The development of Euclidean axiomatics
Abstract
The paper lists several editions of Euclid’s Elements in the Early Modern Age, giving for each of them the axioms and postulates employed to ground elementary mathematics.
Keywords
Thirteenth Century Twelfth Century Book Versus Greek Text Equal CircleIn the following pages I give a list of the systems of principles adopted in the main editions of Euclid’s Elements in the Early Modern Age, as well as some information about the axioms in Antiquity and in the Middle Ages. The aim of such a list is to provide the reader with a quick survey of the development of mathematical axiomatics, stressing the most important turning points in the foundations of geometry.
As is well known, the Elements of Euclid represented for centuries the very model of scientific and deductive reasoning, and their diffusion and influence in Europe were only matched by the Bible and by a few other writings of the Fathers of the Church. They were translated, edited and commented upon hundreds of times, and these editions and commentaries shaped the scientific tools, the methodological standards and the mathematical language of many centuries. Euclid’s theorems were used to build further and more daring mathematical theories, or applied in the physical sciences, while the deductive structure of the proofs was studied by mathematicians, logicians and epistemologists as the ideal of reason itself. In this connection, a special historical role was played by the principles employed as grounds and foundations of the entire construction. Euclid had begun the Elements with a few unproven assumptions which had to be granted in order to establish his grand mathematical system. Since Antiquity, however, Euclid’s system of principles has been repeatedly discussed and challenged: A few gaps in the proofs were found and the missing arguments provided with additional axioms; other Euclidean principles were in turn proven from more elementary statements and thus removed from the list of axioms; still others were worded in simpler ways, or changed to accommodate philosophical qualms or to satisfy architectonic considerations; and a few more were added in order to extend the geometrical results beyond the boundaries fixed by Euclid. These changes in the system of principles grounding elementary mathematics were among the most important outcomes of the foundational studies carried out during the Early Modern Age. From the Middle Ages up to the beginning of the nineteenth century, hundreds of different editions of the Elements provided their own principles for the same corpus of mathematical theorems; while the statements of the Euclidean propositions remained more or less the same, and the proofs themselves were seldom changed, we witness, in this period, an remarkable explosion of new principles aimed at rigorizing, systematizing and improving Euclid’s work. The enormous creativity of these foundational attempts is attested by the invention of some 350 different axioms for elementary mathematics, with some works employing as many as fifty of them to ground the theorems of the Elements (Euclid himself probably only had ten principles). Many of these new axioms have great mathematical significance and introduced into the foundations of mathematics several important ideas that were to be given a full development only in the nineteenth and twentieth centuries. They concern continuity issues in the Euclidean plane, properties of incidence in two or three dimensions, Archimedean and nonArchimedean quantities, mereological relations, congruency and rigid motion, the Parallel Postulate and the Euclidean metric, or the first attempts at an axiomatization of arithmetic and algebra. In this connection, it is very important to stress that the bestknown works on the foundations of mathematics produced in the modern age, such as Dedekind’s, Frege’s and Peano’s works on the principles of numbers, or Pasch’s, Pieri’s and Hilbert’s famous axiomatizations of geometry, by no means represent the sudden discovery of the need for rigorous foundations, but are rather the late and highly perfected outcomes of a long tradition that had been developing almost continuously from the time of Euclid onward. Even if it may be difficult to precisely ascertain a direct influence exerted by these early modern programs on, say, Hilbert’s Grundlagen der Geometrie, there can be no doubt that these important mathematical works of the previous centuries did indirectly affect modern, formal research on axiomatics. I will not pursue here, however, the search for the sources of nineteenthcentury foundational studies, as this would require an altogether different approach. In any case, while the development of the foundations of mathematics in the last two centuries has already been studied in some depth, the history of axiomatics in the Renaissance and the Early Modern Age is still largely neglected, and I hope that the present essay can represent a beginning on, and an invitation to, this rich field of research, as well as a useful tool for further investigations.
I take this opportunity to express my heartfelt gratitude to Fabio Acerbi and Bernard Vitrac for their accurate reading of a draft of this paper that largely contributed to improve the present version. I greatly benefited from further suggestions offered by Niccolò Guicciardini, Jens Høyrup, Alexander Jones, John Mumma and Victor Pambuccian. I thank Sonja Brentjes for her invaluable help with the Arabic sources and for having helped me with the pseudo Tūsī text, and Philip Beeley, who informed me about Wallis’s unpublished notes on Euclid and provided me with a copy of them. My special gratitude goes to the Library of the Max Planck Institute for the History of Science, for having provided me with several difficulttofind editions of the Elements, and to Massimo Mugnai for his gift of a copy of Grandi’s version of Euclid. Any further suggestion will be very welcome and will improve future editions of this essay.
1 Introduction
1.1 The Euclidean tradition
In the present essay I consider the axiomatic systems of about a hundred editions of the Elements, ranging from Byzantine manuscripts down to early nineteenthcentury printed volumes. In drawing up this list, I did not strive for completeness but rather for perspicuity, and I have selected a few important works among the enormous mass of manuscripts and books dedicated to the recension, translation and interpretation of Euclid. More exhaustive inventories of the editions of the Elements may be found in the bibliographical catalogues drawn up by Riccardi (1974) and Steck (1981); these catalogues are, however, themselves quite incomplete (in fact, a few of the books that I consider here are not even quoted in these accounts). I have mainly taken into consideration the early modern editions, although I have mentioned a few medieval manuscripts which are especially important for the history and the reception of the Elements. I also concentrated on Western mathematics, even though a few important developments in the Euclidean tradition first appeared in the medieval Islamic world; these works are only mentioned for the echoes that they have produced in the Latin tradition. Although a complete recension of the Arabic editions of Euclid is still missing, it should be noted, in any case, that it does not seem that they offered a great diversity of axiomatic systems (see for instance Brentjes 1997). Among the early modern editions, I have normally only listed the printed and published works. In this respect, however, I have made a few exceptions, including in my list Maurolico’s, Torricelli’s, Wallis’s, Pascal’s, Roberval’s and Newton’s handwritten work on the principles of geometry, which seemed to me too important to be ignored. Several other similar essays (such as Leibniz’s unpublished papers on the analysis situs) were not suitable to be easily included in the list and in any case did not exert any historical influence on the respective authors’ own contemporaries.
As a general rule, I have tried to include all those editions which proposed new axioms and principles, as well as those editions produced by mathematicians important in their own right (even if these editions did not represent any advancement in the axiomatic field), and the editions that possessed some special historical significance (by reason of their broad diffusion, for instance, or because they were the first editions in a particular modern language). I have only quoted the first edition of each work, unless the editions that followed introduced important changes in the system of principles (as is the case, for instance, of the second edition of Clavius’s commentary, or of Arnauld’s Nouveaux Élémens).
A serious limitation of the field examined here is the fact that I have only considered editions of the Elements themselves, leaving aside all other material on the foundations of geometry. It should, however, be borne in mind that several important changes in the axiomatic systems were proposed in monographic essays dealing with special problems of the Euclidean text, as is the case, for instance, with several Arabic treatises, Wallis’s or Saccheri’s studies on the Parallel Postulate, or Galileo’s new theory of proportions. A few more principles of mathematics were discussed in general mathematical works (such as certain treatises on algebra, or an essay on fluxions) which did not comment at all on Euclid. That development of the axiomatics, then, which is to be found in the editions of the Elements only gives a part of the whole story: and yet, this is (by far) the largest part, as the main discussions on the foundations of geometry in the Early Modern Age did indeed take place in the Euclidean commentaries. Some of these commentaries, moreover, introduced innovations and alterations with respect to the Greek text of the Elements to such an extent as to become almost new treatises on the foundations of geometry. There is, therefore, a certain degrees of arbitrariness in the decision to include certain works in the list of the Euclidean commentaries, even though they departed very far from the original structure of the Elements, while excluding certain others as moving really too far away from the original to be still considered discussions of Euclid’s work itself. This is especially true as regards eighteenthcentury manuals on elementary mathematics, which often offered a complete reworking of the content of the Elements and no longer had the form of a commentary at all. I have reproduced here the systems of axioms contained in a few of these manuals, choosing the most important representatives of the age (such as Simpson’s or Kästner’s books); the reader should, however, be aware that dozens of such volumes were produced in Europe at this time—and in eighteenthcentury Germany in particular.
It may perhaps be useful to provide, at this point, a short synoptic sketch of the general development of the systems of axioms employed in the various editions of the Elements from Antiquity up to the modern era.
1.1.1 Antiquity and the Middle Ages
Euclid’s Elements (in thirteen Books) were probably written in the third century BC, drawing together materials from a more ancient mathematical tradition, and surely underwent several changes and modifications already during the Hellenistic Age. Several centuries later, the mathematician Theon of Alexandria (fourth century AD) prepared an edition of the Elements which has had a great importance in the following tradition of the text. The oldest extant copies of the Elements are, however, Byzantine manuscripts dating back to the beginning of the ninth century; later Greek manuscripts were in circulation among scholars during the Renaissance and in fact provided the Greek text for the first editions of Euclid in the modern world. One further important manuscript was discovered only at the beginning of the nineteenth century by François Peyrard, and it apparently contained a version of the Elements older than any other, and possibly predating Theon’s editing of Euclid. At least, it seems that its author had access to both Theonine and nonTheonine manuscripts (it should be remarked that this statement, endorsed by many scholars, has been challenged by some others; see Knorr 1996). It is still considered the best guess which we possess regarding the original form of Euclid’s work, and has been employed by the Danish philologist Johan Ludvig Heiberg to prepare the only critical edition of the Elements (1883–1916) that we currently have. For a history of the Greek text of the Elements, see for instance Rommevaux et al. (2001).
All these Greek manuscripts (both those pertaining to the “Theonine” tradition, and Peyrard’s text) show many definitions prefacing several books of the Elements, and a set of fourteen of fifteen other principles (depending on the manuscripts) at the beginning of Book One. The latter principles were considered to be the only statements on which the mathematical construction of Euclid was to be based, and are divided into a first list of postulates Open image in new window and a second list of common notions Open image in new window or, as they were later to come to be called, axioms. The rationale behind this division of the principles into ‘postulates’ on the one hand and ‘common notions’ on the other was a subject of debate and controversy already in ancient times, and several philosophers and mathematicians advanced their own opinions on the epistemological status of these two distinct kinds of assumptions. We will not discuss this difficult topic here, but will note that all the editions of the Elements in the Early Modern Age, with no exception, were to preserve this distinction (even if they interpreted it in different ways).
Some principles among the fourteen or fifteen contained in the oldest manuscripts, however, came to be considered spurious already in Late Antiquity, and Proclus (fifth century, the author of a crucially important commentary on the First Book of the Elements) informs us that a number of them were in fact later interpolations. Modern philology inclines to accept Proclus’s opinion and attempts to prove (even if on different grounds) that some of the principles must have been added in the ancient transmission of the text. A largely (but not unanimously) endorsed opinion among scholars is that Euclid originally had only ten principles (five postulates and five common notions), while the other axioms are Hellenistic additions. Some further doubts are usually raised about two common notions (CN4 and CN5) that some scholars would like to regard as interpolations as well; if this latter view holds true, then Euclid only had eight principles. We know, in any case, that several ancient mathematicians discussed the Euclidean system of principles, and began to add, change or remove a number of axioms and postulates; this seems to have been the case, for instance, of Apollonius, Geminus, Ptolemy and Pappus. In particular, Heron of Alexandria (probably first century AD) wrote a commentary on the Elements in which he addressed a few gaps in the Euclidean proofs and tried to fill them introducing new assumptions (which he may, however, have regarded as provable). While Heron’s commentary is now lost, it was still known in the Islamic Middle Ages (see for instance Brentjes 1998, or the introductory essay to Acerbi and Vitrac 2014). A few fragments of it, moreover, circulated in the commentary to the Elements written by Proclus, whose text enjoyed only a very limited circulation in the Middle Ages, but was rediscovered in the Renaissance. Similarly, Simplicius’s commentary to the Elements (sixth century) was extensively read in the Arabicspeaking world, but was later lost. A few of the axioms added in the Middle Ages and the Renaissance, in fact, derive from these Greek commentaries, and from Heron in particular, as the philosophers Proclus and Simplicius did not dare to add their own principles to the “most perfect” work of Euclid.
