1.1. Review by: Pierre Brunet.
Revue d'histoire des sciences et de leurs applications 1 (2) (1947), 188-189.

This very short volume - like all those in the "Que sais-je?" series to which it belongs - is rich in substance; its methodical organisation of the subject matter and its straightforward presentation enable readers seeking an introduction to find exactly what they are looking for. Like the others, it is likely readers will find this summary interesting, as it conveniently groups together easily usable information. Undoubtedly, the author has given much thought to the historical problems whose elements he outlines here, and his work has not been limited, as is too often the case with popularisers, to compiling more or less obscure notes borrowed from more or less dubious sources; on the contrary, he has been able to rethink the whole himself, and thus obtain a homogeneous and solidly constructed whole. The historical overview to which the first chapter is devoted allows for a kind of internal unity in the other chapters, where the author successively deals with number, the technique of numerical calculation, the algebraic sciences, and mathematical analysis.

Perhaps it is to avoid upsetting philosophers that René Taton writes that "Géométrie is an appendix to the Discours de la méthode." In reality, both in Descartes's thought and in the physical presentation of the work, the Discourse is, on the contrary, an introduction to the scientific works that follow it. This minor clarification is of little importance; and there is no reason to attribute more to a remark suggested to me by a passage in the text of this interesting little volume. But I believe the author has been too influenced by certain historians when he supposes that Gerbert used "tokens marked with Roman numerals from I to IX." From the information given to us about Gerbert's teaching by a contemporary of the learned monk, his student Richer, it seems clear that he used Arabic numerals. This fact is all the more plausible because, while one may rightly consider Gerbert's supposed journey to Cordoba a legend, it is nonetheless true that he stayed for some time on the northern borders of Spain and could have become familiar with the numerals brought from the East by the Arabs.

However, as I have said, I would not want this to be taken as a criticism of a work in which I am pleased to recognise qualities that rank it among the best in this collection in the field of the history of science.

2.1. From the Introduction.

Among the great names that illuminated French science at the end of the 18th and beginning of the 19th centuries, Monge occupies a prominent place.

By systematising and refounding the old methods of architectural drawing, Monge succeeded in creating a new branch of geometry: descriptive geometry. In addition to the new and elegant methods it introduced, methods very close in many cases to those still in use today, this science, through its immediate development, revived faith in the study of pure geometry, which the successes of Cartesian geometry and infinitesimal analysis had almost extinguished, and the magnificent flowering of modern geometry and projective geometry during the 19th century stemmed from this current.

But the establishment of descriptive geometry as a new branch of science is only one aspect of Monge's mathematical work, and various chapters of his treatise form a transition to his more analytically oriented work. Cartesian geometry, as developed by Euler in the mid-18th century, still almost entirely excluded lines and planes from its scope; by showing the necessity of restoring to first-order elements the essential role that should be theirs, Monge laid the foundations of modern elementary analytic geometry.

Until him, work in infinitesimal geometry had focused primarily on plane problems, and the brilliant work of Clairaut and Euler had only begun the application of this science to space. With Monge, infinitesimal geometry of space developed rapidly; the study of curves and surfaces began to take shape as a body of doctrine that his own work and that of his students would quickly enrich. Despite its profound originality, Gauss's rich contribution to this field simply continued this fruitful current, employing a new technique.

Moreover, Monge understood almost from the beginning of his research that problems of space were as intimately linked to the theory of partial differential equations as infinitesimal plane problems are to differential equations. And, clarifying his idea, he highlighted the link that unites partial differential equations with families of surfaces defined by the same mode of generation; this link constitutes the guiding thread of his studies in the two associated fields of infinitesimal geometry in three-dimensional space and the theory of partial differential equations.

Thus, despite its apparent diversity, his mathematical work possesses a very solid internal unity. Monge's exceptionally profound sense of space illuminates all his studies, whether they appear purely analytical or oriented towards geometry. In his work, geometry and analysis constantly lend each other mutual support, each of these sciences illuminating reasoning that, viewed from a single perspective, might have seemed too abstract or too strictly descriptive. His thought seems to embrace simultaneously the various aspects of the same problem, and his skill allows him to choose at every moment the viewpoint that best illuminates the question under study.

While Euler, who codified almost all mathematical knowledge in the mid-18th century, personifies the analytical aspect of mathematics at the time, Monge, on the contrary, appears as the first truly geometrically inclined innovator since Desargues. And while Euler's analytical method, characterised by rigour and abstract calculation, is reflected in the work of Lagrange, Legendre, Laplace, Cauchy, Gauss, and the greatest analysts of the 19th century, Monge's method, characterised by the intimate union of analysis and concrete geometric reality, is at the origin of the many currents which, passing through Poncelet, Steiner, Plücker, Klein, Lie, Darboux, and so many other esteemed geometers, will allow the development of the various geometric branches of mathematics to resume and analytical abstractions to be illuminated by concrete examples.

In his famous study on Intuition and Logic in Mathematics, Henri Poincaré distinguishes two main types of mathematicians: analysts concerned above all with logical rigour and geometers who create new concepts than by associating them with concrete images. The examples he chooses - Méray, Hermite, Weierstrass, and Sophie Kovalewski as analysts, Klein, Joseph Bertrand, Riemann, and Sophus Lie as geometers - are they not, respectively, in their research methods and their manner of presentation, disciples of Euler and Monge?

Living in an era when the physical and chemical sciences were undergoing a revolution, Monge, with his inquisitive, enthusiastic, and concrete mind, could not fail to take a keen interest in them. Thus, we will see him participate in the rapid progress of physics, anticipate certain points, experiment while taking almost perfect precautions, measure with the greatest precision for the time, and finally join the efforts of the scientists of the French school who ensured the triumph of the new chemistry.

We will also see that, driven by his profession as a professor in a technical military school, by his personal interests, and by his manual dexterity, he turned very early on to practical application and closely followed the evolution of the various theoretical and technical problems posed by the rise of metallurgy and the birth of machinery. Was not his course in descriptive geometry, moreover, essentially intended to provide technicians and artists with a flexible, convenient, and rigorous method of representing figures in space? His greatest desire was to see French industry develop, and he was convinced that this development could only come from an adaptation of scientific methods to the various fields of technology. Descriptive geometry, machine theory, physics, and chemistry were for him not only sources of captivating study in themselves, but, even more so, tools enabling, through the progress of industrial methods, the improvement of general living conditions.

Monge was also a fervently democratic spirit, inspired by a very active social conscience; the brilliant success of his career did not make him forget the difficult lives of the humble people he knew in his early days. Thus, the Revolution found in him one of its most resolute supporters. While the scholar and technician briefly gave way to the administrator during a short stint at the Ministry of the Navy, the Committee of Public Safety subsequently knew how to utilise the breadth of his skills and his total devotion to the cause of the Republic. Alongside Carnot, he played a leading role in organising the victory during Years II and III, whether in the technical direction of arms and gunpowder manufacturing, in the development of new working methods, or in the technical training of officers.

Then, once the danger had passed, Monge revealed himself in a new light. His long experience as a professor at the Engineering School of Mézières and as a naval examiner, his keen pedagogical sense, and his knowledge of the nation's practical needs allowed him to play an important role in organising the new education system, in particular, in launching the École Polytechnique, which he intended to be the training ground for all of France's most capable engineers. The School's immediate success amidst difficulties of all kinds proved the excellence of his ideas; but it did not deter all those who wished to see its direction changed, and, until the end of his career, Monge had to fight to preserve, as far as possible, the spirit of this institution.

One cannot speak of the École Polytechnique without considering Monge's exceptional prestige as a professor. The testimonies of his students are unanimous on this point: rarely has a professor been able to interest and galvanise his students with such success and attract such a number of valuable disciples to his favourite fields of study. Therefore, one cannot judge Monge's scientific work without taking into account his oral teaching and direct influence as much as his published works. Moreover, he was too interested in the most varied fields of research to agree to concentrate his attention and sacrifice a large part of his time to writing up the results of his research and reflections. And, if a rapid outpouring of new results appears around him and in his wake, it is certain that most of the ideas underlying them belong to him.

To complete this brief portrait, we would still need to outline Monge's administrative career, mention his various missions, and consider how this ardent republican was so quickly seduced by Bonaparte, to whom he devoted a constant friendship and whom he admired even in some of his most counter-revolutionary actions.

This brief overview of Monge's work already gives a first glimpse of his rich and original personality, but only a more detailed study can allow us to grasp its full range and judge it in terms of its immediate results and longer-term consequences. This is the goal we have set for ourselves in this study. While there are numerous monographs devoted to Monge - those by Brisson, Dupin, Arago, and Louis de Launay -, but unfortunately, none of them deals with his scientific and technical work in a sufficiently complete and detailed manner, and many of his contributions have remained unnoticed or underestimated. It therefore seemed necessary to us to try to paint a fairly precise and faithful picture of the work of this great geometer, as well as of his influence on the progress of science. Our study begins with a biographical sketch that situates Monge's various works within the context of his successive activities. Then come a series of chapters which, constituting the core of this work, examine the various aspects of Monge's mathematical work: descriptive geometry, analytic geometry, infinitesimal geometry, pure geometry and modern geometry, and mathematical analysis. This division, imposed by the needs of a systematic study, could have the drawback of obscuring the richness of Monge's thought, which never concentrates on a single area of ​​mathematics but simultaneously perceives the various aspects of each question he studies. In order to avoid this drawback as much as possible, we have had to study the various aspects of the same work in different chapters on several occasions; but we have always striven in this case not to break the profound unity of each of these works.

Monge's work in mechanics, physics, chemistry, and technology is less well known; we have attempted, with the help of new documents, to paint a more complete picture of it, and this has allowed us, it seems, to better highlight the rather considerable importance of his role in the development of these various branches of science.

In the concluding chapter, a brief preliminary study of the scientific milieu of the time and of the various aspects of Monge's personality, his human qualities, and his highly personal gifts as a creator and teacher, allows us to better situate the main lines and internal unity of his work and to glimpse the richness of the various currents which, originating from his different works and his teaching, would contribute to the renewal of science during the 19th century.

The work concludes with an overview that situates Monge's works, published or unpublished, in relation to the most significant events of his life, through a bibliography of the main reference works related to our study and an index of the names of people cited in the work.

In preparing our work, we consulted and studied all of Monge's publications as well as numerous unpublished manuscripts and a fairly extensive correspondence.

In several previous studies, we were able to publish some of these unpublished works, which will allow us to streamline the presentation of this overall study; however, there are still many unpublished manuscripts and letters by Monge that are of undeniable interest to the history of science. The publication of the Complete Works of the great geometer would allow us to bring them together while also regrouping the various works scattered in journals or academic collections that only a few libraries possess in their entirety. May this study highlight the value of such an undertaking, which the scope of Monge's work in several important areas of science certainly justifies.

