/La Divina Commedia - Parallel Italian/English Translation. Benediction Classics, UK 16. De Young G (2016) The Latin Translation of Euclid’s Elements attributed to Adelard of Bath: Relation to the Arabic transmission of al-Hajj¯aj. In: Zack M, Landry E (eds) Research in history and philosophy of mathematics. Proceedings of the Canadian society for history and philosophy of mathematics 2015 annual meeting in Washington, Springer International Publishing Switzerland, Cham 17. Di Giorgio Martini F (1870) Trattato di Architettura Civile e Militare. Tipografia Chirio e Mina, Torino 18. Di Giorgio Martini F (1967) Trattati di Architettura Ingegneria e Arte Militare: Tomo I, Architettura Ingegneria e Arte Militare; Tomo II, Architettura Civile e Militare. Il Polifilo Editore, Milano 19. Dohrn-van Rossum G (1996) History of the hour: clocks and modern temporal orders (translated by Thomas Dunlop). The University of Chicago Press, Chicago 20. Dondi dall’Orologio G (1988) Astrarium. In: Poulle E (ed). CISST 21. Dondi dall’Orologio G (2003) Tractatus Astrarii, with translation in Italian in front of A. Bullo, 2 vols. Editrice Nuova Scintilla, Chioggia 22. Galilei G (1632) Dialogo sopra i due massimi sistemi del mondo tolemaico e copernicano. Battista Landini, Fiorenza

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23. Hall AR (1976) Guido’s Texaurus-1335, Humana Civilitas: Sources and studies relating to the middle ages and the renaissance, vol 1, On pre-modern technology and science, a volume of studies in Honor of Lynn White Jr, Hall BS, West DC (eds). Undena Publications, Malibu, CA, pp 10–52 24. Hall BS (1982) Guido da Vigevano’s Texaurus Regis Francie, 1335. In: William E (ed) Studies on Medieval Fachliteratur, Brussels, pp 33–44 25. Hill DR (1984) A history of engineering in classical and medieval times. Croom Helm & La Salle, London 26. Hill DR (1994) Islamic science and engineering (the new Edinburgh Islamic surveys). Edinburgh University Press, Edinburgh 27. Innocenzi P (2019) The innovators behind leonardo: the true story of the scientific and technological renaissance. Springer International Publishing AG, Cham 28. Isaacson W (2017) Leonardo da Vinci. Mondadori Libri SpA, Milano 29. Knorr WR (1993) Arithêtikê stoicheìoôsis: On Diophantus and Hero of Alexandria. Historia Mathematica 20:180–192 30. La Rocca C (2000) A man for all seasons: Pacificus of Verona and the creation of a local Carolingian past. In: Hen Y, Innes M (eds) The uses of the Past in the Early Middle Ages. Cambridge University Press, Cambridge 31. Lallemend M, Boinette A (1884) Jean Errard de Bar-le-Duc, premier ingesnieur des tres chrestien roy de France et de Navarre Henri IV: sa vie, she’s oeuvres, sa fortification. Thorin, E. Libraire and Dumoulin, J.B. Libraire, Paris, VI-332 p 32. Lang UM (1997) Nicetas Choniates, a Neglected Witness to the Greek Text of John Philoponus, Arbiter. J Theol Stud 48(2):540–548 33. Lindberg DC (1980) Sciences in the Middle Ages. University of Chicago Press, Chicago 34. Manna F (1998) Uomini e Macchine, vol I, vol II, prima parte, and vol II, seconda parte, self-published in Casalnuovo di Napoli, and printed by Offselit, Cava dei Tirreni 35. Massera AF (1958) Roberto Valturio, omnium scientiarum doctor et monarcha (1405–1475). Lega, Rimini 36. Mayr O (1970) On origins of feedback control. MIT Press, Boston 37. McKenzie J (2007) The architecture of Alexandria and Egypt c. 300 BC to AD 700. Yale University Press, New Haven 38. Merrill EM (2013) The Trattato as Textbook: Francesco di Giorgio’s Vision for the Renaissance Architect. Arch Hist Open Access J Eur Arch Hist Netw 1(1), Art. 20 39. Moon FC (2007) The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer, Dordrecht 40. Morello G (ed) (2000) Le Macchine del Rinascimento. Eurografica snc, Roma 41. North JD (1976) Richard of Wallingford: an edition of his writings with introductions, vol 1 and vol 2 (English translation and Commentary by John David North). Clarendon Press, Oxford 42. Pedretti C (1988) Leonardo architetto, Re-ed. Electa Spa, Milano 43. Pedretti C (1999) Leonardo: the machines. Giunti Editore SpA, Florence-Milan 44. von Prager FD (1974) Introduction in: Philo of Byzantium, Pneumatica. Ludwig Reichert Verlag, Wiesbaden 45. von Prager FD, Scaglia G (1971) Mariano Taccola and His Book De Ingeneis. Massachusetts Institute of Technology Press, Cambridge 46. Préaux C (1966) Stagnation de la pensée scientifique á l’époque hellénistique. In: Essays in Honor of C. Bradford Welles. American Society of Papyrologists, New Haven 47. Price de Solla DJ (1975) Science since Babylon. Yale University Press, New Haven 48. Price de Solla DJ (1974) Gears from the Greeks. The Antikythera mechanism—a calendar computer from ca. 80 B.C. Trans Am Phil Soc 64, Part 7 (and reprint in 1975, Science History Publications, New York) 49. Ramelli A (1976) The various and ingenious machines of Agostino Ramelli. A classic sixteenthcentury illustrated treatise on technology (translated by Martha Teach Gnudi, with annotation and glossary by Ferguson ES). Dover, New York

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50. Rashed R (1993) Géométrie et dioptrique au Xe siècle. Ibn Sahl, Al-Qui et Ibn Al-Haytham. Les Belles Lettres, Paris 51. Reti L (1963) Francesco di Giorgio Martini’s Treatise on Engineering and its Plagiarists. Technol Cult 4(3):287–298 52. Richter JP (1970) The notebooks of Leonardo da Vinci, Compilated and Edited by Jean Paul Richter, London, reprint of 1883 edition. Dover, New York 53. Riemen R (2006) Prologo. In: Steiner G (ed) Una certa idea di Europa. Garzanti, Milano 54. Russo L (2015) La rivoluzione dimenticata. Il pensiero scientifico greco e la scienza moderna, 9th edn. Giangiacomo Feltrinelli Editore, Milano 55. Scaglia G (1971) Mariano Taccola, De Machinis: The Engineering Treatise of 1449, vol 1, vol 2. Dr. Ludwig Reichert Verlag, Wiesbaden 56. Scheller RW (1995) Exemplum: Model-book drawings and the practice of artistic transmission in the Middle Ages. Amsterdam University Press, Amsterdam 57. Stoimenov M, Popkonstantinovi´c B, Miladinovi´c L, Petroci´c D (2012) Evolution of clock escapement mechanism. FME Trans 40(1):17–23 58. Tannery PL (1893) Diophanti Alessandrini Opera Omnia: cum Graecis commentariis, vol 1. In aedibus B.G. Teubneri, Lipsiae 59. Tannery PL (1895) Diophanti Alessandrini Opera Omnia: cum Graecis commentariis, vol 2. In aedibus B.G. Teubneri, Lipsiae 60. Tribbetts GR (1975) Review: Donald R. Hill, The book of knowledge of ingenious mechanical devices. Bull Sch Orient Afr Stud Univ Lond 38(1):151–153 61. Uccelli A (1940) I Libri di Meccanica di Leonardo Da Vinci (Chap. XI). In: De Moto. Ulrico Hoepli, Milano 62. Valturio R (2018) De re militari, Editio MMXVIII. Guaraldi M, Demarchi P (eds) Guaraldi Editore, Rimini 63. White M (2000) Leonardo The First Scientist. Little Brown & Company, London 64. Zöllner F, Nathan I (2003) Leonardo da Vinci: tutti i dipinti e disegni. Taschen GmbH, Köln 65. Zonca V (1607) Novo teatro di machine et edificii. Pietro Bertelli, Padova

Chapter 5

The Second Scientific Age: From Galileo to Today

Abstract In this chapter, the initially slow and then increasingly rapid awakening of the second scientific age, which goes from Galileo to the present day, is described. Attention is first focused on the first two centuries of scientific activity, from Galileo to Lagrange, because in this time the mechanics establishes itself as queen of the sciences. However, it is emphasized that, in these two centuries, stagnation continues to prevail in the field of gears, which continue to look to the past again and do not immediately benefit from new scientific achievements. Subsequently, attention is focused on the slow detachment from pure empiricism and the equally slow penetration of science into the field of gears. The main scientific contributions are described, which initially concern only the geometric–kinematic aspects. It is also highlighted that, gradually, these aspects begin to be seen from the perspective of the new cutting processes that are gradually conceived and developed. Still later, the attention is focused on the mechanics of solids and on material strength theories that, even in this case, first gradually and then with forced charge, enter the field of gears. In this regard, this brief history only describes the contributions of the pioneers, who have begun new lines of research on gears, starting from Lewis to the present day. Therefore, only the milestones on the state of knowledge concerning the load carrying capacity of the gears in its most diverse aspects are described. To give a concrete demonstration of the fact that gears are ancient science in continuous updating, in the final part of this chapter, brief considerations are presented on the historical aspects of three particular types of damage, which are related to the load carrying capacity of the gears in terms of abrasive wear, micropitting and tooth flank breakage. The chapter closes with epigraphic news on the main monographs and textbooks on gears from 1900 to the present, with a due exception to Olivier’s 1842 treatise, which is probably the first monograph that was written and published on the gears.

© Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_5

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5.1 The First Two Centuries, from Galileo to Lagrange: Mechanics Establishes Itself as the Queen of the Sciences, but the Gears Still Look to the Past At the end of the 16th century, the study of the Hellenistic science was intensified. This study had been started long before (see previous chapter). Even Leonardo felt the need to become aware of the Hellenistic technology and, in this regard, as we learn from his record in the Codex L of the Institute de France, he studied an Archimedes’ work, now lost, from which he drew the inspiration to draw a steam cannon, called Architronito, attributing the invention to Archimedes. It is noteworthy that we have reason to believe that not all the Hellenistic works, which were available in the Renaissance, are coming to us, and that not all those that have come down to us are all known. Fortunately, every so often, we learn with satisfaction that some surviving manuscript was found in some palimpsest or in some remote ravine. Today, however, it is necessary to shed new light on the process that in the two centuries between the end of the sixteenth century and the end of the eighteenth century led to profound transformations in the study of nature, with the emergence of mechanistic philosophy, new mathematical methods of analysis, and the establishment of the experimental approach. There is no doubt that modern European science originated from Hellenistic science. Some scholars argue, however, that this interconnection is not so much due to the discovery of Hellenistic science, but rather to the fact that the applied components of the latter, namely mechanics, optics, acoustics, and astronomy as well as their technological processes, continued to be transmitted through the Middle Ages, without serious interruption (see, for example, [64]). Other scholars, such as the aforementioned Russo [321], argue instead that science had a drastic hiatus in the middle ages, as the scientific method, which is the soul of science to the point of identifying with it and that is typical of Hellenism, had been completely lost in the second half of the 2nd century B.C. Without entering into a problem that is too complex to unravel, in accordance with the concepts of science and scientific method, the author allows himself to disagree with the statements of Capechi and to marry the Russo’s thesis. All historians of science have agreed to start second scientific age with the Pisan Galileo Galilei (1564–1642). In fact, Galileo, after absorbing the treatises of Archimedes, on which he wrote several comments, was the first to groped to build new scientific theories, with the publication in 1638 of his treatise entitled Discorsi e Dimostrazioni Matematiche intorno à due Nuove Scienze attinenti alla Meccanica & i Movimenti Locali (i.e., Discourses and Mathematical Demonstrations Relating to Two New Sciences, concerning Mechanics and Local Movements), which for the sake of brevity we will call from now on Two New Sciences [146]. Figure 5.1 shows the title-page of this treatise, which constitutes the pillars of the scientific method, for which even Einstein called Galileo the father of modern science. However, the scientific contributions on mechanics, collected in the Le Meccaniche (i.e., Mechanics), which is one of the early Galileo treatises, are not to be forgotten.

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Fig. 5.1 Title-page of the Galileo treatise “Discorsi e Dimostrazioni Matematiche intorno à due Nuove Scienze…”

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As also pointed out by Drake [110], Galileo clearly and explicitly set itself to achieve a specific goal: to recover the Hellenistic science. In this, he followed the footsteps of his patron, Guidubaldo Bourbon Del Monte of Pesaro (1545–1607), in respect to which we are indebted for the first enunciation of a basic principle of theoretical mechanics, as it is the Principle of virtual work. Indeed, this principle appears correctly formulated in his Mechanicorum liber (i.e., Book of Mechanics), published in 1577 [97]. Guidubaldo del Monte also made a number of experiments on the dynamics of the graves, not entirely different from those made by Leonardo da Vinci, however without departing much from the Aristotelian dynamics. He also racked his brains to understand the principles of operation of Archimedes’ cochlea used as a water lifting mechanism, but without success. However, we should not be surprised by this, because other scholars of that time could not do better. Finally, despite their friendship, Guidubaldo criticized some of the Galileo’s discoveries, such as the one about the isochronism law of small oscillations, perhaps because he fundamentally did not understand them. Already in the second half of the 16th century an intense cultural activity was to animate the major Italian centers, and then the European centers, and a new working mode was stated, albeit gradually, with engineers and technicians who increasingly collaborated with mathematicians, in order to develop new scientific theories capable of facilitating the progress. It was finally abandoned the idea of a knowledge essentially contemplative. Instead, the idea was generated and progressed, which considered technology not as a by-product of scientific research, but as an equally noble activity, worthy of being included in the daily life of a fruitful and happy exchange of studies and observations. So, a new kind of scientist was outlined, who not only knew and possessed the bases of knowledge, but also knew how to do, getting his hands dirty in the laboratories to develop and build the products of knowledge. The recovery of the ancient Hellenistic science was long. However, the road to the success of scientific method was even longer, and in some cases problematic, so much so that the great Newton (1642–1727) not never definitively god rid of the Aristotelian concepts of natural philosophy, inherited by scholastic philosophy, as evidenced by the title of his famous work Philisophiae Naturalis Principia Mathematica (i.e., Mathematical Principles of Natural Philosophy), whose first edition was published in London in 1687 [272]. However, Galileo is the real point of discontinuity between the old way of thinking, closely related to the Middle Ages, and the new way of thinking, that begins with the Age of Reason. Galileo is a giant of rationality. He is to be considered the first great interpreter of this new way of thinking, and has the great merit of having initiated to release mechanics from the shackles of Aristotelian and scholastic philosophy. With his writings, and especially with the integrated final edition of Dialoghi delle Nuove Scienze (i.e., Dialogues of the New Science), edited by his son, Vincenzo Galilei, and published posthumously in 1718, Galileo offers the exact measure of his speculative sagacity and his inventive genius, in purely mechanical field. In this regard, just remember that Galileo, already in the introduction to the above-mentioned work, highlights “le utilità che si traggono dalla scienza meccanica e dai suoi instrumenti”

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(i.e., “the utilities that are drawn from mechanical science and its instruments”), then reiterating the primary importance of the theoretical investigations, outside of which a solution of mechanical problems often does not exist. Using very well his mechanical genius, Galileo was able to conceive and built several measuring instruments, through which he promoted the affirmation of scientific instruments as essential auxiliary means of scientific research, thus establishing an indissoluble union between theoretical and experimental worlds. Among the many instruments designed by Galileo, we must mention here the pendulum clock, which he designed as a variation of Pulsilogio (a pendulum instrument for measuring the pulse beat, made by the Venetian Santorio Santorio (1561–1636), according to an idea of Galileo, of which was a friend). This clock was characterized by a cylindrical spur gear train and by a new ingenious escapement mechanism, called “a doppia virgola ed a riposo” (i.e., “with double comma and at rest”). Figure 5.2 shows the Böttger reconstruction of the pendulum clock conceived by Galileo; it highlights the arrangement of the entire system as well as the details of its regulation and control mechanism. Both the pulsilogio and Galileo pendulum clock used pendular motion,

Fig. 5.2 Pendulum clock by Galileo according to the reconstruction of Böttger: a configuration of the entire system; b details of the regulation and control mechanism

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which was known for some time and of which there are clear references in Leonardo’s drawings as well as in the works of the aforementioned Besson (see Sect. 4.3). As we learn by Galileo himself, through the letter he sent to Lorenzo Rèal, governor of the Dutch East Indies, the structure of his “misuratore del tempo” (i.e., “time measuring instrument”), as Galileo called his pendulum clock, was designed in such a way as to be operated … “non del solo peso pendente da un filo, ma anche di un pendolo, che così diventa l’esattissimo compartitore, in minutissime particelle, del tempo medesimo” (i.e., “not only by a weight hanging from a wire, but also by a pendulum, so that it becomes the very exact splitter, into very small particles, of the same time”). The prototype of this pendulum clock was built thanks to the valuable contribution of his son, but was completed in 1649 after the death of Galileo. The Galileo’s invention was based on the discovery of the isochronism of small oscillations, made by Galileo himself, observing the oscillations of a lamp of the Cathedral of Pisa. There is historical evidence that Galileo had already started to do researches concerning pendulum movements in 1603 [61]. The pendulum clock of Galileo had superior characteristics compared to those of the first version of the Huygens pendulum clock, moreover developed later. This last pendulum, invented by Huyghens in 1656, was characterized by ten or more sets of gears; however, it ensured a higher accuracy than previous mechanical clocks. It also had the peculiarity of replacing the weight drive with a mechanical system consisting of a balance wheel and a spring; however, it kept the old defective escapement mechanism, typical of pre-Galilean clocks. We can deduce this fact by a communication of the same Huyghens at Paris Academy of Science, of which he was a member. Galileo’s priority concerning the pendulum clock invention is witnessed by Vincenzo Viviani (1622–1703), the last follower of Galileo, who, defending the master, accused Christiaan Huygens (1629–1695) of plagiarism. After having heard about Huygens’ invention, in his letter of August 1659 to Prince Leopoldo, Viviani explains in detail how Galileo had designed for the first time a pendulum regulated clock and how his son Vincenzo had tried to construct this orivuolo (clock) with the help of a locksmith. In the same latter, Viviani describes Vincenzo’s enthusiasm when he finally, in April 1649, touched the prototype of the clock, made in accordance with the conception that his father had already set to him in the presence of Viviani himself. Today, however, it is a proven fact that Huygens was not the first to conceive a pendulum regulated clock, also because, by his own admission, in his treatise Horologium oscillatorium: sive de motu pendulorum ad horologia aptato demonstrations geometricae (i.e., The pendulum clock or geometrical demonstrations concerning the motion of pendula as applied to clocks), published in 1673, he reveals that his invention was based on Galileo’s invention of the principle of isochronism, however expressing in the preface of the same textbook “gratissima testimonianza a favore di Galileo”, i.e. “very grateful testimony for Galileo” [189, 190]. Figure 5.3 shows the pendulum clock of Huygens as described in the aforementioned treatise, in which the claim was made that it was strictly isochronous, so that it constituted one of the first precision chronometers. This remarkable characteristic, which distinguished it from the previous version of the pendulum clock developed by Huygens himself, derives from the fact that it oscillated on a cycloid arc rather

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Fig. 5.3 Details of the Huygens’ pendulum clock from his treatise of 1673

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than on an arc of circle. To get to this solution, Huygens had preliminarily addressed and solved by geometrical methods the so-called tautochrone problem, that is to find the curve down which a mass will slide under the influence of gravity in the same amount of time. Anyway, the pendulum clocks were a big step forward in technology and manufacture of gears, and found wide use in public buildings. About the gears, however, we must recognize Galileo an even greater merit, albeit indirect. He is in fact the founder of the Mechanical Science, as his famous treatise Le Meccaniche (i.e., Mechanics) published in 1594 testifies [145]. In this treatise, various static problems were treated on the basis of the principle of virtual displacements. The problems of mechanics of materials are however included in the first of two dialogues of his already mentioned treatise Two New Sciences, where Galileo lays the foundations of the bending strength theory of the beams. In particular, Galileo gives a complete derivation of the parabolic shape of a cantilever beam of equal strength, the cross section of which is rectangular [341]. Only 260 years later Wilfred Lewis (see Sect. 5.3) would have had the happy idea of using this Galileo’s theory to lay the foundation of the theory concerning the bending load carrying capacity of the gears. About other Galileo deductions concerning beam bending theory, see Sect. 5.3. Many of the scholars and scientists of the Age of Reason deserve to be mentioned here, because with their contributions they determined the rebirth of science. However, this is a brief history of the gears, included in a monothematic treatise on the many aspects of their calculation, so we must focus our attention on the contributions regarding this topic. In this regard, we must remember Johannes von Kepler (1571–1630). The reason of this is not because he contributed to the affirmation of the heliocentric theory, rediscovered by Nicholas Copernicus (1473–1543), the author of De Rivolutionibus Orbium Coelestium (i.e., On the Revolutions of the Celestial Spheres), published in 1543 shortly before his death. The reason instead is that he was not just an abstract thinker, solely devoted to astronomy, but he was also a distinguished inventor of mechanical gearing devices. In fact, in addition to the telescope, his inventions include a planetarium, with fine gear trains, and a gear pump, with a strong flow rate [201]. He also developed, inspired by Hero, some hydraulic machines used as dewatering devices of mines. Anyway, before focusing our attention on the scholars and scientists whose contributions led to significant advances on the gear knowledge, we must dwell briefly on the three giants of the scientific revolution, which were René Descartes (1596– 1650), Gottfied Wilhem von Leibniz (1646–1716) and Isaac Newton. In fact, they gave solid mathematical and scientific basis to the mechanical knowledge, including those regarding the gears, and constituted a kind of holy Trimurti of science of the 17th century and the beginning of the 18th century. From Dioptrique (Dioptrics), which is included in the Discours de la méthode (for the complete title, see below), of 1637, and Principia Philosophiae (Principles of Philosophy), of 1644, we can deduce that Descartes aimed to establish, on solid, clear and evident bases, a science whose special laws governing individual phenomena can be deduced with mathematical rigor. Moreover, he assigned to experimentation the task of confirm the result obtained by theory, or at least achievable by rational procedure, rather than to proving the same result. He then considered the role of

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the experimentation less significant than that assigned to it by Galileo. Moreover, in the first and most famous of his two aforementioned treatises, entitled Discours de la méthode pour bien conduire sa raison et chercher la vérité dans la science (i.e., Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences), published in 1637, Descartes postulated a rationalistic and mechanicalmathematical science of nature, affirming that scientific knowledge provided by it are the ones to be used to dominate the same nature [100, 99]. As for the other two members of that Trimurti, who were almost contemporary, we must recognize first to be indebted, with both of these scientists, the fact of giving us the sublime calculus or sublime géometrie. In fact, the differential and integral calculus of mathematical analysis, born from the fertile mind of these giants of science (almost simultaneously and independently of each other), was called in this way until the end of the 19th century [243]. Leibniz was a precocious and extraordinary genius, which already at eight years knew the Greek and Latin and, at twelve years, he wrote poems in Latin. Soon, however, his spirit was addressed, like Descartes, to the mathematical-mechanistic nature interpretation, making himself irresistibly attracted by the differential calculus and combinatorial analysis, which are notoriously two opposite branches of mathematical thinking. The concentration of interest in Leibniz toward two such contrasting manifestations of mathematical thinking, that is, towards the continuous mathematics and discrete mathematics, is a unique case in the history of mathematics, which has never registered nor previous cases or subsequent cases (see [139, 235]). With his dream of a symbolic reasoning, defined by the well-known characteristica universalis, Leibniz found himself in advance of more than two centuries than it had to be happen. In fact, his idea of an ideographic language, capable of expressing the concepts of the different sciences and their logical relationships, became the guideline of the subsequent development of logic and symbolic mathematics as well as of applications that have been made since the mid-19th century until today’s Information Age, in a sense prophesied by Leibniz. In reference to the combinatorial analysis and probability theory, the priority and advances of Leibniz are to be considered even more significant, especially in today’s world where everything is considered uncertain. Subsequent applications in testing and especially in quantum mechanics have unequivocally demonstrated the exceptional foresight of Leibniz. Leibniz was not only an outstanding mathematician and great philosopher (he is the author of the well-known and ingenious theory of monads), but he was also a jurist, diplomat, theologian, logician, glottologist, historian, naturalist, geologist, chemist, politician, manager, and inventor. He was a universal spirit: his friends Antonio Migliabecchi, Frederick I of Prussia, and Voltaire (François-Marie Arouet) called him, respectively, Bibliotheca Magna, Complete Encyclopedia, and Wise Illuminator. To satisfy his anxious search for income, but also to demonstrate that he knew how to make practical products, Leibniz began to do the serving inventor for principles and government men, developing an incredible amount of ideas and designs of all kinds, which are now stacked in powerful packaging in the former library of Hannover, and have jet to be classified and examined.

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Although the contribution of Leibniz in mechanics cannot compete with those he gave in mathematics and philosophy, those relating to the concepts of force, momentum, and kinetic energy as well as the impact study cannot be kept under silence. As an inventor, Leibniz designed interesting machines, but here we want to remember only his mechanical calculator, known as Leibniz’s Calculating Machine, presented in 1673 to the Royal Society, of which later he became a member. This machine, called Stepped Reckoner, invented in 1671 and gradually improved starting from the first prototype built in 1673, was able to perform all four arithmetic operations, thanks to the use of a stepped reckoner, which attracted attention and was the basis of his election to the Royal Society. Leibniz’s calculating machine was characterized not only by a gear transmission system with cylindrical spur gears, more improved with respect to that adopted by Blaise Pascal (1623–1662) for his machine automatique, called Pascaline, but also by the introduction of bevel gear pinions of varying length, for which sizing Leibniz was engaged for many months. The stepped reckoner, based on Pascal’s ideas, multiplied by repeated addition and shifting, using the binary system, of which Leibniz was a stronger supporter. This binary system was mechanically implemented using the Leibniz wheel. This wheel, also called stepped drum, was a cylinder with a set of teeth of incremental face widths that, when coupled to a counting wheel, could be used in the calculating engine of a class of mechanical calculators. Wheels of this type were used for about three centuries, until the advent of the electronic calculator in the mid-1970s. Figure 5.4 shows one of the old machines (Fig. 5.4a) and a replica from 1894 of the Leibniz’s stepped reckoner without the cover (Fig. 5.4b), while Fig. 5.5 shows an exploded view of the stepped-drum mechanism (Fig. 5.5a) and a detail of it, also in exploded view (Fig. 5.5b). Moreover, Leibniz invented other computational machines, based on a mechanics similar to that of the afore described machine. Couturat [90] reports having found an unpublished note of Leibniz, dated 1664, describing a machine capable of performing some algebraic operations. In 2010, Rescher [310] discovered that Leibniz also devised a cipher machine. In 1693, in an article published in Acta Editorum de Leipzig, Leibniz himself described the design of a machine that in theory could integrated differential equations, which he called integraph (see Leibniz, translation [218]). Unlike the Pascaline, which had the merit of opening the calculator era, the new and innovative gear system of Leibniz was essentially a transposer, which was used to store a number and sum it repeatedly. Another great insight of Leibniz was the basis of the first attempt to build a computer that used the binary number system, already introduced by Juan de Caramuel of Lobkowitz (1606–1682), and developed by him. The Leibniz’s stepped reckoner has however as a precursor the Pascaline, of which Figs. 5.6 and 5.7 show the outward appearance and, respectively, the gears with their operating scheme. Pascal, however, has to be credited as the initiator of the era of computing mechanical machines, which opened up new horizons including those of the scientific investigation. In fact, before Pascaline only devices of simple positioning were available, such as the Chinese Souan-pan, which is basically to be considered a variation of

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Fig. 5.4 a One of the old Leibniz’s calculating machines; b replica of the Stepped Reckoner at the Hannover Landesbibliothek

the Roman abacus. The Pascaline (also known as Pascal’s calculator or arithmetic machine) can be also considered as the first device that has synthesized a mental process of calculation in the exact reality of a machine. The great French philosopher built the first prototype of Pascaline in 1642, when he was just nineteen, also to alleviate the burdensome job of his father who was an accountant. After as many as 50 prototypes, he developed the first machine, which he presented to the public in 1645, but which continued to improve with subsequent modifications up to the 1649 version; this allowed him to obtain the Royal Privilege, which guaranteed him the exclusive rights to build and sell these machines in France.