During the early Middle Ages knowledge of Euclid disappeared from the West (except for a few fragments preserved in a very elementary mathematical work attributed to Boethius), but flourished in the Islamic world. The Elements were translated into Arabic twice in the early ninth century by AlHajjāj, and again a few years later by Isḥāq ibn Ḥunain (a better version than AlHajjāj’s, revised by the great mathematician Thābit ibn Qurra). These editions were translated from Greek manuscripts that were not very dissimilar to those which we still possess, and thus presented the extended system of principles (fourteen or fifteen of them) which were already common in Greek Late Antiquity. In the tenth century, the mathematician AnNayrīzī wrote an important commentary on Euclid, drawing on Simplicius’s and Heron’s (and those by others) new additional assumptions. Even though this was not an edition of the Elements themselves, but rather a commentary thereon, it was clear that Euclid’s system of principles could, and probably should, be enlarged to fill the gaps present in several proofs. AnNayrīzī’s commentary on Euclid survived as regards Books I–VI and X, with a very short fragment on Books VII and VIII and a longer fragment on Book IX.
The Elements of Euclid reappeared in the West in the first half of the twelfth century, when Adelard of Bath translated into Latin an Arabic manuscript containing (a modified version of) AlHajjāj’s translation. Adelard’s edition was followed by several other translations made from similar Arabic sources (and thus with similar principles), and by the different translation of the Isḥāq–Thābit’s version made by Gerardo da Cremona. Gerardo, however, also translated AnNayrīzī’s commentary, and this paved the way for a reworking of the Euclidean axiomatics. Already in the edition of John of Tynemouth a few new principles were added at the beginning of Book VII of the Elements, which were partly drawn from AnNayrīzī. In the thirteenth century, Campano da Novara edited a final version of the Euclid of medieval tradition, collating together several previous translations and even an independent work (the Arithmetica by Jordanus de Nemore) to fashion a highly successful edition of the Elements. It contained sixteen geometrical principles in Book I and fourteen arithmetical principles in Book VII. Campano’s version was widely used in the following centuries, and it was the first edition of the Elements to be printed, in 1482.
1.1.2 The Renaissance
Campano’s edition and its axiomatic system were used several times in the following century. Luca Pacioli produced an improved version of it, and Niccolò Tartaglia translated it into Italian in 1543 (altering a few axioms thereby). Tartaglia’s edition inaugurated the translations of Euclid into modern languages and was followed by Xylander’s translation into German (1562), Forcadel’s into French (1564), Billingsley’s into English (1570), Zamorano’s into Spanish (1576) and Dou’s into Dutch (1606). In the meantime, Greek manuscripts had begun to arrive in Europe, and Bartolomeo Zamberti had translated the Elements into Latin directly from the Greek (1505), thus opening a new era in the philological reception of the text. In 1533 Simon Grynaeus published the editio princeps of the Greek text. The latter editions produced on the basis of the Greek text once again proposed the ancient system of principles which featured no arithmetical axioms nor any of the further corrections made by Campano. Grynaeus’s edition, however, was based on a Greek manuscript that (probably following Proclus’s hint) changed the division between postulates and axioms, and only admitted three postulates (P1, P2, P3, i.e., the clearly constructive ones), listing the others (P4, P5, P6) among the axioms. Grynaeus, in fact, even appended an edition of Proclus’s commentary to Euclid’s text: so that the system of principles contained in the editio princeps of the Elements was in fact a Neoplatonic philosophical construction. In any case, Grynaeus’s edition of the Greek text remained the only one available for a very long time, and most of the “philological” editions of Euclid in Latin or in modern languages which were prepared in the following two centuries simply followed Grynaeus’s altered system of axioms and postulates (e.g., Caiani, Camerarius, Scheybl, Gracilis, Xylander, Dasypodius, Forcadel, Candale, Dou, Henrion, Keill).
A debate on the superiority of Zamberti’s and Grynaeus’s Greek sources over Campano’s Arabic ones then ensued; comparative editions were prepared; and even if, in the end, the Greek tradition prevailed in the hearts of the humanists and Campano’s version was abandoned by the end of the sixteenth century, this latter’s richer system of principles continued, nonetheless, to charm several geometers who prized mathematical perspicuity (and soundness) above philological exactitude.
A clearly recognizable line of French mathematicians dared to modify Euclid’s system of principles to comply with mathematical correctness. Jacques Lefèvre d’Étaples published a new, enlarged system of arithmetical principles (drawing on Campano); Oronce Fine began to add a few new geometrical axioms; and Jacques Pelletier showed no hesitation in criticizing Euclid for his use of rigid motion in geometrical proofs, thus changing the system of principles in such a way to be able to dispose of these kind of demonstrations. A few years later, the mathematician Pedro de Monzón, who had translated Fine into Spanish, collected the fruits of this innovative tradition and published an edition of Euclid that comprised fortythree principles (both geometrical and arithmetical). Among the studies on the foundations of geometry in the sixteenth century, we should also mention the edition of Euclid produced by Herlinus and Dayspodius in 1566, which rewrote all the proofs of the Elements, molding them into syllogistic form; this important advancement in the logical understanding of geometrical reasoning and the deductive structure of the Elements, however, simply accepted Grynaeus’s system of principles, and added nothing to the science of axiomatics in the strictest sense. The culmination of the sixteenthcentury debates on the Elements was attained in the 1570s, when Commandino and Clavius printed their important editions of the Elements.
Federico Commandino published in 1572 his Latin (and Italian) version of Euclid, which was based on different Greek manuscripts and was in many respects more correct than Grynaeus’s editio princeps (even though Commandino produced no edition of the Greek text itself). Even if Commandino clearly had important philological aims, he was also an outstanding mathematician and endorsed several changes in the axiomatic system which had been found necessary by the preceding tradition, adding a few further modifications himself; he added for the first time certain axioms in Book V and in Book X of the Elements. The very remarkable outcome is an edition of Euclid which can certainly be trusted for its philological accuracy but which nonetheless appeared much richer in foundational discussion (with a total of thirtynine axioms and postulates), and many subsequent editions, both in Latin and in other, modern languages, in the course of the seventeenth and eighteenth Centuries, were simply to be based on Commandino’s text.
In 1574 the Jesuit mathematician Christoph Clavius published another crucially important edition of Euclid in Latin. Clavius’s edition was by far the longest and most complete edition of the Elements ever produced, and was accompanied by a very long commentary that collected almost everything that had been published about the Euclidean text in the previous centuries. For this reason it quickly became the reference work for any further discussion on the foundations of geometry in the following two hundred years, and easily the most important edition of Euclid that was ever published. Clavius believed that the original Greek text was important and had to be preserved but that mathematical perspicuity was a higher aim; his compromise went further than Commandino’s in correcting Euclid. Clavius himself envisaged a few new axioms that were useful, in his view, in order to rigorize the proofs of the Elements; given the enormous success of this edition (and the fact that it immediately became the model for all the further Jesuit editions of Euclid), these new Clavian axioms were successively endorsed by almost the whole community of mathematicians. Another important aspect of Clavius’s work on elementary geometry is that, in the 1580s, he came into contact with an Arabic edition of the Elements attributed to Nasīr adDīn atTūsī (thirteenth century), and here he discovered the great difficulties that the Arab mathematicians had faced in attempting to prove the Parallel Postulate. In the 1589 edition of his commentary Clavius added Nasīr adDīn’s alleged proof of the Parallel Postulate, together with a few reflections of his own, and this episode marked the beginning of the rich epoch of the attempts to prove the Parallel Postulate in Europe—an epoch that culminated in the eighteenth century, with the subsequent discovery of nonEuclidean geometry. Clavius’s complete system of principles, which was to remain a milestone in the following developments, contained four postulates and twenty axioms in Book I (elementary plane geometry), one axiom in Book V (the theory of proportions), three postulates and twelve axioms in Book VII (arithmetic), and one postulate and three axioms in Book X (irrational magnitudes).
1.1.3 The seventeenth century
The seventeenth century was the most creative age in the invention and development of new principles for geometry and arithmetic, and in many respects the highest point of research into the foundations of mathematics in the Early Modern Age. Following an age in which the main goal seemed to be that of the restitution of an accurate text of the ancient mathematical work after the corruptions of the Middle Ages, the new century decidedly aimed at the reform and improvement of the Elements beyond what Euclid himself had been able to do. To this effect, several completely new systems of axioms were devised in order to ground elementary mathematics. This innovative line of attack to the ancient text was especially popular in France and Italy.
French mathematicians often endorsed an epistemology of Ramist origins (that was later contaminated by similar stances emerging out of the new Cartesian tradition) which was intent on designating as an “axiom” any clear and evident mathematical proposition, and in any case any proposition the proof of which could be considered more complicated, and thus less evident, than the statement itself. Such an epistemology, which was stated in its most classical form in the Logique de PortRoyal (1662), led to a flourishing of “evident” principles in this century’s editions of Euclid which was without precedent in previous ages. Several mathematicians simply wrote down dozens of axioms and postulates at the beginning of the Elements without even troubling themselves about whether they were actually needed for the proofs or whether they could be deduced one from the other. Into these highly pleonastic systems of principles, which were not inspired by any consideration of economy of means, a few good mathematicians also, indeed, inserted several important axiomatic principles which had been overlooked in the past and which were to play a central role in future axiomatizations. We may mention in this connection Herigone’s edition of the Elements (1634), which contains fortynine axioms and postulates, the greater part of them being extensions of the Euclidean common notions (i.e., principles on mereology and equality), and also the edition by Claude Richard (1645), with fortysix principles mostly dealing with geometry and introducing a few “topological” axioms to guarantee the continuity of the figures and the existence of their intersection points (the socalled Line–Circle or Circle–Circle continuity axioms). Richard was a Jesuit, and, like him, other French Jesuit masters continued Clavius’s work on the foundations of mathematics, with the additional aim of producing pedagogically efficient handbooks of geometry; among their most remarkable outcomes, we may mention Tacquet’s radical reform of the Elements and their principles, as well as the work by Dechales and Fabri. On the other side of the political–religious divide of the day, the Jansenist theologian Antoine Arnauld wrote an important new reworking of elementary mathematics (1667, 1683), which also included many new axioms (fortytwo in total) and was followed by several editions loosely inspired by it (by Lamy, Malézieu, Sauveur, and others). The most important breakthrough was probably achieved, however, by Pascal and Roberval, who wrote two very important essays on the foundations of geometry, both of which clearly show their mathematical genius. Neither of them, however, was published. Roberval’s work survived in its entirety, while we only have the introduction (and the axiomatic basis) of Pascal’s essay on elementary mathematics. The latter text was copied by Leibniz in Paris in 1675, and largely inspired Leibniz’s own work on the foundations of geometry (the analysis situs) which also offers a thoroughgoing discussion on axiomatics, but remained unpublished (and thus historically ineffectual) among Leibniz’s papers in Hannover (on Leibniz’s analysis situs, see De Risi 2007, 2015b).