There is, however, one difficulty that we would not wish to conceal, especially since we ourselves have encountered it; due to the numerous administrative positions and various missions entrusted to Monge, only a small part of his work and the vast majority of his correspondence are of interest only to documentary and political history, and not to the history of science. While it was relatively easy for us in this study devoted to the scientific work of the great French geometer to eliminate all documents extraneous to the specific subject of our work, this selection would certainly be more difficult in the preparation of so-called complete works. But this obstacle is not such that it should cause us to abandon this undertaking.

2.2. Review by: S Issmann.
Revue Internationale de Philosophie 6 (22) (4) (1952), 455-456.

The rise of modern geometry and projective geometry owes much to the work of Gaspard Monge (1746-1818).

A remarkable theorist, he never lost sight of the applications of mathematics to the most diverse technologies and also contributed to the progress of physics and chemistry.

He devoted a large part of his life to teaching and put his many talents at the service of the First French Republic. He gave new impetus to the metallurgical industry in France, at a time when the country was threatened by foreign invasions, and played an essential role in the creation of the École Polytechnique.

He later rallied to Napoleon, who showered him with honours, and died during the Second Restoration without any official tribute being paid to him.

Monge is above all the creator of descriptive geometry.

Descriptive geometry stems from practical methods used in various techniques, notably architectural drawing, practical perspective, and gnomonics, or the art of drawing sundials on which the time is indicated by the movement of a point shadow.

Certainly, Monge had predecessors: Desargues, who solved various problems in architectural drawing and developed important elements of the projective theory of conic sections; Albrecht Dürer, who published a remarkable treatise on geometry in 1525 oriented towards applications to art; and many others. But it was Monge who extracted the fundamental principles from earlier techniques and thoroughly studied the problems of representing objects in space on the plane.

But it would be arbitrary to isolate Monge's work on descriptive geometry from his studies in other areas of mathematics.

His contributions to analytic geometry are essential.

He broadened the perspective of Fermat and Descartes, the two great founders of this science, through the analytical study of the point and the line, which had been neglected and which made it easier to approach more complex problems. Following Clairaut, Euler, and Lagrange, he succeeded in definitively extending analytic geometry to three-dimensional space and making it a coherent system.

In the field of infinitesimal geometry, his work is even more important.

He studied the evolutes of double-curved curves and developable surfaces using the analytic geometry of space and highlighted the relationships between partial differential equations and families of surfaces defined by the same mode of generation, relationships which he used extensively in his later work on families of surfaces.

Monge can also be considered the direct precursor of modern geometry, whose impetus was given by Poncelet at the beginning of the 19th century, not only for his numerous works in various fields of mathematics but also for his contributions to pure geometry.

In significant parts of his work, he used mathematical analysis with great success and contributed to its progress through his studies on partial differential equations and total differential equations.

In this remarkable work, René Taton has shown the important place Monge occupies in the evolution of the mathematical sciences and revealed the profound unity of his diverse work, which was always concerned with practical applications.

2.3. Review by: Pierre Poumier.
Revue Historique 208 (2) (1952), 331-334.

Taton's book is a valuable contribution to the history of mathematics, highly methodical and very clear. After outlining the history of the questions Monge addressed, he examines Monge's contributions and then traces their influence. In doing so, he reveals the originality and unity of this powerful mind and explains its impact. He organises his analysis of Monge's work according to its subject matter, progressing from the simple to the complex. In each of the areas he explores, he reconstructs the genesis of Monge's thought, a difficult task, as Monge was rather indifferent to the publication of his discoveries. An overview of Monge's scientific work, a general bibliography, and an alphabetical index of cited names complete Taton's book. We will attempt to provide an overview of this rich body of work, avoiding, for the benefit of readers of the Revue historique, details that would require highly specialised knowledge.
...
Monge was not one of those abrupt geniuses, misunderstood when they first appeared, like, for example, Évariste Galois. Many of his discoveries were adopted almost immediately, unchanged, into teaching; thus, he had a fruitful influence on the minds of his time. But, when the initial impetus of his research subsided, the very perfection of his formulations came at a price. It engendered a routine entirely contrary to his spirit. It was too convenient to use his results without rethinking the correspondences that gave them all their meaning. Exercises in analytic geometry too often became a blind game of formulas, and there were professors of descriptive geometry who taught students to draw diagrams while forbidding them to think in three dimensions. From this, perhaps, stems the sort of sclerosis that afflicted the studies of the preparatory classes for the Grandes Écoles and the École Polytechnique, the "taupin" mentality from which it was so difficult to break free (this is a former professor of preparatory classes speaking).

2.4. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 5 (2) (1952), 183-185.

This work is the author's principal thesis for his doctorate in Letters. Monge is one of the purest glories of French science. A regrettable neglect by the authorities, from Louis-Philippe to the present day, means that we are still awaiting the publication of his complete works. The work we are analysing fortunately fills, to some extent, the void created by this neglect, but it also makes us all the more aware of the magnitude of this void.
...
The work is complemented by an overview of the scientific work: printed works classified in chronological order of publication, scientific manuscripts, scientific correspondence, academic memoirs in chronological order of presentation.

2.5. Review by: R Blanché.
Revue Philosophique de la France et de l'Étranger 148 (1958), 98-99.

Note. The review covers both Taton's book on Desargues and his book on Monge.

To restart geometry, which had stalled at the point where the Greek and Alexandrian authors had brought it, two things were necessary. First, to break with the set of prejudices that erected a barrier between pure speculation and technical problems, devaluing the latter and relegating the study of transformations - so essential to the spirit of modern geometry - to the latter because of its practical applications. Second, to possess an exceptional aptitude for seeing in space, for thinking like a geometer, and thus for resisting the current that, especially since the beginning of the 17th century, was increasingly leading mathematicians toward analysis. These two conditions were equally met in Desargues, the brilliant precursor of modern geometry, and in Monge, who, through his students as well as his writings, was its true initiator.

Subject to the hostility of practitioners whose empiricism criticised his claim to universal methods, and little understood by the geometers of his time, despite the esteem in which Descartes and Fermat held him and the influence he exerted on Pascal, Desargues, rediscovered by Servois and Poncelet, remains little known even today. His Leçons de Ténèbres (Lessons of Darkness) are lost, and his Brouillon Project (Draft Project), printed in 1639 in fifty copies, was already untraceable in Leibniz's time. Chasles discovered only a manuscript copy made by La Hire, and M Taton, after fruitless searches, was resigned to publishing this text after Poudra when, at the last moment, he had the opportunity to consult an original copy which M P Moisy had very opportunely just acquired. Besides the Draft Project on conics and the documents appended to it, Mr Taton provides a long letter to Mersenne dated 4 April 1638, and three propositions (including the famous "Desargues' theorem") included in the Treatise on Perspective published by Bosse in 1648 - reserving for a possible future collection the writings relating to applications. Introductions and notes illuminate the texts, which are preceded by a study on the life and work of Desargues.

The other book, framed by a biographical chapter and a conclusion on Monge's personality, provides a detailed and precise analysis of the various aspects of his scientific work. Keen to convey its originality precisely, Mr Taton never fails to place it in its historical context, so that his book, in which erudition is always placed at the service of intelligence, transcends the interest of a simple monograph. Two characteristic traits of Monge's genius, in which the concrete and the abstract constantly unite to illuminate each other, are particularly emphasised. First, the connection he establishes between theoretical and technical problems. Often consulted by these individuals, and always attentive to applications likely to alleviate human suffering, he was no less concerned, on the other hand, with explaining and generalising practice through theory. Thus, disparate graphic methods were brought together by him into a general and uniform technique, or rather, reduced to the unity of a rational doctrine of which descriptive geometry is, as Darboux says, merely the graphic expression. No less remarkable is the correspondence that exists in his mind between geometric representation and analytical construction. Monge, undoubtedly, unlike Euler or Lagrange, was essentially a geometer; he had the gift of seeing not only surface but also volume, and this is why, with him, analytical geometry and infinitesimal geometry made up for the lag they had experienced in the study of space compared to that of the plane. Nevertheless, it is only in a very refined sense that he can be described as intuitive. This geometer makes little use of diagrams. Perhaps partly because three-dimensional forms are better suggested by gestures than by flat representations on a board, and Monge was famous for his ability to evoke the most complicated spatial relationships before his students with no other aid to illustrate his words than the movements of his hands. But also because he grasps, in a way, the concrete through the understanding of the abstract operations he represents. One of the characteristic features of his mathematical genius is the constant cooperation he establishes, for their mutual illumination, between geometry and analysis. For example, he finds a geometric interpretation for partial differential equations and, conversely, translates into their language the infinitesimal properties of surfaces and curves in space, thus attributing to them, for three-dimensional geometry the role that is recognised, for the plan, by differential equations. Let us hope that scientists, ordinarily focused on the infinitesimal increase that leads from the present to the future and scarcely having the time, and not always the inclination, to look back on the past of their science, will often give us works of this quality.

2.6. Review by: George Sarton.
Isis 43 (1) (1952), 77-79.

Note. The review covers both Taton's book on Desargues and his book on Monge.

The Editor of Isis has always discouraged the writing of omnibus reviews. From his point of view, it is not well to deal in a single review with books which do not form a group of great homogeneity, for example, with books which refer to different periods. Yet, an exception must be made in this case, in spite of the fact that Monge lived one century and a half later than Desargues. The exception is justified not only because both books were written by the same scholar, but chiefly because Monge was the continuer of Desargues. These two men were two outstanding links in a golden chain which represents one of the most glorious traditions of the French genius. Think of it, Desargues, Pascal, Monge, Poncelet, Darboux. The two books are equal in their excellence; for they illustrate the same method, cautious and rigorous as to facts, intelligent and generous in the interpretations. Moreover, the fundamental requisite is satisfied, because R Taton has a deep knowledge of the mathematical ideas which Desargues and Monge invented or developed.

The two books differ in one respect. The first might be described as a scientific edition of Desargues' geometrical writings with an elaborate introduction. On the other hand, the Monge volume, though larger, does not contain any text and might be considered as an introduction to the Oeuvres complètes, the realisation of which will not be, let us hope, delayed too long.

Girard Desargues was born at Lyon on 2 March 1591 (not 1593) and he died, one does not know where, in 1661 (not 1662), before 8 October. It is seldom that the date and place of death of an illustrious man are known with less precision than the date and place of birth. The discrepancy is explained in this case, because Girard was the son of a man of substance who was "notaire royal" in Lyon, and secondly because his fame was tardy. We know relatively little about his youthful development, and he was already 39 when his presence in Paris is reported for the first time, and 44 years old (in 1635) when we find him in the little academy established by Father Mersenne, where he could associate with such men as Gassendi, Etienne Pascal, Mydorge, Roberval, etc. His first geometrical writing, Exemple de l'une des manières universelles (12 pp., 1 double plate, Paris 1636), was already written two years before, for Mersenne seems to refer to it in a letter to Peiresc of 24 August 1634. It is significant that that short text illustrates already two of his fundamental (and complementary) ideas: geometry is not simply a theoretical science in the Greek sense, it is also a graphical method for the solution of practical problems; on the other hand, perspective is more than a collection of technical tricks, it is a branch of geometry. He enriches and vitalises these two concepts, geometry and perspective, by mixing them and enabling each to fertilise the other. In 1639, Desargues published his Brouillon project, and the following year received the finest tribute which could be given to him so early in the Essay pour les coniques, a broadside written by the son of his friend, Etienne Pascal, the boy Blaise, then 17 years old. [See facsimile of the Essay and his English translation by Frances Marguerite Clarke (Isis 10, 16-20, 1928).] Blaise Pascal remarks, "We can also demonstrate a property stated below, due to M Desargues of Lyon, one of the great geniuses of his time and well versed in mathematics, particularly in conics ... ." Blaise was Desargues' first disciple and for a long time the best one.