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Fig. 5.5 Exploded views of: a the stepped-drum mechanism; b a detail of this mechanism

Fig. 5.6 Outward appearance of Pascaline

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Fig. 5.7 Gears of the Pascaline and operating scheme

The Pascaline was able to carry out the addition and subtraction of two numbers as well as the multiplication and division by repeated addition and subtraction. Leaving aside the working principle of Pascaline (in this regard, see [118, 290]), as far as we are concerned, it should be emphasized the Pascal’s calculator consisted of a digit input similar to the rotary dial of an old telephone, which operated a series of metal gears, configured and arranged in such a way that a complete rotation of each gear wheel would advance the one on the left of one unit. The gears of the mechanical computer rotated in only one direction, so the negative numbers could not be directly added. Therefore, to subtract one number from another, the nine’s complement method was used. The heart of the calculator consisted of a carry mechanism and a particular component, called sautoir (i.e., sauter, jumping element) by Pascal himself. Figure 5.8, taken from Pascal, shows a schematic diagram of the carry mechanism and the sautoir. To resist the load applied to the digit input dial by an operator, without the frictional forces increasing excessively, Pascal used lantern gears, of the kind he had seen work in the turret clocks and water wheels. He also devised the sautoir, which is the centerpiece of the Pascaline’s carry mechanism. It is a pawl and ratchet mechanism that, on the one hand, avoided a wheel rotating in the opposite direction during an operator input, and on the other hand guaranteed a precise position of the display wheel and the carry mechanism for the next digit when it was pushed up and brought into its next position. Leibniz is one of the leading exponents of Western thought, and one of the few thinkers called universal genius by posterity. On the contrary, this title was never given to Newton, his great rival, despite the fact that Newton has surpassed Leibniz

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Fig. 5.8 Schematic diagram of the carry mechanism and sautoir of the Pascaline

and many more, in natural philosophy. We here do not want to go into the unpleasant controversy, to which the Royal Society did not remain extraneous, as it should have done. With this controversy, the two contenders were disputing the priority on sublime calculus. This controversy had no reason to be, and therefore it never should have taken place, since the two contenders reached the same result almost simultaneously, using different criteria. In fact, Newton made use of the well-known and famous fluxion, which is why, considering each variable quantity as a “quantitas fluens”, he called fluxion of that quantity the velocity of its variation. He thus reduced the problem of differential calculus in search of the law able to express the variational gradient of the same quantity, and that of the integral calculus to the reverse operation, that is the definition of the primitive law. Leibniz instead assumed as a criterion the continuity principle, according to which, when two quantities are approaching to the point that to be confused with one another, also the events that derive from them must necessarily do the same [217]. In doing so, Leibniz established firmly the concepts of infinitesimal and infinity (it is to be noted that these concepts lacking in Newton) by means of an uninterrupted extension, i.e. continuous, reaching in this way to its fundamental theorem of calculus, which

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Fig. 5.9 Internal title page of a copy of Newton’s treatise Philosopliae Naturalis Principia Mathematica

specifies the relationship between its two basic operations, i.e. differentiation and integration. This theorem, ironically, is now known as Newtonian reverse, as the first part of it shows that an indefinite integration can be reversed by a differentiation. The works of Leibniz on differential and integral calculus appear more convincing and clearer than those of Newton, but the latter is credited with having writing a masterpiece, like the afore-mentioned Philosopliae Naturalis Principia Mathematica, which laid the foundations of classical mechanics and is considered the most important scientific test of the world. Figure 5.9 shows the internal title page of a copy of the first edition of this treatise, with handwritten corrections by Newton himself for the second edition. Newton was also a genius, not because of the universality of thought, but rather for depth and sharpness of views. Thanks to Newton, dynamics passed from the embryonic stage to the stage of well-defined science. In fact, Newton defined its fundamental principles, by codifying them in the concepts and relationships that are displayed in detail and clearly in the Principia. In addition, Newton extended the validity of the laws on the motion of bodies to celestial phenomena, proceeding with rational and experimental methods at the same time, and using precise rules for the interpretation and study of physical phenomena. Just using these rules, Newton was able to clarify the procedure to formulate his famous law of universal gravitation, described in his treatise De motu corporum in gyrum, i.e. On the motion of bodies in orbit, published in 1684 [361]. Newton did not write anything specific about gears, but his classical mechanics was the basis of the acquisition of scientific knowledge that make up the highest conceptual aspect of the subject we have dealt with in this

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textbook, and still continues to be the source of ever new knowledge that they never ceases to amaze. Before returning to deal more particularly with the gears, of which we are making a brief history, we cannot fail to mention one of the incomparable 18th century scientists, who ennobled mechanics, making it a queen science: Leonhard Euler (1707–1783). He was a giant in all the sciences of his time, and then was compared by many to Leibniz, for the breadth of disciplines and problems he faced and studied. However, actually he was certainly much deeper than Leibniz in physics, especially in mechanics. In fact, he was able to transform the latter discipline in an analytical science, to the point that Bell [31] asserted that Archimedes could have written the Principia of Newton, but that no Greek scientist could ever make up the Mechanica of Euler [121]. It is to be noted that Bell referred to the draft in a first succinct form in 1736, with the title Mechanica sive motus scientia analytice exposita, and then chiseled after he was named by Frederick the Great of Prussia to teach in the Berlin Academy. Figure 5.10 shows the title pages of first edition of Tomus I and Tomus II of the Euler’s Mechanica. We are indebted to Euler mechanics as we know it today. Already in that first editions of his Mechanica, published in Petersburg, the fundamental principles of mechanics appear set out, for the first time, so that all relevant issues can be addressed and treated analytically. In particular they concern: mechanics of point; mechanics of rigid body; geometry of masses, with the correct definition of centrum massae or centrum inertiae, moment of inertia, principal axes of an area, etc.; equations defining the general motion of a mechanical member subject to any system of forces; etc. He

Fig. 5.10 Title pages of the first edition of Tomus I and Tomus II of Euler’s Mechanica

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also gave a great contribution to the development of celestial mechanics, which is the most complex mechanics, especially when a problem of planetary perturbations must be addressed and solved, such as Euler did brilliantly. There was no branch of mathematics, pure and applied (many mathematicians and scientists believe that Euler was the greatest mathematician of all time), and physical and mechanical sciences, in which Euler has not given a substantial contribution, leaving an indelible mark of his mental rigor, analytical rationality, and inventive genius. Here it is impossible to summarize the innumerable contributions of Euler in the above fields of pure and applied sciences. It is not even possible to give an idea of his immense scientific production, which is scattered in about twenty textbooks, all very important and some of which are ponderous volumes, and several hundreds of memories, all of significant value, published in the Commentaries of the Petersburg Academy, and Memoirs of the Berlin Academy. Already more than two centuries before, Leonardo da Vinci had defined Mechanics as Paradise of all sciences. Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet, known as Nicolas de Condorcet (1743–1794), made to Euler, with whom he worked, the greatest praise, recognizing that, when the Leonhard of the physical sciences, i.e. Leonhard Euler, ceased to live and calculate, he had already transferred such an abundance of sublime calculus in mechanics, as to make it indeed the paradise dreamed by Leonardo da Vinci [94]. We fully share this magnificent commendation made by de Condorcet, adding that Euler gave scientific dignity to mechanics in all its aspects and, with it, as we shall see later, to gears that cover several parts of the subjects in which mechanics is subdivided. Joseph-Luis Lagrange (1736–1813), born Giuseppe Luigi Lagrangia in Turin, also gave a significant contribution to mechanics. It should first be remembered that, since from the beginning of his work, he dealt with the calculus of variations, which is the basis of modern numerical methods, such as FEM. In this regard, he further developed this branch of the mathematical analysis, which had been invented by Euler, up to the definition of the best-known result, i.e. the Euler-Lagrange equation. However, we are especially indebted to Lagrange, because he has bequeathed us the treatise Mécanique Analitique (i.e., Analytical Mechanics), published in Paris in 1788, which is to be considered a pillar of mechanics as well as an absolute scientific masterpiece [209]. Figure 5.11 shows the title page of the first edition of the Mécanique Analitique. In this treatise, Lagrange expands and extends the Newtonian assumptions of the free body, in order to be able to analyze the interactions between the bodies of any general mechanical system, subject to constraints or, in any case, subject to stringent conditions. The general mechanical system is supposed also in compliance with the Principle of d’Alembert, which had been enunciated by d’Alembert (1717–1783) in 1743, in his treatise entitled Traité de Dynamique [115]. This principle, which is to be considered as an extension of the principle of virtual works for non-inertial systems, establishes that at any moment any state of motion of a mechanical system can be considered as a state of equilibrium in which all the forces involved, including the inertia forces, are simultaneously in action. With this principle, the dynamics of any mechanical system can be studied and analyzed as a succession of equivalent static conditions where all the forces acting on it, including the forces of inertia, considered

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Fig. 5.11 Title page of the first edition of Lagrange’s treatise Mécanique Analitique

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as fictitious forces, are added. Figure 5.12 shows the title page of the first edition of the Traité de Dynamique. It should be pointed out that Lagrange in the aforementioned treatise Mécanique Analitique, for the first time expresses explicitly on analytical basis the already known principle of virtual works (we have already said that the first statement of this principle

Fig. 5.12 Title page of the first edition of d’Alembert’s treatise Traité de Dynamique

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is due to Guidulbaldo Bourbon del Monte), and introduces the generalized coordinate method, which is to be regarded as the most brilliant result of his analysis as well as the key contribution to mechanical science. With the latter method, which transcends those used by d’Alembert and Euler, Lagrange determines the configuration of any material system by a sufficient number of variables, the number of which is the same as its degree of freedom. In this way, the kinetic and potential energies of the system can be expressed in terms of these generalized coordinates, and the differential equations of motion thence deduced by simple differentiation. The Mécanique analitique of Lagrange is a great work of methodical and unifying synthesis. It arises from the conviction of the author according to which, as he himself writes, it is possible to “reduce the mechanical theory and the art of solving problems that belong to it in the general formulae, whose simple development provides all the equations necessary to achieve this goal”. With Lagrange, mechanics actually becomes a new branch of mathematical analysis, thus rising to the highest dignity of pure science, which is added to that already recognized of applied science. So far, we have done an overview of the major events that occurred in the first two centuries of the second scientific age, which saw the rebirth of science and especially the affirmation of mechanics as the queen of the sciences. Of course, we have focused our attention on the contribution of the most representative scientists, who defined the scientific foundations of mechanics. When appropriate, we have also highlighted the specific contributions of the scientists themselves, regarding the gears and, if we do not had done, we propose to do below. However, we must not forget the valuable contributions that other scholars gave for the advancement of scientific and technological knowledge for the mechanics in general, and for gears in particular. In this latter regard, we refer the reader to more encyclopedic textbooks. However, it is necessary to bear in mind that the scientific reawakening and fervor of studies that characterized Europe from Galileo onwards did not have immediate repercussion on the gears. Gearing transmissions continued to be designed according to empirical criteria, at most benefiting from even empirical experiences that were gradually accumulating, concerning metallic materials and their manufacturing processes. In this regard, the most significant contributions were certainly those given by the clockmakers and manufacturers of measuring devices and scientific instruments. This empirical way of proceeding with the conception and design of the gears did not however prevent the development of new gearing mechanisms. Among the various examples revealing this specific topic, the one described here deserves to be mentioned. It consists of the sun and planet gear mechanism, used to convert reciprocating motion to rotational motion. This mechanism was designed by Scottish engineer William Murdoch, an employee of Boulton and Watt, to circumvent James Pickard’s patent concerning the slider crank mechanism. It was patented in the name of James Watt in 1781 [360], and played an important role in the development of rotational gearing devices in the Industrial Revolution. As Fig. 5.13 shows, the Watt’s mechanism converts the reciprocating motion of a rod, driven by a steam engine and connected with the connecting rod, into a circular motion of the sun gear that rotates around its axis and meshes with a planet gear, which instead is not rotating, as rigidly fixed to the end of the same connecting rod.

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Fig. 5.13 James Watt’s sun and planet gear mechanism

In comparison with a slider crank mechanism, this type of mechanism enjoys an interesting feature. In fact, when both sun and planet gear have the same number of teeth, the driven shaft rotates twice around its axis for each round-trip stroke of the rod instead of one. Of course, the axes of sun and planet gear are kept at a fixed distance by a link, which freely rotates around the axes of the sun, and holds the planet gear in mesh with the same sun, without contributing to the drive torque. Clockmakers were increasingly involved in the miniaturization of clocks, first with the transition from tower clocks used for monasteries and public buildings to alarm clocks and, later, with the construction of table clocks and, finally, pocket watches. Figure 5.14a–c, which intends to highlight the not high technological leap between the fourteenth and the seventeenth centuries, show respectively the gear trains of: • an English tower clock from 1348, fully operational until 1872 in the castle of Dover (Fig. 5.14a); • a clock with striking mechanism dating back to around 1390, attributed to the Florentine Laurentium da Valpuria, which can be considered as a primitive alarm clock (Fig. 5.14b); • a portable timekeeper with spiral balance spring (or hairspring) coupled with a balance wheel, used for the first time by Huygens in 1675, which ushered in a new era of portable mechanical watches, with precision comparable with that of pendulum clocks (Fig. 5.14c).

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Fig. 5.14 Gear trains of: a a tower clock from 1348; b a primitive alarm clock dating back to around 1390; c the Huygens’ portable mechanical watch from 1675

The portable timekeepers benefited from the expertise of the manufacturers of precision instruments, which were distributed on the Netherlands-Italy axis, with their epicenter in Nuremberg, supposedly the city of origin of the so-called Nuremberg eggs, due to their shape. On this axis, portable timekeepers were produced, which are to be considered masterpieces of style, refinement and precision, this last guaranteed by the use of regulation systems connected with the driving mechanisms, both spring-loaded or counterweighted. Figure 5.15a, b show the mechanisms of two of these portable timekeepers built in the 18th century, the first with a spiral spring

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Fig. 5.15 Two specimens of portable timekeepers with: a spiral spring mechanism; b counterweight mechanism; c a German tower clock of the late 14th century as a comparison

and the second with a counterweight. Both specimens are characterized by gear trains that, with the exception of dimensions and accuracy, do not differ substantially from those of a 1392 German tower clock, which is shown in Fig. 5.15c. The reduction in the size of clocks and, consequently, that of the corresponding gearing mechanisms was possible thanks to the introduction in the early 15th century of mainspring. This allowed the development of portable clocks; from evolution of these portable clocks the first pocket watches were born in the 17th century. Their time measurement was not very accurate, at least until the balance spring was added to the balance wheel in the middle of the 17th century. The addition of the balance spring made the balance wheel a harmonic oscillator like the pendulum in the pendulum clock, which oscillated at a fixed resonance frequency, resisting the oscillations at other frequencies. This innovative solution greatly increased the accuracy of time measurement of table clocks and pocket watches, making it comparable to that of pendulum clocks. This is not the place to delve on the dispute over who was the first to introduce the balance spring. Just remember that, according to some authors, the British scientist Robert Hooke (1635–1703) was the first to introduce it, although in the form of straight spring, while other authors argue that the aforementioned Dutch scientist Christiaan Huygens was the first to introduce the balance spring, in the form of spiral balance spring.

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The manufacturing accuracy of metal gears benefited greatly from the development of increasingly precise measuring instruments. In this regard, the following measuring instruments developed in the 16th century are to be remembered: the holometer (an instrument for making of angular measurements for surveying) by Abel Foullon (1513–1563 or 1565), conceived in 1551 when he was director of the Mint for Henry II of France and also an engineer of the king of France after Leonardo da Vinci [137]; the surveying instruments made in 1557 by Baldassare Lanci (1510– 1571) for the Grand Duke’s collection, when he entered the service of Cosimo I de’ Medici. Figure 5.16 shows three surveying instruments of this period, namely an azimuthal quadrant, a holometer, and a planimetric circle with connected altimetric circle. In the late sixteenth century, mathematical methods were introduced in the surveying practice, based on the use of simple measuring instruments, such as azimuth theodolite, while more complex instruments, such as altazimuth theodolite, recipiangle and trigonometer, found strong resistance from surveyors. Nevertheless, these more sophisticated measuring instruments gradually became established, until they became commonly used in the mid-seventeenth century [33]. In the first decade of the eighteenth century, new astronomical instruments began to spread in Europe from the Nordic countries, as the meridian circle (it identified the position of stars, determining their height at the moment of their passage through the meridian), which was designed and built by the Danish astronomer Ole Christensen Römer (1644–1710). This instrument represented a remarkable progress compared to the quadrant even in the improved version introduced by the Danish astronomer, astrologer and alchemist Tycho Brahe (1546–1601). In the same decade, the demand for instruments able to artificially reproduce the positions and movements of the various celestial bodies, in particular the sun and moon motions, was very intense. The ancestors of the modern planetaria were so developed. They were called orreryes,

Fig. 5.16 Surveying instruments of the 16th century: a azimuthal quadrant; b holometer; c planimetric circle with connected altimetric circle

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named after the Englishman Charles Boyle, Count of Orrery (1676–1731). One of the many specimens of orreryes, made of wood and brass and built to order of Boyle by the Scottish astronomer James Ferguson (1710–1766), who was inventor and improver of scientific instruments, gaining Boyle’s trust, is shown in Fig. 5.17. Finally, it is worth mentioning the great contribution that the metal gears construction technologies had from the marine chronometers, which were developed to meet the needs of precise time-keeping for navigation and are to be considered as significantly improved clocks compared to those previously designed for other needs. In this framework, the activity of the Yorkshire carpenter John Harrison (1693– 1776) is to be remembered; he in 1735 built his first chronometer, which he steadily improved on over the next thirty years, up to the most refined models built in sixties of that century. These models had many innovations, including the use of bearings to reduce friction, weighted balances to compensate for the ship’s pitch and roll in the sea, and the use of two different metals to reduce the effects of thermal expansion.

Fig. 5.17 Mechanical orrery of the mid-18th century, which simulated the motions of the Sun, Mercury, Venus, Earth and Moon with the simple rotation of a crank

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Fig. 5.18 a Marine chronometer built by Harrison at the age of 17; b drawings of the gearing mechanism of the Harrison’s H4 marine chronometer of 1761

Figure 5.18a shows a marine chronometer built by Harrison at the age of 17, while Fig. 5.18b shows the drawings of the Harrison’s H4 marine chronometer of 1761 [170].

5.2 From Desargues to Today: From Empiricism to the Slow Penetration of Mechanical Science into Gear Design After the long, dark medieval night, the awakening of the technical-scientific interest on the gears takes place between 17th and 18th centuries, when same scholars, steeped of Renaissance culture, begin to study the possibility to making gears with appropriate kinematics, by the use of suitable profiles. The empirical criteria of designing the gears were however recalcitrant to die, and the affirmation of scientific criteria in the specific field that interests us was very slow. Here we will discuss the subject by dwelling on the description of the earliest scientific contributions concerning

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the gears, which were initially greatly diluted over time. When these contributions intensify, given the limited nature of this brief history, we will focus our attention on the contributions due to the initiators of specific branches of knowledge on the gears. The first entry of science in the field of gears occurred in 1639, when one of the founders of projective geometry, Girard Desargues (1591–1661) reinvented the cycloid and showed its application to the construction of cycloid profiled gear teeth. He indeed designed and built a system for raising water based on the use of epicycloid gear wheels, whose operating principle was unknown at the time. His most important work is entitled Brouillon projet d’une atteinte aux événemens des rencontres d’un Cône avec un Plan (i.e., Draft of an essay on what is obtained by intersecting a Cone with a Plane). Only a few copies of this essay were printed in 1639, of which only one manuscript copy survives, made by Philippe de La Hire (1640–1718), and discovered by Michel Chasles (1793–1880). Actually, it is difficult to say with certainty who first introduced the cycloid. Certainly, this curve had already been studied and described in the 15th century by Nicolaus Cusanus (1401–1464), in his attempts to square the circle, which he exhibited in the De circuli quadrature (On the quadrature of circle) of 1450. However, the rigorous definition of the cycloid is due to Marin Marsenne (1588–1648), who established the first obvious property, that according to which its base is equal to the length of its generation circle. Marsenne then tried to find the area under the curve, but without success. Galileo gave name to this curve in 1599; he too tried to find the area under the curve, even without success. Girolamo Cardano (1501–1576) made significant contributions to hypocycloids, which led him to develop in 1557 the Cardan gear mechanism that is used to convert rotation motion into a reciprocating linear motion, with greater efficiency and precision than other mechanisms (for example, Scotch jokes), as it does not use linkages or slide ways [66]. This result was achieved using an external cylindrical spur gear pair, where the quotient of the pitch diameter of the larger gear wheel divided by the pitch diameter of the smaller gear wheel must be equal to two. However, for the definition of teeth of this gear pair, Cardano limited himself to giving only empirical rules. Figure 5.19 shows an axonometric view of the Cardan gear mechanism. Albrecht Dürer is credited to have discovered and investigated the shape of the epicycloid [114]. The wonderful engravings made by Dürer in 1515, entitled Triumphzug Kaiser Maximillian, two of which are shown in Fig. 5.20a, b, do not yet suffer from the application of mathematical principles in his masterpieces of art, of which the same Dürer became a promoter. The figure in fact represents two wheeled coaches with hand-operated gear trains, the first of which (Fig. 5.20a) includes orthogonal lantern pairs and cylindrical gear pairs, the latter having teeth with a rectangular profile, while the second coach (Fig. 5.20b) includes worm gear pairs, consisting of spiral worms that mesh with unlikely worm-wheels also having teeth with a rectangular profile. Evidently, at that time, the cycloid mathematics and curved tooth profile had not yet entered the gear concept. The interest of mathematicians for the cycloid, hypocycloid and epicycloid had been aroused very early. The cycloid was studied not only by the aforementioned

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Fig. 5.19 Axonometric view of the Cardan gear mechanism

mathematicians, but also by Evangelista Torricelli (1608–1647), Pierre de Fermat (1601–1665) and Descartes, who all found the area subtended by this curve. Moreover, it was studied by Gilles Personne de Roberval (1602–1675) in 1634, Christopher Wren (1632–1723) in 1658, Huygens in 1673, and Johann Bernoulli (1667–1748) in 1692 [35]. More or less, they discovered most of the important properties of this family of curves. In particular, Roberval and Wren found the arc length. It is here to remember the famous controversy triggered by Pascal in 1658, with his six Dettonville Problems (Pascal is hiding under the name Amos Dettonville, an anagram of the pseudonym Louis de Montalte, in turn anagram of the motto Talentum Deo Soli, i.e. My talent for God alone, which he used for a previous writing), to whose solution John Wallis (1616–1703), Huygens, Wren, Lalovera (i.e., Antoine de Laloubère, 1600–1664) and others were interested. Lalovera is chiefly known for an incorrect solution of Pascal’s problems on the cycloid [211]. In 1671 Leibniz, by putting an end to the controversy, proved the brilliance of Pascal’s demonstrations on the cycloid (as well as those on conics), concluding that were it not for an evil fate Pascal would have almost certainly gone on to make further and deeper mathematical discoveries. The studies described above show that, already in the middle of the 17th century, the mathematics of cycloid had been very well worked out from the standpoint of purely mathematical interest. The results of the mathematicians, however, did not penetrate the practice of engineers. To tell the truth, in the 17th century there were close relationships between scientists and manufacturers of scientific instruments and clockmakers (they were called the mechanicians), but their interests diverged about the gears and it seems that the acquired knowledge were not extended to the scientific study of gears. The mechanicians continued to favor empirical solutions with satisfaction. Sometime later Desargues, Philippe de La Hire, in 1694, taken up and elaborated the subject concerning gears. In 1690, de La Hire developed the geometrical principles of gear design, by defining systematically the teeth profiles on solid mathematical bases. In the second volume of the Mémoires de l’Académie Royale des

Fig. 5.20 Two wheeled coaches engraved by Dürer, with hand-operated gear trains consisting of: a orthogonal lantern pairs and cylindrical gear pairs; b worm gear pairs

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Sciences, he published four treatises, the first of which entitled Traité des Epicycloides (Treatise on epicycloids), which is to be considered the first mathematical treatise on gear design. The conclusion reached by de La Hire, according to which the involute curve was the best shape of the teeth profile, is to be considered of great design value. In this regard, he gave a full description of the involute tooth profile, but it was necessary to wait about 200 years before it gained wide acceptance. De La Hire emphasized the need to maintain uniform contact pressure and motion, but he missed the sliding contact and corresponding friction losses, talking of pure rolling. Finally, de La Hire focused his attention on epicycloid tooth profile, and was the first to make systematic applications of this profile, which he described in his aforementioned treatise [95, 96]. He was the first to laid the basic geometrical principles of epicycloid gears, having understood that if the tooth of a member of the gear pair was formed by a portion of an exterior epicycloid described by any generation circle, the tooth of the mating member be a portion of an internal epicycloid described by the same generation circle. He also showed how to theoretically find the corresponding shape of tooth profile able to mesh correctly with a given profile shape, but came to the conclusion that, for some profile shapes, this goal could be impossible to achieve in practice. Finally, de La Hire considered the involute as the best of the exterior cycloids, having identified it as a special limit case in which the generation circle’s radius becomes infinite. He then came to the conclusion that the involute of a circle of an infinite radius is a straight line, so the involute teeth of a rack are straight sided. Charles Étienne Louis Camus (1699–1768) was the first mathematician to develop the theory of gear teeth, framing it in the more general theory of mechanisms. He repeated and reworked everything that de La Hire had already done, but added his own important elements, making a detailed analysis of the best tooth shapes to use for combinations of spur and lantern gears. He also corrected de La Hire, recognizing the sliding contact between teeth with epicycloid profile (he focused his attention exclusively on these profiles, leaving out the involute profiles), and believing that it was the main source of friction and wear in gearing [62]. Having in mind the wooden gearing of mills, but also influenced by clockmakers, he also dealt with orthogonal lantern transmissions, consisting of crown wheels with pin teeth and cylindrical or bevel lantern pinions. Successively, Camus analyzed the action of mating teeth in comparison with the center distance, and concluded that the condition of contact between the mating teeth are better when the tooth of the driving gear wheel works in the recess portion of the path of contact. Furthermore, according to the aforementioned preliminary studies of de La Hire, Camus concluded that the line of action had to pass through the points of tangency with the two pitch circles, being normal to the two mating profiles at their point of contact. It is not to be forgotten that we have inherited from Camus the theorem that bears his name (the Camus’ theorem), which formulates the conjugation conditions of two cycloid curves (for a description, see [105, 106, 228]). Camus also faced the problem of the minimum number of teeth, and the one of the best shapes to give to the tooth tips. He was the first to deal with bevel gear pairs (albeit made up of a pinion and crown bevel gear wheel), although, at least in nuce,

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de La Hire had already raised the issue of these gears. To this end, he studied the bevel gear pair simulating it with a rolling cone on a plane. Having in mind only clockmaking applications, Camus does not pose the problem of the tooth shape in a gear set consisting of three or more gear wheels, a problem that was however very important to solve for the mills of his era. However, he laid the bases of a theory of the geared mechanisms. Subsequently, it was Willis to develop the topic in a systematic and comprehensive manner. In 1751, Leonhard Euler defined for the first time the conjugate action law and showed that gears designed according to this law have a steady speed ratio. In 1754/55, still Euler, and always for the first time, defined the involute of circle, providing the mathematical equation of this curve, as well as the basic design criteria. He also found and defined precisely the conditions that must be met so that the gear teeth are operating satisfactorily, and showed that either the cycloid or the involute profiles were able to meet the above conditions. He also proposed a method for obtaining these profiles. Euler’s contributions to the evolution of a scientific theory of gearing were so significant as to constitute a benchmark between before and after Euler. Although the involute curve of a circle was known for a longtime, Euler was certainly the first to apply it to the gear tooth profile, and to demonstrate that this type of tooth profile to the best fitted the need for a correct kinematic operation. He also showed that the involute profiles were able to guarantee a uniform rotational motion of the two members of a gear and determined the conditions of non-interference between the same conjugate profiles to be respected. According to some authors, the basic law of conjugate gear-tooth action (it states that, when the gear members rotate, the common normal to the tooth flank surfaces at the point of contact must always intersect the line of centers at the same point, which is the pitch point) was known to Euler and also to Felix Savary (1797–1841). However, not all authors agree with this statement. In any case, Euler and Savary are credited of the Euler-Savary equation, which establishes a correlation between radii of curvature of the centrodes of gear members and the corresponding radii of curvature of the interacting tooth profiles (see [113, 149]). Euler understood the principle of the common tangent, and showed how to determine the mutual interaction between the teeth, assuming a pressure angle of 30 degree. Euler has left us two memorable works concerning the gears, the first titled De Aptissima Figura Rotarum Dentibus Tribuenda (i.e., On finding the best shape for gear teeth), written in 1751, but presented and published in 1754 [122], and the second titled Supplementum de Figura Dentium Rotarum (i.e., Supplement on the shape for gear teeth), written in 1762, but presented and published in 1765 [123]. The contributions of novelty and originality are mainly described in the second of these two works [53]. Figure 5.21 shows the first pages of the two aforementioned works by Euler. Unfortunately, Euler’s works were too mathematical and theoretical works to be of practically use to clockmakers and millwrights. Moreover, like the works of other scientists of that time, they were written mainly in Latin, an incomprehensible language to essentially empirical mechanicians.