Italian studies on Euclidean axiomatics were mostly carried on by the school of Galileo and concentrated especially on the reform of the theory of proportions, since Galileo himself had attempted such a construction (in the unpublished Fifth Day of the Discorsi) and the topic was central to his project of the mathematization of mechanics. It is to Torricelli (1647), Borelli (1658), Viviani (1674), Giordano (1680) and Marchetti (1709) that we mainly owe the progress made in this field. As compared to the French tradition, these Italian axiomatizations show less creativity in producing new axioms, but a greater attention to their systematic function and the economy of principles. From a purely logical point of view, some of these books may be considered to be the culmination of the logical reflection of the Elements, and Borelli’s Euclides restitutus, in particular, marks a high point in researches on the foundations of mathematics.
1.1.4 The eighteenth century
In the eighteenth century, the foundational discussion on mathematics shifted for the most part outside the scope of the Elements. Most efforts were now dedicated to the many issues raised by the new Calculus, and reflection on the foundations of mathematics was often aimed rather at justifying the employment of infinite or infinitesimal quantities; these discussions, however, are seldom to be found in an edition of Euclid. Even elementary mathematics is from time to time grounded on different principles, and algebra (for instance) offered the possibility of a dissimilar foundational approach, grounded in symbolic manipulations rather than simple geometrical constructions. The Elements were no longer regarded as the proper place for advancing a scientific discussion, but rather just as the most classical handbook for a basic schooling in mathematics. The outcome of this modified approach was, on the one hand, a growing attention to pedagogical improvements in the teaching of elementary geometry, and thus a proliferation of simplified editions for students, and on the other hand a renovated search for a more reliable and philologically accurate edition of the Elements.
The former attitude was especially widespread in France and Germany, where a large number of Euclidean textbooks were produced, often largely departing from the original text of the Elements. These systematizations of elementary mathematics do not offer much, however, as regards the foundational domain, and their axiomatizations are often oversimplifications of older editions. Whereas the epistemology of the “clearness and evidence” of mathematical principles had resulted in France, in the previous century, in an overproduction of axioms, the same epistemology in the eighteenth century often yielded axiomless editions of the Elements, which merely made mention of the fact that several evident assumptions were implicitly employed in the development of the proofs (this is the case, for instance, of the celebrated book on elementary geometry written by Clairaut in 1741); a few other editions listed a small number of axioms, claiming that these were just examples of an infinite array of principles that cannot but remain implicit (see Deidier, 1739; or Legendre, 1794). In Germany the surviving Scholastic idea that all the axioms should ultimately be proven from the definitions also engendered several axiomless textbooks which enjoyed a wide diffusion (the most important representatives of these being Christian Wolff’s, Andreas Segner’s and Wenceslaus Karsten’s German and Latin treatments of elementary mathematics). Nevertheless, there can be noted a few interesting developments in the area of axiomatization in this period, such as Samuel König’s and Abraham Kästner’s axiomatizations from 1758.
The philological attitude toward the Elements, on the other hand, was especially common in Britain, where David Gregory produced an emended edition of the Greek text (1703) which may be considered the first improvement on this latter since Grynaeus. Robert Simson, likewise, published a highly successful edition of the Elements (1756) which aimed at restoring the Euclidean text to the state in which it existed before the interventions of Theon of Alexandria; this was in fact later to be accomplished by Peyrard through his discovery of an actual preTheonine manuscript (or, at least, a manuscript in which nonTheonine elements are to be found); Simson’s attempt was based merely on some lucky guesses. If these philological editions offer nothing in the way of an actual development of axiomatics, the commentaries by Thomas Simpson (1747) and, to a lesser extent, John Playfair (1795), nonetheless added several important principles, and attempted a new foundation of elementary mathematics. It may be mentioned that the axiomless attitude toward elementary geometry is well attested in Britain as well, with John Leslie’s celebrated Elements of Geometry from 1809, and a famous Geometry without axioms, published by Perronet Thompson as late as 1833.
We end our survey with Peyrard’s edition of Euclid, published in 1814. This book, in which the Greek text was established from several manuscripts, marked the most relevant advancement in textual analysis so far and opened the rich era of nineteenthcentury Euclidean philology which eventually found its culmination in Heiberg’s 1883 edition of Euclid (largely based on the manuscript found by Peyrard). The discussion of the system of principles in the nineteenth century mostly developed in the form of monographic essays or independent treatises (such as those by Pasch, Frege, Peano or Hilbert) which completely abandoned the literary form of commentary on the text of Euclid. In the same years in which Peyrard was searching for Greek manuscripts, and in the following decades, when his recension of the text begun to acquire authority and to be translated into several languages, Poncelet and Staudt were developing new foundations for geometry starting from the projective properties of space, and Lobachevsky and Bolyai were publishing their treatises on hyperbolic geometry. The gradual development of projective geometry and the nonEuclidean systems, as well as the subsequent birth of abstract algebra and set theory, shifted most of the foundational attention to these new fields. It became clear that the Elements of Euclid could no longer stand as the systematic ground of modern mathematics, since the ancient text could not bear the weight of these new axiomatizations. A completely different approach to the foundations of geometry was then in need.
1.2 Axioms and postulates
I cannot attempt to present here a whole history of the epistemology of mathematics; nor to discuss of the meaning of the notions of axiom and postulate from Antiquity down to modern times; nor to give a list of the many and diverse goals that mathematicians may have aimed at in reshaping the system of the Euclidean principles (for a few contemporary suggestions, see for instance Schlimm 2013). It should at least be noted, however, that many mathematicians in the Early Modern Age believed that axioms and postulates were in fact provable from definitions. They were not so much considered to be statements which admitted of no proof as statements which required none (since they were sufficiently evident already in themselves). In fact, however, a few of the most important editions of Euclid did explicitly prove the axioms and postulates; the commentary by Clavius, for instance, gave demonstrations for all mathematical principles except definitions. It then became a point of controversy whether or not it was useful to prove the axioms at all; many French authors during the seventeenth century, for instance, claimed that Clavius’s exercise had plainly been futile. A further disagreement concerned the criterion which was to be applied in distinguishing a principle from a theorem (as both of these latter are provable); the most common (and quite generic) answer given here was that, whereas an axiom or a postulate is immediately proven from the definitions, a theorem is only mediately proven from these latter (and from other theorems, and from other axioms); the notion of “immediacy” evoked here, however, remained quite vague. As the number of principles grew their demonstration was often omitted, and in many cases it is not easy to understand how such principles could have been proven or explained away. It has already been noted that many authors also claimed that the proper number of axioms should be regarded as infinite, and that the explicitly stated principles should be considered as merely examples and instances of the innumerable statements which immediately follow from the definitions themselves (or from the “essences” of the geometrical entities defined). I have found the first trace of this claim in a few medieval manuscripts of the Elements in the tradition of Robert of Chester (even though previous, more subtle traces of this attitude may be found in Ibn alHaytham and the Arabic tradition), from whom the idea passed to Campano, and thus on into the mainstream epistemology of geometry. In any case, it should be clear that there is a sense in which definitions alone were considered the proper principles of mathematics, while axioms and postulates were often regarded as derivative statements.
All this notwithstanding, I have, in the following pages, only listed axioms and postulates, leaving aside the definitions. The omission, however, is not as serious as it may at first appear. It should be remarked that the system of definitions is much more stable than the system of axioms and postulates, and the definitions found in the early modern editions of the Elements are quite close to the list found in the Greek manuscripts. When the moderns add or modify a few definitions, moreover, this seldom has any effect on the further mathematical development of the treatise. Some definitions were only changed in order to dispel certain purely epistemological qualms (for instance, some authors define a circle as the trace left by the endpoint of a rotating segment rather than as a line equidistant to the center, with the aim of giving an explicit construction and thus a “real definition” of the geometrical entity in question); other definitions were added in order better to explain a few passages in Euclid, or so as to be able to add a few theorems. In most cases, the original Euclidean definition is simply placed side by side with the new definition, which is then regarded as equivalent to this latter, without any proof being given for this assumption. In short, then, the system of definitions was usually only modified in respect of foundational preoccupations which did not affect the further mathematical treatment. The axioms and postulates which are derived (or which should be derived) from these definitions are the statements which are properly employed in the proofs and which therefore affect the deductive structure of the Elements.
Nevertheless, it should be noted that a number of new definitions had real consequences for the axiomatics: This is especially true for the definition of a straight line (which troubled several mathematicians over a period of many centuries), for the definition of parallel lines (as several such definitions were conceived in order to prove the Parallel Postulate) and for the whole definitional system of Book V of the Elements on the theory of proportions. It is still impossible, however, to account for them in a synoptic survey like this one, and the reader is simply warned that definitions may play a supplementary role in the system of principles that does not always appear in an entirely evident manner in the list of axioms and postulates. In order, then, to follow the developments consisting in the important attempts to prove the Parallel Postulate or to reform the Eudoxian theory of proportions, further references may be required.
I have also omitted a small number of principles which are to be found in a few encyclopedic works on mathematics which also contain a section on the Elements inasmuch as these principles are irrelevant from a foundational point of view. Some of them are simply logical or metaphysical statements, and have no mathematical meaning. For instance, it is not uncommon in some Scholastic works to find stated, as an axiom, the Principle of Contradiction, or the Aristotelian claim that mathematics is a discipline abstracted from sensible magnitudes, or the claim that it is possible to consider the surface of a thing while abstracting from its body. Some other principles are properly mathematical, but have no foundational meaning. For instance, a few handbooks list, among the arithmetical axioms, certain principles explaining the working of the positional notation of numbers. These kinds of principle may be important for the history of notations and mathematical culture, but their relevance is one that lies far wide of the aim of the present essay. An example of both kinds of principles is to be found in the Arithmetica perfecta by Georg Henisch (1609, not included in the following list of mathematical editions), which contains sixteen axioms and seven postulates for arithmetic, none of which has anything to do with the theory of numbers in the modern sense; the first axiom states that numbers are potentially but not actually infinite, the third that a number cannot perform any action, the seventh that the natural order of numbers is from the left to the right, the ninth that zero means nothing in itself but adds much value to another number if it is placed at the end of it, while the second postulate states that it is possible to distinguish a whole number from its decimal part by placing a comma in the right place. Interesting as they are, I have excluded these kinds of principle from my list. It should always be remembered, in any case, that a mathematical treatise of the Renaissance or of the Early Modern Age (even an edition of Euclid) may include an assortment of topics which look very odd from a contemporary perspective.