Desargues earned his income, apparently a good one, as architect and engineer. His architectural contributions, many of which are listed, were mostly of the technical kind, the building of staircases, vaults, etc. His work as an engineer and technical adviser is but vaguely known; we have definite knowledge only about one project, the water pump which he set up in the castle of Beaulieu and which was described by Huygens in 1671.

The texts included in this edition are as follows. I. Letter of Desargues to Mersenne on the problem of tangents, 1638. II. Brouillon project, 1639. Taton was able to use one of the rare original copies recently discovered in a mixed lot of papers of the Bibliothèque Nationale, hitherto uncatalogued. Many documents are added to this opus. III. The geometrical propositions of 1648.

Desargues' outstanding disciple was Blaise Pascal, but whatever fame he had in his time was established by another, humbler, disciple, Abraham Bosse (1602-76), who published in Paris 1643 two books vulgarising Desargues' ideas. I. La pratique du trait a preuves de M Desargues pour la coupe des pierres en l'architecture. II. La maniere universelle de M Desargues pour poser l'essieu et placer les heures et autres choses aux cadrans au Soleil. The first was translated in German (Nuremberg 1699), the second in English (London 1659). The French texts were reprinted in Paris (1648, 1653), and translated into Dutch (Amsterdam 1664). The Dutch translation was one of the first western geometrical treatises to reach Japan. I have tried to obtain some information concerning its influence upon Japanese mathematics, but in spite of the kind efforts of Japanese friends, no definite knowledge is yet available (Isis 41, 300, 1950; including facsimile of the title page of the Dutch translation).

So much space has been devoted to Desargues, that I must be briefer concerning Gaspard Monge (1746-1818), and yet there would be much more to say about the latter. While Desargues was a Lyonnais, Monge was a Bourguignon, being the most illustrious son of Beaune. Monge belonged to a poorer family, yet the father was ambitious for his sons and managed to have them thoroughly educated in the Beaune College of the Oratory; Gaspard was sent for a while to the larger college of the same order in Lyon. In 1764, he returned home, drew a map of Beaune which attracted some attention to him, and soon afterwards joined the military school (Ecole du génie) of Mézières in the Ardennes. He spent there twenty years of hard and inspired work. We cannot continue his biography which involves not only his own work but the Revolution, the conquests of Italy and Egypt, the Institut, the Ecole Polytechnique, etc. Read all that in Taton.

Monge's personality was far more diversified than Desargues' and though he did not live much longer than his predecessor (72 vs. 70 years), his life was switched repeatedly on new tracks. Desargues was annoyed by the disorders of the Fronde, but what were these as compared with the Revolution and all the vicissitudes which followed? Remember that Monge lived until 1818 and thus witnessed the glories of the Empire and also, to the bitter end, the calamities which followed. Monge's genius was deeper and more complex than that of Desargues'. Everybody knows him as "the founder of descriptive geometry"; that description is correct as far as it goes, but so incomplete that it is very misleading. A simple list of Monge's mathematical achievements would be so long that we could not insert it here. Inventing descriptive geometry implied a revival of the geometrical methods which had been thrown into the background by the development of infinitesimal analysis and its prodigious fertility. Monge did not found only descriptive geometry, he was one of the founders of analytic geometry, and he was one of the first to extend infinitesimal geometry to the third dimension. Moreover, he realised that such three-dimensional problems were intimately connected with equations of partial derivatives, while plane problems were connected with differential equations. His intuitions were geometric rather than analytic like those of Euler, Lagrange, Legendre, etc., but he was attacking the same problems and throwing a new light upon them.

Though his mathematical activities were prodigious, his mind overflowed the mathematical field in every direction. He was interested in mechanics, physics and chemistry, in technical progress; his long experience as a teacher in the school of Mézières enabled him to reorganise the scientific and technical training of the whole nation. He was one of the founders of the Ecole Polytechnique, a brilliant teacher, and administrator. In spite of his democratic tendencies and his enthusiasm for the Revolution, he let himself be mesmerised by Bonaparte, and worse still, by Napoleon. He was one of the men who voted for Bonaparte's election (1797) as member of the Institut in the place of the illustrious Lazare Carnot "l'Organisateur de la victoire," driven into exile. Many of the non-scientific aspects of Monge's personality have been described in earlier books, e.g., by Louis de Launay (Paris 1933; Isis 22, 242-45) and Francois Charles-Roux: Bonaparte, gouverneur d'Egypte (Paris 1935; Isis 26, 465-70); mathematical aspects have been discussed in mathematical histories such as Cajori's History of mathematics (2nd. ed. New York 1919) and Julian Lowell Coolidge: History of geometrical methods (Oxford 1940; Isis 33, 347-50), but Taton is the first to give us a full account of his scientific work in every direction.
...
The two volumes include elaborate bibliographies and indexes. They are both important additions to the history of mathematics. The author is still very young, and we have good reason to hope that our debt to him, which is already great, will be considerably increased in the years to come.

3.1. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 5 (1) (1952), 94-95.

This work is the author's supplementary thesis for his Doctorate in Letters.

The first part forms the introduction to the works published subsequently.

The first chapter, devoted to the early life of the mathematician Girard Desargues, specifies for the first time his year of birth, 1591. This is thanks to the baptismal certificate, found in the Lyon Archives, and dated 2 March 1591. A second, substantial and well-documented chapter examines the years of scientific production, which, as far as we know, extend roughly from 1636 to 1648, and during which Desargues was involved in the Parisian literary society where Mersenne, Beaugrand, Roberval, and the Pascals moved, and which corresponded with Descartes, Fermat, and many others.

Since Desargues was not a scholar confined to his study, Chapter III examines his career as an architect and practitioner more briefly. Finally, Chapter IV of this introduction provides a bibliography of his works and writings. It is known that in 1864, Poudra published everything he could find of this body of work in a two-volume work. Unfortunately, he was unable to locate all of the great geometer's writings, and even for those he was able to publish, he did not always have access to the best sources.

The author of the work we are analysing devotes the second part of his work to the annotated publication of the more purely mathematical works. He was unfortunately unable to get hold of the Leçons de ténèbres (Lessons of Darkness), whose existence he firmly establishes and which a second fortunate coincidence, sought as patiently as the first we are about to discuss, may perhaps lead him to discover in one of our great libraries.
...
Desargues' geometric work, difficult to grasp on a first reading, is of paramount importance in the history of geometry. The great geometers of the 19th century, such as Poncelet and Chasles, recognised this even before fully understanding it. This new edition, the first revised and corrected edition, since it has been possible to include Desargues's own correction sheet, will be invaluable to researchers. The numerous accompanying notes greatly facilitate its reading, and the editor's mathematical and historical expertise lends them considerable authority.

3.2. Review by: R Blanché.
Revue Philosophique de la France et de l'Étranger 148 (1958), 98-99.

See review above.

3.3. Review by: George Sarton.
Isis 43 (1) (1952), 77-79.

See review above.

3.4. Review by: S Issmann.
Revue Internationale de Philosophie 6 (22) (4) (1952), 456-457.

The work of the great French geometer Girard Desargues (1591-1661), contemporary of Fermat and Descartes and the teacher of Blaise Pascal, was largely unknown during his lifetime and was only appreciated for its true worth at the beginning of the 19th century.

His works in geometry, too audacious for his time, provoked fierce criticism and remained, to a large extent, misunderstood. Neither Fermat nor Descartes gave them the attention they deserved.

While Descartes aimed to develop a universal method of reasoning and apply it to the most diverse fields of science, and gave mathematics an algebraic and analytical orientation, Desargues strove above all to renew geometry and extend its results to the most varied fields of application.

He had a profound knowledge of various techniques, such as practical perspective, gnomonics, and stonecutting, which since antiquity had been used almost exclusively by artisans, and his principal merit was to have extracted some fundamental theoretical principles from them. He showed that the improvement of these techniques depended essentially on the progress of geometry, of which they are merely applications.

He studied more particularly the properties of certain conics, which are, in a way, circles seen in perspective, and developed the first elements of the theory of geometric transformations, the full importance of which would only become apparent in the second half of the 19th century, with the work of Felix Klein.

The author of this work, René Taton, after situating Desargues's work within the evolution of mathematical thought, presents us with an annotated and commented edition of the French geometer's principal work, the Brouillon Project, on conics.

The original text of this work was very recently discovered by Mr Pierre Moisy at the National Library of Paris. Until now, only a rather incomplete copy, made in 1679 by the geometer Philippe de la Hire, was known.

René Taton added an appendix devoted to mechanics and a "Foreword," a sort of mixture of errata and addenda, both of which had been published in 1639 in Paris at the same time as the Draft Project. These two writings, also rediscovered by Mr Pierre Moisy, had never been republished since then.

René Taton has added to the preceding texts various documents that demonstrate the importance of Desargues' work, the influence it exerted, and the reactions it provoked, including a letter from Desargues to Mersenne, Pascal's Essay on Conics, and three geometric propositions extracted from Abraham Bosse's Treatise on Perspective (1648) that should be attributed to Desargues. The first of these propositions is the "Desargues Theorem," which D Hilbert used in his axiomatisation of geometry.

4.1. From the Foreword

The progress of science results from a continuous series of discoveries of widely varying importance and significance. Each of these, building more or less directly on previous achievements, in turn paves the way for further progress. But this advance is far from steady. Its general direction is often obscured by periods of relative stagnation, or even apparent regression; while at other times progress seems, on the contrary, to accelerate, leading to the overall renewal of vast fields of science or creating fruitful connections between different sectors of knowledge.

Scientific discovery, in fact, presents itself in quite varied forms, depending on the field to which it applies: mathematics, theory, or experimentation; on the temperament, training, and experience of its author; and finally, on the more or less favourable circumstances in which it takes place.

Making a discovery requires, on the part of its author, undeniable qualities of scientific method and intuition, even a very special kind of genius, necessary for developing major syntheses or audacious theories. Moreover, every discovery of any significant magnitude encounters resistance sometimes very strong; to engage in the necessary struggle against routine and prejudice, the scientist must demonstrate undeniable intellectual audacity. But, if, in this struggle, he sometimes finds himself somewhat isolated, he remains nonetheless in solidarity with his predecessors and contemporaries. The science of an era results, in fact, from the successive contributions of numerous generations of researchers, whether geniuses or simple and obscure servants of science. And, in its presentation, its objectives, and its applications, this science reflects the concerns of the civilisation of the time.