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Fig. 5.21 First pages of Euler’s works De Aptissima Figura Rotarum Dentibus Tribuenda and Supplementum de Figura Dentium Rotarum

A very brief mention must be made to the writings of Abraham Gotthelf Kästner (1719–1800), who is known for writing textbooks and compiling encyclopedias rather than for original research. He in 1781 described a simple practical method for calculating epicycloid and involute tooth shapes, which can be considered as the first step to make practically usable the theoretical works done by other authors. Then studying the epicycloid pinion-rack pairs, he gave suggestions regarding the teeth length and the minimum acceptable pressure angle (about 15°), concluding that pinion and rack had both epicycloid teeth. Finally, he provided a practical way to describe and apply the involute to the gear teeth. All his suggestions, in truth not very significant, turned out to be unusable by the gear makers, also because they were written in Latin [91]. In the wake of Radzevich [303], here we must mention James White who first, in 1812, to produce spiral bevel gears, had the idea of using the well-known Hooke’s stepped wheel concept. This scientist, to whom we owe the formulation of Hooke’s law, in 1666 had developed the stepped wheel, which is to be considered the predecessor of the helical gears, moreover dating back at least to Leonardo da Vinci [186]. Hooke took a number of identical gear wheels with a small face width, arranging them side by side, but rotated with respect to each other by the same amount in the same direction, i.e. with an angular shift of a predetermined angle. Figure 5.22a, taken from Hooke, shows his stepped gear wheel, while Fig. 5.22b shows a schematic

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Fig. 5.22 a Hooke’s stepped gear wheel; b a schematic perspective view of the Hooke’s stepped gear wheel

perspective view of the same Hooke’s stepped gear wheel (see also Section 8.1 of Vol. 1). Regarding White, Radzevich reports that he would have given an interesting example of a practical man who knows how to apply the theoretical bases when he thought of extending the concept of Hooke’s stepped gear wheel to straight bevel gears in order to obtain spiral or curved toothed bevel gears. Figure 5.23 shows a schematic diagram of the intuitive procedure used by White to obtain a stepped bevel gear wheel, from which it is possible to obtain a spiral or curved toothed bevel gear wheel, through an operation of tendency to the limit (see also Section 12.1 of Vol. 1).

164 Fig. 5.23 A schematic perspective view of the White’s stepped bevel gear wheel

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(a)

(b)

Experimenting prototypes of these types of gears, White found that their two members engage perfectly, whatever the relationship between their diameters or the angle between their axes. Let’s leave out here the conclusions drawn by White in terms of teeth wear (in this regard, see [303]), based on old incorrect notions. Certainly, they would been avoided if White, as an intelligent engineer as he was, had analyzed the action between the mating teeth using appropriate mathematical knowledge, based on physical principles. White, however, like almost all the engineers of his time, was completely devoid of this mathematical knowledge.

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In 1841, Robert Willis (1800–1875) made a systematic analysis of the gears, extending as much as possible the theoretical basis related to them, and presented the results in a form usable for engineers and designers. In the broadest possible vision for his time, Willis framed the study of the gears on a mathematical basis, in the most general science of the mechanisms [364]. Therefore, he provided a comprehensive analysis of the gears, based on the physical-mathematical laws governing their motion conditions, leaving aside, by his own admission, all the questions concerning their dynamic aspects and strength problems. Willis reworked the method of the intersecting cones, already introduced in 1787 by the English technician and printer, John Imison (?–1788) for the analysis of bevel gears (so called for the first time by the same Imison), and used the hyperboloid of revolution for the analysis of spiral gears as well as of worm gears, which are a special case of crossed-axes helical gears [192]. Willis showed that the bevel gears are a particular case of spiral gears, and spiral gears are a particular case of hypoid gears with a distance between the axes equal to zero. He also proved that when this distance increases, the rolling motion progressively becomes less and less perfect. The case of axes that are neither intersecting nor parallel is then discussed by Willis by introducing two pairs of cones. Willis does not pose the problem of comparing cycloid tooth profiles and involute tooth profiles, but he made a through and systematic analysis of all cycloid tooth profiles (cycloid, epicycloid and hypocycloid profiles), gradually studied previously by other authors reducing them to a single general case. Already de La Hire had overshadowed this possibility, but he presented an imperfect method in this regard. Thomas Young (1773–1829), too, had proposed his method in 1807, but he did not work it out fully [372]. Finally, according to Woodbury [370], George Biddel Airy (1801–1892) gave the general solution of the problem in 1827 [2], with the formulation of what we know today as law of conjugate gear-tooth action (see also [303]). Notoriously this law establishes that a common normal to the tooth profiles at their point of contact must pass through a fixed point on the line of centers, i.e. the pitch point, whatever the position of the contacting teeth. It can be satisfied by various tooth profile shapes, although actually the only tooth profile shape of current importance is the involute of the circle. Starting from Olivier (see below), the kinematic geometry of gearing (that is, that particular branch of the gearing kinematics that interrelates the displacements) developed more quickly [105]. In this framework, in addition to the aforementioned Euler-Savary equation, it is necessary to mention the Arhnold-Kennedy instant center theorem, which states that if any three bodies have a relative motion to each other, their instantaneous centers lie on a straight line. When this theorem is applied to a gear pair, it implies that the pitch point must always lie on the line that connects the two centers of rotation of the driving and driven members of the gear under consideration. This theorem is a special case of the vector loop equation, developed later, which defines the relative motion between the three-link 1-dof kinematic chain, constituting the gear pair. Even the law of gearing was soon generalized, to extend it from planar gearing to spatial gearing, for a given transmission function, understood as the relationship between the angular position of the input member of a gear pair and the corresponding

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angular position of the output member. Therefore, in the most general case of axes in any way arranged in the Euclidean three-dimensional space, for a given position of axes and a given transmission function, the laws of gearing became three, and were formulated as follows: • A unique relationship exists between the instantaneous displacement of the output member and the instantaneous displacement of the input member (first law of gearing). • A unique relationship exists between the spiral angle and the pressure angle at the contacts between conjugate surfaces in order to provide motion transmission as defined by the first law of gearing (second law of gearing). This second law results from applying Ball’s reciprocity relation to direct contact mechanisms [20, 21]. • The conjugate action requires a unique effective curvature at the contacts that satisfy the second law of gearing (third law of gearing). Knowledge of the effective curvature between two conjugate surfaces in mesh enables the distance between the two surfaces to be determined. These three laws of gearing are equally valid for any direct-contact mechanism. The third law of gearing is the spatial equivalence of the Euler-Savary equation for planar gearing. Several attempts have been made to generalize the Euler-Savary equation for planar gearing in a similar relationship for spatial motion. In this regard, it is worth mentioning the very appreciable work of Disteli [103]. Each of these attempts, however, provided results other than a unique relationship for the effective curvature of two conjugate meshing surfaces (see [50, 350]). The differences between the Euler-Savary equation for planar gearing and the third law of gearing have proved necessary to take into account the non-degenerate relationships that spatial motion exhibits over planar motion. Returning to Willis, it is however to be recognized that he came to advocate the involute tooth profile on the basis of a study regarding the path of point of contact and the lowest possible number of teeth for spur gears, external and internal, as well as for pinion-rack gear pairs. This study led Willis to consider the ideal working tooth depth and addendum, as well as the tooth thickness and tooth space. The complexity related to the use of epicycloid tooth profiles, especially for cast teeth, common at that time (separate molds were need to make the two members of the geared pair, when a good fit between them was required), led Willis to recognize their limits for an interchangeable gear set. In this regard, Willis demonstrated the advantages of the involute tooth profiles also from the point of view of the bending strength (in a special way as against the epicycloid tooth profiles with radial flanks). Furthermore, he showed how it was easy to minimize the backlash between involute teeth by means of a simple adjustment of center distance. However, as Thomas Young in 1807, Willis also erroneously claimed that the pressure angle of the involute tooth profiles tends to force apart the centers, more than what happens for the epicycloid tooth profile [372]. On this topic, see also below. A study of the meshing between the teeth of a pinion-rack pair brought Willis to note that the involute curve of the rack tooth profile became a straight line, and that

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the contact was a line contact, rather than a point contact, as it happened in the case of an epicycloid rack tooth. Willis also gave several contributions to the theory of the worm gears, suggesting the shape to be given to the worm threads, and how to make the screw tool to cut them. This way of cutting the gear teeth through a screw was already introduced by Ramsden [307], who first had cut a gear wheel using a hob as early as 1768. Willis also made some contributions to the Hindley’s worm controversy; on this controversy, see McDonnell [249]. Willis also studied, in addition to the one-threaded worm, the double- and triplethreaded worm, showing that the meshing of a worm so made with the corresponding worm wheel is a special case of hypoid gears, where the number of threads is one, two or three. In this way, he was able to provide a theoretical basis for the spiral gears of the Pedmont silk mill of 1724 [303]. Willis was the first to highlight the possibility of combining any gearwheel with the same module, and to introduce the geometric modular sizing and diametral pitch sizing, the advantages of which were shortly recognized by Bordmer [49]. For the above reasons, we can say that Willis was more than a mere systematizer of the gear science of his time, a true innovator in that he gave very substantial contributions of the gear theory, moreover presenting them in a form readily usable by gear designers. Just four years before Willis, in 1837, John Isaac Hawkins (1772–1855) showed experimentally the fallacy of what theoretically supported by Thomas Young in 1807, and subsequently endorsed by Willis in 1841, highlighting that the separation force was not significantly influenced by the pressure angle, and attributing this to the fact that the sliding friction between the mating teeth counteracts this separation force. However, the most significant contributions of Hawkins to the gear theory is to have shown clearly the advantages of the already known involute tooth profile with respect to the cycloid tooth profile (as usual, in the more general meaning of cycloid, epicycloid and hypocycloid tooth profiles), even with the modifications gradually proposed to optimize it [63]. In fact, Hawkins pointed out that, compared to the epicycloid tooth profile, the involute tooth profile guaranteed at least the following considerable advantages: • the kinematic operating mode was perfect, even for displacements between the axes of the two members of the gear pair, compatible with the tooth depth; • more than one tooth pair was simultaneously in meshing, with consequent benefits in terms of mechanical strength and load carrying capacity; • under the same operating conditions, the rolling velocity between the mating teeth increased, and the sliding velocity decreased, with consequent decrease of the (slide/roll)-ratio, which was almost halved. Hawkins is also to be remembered for his first English translation of the Camus’ works, made in 1806, which is considered a significant transition from mathematics to engineering, despite the fact that his zeal to make the work of mathematicians easily understandable to mechanicians led him to make a major error. This error consisted of the addition, to the translation of Camus, of a part of the new edition of the Imison’s work, which unfortunately contained an error concerning the ratio between the diameters of the rolling and base circles for the generation of epicycloid

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profiles. This error, rectified in the already mentioned edition of 1837, had serious consequences in practical applications, and two generations were necessary to remedy it. On the other hand, it stimulated in England a lively interest on the question concerning the tooth profile shape. The error due to Imison, according to which the two aforesaid diameters had to be equal, determined a weakening of the tooth at its root as well as an insufficient tip and root clearance, so a further operation was necessary, aimed at ensuring the appropriate value of clearance. This cut-back operation was not carried out; therefore, the teeth were worn quickly. The technicians brutally copied the worn shape of the tooth when they built new gears, but did not understand the theoretical reasons underlying this phenomenon of deterioration of tooth profiles due to accelerated wear. Hawkins, on the other hand, after having rethought the initial error, clearly demonstrated that the aforesaid drawbacks were eliminated with cycloid profiles generated by a rolling circle having a diameter equal to half of that of the base circle. Hawkins corrected other errors of Imison, such as those concerning rack teeth and, on the basis of the already anticipated advances of Hooke, he understood that the spiral gears constituted the right solution to eliminate the impact contacts deriving from the wear of poor-quality teeth profiles. Finally, it must be remembered that Hawkins outlined briefly how the principles described by Camus could be applied to the teeth of the bevel gears, both with cycloid and involute teeth profiles. Some of his statements (see [303, 370]) do however understand how he was completely unaware of the difficult problem related to cutting these gears, whose solution will have to weit until 1885, when Hugo Bilgram (1847– 1932) introduced the octoid tooth. The most significant contributions of Hawkins, which make him one of the most important names in gear history, are certainly those concerning the involute profiles, but it should not be forgotten that he marks the beginning of the transfer of knowledge from mathematicians to engineers. However, both Hawkins and Willis are indebted to Robertson Buchanan (1770– 1816), a working engineer in mills who was also interested in their theory. As he writes in his work of 1808 [55], Buchanan drew heavily on the already mentioned writings of de La Hire and Camus, and treated the gear teeth in a form that was understandable and therefore usable by those who did not have a mechanical education supported by adequate mathematical bases. He simulated the cycloid teeth profiles using suitable arcs of circles able to give a sufficient approximation of their geometry, and introduced tables and figures useful for the design, through which it was possible to obtain good gears. In 1814, Buchanan published another treatise on mill-work and other machinery [56], in three volumes. The second two-volumes edition of this 1823 treatise, edited by Thomas Tredgold (1788–1829) with his notes and additional articles, is of great importance for the analysis of bevel gear teeth. In fact, this edition describes for the first time Tredgold’s idea of 1822, according to which the back cones of a bevel gear pair were developed into plane and a bevel gear pair was reduced to an equivalent cylindrical gear pair. Figure 5.24 shows the title pages of the two volumes of the Buchanan’s treatise, edited by Tredgold [57]. Instead, Fig. 5.25, taken from page 103 of the first volume of the aforementioned treatise, highlights the two cylindrical spur

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Fig. 5.24 Title pages of the Buchanan’s treatise of 1823, edited by Tredgold

gears introduced by Tredgold that, according to the approximation that bears his name, are kinematically equivalent to what happens at the back cones and inner cones of the bevel gear pair under consideration. Undoubtedly the writings of Hawkins, Willis and Buchanan constituted a turning point in the history of gears and gear-cutting machines. All three of these authors contributed to the transfer of the mathematical bases concerning the geometry and kinematics of the gears to the designers, who thus became able to conceive more efficient and precise gear drives, able to best satisfy the growing needs dictated by practical applications, which were progressively expanding considerably. Almost to summarize the knowledge of that era concerning the gears, in 1842 Théodore Olivier (1793–1853) published what may be considered the first monograph on gear theory. Olivier demonstrated that in rigid-body kinematics the locus of the instantaneous axes of any time-dependent motion is a ruled surface and, to better explain to his pupils of the École Polytechnique the mathematical formulation of the problem in the three-dimensional space, he built several models of ruled surfaces. Then applying these concepts to the gears, he built numerous models of various kinds of gear pairs aimed at their more in-depth study, and he also devised machines

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Fig. 5.25 Cylindrical gear pair equivalent to a bevel gear pair at the back cones

for their manufacture. Olivier summarized the gear theory by writing his extensive treatise on the subject, of which the Fig. 5.26 represents the title page [283]. He must therefore be regarded as a great pioneer in the science of gears. Despite the Hawkins’ contributions regarding the involute tooth profile constituted a real turning point in gear theory, we have to expect almost two generations before these profiles were widely used in practice. In fact, until the end of the 19th century, the cycloid gears were the ones most commonly used, especially for heavy gearing as well as for clock and watch gears. This despite the fact that in 1899 George Barnard Grant (1849–1917) proved unequivocally, on a mathematical basis, that the involute tooth profile was superior to the cycloid tooth profile in terms of [163]: • • • • • •

adjustability; uniformity of distribution of the pressure of contact; sliding friction and efficiency; thrusts on the bearings; mechanical strength and load carrying capacity; appearance.

In other words, Grant proved an overwhelming superiority of the involute tooth profile from any point of view of the designer’s interest. His conclusions are also supported by other considerations made by the same Grant in two of his previous works [161, 162]. Interpreting well this not reasonable situation, Woodbury [370] wrote that the general favor at that time granted by users to the cycloid curve compared

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Fig. 5.26 Title page of Théodore Olivier’s treatise on gears

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to the involute curve was due to a prejudice based on the long-continued custom, but not on intimate knowledge of the geometrical properties of the two curves. The involute tooth profile began to assert itself, gradually replacing the cycloid tooth profile, only in the last quarter of the 19th century. This gradual affirmation of the involute tooth profile was mainly due to the Brown and Sharpe company. In fact, according to a patent obtained by Joseph R. Brown (1810–1876) in 1864, concerning a generating cutter tool for gears, the Brown and Sharpe company demonstrated that, for cutting cycloid gear wheels of given diametral pitch, with z-number of teeth, included in the range 12 < z < ∞ (i.e., from a pinion with 12 teeth to a rack), needed a set of 24 gear cutters. Indeed, to cut involute gear wheels, having the same diametral pitch and the same quality grade, and number of teeth in the same range, enough a series of 8 gear cutters. The same company showed that, to cut involute gear wheels having the same diametral pitch and tooth numbers in the same range, but with better quality grade, was enough a set of 15 gear cutters, and that this was due to the fact that the involute cutters had less tendency to drag compared with cycloid cutters. All this brought about a wide acceptance of the involute tooth profile towards the end of the 19th century and the dawn of the 20th century. Actually, the gear generation principle was already introduced in 1842 by Joseph Saxton (1799–1873), who invented the first generation-machine to cut clock gear wheels to the true cycloid shape. This machine constituted the most amazing advance in gear cutting, since at that time cycloid tooth profiles could not generally be produced with sufficient accuracy by means of a cutting process using formed milling cutters. Saxton also invented a straight-sided milling cutter for the generation of involute teeth and developed the corresponding gear cutting machine. This machine was the first to work on the basis of the generation principle. It represented an important innovation compared to gear cutting machines with indexing devices then used in Europe, but required an engineer who understood the gear geometry. In this new type of gear cutting machine, instead of having an indexing device on the work table, the gear blank rotated continuously, synchronized with the cutter, which actually operated as a master gear reproducing itself in the blank. Given the complexity of the method, however, it began to assert itself only after 1850 (see [91, 303]). It should not be forgotten that the gear generation principle as well as the gear cutting machines that implemented it gradually contributed to today’s almost generalized use of profile shifted gearing. From the point of view of the history of technology, it is to be considered that the knowledge related to the profile-shifted gears were known from some time, and so, at least partially, the benefits that could be obtained with their use. However, as long as the only technology of gear cutting was the non-generation process, as the milling cutting, the profile-shifted gears could not actually find wide application, as, for the teeth cutting, it was necessary to construct two special milling cutters for each gear pair. The profile-shifted gears entered instead in the current technological practice when the gear generation cutting with rack-type cutter was introduced [370, 112]. However, even when the gear generation cutting began to spread, a lot of time was needed to understand the full potential of this technology in relation to the optimal design of the gears. In fact, initially, the profile shift was used for involute

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gears designed to mesh at the standard center distance, and thus emulating what happened in the cycloid gear pairs, which notoriously may mesh only at their standard center distance. In this way, the standardization, and thus the interchangeability and production of the catalogue gears, were privileged. Only in a second time the designer was able to understand the enormous benefits that could be drawn from the flexibility of use the profile shift, which, though within certain limits, allows to choose the values of the tooth thicknesses, center distance, and gear blank diameters so as to satisfy as much as possible the design requirements. The only possible restriction, consisting in the use of standard cutters for cutting profile-shifted gears, is not valid for the gear generation cutting, as one of the main advantages of the involute profile is to allow the cutting of non-standard gears by standard cutters. The new cutting processes of gear wheels, especially those based on the gear generation principle, posed new and unusual problems to be solved, giving a great impulse to the deepening of the theories already introduced as well as the development of new gearing theories. In this technological framework, it is worth mentioning the well-known hobbing debate of the end of the first decade of the 20th century, concerning the hobbing cutting processes that had become common at the beginning of the same decade. It is here to underline the double contribution of Ralph Flanders (1880–1970). In his work as a Machine magazine editor in New York City, between 1905 and 1910, he had intense relationships with most gear manufacturers, actively working on development in the machine tool industry. In this period, which preceded his direct occupation in one of these industries, he wrote several works on geared mechanisms, gear drives, more complex gear systems, gear cutting machinery, hobs, as well as on manufacturing processes of endothermic engines and automobiles. In this framework, he took part in the debate, then very controversial, lively and heartfelt, on the cutting technologies of gears using hobs as well as on the interchangeability guaranteed by involute profiles. Using well-known experimental results, as a first contribution he clarified the ideas to those who argued about the hobbing process, showing what was the exact shape of the tooth profiles obtained by this cutting process. As a second contribution, he promoted gear standardization, which became a topic of lively discussion, and was a significant architect for the definitive affirmation of involute profiles with respect to cycloid profiles. These contributions are evidenced by a large number of works from 1909 and 1910 (see, for example, [133–135]). Fundamental contributions to the affirmation of the gear generation processes and gear-cutting machines that implemented the same processes, are certainly those given by Edward Sang (1805–1890). He has the merit of having developed a fascinating general theory of gear teeth, able to give design value responses to the long-standing problem of gear interchangeability [325, 326]. Considering the case of gears characterized by more than one simultaneous point of contact, he defined the geometric locus of the tracing point, thus arriving at the principle of the hour-glass curve. This way of proceeding allowed Sang to put, in its most general terms, the problem of the minimum number of teeth and the related quantities for the involute and cycloid tooth profiles of the pinion. He was thus able to define the optimal tooth shape not only from the geometric point of view, but also from that of the minimum effect of wear

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on the variation of the teeth profile. Really lying the foundations of general analytical gear treatment, he also pointed out that gear teeth could be designed according to different criteria, such as: interchangeability and correct kinematic operation of the teeth profiles ; ease of manufacturing; minimization of friction and, consequently, of the teeth wear; insensitivity to the inaccuracy of center distance; etc. In this regard, he provided a detailed mathematical analysis of the design of gears characterized by minimum friction between the mating tooth profiles as well as the minimum effect of the wear on their operating conditions. However, the most important analytical-mathematical contributions by Sang on this topic concern the problem of gear manufacturing. In this regard, he first pointed out any significant correlation between theory and manufacturing practice of metal gears, hitherto ignored, thus contributing significantly to the success of gear cutting machines based on generation principles. He classified the processes of transformation by cutting from blanks to gear wheels, according to the following four classes, in order of increasing complexity: milling with formed milling cutters, broach, or cutting with single-point tool; cutting by generation with rack-type cutter or pinion-type cutter; cutting by generation with circular cutter following a predetermined curve; cutting by generation with cutter following the combination of the tracing point and the angular motion of the wheel. Only the first two of these four gear cutting processes gained real practical importance. The other two classes of cutting processes, for his own affirmation, allowed Sang to formulate and analyze the problem in its more general mathematical terms, for which they had no practical, but only speculative importance. Sang highlighted several features of the aforementioned cutting processes. For example, with regard to the non-generation cutting process by formed milling cutters, he focused on the original expense, the difficulties in making and sharpening the cutter, and the need for a huge set of cutting tools to cover each pitch and diameter. He also noted that all the formed cutting tools, of whatever nature they were, required checking by means of a template, and for this purpose he conceived the first gear comparator, which he called miglioscope, and described how to use it to guarantee the proper alignment of the tool with the gear blank axis. With this instrument, he demonstrated clearly that he learned well from instrument makers. Sang also faced the problem of cutting by generation with a rack-type cutter and described the use of this type of cutting tool in equally general terms, demonstrating its practical applicability in the production of gears. In this regard, he pointed out that all the gear wheels of a given pitch are cut with a single cutter and that any desired tooth-shape rack can be used. He also showed how to define and calculate the curves connecting the outermost parts of the involute profiles with the top lands of the teeth for an involute rack and how to make the cutting machine easily self-feeding, also demonstrating that this cutting process cannot be applied to internal gears. Moreover, Sang sensed the practical problems posed by the cutting process by generation with a pinion-type cutter, but thought that they were of such a size as to compromise the usefulness of the method. It was then Edwin R. Fellows (1865–1945) to demonstrate, in 1896, how this method could be applied in practice. Fellows in fact conceived a method whereby a pinion-type cutter (today also known as Fellows