In listing the various principles, I have divided them into thirteen different classes, in accordance with their different mathematical meanings. It should be noted that this division was only carried out in order to facilitate the comparison between principles and has no roots in the real historical development: The actual classification and grouping of principles in the various Euclidean editions is given under each entry, and this classification and grouping is normally only concerned with a division in books and between postulates and axioms. My own arrangement is loosely based on Hilbert’s work on the foundations of geometry (which offers a classification of principles that is still useful and fully perspicuous), but I was obliged to incorporate a few changes to Hilbert’s list in order to accommodate the incredible variety of ancient axioms. In particular, it will be noted that Hilbert’s “axioms of order” are absent from our list: This is an important feature of ancient and early modern mathematics which never thematized relations of order as such. These began to be studied in projective geometry and had to wait until a still later period to be incorporated into an axiomatic system. Hilbert himself attributed the introduction of this kind of axioms to Pasch (1882), and this event should be regarded as an important breakthrough in the history of the foundations of geometry; Pasch’s axiom is nonetheless hinted at in a few early modern axiomatizations (see below). To facilitate a comparison with the modern discussion on axiomatics, I include Hilbert’s system of axioms in the Grundlagen der Geometrie (1899, with a few revisions in the following editions) after the other lists of principles. For Hilbert’s own historical sources, see Hallett and Majer (2004), and Volkert (2015) (which also offers a short history of geometrical axiomatics).
A few remarks are in order to explain some of the principles and their classification.
1.2.1 Ancient principles (Groups 1 and 2)
The Euclidean list of five Postulates and five Common Notions follows Heiberg’s edition and finds its main foundation and justification in a number of comments made by ancient scholars, who had in their hands editions of the Elements with lengthier systems of principles but claimed that some of them had been added in the centuries following Euclid. The additional principles are here listed as P6* and CN6*–CN9*. Axiom CN8*, in particular, is attributed by Proclus (In Euclidis 197) to Pappus (along with M1 and M2, see below). Besides the principles which can be found in the Greek manuscripts of the Elements, we also mention a few other axioms and postulates which are explicitly stated in ancient commentaries and which clearly influenced the following editions, as well as a few important variants of these. Among these axioms and postulates, the extra assumptions EX1, EX2 and EX3 are attributed by Proclus (In Euclidis 198) to Pappus as well, and traces of them are evident in several further axioms, as well as in the definitions of a straight line and a plane; they are, however, nowhere formulated as axioms in a system of principles (even though Albert the Great mentioned them as possible additional axioms in his commentary to Euclid, see Tummers 2014). Other ancient assumptions which did find their way into medieval or early modern axiomatizations will be listed under the respectively appropriate category.
A few postulates changed their formulation in different editions of the Elements. In particular, Postulate 2 on the extendibility of a straight line states, in its original form, that the straight line may be produced continuously Open image in new window . Already in the translation of Adelard of Bath (from the Arabic), however, the postulate is altered so as to make it into a statement about the extendibility of a straight line to an arbitrary length (“assignatam lineam rectam quantolibet spacio directe protrahere”), dropping the aspect of continuity and stressing rather the length of the extension. In Gerardo da Cremona we also find a version stating the indefinite extendibility of the line, the original meaning of which may have been the same as that conveyed by Adelard’s translation. Both versions of the axiom were to be much used in the Renaissance and the Early Modern Age. The modified statement was employed by Campano, and it is common in the tradition stemming from him (in Tartaglia, for instance, or in some Jesuit textbooks), while the Greek formulation is to be found in Zamberti, Grynaeus, Commandino, Clavius and other authors more attentive to philological qualms. This notwithstanding, the two traditions mixed very soon, and it is not uncommon to find the statement about indefinite extension also in editions translated from the Greek; early instances of this attitude are the Elements edited by Xylander, Forcadel and Grienberger (who was normally copying his principles from Clavius), but many more may be found in the seventeenth and eighteenth Centuries. I have not signaled these oscillations, as the various spelling of the principle were clearly considered as equivalent, and in a few volumes their equivalence is in fact explicitly stated. It should be remarked that the version of P2 stating the indefinite extendibility of the line was sometimes transformed into a claim regarding the infinite extendibility of it (instances are Herlin & Dasypodius, Richard, Arnauld and others). Even though the latter spelling may have played a role in establishing the modern nonconstructivist theory (i.e., that straight lines are just infinite), the early modern authors that employed the word “infinite” in this connection always added that it was a potential infinite (or a “syncategorematic infinite,” following Richard’s Scholastic wording), and therefore it could be equated with the standard notion of an “indefinite” extension.
The most important geometrical principle added in Antiquity, however, was the one stating that two straight lines do not enclose a space (P6*). This axiom is often stated in this form, even though it is also found from time to time (and more frequently in the Early Modern Age) as the simpler statement that two straight lines have at most one point in common (I17; the first instance of it seems to be in Arnauld, 1663), or a variation of the latter (e.g., I31, I32). While principles P6* and I17 are not equivalent in general (in elliptic geometry I17 obtains and P6* does not), the counterexamples were too abstract to be made specific note of by early modern mathematicians, who regarded the two axioms as interchangeable.
1.2.2 The theory of magnitudes (Groups 3 and 4)
I include under this heading several principles which are obvious extensions of the Euclidean Common Notions, and which deal with mereology, mereological addition and subtraction, equality, and relations of order among magnitudes. These principles were often looked upon as the main axioms of the mathesis universalis, that is to say, of a mathematical theory which aspired to encompass both geometry and arithmetic (or, in other words, a theory of both continuous and discrete quantities). Many of them were later to be applied in the theory of proportions, which has also been conceived by some authors as a form of universal mathematics. A number of them are especially connected with the arithmetical axioms and may in fact be conflated with them; the reason is that numbers themselves were often conceived of as collections of unities (this is the Euclidean definition in Elements VII, def. 2), and therefore the rules of composition and division of numbers were stated as mereological operations (cf. Elements VII, def. 3–4). The axioms which I have listed under the category of the theory of numbers (Group 7) should therefore be integrated with the principles regarding sum, subtraction and multiplication of magnitudes which are also presented here.
Many of the principles listed here concern the transitivity of some relation or operation (taking as a model CN1), and usually aim at establishing an equivalence relation of some kind. Other principles are directly about the substitutivity of equivalent magnitudes in several contexts (see especially A18, A19). Still other principles are aimed at providing an ordering structure for magnitudes (although these should not be confused with the missing geometrical axioms of order), and, to this effect, there were also conceived and stated certain axioms on trichotomy (see A3, A21, A24).
A corruption of the medieval text appears to be the insertion of the Common Notion A1, which seems to be a useless reduplication of CN1. This first appeared in Adelard’s version, which, in turn, lacks axiom CN8*; it is very likely that a trivial translation slip from the Greek into the Arabic, to be found in the AlHajjāj tradition, changed the original CN8* into A1 (such a mistake is probably absent from the Isḥāq ibn Ḥunain translation, and therefore also from Gerardo da Cremona’s Latin version, which was drawn from this Arabic source). The mistake, however, spread in the whole tradition following Adelard (Robert of Chester, John of Tynemouth, Campano da Novara). The spelling of CN1 and A1 is in fact slightly different, but it is hard to figure out a proper difference in meaning: in Adelard’s prose, CN1 reads “Si fuerint alique due res alicui rei equales, unaqueque earum erit equalis alteri,” while A1 reads “Si fuerint due res alicui uni equales, unaqueque earum equalis erit alteri”; Campano changed the formulation, and his CN1 is “Que uni & idem sunt equalia, & sibiinvicem sunt equalia,” while his A1 is “Si fuerint due res uni equales ipse sibiinvicem erunt equales.” Luca Pacioli, in 1509, tried to preserve the received text and explained the difference between the two axioms in terms of their being an abstract and a concrete version of the same principle. In 1543, Niccolò Tartaglia simply eliminated A1 from the list of axioms, reintroducing CN8* (following, in this one instance, Bartolomeo Zamberti’s translation from the Greek, which had not suffered from the Arabic textual corruption). On the other hand, Pierre Herigone’s principle A15, stating that things equal to equal things are also equal to one another, may be regarded as a axiom different from CN1, as it expresses a more complex relation than simple transitivity (i.e., if \(a=b\) and \(c=d\) and \(b=d\), then \(a=c\)). The same holds for Herigone’s M20 and M21 compared with CN8*, and M23 with CN9*.
Axioms M1 and M2, added by Gerardo da Cremona, are taken from AnNayrīzī’s commentary; Proclus (In Euclidis 197) attributes them to Pappus. Principle A4 is almost classical and will be repeated very often in the Euclidean tradition, but it is first stated as an axiom in Bradwardine’s geometry.
It may be remarked that many of the principles under this heading were introduced during the Renaissance (by Lefèvre, Commandino, and Clavius), and that they were in fact among the first newly conceived axioms for the Elements. Herigone also contributed much to this category, in an attempt to completely formalize the relations of equivalence. In many respects these principles may be considered to constitute the prehistory of the attempts later made toward an axiomatization of algebraic relations.
1.2.3 The theory of proportions and the Axiom of Archimedes (Groups 5 and 6)
The theory of proportions, to be found in Book V of the Elements, is probably that part of elementary mathematics which has undergone, in the history of the Euclidean corpus, the greatest amount of modification. It was entirely based on a complex system of definitions, possibly conceived by Eudoxus, which suffered serious textual corruptions during the Middle Ages, and which affected all the medieval Latin translations depending on Adelard (including Campano): see Murdoch (1963) and Molland (1983) for a first survey. On the other hand, an entire new theory was developed in the Arabic world (from a few hints in classical sources), the socalled anthyphairetic theory of ratios, which however (as far as I know) was never axiomatized and reduced to principles (see, for instance, Itard 1961). The original meaning of the theory expounded in Book V of the Elements was recovered during the Renaissance, and the restitution of Euclid on this topic was in fact the aim of many philological editions. Once the text was established, several criticism were raised against it, as its meaning remained for a long time quite obscure. Later on, it was widely accepted and began to be regarded as the highest point of the mathematical construction of the Elements. In fact, the theory of proportion was adopted as the main mathematical tool to express the laws of Nature at the dawn of the scientific revolution. This, in turn, produced several attempts to reform the Eudoxian theory so as to adapt it to the needs of the new science, and in the first half of the seventeenth century many new principles were envisaged in order to fashion theories of ratios which were grounded on very different foundations. The theory gradually waned in the second half of the century, when algebra and analysis took its place as the tools with which Nature might be mathematized.
Given that the theory was originally founded on a system of definitions, many of the further efforts were also aimed at a reformation of the Euclidean definitions, and the list of axioms presented under this heading do not faithfully represent the actual debates on the matter. The first mathematician to substitute axioms for definitions in the theory of proportions was Francesco Maurolico (in 1567), whose work remained unpublished but probably exerted a certain influence on further studies thanks to its manuscript circulation. We know, for instance, that Clavius studied Maurolico’s notes (even though this is not apparent in Clavius’s own work on the theory of proportions), while we find traces of Maurolico’s axiomatization in Giovan Battista Benedetti (1585), who in turn had a strong influence on Galileo and his school. Borelli himself gained access to Maurolico’s manuscripts before writing his Euclides restitutus (1658). It will be noted, in any case, that most of the axioms in the theory of proportions come from editions produced in the second half of the seventeenth century (i.e., after Galileo’s death), when the theory itself had but very little foundational significance as it had been already superseded by more advanced developments in analysis and algebra. These axiomatizations may nonetheless stand as testimony to the fact that the axiomatic approach (here contrasted with the definitional approach) to the foundations of mathematics was, by the end of the seventeenth century, on the point of triumphing. It should also be noted that many among the first propositions of Book V of the Elements were employed as axioms; the general idea was that they were immediate consequences of the difficult, or debatable, Eudoxian definitions and (in this case) could perhaps be substituted for them as more perspicuous principles. Thus, even if we may regard some of the new foundations of the theory of proportions as very remarkable logical accomplishments, it is difficult to find any single axiom of the theory which might be considered as a substantial progress as regards formalization. We should, however, mention the axiom of the existence of the fourth proportional (R1, R17, R25), which was implicitly employed by Euclid in proving Elements V, 18, and engendered a wide debate on the very notion of mathematical existence and existential assumptions (for instance, Clavius claimed that a fourth proportional to three given magnitudes has to exist, since its existence does not entail any contradiction—thus equating mathematical existence and logical consistency). The missing principle had been already noticed in the Middle Ages, and Khayyām, for instance, had attempted to prove it (see Rashed and Vahabzadeh 1999, pp. 350–351).