The study of the origins, conditions, circumstances, and characteristics of scientific discovery presents very varied epistemological and psychological aspects, and numerous studies with often divergent conclusions have been devoted to it.

Avoiding venturing into the dangerous territory of overly systematic interpretations, we prefer to undertake a description of the various domains of discovery, its principal factors, and its essential aspects. Numerous examples drawn from different sciences will aid us in this task of exploration. In order not to risk diminishing their demonstrative value, we will strive to present them as objectively as possible, foregoing convenient schematisations that only offer a distorted image of the ever-complex reality.

Therefore, our conclusions will often appear less clear and less categorical than those of some authors; but is not this complexity of aspects of discovery a faithful reflection of the variety of temperaments and the diversity of circumstances?

4.2. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 8 (4) (1955), 357-359.

This work comprises a foreword, three parts, and a conclusion.

I. The Different Domains of Discovery.

Mathematics. Theoretical Sciences. Observational and Experimental Sciences.

II. The Factors of Discovery.

Methodical Discovery. The Fruitfulness of Certain Discoveries. The Flash of Thought. The Role of Chance. The Role of Error.

III. Various Aspects of Discovery.

Originality of Discovery. Failed Discoveries. Audacity versus Routine. Scientific Discovery, a Reflection of the Civilisation of an Era.

This is followed by a short conclusion, an index of personal names (approximately 300 names with their earliest and latest dates of birth and death), and an analytical index by subject: Mathematics, 25 citations; Astronomy, 36; Physics, 52; Chemistry, 16; Instrumental Technology, 19; Biology, 15; Medicine, 14.

The extensive bibliography is provided in footnotes throughout the book.

4.3. Review by: Jane Oppenheimer.
The Quarterly Review of Biology 31 (4) (1956), 288-289.

Causalités et Accidents de la Découverte Scientifique is one of a series of original monographs entitled Évolution des Sciences: Liaisons et Synthèses-Methodes et Techniques, addressed to a scientific audience and dealing with the essential nature of scientific research. This particular member of the collection, contributed by a mathematician, discusses the development of scientific concepts by considering the psychological processes of the individual investigator as a function of his moment in scientific history.

It is broad in its scope, in so far as it has chosen its examples from diverse fields: mathematics, astronomy, physics, chemistry, biology, and medicine. There is however a strong undertone throughout of emphasis on the more technological aspects of science. For instance, a subsection, The problem of precursors, of a larger section on Originality of discovery devotes slightly less than two pages to paludism; two to musical timber; a little over three pages to the prehistory of antibiotics; and eleven to a detailed and most fascinating description of the invention and development of lighter-than-air flying apparatus. The examples chosen from biology, throughout the volume, bear further witness to this tendency. The discovery of the circulation of the blood and the development of the evolution concept are barely mentioned (as examples of what the author considers revolutionary discoveries accepted only after considerable controversy). While three pages are devoted to the laws of Mendel as instances of forgotten discoveries, and one to Spallanzani's discovery of the role of the spermatozoa as an example of a découverte manquée, by far the greater part of the author's attention to matters biological is concentrated on the discovery of the antibiotics and on strictly medical discoveries concerned rather with improving methods of combat against disease than in elucidating more theoretical scientific principles.

From the specifically historical point of view Taton limits himself on the whole to tracing the interrelationships of ideas which are rather intimately related to one another, within the bounds of a single or of closely related sciences; he turns less attention to the relations of the various sciences to one another and to other disciplines of thought.

It is a strength of his volume, however, that it is original in its arrangement and organisation. The material is presented in other than chronological sequence, and the historical data are included as subsidiary to the treatment of generalised problems which characterise many historical periods: the factors responsible for the fate of discoveries, unrecognised as such and made before their time; the function of chance, and of error, in scientific discovery; the place of logic and intuition in scientific method. The author is distinctly at his best when he considers, as he does in the bulk of his book, the creative thought processes of individual contributors. In fact, he has a number of original suggestions relating to these. Among them may be mentioned his interest in the significance of an aesthetic sense in directing mathematical discovery, and in the importance, in a number of discoveries, of the confluence of two currents of thought within the mind of a single investigator. These portions of the book, in particular, should prove of consummate interest to those who are concerned with psychological factors involved in the production of scientific creativity. The writing throughout is very good, and the pictorial illustrations are excellent.

4.4. Review by: P Le Corbeiller.
Science, New Series 122 (3181) (1955), 1193.

The author of this interesting and unusual book has collected a large number of documents concerning scientific discoveries. The book has the positive merit of having a wide scope, ranging through mathematics, astronomy, physics, chemistry, instrumental techniques, biology, and medicine, and the negative merit of supporting no specific theory, historical, psychological, or other, but presenting each individual item in its variegated collection for the appreciation and judgment of the reader.

The author remarks in his preface that his objective presentation avoids forcing the complexities of actual scientific research into a systematic schematism. Unfortunately, this avoidance of rigidity has led him into the pitfall of disorder. We are made to jump from antiquity to modern times and to the Renaissance, from spectroscopy to hydrostatics and to typhus, from great names to obscure ones, from basic discoveries to chance observations and to dead ends, from original quotations to amusing anecdotes, so that the very richness of the collection leaves the mind in a whirl. In spite of this, the book will find its place on the shelves of the teacher, the historian, or the philosopher of science; each one will find recorded in it several cases that he has not met before as well as illustrations for his own pet theories. In this he will be aided by two excellent indexes, one of names and one, analytic, of subjects. The plates are also very good.

4.5. Review by: Roger Hahn.
Isis 48 (3) (1957), 352-353.

The phenomenon of scientific discovery is one which may never be comprehended fully by historians of science. At root it is an artistic creation which cannot be explained entirely by analytical and rational means. Fully aware of this, Professor René Taton has chosen to discuss in his new book different phases of scientific discovery. At the outset, Taton declares that he does not pretend to formulate or test any theory concerning the technique of scientific discovery. In fact, the subtitle of his book, "illustrations of several characteristic periods of the evolution of the sciences" is more revealing of its contents than is the main title.

The book is divided into three sections, each selectively covering the full breadth of the sciences throughout time. Section one divides the subject according to the different disciplines, emphasising the fundamental similarities and differences among the mathematical sciences, the theoretical sciences, and the sciences of observation and experimentation. This subdivision, establishing guide lines rather than strict delineations between the individual sciences, gains from the imprecision of definition of terms. In this treatment Taton evades the inherently polemical problems of the apparent or true individualism of each science. His approach is more expository and historical in nature than philosophical. Thus, each subsection is predominantly devoted to illustrative case histories. The second section is entitled "The factors of discovery," and covers such widely different subjects as discovery by method, the fecundity of certain discoveries, and the role of chance and error. Here again examples are abundant and highly suggestive, though frustrating to the reader who seeks a general theory explaining the mechanism of scientific discovery. In the third section, the author takes up other aspects of a more general nature, such as the question of precursors, priority, premature discoveries, near-discoveries, the resistance to new ideas, and the relations between scientific discoveries and the times in which they are made.

In its essentially historical approach, this book resembles the Harvard Case History series, initiated by the former president of Harvard, James B Conant. But it is more ambitious in scope and less detailed, though historically as valid. The volume would be an excellent medium for stimulating an interest in the history of science among young students, and a forthcoming English translation should make it valuable as a teaching tool in the United States.

The choice of examples reflects the range of interests and the variety of the author's knowledge. Though it is obvious that Taton has a superior grasp of the history of mathematics and its sources, he is careful to incorporate examples from all the sciences and from most of the scientifically productive countries of the West. The American reader not familiar with the current French literature will gain much from the references cited by the author, which are predominantly, though not by any means exclusively, French. Some discussion of the choice of illustrative examples could easily be made. The pages of history are endless. In the section concerning the role of chance in discovery, for example, this reviewer would have preferred the inclusion of Oersted's discovery rather than the somewhat questionable role of chance in Nicolle's discovery of the existence of typhus carriers. In illustrating the role of simplificatory errors, the author spends almost ten pages discussing the example of Kepler. The case of Gay Lussac and the law of small whole numbers might have proved as effective, and certainly shorter. In all fairness to Taton, it must be admitted that possible substitutions are endless, and that examples picked by him are well chosen to illustrate general ideas, and usually well documented with references, diagrams and photographs. Certain omissions in the footnoted bibliography are noteworthy: the absence of the work of Doris Hellman in the discussion of comets, and of Carl Boyer's work on the foundation of calculus.

Taton has written this book as one in a series entitled "Evolution des Sciences." This series addresses itself to a learned public not necessarily well versed in the history of science. For the specialist, the book may appear inconclusive in its presentation of the major underlying issues in the development of the sciences. Taton evokes many issues, without ever stating them very clearly, or giving a specific interpretation of his own. The examples are left to speak for themselves. One implicit element in the book is the equilibrium between the role of the total environment and that of the creative scientist in scientific discovery.

In several instances, Taton points to the importance of sociological influences in guiding a particular researcher toward an area of investigation. At other times, he points to the often delaying and conservative effect of religious, philosophical, or even scientific dogmas on the progress of science. Taton also cites examples of the inadequacy of experimental techniques in solving certain problems. Yet throughout his book the equilibrium between the role played by the environment and by the individual is maintained by examples highlighting genius, intuition and chance. The problem is itself worthy of another book which Taton might write for another public. In this one, however, the main issues are merely presented for discussion.

The author's rather non-committal attitude seems abruptly contradicted by the inclusion of two pages in italics at the end, under the title of "Conclusion." Here the author pleads for disinterested research and warns of the evils surrounding team-research work. Laudable as these pages may be, they seem completely out of place as a conclusion to a seemingly non-partisan work, and might more readily serve as an introduction to another book rather than as a conclusion to this one.

In view of its general usefulness, the forthcoming English translation (London: Hutchison) ought to find its way to the book shelves of all teachers of the history of science.

4.6. Review by: Paul Chalus.
Revue de synthèse 76 (1955), 237-240.

R Taton, compelled by the facts, constrained by the subject matter he addresses, presents views that would be easy to compile; he develops a philosophy - we would not say a system - which is all the more interesting to note because this philosophy was not intentional. It demonstrates both the intrinsic philosophical value of the evolution of science and the qualities of the thinker who speaks to us about it. ...
...
...starting from the facts, always attached to the facts, he will gradually be able to extract from the history of science some of the great lessons it offers on the process, and consequently, on the very nature of the mind. With the magnificent example given by Gaston Bachelard, we can no longer doubt the value of this path, which Henri Berr had already advocated for a very long time.