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gear shaper, in honor of its inventor), which was itself a gear with hardened cutting edges, and the gear blank to be cut revolved together as if they were a gear pair. This cutting process, of course implemented with Fellows shaping machines, was successful because it did not require intermediate template or other type of mechanical guide for shaping the gear wheel. The innovation introduced by Fellows gave a vital contribution to the mass production of effective and reliable gear transmissions for the nascent automobile industry. Sang also noted that, whatever the cutting process used, the resulting teeth accuracy depended on the accuracy of the cutting-edge shapes. He however failed to note the easy of producing this accuracy in the involute rack. At that time, it was practically possible to get a truly hard cutting edge only on straight and circular edges. This is not the case to dwell on the last two classes of gear cutting processes introduced by Sang, since, as we have already said above, they had no practical application. This is because the special shapes of tooth profiles presented by Sang were, by his own admission, too complex to have practical application, albeit valuable for the breadth of analytical-mathematical treatment that they required. This laudable admission of Sang, which demonstrates practical sense of mechanical possibilities combined with the ability to analytically discuss gear teeth problems in the most general way possible, makes his contributions constitute the climax of all that had gone before and a transition to what was to follow [303]. Another great name to be mentioned is that of Oscar James Beale, whose contributions to the development of gear cutting processes are to be considered of fundamental importance. What little we know of Beale’s life only concerns his activity as a gear cutting machine designer. At the end of the 19th century, he was head of engineering of Brown & Sharpe company and, in this role, he distinguished himself as an excellent designer of new and innovative machines for cutting gear teeth. The gear cutting machines he designed were futuristic: just think of the well-known bevel gear cutting machines with two milling cutters, which are a masterpiece of mechanics, one of which is shown in Fig. 5.27. He also took an active interest in metrological problems, especially those related to gears, so much so that he designed and built the well-known odontograph that bears his name, which he used to realize and control the outlines of the gear teeth. Of course, this odontograph was designed to accurately describe the outlines of doublecurve cycloid gear tooth profiles we have already discussed, since until then the involute tooth profiles had not yet established themselves. Beale, however, was aware of the superiority of the involute tooth profiles compared to the cycloid ones, so much so that Joseph R. Brown, by marketing the tools for tooth cutting, championed the use of involute gear teeth, which were characterized by higher mechanical strength and minor losses due to friction between the teeth. However, since an inveterate habit, which was recalcitrant to take account of the considerations arising from new scientific knowledge, still favored the use of cycloid tooth profile until the late 19th century, the firm he worked for continued to make cycloid gear teeth. In the evening, Beale kept lessons at his house on the gears, but teaching he continued to learn, eventually inventing gear cutting machines that helped to satisfy

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Fig. 5.27 Beale’s bevel gear cutting machine with two milling cutters

the increasingly complex demands of the automobile market, which was already in overwhelming development. Beale wrapped his lessons in a book, first titled Beale’s Blue Book on Gear Wheels. With an addendum of the same Beale, at the request of the company for which he worked, this book was transformed into A Treatise on Gearing. Through subsequent revisions and expansions, this treatise became A Handbook for Apprenticed Mechanists, whose first edition was printed by J.W. Pratt in New York [26, 65]. Returning to the geometric-kinematic aspects of the gears, we must dwell on Franz Reuleaux (1829–1905), who is considered the father of the kinematics. He investigated the kinematic aspects of the gears, laying the foundations of a more in-depth understanding of how they operate [312]. Reuleaux is credited with the idea

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that a mechanism is a chain or a network of geometrically constrained bodies, which he also applied to gears. This idea, already present in Willis (see [263]), although in an embryonic manner, surpassed the theory of the so-called six classical simple machines (lever, wedge, screw, wheel and axle, pulley and inclined plane), defined by Renaissance scientists, but inherited from the aforementioned pseudo-Aristotelian text Mechanical Problems or Mechanica, still surviving and dominant until the late 18th century (see Chap. 2). In addition, Reuleaux developed what are today called Reuleaux’s thumb-shaped tooth gears [311]. In fact, Reuleaux elaborated upon asymmetric shape proportions, using tooth profiles that he called thumb-shaped teeth. These teeth had epicycloid drive tooth flanks and involute coast tooth flanks with pressure angle of 53°. Reuleaux himself wrote that, by combining epicycloid and involute, i.e. using the two curves for opposite flanks of the same tooth, a profile of great strength was obtained, and that this particular shape of the teeth was suitable for especial heavy-duty gear drives when motion was constantly in the same direction. As always happens in the evolution of technology and science, the idea of curved and different teeth profiles for their two opposite flanks was not new. As we saw earlier, in the section regarding the first scientific age, since the 3rd century B.C. the gear art included metal and wooden gears, with triangular teeth, buttress teeth, and pin teeth, while gear science almost certainly included curved toothed gears, with teeth profiles having geometry according to the Archimedes spiral. Even in Leonardo da Vinci notebooks we find many gear drawings, one of which has teeth like buttress teeth (see [370]). Willis was also a precursor of Reuleaux, as he had already stated that, in a machine whose wheels were only required to rotate in one direction, the teeth strength could be doubled through an alteration of the teeth shape, i.e. using asymmetric buttress gear teeth. To this purpose, he had also chosen to use epicycloid profiles for the drive tooth flanks and involute profiles for the coast tooth flanks, as Reuleaux did later. However, it is important to underline a specific contribution given by Reuleaux. In fact, Reuleaux’s triangle is a fascinating curve having constant width. Among other interesting geometric features, this closed curve has the property of being able to roll as smoothly as a circle on a flat surface. Therefore, with Reuleaux Gear Triangle related to this closed curve, it is possible to design and built planetary gear trains with interchangeable gear wheels, some of which may be circular gears, while others may be Reuleaux Gear Triangle, with the interesting peculiarity that both types of gear wheels can rotate at the same rotational speed. Therefore, the Reuleaux Gear Triangle is an unusual meshing of three gears in a triangle, were the constant width property of the Reuleaux triangle ensures that the teeth are in meshing each other at all times. Here we do not want to leave out the Reuleaux’s machines, many of which recall Leonardo da Vinci machines. Moon [263] establishes an interesting parallel between Leonardo and Reuleaux, and thus between Renaissance and late 19th century Industrial Revolution, with the aim of highlighting the scientific study of any machine, its codification into a language of invention, and its deconstruction into basic machine elements, with the gradual transition from an empirical knowledge, even if ennobled

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Fig. 5.28 Globoid worm gear pair according to Leonardo (a) and Reuleaux (b)

by an artist-engineer of absolute genius, to a scientific knowledge that is based on physical principles and makes use of mathematics. Moon also emphasizes the great inventive capacity of Leonardo da Vinci and Reuleaux concerning gears and their combination, enslaved to the achievement of well-defined design goals. A typical example of the curiosity about unconventional kinematic mechanisms that Leonardo and Reuleaux had in common is that concerning the globoid worm gear pair, shown in Fig. 5.28; it compares the two solutions proposed by Leonardo and Reuleaux for this innovative type of mechanism. About the globoid worm gear pair, it should be remembered that the doubleenveloping gear concept is very ancient, going back to the Renaissance. The development of the worm gearing principle progressed along conventional lines from the era of Archimedes, which is universally accredited as the inventor of worm gears, until the 15th century, when Leonardo da Vinci evolved the double-enveloping gearing concept. The generating process of globoid worms has also evolved over time, staring from the one introduced, in 1765, by the famous English clockmaker Henry Hindley (1701–1771) ). With this process (see [234]), which is considered the first process historically introduced in this specific field, the worm is cut by means of a trapezoidal-edged cutting blade, while the worm wheel is cut by a worm-type cutter which is identical to the Hindley’s worm. Today, several cutting processes are used, with different gear cutting machines and different cutters. The methods gradually introduced for the geometric and kinematic study of gears having non-parallel and non-intersecting axes, that is arranged in any way in the three-dimensional space, deserve our attention here. The first fundamental theoretical contribution on this topic is that of Mozzi del Garbo (1730–1813), of 1763, almost immediately followed by that of Euler, of 1765 (see Sect. 10.1 in Vol. 1). These scientists developed the theory of the screw motion, which constitutes a very elegant and refined way of analyzing the motion between two bodies with axes arranged in the three-dimensional space in any direction. Many subsequent researchers contributed to the application of this theory to gears, but the first to offer an in-depth treatise

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on this subject was Robert Stawell Ball (1840–1913), with his two monographic publications of 1876 and 1900 [20, 21]. Cormac’s contribution must also be noted. He in fact, in 1936 [86] published a monographic treatise on screws and worm gears, characterized by a high-level mathematical discussion, which also included an accurate description of their manufacturing processes and practical applications. An extensive part of this treatise was dedicated to the cutting processes of various types of worm gears and shapes of hobs as well as to the mathematical analysis of the shapes required to satisfy specific design conditions in relation to the given applications. The worm gears are, however, a special case of non-parallel and non-intersecting gears, so they introduce us to the topic concerning spatial gearing. The fundamentals of spatial gearing theory had already been introduced by Olivier [283], who considered the generation of conjugate surfaces as an enveloping process and for this purpose introduced the concept of an auxiliary generating surface. These fundamentals, based on projective geometry, led Olivier to define the conditions of line and point contacts. However, projective geometry has well-known limitations. Aware of these limitations, Gochman [157] sensed that in order to overcome them, it was necessary to face gearing theory with the methods of analytical and differential geometry. The mixed method, which simultaneously uses the projective geometry and the analytical and differential geometry for the generation of tooth profiles, is now referred to as the Olivier-Gochman generation approach. This mixed approach, later generalized by Davydov [93] in order to apply it to the case of non-rigid and non-congruent pairs of generating surfaces, is today the basis of any method of the generation gear cut by enveloping process. The generation process of the Novikov gears is different from that corresponding to the aforementioned approach, but it shares the same bases, since its industrial implementation is only possible applying the basic rack profile of a pair of non-congruent generating racks [160]. Even the recent outlined trend of cutting gears by pin tools with CNC machines does not save the design engineers from the need to cut teeth using virtually the Olivier-Gochman generation approach. Further developing Gochman’s methods, which require a remarkable mathematical apparatus since they are based on analytical and differential geometry, Kolchin [204, 205] refined them in such a way as to make them suitable for solving important practical tasks, such as: bevel gears, non-orthogonal worm gears with cylindrical worms, double-enveloping worm gears, profiling of cutting tools, analysis of errors in gearing, etc. Kolchin also proposed a parabolic tooth flank profile modification in order to make the gears less sensitive to misalignment of the axes. The methods conceived by Olivier-Gochman and Kolchin were, however, too burdensome when they were used to solve the increasingly complex geometric and kinematic problems related to spatial gearing. New methods of analysis were therefore necessary for the solution of these problems. They did not take long to be developed and were based on the use of matrix algebra, as this showed to be suitable for automatic computation on high-speed digital computers. Matrix algebra had already been developed and refined in 1858 by the English mathematician A. Cayley

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[72]. The study of gears, which is fundamentally based on a repeated transformation of equations describing curves and surfaces in three-dimensional space from one coordinate system to another, almost immediately made a treasure of these new and powerful methods of analysis. Litvin [225] credits S.S. Mozhayev with being the first to use, between 1948 and 1951, the matrix algebra for the analytical representation of coordinate system transformation (see also [303]). Both multiplication and addition of matrices are necessary in matrix operations where any position vector in the three-dimensional space is represented with the usual Cartesian coordinates. If instead homogeneous coordinates are used for its representation, only multiplication of matrices are necessary. The use of such coordinates for coordinate transformation in the theory of mechanisms, including gears, has been proposed almost simultaneously by Denavit and Hartenberg [98] and Litvin [225]. The next step in the development of a gearing theory was the introduction of a kinematic method of surface generation, based on the simple consideration that, at any point of contact between two mutually enveloping surfaces, their common normal of contact, given by vector n, had to be perpendicular to their relative velocity vector, v. This kinematic approach to the study of gears, which required a remarkable use of matrix algebra, was proposed between the end of 1940s and the beginning of 1950s by three researchers of the Russian School, independently of each other: V. A. Shishkov, J. S. Davydov and F. L. Litvin. The contributions of these three researchers are not clearly definable. According to Lagutin and Barmina [210], Shishkov would have described such an approach verbally in a conference held in 1949, while Davydov in 1950 would have formulated the same approach, expressing it through projections of the two aforementioned vectors. Finally, in 1955, Litvin would have formulated the equation of contact, or meshing equation, in the concise form of scalar product of the two vectors, v and n, that is in the form v · n = 0, which today has wide application in the field of gears. Instead, according to Radzevich [303], this meshing equation would be attributed only to Shishkov, who would have published in 1948 the results of his research concerning the development of a kinematic method of surface generation, largely based on the inter-correlations between the normal of contact to interacting surfaces and the vector of relative velocity. Radzevich himself calls this meshing equation Shishkov equation of contact Therefore, Davydov and Litvin would only have refined the kinematic method proposed by Shishkov and the related meshing equation. A particular historical mention deserves to be done on non-circular gears, which are notoriously used for the generation of a prescribed transmission function or as a generating driving mechanism to modify the displacement function or the velocity function. It is not possible today to know with certainty who first introduced these non-circular gears. We have already said in Sect. 4.2 that the Dondi’s manuscript of the Tractatus Astrarii contains one of the earliest documented procedures for design of non-circular gears (more precisely, elliptical gears) included in gear trains used to account for the apparent irregular motion of the planets [1]. Holditch [183] was certainly one of the first authors to describe non-circular gears in a publication.

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Grant [163] was instead the first to envisage a practical method of non-circular gear manufacture that, however, did not always guarantee a conjugate action between the mating tooth profiles, as it approximated the pitch curve using an osculating circle. Previously, we also made a quick mention of the Reuleaux Gear Triangle coupling mechanism, which is also a non-circular gear. The analytical discussion of non-circular gears dates back to very recent times, for at least three reasons: first, these types of gears received little attention from designers, who did not recognize their potential as components of geared mechanisms; second, few gear manufacturers were able to build them with due accuracy; third, large amount of computations were necessary for their accurate design and manufacture. The most widespread analytical discussion concerns non-circular gears between parallel axes and, particularly, elliptical gear pairs, which represent the most common types of non-circular gears used for practical applications. Gobler [156] and Ollson [285] describe interesting applications of these gears concerning motion transmissions between parallel axes in printing presses and other industrial machines (with gear wheels that rarely exhibit a complete rotatory motion) and, respectively, between intersecting axes. Ollson himself [284] had used the wealth of knowledge concerning elliptical gears to demonstrate the design and manufacture of non-circular gears. To this end, he had simulated the pitch curve of any non-circular gear using a sufficient number of elliptical segments, thus reducing the analysis of non-circular gears to that of elliptical gears. Other interesting contributions for their understanding and development of accurate theories for their definition, are those of Litvin [226], mainly concerning the design and manufacture of non-circular gears, and the subsequent ones of Cunningham [92], Bloomfield [46] and Benford [32], concerning various methods of synthesizing pitch curves of these types of gears. The gears with circular arc tooth profiles also deserve to be mentioned. They were introduced independently in the USA and USSR, respectively in 1926 by Wildhaber [363] and in 1956 by M.L. Novikov (he is the owner of the U.R.S.S. Patent no. 109,750 of 1956), to realize particular helical gears by means of generation processes using circular-arc rack type cutters. These gears are commonly referred as WildhaberNovikov gears. However, even if they have a common denomination, these gears present a significant difference, since the mating tooth surfaces of Wildhaber gears are in line contact, while those of Novikov gears are in point contact. This point contact in Novikov gears is obtained using two mismatched rack type cutters for generation of pinion and, respectively, gear wheel. The gears with these types of profiles have the advantage of having a reduced effective curvature, since the centers of curvature of the two mating surfaces at the common point of contact are localized in the same half-space with respect to the plane of tangency through the same point of contact. This implies, in turn, a reduction of contact stresses and a higher strength to pitting, wear, scuffing and tooth root bending fatigue. Other features of these gears, whether they are advantageous or disadvantageous with respect to the involute gears that are those considered in Vol. 1 and Vol. 2 of this monothematic textbook, are described with greater details in wider textbooks, such as for example Dooner and Seireg [106] and Litvin and Fuentes [228].

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We consider it useful to conclude this section with a succinct mention of new practical applications of gears which, albeit with the due changes gradually introduced over time, were conceived centuries ago or millennia ago. This demonstrates what we have repeatedly said that the gears are ancient science in continuous updating. Without losing generality, we limit ourselves here to considering the planetary gear trains and face gears. Furthermore, we describe a futuristic application, which will become history I cannot specify in how many gears, when the theoretical analyzes concerning geometric-kinematics problems, including those inherent to cutting technologies, and experimental research will have demonstrated its technical feasibility: we are talking about the double-enveloping recirculating ball worm drives. About the planetary gear trains, we leave aside their distant origins, whose first historical documentary information and first archaeological evidences date back respectively to the third century and the first century B.C. (see Chap. 3). Instead, we focus on what happened in the first half of the twentieth century of our era, driven by the automotive industry. It was indeed this industry that promoted the birth and development of the first traditional automatic geared transmissions for motor vehicles, consisting of the combination of several planetary gear trains, whose main members were differently connected to each other and with the housing in order to obtain the desired number of transmission ratios. The first automatic transmission, which switched the optimum gear without driver intervention, except for starting off and going in reverse, was developed by General Motors, based on ideas and patents dating back to the first thirty years of the 20th century (see [39, 257, 359]). The same General Motors introduced the related technology in the 1940 Oldsmobile model, as Hydra-Matic Transmission. Figure 5.29 shows a schematic diagram of this automatic transmission, which consisted of a fluid coupling, three brakes, two clutches and three planetary gear trains, the first to responsible for obtaining four forward speeds and the third for reverse velocity. With the exception of CVTs-(Continuous Variable Transmissions), current automatic transmissions are basically very similar to this automatic transmission, as the few differences are limited to the use of more sophisticated hydraulic and electronic systems, which are responsible for their operation and control. Of course, automatic transmissions were developed on the basis of manual or semi-automatic gearboxes, which used planetary gear trains. The Wilson gearbox, which relied on a number of epicyclic gear trains, coupled in an ingenious manner (it is not to be confused with the pre-selector or self-changing gearbox, due to the same inventor, Wilson, W.G.), can be considered to be the precursor of these automatic transmissions. It had a planetary gear train for each intermediate gear, with a cone clutch for the straight-through top gear, and a further planetary gear for going in reverse (see [129, 216, 257, 268]). As we have already seen in Chap. 2, face gear drives are among the oldest types of gears of which we have historical memory. Nevertheless, they can be considered as futuristic gears, because we do not know well yet all their features, and above all their potential in relation to possible practical applications, in competition with other types of gears. These types of gear drives, albeit in a rudimentary form, have been used by the Chinese (see, for example, the chariot compass, described in Chap. 2, as

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Fig. 5.29 Schematic diagram of Hydra-Matic automatic transmission

well as in ancient Egypt, Mesopotamia, and in the Roman world, for waterwheels, watermills and windmills (see also Chap. 2). A face gear, consisting of a face gear wheel with pins, and a lantern pinion or a pinion with pins, in fact characterized these ancient practical applications. The revival of interest in these types of gears came from the middle of the past century [45, 74, 75, 138]. This interest was intensified towards the end of the same century, and the beginning of this 21st century [3, 24, 165, 169, 171, 202, 221, 229, 230, 376]. This interest is also witnessed by specific research programs, in the USA, in Europe and perhaps in other countries. The reasons of this delayed interest on these types of gears are to be found in the fact that the benefits of using face gears in the helicopter transmissions also began to be obvious with some delay. Early studies on the subject clearly showed that the use of face gear technology would have allowed a power density improvement and lower cost when applied to a helicopter transmission. In fact, the ability of face gear pairs to provide high reduction ratios as well as a self-adjusting torque splitting, allows to replace conventional multi-stage gearboxes with gear units requiring a lower number of stages. This leads to a better power/weight ratio, a small number of parts of the gearbox, and a reduction of volume. In addition, the split torque design of this type of transmission offers improved reliability and reduced costs against existing conventional gearing designs used in large horsepower applications.

184 Fig. 5.30 Split of torque for a helicopter transmission, with: a spiral bevel gears; b face gears

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(a)

(b)

A helicopter transmission must meet, among other things, the following two requirements: it must allow a split of torque from the engine shafts to the combining gear; it must be able to ensure a drastic speed reduction from that of the engine shafts (the turbine shafts) to that of the main rotor. The combining gear is rigidly connected with the input member, the sun gear, of a planetary gear train, whose output is the planet carrier, which drives the main rotor shaft. Figure 5.30 highlights the concept of split of torque. Figure 5.30a shows a more traditional design solution, which uses two spiral bevel pinions forming a single rigid body, which drive two mating spiral bevel gear wheels that transmit power to the combining gear. Figure 5.30b shows instead a far more advanced design solution, in which a single spur (or helical) pinion is in mesh with two face gears. In both figures, the transmission part made up of the planetary gear train, which drives the main rotor, is omitted. The same Fig. 5.30 shows that the solution with face gears has at least two main advantages (with all that follows) over the one utilizing spiral bevel gears. On the one hand, the torque split configuration allows to transmit reduced loads on bearings that support the various shafts, and, on the other hand, the pinion is single and consists of a simple spur (or helical) pinion, compared to the double spiral bevel pinion, which in itself is much more complex. It results in greatly reduced weight and cost. This is another very important design goal, also because the transmission is called to support the weight of the entire aircraft, including that of several flight accessories. As an example of new applications that, in due course, will become history, here we want to briefly describe the double-enveloping recirculating ball worm drives. They belong to the extended family of special worm gear drives described in Section 11.15 of Vol. 1. They are based on the combined use of two mechanical operating principles, that of worm drives described in Chapter 11 of Vol. 1, and that of recirculating ballbearing screws (see [207]). The basic theoretical concepts of worm drives date back, as we already know, to Archimedes, while those of double-enveloping worm drives are more recent. The principle of operation of these latter geared mechanisms was

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certainly known to Leonardo, who made extensive use of it, as clearly shown in Figs. 4.33 and 4.40, and most probably, until proven otherwise, he was also their inventor. The ball-bearing screws date back to 1898, when Stevenson, H. M., and Glenn, D., obtained respectively US patent 601451, dated 1898.03.29, and US patent 610044, dated 1898.08.30. Simultaneous use of the aforementioned mechanical operating principles allows to have a worm mechanism where the surfaces of worm threads and worm wheel teeth are no longer in direct contact, and motion and power are transmitted through the balls rolling between them. The key feature that can determine the success of such mechanism is the replacement of the sliding friction between the meshing surfaces of the worm threads and worm wheel teeth by the rolling friction of spherical balls complemented by a smooth recirculation path implemented into these two members. Unfortunately, however, despite studies and research on these special worm drives, which have resulted in the release of numerous international patents, nobody has even been able to implement and fine-tuned a double-enveloping recirculating ball worm drive capable of ensuring correct operating conditions. These conditions include a proper kinematic coupling between worm and worm wheel, mediated by balls, as well as the smooth ball recirculation in the appropriate raceways, and the appropriate and continuous contact between the same balls and between the balls and side surfaces of the raceways. For a traditional worm drive, where contact between the active mating surfaces is a direct sliding contact, the cutting process of the two worm drive members are those described in Chapter 11 of Vol. 1. When contact between the active worm and worm wheel surfaces is an indirect rolling contact, because it is mediated by the interposition of rolling balls, especially when worm and worm wheel have both hourglass shape, these cutting processes are much more complicated. This is because, in the axial plane of the worm, for the worm, and in the transverse plane of the worm wheel, for the worm wheel, the sections of the raceways cannot be simple circumference arcs for both these two worm drive members. In other words, the traditional cutting motions for the generation of a helicoid thread on a toroidal surface using a spherical head-cutting tool (rotating motion about the worm axis and simultaneous rotation of the cutter about the torus axis) do not generate an appropriate envelope. In fact, the surface obtained by envelope do not have geometry such to avoid both the interference (this would result in balls jamming inside the raceways), and excessive backlash (this would result in loss of contact). The appropriate coupling on all contact surfaces between balls, raceway on the worm, and raceway on the worm wheel are not guaranteed. The problems still to be solved in order to realize a double-enveloping recirculating ball worm drive that has a correct kinematics are still numerous and complex. Further theoretical and experimental studies are needed. The author, in conjunction with his research team at the University of Rome “Tor Vergata” (see [258]), made a small contribution in this regard, as well as the two patents RM 2004A000138 (2004) and PCT/IB2005/050898 (2005) demonstrate. However, the road to solving all the aforementioned problems is still very long and bumpy, and the time to talk about the history of this particular geared mechanism in the current state of knowledge is not

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Fig. 5.31 Conceptual drawing of speed jack with double-enveloping recirculating ball worm drive

predictable. Figure 5.31 shows an example of conceptual drawing of speed jack with double-enveloping recirculating ball worm drive studied at the University of Rome “Tor Vergata”.