Strictly connected with the theory of magnitudes and the theory of ratios is the socalled Axiom of Archimedes, and the notion of Archimedean quantities, which was first formulated by Euclid (or by Eudoxus) as a foundational definition in the theory of ratios (Elements V, def. 4), and later also stated, in a different version, as an assumption Open image in new window of Archimedes’s De sphaera et cylindro. This principle was introduced among the axioms of the Elements by Commandino (axiom AA1) and was restated in several other editions. Its immediate application in a geometrical context was the debate on the angle of contact that had classical sources (Proclus, Simplicius) but especially flamed in the Renaissance and the Early Modern Age, when Clavius, Wallis and many others discussed whether a curvilinear angle (which has no finite ratio with a rectilinear one) might still be considered an angle (and an array of related questions). A few decades after Commandino’s axiomatization, in any case, the theory of indivisibles was developed in Italy (by Cavalieri, Torricelli and others), and these nonArchimedean quantities gave rise to a further, wider debate on their admissibility in mathematics; still later, similar concerns were raised about the infinitesimals of the new Calculus. The debate itself, then, on the Archimedean Axiom is mainly to be found outside of the domain of elementary mathematics, and many eighteenthcentury treatises on analysis deal with the topic. Nevertheless, it continued to contribute to enlarging the number of the axioms in the Elements in various forms. Indeed, among the editions of the Elements, the only (very partial) axiomatization of a theory of nonArchimedean quantities is to be found in the edition of the Elements produced by Bernard Lamy (1685), and perhaps in the unpublished papers by Roberval, whose axioms I18–I21 (that I have arranged in Group 9 following their most immediate meaning) might also be read as a foundation of a theory of nonArchimedean magnitudes.
1.2.4 The principles of arithmetic (Group 7)
As is well known, Euclid included no principles at all (except definitions) in the arithmetical books of the Elements. The reasons for his treating arithmetic differently from geometry in this way are hard to envisage precisely. My guess is that the epistemology of Euclid’s age neither required nor allowed the use of subjectspecific axioms in any form. The notion of this age was that a perfect science ought to be grounded in definitions and general logical laws alone (that may include laws on quantity, i.e., “common notions”) without recurring to any further specific principle. Postulates were allowed in the geometrical books to answer a few sophistical objections and were intended to warrant those auxiliary constructions which were needed in the proofs, but since the arithmetical proofs were considered to have no need of any preparatory Open image in new window , and to be susceptible of being carried out in a purely demonstrative way (that is, through a simple Open image in new window ), no postulates were added in Book VII (note, however, that a few propositions of this book require, indeed, that the mathematician is allowed to take or construct certain numbers; from these few instances, the notion of an arithmetical postulate will later arise). In sum, the very possibility of arithmeticspecific axioms was inconceivable to Euclid (or was, at least, considered by him to be simply wrong), and their first appearance became a possibility only in the late Hellenistic age, if only in the form of assumptions that should (and could) be proven through a further analysis of the matter. In any case, it must be stressed once again that the ancient conception of a system of axioms was very different from our own, and that the need for principles in a given domain was not seen, in Antiquity, as a logical necessity (to rule and define such domain), but rather as a shortcoming characteristic of an imperfect science which needed to make assumptions in order to justify its development; from this point of view, it seems clear that no Greek could ever have had the sense that anything was missing from the arithmetical books of the Elements simply by reason of the fact that they contained neither axioms nor postulates.
Be it as it may, however, the conspicuous absence of arithmetical principles in the Elements had paramount consequences for the later history of mathematics and epistemology. Whereas the axiomatization of geometry was, in fact, discussed for centuries and hundreds of new principles were invented and tested for it, the foundations of arithmetic remained an issue which was almost entirely neglected until the great formal constructions of the nineteenth century (by Dedekind, Frege and Peano, for instance), with geometry remaining during all this preceding period the grounding discipline for the whole of mathematics. We may recall that Kant himself, in the Critique of Pure Reason, still attempted (at the very end of the eighteenth century) a philosophical justification of the fact that geometry has axioms, while arithmetic has no proper principles of its own (KrV, A163–165/B204–206). These common epistemological views, which in many respects were just the outcomes of a mathematical tradition (possibly consolidated by the loss of the arithmetical axiomatizations of Late Antiquity), were a further cause of the long delay in the discovery and outline of a transparent structure for the theory of numbers.
It should not be thought, however, that attempts at moving toward an axiomatization of arithmetic were completely absent from the Early Modern Age or in the tradition of Euclid’s Elements. In fact, the first printed volume of the Elements (1482) was based on the medieval edition of Campano da Novara (thirteenth century), who had, in his time, depended on several other Arabic sources. Campano’s crucially important edition prefaced Book VII of the Elements with four postulates and ten common notions bearing on arithmetic, thus constituting a first attempt at an axiomatization of the theory of numbers. When, a few years later, a sounder text of Euclid’s Elements was translated into Latin by Bartolomeo Zamberti—which translation, of course, being based this time on a Greek original, presented no principles in Book VII—the new humanistic environment of the Renaissance was quick to reckon the interpolated arithmetical axioms as examples of the barbarisms of a dark age; the greater part, therefore, of the subsequent editions of Euclid (up to the age of Kant) simply stuck to philology and ignored these principles. This notwithstanding, Campano’s edition, which was so important in the mathematical studies of sixteenth century, proved able to assert itself in foundational research. The important editions of Euclid by Commandino and Clavius followed the Greek text (rather than Campano’s Arabized Latin) but found it useful, for foundational purposes, to interpolate Campano’s arithmetical axioms into Book VII. Even if their choice was not followed by several other important editions (which eschewed any textual addition), a minority tradition regarding the Elements continued to include arithmetical axioms, and these were sometimes discussed, modified, extended or reduced depending upon the particular foundational opinions of the mathematical editor. In this way, a persistent practice of axiomatic discussion on the theory of numbers was established and, although this discussion remained far less developed than that regarding the foundations of geometry, an unbroken line may be observed to run from Campano down to those important contributions of the nineteenth century which, in many respects, grounded contemporary foundational research.
We may well ask, however, what were Campano’s own sources. It is easy enough to establish that Campano, who was more a compiler than a translator, took his fourteen principles of Book VII from two different sources: the Arithmetica of Jordanus de Nemore (a treatise which is not immediately connected with the Euclidean tradition) and the translation of the Elements, from the Arabic, of John of Tynemouth (both thirteenth century).
Jordanus’s life, education and culture (even his nationality) are unknown, and so are his sources (on him, see Høyrup 1988). He was, however, undoubtedly a very good mathematician and may have invented his system solely with a view to providing principles for arithmetic. Two of his principles are already to be found in Proclus (whose work Jordanus probably could not have known, since Proclus’s commentary to Euclid is believed to have been rediscovered in the West only at the end the fifteenth century, and to have not been know in its entirety to the Arabic world); these, however, are just interpolations from the text of Euclid (see below), and Jordanus may have reinterpolated them as Proclus (or someone else) did before him. A few others of Jordanus’s principles have undoubtedly a Greek flavor, but some others are decidedly formulated in medieval terms (one of them, for instance, refers to the medieval notion of denominatio). In any case, it is very unlikely that Jordanus could have had access to any Greek text, and we are obliged to hold that, if he had sources at all, they were probably Arabic. But whatever Jordanus’s sources were (if any), it seems likely that he had modified and refashioned them so as to better fit his own use of them and his own conceptual instruments; we cannot say for certain whether he himself conceived the idea of introducing axioms into the domain of arithmetic, or whether found it in an older text. It should lastly be noted that Jordanus fashioned a few arithmetical definitions as well that Campano added in his Elements.
Tynemouth’s sources, on the other hand, are relatively clear: The arithmetical axioms given by him all come from the Arabic commentary on the Elements (which was not itself an edition of Euclid) authored by the Persian mathematician and astronomer AnNayrīzī (tenth century), who had had, in turn, some access to Heron of Alexandria’s lost commentary on Euclid. We do not know whether Heron had provided a proper set of axioms for arithmetic: From AnNayrīzī’s wording, it seems rather that he simply took note of a few assumptions which seemed necessary in order to complete this or that proof of Euclid’s, and he may even have proven these further assumptions (see Vitrac 1990–2001, vol. 2, pp. 292–296); from Proclus (In Euclidis 196), we know that Heron had in fact reduced the number of the Euclidean common notions to three, and it would be strange if Proclus did not mention in this connection Heron’s addition of several arithmetical axioms, if indeed this had really occurred. It is possible, but unlikely, that an Arabic edition of the Elements had already inserted these assumption into the Euclidean text labeling them as axioms of Book VII. In any case, AnNayrīzī’s commentary had already been translated into Latin by Gerardo da Cremona in the twelfth century, and Tynemouth may have been able to interpolate these Heronian assumptions, as proper arithmetical axioms, into his edition of the Elements—something that had not been done by either AnNayrīzī, or Gerardo. In this way, Tynemouth inaugurated the modern practice of regarding the Euclidean theory of numbers as an axiomatic domain.
The question of a possible Greek axiomatization of arithmetic is, in any case, relevant indeed. No Greek arithmetical work (such as Diophantus’s, for instance) has come down to us which explicitly mentions any principle besides the definitions. Even if, of course, the greater part of the ancient mathematical texts are lost, this may at first appear to be a confirmation that Euclid’s practice in Book VII was followed by the entire Greek tradition. There has come down to us, however, a (highly problematic) contrary instance in Proclus. The Neoplatonic philosopher, in his commentary on Book One of Euclid’s Elements, discussed several opinions regarding the basis of the distinction between axioms and postulates. One of them consisted simply in the contention that axioms (or “common notions”) are principles shared by all mathematical sciences, and thus by both geometry and arithmetic in particular (that is to say, they form the principles of what will be later called the mathesis universalis). Another way of spelling out the difference between axioms and postulates, however, is that axioms express states of affairs (e.g., “all right angles are equal”), while postulates license constructions (“to draw a straight line between two points”). If the latter distinction is accepted, Proclus said, we may still conceive that there are general principles of a mathesis universalis, and these will be, in their turn, distinguished into general axioms and general postulates, and we will also have specific geometrical postulates and axioms, and specific arithmetical postulates and axioms. Proclus (In Euclidis 184) gave as examples of these arithmetical principles such propositions as “Every number is measured by the number one” (axiom), and “A number can be divided into least parts” (postulate). This is possibly the only mention of arithmetical principles that is to be found in the whole Greek mathematical corpus.