5.1. Review by: Robert Multhauf.
Science, New Series 127 (3303) (1958), 870.

The history of science, in addition to contributing to what might be called the liberal scientific education, has an undoubted potentiality in providing a record from which scientists can obtain methodological guidance through the experiences of their predecessors. The realisation of that potential depends upon an increased cultivation of the field, especially in the more recent period. The product may be something like the present work, which cites and analyses examples of chance, error, "flashes of thought," and other incidents rare and commonplace in the history of discovery and invention.

This book should interest scientists for the picture that it gives of this product of the history of science. The case histories selected are as various as 16th century astronomy and 20th century medicine, and an interesting attempt is made to correlate them. Unfortunately there are too many cases for the size of the book, and the resultant oversimplification sometimes jars the sensibilities of the historian. Neither scientists nor historians will appreciate the evidences of careless writing, such as the confusion between the terms discovery and invention (worsened by the translator) and the looseness of the use of the terms chance, error, and others.

The original French edition was more modest in format, title, and price.

5.2. Review by: Cyril Stanley Smith
The Journal of Modern History 32 (3) (1960), 261.

This is a translation of the author's Causalités et accidents de la découverte scientifique, published in Paris in 1955. It is unfortunate that R Taton, one of the very best historians of science, could not have been represented in English by a more significant book. As a description of the systematic side of the "art of scientific discovery" this book is but a small advance over the one of this name by Gore published in 1878, though it is written in a more personal style and includes many interesting case histories. Successful chains and broken links in the sequence of discoveries are discussed, with particularly good treatment of recent events in the biological sciences. The book contains enjoyable descriptions of the manner in which scientists go about their work and of the environmental factors that have influenced some of them. The sequential nature of scientific development is emphasised, as well as the essential role played by the interaction between experiment and theory, and by technology - the latter sometimes being carried to the point of missing the distinction between technology and science. The first balloon ascents belonged even less to the domain of science than did the first Sputnik, but like them it was a superb use of science which properly caught the public imagination and indirectly served to foster science itself.

The author rightly stresses the role of precursors and notes that "most discoveries were made on soil that had already been prepared to some extent, and that however revolutionary a scientist may have been, his work is only a more or less direct continuation of the work of his predecessors". His final comments on the values and dangers of politics in the administrative organisation of science are very apposite today. Yet with all its good points the book has a pedestrian quality and somehow reflects neither the hard work and real excitement of scientific research nor the essential beauty of it, and it does not lead to any new understanding of the intellectual process. The illustrations are well reproduced and the book is well printed, though grossly over-priced at over six cents per page. Readers should get their copies from an English bookseller for 30s. - less than half the American price.

5.3. Review by: Pincus Schub.
The American Mathematical Monthly 65 (7) (1958), 536.

Korzybski in his general semantics defines plants as energy binders, animals as space binders, and man as a time binder. One is tempted to extend this epigrammatic definition to that rare subspecies of Homo sapiens, the geniuses, as idea binders. Taton's book is a survey of the subtlest form of idea-binding: invention and discovery.

What leads to an invention? Can we discover the laws of discovery? These are much-mooted questions, and the answers vary widely. Some writers maintain that "the act of discovery is purely an accident and owes nothing to reason or logic" (Nicolle). This view is rejected by Taton. Especially with regard to mathematical discovery he follows Poincaré and Hadamard in assuming the existence of two more-or-less distinct types of mind: the logical and the intuitional, or the analyst and the geometrician. The former proceeds by sustained and methodical effort, while the latter, as described vividly in Poincaré's self-analysis, may arrive at his results in three stages: first, conscious but often fruitless concentration ("stirring of ideas"), followed by a period of rest and incubation (subconscious processing of ideas), which stage may end happily by sudden illumination and insight, requiring merely verification for its completion.

In the observational and experimental sciences, however, the role of chance in discovery cannot be denied, but even here Taton holds with Pasteur that chance helps those whose minds are well prepared for it. Another point that he develops and illustrates convincingly by historical examples is the fact that a discovery must be attributed not only to its author, but also to its epoch. Scientific discoveries usually appear at the right time and proper background. Premature discoveries are not appreciated by their contemporaries, and even their authors are frequently unable to perceive their full significance. Still another fact: error has occasionally played a paradoxically fruitful role in discovery.

Taton concludes his compact and erudite survey with a word of warning and apprehension regarding the increasingly collective character of modern research. While the advantages of this development are unquestionable, the dangers of putting the accent on the immediately profitable and of reducing individual initiative are undeniable.

The book is handsomely illustrated and the translation reads smoothly, disregarding an occasional solecism like "this phenomena."

5.4. Review by: O T Benfey.
Journal of Chemical Education 35 (7) (1958), A320-A321.

To those interested in case histories describing the varying paths of scientific discovery, this book will come as a welcome addition since it brings to the attention of the reader a number of important pieces of continental European work not generally known in this country. With the increasing concern for the right training of scientists must come a greater interest in the actual ingredients that make for greatness in science. There is an impressive and growing literature on the creative process in mathematics, and a number of studies in the area of physics have been made. Chemistry has barely been touched and unfortunately this book does little to fill the gap. Of the sciences tabulated in the index (astronomy, biology, chemistry, instrumental techniques, mathematics, medicine, and physics) chemistry is the subject most sparsely treated, only the discovery of the elements through studies of radioactivity and spectrum analysis being dealt with in detail.

Furthermore, apart from the classification of factors in discovery implied by the chapter headings in part 2 (Systematic Discovery, The Fruitfulness of Some Discoveries, Flashes of Thought, The Role of Chance, The Role of Error) the author, according to the preface, has intentionally desisted from theorising about the nature of scientific discovery. This has its merits but it leaves the reader bewildered in the face of the multitude of examples.

In part 1, the author differentiates between three realms of discovery, those in mathematics, theoretical sciences, and observational and experimental sciences. Part 3 deals with the problem of simultaneous discovery, of missed discoveries, and with the distressing struggles many valuable ideas and discoveries face before they are accepted and utilised. Throughout the book the qualities of character required by innovators are well brought out.

The book ends with a discussion of the increasingly collective character of modern scientific research and the dangers to creativity that such an organisation of science must guard against.

5.5. Review by: Arthur H Bragg.
Bios 29 (1) (1958), 24.

The substance of this book is grouped under three subheadings: 1. Realms of Discovery (mathematics; theoretical science; observational and experimental science), 2. Factors in Discovery (systematic discovery; fruitfulness of some discoveries; flashes of thought; chance; role of error), and 3. Aspects of Discoveries (originality; missed discoveries; struggle against routine and specific discovery or a reflection of the civilisation of an epoch).

The majority of works discussed are taken from European mathematics, aeronautics, physics, and astronomy. Very little biology as such is covered but the borderlines between physiology, bacteriology, and virology on the one hand and medicine on the other are well represented. The illustrations are interesting to any scientist and the principles discussed pertain to all science. Biologists (especially young ones) can profit from the reading of this book.

My only adverse criticism is the price which seems excessive, even in these times, for so small a volume.

5.6. Review by: R F J Withers.
The British Journal for the Philosophy of Science 12 (47) (1961), 258-259.

Most British and American writers on the History and Methodology of Science have merely described when the discoveries in their sciences were made or have concentrated upon the logic of discovery. They have been interested in the formal connection between the discoveries they have chronicled, either temporally, or philosophically as within one world system or another. In either case they have regarded the act of discovery itself as outside their province. The logic of deduction and induction is discussed in philosophical terms, and lip service is paid to the psychological aspects of the processes involved.

Dr Taton refreshingly introduces his readers to the psychological problems involved in the act of discovery. His work has been well translated by A J Pomerans. He describes, with copious examples, some of the factors involved in discovery - flashes of thought, the rôle of chance and the rôle of error. He also describes what is involved in an original discovery and how some discoveries have been missed by great scientists working so close to the discovery that future generations have been forced to wonder why it was missed. The examples chosen are from many different fields and form a useful starting point for thought. One is struck by the number of continental references to writers also interested in this aspect of the philosophy of science. It would be of considerable importance to have other works in this field made more easily available for English speaking readers.

How successful is Taton's account? I think its success and its defect lies in its superficiality. Reading his account one can arrive at certain morals for the would-be scientist. One pictures the perseverance of the Curies, and can see that perseverance is, morally, a good thing in a scientist. Or one can see that without a broad theoretical grasp of a scientific situation one is going to be unable to see the relevance of a particular observation. There are many such moral precepts embedded in this book. However, this does not add up to the psychology of genius or the psychology of invention. We need to know much more about the psychology of the people described before we can be convinced of the real steps by which they associated previously unassociated phenomena. We need to know more about the psychology of explanation and the motivations for it in a particular scientist's case before we can more than scratch at the surface of scientific theorising. One gets the impression that Dr Taton believes that the unconscious behaves as though its logic is that of the conscious. We have known since Freud that this is not so, and it is precisely in this field of unconscious associations that flashes of thought have their psychological coherence. Nowhere in the book is there a discussion of the obsessionalism that makes one worker wash his mistakes down the sink and that makes another notice a bactericidal fungus. Fleming is discussed, but not as a person.

Dr Taton is to be congratulated in making a step in the direction of the psychology of discovery. However, one wonders whether his goal can ever be achieved by writing history, for it is impossible to build up a science of the psychology of discovery by reference to untestable historical data. Much more needs to be done in the present before we can assess the correctness of his historical psychological theorising.

5.7. Review by: Anon.
The Military Engineer 50 (333) (1958), 86.

In this British text are discussed the different domains of discovery, the factors involved and the various aspects of discovery.

5.8. Review by: F I G Rawlins.
Bulletin of the Institute of Physics 9 (1958), 38-39.

The value of this book lies not only in its theme, as indicated by the title, but more particularly in the position which the author takes up with regard to the epistemology and psychology of the facts of discovery. The usual tendency in dealing with the subject is nearly always towards over-simplification. Here, each major advance is seen as the outcome of a chain of events, whether it be in astronomy, biology, chemistry, instrumental techniques, mathematics, medicine or physics. The predominant factors may be historical, social, personal, but in essence they are aesthetic. The mind of the born discoverer is in advance of its time in a capacity to apprehend hitherto unknown harmonies and synthetic elements.

To British readers, the balance of power may seem a little tipped in favour of Europe, especially France. This is no bad thing if it illustrates how variable the density of invention has been at various epochs. Genius is certainly not the prerogative of any one race. But the geistesblitz, or flash of intuition, is capricious and seems to strike when people of exceptional ability are off duty. And this train of thought leads the author to utter a word of warning-very wisely-against the dangers inherent in the fashionable team-work of today. The really valuable scientist must be free to break out on his own, however inconvenient it may be for planned researches and useful targets. Effective as organisation is, it may become the foe of creativeness.

The well-known coincidences of discovery are discussed well and with freshness - analytical geometry (Descartes and Fermat), the calculus (Newton and Leibnitz), non-Euclidean geometry (Lobachevsky, Bolyai, Gauss) and elliptic functions (Abel and Jacobi). It would seem that periodically a species of logical pressure develops, such that, concentrated on two or more minds simultaneously, the valves are bound to lift, and discovery follows. Conversely, a "near-miss" is occasionally recorded, as when Helmholtz invented the ophthalmoscope, though Brucke could equally well have done so at the same moment.