5.3 From Galileo to Today: Entry of Solid Mechanics and Material Strength Theories in Gear Design So far, we have focused our attention on the geometric and kinematic aspects of the gear theory as well as on the main aspects of gear cutting processes, and we have left aside those concerning their mechanical strength and load carrying capacity. With wooden gears, speeds and loads were very low, so that the problem of the gear strength remained an empirical problem, which was solved using equally empirical and roughly approximated methods, depending on the various types of wood and operating conditions. We do not know anything about the procedures, more or less empirical, used in antiquity to take into account the mechanical strength of the wooden gears, as nothing has been handed down in this respect. In fact, the literature we inherited only tells us how gears were used to meet specific needs and practical applications. The study and analysis of the mechanical strength and load carrying capacity of the gears are closely related to the development of the theory of elasticity. The problem of the bending strength of prismatic and cylindrical beams was first dealt with by Galileo Galilei, in 1638 [146]. Figure 5.32 shows a cantilever beam with load at the end, according to Galileo’s illustration of a bending test. However, while clarifying many aspects of beam bending, Galileo fell into the error of considering

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Fig. 5.32 Bending test with a cantilever beam according to Galileo’s illustration

the fibers all stretched, so he lacked the concept of neutral axis. As we have already highlighted in Sect. 5.1, Galileo has the merit of having laid the foundations of the beam bending strength theory and giving us a complete derivation of the parabolic shape of a cantilever beam of equal strength, whose cross-section is rectangular [341]. The theoretical bases of the bending strength of gear teeth and their root load carrying capacity are therefore to be attributed without any doubt to Galileo [179]. The next contribution on this topic was that of Hooke who, in 1678, based on systematic experimental research, enunciated the law of proportionality with the well-known aphorism “ut tensio sic vis”, i.e. “as the extension, so the force” [185]. Therefore, Hooke discovered the law of elasticity that bears his name and describes the linear variation of normal stress with extension of an elastic body. Despite its limits of approximation, this law greatly expanded the boundary of statics, expanding them from rigid to deformable bodies. However, it is here to remember that the prevailing consensus that Galileo was the first to attempt to develop a beam theory [341] must now be reviewed as recent studies have shown that Leonardo da Vinci was the first to make crucial observations on this topic. Leonardo in fact, about one century before Galileo, made a fundamental contribution to what is commonly referred as Euler-Bernoulli beam theory. Galileo is not mentioned in the beam theory together with Euler and Bernoulli, given his well-known error according to which the stress distribution in any cross section of the transversely loaded bent beam was a constant. The Euler-Bernoulli beam theory is so called because first Jakob Bernoulli (1654–1705) made significant discoveries

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and, subsequently, Euler and Daniel Bernoulli (1700–1782) were the first to put together a useful theory around 1750 [150]. Notoriously, the beam theory is based on the assumption made by Jakob Bernoulli at the end of the 17th century according to which the strain is proportional to the distance from the neutral surface, the constant of proportionality being the curvature. Today we are sure that Leonardo hypothesized this type of deformation two centuries earlier. This certainty derives from the folio 83 of the Codex Madrid I, one of the two notebooks that were discovered in 1967 in the National Library of Spain in Madrid. Examining the strain distribution in the beam/spring with rectangular cross section represented in this folio and the translation made by Zammattio [375] of what Leonardo wrote around the drawing, Ballarini argues that in this drawing of Leonardo, who meditates on the elastic strain of spring, all the essential features of the strain distribution in a beam are already established [22]. Figure 5.33, taken from the aforementioned work by Zammattio, shows the straight beam/spring considered by Leonardo, the underlying inflected configuration with related strain distribution and the still underlying Leonardo’s comment, in its typical specular writing, from right to left, readable by reflection with a mirror. The Leonardo’s comment, in the translation made by Zammattio himself, is as follows: “On the bending of the spring: if a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring. You shell discover, if you consider all of the aforementioned modifications, that by taking part ab in the middle of its length and then bending the spring in a way that the two parallel lines a and b touch at the bottom, the distance between the parallel lines has grown as much at the top as it has diminished at the bottom. Therefore, the center of its height has become much like a balance for the sides. And the ends of those lines draw as close at the bottom as much as they draw away at the top. From this you will understand why the century of the height of the parallels never increases in ab nor diminishes in the bent spring at co”. The pyramidal stress distribution represented by Leonardo in his drawing as well as his commentary accompanying the same drawing show how Leonardo has clearly inferred, for equal tensile and compressive strains at the outer fibers of the bent beam/spring with rectangular cross-section, the existence of a neutral surface and a linear strain distribution. In agreement with Ballarini, we can conclude that if Leonardo had available Hooke’s law and the calculus, it is conceivable that he would have derived the formulae that correlate the normal stress with bending moment and curvature for slender linear elastic beams, which express what is universally referred to as Euler-Bernoulli beam theory. So today this theory would be referred to as da Vinci beam theory. As a complement to what Ballarini argues, we want to add here that if it is true that Leonardo did not have available the Hooke’s law (i.e., the extension is proportional to the force), it is equally true that the physical principle of proportionality of the linear elasticity theory was very clear to him. It is clearly expressed in the sentence … “the distance between the parallel lines has grown as much at the top as it has diminished at the bottom”. Moreover, he had equally clear in his mind what we now call Poisson

5.3 From Galileo to Today: Entry of Solid Mechanics and Material … Fig. 5.33 Strain distribution in a bent beam/spring and Leonardo’s comment: Codex Madrid I, folio 83

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effect, i.e., the transverse strain phenomenon according to which a material tends to expand (or contract) in direction perpendicular to the direction of compression (or stretching): it is clearly expressed in the first sentence of the aforementioned Zammattio’s translation. For almost two-thirds of the 18th century, only a few conceptual extensions and partial corrections to the results of Galileo and Hooke were introduced by Leibniz [217], Mariotte [244], and Bernoulli [34], concerning respectively the elastic response of loaded beam, the position of the neutral axis, and the conservation of the flat sections with strains proportional to the distance from the neutral surface. However, substantially, the scientific community persisted in Galileo’s error until around 1750, when Leonhard Euler, Jakob Bernoulli and Daniel Bernoulli finally resolved the bending problem in the correct terms that we know today. A few years later, Coulomb [87] defined the basic criteria for the study of bending and shearing and, subsequently, also those for the study of torsion. As for the bending, Coulomb developed the equation to determine the position of the neutral axis, according to the setting given by Parent [289]. Still for some years after Coulomb, the right direction he had marked was not followed, not even by Young [372], who however introduced the modulus of elasticity or Young’s modulus, and developed all the cases of simple resultant stresses. Navier’s contribution [269], which is considered the founder of today’s theory of elasticity, is fundamental. Navier gathered all previous research and gave them consistency and unity, thus establishing the foundations of the mathematical theory of elasticity and setting for the first time the general equations of equilibrium. The equally decisive contributions of the same period, with extension to the study of the hyperstatic problems, are also to be remembered. They are due to mathematicians and engineers working in the French cultural sphere of the École Polytechnique and the École Nationale des Ponts et Chaussés, as Cauchy [68–71], Poisson [295], Lamé [212], Clapeyron [77] and finally Barré de Saint-Venant [323], whose contribution to the study of linear structural elements is to be considered fundamental (see also [60, 208, 337, 345, 346]). The following studies were directed to the solution of particular problems of the continuum mechanics. Among these studies, those related to the work of deformation must be remembered and, specifically, those carried out by the Italians Menabrea [252] and Castigliano [67]. To Menabrea we owe the theorem of minimum work, while with Castigliano we are indebted to the theorem of work derivatives. The contribution of Mohr [261, 262] must also be highlighted, as Mohr was the first to apply the principle of virtual work to the problems of structural calculation, which are of interest here. All the aforementioned studies, mostly aimed at solving hyperstatic problems, have been increasingly expanded and deepened, with further noteworthy contributions (e.g.: [80–84, 316–318, 353], etc.). The subsequent tendency was to investigate the field of plastic deformations, for a better exploitation of the strength properties of the materials. In this framework, the theories of plasticity, elasto-plasticity and viscosity-elasticity have been developed and, more generally, theories on the coaction states [136, 182, 199, 332].

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The following scientific contributions are to be considered of particular importance for the gears: the theory of Hertz [178], for the definition of contact stresses over the contact area; the extension of the Hertz theory made by Belajev [29, 30], to define the three-dimensional stress state below the contact area; the strength theories and strength criteria to define an equivalent stress to be compared with the limit strength of material; the hydrodynamic lubrication theory developed by Reynolds [314] and, above all, the elastohydrodynamic lubrication theory that makes Hertz theory and elastohydrodynamic lubrication theory interact; the tribology and wear criteria; the fracture mechanics and damage criteria. The strength theory, which is at the basis of the calculations concerning the load carrying capacity of the gears, starting from the aforementioned intuitions of Leonardo da Vinci, made the first steps with Galileo Galilei and established itself as such with the contributions of Coulomb (1736–1806) and Mohr (1835–1919), so much so it is still defined as Mohr-Coulomb strength theory (see [87, 261, 262]). Its boundaries were progressively extended to include, today, yield criteria and failure criteria, multiaxial fatigue criteria and multiaxial creep conditions as well as material models in computational mechanics and computer programs [373]. Considerable theoretical and experimental research has been carried out on the strength capacity of materials under complex three-dimensional stress state in the 20th century, in order to formulate strength theories congruent with the test data. Hundreds of models or criteria have been proposed, but so far, no single model or criterion has shown itself fully suitable, so the subject remains substantially open. Regarding the strength criteria, the criterion universally known as von Mises criterion should be particularly mentioned, although it would be better to call it (Maxwell/Huber/von Mises/Henchy)-criterion or briefly MHMH-criterion. This is because, starting from the Maxwell’s initial contribution dating back to 1856 [248], it was progressively enriched by the contributions of Huber [188], von Mises [353] and Henchy [173]. This criterion was predominantly interpreted as distortion energy criterion, according to what was done by Henchy. The denomination of distortion energy criterion, introduced by the same Henchy, remedies an error of historical perspective, which removed the due merit to Maxwell, who was the first promoter of the criterion, as well as that of the other two scientists mentioned above who, together with von Mises, enriched it with others more than appreciable contributions (see [182, 332, 341, 356, 357]). In 1933 Nadai [264–267] conceived the octahedral shear stress criterion and demonstrated that it can be traced back to the distortion energy criterion. Subsequently, in 1952, Novozhilov [281] demonstrated that the distortion energy criterion was interpretable as the root mean square of the shear stresses for all intersection planes. Still later, in 1968, Paul [291], in the wake of Nadai, interpreted the distortion energy criterion as the root mean square of the principal shear stresses. However, it should be remembered that at the time when beam theory and theory of elasticity were developed, science and engineering were generally seen as very distinct fields, and considerable doubts existed that an academical mathematical product could have practical applications, useful and sure. Bridges and buildings continued to be designed in an empirical manner, drawing on the only accumulated

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experience, until the end of the 19th century, when the Eiffel Tower and Ferris Wheel demonstrated the validity of theories developed on a large scale. Naturally, the gears did not have a different story. They also continued to be designed according to empirical criteria, as if strength theories already conceived and in the course of refinement did not exist. In this regard, Polder [296] highlights that the beginning of the studies concerning the load carrying capacity rating of the gears, based on concepts of the theory of elasticity, can be traced back to the systematic tests of Pieter van Musschenbroek (1692–1761) of the University of Leyden, which are described in his two works Elementa Physica (reworking of the previous work of 1726, titled Epitome elementorum Physico-matematicorum conscripta in usus academicos) and Dissertationes physicae experimentalis et geometricae de magnete, respectively published in 1726 and 1729. According to Polder in fact this Dutch scientist systematically analyzed and checked the wooden teeth of windmill gears, applying the bending strength formula published by Galileo about one century earlier [146]. From the second of the aforementioned publications [351], it is however certain that he performed pioneering work on the buckling of compressed structures and provided detailed descriptions of testing machines for tension, compression, and flexure testing. Throughout the following century, many scientists took care of the load carrying capacity of the gears, trying to improve or extend the Galileo’s formula. However, only at the beginning of the 19th century, the strength of the teeth and gears became a significant topic of discussion and analysis. In those years, it was clear that the geometric-kinematic perfection of tooth profiles was not enough, and that it was necessary to associate to it the mechanical perfection. The awareness of the need for mechanical perfection of the gears is to correlate closely with the appearance of cast-iron gears, which lent themselves better to use for more high speeds, and more heavy loads. On the other hand, the uniform mechanical characteristics of the metallic materials compared to those of the wood allowed a more systematic approach of the problem of mechanical strength, in the light of theory of elasticity, which was establishing itself as an autonomous science. A first rough attempt to calculate the mechanical strength of structural elements, including gear teeth, can be found in a Tredgold’s work published in 1820, which contains a chapter dealing with strength of materials [343]. This initial attempt was based on the already mentioned Young’s work [372], but it turned out very inaccurate, as in Tredgold lacked the mathematical basis of the theory of elasticity. In a later Tredgold’s work, published in 1822 [344], the results of experiments by the author are described and many practical rules for designers of cast-iron structures are given. Previously, engineers relied on the formulae and results found in authors of even coarser writings, including the aforementioned Peter van Musschenbrock’s Dissertationes. In the already mentioned 1823 edition of the work of Buchanan [57], edited and integrated by Tredgold, the latter included a very elaborate calculation of the tooth strength of the main types of gears, made of wood or cast iron, with charts, graphs and tables for its more user-friendly. However, this calculation procedure was essentially a still empirical calculation, along the line of that previously developed and used

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for the wooden gears. However, William Fairbairn (1789–1874), showed that the results obtained with the coarse theoretical procedure proposed by Tredgold roughly coincided with those obtained with the empirical procedures of that time [124, 125]. Even Franz Reuleaux (1829–1905) dealt with the strength of the gears. For this purpose, he used the analytical mechanics much more than his predecessors had done, but also the results obtained by him are not to be considered very accurate and reliable, partly because they depend largely on the elastic constants of the materials, which at that time were not known with sufficient accuracy [312]. This is because the material science was not sufficiently developed, and material testing machines were not existing or very coarse [351]. All the above-mentioned models used for the analysis of the tooth root strength simulated the tooth as a cantilever beam having a constant resistant section, clamped at one end at the fixed support, consisting of the gear crown rim, and loaded at the free end, by a force vector acting parallel to the clamped section. This section was considered as the most stressed resistant section. These models, which also considered the cantilever beam subjected to pure bending, and therefore neglected the shear stress, are here called primitive models to distinguish them from the Lewis’ model, and its variations. They go from van Musschenbroek’s model to the Reuleaux’s model. With these models, the stress distribution at the clamped section is linear, with a maximum tensile stress at the side of the load application, and a maximum compressive stress at the opposite side, and equal in absolute value to the maximum tensile stress. However, the real turning point on the calculation of the load carrying capacity of the gears occurred only towards the end of the 19th century. In fact, the first universally recognized analysis of the stress state in the gear teeth was the one made in 1892 by Wilfred Lewis (1854–1929). This is confirmed by the fact that, with the necessary adjustments resulting from new knowledge gradually acquired over time, the Lewis’ method still serves as a conceptual basis of the analysis of the gear tooth bending strength [222]. Lewis considered the gear tooth as a cantilever beam built-in at the gear crown rim, and loaded at its free end by a concentrated force vector, acting at pressure angle and distributed equally across the tooth face width. However, the real revolutionary idea proposed by Lewis was the introduction of the uniform strength parabola, the Lewis’ parabola, inscribed within the two profiles delimiting the tooth externally. Figure 5.34, taken from the original work of Lewis, shows at the top two parabolas that simulate the Galileo’s constant strength cantilever beam already described in Sect. 5.1. Points of tangency of these parabolas with the opposite tooth flank profiles allows to identify the critical cross section of the tooth, since it is evident that the tooth is everywhere stronger than uniform strength parabola, with the exception of points in which the parabola is tangent to the tooth outline. However, we must not forget the significant and specific contributions on the same topic, given in the late 19th century by Richard Stribeck (1861–1950) and Oscar Lasche (1868–1923), as well as the contributions, although non-specific, given by von Bach (1874–1931). This latter at that time was specializing the applications of the theory of elasticity and strength of materials to machine design, not disdaining

Fig. 5.34 Lewis parabola and critical cross section of the tooth, from the original work of Lewis

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to contrive experimental solutions to stress analysis problems concerning machine elements and parts (see [353, 215, 338, 339, 352]). By the early 20th century, studies and researches on the gears are intensified. The gear teeth in action begin to be considered, while the problems are deepened concerning both the mechanical strength characteristics of materials, gradually introduced for their manufacture, and the cutting processes of the teeth and production technologies of the gears. Already Lasche [215] foresaw the likely effect of manufacturing errors of the teeth as well as the high increases of load that could result from them, especially at high pitch-line velocities. Ralph Edward Flanders discussed then the problem of the likely nature of the dynamic load on the teeth [134, 135]. According to Flanders, the dynamic loads were correlated not only to manufacturing and mounting errors and to the initial masses in rotational motion, but also to dynamic effects able to cause an imperfect meshing even with perfectly shaped teeth, however deformed by the load transmitted by them. To shed light on the above subject, and to give a first comprehensive response to questions posed by Lasche and Flanders, Marx and Cutter [246] conducted an extensive series of tests with cast-iron gears, with pitch-line velocities up to 10 m/s. For this purpose, they used an experimental apparatus, previously developed by Marx himself, which also was the initiator of the same series of tests, begun in 1911. Using the same apparatus of Marx, Franklin and Smith [140] made another series of tests, always on cast-iron gears, however varying their accuracy grade. Fisher [130] reports that, in reality, the first to suggest the introduction of a dynamic load factor in calculations of the gear strength was Walker, E. R., in 1868 (see also [287]). This factor, originally called speed factor, was defined as the quotient of the static load divided by the dynamic load. It was determined empirically, by comparing the values of the tooth fracture strength at different operating speeds. Towards the late 19th century, based on Walker’s factor, Barth, C. G., (see [59]) proposed a new empirical equation to calculate the dynamic factor. In this equation, subsequently called Barth’s equation, together with a constant term depending on the material and accuracy of the gear cutting technology, the pitch-line velocity appeared explicitly, in the form of a linear term. Franklin and Smith [140] confirmed the Barth’s equation for gears characterized by a certain pitch error, but revealed that the constant term appearing in this equation had to be changed, to take account of different pitch errors. On the other hand, also the tests performed by Marx between 1910 and 1915 showed that the values of the dynamic factors calculated with the Barth’s equation where not sufficiently accurate, especially for gears operating at higher speeds. Based on the results of these tests, Ross [320] proposed a new empirical equation, in which the pitch-line velocity appeared under square root, rather than in a linear term. The Barth’s equation and Ross’ equation, as well as those developed later, which introduced more or less considerable changes in such equations, also including the constant term, constituted a landmark for the AGMA Standards, and are still mentioned in some design handbook of the gears.

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The result of the aforementioned tests, together with those due to a preliminary study made by Wildhaber [363], which took into account at least partially of previous Lasche suggestions, allowed Lewis to design and build a gear-testing machine, with the support of the ASME Special Committee on the Strength of Gear Teeth. With this gear-test machine, tests were made over a period of several years at the Massachussetts Institute of Technology, under the leadership and direction of the same Lewis until his death, in 1929, and subsequently to him under the leadership and direction of Buckingham. The results of this relevant multi-year series of tests were published in 1931, in a report entitled Dynamic Loads on Gear Teeth. They constituted a historical turning point in gear design, since they allowed defining the dynamic load factor, which still characterizes the calculation formulae currently used. The variations due to subsequent insights, which often also led to different definition of this dynamic load factor, with consequent different formulae to calculate it, do not in any way invalidate what we are saying here. It should be noted that the results of these tests showed that the load carrying capacity of the gears very little changed for pitch-line velocities above 25 m/s. A thorough analysis of these results pointed out that the errors due to inaccuracies of manufacture (pitch errors, tooth profile errors, etc.) or elastic deformations, or both, caused a variation of the relative velocity of the mating members of the gear pair, with a consequent variation in the load cycle on the gear teeth. The same analysis allowed concluding that this load variation during the meshing cycle depended largely on the effective masses, effective profile errors and gear speed. According to this analysis, a new dynamic load equation was developed, which is known as Buckingham’s equation (see [287]). Also, this equation, so called because it gathered the fruits of the work of Buckingham [58, 59] is an empirical equation that, however, has enjoyed a more extended popularity in the time compared to the aforementioned empirical equations. In the same historical period in which the above-mentioned problems concerning the gear dynamic load were studied and deepened, another subject drew the attention of researches. This subject concerned the point on the tooth working profile where the load was to be applied, in order to obtain a reliable value of the stress in the most stressed section, i.e. the critical section. In this regard, Lewis, in addition to introducing the uniform strength parabola to identify the critical section, had assumed to apply the entire load to be transmitted at the tip end of the working profile. In relation to this same subject, the proposal of McMullen and Durcan [250] is certainly to be considered another turning point. In fact, based on the consideration that for actual gears we have the necessity to have a value of the transverse contact ratio higher than the unity, they proposed to apply the full load to be transmitted at the outer point of the single pair tooth contact, instead at the tooth tip. Thus, a question closely related to the identification of the outer point of the single pair tooth contact originated: that of the load distribution between pairs of teeth in simultaneous meshing. This question stimulated the interest of many scientists and researchers. In this regard, it is here to remember the work of Karas [198],

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which is to be considered as one of the most complete theoretical works on this topic, despite the fact that the effect of lubrication was neglected. Of course, the problem became even more complex in the event that the effect of lubrication was considered, but a theoretical solution was still possible. In fact, in the early 1940s (see [153]), the problem was solved in the ideal case of perfect gearing and perfect assembly, imposing the following two conditions, which uniquely define the statically indeterminate problem to be solved: • At each point of contact between the various pairs of teeth in simultaneous meshing, the sum of the elastic deflection in the direction of the line of action, due to the load acting therein, and lubricant film thickness at the same point, evaluated in the same direction, is a function of the partial load, Fti . This partial load m is applied to Fti , where the i-th pair of teeth in simultaneous meshing. Of course, Ft = i=1 m is the number of teeth pairs in simultaneous meshing. • All aforesaid sums constituted by elastic deflection plus lubricant film thickness must have the same value for all the pairs of teeth in simultaneous meshing, because the gears that we are analyzing have involute tooth profiles. Unfortunately, the results obtained with the aforementioned theoretical methods had a very limited interest, in relation to practical applications. This was due not only to the complexity of the same theoretical concepts on which these methods were based, but also and above all to small transverse pitch deviations or other similar errors, of which these methods did not take into account. These deviations and errors could greatly alter the actual distribution of the transverse tangential load on the various pairs of teeth in simultaneous meshing. In reality, the ideal conditions do not exist, for which we must always consider that the gears to be designed will be in any case characterized by deviations from the ideal shape, to which the elastic deformations and displacements of all the structural members of a gear drive overlap. All these deviations, deflections, and displacements, significantly alter the transverse tangential load distribution in the regions of double or multiple pair tooth contact, where two or more pairs of teeth are simultaneously in meshing, while they have no influence in the region of single pair tooth contact, where only one pair of teeth is in meshing. In this last case, the problem is in fact statically determined. Given the practical impossibility of solving the statically indeterminate problem by means of an accurate theoretical method, for calculations of the load carrying capacity of the gears the convention was used according to which the total transverse tangential load, in the double or multiple contact regions, is divided equally between the various pairs of teeth simultaneously in meshing. This convention was once again based on an empirical criterion that mediated results acquired in the field, but characterized by a wide scattering. In the same first three decades of the 20th century, the problems of wear and surface durability of the gears began to emerge and to be addressed. Logue, in 1910, suggested the use of the radius of curvature of the gear tooth profile as a measure of contact stress that influences the wear. Unfortunately, the first edition of Logue’s work is part of the books now considered lost. However, the third edition is still available, revised

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by Trautschold [232]. Buckingham [59] reports that, in substantial agreement with Logue, between 1920 and 1927, Joseph Jandesek elaborated numerous diagrams, formulae and calculations, based on Hertz theory developed by Heinrich Rudolph Hertz (1857–1894) in 1882 (see [178, 155, 195]); all these diagrams and formulae assumed the maximum compressive surface stress as a measure of surface durability of the gear teeth. However, the turning point on this issue was still represented by Buckingham who, between 1920 and 1926, applied the Hertz theory adapting it to the surface contact between the mating profiles of the meshing gear teeth. He noticed in fact that the contact conditions between the mating profiles of cylindrical spur gears were similar to those between two cylinders, with the only variation that the radii of curvature of the two mating profiles varied continuously during the meshing cycle. The results of these analyses were summarized by Buckingham in his paper “The relation of load to wear of gear teeth”, which was submitted to the American Gear Manufacturing Association (AGMA) in 1926 [37]. In this paper, another Buckingham’s equation was formulated, having much more substantial theoretical bases than the one previously described, as based on the Hertz theory. It expressed for the first time the so-called wear strength of the gears. In this regard, it should be noted that, at that time, a clear distinction between pitting and wear did not exist jet, and as wear strength was understood what we now call load carrying capacity against pitting. On this basis, Buckingham developed a method to calculate the Hertzian contact pressure to be taken as a reference measure of surface durability of the gears. This method, with the obvious variations due to studies and analyses gradually carried out until today, constitutes the basis of calculations of surface durability. We can say, without any doubt, that Buckingham is the basis of calculation of surface durability, as well as Lewis is the basis of calculation of tooth root strength. The next milestone in the strength calculation of gears was certainly the one established by Almen [8], and Almen and Straub [11], who underlined the need to take account of material fatigue strength. Already in 1870 August Wöhler (1819–1914), based on the well-known experimental tests carried out by himself, had highlighted the fact that the cyclic stress range is more important than peak stress, and together with the S-N curve, or Wöhler curve, introduced the concept of endurance limit (see [154, 371]). As we infer from Schütz [331], the history of fatigue began with Wilhelm Albert (1787–1846), in 1837. Almen and Straub, using knowledge so far accumulated on the fatigue strength of materials, performed numerous fatigue tests on cylindrical and bevel gears. These researchers elaborated results obtained from these tests, and obtained for the first time S-N damage curves specifically valid for gears. Therefore, they got a new relationship to be used for calculations of the bending strength of the teeth. The contributions of the many subsequent researchers, who continued and continue to work on this subject, brought and continue to bring refinements and updates to the relationships that we use to take account of the bending and pitting fatigue stresses. In this regard, we refer the reader to the two review papers of Chauhan [76] and Aziz et al. [19], as well as the work of Kramberger et al. [206].

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Meanwhile, studies and researches on the stress concentration due to notches were intensifying. Wöhler was certainly the first that did experiments on the effects of stress concentration on fatigue strenght (see [341]). Bauschinger [25] continued the experimental work started by Wöhler, while Föppl [141–143] and Föppl and Föppl [144] made a more systematic study of stress concentration, which then culminated in the famous work of Neuber [270]. In particular, the great contribution made by August Otto Föppl (1854–1924) to the adoption of more rigorous methods of stress analysis in engineering practice, which had considerable influence in the growth of strength of materials, made school in Europe and outside Europe. Continuing to perform fatigue experiments, in the wake of Bauschinger, of which he was the heir to Munich, Föppl studied the effects of stress concentrations in specimens with grooves. Even Peterson [293] and Peterson and Wahl [294] gave significant contributions on the influence of stress concentrations on fatigue. Such fertile ground for the accumulated knowledge on this subject bore fruit almost immediately in relation to gears. In fact, Dolan and Broghammer [104], using experimental photoelastic techniques, focused their attention on the problem of stress concentration at the tooth fillets as well as in the areas of points of contact between the mating profiles. For the first time they determined the values of the stress concentration factors to be used for calculations of tooth root strength, taking into account also of the radial component of the load applied at the outer point of single pair tooth contact. Figure 5.35, taken from the original work of Dolan and Broghammer, shows the fringe patterns of the

Fig. 5.35 Fringe patterns of four different bakelite models of gears tooth

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stress state of four different bakelite models of gear tooth (they are designed with ( a) , ( b) , (c) and ( d) ), loaded in the photoelastic polariscope at given points of the transverse tooth profiles, with loads applied along the local normal to the same profiles. Even these two researchers must be considered as the initiators of a research line on the gears, still lively and even changing. The results obtained by Dolan and Broghammer had a great impact on scholars and experts who dealt with gears, and the specific national and international standards did not take long to build on them. Soon AGMA Standards accepted the results obtained by the two researchers, albeit with some modifications consisting in the introduction of other coefficients to take account of the various operating conditions of the gear, such as deflections of shafts, manufacturing and assembly errors, load distribution, rotational speed, etc. Almost simultaneously (in the 40s of the 20th century), the British Standard Specifications for machine cut gears, B.S.S., introduced the total compressive stress, calculated at the compressed tooth root fillet of the critical section. It was obtained by adding the compressive stress, due to the tangential load component (i.e., the bending load component), and the compressive stress due to the radial load component, both calculated in the load conditions related to the outer point of the single pair tooth contact. In the second edition of his well-known treatise, which followed the first edition of 1942, Merritt [255], however, noticed that in practice the fatigue failures in gear teeth began at the fillet under tensile stress, and therefore proposed that this tensile stress was considered as the basis of a calculation criterion. To take account of this fact, in the third edition of his treatise, Merritt [256] suggested that the B.S.S. method was modified by introducing a concentration factor determined experimentally. We must also give another great credit to Merritt: that of having been the first to highlight the effect of the rim flexibility on the tooth root bending strength as well as on the dynamic behavior of any gear. This insight from Merritt has been demonstrated by subsequent researchers (see [148, 194]) and afterwards has been incorporated into the ISO formulae regarding the assessment calculations of tooth root bending strength of involute external and internal cylindrical spur and helical gears. Using the calculation formulae and knowledge available at that time, Kelley and Pedersen [200] found a significant dispersion of results related to the cutting process made with hobs having full tip radius, which then constituted a novelty. They therefore believed to carry out an in-depth experimental analysis, based on the Heywood photoelastic method [180, 181]. This analysis addressed, for the first time, the problem of the influence of the cutting technology on the gear strength. It was aimed either to a more reliable identification of the weakest section, i.e. the critical section, with respect to what the uniform strength parabola allowed, or to determination of an equally more reliable stress concentration factor than that proposed by Dolan and Broghammer. The results achieved by Kelley and Pedersen were praised for their high reliability, and were adopted by many qualified USA and European Industries.