It is not easy to understand the actual historical relevance of Proclus’s remark, or whether it should be considered as a piece of evidence for the existence of a welldeveloped (but now lost) system of axioms for arithmetic already in ancient times. On a first reading, this seems not to be the case: Proclus’s argument here is clearly dictated by a concern for symmetry (if geometry has postulates and axioms, then arithmetic must have its own as well) and may just represent the philosophical hobbyhorse of a metaphysician trying to introduce some order into certain disordered mathematical writings which did not fit well with his elegant epistemological design. If this is the case, we should not expect to find that any Greek mathematician ever developed an axiomatic approach to arithmetic. And yet the examples that Proclus gives are grounded in real mathematical needs: His axiom is an assumption (explicitly) taken for granted by Euclid in Elements VIII, 6, while his postulate is assumed in the proof of Elements VII, 31. They thus appear to enjoy the same status as those other principles which, already in ancient times, were extrapolated from Euclid’s demonstrations, where they appeared as unwarranted, and transformed into axioms (for example, the axiom that “two straight lines do not enclose a space”, which Proclus himself considered spurious, seems to have been taken from a similar statement appearing in the proof of Elements I, 4). It would be odd if Proclus, when inventing two principles exemplifying the existence of arithmetical axioms and postulates, had happened to hit upon exactly these propositions; it seems more likely that he was taking them from an existing edition of the Elements, or a commentary thereon, where they had already been inserted as principles of the arithmetical books. It is hard to guess at Proclus’s own source (if any) in this respect: It might have been Heron himself, but several other sources, Hellenistic, Roman, or belonging to Late Antiquity, are also possible. Odd as it may seem that we find no further trace of arithmetical principles in any of the extant mathematical works of Antiquity, we do at least find, for the reasons given, a hint in Proclus that an axiomatization of arithmetic (be it in a systematic or unsystematic form) had been attempted already in ancient Greece. We know that Proclus had planned, and perhaps realized, a commentary on the other books of the Elements, and it is possible that much more information about these arithmetical principles was offered in these projected or lost commentaries, but we cannot say more (we have no extant commentary dating from antiquity on Elements VII, 31 and VIII, 6). But it should be clear, in any case, that in Proclus’s times (and possibly already in the age of Heron) mathematical epistemology had already changed, with respect to Euclid, to such an extent as at least to allow the possibility of an arithmetical axiomatization.
As for those principles themselves which were envisaged in the centuries after Euclid, their relevance is very obvious, and even if they do not attain to (nor aim to attain to) the simplicity and effectiveness of Peano’s axioms, they nonetheless formalize a vast array of those properties involved in basic operations on numbers, often with a very high degree of logical abstraction. It has already been noted that they should be read together with most of the general principles on the mereology of magnitudes (Groups 1–4).
1.2.5 Principles on space and situation (Group 8)
I have placed under this heading a few very general principles which are more philosophical than mathematical but which nonetheless enjoy an important geometrical significance. Many of them come from Francesco Patrizi’s Della nuova geometria (1587) and Pascal’s Introduction à la géometrie (1655). The importance of these two works consists in their claim that geometry should be regarded as the science of space (S1, S7), a contention which represented a rather new epistemological position in the sixteenth and seventeenth Centuries. Since the days of Eudoxus and Aristotle, throughout antiquity and for a long time afterward, the science of geometry had tended to identify its proper and specific object rather as the notion of magnitude, or continuous quantity. As a consequence, spatial notions tended not to be represented in classical mathematical works and are not employed in the Elements. This situation persisted on into the Middle Ages, and indeed even on into the Renaissance and the Early Modern Age; the axioms of Groups 2–6 bear witness in fact, to the preeminence of the notion of magnitude over any other general concept. The new view that it was the proper and essential task of geometry to investigate space and spatial relations is a view which is first to be found in Patrizi’s work (a book on elementary geometry which is otherwise unremarkable from the mathematical point of view); several decades later it resurfaces in Pascal’s unpublished introduction to geometry, which had been, in its turn, inspired by the introduction of a few spatial terms into the mathematical treatises on perspective. Pascal’s work influenced Leibniz’s attempt to establish a new geometry of space, which the latter named analysis situs; finally, this Leibnizian project was among the motivating factors behind a proper development of spatial notions in lateeighteenth and nineteenthcentury geometry. For a survey of this important transformation of geometry, see the collection of essays published in De Risi (2015a); on Patrizi’s new geometry, see in particular (De Risi 2014).
It should be noted that projective geometry and nonEuclidean geometry are simply inconceivable without the notion of a geometrical space, and several principles grounding modern geometry in the nineteenth century cannot but refer to spatial or positional concepts (including the abovementioned axioms of order). The editions of the Elements which we consider below do not show evidence of these later outcomes, which fell outside of the domain of commentaries on Euclid, and were largely devised at periods situated some years beyond the chronological boundary we have set. Nevertheless, it is important to observe how spatial notions were indeed employed in a number of works on the foundations of geometry well before the nineteenth century. In particular, we may note that the notion of situation (Latin: situs), meaning the basic spatial relation obtaining between two objects, is the main tool used by Pascal to build up his own axiomatics (Patrizi also employed it in his definitions of geometrical entities). This notion was derived from the Euclidean treatise on Data, that is, from one of the very few essays belonging to ancient mathematics which (very briefly) discusses a spatial concept (i.e., position, Open image in new window ). The latter work had been translated into Arabic during the Middle Ages, and into Latin (from the Arabic sources) by Gerardo da Cremona; another Latin translation, made directly from the Greek, circulated in the Middle Ages, and was later superseded by Zamberti’s translation; the first important editions, though, came only with Hardy (1625) and Gregory (in his Euclidean edition from 1703).
We may also note that, in this Group of axioms, a number of principles also state the tridimensionality of space, an axiom which had been never explicitly assumed before the late Renaissance.
1.2.6 Principles of incidence and continuity (Groups 9 and 10)
Dimensional considerations are nonetheless central in the axioms of incidence, since these latter deal with the intersections of various geometrical figures. The axioms in this category are among the most interesting of our survey, and have survived in more intact forms into modern axiomatizations (such as Hilbert’s). Their importance lies in their characterizing very basic geometrical entities, such as a straight line and a plane, the geometrical behavior of which had hardly been adequately captured by ancient or early modern definitions; in fact, in the absence of exact definitions of these notions, the axioms here in question formed the true principles employed in the proofs. While the origin of a number of these axioms is to be found in the Greek commentaries, many more were added by Clavius (who achieved important advances in this connection), and by Borelli, who introduced several axioms which were later to be employed by many other editions.
We may note that Clavius’s axioms I5 (“Two straight lines do not have a common segment”) is from time to time also spelled as “Two straight lines which coincide in a segment, will coincide in their entirety (if extended far enough),” which is, however, an equivalent formulation of the former.
It should also be noted that, contrary to Hilbert’s neater axiomatization, the present category has no clear boundaries, and a few principles classed here might well also have been classed among the principles of congruence: see, for instance I10, I24, or the series of axioms by Lamy regarding perpendicular lines; postulate I4, which is a simple extension of P2 to a higher dimension, is not, properly speaking, a principle of incidence either; nevertheless, it may legitimately be placed here since it expresses the indefinite (or infinite) extension of planes (it may also be regarded as a specification of EX1). The lack of sharp boundaries is especially evident with respect to the group formed by the principles of continuity, some of which can be distinguished only with difficulty from those of incidence; my basic criterion of distinction here was to include all principles which dealt with intersections with bounded figures among those of continuity (the only exception being C11).
Rather than assuming a single allencompassing continuity principle (as did, for example, Hilbert) early modern axiomatizations attempted to solve the problem of spatial connection by means of a vast array of properly geometrical principles. Several of them were developed in the course of discussing the proof of Elements I, 1, in which a number of mathematicians believed they could identify a demonstrative gap, since Euclid did not prove that two drawn circumferences actually meet in two points. Borelli is, once again, a good example of this kind of discussion, which fills some pages of his Euclides restitutus; the first mathematician to conceive an axiom to fill this gap, however, had been Richard in 1645. Other principles were conceived for broader purposes, and in fact they may be considered topological axioms in the proper sense. A French line in the evolution of the continuity axioms started with Fine and was further developed by Richard, Pascal and Roberval. Many axioms deal with the intersections between the boundary of a closed figure and a line which has its extremities inside and outside of the latter figure; some of them may be regarded as early modern forms of the socalled Line–Circle continuity principle, which states that an (infinite) line with a point in a circle cuts the circle in two points. This principle is, in fact, equivalent to the Circle–Circle continuity axiom which is tacitly employed in Elements I, 1 in order to establish the assumption that the two circles actually meet (cf. C13). The latter principles are both much weaker than Hilbert’s full continuity axiom, but are still demonstrably strong enough to ground all theorems of elementary geometry, as far as continuity claims are concerned. A perspicuous and historically sensible proof that Hilbert’s axioms of incidence, order and congruence, together with the Parallel Postulate and Line–Circle continuity, are a viable basis for the performance of all the demonstrations of Euclid’s Elements is given in Hartshorne (2000). I have placed among the principles of continuity also Roberval’s axiom C18, which is a fairly good statement of that famous Axiom of Pasch (1882) which was employed by Hilbert to complete his axioms of order. Even if it is stated by Roberval himself to be a proposition about continuity and intersections (and already hinted at by Richard in his C9, a version of the axiom more similar to the socalled “Crossbar Principle”), this may well be the only instance of a deep principle of order making its appearance before the nineteenth century. It should also be noted that Pasch’s Axiom is itself equivalent to the plane separation axiom, stating that an (infinite) straight line cuts a plane into two disjoint parts; the axiom was in fact stated in this form by Kästner: see his I38 (which I have listed among the principles of incidence). Note that Kästner also provides a very neat statement of the Line–Circle continuity axiom C20, thus forming a quite remarkable system of principles in this respect.
1.2.7 Principles of congruence and equality (Groups 11 and 12)
These axioms are concerned with the equality of measure of different figures and thus with the notion of congruence.
The axioms of Group 11 are especially concerned with the rigid motion of figures and provide a (very partial) view on those innumerable debates regarding the use and abuse of movement in geometry which had already begun in the Arabic Middle Ages but which found their official inception in the West with Pelletier’s criticism (1557) of Euclid’s proof of Elements I, 4 by superposition. The group also includes a few postulates which directly allow the reproducibility of a given figure in another place, and may therefore be closely related to the axioms on situation (Group 8), as the latter may also concern the different spatial positions of congruent figures (cf. T5). I have listed principle T15 in this group as well, which states that one may perform constructions keeping the opening of the compass—something that was not allowed by Euclid, but does not increase the deductive power of the Elements, as it amounts to assume Elements I, 2–3 as an axiom (cf. T13, T22, T26). It may be noted in this connection that I have omitted from the list of Euclidean editions a few geometrical works dealing with issues on constructability and geometrical instruments, such as Giovan Battista Benedetti’s Resolutio omnium Euclidis problematum (Venezia, 1553), Georg Mohr’s Euclides danicus (Amsterdam, 1672, in both Danish and Dutch; with the addition of the essay Euclides curiosus, Amsterdam, 1673), or Lorenzo Mascheroni’s La geometria del compasso (Pavia, 1797). All these treatises show how to perform classical Euclidean constructions by means of a compass alone (without straightedge), or with a compass with fixed opening. Despite their clear foundational significance, however, none of them lists any axiom, and their approach to the subject is more practical than axiomatic.
The axioms in Group 12 do not employ constructions or displacements but rather establish abstract criteria for the equality of measure. Clearly, the divide between the two groups is rather artificial, and, looked at from an abstract point of view, they be considered to state very similar propositions; nonetheless, their foundational strategies were clearly different in the Early Modern Age, and I have therefore distinguished between them.