It may well be that we are not without witness or excitement in our own day. For example, as recently as 9 November 1957, the discovery of a proof of Fermat's last theorem (1637) was reported from Lahore, after some 320 years of attempts. Only time will show whether this one will achieve universal acceptance. Not long before, there was a false alarm, as indeed sounded in 1845, when a supposed solution was negatived by Dirichlet. And thus, the fascinating story of discovery goes on, so well treated in this book.

It only remains to add that the translation is adequate and idiomatic. If only time would allow, the six-forms of schools, and many undergraduates, would delight in it.

5.9. Review by: Nicholas Rescher.
American Journal of Physics 26 (6) (1958), 413-414.

This slender volume (in size though not in price), enriched with over thirty handsome plates, is a very readable translation from the French, by A J Pomerans, of R Taton's study of what might be called the natural history of scientific discovery. Its plan is that of a series of case studies of discoveries from various natural sciences, selected to illustrate diverse aspects of the way in which discovery comes about in this domain.

In this Preface, Taton writes, "In order to avoid the dangerous territory of interpretations that are too systematic, I have chosen to do no more than give a description of the different realms of scientific discovery, its principal factors and its essential aspects, with examples drawn from the various fields." In this goal of avoiding being overly systematic, Taton succeeds all too well; but unfortunately at the cost of virtually all interpretation whatsoever. This makes it impossible for him to draw any really important or meaningful conclusions from the material he is studying. The summaries in which the chapters result have all of the novelty and theoretic interest of the financial pages of the daily newspaper. Thus the chapter on "Theoretical Sciences" culminates in the observation that "every new theory appears to be constructed somehow as a reaction against the faults and imperfections of a previous one." And the chapter on "The Originality of Discoveries" is climaxed with the point that, "This frequent convergence of the efforts of different research workers by no means reduces the personal worth of the individual [contributions]." The reader in search of fresh thought on the theory of creativity in science will meet with keen disappointment.

The strength of this book is with its presentation of concrete illustrations, and not with theory. But even here, the selection of examples suffers from the absence of the underpinnings of interpretation which alone can make for firm continuity and meaningful order. The coherence and unity among the many examples of scientific discoveries that are discussed is roughly equivalent with that of the elder Disraeli's Curiosities of Literature. The examples are sometimes even virtually pointless.

5.10. Review by: M B Hesse.
History 44 (151) (1959), 193-194.

This is a translation from the French edition of 1955, in which M Taton collected a number of episodes from the history of the sciences to illustrate various psychological and other factors in discovery and the problems of assigning priorities and of missed discoveries. Some of these episodes are well known, others not so well known, and all are presented interestingly and in a simple and non-technical style. The book is illustrated by several plates.

It is not easy, however, to see just what is accomplished by this sort of history of science, for none of the accounts can be developed in sufficient detail for the reader to feel that he thoroughly appreciates the situation in which the discoveries were made. This might be less important in a work for the general reader if Taton were using his examples to illustrate a thesis independently argued, but he explicitly repudiates this intention in his Preface. The book is marred by an inductivist view of scientific discovery such as one hoped had been banished from modern histories by more profound understanding both of the logic of science and of its conceptual inheritance and environment. If therefore my subsequent remarks are critical, it is in the conviction that Baconian presuppositions have bedevilled the history of science for too long, and ought not to go unchallenged.

That M Taton is a convinced (though perhaps unconscious) Baconian is shown by his persistent references to the damaging effect of philosophical preconceptions upon scientific discovery, by his favourable report of Claude Bernard's insistence upon 'complete mental freedom when confronted by generally held theories', and by his pervasive assumption that 'facts' reveal themselves to the most empty mind and the most accurate instruments. Thus, the beginning of modern astronomy is dated from the day on which Galileo set up his telescope in his garden at Padua. But this is to simplify past the point of distortion. It neglects even the fact that the attitude of Galileo himself to his observations was ambivalent: he uses them to argue against Aristotle, but he knows that theories do not arise from, nor do they rest on, grounds of experiment alone. Galileo is sure what will be the motion of falling bodies without performing the experiment; he points out the 'sublime intellect' which led Copernicus resolutely to affirm 'what sensible experience seemed to contradict' in respect of the orbit of Venus, and here Copernicus was right without benefit of the telescope; and he shows how even without a telescope observations could have been designed which would have confirmed Copernicus's assertion about Venus, though it is impossible to believe that these naked-eye observations could ever have been made unless they had been first suggested by Copernicus's theory. Again, Taton describes Ampère's failure to interpret an experiment which actually showed the effects of induced currents, and explains it by Ampère's obsession with different theoretical concerns, but the opposite and more fortunate effect which Faraday's theoretical preconceptions had upon his researches into the same phenomena is not mentioned. One more example of Taton's philosophy of history must suffice: he apologises for Kepler's mysticism, and explains that 'fundamental errors were the basis of a great deal of progress during this time'. But when is it ever otherwise in the development of science? Were not Newton and Dalton in fundamental error, and is it not very likely that we are also? But we may hope to be spared the judgements of future inductivist historians.

On the whole the translation reads well. There is an unfortunate rendering of 'connaître' as 'understand' on p. 158, where Taton clearly means that in order to interpret a phenomenon correctly we must be able to perceive or recognise it, not understand it, for this would imply that before we can interpret a phenomenon we must have a theory about it, and this would be a denial of the view he maintains throughout.

5.11. Review by: Anon.
The Quarterly Review of Biology 33 (3) (1958), 217.

Reviewed in the original French edition, Quarterly Review of Biology 31 (1956), 288. A work of the greatest interest to all concerned with the history of science and the nature of scientific discovery.

5.12. Review by: Marie Boas.
American Scientist 46 (1) (1958), 62A.

This interesting work is designedly a sketch rather than a full discussion. The author, an able and distinguished French historian of science, has been fascinated by the complex causation inherent in the development of science. As an historian devoted especially to the 18th century he wishes to avoid the setting up of systems. Consequently he has, on the whole, chosen to give examples illustrating the complexity of scientific discovery, rather than try to develop a systematic theory of scientific causation. This is refreshing. Taton has been careful to include in this work examples from all the sciences, from mathematics (his special interest) to medicine. He has been particularly interested in those cases where a scientist has been stimulated by considerations outside pure science, technological, artistic, mystic, for example, and there is a suggestive discussion of the rôle of error.

The book is divided into three parts, "The Different Realms of Discovery," "Factors in Discovery" and "Various Aspects of Discovery," the last two being essentially illustrative of the general considerations of Part I. The French edition (published in 1955) was designed for the general reader; this edition (handsomely presented) should make further appeal to the scientist interested in the problems of scientific discovery, past and present, especially to those scientists who are not attracted by the more usual mathematically oriented philosophy of science. The range of presentation is particularly appealing, and though some of the examples are familiar (like the discovery of penicillin) most are better known to the historian than to the scientist. There are numerous references to the French literature, which many readers will wish to pursue. The translation is adequate.

5.13. Review by: John Chapman.
Southwest Review 44 (1) (1959), 86-87.

The problem of Dr Taton's book is to show how the minds of men who have made major scientific discoveries operate. Wherever possible - the possibilities have been more numerous than one would think - he relies on the accounts offered by the discoverers themselves. Specifically and particularly he has drawn on the answers to a questionnaire circulated among mathematicians of France early in this century. But for his most enlightening and interesting information he relies upon Henri Poincaré, whose experience in mathematics seems to summarise and epitomise that of discoverers in most other scientific fields.

Compressing this experience of major discovery into a sequential narrative, one obtains somewhat the following picture. The scientist, after spending a number of years in mastering his discipline as it currently stands, detects inconsistencies that he cannot accept. He then undertakes to find an explanation for these deviations. The result is usually a much more attentive study of the known aspects of the particular problems and a resultant further dissatisfaction. Consciously the scientist then begins to undertake new constructions. These frequently prove fruitless. Then some other question arises - more often the deflecting force is some ordinary incident such as the necessity to produce some lectures or to make a trip or to move one's household. Meantime, while the ordinary business is proceeding, it would seem that extensive activity is taking place subconsciously. Suddenly a complete solution to the problem strikes the conscious mind. After that it is only necessary, when time and occasion allow, to carry out the proper confirmatory experiment or to write the necessary conclusive formulas.

Such an account, which is partly Poincaré's, seems to fit the pattern of many another discovery Dr Taton studies. Not only is this true, but the process of revelation or discovery is so similar to that described in Aesthetics by Feibleman that one is almost forced to the conclusion that the operations of the discovering or inventing mind in any field, scientific or literary, are almost, if not completely, identical.

Dr Taton then goes on to examine the milieu in which discoveries take place, leaving aside the act or thought of the individual discoverer. It should be obvious that a discovery cannot be entirely independent of the intellectual environment in which the discoverer lives. One epoch may see major concern with chemistry, another with physics, a third with mathematics, a fourth with biological sciences. Not only is this true; it is certain also that at this time scientists in several different places are almost sure to possess the same kind and amount of information, so that simultaneous and coincidental discoveries are likely to occur.

Other circumstances intervene to which the name of accident may be applied. Of these the most familiar is the chance that happened to deposit a penicillium mould on the culture of staphylococci with which Alexander Fleming was conducting some experiments. To what extent was the discovery of penicillin accidental? asks Dr Taton. Fleming was already considerably concerned with substances which combat bacterial growth. An early step in the discovery of antibacterial products of moulds had already occurred. Penicillium is a common laboratory contaminant. If the accident had not come to the attention of Fleming, almost certainly it would have been noticed by someone else in a comparatively short time. The perception of meaning of the accident - which is by far more important than the accident itself - was possible only to a prepared and reasoning mind.

And so it goes with a large number of other discoveries. In a sense they are the result of the time: a similar discovery in 1870 could have led and did in fact lead to nothing, since - as one might say - the world wasn't ready. Similarly the pursuit of a certain predetermined experiment may turn up as a by-product another line of investigation much more important and fruitful than the original research. The only deterrent to such a discovery may be a too single-minded devotion to the testing of the original hypothesis. And, under special conditions, a theorem that eventually is shown to be false may stimulate a series of discoveries in themselves fully as valid as if the original assumption had eventually been substantiated.

The general impression Dr Taton leaves, then, is that chance plays a comparatively small part in the discovery of new knowledge; that most of it is the result of hard work, extensive thought, and an open mind. The subconscious activity that proceeds following the period of fertilisation and questioning plays a major role in discovery, but this activity is not to be forced and can be controlled indirectly only to a very slight degree. At the level of conscious activity the most serious obstacle to major discovery is a narrow purpose or a limited interest that permits one to overlook what lies before him. Unfortunately, though Dr Taton does not specifically make this point, much current research is straitjacketed both by a desire for immediate and applicable result and by the very character of the grants for specific projects which have become so much a part of all scientific investigation since 1940. Discovery may further be thwarted by the tendency to set up research teams the major purpose of which may be the rapid accumulation of data or facts pertinent to a single end.