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The uniform strength parabola introduced by Lewis was causing some problems in relation to the identification of the critical section of the tooth subjected to bending load. The position of this critical section was in fact to be dependent upon the point of load application, for which the Lewis’ parabola was not always tangent to the root profile at the tooth root fillets. It is in fact easy to show that cases occur in which this parabola is tangent to tooth profile at points remote from the section where the tooth is built-in on the gear rim (see Figs. 3.4 and 3.5 of Vol. 2). On the other hand, the experimental evidence showed unequivocally that fatigue failures in the gear teeth always started in the tooth root fillet under tensile stress. For this reason, another method of identification of the critical section was necessary to be introduced. Niemann and Glaubitz [275, 276], based on analytical studies and experimental tests, proposed a new method for determining the critical section, the one now called the 30°-tangent method (or 60°-tangent method for internal gears), because the critical section is the one corresponding to the contact point of the tangent to the tooth root fillet arranged at 30° (or 60°) compared to the tooth centerline. The tooth root fillet, with respect to which the 30° tangent (or 60° tangent) had to be drawn, was conventionally the one generated by the root fillet of the basic rack. This mixed method, either theoretical and experimental, was an approximate method, because it did not exactly match the experimental results of the photo-elastic tests, but the differences were very small. Compared with the uniform strength parabola of the Lewis’ method, this method had however the great advantage consisting in the fact that the critical section was no longer dependent on the point of load application, but only on the tooth shape. This advantage as well as other contributions given by Niemann [273], Niemann and Rettig [277] and Niemann et al. [280], of the German school, gave a great impetus to the development of conventional calculation methods of the load carrying capacity of the gears. This 30°-tangent method was adopted by DIN Standards towards the late 50s of the 20th century, and later by ISO Standards in their various editions gradually introduced. Subsequently, based on the results of studies and experimental tests carried out by Winter [366, 367] and Winter and Hösel [368], the ISO Standards were differentiated with respect to the DIN Standard. The differentiations regarded either the introduction of new stress concentration factors, more reliable than previous ones, or the introduction of several other coefficients, in order to parameterize all the factors affecting the gear lifetime. In this context, even the contribution brought by Henriot [176] are to be considered and appreciated. Starting with Tuplin [348], new mathematical models for dynamic analysis of the gears were introduced. These models, even those initial, considered the entire vibrating system of which the gear was a part, for which, at least in the intention, they allowed to evaluate not only the dynamic load acting on the single gear, but also to study other dynamic properties of the whole system. The initial models were, however, very simple and therefore they allowed to obtain only little additional information beyond the dynamic load. The Tuplin’s model was the first of these types of models. It was a simple springmass vibrating system, with a spring having equivalent tooth mesh stiffness assumed

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as a constant, positioned between the equivalent mass on one side, and the fixed frame on the other side, with interposition at the base of the spring of suitable retracting wedges, simulating the gear errors. This model was used only to estimate the dynamic factor at operating conditions well below resonance. Other numerous models for the dynamic analysis of the gears have been gradually introduced until today, more and more advanced and complicated, to calculate more accurately the dynamic loads, also as a function of the dynamic transmission error, or in order to take account of specific influences, or still to achieve predetermined goals, such as: • periodic excitation, as for example the one related to step changes in mesh stiffness, due to the change from single pair to double pair tooth contact; • transient excitation, as for example the one related to different types of errors, some of which with random distributions; • types of gear, as cylindrical spur and helical gears, bevel and spiral bevel gears, crossed-axes gears, including worm gears and hypoid gears, or various combinations of these gears; • areas of application, including aerospace and automotive gearings, industrial gears, gears for marine applications, etc.; • speed regions, i.e., subcritical, main resonance, and supercritical speed ranges; • vibratory behavior, in terms of modal shapes and related frequencies, and noise control; • acoustics of cavity and comfort control; etc. This is not the case of mention specifically the very numerous models developed by the many scientists, researchers and scholars who progressively became interested in this subject; also, the author gave a small contribution regarding this topic (see [148]). This is because here the attention is mainly focused on turning points related to initiators of specific lines of research on the gears. The reader can however draw on the desired news in the scientific and technical literature concerning this topic, which is however very rich as well as continuously updated. In this regard, at the beginning of the 20th century, with the introduction of new materials and advanced manufacturing technologies, which together allowed to make small gears, but also with high performance characteristics, in terms of load carrying capacity and operating speed, a new phenomenon was detected, the one of scuffing. Various researchers, mainly on an empirical basis, studied this phenomenon intensively. The cornerstone of this important subject is to be found in Blok [41–44], which was certainly the scholar who laid the foundation for its understanding and solution. In fact, we are indebted to Blok the introduction of flash temperature criterion, which has a solid foundation, because it is based on a realistic thermodynamic theory, confirmed by numerous tests, specifically made by Blok himself. Due to the well-known war events, related to the Netherlands occupation by the Nazis in World War II, it was not possible to give immediate application to the Blok’s theory, which recognized the lubricant as the third gear material. This fact led to a major advance in the development of additives to mineral oils. Meanwhile, to try to solve the problem in some way, empirical expressions were development,

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having non-general validity, and mainly based on tests, in some cases supported by theoretical considerations. The Almen’s factor PVT [10], introduced six years after the Almen’s factor PV [9], is one of these empirical expressions. Despite its many limitations, the Almen’s factor PVT, which would be more correct to call factor of Almen and Straub, must be regarded as the best empirical criterion, so it enjoyed some luck, until the flash temperature criterion was rediscovered and applied. The latter was revised in terms of accumulation of energy along the path of contact, so in the second half of the 20th century the new concept of integral temperature was introduced. The development of the elastohydrodynamic lubrication theory, usually referred to as EHL or EHD lubrication, actually began at the end of the 40 s of the 20th century. In reality, even before Martin [245] attempted to theoretically assess the possibility that between the tooth mating flanks of the gears, taken as rigid cylinders, hydrodynamic lubrication conditions were generated, but the results were disappointing. In fact, the calculated thickness of the lubricant film was much less than the value of surface roughness. On the other hand, experiences made mainly with disk machines, introduced by Merritt [253, 254] clearly demonstrated that the real conditions of the lubricated contact were much more favorable. These machines, specially built to simulate the contact between the gear mating teeth, made it possible to measure, by means of a dynamometer, the torque due to the friction resistance developed between the disks, pressed one against the other with a known force, and rotating at a given speed. The first satisfactory results concerning this subject were obtained, however, only later, by Ertel [119, 120] and, above all, by Grubin and Vinogradova [164], which made use of a number of insights regarding the shape of lubrication film and contact area and the pressure profile, which allowed them to get a realistic estimate of the lubrication film thickness in the middle of the contact area. The Grubin and Vinogradova work is to be considered in all respects the real turning point of the application of elastohydrodynamic lubrication theory to the gears. The theoretical solution of this problem is extremely complicated, and constitutes, in fact, a utopia. A fortiori, numerical solutions were therefore necessary, supported by suitable experimental tests. The development of sufficiently accurate methods of numerical solution of the EHD lubrication problem is due to Dowson and Higginson [107, 108], for the case of line contact, and Hamrock and Dowson [167, 168], for the cases, theoretically and numerically more complicated, of point contact and elliptical contact. Even these two contributions must be considered as turning points on the subject. We have said many times that this brief gear history is confined deliberately to the pioneers, that is, to the initiators who have opened new lines of researches, to be considered as milestones on the state of knowledge concerning tooth root strength and load carrying capacity of the gears. Also, for reasons of space, it is not possible to mention here all those who have helped to widen the horizon of a specific line of research, through theoretical analyses and experimental tests certainly worthy of the highest praise.

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However, this brief history on the gear strength calculation cannot be completed without mentioning what happened in the 1940s and 1950s. In these two decades of the 20th century, to meet the increasing demands in the design of structures, especially aeronautical and nuclear structures, which required greater accuracy and speed for their analysis, new methods of structural analysis were developed, since conventional methods, although fully satisfactory when used for simple structures, have been found inadequate when applied to complex structures. Only new and more general and efficient calculation methods could capture the interaction of inertial, aerodynamic, seismic, elastic, aeroelastic, thermal and other loads, and their timedependence, as well as physical and geometric nonlinearities in order to predict structural instabilities and determine large deflections. Methods of structural analysis based on the use of matrix algebra were thus developed, as they proved to be suitable for automatic computation on high-speed digital computers. The analysis of complex structures thus benefited from the formulation of general matrix equations describing a mathematical model, including initial input data and final output, represented by force and stress distributions, strains, deflections, influence coefficients, natural frequencies, and mode shapes. Studies and research concerning these new methods of structural analysis culminated with the revolution that took place between the years 50s and 60s of the 20th century. This revolution was defined by Clough [78] as the FEM (Finite Element Method) revolution. In fact, before these two decades, various methods to solve continuum problems had been proposed (see [193, 247]). Among these methods, it is enough to recall that proposed by Courant [89], who suggested a piecewise polynomial interpolation over triangular sub-regions as a way to obtain approximate numerical solutions. As Courant’s own admission, this method was using an approach such as a Rayleigh-Ritz solution of a variational problem. However, none of these methods was successful, because there were no computers to do the calculations. They were engineers, and not mathematicians, to promote and determine the explosive development of FEM, with its application to the solution of continuous problems. This happened in the four years 1953–1956, and the pioneers were Levy [220], Argyris [14–16], Argyris and Kelsey [17, 18], and Turner et al. [349], who wrote the stiffness equations in matrix notation, and solved them with digital computers. Just ten years after the introduction of this calculation method, Melosh [251] recognized that the FEM had rigorous theoretical foundations, based on continuum mechanics, so it became a respectable area of study for academicians. The calculation of the gears, in its most varied aspects, was almost immediately invested by the revolution of the FEM (see, for example, [362]). FEM and BEM (Boundary Element Method), which developed and established immediately after the FEM, although the mathematical basis of the integral equation techniques for problems of continuum mechanics on which BEM is based were already long time known, gradually became an indispensable tool for study and analysis of the gears. These two methods, either separately or together for comparative purposes, were used with gradually increasing intensity and, since the 70s of the 20th century, they became an indispensable and irreplaceable calculation method. For some particular problem concerning the gears, the BEM was preferred to the

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FEM, given its ability to reduce the dimensionality of the problem of one, with the consequent advantages in terms of economy of calculation time. In fact, with BEM, a three-dimensional volume problem becomes a two-dimensional surface problem, while a two-dimensional plane problem becomes a one-dimensional line problem, because it involves only one-dimensional line integrations. This is not the case of mention the early works on the gears, based on the use of these two numerical methods, because often their originality was limited to the effort to adapt them to address and solve particular problems, in the best possible way, and with the maximum reduction of the amount of computation time. It should however be point out that the results obtained by these two methods had led to hone formulae already known, or to propose new and more satisfactory formulae, to be used in conventional methods of gear calculations. For this reason, here we limit ourselves to mentioning the first and most well-known textbooks of general character. Three of these textbooks are related to FEM: Przemieniecki [301], Zienkiewicz [378], and Cook [85]. The other three are related to BEM: Brebbia, [54], Banerjee and Butterfield [23], and Becker [27]. The seventh textbook, due to Fennet [128], is more general, because it, together with analytical techniques, summarizes numerical methods, including the FDM (Finite Difference Method). To conclude this brief historical overview, covering the milestones on the mechanical strength and load carrying capacity of the gears, we cannot forget the mixed methods, which use techniques of direct numerical integration and continuum discretization methods, such as FEM and BEM. These mixed methods have the advantage of adding the peculiarities of the two different numerical techniques that they use, with a synergistic effect that enhances the calculation performance of both. The use of a direct numerical integration method allows to overcome the limitations of a static model, since it allows to change step by step the stiffness matrix of the entire gear system examined, which is characterized by a geometric non-linearity resulting from the variation of relative position between the mating members during the meshing cycle. In the framework of this particular subject, the writer author made a small contribution [147, 148], consisting in the development of a general model of step by step numerical integration, which uses alternatively four integration methods, i.e.: the Houbolt’s method [187], Newmark’s method [271], Hamming’s method [166], and Wilson’s method [365]. These numerical integration methods are combined with a bi-dimensional FE-model of a cylindrical spur gear pair, which takes into account the main parameters of influence, such as: • • • • • • •

effective profiles and teeth shape; gear body geometry; materials and their heat-treatment; lubrication and cooling conditions; tooth cutting errors and assembling gear errors; friction forces and specific sliding between profiles in contact; hardness stratifications due to heat-treatment; etc.

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This combined model is used to determine step by step, throughout the meshing cycle, the changes in: • load distribution on various teeth pairs in simultaneous meshing; • change in this load distribution under different meshing conditions; • tooth root stress as a function of the position of points of contact along the path of contact; • Hertzian stress on the active tooth flanks, as a function of the position of points of contact along the path of contact; • maximum stress and its distribution at tensile and compressive tooth root fillets, to be used for fatigue lifetime calculations; • distributions of the contact stress. Finally, it should be noted that, in the last twenty of the last century and at the beginning of the third millennium, valuable works have been published on topics of primary importance for gears, such as those concerning pitting and micropitting, tooth root fatigue strength, tooth flank fatigue fracture, tooth interior fatigue fracture, spalling, scuffing, etc. The state of art concerning these topics is described in the various chapters of Vol. 2 where these topics are discussed, but also in other chapters of the same Vol. 2 where related topics are discussed. This author did not consider it necessary to recall these topics in this brief history of gears, as they rather than being history are recent news or even chronicle in progress. In this regard, the reader is referred to the reading of the aforementioned state of art, which in a certain sense represents their history in progress that will become history with the due times. Actually, this history will continue for a long time, because real black holes of knowledge still exist on the gears. They concern not only the still very broad possibilities of their practical applications, but also important topics such as those concerning tooth fatigue breakage. It is sufficient to remember that we know almost nothing about spalling and that new knowledge is needed to better understand the phenomena of micropitting and macropitting, and especially those of tooth flank fatigue fracture and tooth interior fatigue fracture. Moreover, the road to a unified science of gears, capable of simultaneously considering the geometric-kinematic aspects, the technological aspects and load carrying capacity and mechanical strength aspects, still appear impervious and nothing can be glimpsed even in the farthest horizon. Before closing this section, we want to consider in more detail the historical aspects of three particular types of damage, which are related to the load carrying capacity of the gears in terms of abrasive wear, micropitting and tooth flank breakage. The problem of abrasive wear takes us back in time, while the problems inherent in the other two types of damage have been placed much closer to us. The three problems have a common denominator: a shared method of calculation for each of them has not yet been found.

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5.3.1 Wear Load Carrying Capacity of Gears The first subject that deserves special consideration is that concerning the abrasive wear of the gears. In this regard, it is first to point out that nether the classical wear theories nor the current most advanced tribological models are able to fully explain the phenomenon of wear, considered in its more general aspects or in those specifically concerning gears. Regarding the gears that are of interest here (but not only for the gears), the experts have not yet reached an agreement and the international standards have not dared to approach this subject. The classical wear theories start from the premise that the rate of material removal is a function of the material hardness, sliding velocity, applied load, and probability of a material to produce a wear particle under a given contact condition, including lubrication regime. In the historical development of these classical wear theories, three different models can be identified, which are respectively based on empirical relationships, contact mechanics approach, and fracture mechanics approach. These models, extremely numerous, differ from each other not only for the method used to approach and study the wear phenomenon, but also about wear effects and governing variables. About 100 influential quantities have been introduced to describe the wear processes, but it is rare that two different models agree equal importance to any of these influential quantities. However, despite these substantial differences, all models agree that friction is an important concept, which is intrinsically correlated to wear. In the Western World, the oldest evidence on friction comes from Greece. In fact, in the 4th century B.C., the philosopher Themistius, exegete of Aristotle, observed that the rolling friction was much lower than the sliding friction, thus justifying the invention of the wheel as one of the biggest steps forward in the field of land transport. The concept of friction remained substantially a mysterious concept throughout the Middle Ages, and only in the Renaissance Leonardo da Vinci devised the basic laws of friction, providing a linear relationship between the forces related to the contact and the vertical load, regardless of the area of contact. However, the classic rules of sliding friction discovered by Leonardo da Vinci remained in his notebooks, unpublished (see [236, 288]). About two centuries later, Amontons [12] revisited the Leonardo da Vinci laws, formulating the two laws that are still called, improperly, Amontons’ first law and Amontons’ second law. According to these two laws, the force of friction is directly proportional to the applied load and, respectively, independent of the apparent area of contact. Amontons formulated the well-known equation, called Amontons’ equation, which correlates the friction force, coefficient of friction, and applied normal force. The direct proportionality between friction force and applied load was unanimously accepted, but academics did not hide their skepticism on independence from the apparent area of contact, despite some experimental evidence of tests made by other researchers in this regard.

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The Amontons’ equation is also called Coulomb’s equation, in honor of Coulomb [88], which is recognized as the main architect of the friction laws. Coulomb confirmed the importance of the contact surface roughness, as emphasized by Amontons, and suggested that friction was due to the work done by dragging a surface on another surface. Coulomb also realized that in reality the contact occurred only in correspondence of the surface asperities, for which introduced the concept of actual area of contact, i.e. the common area between the mutually contacting asperities, as the areas concerned are characterized by a complex topography of asperities. Furthermore, Coulomb assumed that the kinetic friction was independent of the sliding velocity. This assumption is known as Coulomb’s law of friction. Coulomb, however, refused to accept the Disaguliers’ theory [101, 102], according to which the frictional resistance between smooth surfaces, obviously not due to roughness, should be attributed to adhesive forces, summarized by the term cohesion. Subsequently, Leslie [219] highlighted the weaknesses of theories of Amontons and Coulomb, and assumed that friction was due to surface deformations induced by the roughness. After this intuition of Leslie, nearly a century was necessary because a unified friction theory had developed, according to which the basic parameters are related to the geometry of surfaces in contact, their elastic properties, the adhesive intermolecular forces, and the energy lost due to surface deformations. The first historical model that correlates friction and wear is due to Reye [313]; therefore, it is known as Reye’s hypothesis (see [129, 154]). This hypothesis led to the Reye’s equation, which states that the volume of material removed from a body due to wear effects in a given time is proportional to the passive work done in the same time by the frictional forces that have produced the wear, i.e. proportional to the energy dissipated into the body during the relative motion of the two contacting surfaces. This Reye’s model constitutes one of the first models that considered the wear phenomenon from a new point of view, the one based on energy considerations. Despite some its well-known limit (for example, the assumed shear stress distribution at boundary of the area of contact does not meet the Betti’s reciprocal theorem), this model is a milestone. In fact, both the refined models, and new models, developed by subsequent researchers, have their roots on Reye’s hypothesis. Based on empirical investigations, Bowden and Tabor [52] showed that the actual contact area between two bodies is really very less than the apparent contact area, since flat surfaces are held apart by small surface irregularities, which form bridges. However, to obtain quantitative results to be used for the calculation of the change of contact area with the load, they used the Hertz theory [178] concerning the elastic contact deformation and considered the probability that surface asperities are to collide under a given contact condition. Moreover, by processing results of experimental measurements of conductance, performed by the method developed by Bidwell [38], they were capable to formulate two theoretical equations. The first equation is based on the assumption of elastic behavior of the material, which provides that the conductivity between the bodies (and thus their contact area) depends on the cube root of the applied load. The second equation is based on the assumption of plastic behavior of the material, which provides that the conductivity between the bodies depends on

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the square root of the applied load. Further experimental measurements substantiated this second assumption, for which Bowden and Tabor stated that the total cross section of the junctions and the tangential force required to break them were directly proportional. With reference to the relative motion between surface asperities, Holm [184] hypothesized that the wear process was due to the collision of individual atoms on opposing asperities in the motion of one towards the other. On this basis, he established that the amount of material removed during these atomic interactions were a function of the properties of contacting materials and load applied over the contact, and so came to an equation that is known as Holm’s wear equation. According to this equation, the volume of material removed per unit sliding distance is proportional to the quotient of the applied normal load divided by the so-called flow pressure of worn surface, which is comparable to the material hardness, the proportionality factor being the probability of removal of an atom for atomic encounter, which depends on the properties of the contacting materials. Archard [13] considered a greater number of variables that influence the wear process, i.e. wear mechanism, area of contact, contact pressure, sliding distance, surface asperities, motion and interaction of these opposing asperities, material properties, etc. Starting from the Holm’s wear equation, and expanding its horizons, Archard assumed that the deformation occurring was of a plastic type, and formulated an equation that is known as Archard’s wear equation. According to this equation, the total volume of material worn away is equal to the product (dimensionless wear coefficient x compressive normal load between the contacting surfaces x total rubbing distance) divided by the hardness of the softest contacting materials. This equation implies the assumption that the volume of material worn away is independent of the area of contact. In the same way as Reye’s equation, also the Archard’s wear equation expresses that the volume of material removed by wear is proportional to the energy dissipated into the material. Both the equations can be converted to express, in a perfectly equivalent manner, the wear velocity as a function of the product pvg (surface interface pressure × sliding velocity) multiplied by a factor equal to the quotient of the dimensionless wear coefficient divided by the hardness of the softest contacting materials. For two rubbing surfaces 1 and 2, both the equations imply that the wear rate of surface 1 is proportional to the wear coefficient of material 1 when in contact with material 2, and inversely proportional to the material hardness of surface 1. Furthermore, under the assumption of a constant coefficient of friction, the same wear rate is directly proportional to the rate of friction work. Actually, the product pvg that appears in these equations expresses, for a given coefficient of friction, the power lost due to friction, which is converted into heat. The Archard’s wear equation represents a simple but extremely effective model to describe the sliding wear between the asperities of two surfaces in relative motion. For this reason, it is the most used as well as the most cited in this field, also for a historical perspective error. Indeed, this equation, in whatever form is expressed, does not differ from the corresponding Reye’s wear equation, proposed almost a

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century earlier. The Reye’s wear equation becomes very popular in Europe, and it is still thought in university courses of applied mechanics, but it was totally ignored in the scientific literature of the Anglo-Saxon school, where other non-new models established themselves, such as those due to Holm [184] and Archard [13], which have been developed much later. However, it should be noted that, with Archard’s wear equation or with the corresponding Reye’s wear equation, whatever the forms in which they are expressed, we have the possibility of obtaining experimentally the values of the wear coefficient for a particular design application, using for testing the same material combination and the same operating conditions of interest. It is also to be noted that the same experimental equipment for measuring the wear coefficient related to abrasive wear can be used to measure the wear coefficient related to adhesive wear. It is not the case here to make a historical mention of the classical wear theories based on contact mechanics and fracture mechanics approaches, as these theories have not yet been applied in reference to wear load carrying capacity of gears. Studies on abrasive wear in terms of delamination wear have been carried out with the contact mechanics approach (see for example the classic review work by Johnson [196]), but without noteworthy repercussions on gears. As we have highlighted in more than one chapter of Vol. 2, the effects of this approach in terms of surface and subsurface damage of the gears have been very significant. Instead, today there are still founded doubts on the correlation between abrasive wear and fracture mechanics, like those already raised in 1990 by Rosenfield [319]. The analysis of the load carrying capacity of the gears against abrasive wear, which occurs when, due to high load values and low pitch line velocities, the minimum thickness of lubricant oil film falls below well-defined threshold values, is performed using calculation relationships obtained using mixed methods. The most used current relationships have in fact been obtained with systematic test plans on test gears combined with the use of one of the aforementioned equations. However, this is a recent history, so we refer the reader to Chapter 9 of Vol. 2 or to more specialized textbooks.