1.2.8 The parallel postulate (Group 13)
The last group of principles lists a few alternatives to the Parallel Postulate which were conceived to replace the infamous Euclidean principle (P5). It should be noted that the list is far from complete, as many attempts to demonstrate the Euclidean Postulate were undertaken by making use of different definitions of parallel lines, the most famous of which is that two straight lines are parallel if they are equidistant (a statement itself equivalent to the Postulate, if one assumes the existence of these equidistant lines). Moreover, several other attempts to prove P5 are contained in monographic essays, such as Wallis’s De postulato quinto (1663), Saccheri’s Euclides vindicatus (1733) or Lambert’s Theorie der Parallellinien (1766). Such works are not included in our list of editions of the Elements, and often offered still more and different statements equivalent to P5. The present list is thus quite insufficient even to sketch a short history of the attempts to prove the Parallel Postulate. A few considerations on the axioms are nonetheless in order. Axiom PP1 may be regarded as a principle stating the existence of equidistant straight lines, and it shows a good understanding (by Fine) of the fact that the definition alone would be insufficient to this effect unless one could prove (or assume) that equidistant straight lines are possible (something that is equivalent to the acceptance of P5). Axiom PP2 is a restatement of Elements I, 30 on the transitivity of parallelism, a property of parallels also depending on P5 and only true in Euclidean geometry. It was first stated by Proclus (In Euclidis 371) and later employed many times in the history of the attempts to prove the Postulate. Axiom PP3 excludes asymptotic parallel lines in hyperbolic geometry (which do not share a common perpendicular), thus establishing Euclidean geometry. Axiom PP4, which originates in the Arabic treatises on this topic, establishes the Euclidean character of a (socalled) Saccheri quadrilateral (these quadrilaterals in hyperbolic or elliptical geometry have the base shorter or longer than the opposite side). Axiom PP5 is taken from Clavius’s proof of the Parallel Postulate, which assumed this statement to show that equidistant straight lines may be constructed, and thus Euclidean geometry be true, arguing that the present proposition about the flow of a point was a selfevident proposition. This argument was employed already in the Middle Ages by Arabicwriting authors (and Ibn alHaytham in particular) and exerted a lasting influence on the attempts to prove P5; explicitly assuming it as an equivalent statement represented a significant step forward toward a full understanding of the subject. Axiom PP6 is a simple restatement of the Parallel Postulate itself, based on the fact that it is possible to prove that, if two straight lines cut by a third line form angles lesser than two rights, the lines will locally approach one another; it is thus possible to state P5 in the form of the proposition that, in the latter situation, the lines will also meet. The confusion between the local approach of the two lines (in a neighborhood of the intersection with the transversal) with their global approach was a serious hindrance to the proper understanding of the behavior of hyperbolic lines. Axiom PP7 is a restatement of PP4 on Saccheri’s quadrilaterals in a different form. Axioms PP8, PP10 and PP11 are further variations on the theme of equidistance. Mercator’s Axiom PP9 had some historical relevance, as it may be regarded as the first instance of the socalled direction theory of parallelism stating that parallel lines share the same direction (here: they do not “incline” toward one another) and therefore the property of parallelism has to be transitive (a theory often employed at the end of the eighteenth century and the beginning of the nineteenth to establish Euclidean geometry). Axiom PP12 is a property which had been implicitly employed by the German mathematician Andreas Segner to prove the Parallel Postulate (as it is false in hyperbolic geometry), and which had been explicitly stated as a principle by his disciple Johann Friedrich Lorenz. It may easily be regarded as an alternative statement of PP2 or PP9 on transitivity. “Playfair’s Axiom” PP13 was also well known many centuries before Playfair’s edition of Euclid, but it had been never stated as such in substitution of P5 in a version of the Elements. Given the immediate perspicuousness of the statement, it enjoyed an enormous success in the following axiomatizations.
I have also included in this category two ancillary principles, PP14 and PP15, which are not equivalent to the Parallel Postulate. The latter is a restatement of Ptolemy’s proof of Elements I, 28 (a converse statement of the Parallel Postulate, which is provable without it) based on symmetry, which Richard found to be so general in its application as to deserve to be included among the axioms. The important axiom PP14 is the socalled Principle of Aristotle (since it was vaguely stated in De Caelo A 5, 271\(^{\mathrm{b}}\)30–32), which Proclus employed in his commentary on the Elements (In Euclidis 371) to prove the Parallel Postulate, and which was later included by Commandino (1572) in the axiomatic system of the Elements with the purpose of replacing the latter postulate by a simpler statement. Proclus’s and Commandino’s proofs were, however, flawed, and in fact it is impossible to prove P5 from PP14 alone. Clavius later (1589) attempted to prove PP14 but gave an insufficient demonstration of it, and only Saccheri (1733) provided a complete and direct proof of the validity of PP14 in absolute geometry. This notwithstanding, Aristotle’s Principle may play an important role in the foundations of geometry, as it may be proven that, if one accepts some very basic statements about geometrical continuity (the Circle–Circle continuity axiom, for instance), the acceptance of Aristotle’s Principle is in fact enough to grant the full Axiom of Archimedes (AA1) for geometrical magnitudes; see Greenberg (1988, 2010), and Pambuccian (1994). Lastly, it may be remarked that the Axiom of Archimedes is also directly involved in the equivalence of the abovementioned principles, which aim to substitute for P5; for instance, PP4 and PP7 on Saccheri’s quadrilaterals are only equivalent to P5 if AA1 (or PP14 and Circle–Circle continuity) are also assumed; the latter fact was, however, only discovered by Dehn at the beginning of the twentieth century and was never referred to in early modern treatments of this topic.
In translating the principles into English, I generally followed the particular formulation which was given to them in their first historical occurrence. This may result in some slight deviation in terminology in related principles introduced by different authors (for instance, someone may say that a magnitude measures another magnitude, while someone else may say, in a related principle, that a magnitude divides another magnitude, or is a part of this latter; the meaning is, however, the same). I have not differentiated between principles which only differ from one another as regards the terminology employed but which agree with one another as regards their mathematical meaning. Thus, for instance, I have sometimes conflated “magnitude” and “quantity” when these two terms were used by different authors to express the same mathematical notion. These terminological differences may have, in some cases, important epistemological consequences and are very useful for understanding the general philosophical outlook of a mathematician, or the geometrical tradition to which he belongs. For these further and deeper studies, therefore, the original works should always be consulted.
The principles are listed in the various editions in the same order in which they occur, separated by a comma. A dash means that a single principle in the work is in fact composed of two different principles in my list (e.g., Adelard of Bath has a First Postulate stating that a straight line can be drawn between two points (P1) and can be further extended (P2); this postulate is thus mentioned as P1–P2). Sometimes, a number of principles are added in later sections of an edition of Euclid (for instance, since the Parallel Postulate is only used to prove Elements I, 29, many editions do not list it at the beginning, but rather in the middle of the First Book, i.e., after Elements I, 28), and these are separated from the others by a semicolon. The symbol (...) after the list of principles means that the author explicitly states that the principles are infinite in number, and that he is only giving a few instances of them. The intended meaning of a few more obscure principles is given in the short commentary to the edition in which they occur for the first time. I have signaled in square brackets those principles which are somehow problematic within the context of the work in which they appear; a fuller explanation is given in the commentary.
2 The principles
2.1 The principles of Euclid
 P1:
To draw a straight line from any point to any point.
 P2:
To produce a straight line continuously in a straight line.
 P3:
To describe a circle with any center and distance.
 P4:
That all right angles are equal to one another.
 P5:
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
 CN1:
Things which are equal to the same thing are also equal to one another.
 CN2:
If equals be added to equals, the wholes are equal.
 CN3:
If equals be subtracted from equals, the remainders are equal.
 CN4:
Things which coincide with one another are equal to one another.
 CN5:
The whole is greater than the part.
2.2 Further principles in the Greek tradition
 P6*:
Two straight lines do not enclose a space.
 CN6*:
If equals be added to unequal things, the wholes are unequal.
 CN7*:
If equals be subtracted from unequal things, the remainders are unequal.
 CN8*:
The doubles of the same thing are equal.
 CN9*:
The halves of the same thing are equal.
 EX1:
Magnitudes are susceptible of the unlimited both by way of addition and by way of successive diminution, but in both cases potentially only.
 EX2:
All parts of a plane, and all parts of a straight line, coincide with one another.
 EX3:
A point divides a line, a line a surface, and a surface a solid.
2.3 Additional common notions and the foundations of the theory of magnitudes
 A1:
If two things are equal to another thing, they are equal to one another. (Adelard of Bath, twelfth century) [cf. CN1]
 A2:
Any part is lesser than its whole. (Jordanus de Nemore, thirteenth century) [cf. CN5]
 A3:
If two equal quantities are compared to a third of the same genus, they will both be equally greater than the third, or equally lesser, or both equal to the third. (Campano da Novara, thirteenth century)
 A4:
The whole is equal to all the parts taken together. (Bradwardine, fourteenth century)
 A5:
Things which are different from the same thing are also different from one another. (Bradwardine, fourteenth century)
 A6:
The container is greater than the contained. (Maurolico, 1567) [cf. CN5]
 A7:
Given any magnitude, it is possible to take a magnitude greater or lesser than it. (Clavius, 1574)
 A8:
First is what precedes everything else in its order. (Patrizi, 1587)
 A9:
Whole is what has parts. (Patrizi, 1587)
 A10:
Every whole is divided into parts. (Patrizi, 1587)
 A11:
Every quantum can be divided, i.e., is divisible. (Patrizi, 1587)
 A12:
Every divisible thing can be divided into lesser things. (Patrizi, 1587)
 A13:
If a thing is greater, or lesser, than another thing, it is also greater, or lesser, than a thing equal to the latter. (Clavius, 1589)
 A14:
If a thing is greater, or lesser, than another thing, something equal to the former is also greater, or lesser, than the latter. (Clavius, 1589)
 A15:
Things which are equal to equal things are also equal to one another. (Herigone, 1634) [cf. CN1]
 A16:
If a magnitude is greater than another magnitude, and the latter is greater than a third magnitude, the first magnitude is also greater than the third. (Herigone, 1634)
 A17:
If a magnitude is lesser than another magnitude, and the latter is lesser than a third magnitude, the first magnitude is also lesser than the third. (Herigone, 1634)
 A18:
The substitution of equals does not change the equality. (Herigone, 1634)
 A19:
The interpretation does not change the equality. (Herigone, 1634)
 A20:
The measure is not greater than the measured thing. (Herigone, 1634)
 A21:
Every magnitude is such as it is said to be, if it cannot be said otherwise. (Herigone, 1634)
 A22:
Parts which are identical with other parts are also identical among themselves. (Herigone, 1634) [cf. CN1]
 A23:
If the same quantity, or equal quantities, are divided into a smaller or a larger number of equal parts, any one of the parts taken in larger number will be lesser than any one of the parts taken in smaller number. (Richard, 1645)
 A24:
A magnitude which is not greater or lesser than another magnitude is equal to it. (Schott, 1661) [cf. the intended meaning of A21].