5.14. Review by: R K Merton.
American Sociological Review 24 (2) (1959), 264.

Translated from the French, this "assembly of commentaries" on patterns of experience in the work of scientists should be of prime interest to the sociologist of science.

It would deserve notice if only because Taton evidently belongs to that small circle of historians who move beyond the mere chronicling of science to show how the experiences of scientists form historically repeated patterns. This would be enough to commend this book to sociologists, but there is more to it than this. Taton has an "instinct for the sociological jugular." His sense for the significant can be glimpsed from a short list of the patterns that he singles out: simultaneous discoveries, the fruitful error, Geistesblitz (or sudden illumination) in scientific work, precursors in science and conflicts over priority, missed discoveries, premature discoveries, and widespread resistance among scientists to pathbreaking discoveries. For each of these, he supplies a complement of historical cases in point, knowledgeably and succinctly summarised.

The merit of the book rests in its selection and arrangement of cases; its limitations, in the author's unwillingness to analyse them. Time and again, after Taton has assembled some cases that, once juxtaposed, begin almost to speak for themselves, he draws away from their direct sociological implications. Instead, he retreats to the position of the historian whose job it is to get the facts straight and to the position of the ethical humanist who applauds or deplores rather than analyses the behaviour under observation. For example, when he comes upon the pattern of nearly simultaneous discoveries, as all who examine the history of science must, he defines the historian's task as that of being "just to precursors" just as, in cases of priority-conflicts, he deplores "the lack of coolness or objectivity which led contending scientists into discussions from which their prestige always emerged impaired" or alludes to "these regrettable manifestations of vanity and exaggerated pride of some scientists." In short, he responds with affect to these historical episodes rather than converting them into problems requiring sociological analysis.

This is a deliberately imposed limitation. From the beginning, Taton rejects any effort at interpretation. He wants to "avoid the dangerous territory of interpretations which are too systematic." By this he presumably means those questionable and inflexible schemes of interpretation which would prematurely exclude facts that may turn out to be strategic. But this self-denying ordinance seems excessive. It assumes that the choice is only between two alternatives: either "schematisations which give nothing but a distorted image" of the complex reality, or presentations of unanalysed cases. As a result of this self-imposed restriction, Taton provides little of the kind of interpretation that is systematic and is faithful to the facts without dragooning them into inappropriate categories of analysis.

But after all this is said, this compact, unpretentious, and informed book remains indispensable to the sociologist of science. In view of this, it is a pity that the American publishers burdened this small book with a price that might be prohibitively heavy for some of those who should have it.

5.15. Review by: Ernest Nagel.
The Annals of the American Academy of Political and Social Science 316 (1958), 191-192.

The study of the conditions under which discoveries and inventions in science come to be made has often been attempted. The results are usually interesting for their historical details, but on the whole they are meagre in their general conclusions. The present book, a translation from the French, is no exception to this rule. Monsieur Taton surveys the familiar factors, both personal and social, which may play a role in scientific discovery; he illustrates them by a variety of examples some of them not widely known from the history of natural science; and he concludes with some brief reflections on the advantages as well as dangers of the increasingly collective character of modern scientific research.

The book leans heavily for its account of the psychology of discovery on the writings of Poincaré and Hadamard. It also makes interesting use of autobiographic descriptions by a number of other distinguished scientists concerning the way they made their discoveries. However, there is no attempt at a systematic coverage of this type of literature, and the few "generalisations" which are drawn from the data included are neither novel nor very informative. Attention is given to the occurrence of sudden flashes of insight which precipitate scientific advances, to the function of general ideas as more or less systematic guides to important research, to the role of chance in the discovery of new phenomena, and to the way in which erroneous ideas may sometimes generate important lines of advance. The author observes quite rightly that even when an unusual occurrence comes to the attention of a scientist by chance, the latter must be intellectually prepared to appreciate the significance of the phenomenon. The obviously just point is also made that almost any initial hypothesis may become a fruitful basis for valuable research, and that a mistaken hypothesis is of greater value to an investigator than the absence of all leading ideas.

One of the most interesting sections of the book deals with "missed discoveries" - with a scientist's failure to observe or to interpret appropriately some phenomenon, even though he was equipped to do so, so that in consequence he lost the opportunity to make a fundamental contribution to his science. The main example presented is Ampère's experiment in which all the clues to electromagnetic induction were made available, though Ampère ignored them and left it to Faraday to make the great discovery. Other sections of the book contain brief discussions of the coincidence of independent discoveries, of conflicts over questions of priority in discovery, of the familiar struggle against scientific orthodoxy which new points of view must often undergo, and of the need for a "ripe" intellectual climate if the importance of a discovery is to be recognised. However, the analysis of these fascinating problems nowhere penetrates beneath the surface. There are some thirty-odd plates, some of them quite handsome; but even so, the price of the book is outrageously high.

6.1. From the Introduction.

The popularisation of written calculation methods is a relatively recent phenomenon, as the use of tokens and calculating tables, still widespread among merchants and bankers in the 17th century, did not disappear until the end of the 18th century. Although universally disseminated through elementary education, these methods gradually declined in importance. Indeed, calculating machines, the first of which was designed by the German astronomer Schickard in 1623, have enjoyed increasing popularity since the beginning of the 19th century. Replacing the old aids of calculation, they have adapted to all types of calculation, from elementary arithmetic operations to the most complex mathematical algorithms. Of very diverse types, they have found their place everywhere from the humblest shops to technical laboratories, where their capabilities are expanding daily into new fields. They are an essential component in the rationalisation of the organisation of accounting services and one of the aids to the development of modern science.

From Schickard's calculating clock to the most recent machines, the various realisations of mechanical calculation appear as the fruit of constant cooperation between inventive genius and the most advanced technology. It is true that mechanical calculation constantly raises very delicate problems of design and development. One need only glance inside a modern calculating machine to see this; the number and complexity of its mechanisms show that its speed and automaticity, resulting from a considerable number of successive inventions, could only be achieved through the increasing complexity of its various components.

In addition to delicate and constantly evolving technical problems, the invention, construction, and use of calculating machines pose problems of adaptation and organisation that the growing role played by mechanical calculation in important sectors of modern life makes more acute every day. Not only must the machines be designed for the operations intended for them, but, moreover, these operations themselves must be adapted in their structure and sequence to the nature of these tools.

The future of automatic computing appears to be largely conditioned by the skill with which it can be integrated into an ever-increasing number of areas of human activity and its techniques renewed to adapt them to its new goals.

The audacity with which, since the end of the 19th century, inventors have adopted concepts that have revolutionised the very essence of this computing bodes well for the future in this respect. And it is certain that, even now, many branches of scientific research and business management see their development depend to a large extent on the progress of new mathematical machines.

Our aim in this elementary work is to provide an overview of the diverse achievements of mechanical computing, from the first arithmetic instruments to the machines of the most advanced office machines. By avoiding overly technical details and limiting ourselves to the basic principles of the major types of machines, it seems possible, despite the large number of existing models and the complexity of each one, to give a fairly complete idea of ​​the astonishing achievements of mechanical calculation. This is what we have attempted, drawing inspiration from the elements of classification and systematisation introduced specifically by M d'Ocagne and L Couffignal.

The study of the evolution of calculating machines will be a valuable guide for us, because the principle of a machine is clearer in a still rudimentary prototype than in a perfected version where the essential components are buried amidst various detailed mechanisms.

Therefore, after having glimpsed the origins of mechanical calculation, we will study the first machines proper, first Schickard's simple and perfect one, destroyed as soon as it was built and completely forgotten since; then those (that have survived to this day) due to Pascal and Leibniz. Next, we will come to the various improvements which, from these machines, lead to the "modern calculators" performing the four fundamental operations of arithmetic. The following chapters will study other current machines grouped according to their specialisation: accounting machines, invoicing machines, etc., each major type of machine being considered from the threefold point of view: operational, mechanical, and practical.

Since the first edition of this work, there has been a prodigious flourishing of the field of numerical calculation, a veritable explosion of its applications, a refinement incessant in its techniques. This expansion is illustrated by the appearance of new terms designating certain aspects of the technique of calculation or new sectors of application: automatic calculation, information processing, operations research, even automatic translation ... It is also illustrated by the creation of specialised courses in several of our Universities or in the annexed Institutes, and by the founding of the French Association for Computing and Information Processing (AFCALTI), which brings together the many specialists and researchers oriented in this direction, and publishes the quarterly journal Chiffres, which publishes studies and works concerning these new techniques.

This rapid evolution of the field and methods of numerical computing has led to an extensive update and a fairly thorough revision of our study. Several recent works in this collection deal with electronic computing and its principal applications; within the now more restricted framework of mechanical computing, our main effort aims to describe the evolution of this technique and to "take stock of current knowledge" on what are commonly called desktop calculating machines.

The illustrations in this new, more extensive edition include, in particular, the most representative types on the current market. We thank the companies or their dealers who provided us with these photographs of machines.

6.2. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 17 (2) (1964), 169.

The first edition, under the sole signature of René Taton, was published in 1949.

Since then, an important historical discovery has necessitated a revision of the first chapters of the work. Before the prototype of Pascal's machines in 1642, Wilhelm Schickard (1592-1635), a professor in Tübingen, had built an original machine, the only model ever constructed, in 1623, which was destroyed by fire shortly thereafter. J-P Flad took a particular interest in this machine, of which he was able to reconstruct several examples.

The very significant progress in electronics in recent years and the rapid evolution of the calculating machine market have, moreover, led the authors to revise the final chapters. As other works in the same series have been devoted to electronic machines, they have focused the bulk of their effort on mechanical calculation itself and on desktop calculating machines. The chapter dedicated to the "Analysis of the Most Representative Types," comprising 37 pages and 31 figures, provides an excellent overview of the current state of the subject.

But philosophers and historians of science and technology will also find much to gain from reading this interesting work.

7.1. From the Introduction.

"We would see nothing if we did not have, with our eyes, the means to surprise, question, and shape an indefinite number of configurations of space and colour" (M Merleau-Ponty, Signes). Although every individual sees spontaneously, a very small number of individuals expose reality. Spontaneous perception (here, visual) has only a distant relationship with the sciences and the visual arts, with objects made to be seen, which are subject to perspective.

The methods that humans have developed to "put things into perspective" are as varied as the societies that have used or still use them. There is not just one perspective. The choice of a suitable perspective depends on what the perspective artist wants to represent, given the impossibility of showing everything; a possibility, moreover, where their representation would fall into the chaos of nature.