5.3.2 Micropitting Load Carrying Capacity of Gears The second subject that deserves special consideration in that concerning micropitting of gears. For this subject, the frame of reference is a real state of the art, so it must be considered as a current chronicle rather than a historical framework. The studies and researches concerning this very interesting subject are in fact too recent, even if for some particular aspects that try to make us understand the mechanism of micropitting not a few connections with scientific bases developed in times that we can consider historical can be found. Micropitting is a pitting phenomenon, i.e. a rolling and sliding contact fatigue damage that, contrary to macropitting (the classical pitting), manifests itself on the roughness scale rather than on the scale of the nominal areas subject to rolling and

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sliding Hertzian contact. Therefore, micropitting and macropitting are two sides of the same coin, being related to the same fatigue damage phenomenon due to rolling and sliding contact, but the former sees only surface asperities interact with each other, while the latter sees interacting the entire macro-areas affected by contact. This phenomenological identity between micropitting and macropitting has been confirmed by SEM (Scanning Electron Microscopy) tests, which unequivocally show that micropitting proceeds with the same fatigue damage process as the classical pitting, with the variation that pits are extremely small. In the gearing industry, the micropitting load carrying capacity of the gears has become a widespread problem with the growing use of hardened and heavily loaded gears. This problem, for some time well known to gear designers [334], has been studied and continues to be studied extensively by researches, who have tried and try to explain the mechanisms of generation and propagation of the fatigue damage induced by it, analyzing more or less general aspects and its particular characteristics (see [4, 36, 191, 223, 224, 231, 233, 322, 377]). The problem is particularly felt especially for specific applications, such as wind turbines gear drives. It should be noted that, at present, there is no a satisfactory general model for the evaluation of the micropitting load carrying capacity of the gears. Such a model should be able to follow the nucleation of micropits and their progressive expansion on the mating tooth flank surfaces until critical threshold values of their size are reached. It should also be noted that such a model, with the appropriate variations, would be able to shed light on the pitting triggering mechanisms, which are quite similar to those of micropitting. The development of such a model involves the solution of several complex problems that , at the present state of knowledge, appear insurmountable (for a more in-depth examination of the main problems to be solved, see Chapter 10 of Vol. 2). In this regard, the flowchart in Fig. 5.36 shows the block diagram of an ideal general model that could allow the achievement of aforementioned goals. The individual blocks of this flowchart describe the main aspects of the calculation procedure, aimed at determining micropitting severity indices capable of providing the gear designer with a reliable reference framework on the design choices to be made as well as possible ordinary and extraordinary maintenance interventions to be carried out. Of course, to optimally solve the problems summarized in individual blocks, prior knowledge must necessarily be used. They are rooted in the heritage of scientific theories we have inherited from the recent and the distant past. In this regard, to give a rough idea of how useful the aforementioned assets can be, we focus our attention on three questions, concerning respectively: simulation of elastohydrodynamic lubrication conditions in transient conditions, with rough surfaces and non-Newtonian rheological behavior of lubricant oil; three-dimensionality of the contact between rough surfaces, lubricated or non-lubricated; determination of average and amplitude stress components and assessment of micropitting damage by means of a multiaxial fatigue criterion. As regards the first question, it is obvious that, in order to deal with the problem in the most general terms, it is necessary to develop an elastohydrodynamic-tribological model that is able to capture the time-depending lubrication conditions that can vary

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Fig. 5.36 Flowchart of an ideal general micropitting model for gears

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from full film lubrication to mixed or even boundary conditions. Furthermore, it must be able to take into account the variability of all the parameters involved along the meshing cycle, the line or point contacts between rough surfaces and the rheological behavior (Newtonian or non-Newtonian) of lubricant oil. To this end, the substantial modification of the Reynolds equation [314] to take into account the pressure dependence of the lubricant density and viscosity [109, 158, 159] is not sufficient. It is in fact necessary to generalize it even more, in order to capture the three-dimensionality of the texture and topography of the contact areas, which varies continuously during the expected lifetime of the gear considered, as well as the Newtonian or non-Newtonian rheological behavior of the lubricant fluid. Some of the necessary prior knowledge to make the Reynolds equation even more general are those developed and/or described, for example by: Ree and Eyring [308, 309], Johnson [195], Wang et al. [358], Ehret et al. [116], Blateyron [40]. As for the second question, it is equally clear that the three-dimensionality of the contact problem cannot be neglected. It implies that the lubricated or non-lubricated contacting surfaces are considered in their factual reality, i.e., as rough surfaces whose texture and topography changes during the expected lifetime of the gear considered. For the determination of the three-dimensional stress and strain states as well as the average and amplitude stress and strain components, on the contact surface and at different depths below the contact surface, more or less refined FEM or BEM models have been introduced by the various researchers who have addressed this problem. It is not the case here that we summarize the theoretical basis of continuum discretization methods with finite elements or with boundary elements, and their limits. We just want to remember the historical theoretical bases that, together or separately with experimental tests, allow to check the results obtained with FE or BE models. They are: Cerruti [73], Boussinesq [51], Flamant [132], Belajev [29, 30]. Finally, as regard the third question, given the three-dimensionality of stress and strain states, for the evaluation of micropitting damage it is necessary to use a multiaxial fatigue criterion. In the general case of mechanical components and structures, capturing the correct damage mechanism of the multi-axial fatigue is essential to define a proper damage quantification parameter for robust multi-axial fatigue life estimation. Multi-axial fatigue damage mechanisms include material microstructure, material constitutive response, non-proportional hardening, variable accumulation loading, mixed-mode crack growth, cycle counting, additional cyclic hardening of some materials under non-proportional multi-axial loading and its dependence on the load path, etc. This is not the case to recall the various multi-axial fatigue criteria so far introduced, which are the basis of the models related to them (stress-based models, strainbased models, fracture mechanics models, non-proportional models, etc.). In this regard, we refer the reader to the scientific literature that constitutes the historical bases of this important topic (e.g. [126, 335]). However, attention should be drawn to the fact that the critical plane damage models considering both stress and strain terms have proved to be the most appropriate models to deal with the subject in question, as they can reflect the material constitutive response under non-proportional loading. In fact, in the cases in which they have been used, these models have proven to be

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able to capture the crack nucleation sites and fatigue lives with the best agreement with the experimental evidences (see [127, 374]). From the historical point of view, it should be remembered that the criteria based on the comparison of the theoretical thickness of the lubricant oil film with the roughness of the tooth flank surfaces were born between the 1940s and 1960s. They were used for an approximate assessment not only of the risk of cold scuffing, but also of the risk related to other possible fatigue damage of the mating tooth flank surfaces. To this purpose, various indices of deterioration of gear tooth surfaces were introduced in succession, starting from the one proposed for the first time by Bodensieck, in 1965 [47]. These indexes are aimed at defining the elastohydrodynamic lubrication conditions between two surfaces in a relative motion, with a lubricant oil film interposed. Their threshold values were in any case determined and continue to be determined with substantially empirical criteria, by means of experimental planning with suitable test gears.

5.3.3 Tooth Flank Breakage Load Carrying Capacity The third subject that deserves special consideration is that concerning tooth flank breakage load carrying capacity. Gears may be affected by various types of fatigue damages. The attention of designers and users so far has been focused mainly on tooth root bending fatigue breakage and pitting and, more recently, on micropitting. However, the types of surface and/or subsurface fatigue damages in gears are not limited to pitting and micropitting. In this regard, for example, spalling should be mentioned, which is a surface contact fatigue damage of the same nature as pitting, but much more dangerous as it results in more rapid deterioration of surface durability than pitting. In fact, pitting appears as shallow craters at contact surfaces, with pits that reach a minimum depth equal to about the thickness of the work-hardened layer. Spalling instead appears as deeper cavities at contact surfaces, with spalls having a depth typically corresponding to (0.25–0.35) half the Hertzian contact width (see Fig. 11.3 in Vol. 2). Furthermore, spalling often induces early failure by severe secondary damage. Consequently, spalling has been considered as the most destructive surface failure mode of a gear. Unfortunately, to date, we do not have reliable calculation criteria of surface durability against spalling. Fortunately, however, in recent times, studies and research have been carried out on two other surface fatigue damages with crack initiation below the surface of the loaded tooth flanks, known respectively as TFF (Tooth Flank Fatigue Fracture) and TIFF (Tooth Interior Fatigue Fracture). Both these surface fatigue damages are also known as TFB (Tooth Flank Fatigue Breakage). These types of flank surface fatigue damages have been especially but not exclusively observed in highly loaded case-carburized gears for power transmission used in many practical applications (spur and helical gears for car, truck and bus transmissions, wind turbines and turbo transmissions as well as bevel and hypoid gears for water turbines). Contrary to other

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surface contact fatigue damages, such as pitting and micropitting, which occur as a result of cracks initiated at or just below the flank surface, these types of damages are due to deeper primary crack initiation, typically located in the area of the case-core interface at approximately mid-height of the tooth (see Sect. 11.2 of Vol. 2). However, they have been found not only in case-carburized cylindrical gears, but also in bevel and hypoid gears used for different applications as well as in nitride and induction-hardened gears. These subsurface fatigue damages were also observed in gears stressed by loads below the rated allowable ones calculated with the load carrying capacity evaluation procedures against pitting, micropitting and tooth root bending strength. This factual circumstance unequivocally demonstrates that their failure mechanism is substantially different from those described for the fatigue damages due to pitting, micropitting and tooth root bending fatigue loads. It should first be noted that the trigger and propagation mechanisms of these two types of damage have not yet been sufficiently clarified. Moreover, even the main factors that determine them are not yet all known. It follows that the need for further study of this topic, through adequate and extensive theoretical and experimental investigations, has become a categorial imperative with the use of the aforementioned case-hardened and especially case-carburized gears, operating at high pitch line velocities under equally high loads. These investigations began in the 1960s (see [292]) and continued until the end of the last century (see [117, 213, 286, 324, 333, 340]), but gradually intensified in the present beginning of the third millennium (see [7, 28, 48, 151, 172, 203, 237–241, 282, 315, 336]), leading in 2019 to a Technical Specification of ISO, which however is limited only to the calculation of the tooth flank fatigue fracture load capacity of cylindrical involute spur and helical core-carburized gears with external teeth. Here we only make a brief summary of the TIFF and TFF calculation methods developed in recent years, which have taken advantage of the aforementioned studies carried out since the 1960s. These methods have a common denominator, as they share the approach procedures referable to the following four stages: calculation of stress history; specification or calculation of residual stresses; calculation of equivalent stresses by means of some fatigue criterion; comparison of the maximum equivalent stress with some threshold value of stress that triggers the fatigue crack, based on experimental results or field experiences. As for the TIFF risk analysis and the related determination of optimum gear microgeometry, material characteristics and case-hardening properties, MackAldener [237] and MackAldener and Olsson [238–241] used two-dimensional FEA models, combined with a specific load distribution analysis program. Using then the torque distribution corresponding to the distribution of the total load on a single tooth during a meshing cycle, after normalizing the same torque with face width, they calculate the stress history. To then estimate residual stresses and material properties, they first used a procedure where transformation strain and material fatigue properties were assumed to be constant throughout the case, while subsequently they introduced a procedure where transformation strain and material fatigue properties were considered to have non-homogeneous profiles along the case thickness.

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Given the complexity of setting up the numerical-analytical model used and the computationally expensive calculation running, the same authors proposed the use of a simpler semi-analytical method. This method allowed faster calculations, design parameter studies as well as a satisfactory optimization of the parameters involved, but at the expense of accuracy of the numerical results, as the crack initiation risk factor was over-predicted by about 20% when compared with that of the previous numerical-analytical model. To assess the influence of gear design parameters in the TIFF risk, the authors used a factorial planning, examining a wide range of variability of the numerous parameters considered, and concluded that the TIFF could be avoided if the slenderness ratio and tensile residual stresses were reduced, the gear non used as a idler gear and optimum case and core properties were used. To reduce the drawbacks, Al and Langlois [6] introduced a modification to the method described above, consisting in the fact that a specialized three-dimensional elastic contact analysis was carried out, which constituted a separate gear loaded tooth contact analysis (LTCA). Results of this separate specialized analysis were then used to determine the load boundary conditions for the TIFF analysis. In this way, the calculation procedure became computationally less expensive, with a reliability of results comparable with that of the method described above. The first model concerning the TFF risk analysis was the one developed by FZG scholars and researchers, such as: Witzig [369], Tobie et al. [342], Boiadjiev et al. [48]. This method is based on the hypothesis of shear stress intensity, introduced by Hertter [177], whose calculation procedure relies on the determination of local stress history. Its field of applicability is however limited, since the equation for calculating the local exposure of the material to the FTT risk is eminently empirical, as it also uses significant empirical contributions. Another limitation is the fact that the method was developed only for single-flank loading, although, at least from a theoretical point of view, it is possible to extend it to double-flank loading (for example, idler gears and planet gears); this extension of the method is however not trivial. Finally, the assumptions underlying the residual stress calculation entail a further restriction of this method, which can only be used for case-carburized gears. This model uses as input the Hertzian contact stress, which is calculated using a gear load distribution program or a simplified analytical procedure like the standardized one, described in Chapters 2 and 5 of Vol. 2. The method neglects the tensile residual stresses within the core, for which it can appreciably underestimate the critical fatigue stress when these residual stresses are not negligible. The assumption of negligible tensile residual stresses within the core is sufficiently valid when the tooth core section is much larger compared to the thickness of the case. Instead, it entails great limitations when the method is applied to slender teeth and extensive case hardening depths. A more general method than the one described above, which can be applied to both TFF and TIFF, is the one proposed by Ghribi and Octrue [151]. It uses a multi-axial fatigue criterion and takes into account the tensile residual stresses within the core, which are very important for the accuracy of the results. Stress history is determined by using Hertzian contact stress together with Belajev’s theoretical analysis (see [29, 30, 195]), described in Sect. 2.4 of Vol. 2 which allows us to derive the distributions

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of the subsurface stress components. The Hertzian contact stress is calculated using the procedures of ISO standards. The influence of bending stresses, which certainly affect the subsurface stress state at different material depths inside the tooth, are not considered. Both of these methods do not use FEA models. They also do not consider the critical effects of the material quality and presence of inclusions, which certainly constitute a key factor of the problem discussed here. This omission represents a non-negligible limit, which could affect the results obtained. To take into account the material quality and presence of inclusions, it would be necessary to introduce a factor that could be applied to characteristic threshold quantities of the material. However, to do this, a systematic campaign of wide-ranging studies and considerable experimental work is necessary as well as an adequate accumulation of experience in the field. Finally, we want to summarize the method developed by Al and Langlois [5] and Al et al. [6], Al et al. [7]. In comparison with the two methods previously described, this method uses the whole stress tensor, including bending stresses, and considers the tensile residual stresses within the core, whose effects increase with torque. It also introduces a different procedure for calculating TIFF and TFF failure thresholds and, in the wake of MackAldener and Olsson [241], is based on the FE analysis. However, unlike MackAldener and Olsson, loading conditions calculated with a specialized load tooth contact analysis (LTCA) are used in place of a full FE tooth contact analysis, too complex to perform and computationally expensive. The procedure is used not only for calculating crack initiation risk factor, but also for investigating the crack propagation mechanism. The specialized LTCA model is a hybrid model, developed by Langlois et al. [214], which combines a Hertzian contact sub-model for the local contact stiffness and a FE sub-model of bending and base rotation stiffness of gear teeth and blank. The model, which also combines 2D and 3D FE sub-models, also considers the extension of the path of contact and the consequent increase in the transverse contact ratio, due to the bending of the non-involute tooth tip part beyond the effective outside diameter, whose effects are particularly important for slender gear teeth that are characterized by a higher risk of TIFF. We would like to point out here that it is used to determine the load boundary conditions at a selected number of time steps through the meshing cycle. At each of these time steps, the load distribution between and across the teeth is calculated and, at each contact line corresponding to the specific time step, load positions, load intensities and Hertzian half widths are determined. The model then assumes that the material properties are variable within the case and core of the gear teeth. These properties play an important role in TIFF, especially for case-hardened gears, which are notoriously characterized by their variability along any local normal to tooth profile. As required by the critical plane criterion, critical shear stress and fatigue sensitivity to normal stress are also assumed to be variable along any local normal to tooth profile. In accordance with what has been done by MackAldener and Olsson [240], the variability law of all these quantities is assumed to coincide with the one corresponding to the hardness profile.

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5.4 Main Monographs and Textbooks on the Gears As we have highlighted in the previous section, it is very probable that the first monograph ever written on the gears is the one published in 1842 by Olivier [283], entitled Théorie Géométrique Des Engrenages Destinée a Transmetter Le Mouvement De Rotation Entre Deux Axes Situés Ou Non Situés En Même Plan, i.e., Geometric theory of the gears intended to transmit the rotational motion between two axes located or not on the same plane. Subsequently, other monographs or very extensive parts of more general textbooks concerning the design of machine elements were gradually written and published. Not all works published on this topic can be recalled here, due to obvious space limitations. Therefore, we here will focus our attention on those considered most significant, published from the early 1900 to today. We apologize for any omissions. The 20th century opens with the monographic treatise on the theory of screws, which was published by Ball in 1900 [21]. This treatise has always received a considerable appreciation from mechanical and design engineers. In fact, it constitutes the definitive reference on screw theory, as it gives a very complete treatment of small movements in the dynamics of rigid body. In recent years, this theory has received renewed interest, because its mathematical treatment has proved to be a novel and effective analytical resource with which to face complex engineering problems, with important applications to mechanical design, multibody mechanics, robotics, hybrid automotive control, and computational kinematics. In 1912 and 1913, Schiebel published respectively the first and second volumes of the first edition of his treatise on gears, entitled Zahnräder. I Teil. Stirn- und Kegelräder mit geraden Zähnen, i.e., Gears, 1st Part, Spur Cylindrical and Bevel Gears and Zahnräder. II Teil. Räder mit schrägen Zähnen, i.e., Gears, 2nd Part, Gear wheels with curved teeth. With these two monographs, the author gave a fairly complete picture of the state of construction of the gears of his time and devoted particular attention to the problems posed by their design, calculation and production [327, 328]. Among the various studies carried out, those concerning the determination of the surface of action of worm gears are to be appreciated in a special way. These two monographic volumes were lucky, as the successive revised and expanded editions demonstrate, up to their final editions, respectively of 1954 and 1957, revised and expanded with the contribution of Lindner [329, 330]. In 1920, Pomini published the fourth volume of his extended and poly-topic treatise Costruzione di Macchine, i.e., Calculation and Design of Machine Elements, dedicated to gears as well as belts, ropes and chain transmissions [300]. In this work, Pomini summarized the design, calculation and manufacturing methods of the main types of gears, used by himself as a design engineer at the Officine Meccaniche Luigi Pomini di Castellanza. In 1936, Cormac published an interesting monographic treatise on screws and worm gears, including their high-level mathematical treatment and an accurate description of their various practical applications, especially those in the aeronautical field (for example, worm gears for retractable chassis) and their manufacturing

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processes [86]. An extensive part of this monograph is dedicated to the cutting methods of various configurations of worm gears, shapes of hobs, etc., as well as the mathematical analysis of the shapes required to satisfy specific design conditions in relation to the given applications of these gears. In 1928, Buckingham published the first edition of his monographic treatise, titled Spur Gears. From a through review of this first edition, which gathered the results of more than twenty years of research on gears, in 1949, Buckingham published his most famous monographic treatise titled Analytical Mechanics of Gears, which is to be considered a reasonably complete mathematical analysis of the mechanics of gearing. This treatise summarized the knowledge of his time, integrated with the results of theoretical and experimental research done between the end of the 1920s and the 1940s by the same author, who declared that a possible subtitle of his work could well be “Final Report of the ASME Special Research Committees on Worm Gears and the Strength of Gears” [58, 59]. This treatise has always been considered as a reference point for research on the gears of the first half of the 20th century. In 1949 and 1950, Henriot published respectively the first and second volumes of the first editions of his monographic treatise on gears, entitled Traité théorique et pratique des engrenages, tome I, Théorie et technologie, i.e., Theoretical and practical treatise of gears, Vol. I, Theory and technology, and Traité théorique et pratique des engrenages, tome II, Étude complète du material, i.e., Theoretical and practical treatise of gears, Vol. II, Complete study of the material. In this treatise he summarized the results of his many years of experience as designer of gear transmissions and expert on their manufacturing techniques. In the era in which he worked, Henriot was considered one of those four to five leading international experts who draw the lines of research development and the guidelines of technical standards concerning gear calculations. His treatise was a huge success, so much so that over the next thirty years, six updated and continuously expanded editions followed the first edition [174, 175]. In 1951, Modugno published the monograph entitled Ingranaggi Cilindrici (Cylindrical gears), which extended the previous monograph of 1940, entitled Teoria e Costruzione degli Ingranaggi ad Assi Paralleli con Applicazione ai Riduttori Marini, i.e., Theory and Construction of Gears with Parallel Axes with Application to Marine Reducers. In these two monographs, Modugno explained the procedures to be followed in the design, sizing and manufacturing of these types of gears. These procedures assumed as the basis of the calculations the geometric elements of the cutters used for cutting their teeth [259, 260]. In 1954, Merritt published the third and last edition of his monographic treatise, simply titled Gears. It supplemented the two previous editions, the first of which had been published in 1942, and summarized the research work of the author on the subject, dating back to 1924. In particular, the last edition addressed the question of rating formulae of the British Standards, re-expressing them in a way that led to a more rational connection between the inevitable assumptions. Merritt, however, reiterated that the proposed rating methods remained empirical, and pointed out that a true understanding of the behavior of tooth surfaces and their lubricant still awaited a full understanding [256].

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In 1951, Giovannozzi published the second edition of the second volume of his treatise, entitled Costruzione di Macchine, i.e., Calculation and Design of Machine Elements, which followed the first edition published in the 1940s [152]. A large part of this volume was reserved for gears, in an interesting vision that took into account the geometric-kinematic aspects as well as those concerning their mechanical strength, load carrying capacity and lifetime, under different static and fatigue loads and under different service conditions, without forgetting the technological aspects of their manufacturing processes. Great attention was also given to the problems related to the involute profile shift of teeth of cylindrical spur and helical gears and bevel gears, as well as the calculation procedures of the gears according to methods of theory of elasticity and various standards under continuous updating. Numerous Italian mechanical engineers have been trained on the sweaty papers made up of the two volumes of Costruzione di Macchine by Giovannozzi, in their various editions, up to the final ones of the 1980s. In 1954, Dudley published the handbook Practical Gear Design, which addressed in a unified vision all aspects of gear design, including geometric and kinematic problems, manufacturing methods and causes of failures [111]. Dudley set himself the goal of doing not a great design, but a good design, that is a practical and economical design to manufacture and well enough thought out to meet all the hazards of service in the field. Dudley then, in 1962, published as editor the first edition of the Gear Handbook: The Design, Manufacture, and Application of Gears, which was very successful [112]. This first edition of the gear handbook and its two subsequent editions, respectively published in 2012 and 2016 [302, 304], have always met the appreciation and satisfaction of gear designers. This handbook came to supplement the one published by Tuplin in 1944, having as a title Machinery’s Gear Design Handbook [347]. In 1957, Pollone published the second edition of his treatise Costruzioni Automobilistiche: Il Veicolo, i.e., Automotive Design: The Vehicle, which followed the first edition published in 1937. This second edition, revised enlarged and enriched compared to the first, was characterized by a wide appendix, where the gears were treated in detail, in particular as regards their geometric-kinematic aspects and their mechanical strength, load carrying capacity and lifetime, under different static and fatigue loads and different service conditions. Great attention was also paid to problems related to automotive applications of gears and their calculation procedures, in compliance with the related international standards. The discussion of this topic was further expanded and deepened with the third edition of the same treatise, entitled Il Veicolo (The Vehicle), published in 1970 [297–299]. In 1960, Niemann [274] published the second volume of his treatise, entitled Maschinenelemente. Entwerfen, Berechnen und Gestalten im Maschinenbau. Ein Lehr und Arbeitsbuch, Zweiter Band: Getriebe, i.e., Machine Elements. Design, Calculation and Construction in Mechanical Engineering. A Textbook and a Workbook. Second Part: Transmissions, whose theme was mechanical transmissions. The topics covered in this volume, as attested by the same subtitle on the title page, mainly concern the gears. Niemann’s work was a huge success among students, designers and gear manufacturers and this success went beyond the borders of Germany, so much

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so that it was translated into various other languages. The treatise was subsequently considerably revised and expanded by Winter and was divided into two volumes, but its diffusion as well as its charm as a complete handbook of design and construction of the gears continue to the present day [278, 279]. With the two editions of the Handbook of Gear Design, respectively published in 1985 and 1994, Maitra [242] wanted to provide design engineers and technicians with a synthetic overview of the many aspects that are encountered in design of different types of gears and gear systems, including spiral bevel gears. In particular, more methods for checking gear teeth and design of gear tooth profiles are provided. Although limited only to the geometrical aspects of the involute gears, mention should be made of the work published in 1987 by Colbourne [79], entitled The Geometry of Involute Gears. Only cylindrical spur and helical gears with parallel axes and cylindrical crossed helical gears are considered. The work is still appreciated because it clearly addresses all the geometric problems of these types of gears, including not only those regarding the working conditions, but also those related to cutting conditions, with or without generation processes. In 1994, Litvin [227], crowning a life of study and research on gears, first in USSR and then in USA, published the first edition of his monographic treatise, entitled Gear Geometry and Applied Theory, which summarized the most modern guidelines of the theory of gearing and the geometric-kinematic aspects of many types of gear drives. This monograph undoubtedly represented a new way of approaching gear design, as the geometric-mathematical aspects were for the first time finalized to the implementation of computer calculations in the widest meaning, including the preparation of numerical models FEM and BEM and the related calculations. The general use of matrix algebra constituted a considerable step forward compared to the more traditional methods of calculation. The second edition of 2004 of the same treatise, with the same title, done in collaboration with Fuentes [228], which came to integrate and expand the first edition, is now an irreplaceable reference for designers, theoreticians, students and gear manufacturers. In 1995, just a year after the publication of the Litvin’s treatise, Dooner and Seireg [106] published the monographic textbook entitled The Kinematic Geometry of Gearing: A Concurrent Engineering Approach. With this textbook, the authors proposed a generalized and integrated methodology for the design and manufacture of various types of toothed bodies. This methodology is also very interesting and innovative, as it solves the intricate interactions between the design, manufacture and assembly processes for the more general toothed bodies in meshing conditions with the analytical and differential geometry procedures. Equally interesting is the discussion that sees the interaction between design aspects and CAD/CAM generation processes with an integrated methodology. With the two editions of the monographic treatise titled Theory of Gearing: Kinematics, Geometry and Synthesis, respectively of 2013 and 2018, Radzevich wanted to provide the scientific community with a synthesis of the knowledge regarding the geometric-kinematic aspects of the gears and, in this regard, formulated a scientific theory of gearing. Drawing on many years of experience in the field of gears, Radzevich has reworked all scientific knowledge concerning the above aspects, presenting

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them in the form of a scientific theory understood in the Euclidean meaning, and therefore in the modern meaning of the term, that is, the position of a given number of postulates and derived concepts that meet these postulates (see Prefaces of Vol. 1 and Vol. 2). The work done by Radzevich [303, 306] is extremely praiseworthy, but circumscribed only to the geometric-kinematic problems of the gears. As we have said elsewhere, gears represent a much more complex subject, which also includes technological and strength aspects. The road to defining an all-encompassing general scientific theory of gearing is still very long and impervious. It should also be noted that Radzevich himself, in 2017, published the second edition of the monograph titled Gear Cutting Tools: Science and Engineering were, to synthetize CAM systems for sculptured surface machining on multi-axis NC machines and for designing gear cutting tools with optimal parameters, DG/K-based methods of surface generation are used. The corresponding mathematical approach of these methods is fundamentally based on the simultaneous use of Differential Geometry of surfaces and Kinematics of multi-parametric motion of a rigid body in the three-dimensional space. DG/K is therefore the acronym of the two approaches, Differential Geometry and Kinematics, which form the bases of the mixed method used [305]. Today, for various well-known reasons, the sizing of the gears as well as the various calculations concerning their load-carrying capacity in actual working conditions are strongly standardized. These calculations are in fact made according to national and international standards, in the awareness that the over-standardization can limit their potential performance, more or less significantly. Many gear drives, especially those for advanced applications in aerospace and automotive [131, 268], are however designed to perform specific functions, so they must be customized and optimized to perform this task in the most technically and economically satisfactory way. In these cases, a sizing and computation approach using direct gear design methods can help to maximize gear drive performance in custom gear applications [197].