Perspective is the inevitable pitfall of any act that seeks to reduce events and objects in space to a flat plane. How can one deprive an object of its corporeal quality while still preserving its meaning? And on the other hand, doesn't a sign on a plane, a simple drawn line, spontaneously hollow out the smooth, white surface, creating a space between the background and the form? Don't two values ​​or two colours naturally appear in front and behind? Doesn't the observer, despite himself, project the spatial experience lived into the artificial space offered to him by the painter, the draftsman?

Aren't these plays of shadow that Plato speaks of in the allegory of the cave this "language for the eyes" that every viewer must learn to read if they want to understand their world?

The universal history of the visual arts is the vast field of application of all kinds of perspectives. Each is rationally grounded, although none is a guarantee of quality. But all are intimately linked to humankind's effort to describe, to take stock of, and understand their universe. Since this effort plays out on both the level of rationality and the level of intuition, perspective and its history fall under both the history of the exact and applied sciences and the history of the fine arts. Hence, in this brief study, the collaboration between the artist and the mathematician.

7.2. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 17 (2) (1964), 169-170.

The History of Technology and the History of Science mutually support each other in this remarkable little treatise, the work of a painter-engraver, a mathematician, two professors, and a historian of science, all in two persons. Euclid is credited with an optics that attempts a systematic exposition of perspective, but in which the choice between the various possible options is never made. Anyone who would seek the history of perspective only in written tradition would consequently be reduced to a fragmentary and therefore distorted view of the efforts of Antiquity.

Albert Flocon, without neglecting this source, relies more on the tradition of pictorial art, vases, frescoes, drawings on papyrus, or bas-reliefs. This investigation continues through the centuries and across continents, culminating in the modern theories of Albert Flocon himself and André Barre.

Throughout history, the techniques and concepts of artists have intersected with the theories of mathematicians, sometimes inspiring them, following them, or challenging them. Moreover, they are not the only ones in this arena, and the role of physicists and physiologists is sometimes paramount. The techniques of photography and cinema also have their say.

Geometers, however, even if they were unable to impose their ideal of a one-eyed and paralysed observer, learned a great deal in this instance. After a first and very brilliant school in the 17th century, which was all too short-lived, the 19th century created projective geometry, which provided modern mathematics, algebra, and topology with "models" that would prove very useful.

This is what René Taton strongly emphasises in the last chapter of this remarkable little treatise.

8.1. Review by: Véronique Boudon.
Revue des Études Grecques 107 (511/513) (1994), 735.

Mention must be made here of the reissue of René Taton's now-classic work. His history of ancient and medieval science - first published in 1957 and subsequently revised and updated in 1966 - is now available to readers in the convenient format of the "Quadrige" series. It is regrettable, however, that no effort was made to update the bibliography, which omits all works published after 1966.

9.1. Contents.

Essor de l'arithmétique et triomphe de la numération décimale.
Les Beaux-Arts, nº spécial, Actualité de l'Humanisme, 15 décembre 1954.

La tentative de Stevin pour la décimalisation de la métrologie.
Cahiers de Métrologie, 1 (1983), p. 5-19 ; réimprimé. Acta metrologiae historicae (III Congrès Int. Métr. Hist., Linz, 7-9 octobre 1983), Linz, 1985, p. 39-56.

La Renaissance et le renouveau mathématique du XVIIe siècle.
A Heinekamp (éd.), Leibniz et la Renaissance, Wiesbaden, Steiner, 1983, p. 202-209 (Studia Leibnitiana Supplementa, 23).

Quelques remarques sur la périodisation en histoire des sciences et sur le concept de XVII siècle.
Revue internationale de Philosophie, 29 (1975), p. 406-419.

Le P Marin Mersenne et la communauté scientifique parisienne au XVIIe siècle.
Actes du Colloque: 1588-1988, Quatrième centenaire de la naissance de Marin Mersenne, Le Mans, Université du Maine, 1994, p. 13-25.

Le rôle et l'importance des correspondances scientifiques aux XVII et XVIII siècles.
Revue de Synthèse, 97 (1976), p. 7-22.

Le lyonnais Girard Desargues (1591-1661) et son oeuvre géométrique et technique.
Actes du 112º Congrès national des Sociétés Savantes, Histoire des Sciences et des Techniques, t. 1, Paris, 1988, p. 75-86.

A la redécouverte des oeuvres de Girard Desargues.
J Dhombres, J. Sakarovitch (éds), Desargues en son temps, Paris, Librairie scientifique A Blanchard, 1994, p. 463-475.

L'Essay pour les coniques de Pascal.
Revue d'Histoire des Sciences, 8 (1955), p. 1-18.

Les origines et les débuts de l'Observatoire de Paris.
Tricentenaire de l'Observatoire royal de Greenwich, Vistas in Astronomy, 20 (1976), p. 65-71.

Picard et la 'Mesure de la Terre'.
G Picolet (éd.), Jean Picard et les débuts de l'astronomie de précision au XVII siècle, Paris, Editions du CNRS, 1987, p. 207-226.

Huygens et l'Académie royale des sciences.
Huygens et la France (table ronde du CNRS, Paris, 27-29 mars 1979), Paris, Librairie philosophique J Vrin, 1982, p. 57-67.

L'Initiation de Leibniz à la géométrie (1672-1676).
Leibniz à Paris (1672-1676), Wiesbaden, Steiner, 1978, pp. 103-129 (Studia Leibnitiana Supplementa, 17).

L'Expédition géodésique de Laponie (avril 1736 août 1737).
H Lacombe, P Costabel (éds), La figure de la Terre du XVIII siècle à l'ère spatiale. Actes du Colloque organisé par l'Académie des sciences (Paris, 29-31 janvier 1986), Paris, 1988, p. 115-138.

Madame du Châtelet, traductrice de Newton.
Archives internationales d'Histoire des Sciences, 22 (1969), p. 185-210.

Sur une pièce nouvelle concernant les recherches de Clairaut sur la théorie de la Lune.
Annales de l'Institut et du Musée d'Histoire des Sciences, 7 (1982), pp. 57-70.

Clairaut et le retour de la comète de Halley.
K Figala, E H Berninger (éd.), Arithmos-Arrythmos. Croquis de l'histoire des sciences. Festschrift pour J O Fleckenstein à l'occasion de son 65e anniversaire, Minerva, Munich, 1980, p. 253-274.

Un épisode significatif de l'histoire de l'optique au XVIIIe siècle: la querelle de l'achromatisme.
Culture, science et développement: contribution à une histoire de l'homme. Mélanges en l'honneur de Charles Morazé, Toulouse, Privat, 1979, p. 345-351.

D'Alembert, Euler et l'Académie de Berlin.
Dix-huitième Siècle, 16 (1984), p. 55-69.

Les relations d'Euler avec Lagrange.
Leonhard Euler, 1707-1783. Contributions à sa vie et à son œuvre, Bâle, Birkhäuser, 1983, p. 409-420.

Lagrange et l'Académie royale des sciences (1763-1793).
La mécanique analytique de Lagrange et son héritage. 1, Supplément au n° 124 (1990) des Actes de l'Académie des sciences de Turin, classe de physique, mathématiques et nature, Turin, 1990, p. 1-23.

La mathématisation des techniques graphiques. Les grandes étapes des origines à Dürer, à Desargues et à Monge.
Technologies, Idéologies et Pratiques, 5 (1986), p. 11-35.

A propos de l'oeuvre de Monge en physique.
Revue d'Histoire des Sciences, 3 (1950), p. 174-179.

Monge créateur des coordonnées axiales de la droite, dites de Plücker.
Éléments de mathématiques, Bâle, 1952, pp. 1-5.

Laplace et Sylvestre-François Lacroix.
Revue d'Histoire des Sciences, 6 (1953), p. 350-360.

L'Ecole polytechnique et le renouveau de la géométrie analytique.
L'aventure de la science, Mélanges Alexandre Koyré, t. 1, Paris, Hermann, 1964, p. 552-564 (Histoire de la pensée, XII).

Diversité et originalité de l'oeuvre scientifique de Lazare Carnot.
J-P Charnay (éd.), Lazare Carnot ou le savant-citoyen, Paris, Presses de l'Université Paris-Sorbonne, 1990, p. 455-470.

Repères pour une biographie intellectuelle d'Ampère.
Revue d'Histoire des Sciences, 31 (1978), p. 233-248.

La formation de Sadi Carnot: Ecole polytechnique et Ecole de Metz.
Sadi Carnot et l'essor de la thermodynamique, Paris, Editions du CNRS, 1976, p. 35-51.

Naissance et développement de quelques communautés scientifiques nationales au XIXe siècle.
Revue internationale des Sciences sociales, 22 (1970), p. 105-122.

Evariste Galois et ses biographes. De l'histoire aux légendes.
Sciences et techniques en perspective, 26 (1993), p. 155-172.

Lobachevski et la diffusion des géométries non-cuclidiennes.
S Garma (éd.), La science en Espagne entre 1750 et 1850. 1er Congrès de la Société espagnole d'histoire des sciences, Madrid, 1980, p. 39-46.

Pierre Sergescu (1893-1954): son oeuvre en histoire des sciences et son action pour la renaissance des 'Archives internationales d'Histoire des Sciences'.
Archives internationales d'Histoire des Sciences, 37 (1987), p. 104-119.

Alexandre Koyré et l'essor de l'histoire des sciences en France (1933 à 1964).
Histoire et technologie, (n° spécial : P Redondi (éd.), Science : la Renaissance d'une histoire), 4 (1987), p. 37-53.

Contingence et rationalité des découvertes et inventions.
Impact science et société, VI (1955), p. 220-255.

Les créateurs scientifiques selon Arthur Koestler.
Le Figaro littéraire, 28 juin 1961, p. 2.

Sur les buts et les méthodes de l'histoire des sciences. A propos d'une intervention du Pr H Guerlac.
Revue trimestrielle d'histoire des sciences et des techniques, 6 (Varsovie, 1962), p. 19-27.

Les biographies scientifiques et leur importance pour l'histoire des sciences.
Texte inédit d'une communication pour l'histoire des sciences faite à l'UIA (Anvers, 27 avril 1982).

9.2. Review by: C.B.
Bruniana & Campanelliana 8 (2) (2002), 564.

The volume brings together a selection of essays published by René Taton between 1954 and 1994, and is enriched by the text of the author's speech given in Antwerp in April 1982, Les biographies scientifique et leur importance pour l'histoire des sciences previously unpublished. The journey traced by the writings collected in the volume, beginning with studies on the affirmation of decimal numeration in the first printed mathematical treatises (Essor de l'arithmétique et triomphe de la numération décimale, pp. 7-11) and the importance of the renewal of mathematics during the Renaissance (La Renaissance et le renoveau mathématique du XVII siècle, pp. 25-33), continues with contributions on some of the major protagonists in the development of science between the 17th and 19th centuries. The volume also includes numerous significant contributions dedicated to reflections on the role, methods and status of the history of science, such as the one dedicated to Alexandre Koyré (Alexandre Koyré et l'essor de l'histoire des sciences en France, pp. 467-480) or the one dedicated to Continence et rationalité des découvertes et inventions (pp. 481-506).