References 1. Addomine M, Figliolini G, Pennestrì E (2018) A landmark in the history of non-circular gear design: the mechanical masterpise of Dondi’s astrarium. Mech Mach Theory 122:219–232 2. Airy GB (1827) On the forms of the teeth of wheels. Trans Camb Philos Soc II:279 3. Akahori H, Sato Y, Nishida Y, Kubo A (2001) Test of the durability of face gears. The JSME international conference, MTP 2001, Fukuoka, Japan 4. Al-Tubi IS, Long H, Zhang J, Shaw B (2015) Experimental and analytical study of gear micropitting initiation and propagation under varying load conditions. Wear 328–329:8–16 5. Al BC, Langlois P (2015) Analysis of tooth interior fatigue fracture using boundary conditions from an efficient and accurate loaded tooth contact analysis. In: British Gears Association (BGA) Gears 2015 Technical Awareness Seminar, 12th of November 2015, Nottingham, U.K. (also Gear Solutions, Feb. 2016) 6. Al BC, Patel R, Langlois P (2016) Finite element analysis of tooth flank fracture using boundary conditions from LTCA. In: CTI Symposium USA, Novi, MI, 11–12 May 2016 7. Al BC, Patel R, Langlois P (2017) Comparison of tooth interior fatigue fracture load capacity to standardized gear failure models. Gear Solutions, July, pp 47–57

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Correction to: The First Scientific Age and Birth of the Science: From the Beginning of Hellenism to the Diaspora of Scientists of the Museum of Alexandria

Correction to: Chapter 3 in: V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_3 The book was inadvertently published with error in Chapter 3 (Figure 3.11 on page 50).

The online version of this chapter can be found at https://doi.org/10.1007/978-3-030-40164-1_3 © Springer Nature Switzerland AG 2020 V. Vullo, Gears, Springer Series in Solid and Structural Mechanics 12, https://doi.org/10.1007/978-3-030-40164-1_6

C1

C2

Correction to: The First Scientific Age and Birth of the Science …

The correct figure should be:

Fig. 3.11 A reconstructive design of the Archimedes odometer with the gear used

The chariot that carries the Archimedes odometer is shown with the two wheels in the air has been corrected by rotating 180 degrees clockwise.

Author Index

A Acerbi, F., 40 Addomine, M., 76, 81, 180 Agius, D., 71 Agricola, G., 91, 94, 97, 117 Airy, G.B., 165 Akahori, H., 183 Al, B.C., 215, 217 Alexander, J.M., 22, 190 Al-Jazari, Ibn al-Raz., 12, 72–75 Almen, J.O., 198, 203 Al-Tubi, I.S., 211 Amontons, G., 207, 208 Angelakis, A. N., 8, 11 Annast, R., 215 Archard, J.F., 209, 210 Argyris, J.H., 204 Asfaw, B., 2 Aziz, I.A.A., 198

B Bach, C. von, 193, 195 Bagci, C., 15 Bakman, D., 10 Ballarini, R., 188 Ball, R.S., 166, 179, 218 Banerjee, P.K., 205 Barmina, N., 179, 180 Barnes, C.F. Jr., 76 Basstein, G., 183 Bathe, K.J., 205 Bauschinger, J., 199 Beale, O.J., 175, 176 Becker, A.A., 205 Beermann, S., 215 Belajev, N.M., 191, 213, 216

Bell, E.T., 146 Benford, R.L., 181 Bennet, J.A., 154 Bernoulli, J., 158, 187, 188, 190 Berthe, D., 211 Bertsche, B., 182, 222 Besson, J., 117, 122, 136 Bhadoria, B.S., 183 Bhandari, V.B., 198 Bidwell, S., 208 Birch, T.W., 182 Biringuccio, V., 91, 94, 97, 117 Blateyron, F., 213 Blok, H., 202 Bloomfield, B., 181, 183 Bo, D., 30, 52 Bodensiek, E.J., 214 Boiadjiev, I., 216 Boinette, A., 122 Bordmer, J.G., 167 Bossler, R.B., 183 Bossler, R.B.Jr, 183 Bottema, O., 166 Boussinesq, J., 213 Bowden, F.P., 208, 209 Boyer, C.B., 70 Bradley, R.E., 161 Brebbia, C.A., 205 Breen, D.H., 215 Broghammer, E.L., 199, 200 Brugh, P., 115 Buchanan, R., 168, 169, 192 Buckingham, E., 195, 196, 198, 219 Burn, R.P., 190 Burnyeat, M.F., 31 Burrows, M., 2, 3 Bur, T., 17

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238 Butterfield, R., 205 Buyse, F., 136

C Camus, C.E.L., 160, 161, 167, 168 Canfora, L., 22, 24 Capechi, D., 132 Carbone, G.M., 176 Cardano, G., 157 Castigliano, C.A., 190 Cauchy, A-L., 190 Cayley, A., 179 Ceccarelli, M., 47, 114 Cerruti, V., 213 Chakraborty, I., 183 Chaphalkar, N., 5 Chauhan, V., 198 Chen, N., 183 Chen, Y-J.D., 183 Chen, Y.S., 183 Clapeyron, B.P.E., 190 Clark, J.G., 82 Clough, R.W., 204 Coates, J.F., 10 Colamarino, T., 30, 52 Colbourne, J.R., 221 Coleman, E., 204 Colin, G.S., 10 Collin, M-M., 143 Colonnetti, G., 190 Conry, T.F., 213 Cook, R.D., 205 Cormac, P., 179, 218, 219 Coulomb, C.A., 190, 191, 208 Courant, R., 204 Couturat, L.L., 140 Coxhead, M.A., 16 Cozzarelli, F.A., 190, 191 Croce, B., 4 Crosher, W.P., 81, 162, 172 Cuneo, P.F., 88, 89 Cunningham, F.W., 181 Cusano, C., 213 Cutter, L.E., 195

D D’Alembert, M., 147, 149, 150 Dalley, S., 47 Dante Alighieri, 18, 80 Davydov, J.S., 179, 180 de Condorcet, M.J.A.N., 147

Author Index DeGusta, D., 2 De La Hire, F., 157, 158, 160, 161, 165, 168 Del Monte, G., 134, 150 De Miranda, A., 9–11 Denavit, J., 180 Descartes, R., 138, 139, 158 de Vaujany, J-P., 183 De Young, G., 60, 71 Di Giorgio Martini, F., 77, 78, 89, 91, 93, 117 Disaguliers, J.T., 208 Disteli, M., 166 Dohrn-van Rossum, G., 81 Dolan, T.J., 199, 200 Dondi dall’Orologio, G., 82 Donovan, S., 22 Dooner, D.B., 160, 165, 181, 221 Dowson, D., 203, 213 Drachmann, A.G., 33, 38, 41 Drake, S., 134 Droysen, J.B., 22 Dudley, D.W., 172, 220 Dupont, P., 161 Durcan, T.M., 196 Dürer, A., 122, 157, 159 E Ehret, P., 213 Elkholy, A., 215 Ellenberger, M., 143 Emmert, S., 211 Ertel, A.M., 203 Euler, L., 146, 147, 150, 161, 162, 166, 187, 188, 190 Eyring, H., 213 F Fairbairn, W., 193 Farboomand, I., 205 Fatemi, A., 213 Fenner, R.T., 205 Ferrari, C., 182, 208 Figliolini, G., 180 Filler, R.R., 183 Fisher, A., 195 Fisher, R., 222 Flamand, L., 211 Flamant, A-A., 213 Flanders, R.E., 173, 195 Flašker, J., 198 Fleck, H.F., 26, 41 Fontanari, C., 40

Author Index Föppl, A.O., 199 Föppl, L., 199 Ford, H., 190 Foucher, D., 211 Foullon, A., 154 Francis, V., 183 Franklin, J., 139 Franklin, L. J., 195 Fraser, P.M., 23, 37 Freeth, T., 44 Fuentes, A., 160, 181, 221

G Galilei, G., 58, 81, 132, 186, 191, 192 Gambiano, G., 46 Garro, A., 200, 202, 205 Gautschi, W., 161 Gere, J.M., 188 Ghazali, W.M., 198 Ghribi, D., 215, 216 Gilbert, H., 2 Giovannozzi, R., 197, 198, 208, 220 Gladwell, G.M.L., 198 Glaubitz, H., 201 Glodez, S., 198 Gobler, H.E., 181 Gochman, Ch.I., 179 Godet, M., 211 Goglia, P.R., 213 Goldfarb, V., 179 Grant, G.B., 170, 181 Grubin, A.N., 203 Guardini, M., 40 Guingand, M., 183

H Hall, A.R., 78 Hall, B.S., 76 Hamming, P.W., 205 Hamrock, B.J., 203 Handschuh, R.F., 183 Harris, O., 217 Harrison, J., 155, 156 Hartenberg, R.S., 180 Hawkins, J.I., 167–170 Heath, G.F., 183 Hein, M., 215 Henchy, H., 191 Henriot, G., 201, 219 Hertter, T., 216 Hertz, H.R., 191, 198, 208

239 Heyman, J., 187 Heywood, R.B., 200 Higginson, G.R., 203, 213 Hill, D.R., 8, 72 Hill, R., 190, 191 Hitchcock, R., 71 Hlebanja, G., 5 Hlebanja, J., 5 Höhn, B-R., 211, 215, 216 Holditch, H., 180 Holm, R., 209, 210 Hooke, R., 153, 162, 168, 187, 188 Hösel, Th., 201 Houbolt, J.C., 205 Houser, D.R., 5, 195, 196 Howell, F.C., 2 Huber, M.T., 191 Huygens, C., 81, 136, 151, 153, 158 Hyatt, G., 5

I Idris, D.M.N., 198 Imison, J., 165, 167, 168 Innocenzi, P., 115 Irgens, F., 204 Isaacson, W., 109

J Jacquin, C-Y., 183 Johns, C.H.W., 10 Johnson, K.L., 198, 213, 216 Jürgens, G., 222

K Kadiric, A., 211 Kahraman, A., 211 Kajale, P., 184 Kapelevich, A.L., 222 Karas, F., 196 Kasner, E., 47 Katchanov, L., 190 Kelley, B-W., 200 Kelsey, S., 204 Keplero, I., 138 Kissling, U., 215 Kleinhenz, O., 5 Klein, M., 215 Knorr, W.R., 59 Kolchin, N.I., 179 Korka, Z., 5 Kramberger, J., 198

240 Kubo, A., 183 Kücükay, F., 222 Kulkarni, S., 184 Kurrer, K-E., 190 L Lagrange, J-L., 147 Lagutin, S., 180 Lallemend, M., 122 Lalovera, A., 158 Lamé, G., 190 Landels, J.G., 31 Langlois, P., 216, 217 Lang, O.R., 215 Lang, U.M., 70, 215 La Rocca, C., 79 Lasche,O., 193, 195, 196 Laufer, B., 10 Lazovic, T., 215 Lechner, G., 182 Leibniz, G.W., 47, 138–140, 143–146, 158, 190 Leslie, J., 208 Levy, S., 204 Lewicki, D.G., 183 Lewis, M., 37, 193, 196, 198, 201 Lewis, M.J.T., 37 Lewis, W., 138, 193 Lindberg, D.C., 70 Lindner, W., 218 Lin, J-L., 44 Linnaeus, C., 1 Li, S., 211 Litvin, F.L., 160, 180, 181, 183, 221 Liu, H., 211 Liu, Q., 214 Liu, Y., 214 Li, W., 8, 11 Lloyd, G.E.R., 31 Logue, C.H., 197, 198 Long, H., 211 Loria, G., 30, 54 Lovász, L., 139 Loveless, W.G., 178 Lyberatos, G., 8, 11 M Mac Curdy, E., 207 MackAldener, M., 215, 217 Maitra, G.M., 221 Mancosu, P., 139 Manna, F., 29, 49, 79, 125

Author Index Mariotte, E., 190 Marquis, G., 213 Martin, H.C., 204 Martin, H.M., 203 Martinze, M.T.M., 211 Marx, G. H., 195 Massera, A.F., 88 Maugin, G.A., 204 Maxwell, J.C., 191 Mayr, O., 72 McDonnell, A., 167 McKenzie, J., 71 McMullen, F.E., 196 Melosh, R.J., 204 Menabrea, L.F., 190 Merrill, E.M., 91 Merritt, H.E., 200, 203, 219 Meyer, P.B., 182 Michaelis, K., 215 Miladinovi´c, L., 76 Minotti, M., 185 Modugno, F., 219 Modugno, F., 219 Mohr, O., 190, 191 Moon, F.C., 31, 42, 45, 78, 82, 125, 154, 177, 178 Morello, G., 91, 109 Mori, M., 5 Morrison, J.S., 10 Murray, W.M., 10 Musschenbroek, P. van, 192, 193

N Nadai, A., 191 Najork, R., 222 Nathan, I., 104 Naunheimer, H., 182, 222 Navier, C.L.M.N., 190 Needham, J., 15 Nenci, F., 41 Netz, R., 40 Neuber, H., 199 Newman, J.R., 47 Newmark, N.M., 205 Newton, I., 134, 138, 143–145 Niemann, G., 201, 220, 221 Nishida, Y., 4 Noel, W., 40 North, J.D., 81 Novak, W., 182, 222 Novozhilov, V.V., 191

Author Index O Octrue, M., 215, 216 Oleson, J.P., 9, 47 Olivier, T., 165, 169, 179, 218 Ollson, U., 181 Olsson, M., 215, 217 Oster, P., 211, 215 Özgüven, H.N., 5, 195, 196 P Pääbo, S., 2 Paipetis, S.A., 47 Panjkovic, V., 207 Parent, A., 190 Pascal, B., 140, 143, 158 Pastore, G., 45 Patel, R., 1, 5 Patil, D.U., 184 Paul, P., 191 Pedersen, R., 200, 215 Pedrero, J.L., 183 Pedretti, C., 104, 109 Pennestrì, E., 44 Peterson, R.E., 199 Petroci´c, D., 76 Petti, F., 44 Piper, M., 5 Poisson, S.D., 190 Polder, J.W., 192 Pollak, B., 222 Pollone, G., 220 Pomini, O., 218 Popkonstantinovi´c, B., 76 Potrˇc, I., 198 Potts, D.T., 8 Prager von, F.D., 86, 88 Préaux, C., 59 Price de Solla, D.J., 43, 67, 78 Przemieniecki, J.S., 205 R Radzevich, S.P., 16, 162, 164, 165, 168, 172, 175, 180, 220–222 Ramelli, A., 117, 126 Ramsden, J., 167 Ramsey, S., 37 Rankov, N.B., 10 Rashed, R., 72 Ree, T., 213 Rescher, N., 140 Reti, L., 89 Rettig, H., 201

241 Reuleaux, F., 125, 176, 177, 181, 193 Reynolds, O., 191 Rice, R.L., 215 Richards, G.D., 2 Richter, J.P., 104 Riemen, R., 76 Ristivojevic, M., 215 Ritter, G.D.A., 190 Romiti, A., 182, 208 Rosenfield, A.R., 210 Ross, A.A., 195 Roth, B., 166 Ruggiu, L., 16 Russo, L., 4, 21, 25, 45, 53, 57, 59, 72, 132 Ryborz, J., 182 Rycerz, P., 211

S Sainsot, P., 215 Saint-Venant, Barré de, A.J.C., 190 Salvini, P., 185 Sandberg, E., 215 Sandifer, C.E., 161 Sang, E., 173–175 Santander, M., 15 Sarton, G.A.L., 31 Sato, Y., 183 Scaglia, G., 86 Scheller, R.W., 76 Schiebel, A., 218 Schütz, W., 198 Seireg, A.A., 160, 181, 221 Shames, I.H., 190, 191 Shamsaei, N., 214 Sharma, V.K., 215 Shaw, B., 211 Sheth, V., 183 Shishkov, J.S., 180 Shotter, B.A., 211 Sijtstra, A., 183 Silvagi, J., 183 Smith, C.H., 195 Socie, D.F., 213 Šraml, M., 198 Stahl, K., 215, 216 Stoimenov, M., 76 Stolze, C.H., 190 Straub, J.C., 198, 203 Stribeck, R., 193 Stringer, C., 2 Sutton, G., 2, 3 Suwa, G., 2

242 T Tabor, D., 208, 209 Tamburrino, A., 8, 11 Tan, J., 183 Tannery, P.L., 59 Taylor, C.M., 213 Tchernetska, N., 40 Tetmajer, L. von, 190 Theodossiou, N., 11 Thomas, J., 215 Timoshenko, S.P., 138, 187, 188, 191, 199 Tobie, T., 215, 216 Topp, L.J., 204 Trautschold, R., 198 Tredgold, T., 168, 192 Tribbetts, G.R., 72 Truesdell, C.A., 190 Tuplin, W.A., 201, 220 Turner, M.J., 204

U Uccelli, A., 114

V Valentini, P.P., 44 Valipour, M., 8, 11 Valturio, R., 78, 88 Veldkamp, G.R., 166 Venci, A., 215 Vico, G., 4 Vinogradova, I.E., 203 Vittorio, A., 46 Vivio, F., 191 von Mises, R., 191 Vullo, V., 191, 200, 202, 205

W Wahl, A.M., 199 Walter, G.H., 215 Wang, J.C., 183

Author Index Wang, S., 213 Warwick, A., 182 Watt, J., 150, 151 White, K.D., 37 White, M., 103 Whiteside, D.T., 145 Wieland, W., 16 Wikander, Ö., 9 Wilcox, L., 204 Wildhaber, E., 181, 196 Willis, R., 165 Wilson, A., 16, 37, 182 Wilson, E.L., 205 Wilson, N., 40 Winter, H., 201, 221 Winter, T.N., 16, 29 Witzig, J., 215, 216 Wöhler, A., 198, 199 Woodbury, R.S., 165, 168, 170, 172, 177 Worlidge, J., 10 Wright, M.T., 42, 44

Y Yan, H-S., 44 Yannopoulos, S.I., 8, 11 Yonkxiang, Lu., 15 Young, T., 165–167 Yu, M., 191 Yu, Z-Y., 214

Z Zammattio, C., 188, 190 Zanzi, C., 183 Zhang, J., 211 Zhou, S-P., 214 Zhou, Y., 211 Zhu, C., 211 Zienkiewicz, O.C., 205 Zöllner, F., 104 Zonca, V., 126

Name Index of Non-authors

A Abraham Gotthelf Kästner, 162 Adelard of Bath, 60 Aelius (or Claudius) Galenus of Pergamon, 60 Aeschylus, 26 Agathocles, 26 Ahmad Ban¯u M¯us¯a, 72, 73 Al-B¯ır¯un¯ı, 53 Alexander the Great, 22 Al-Hajj¯aj ibn Yüssuf ibn Matar, 71 Al-Hasan Ban¯u M¯us¯a, 72 Al-Husayn, 74 Al-Shaghani, 74 Ammianus Marcellinus, 88 Anaxagoras of Clazomenae, 23 Anaximander, 23 Anaximenes of Miletus, 23 Andrea Verrocchio, 84 Anthemius of Tralles, 69, 70 Antigonos I Monophthalmos, 49 Antoine de Laloubère, 158 Antonio Migliabecchi, 139 Apollonius of Perga, 24, 25 Archelaus of Miletus (or likely of Athens), 23 Archias of Corinth, 46 Archimedes, 24–26, 28–31, 39–43, 45–51, 53, 58, 59, 61, 62, 68, 70–72, 74, 86, 89, 122, 132, 134, 146, 177, 178, 184 Archytas of Tarentum, 16, 28 Aristarchus of Samos, 24, 46, 58 Aristomachus of Soli, 61 Aristotle, 15–17, 22, 24, 41, 59, 69, 70, 207 Arkhias, 26 Athenaeus of Naucratis, 10, 33, 46 Aulus Gellius, 28

Azzone Visconti, 81

B Bacchylides, 26 Baldassare Lanci, 154 Barth, C.G., 195 Berenice II, 24 Bernardus Cartonensis Chartres), 76 Blaise Pascal, 140 Böttger, 135

(Bernard

of

C Caesar, 60 Callixenus of Rhodes, 10 Chares of Lyndos, 49 Charlemagne, 79 Charles Boyle, 155 Christopher Wren, 158 Claudius Claudianus, 42 Conon of Samos, 24, 25, 47 Cosimo I de’ Medici, 154 Ctesibius, 24, 31, 33–38, 53, 54, 59, 62, 74

D Daniel Bernoulli, 188, 190 Demetrius I of Macedon, 49 Demetrius Phalerum, 34 Demetrius Poliorcetes, 46 Democritus, 40 Diodorus Siculus, 27, 31 Diogenes Laërtius, 23, 28, 31 Dionysius I the Elder, 26 Dionysius II the Young man, 26

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244 Dionysius of Alexandria, 37, 38 Diophantus of Alexandria, 58 Dositheus of Pelusium, 25

E Ecphantus the Pythagorean, 28 Edwin R. Fellows, 174 Einstein, 132 Epicharmus, 26, 27 Eratosthenes of Cyrene, 24 Eric Watson, 82 Euclid, 24, 31, 32, 40 Eudoxus of Cnidus, 28, 31 Eupalinus of Megara, 28 Eutocius of Ascalon, 69 Evangelista Torricelli, 158

F Felix Heinrich Wankel, 117 Felix Savary, 161 Filippo Brunelleschi, 84 Francesco Maria della Rovere, 115 Frederich II of Sweden, Emperor, 71 Frederick I of Prussia, 139 Frederick the Great of Prussia, 146

G Gaius

Aurelius Valerius Diocletianus Augustus, 61 Gaius Suetonius Tranquillus, 67 Gaius Sulpicius Gallus, 42 Gaius Verres, 27 Galen, 71 Gilles Personne de Roberval, 158 Giovanni Battista della Valle, 115, 122, 125 Girard Desargues, 157 Glenn, D., 185 Guido da Vigevano (or da Pavia), 76–78

H Hannibal, 77 Harum el-Rashid, Caliph, 79 Henry Hindley, 178 Henry II of France, 154 Henry VIII, 81 Heraclides Ponticus, 28 Heraclitus of Ephesus, 23 Herodotus, 28, 50 Hero of Alexandria, 33, 53, 54, 70 Herophilus of Chalcedon, 24

Name Index of Non-authors Hicetas of Syracuse, 28 Hieron I, 26 Hieron II, tyrant of Syracuse, 46 Hipparchus of Nicaea, 25 Hippolytus of Rome, 28 Homer, 31 Hugo Bilgram, 168 Hypatia, 22

I Ibn Sahl, 71, 72 Ioannes Philoponus, 69 Ion of Chios, 23 Isidorus of Miletus, 69, 70 Iustus Byrgius (Jobst Bürgi), 127

J Jacques Andruet du Cerceau, 117 Jakob Bernoulli, 187, 188, 190 James Ferguson, 155 James Pickard, 150 James White, 162 Jandesek, J., 198 Jean Errard de Bar-le-Duc, 122 John Wallis, 158 Joseph Jandesek, 198 Joseph R. Brown, 172, 175 Joseph Saxton, 172 Juan de Caramuel, 140

L Laurentium da Valpuria, 151 Leonardo da Vinci, 6, 58, 78, 82, 85, 89, 91, 97, 134, 147, 154, 162, 177, 178, 187, 191, 207 Leon Battista Alberti, 84 Leopoldo, Prince, 136 Leo the Mathematician, 71 Lorenzo Rèal, 136 Lucius Annaeus Seneca, 60 Lucius Apuleius Madaurensis, 72 Lucius Caecilius Firmianus Lactantius, 42 Lucius Mestrius Plutarchus of Chaeronea, 60 Ludovico il Moro, 89 Ludovico Sforza, 91 Lysippus, 49, 50

M Ma Jun, 15 Marcus Claudius Marcellus, 24, 41

Name Index of Non-authors Marcus Claudius Marcellus Jr., 42, 45 Marcus Terentius Varro, 60 Marcus Tullius Cicero, 27, 41, 45 Marcus Vitruvius Pollio, 28, 33 Mariano di Jacopo (Mariano Taccola), 78, 86 Marin Marsenne, 157 Martianus Mineus Felix Capella, 42 Matteo de’ Pasti of Verona, 89 Melissus of Samos, 29 Michelangelo, 84 Michel Chasles, 157 Moschus (or Moschionus), 46 Mozhayev, S.S., 180 Mozzi del Garbo, 178 Muhammad Ban¯u M¯us¯a, 72

N Nicholas Copernicus, 138 Nicolaus Cusanus, 157 Novikov, M.L., 181

O Ole Christensen Römer, 154 Olimpiodorus the Younger, 72

P Pacificus of Verona, 79 Pappus of Alexandria, 27, 42, 46, 61 Paracelsus, 91 Parmenides of Elea, 29 Paul I, Pope, 79 Paul III, Pope, 91 Pausanias, 50 Pepin de France, 79 Phidias, 50 Phileas of Tauromenion, 46 Philip of Opus, 31 Philiscus of Thasos, 61 Philistion of Locri, 28 Philolaüs of Croton, 28 Philo of Byzantium, 24, 33, 37, 38, 49, 53, 62, 86, 88, 91 Pierre de Fermat, 158 Pindar, 26 Plato, 24, 26, 31 Pliny the Elder (Gaius Plinius Secundus), 31, 33, 49, 52, 54, 60, 62 Polycrates, 28 Proclus Lycaeus Diadocus (Proclus of Athens), 46 pseudo-Archimedes, 74

245 pseudo-Aristotle, 16–18, 41 Ptolemy, 54, 58, 71, 82 Ptolemy I Soter, 23, 31 Ptolemy II Philadelphus, 23, 34, 37 Ptolemy III Euergetes, 24, 46 Ptolemy IV Philopator, 10 Ptolemy VIII Kakergetes, 23 Publius Flavius Vegetius Renatus, 88 Publius Ovidius Naso, 42 Publius Vergilius Maro, 59 Pythagoras of Samos, 28 Pytheas of Massalia, 30 Q Quintus Haterius Thychicus, 66 Quintus Horatius Flaccus, 30, 52 R Rhoecus of Samos, 50 Richard of Wallingford, 81, 82 Roberto Valturio, 78 Rutilius Taurus Aemilius Palladius, 67 S Salviati, 58 Santorio Santorio, 135 Sextus Empiricus, 60 Sextus Julius Frontinus, 62, 88 Shishkov, V.A., 180 Sigismondo Pandolfo Malatesta, 88 Simonides, 26 Simplicius of Cilicia, 69 Socrates, 23 Stevenson, H.M., 185 Strabo, 31, 54 Strato of Lampsocus, 54 Sylvester II, Pope, 79 T Teleches of Samos, 50 Th¯abit ibn Qurra, 72 Thales of Miletus, 23, 25 Theaetetus, 24, 31 Themistius, 207 Theocritus, 27 Theodorus of Samos, 50 Theon of Alexandria, 22, 68, 72 Theophrastus, 59, 60, 91 Thucydides, 26 Titus Flavius Domitianus, 67 Titus Flavius Iosephus, 88

246

Name Index of Non-authors

Titus Flavius Vespanianus, 67 Tycho Brahe, 154

William of Walsham, 81 Wilson, W.G., 182

V Villard de Honnecourt, 76, 77, 79 Vincenzo Galilei, 134 Vincenzo Viviani, 136 Voltaire, 139

X Xenophanes, 26

W Walker, E.R., 195 Wilhelm Albert, 198 William Murdoch, 150

Y Yi Yin, 8

Z Zenodorus, 50 Zeno of Elea, 29