 A25:
A part of a greater magnitude is equal to a given lesser magnitude. (Roberval, 1675)
 A26:
Every magnitude is equal to itself. (Prestet, 1675) [cf. also A21]
 A27:
A proposition, taken in one of its meanings, cannot be both true and false. (Prestet, 1689)
 A28:
If the legitimate and natural consequence of a proposition is false, the proposition itself is false, and its contradictory proposition is true. (Prestet, 1689)
 A29:
A quantity is equal to another quantity when it is neither greater nor less than that quantity. (Hill, 1726)
 A30:
A quantity is greater than another quantity when it is neither equal nor less than that quantity. (Hill, 1726)
 A31:
A quantity is less than another quantity when it is neither equal nor greater than that quantity. (Hill, 1726)
 A32:
If a magnitude is smaller than another magnitude, it is possible to take in the latter one or more parts equal to the former. (König, 1758) [cf. A25, unpublished at the time]
2.4 Principles on mereological composition and multiples of magnitudes
 M1:
If unequal things are added to equals, the wholes will exceed one another as the added unequal things. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century) [cf. Pappus in Proclus, In Euclidis 197]
 M2:
If equals are added to unequal things, the wholes will exceed one another as the initial unequal things. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century) [cf. Pappus in Proclus, In Euclidis 197]
 M3:
The equimultiples of the same thing, or of equal things, are equal to one another. (Jordanus de Nemore, thirteenth century) [cf. CN8*]
 M4:
Things the equimultiples of which are equal are also equal to one another. (Jordanus de Nemore, thirteenth century) [cf. CN9*]
 M5:
If two equals are the halves of two things, the two things are equal. (Lefèvre d’Étaples, 1496)
 M6:
If unequal things be added to equals, the wholes are unequal. (Lefèvre d’Étaples, 1496)
 M7:
If unequal things be subtracted from equals, the remainders are unequal. (Lefèvre d’Étaples, 1496)
 M8:
If two greater things are added to one another, and two lesser things are added to one another, the sum of the greater things will be greater than the sum of the lesser things. (Lefèvre d’Étaples, 1496)
 M9:
Any magnitude that measures another magnitude also measures any magnitude that is measured by it. (Commandino, 1572) [cf. N3]
 M10:
If a magnitude measures both another magnitude and a magnitude that is subtracted from it, it will also measure the remainder. (Commandino, 1572) [cf. N1]
 M11:
Any magnitude that measures two or more magnitudes also measures the magnitude composed by them. (Commandino, 1572) [cf. N2]
 M12:
If unequal things are subtracted from equals, the differences will exceed one another as the subtracted unequal things. (Clavius, 1574)
 M13:
If equals are subtracted from unequal things, the differences will exceed one another as the initial unequal things. (Clavius, 1574)
 M14:
If a whole is the double of another whole, and the subtracted part is the double of the corresponding subtracted part, the remainder will also be the double of the other remainder. (Clavius, 1574)
 M15:
If unequal things are added to unequal things, the greater to the greater and the lesser to the lesser, the wholes will be unequal, and in fact the former greater than the latter. (Clavius, 1589)
 M16:
If unequal things are subtracted from unequal things, the lesser from the greater and the greater from the lesser, the remainders will be unequal, and in fact the former greater than the latter. (Clavius, 1589)
 M17:
If two equals are the doubles of two things, the two things are equal. (Clavius, 1589) [cf. M4, M5]
 M18:
If a half is subtracted from a whole, the remainder will be a half; if more than a half is subtracted, the remainder will be less than a half; if one third is subtracted, the remainder will be the two thirds, etc. (Herigone, 1634)
 M19:
The double of the greater is greater than the double of the lesser. (Herigone, 1634)
 M20:
If a thing is the double of a thing equal to another thing, it is also the double of the latter thing. (Herigone, 1634)
 M21:
If a thing equal to another thing is the double of a magnitude, the latter thing will also be the double of that magnitude. (Herigone, 1634)
 M22:
The half of the greater is greater than the half of the lesser. (Herigone, 1634)
 M23:
If a thing is the half of a thing equal to another thing, it is also the half of the latter thing. (Herigone, 1634)
 M24:
If a thing equal to another thing is the half of a magnitude, the latter thing will also be the half of that magnitude. (Herigone, 1634)
 M25:
If the parts of a whole are equal to one another, the whole will be a multiple of each part as many times as there are parts. (Herigone, 1634)
 M26:
If each part of a whole is the double of each part on another whole, the former whole will be the double of the latter. (Herigone, 1634) [cf. M14]
 M27:
Equimultiples of equimultiples are equimultiples to one another. (Herigone, 1634) [Elements V, 3]
 M28:
A multiple of a multiple of a magnitude is also a multiple of that magnitude. (Richard, 1645)
 M29:
If a magnitude measures two other magnitudes, it also measures their sum and their difference. (Borelli, 1658) [cf. N2 and N1]
 M30:
If a magnitude is contained a certain number of times in another magnitude, a magnitude equal to the former will be contained the same number of times in a magnitude equal to the latter. (Borelli, 1658) [cf. N30 and N31]
 M31:
Equimultiples of two magnitudes to which other equimultiples of the same magnitudes are added or subtracted, are still equimultiples of the two magnitudes. (Arnauld, 1667) [Elements V, 5 and 6]
 M32:
If a magnitude is subtracted from itself, the result is nothing. (Prestet, 1675)
 M33:
Many nothings are equal to nothing. (Prestet, 1675)
 M34:
A multiple of a greater magnitude is greater than the same multiple of a lesser. (Simson, 1754)
 M35:
That magnitude whose multiple is greater than the same multiple of another is greater than that other magnitude. (Simson, 1754)
 M36:
To take any multiple of a given magnitude. (König, 1758)
 M37:
The sum of the equimultiples of the parts of a magnitude are equal to the equimultiple of that magnitude. (Simpson, 1760) [cf. Elements V, 1]
2.5 The theory of ratios and proportions
 R1:
As a certain quantity is to any other quantity of the same genus, so will any third quantity be to a certain fourth quantity of the same genus. (Campano da Novara, thirteenth century) [existence of the fourth proportional]
 R2:
Ratios resulting from equal ratios are equal. (Lefèvre d’Étaples, 1496)
 R3:
Equal magnitudes have the same ratio to the same magnitude. (Maurolico, 1567) [cf. Elements V, 7]
 R4:
If two magnitudes have the same ratio to a third magnitude, they are equal to one another. (Maurolico, 1567) [cf. Elements V, 9]
 R5:
A magnitude has the same ratio to two equal magnitudes. (Maurolico, 1567) [cf. Elements V, 7]
 R6:
If a magnitude has the same ratio to two magnitudes, the latter are equal to one another. (Maurolico, 1567) [cf. Elements V, 9]
 R7:
If two magnitudes are equal to two other magnitudes, their ratios are equal. (Maurolico, 1567)
 R8:
A magnitude greater than another magnitude has a greater ratio to a third magnitude; and vice versa. (Maurolico, 1567) [Elements V, 8]
 R9:
Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less. (Maurolico, 1567) [Elements V, 10]
 R10:
Two ratios which are equal to a third ratio are equal to one another. (Maurolico, 1567) [Elements V, 11]
 R11:
If equal ratios are composed with equal ratios, the composed ratios will also be equal. (Maurolico, 1567) [cf. Elements V, 22]
 R12:
If equal ratios are taken away from equal ratios, the remaining ratios will be equal. (Maurolico, 1567)
 R13:
Equimultiples of two magnitudes are to one another as the magnitudes themselves are to one another. (Maurolico, 1567) [Elements V, 15]
 R14:
If unequal ratios are added to equal ratios, the composed ratios will be unequal. (Maurolico, 1567)
 R15:
If unequal ratios are subtracted from equal ratios, the remaining ratios will be unequal. (Maurolico, 1567)
 R16:
Equal ratios are both equal, lesser, or greater than another ratio. (Maurolico, 1567)
 R17:
The ratio that holds between two magnitudes will also hold between a given magnitude and some other magnitude. Moreover, the same ratio will also hold between some magnitude and the given magnitude. (Clavius, 1574) [a different version of R1]
 R18:
The ratio of a magnitude to another magnitude is composed by the ratios of the parts of the first magnitude to the second magnitude. (Benedetti, 1585) [cf. Elements V, 12]
 R19:
If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth. (Benedetti, 1585) [Elements V, 13]
 R20:
The multiple of a magnitude is to this magnitude as the equal multiple of another magnitude is to the latter magnitude. (Benedetti, 1585)
 R21:
The aggregate of all magnitudes in a continued proportion is a multiple of each of the ratios. (Benedetti, 1585)
 R22:
Any ratio, divided in any way, is composed by the parts in which it is divided. (Benedetti, 1585)
 R23:
If two magnitudes have a different ratio to a third, the greater among them will have a greater ratio. (Torricelli, 1647)
 R24:
A ratio is greater than another ratio if the first magnitude of the first ratio is greater than it had to be to have the same ratio to the second as the third to the fourth. (Torricelli, 1647) [from Galileo]
 R25:
Given three magnitudes A, B and C, a fourth one, Z, can be given, that has with C the same ratio as A to B. (Tacquet, 1654) [a different version of R1 and R17]
 R26:
Two ratios in proportion may have (at least) three terms: if one of them is taken twice as an antecedent, or as a consequent, or one time as an antecedent and the other time as a consequent. (Borelli, 1658)
 R27:
If four quantities are such that the first is greater than the second, but the third is not greater than the fourth, the ratio of the first to the second will be greater than the ratio of the third to the fourth. (Borelli, 1658)
 R28:
Ratios which have a common consequent, are to each other as are the antecedents. (Arnauld, 1667)
 R29:
Different multiples of a magnitude are to one another as the equal multiples of any other magnitude. (Arnauld, 1667)
 R30:
Equimultiples of two magnitudes are to one another as different equimultiples of the same magnitudes. (Arnauld, 1667)
 R31:
If two ratios are equal, the inverse ratios are also equal. (Arnauld, 1667)
 R32:
Two continuous magnitudes F and G have the same ratio as the ratio F to H to the ratio G to H, with a common consequent H. (Gottignies, 1669)
 R33:
The double is to the double as the half is to the half, and so for the other multiples and submultiples; and vice versa. (Fabri, 1669) [cf. R13]
 R34:
If a magnitude has some ratio to another magnitude, the latter also has some ratio to the former. (Fabri, 1669)
 R35:
If a magnitude is substituted in a ratio with an equal magnitude, the ratio does not change; and vice versa. (Fabri, 1669) [Elements V, 7]
 R36:
If a magnitude is substituted in a ratio with an equal magnitude, but the other magnitude is changed to a different magnitude, the ratio also changes. (Fabri, 1669)
 R37:
If a third ratio is greater or lesser than one of two equal ratios, it is also greater or lesser than the other equal ratio. (Viviani, 1674)
 R38:
If four magnitudes are in proportion, an equimultiple of the first magnitude is to the second as the same equimultiple of the third magnitude is to the fourth. (Viviani, 1674) [cf. Elements V, 4]
 R39:
If four magnitudes are in proportion, the first magnitude is to an equimultiple of the second as the third magnitude is to the same equimultiple of the fourth. (Viviani, 1674) [cf. Elements V, 4]
 R40:
If a ratio is lesser than another ratio, which is itself lesser than a third, the first will also be much lesser than the third. (Viviani, 1674)
 R41:
A magnitude has a greater ratio to a certain magnitude than to a magnitude greater than it. (Mercator, 1678) [Elements V, 8]
 R42:
If magnitudes be proportional separando, they will also be proportional componendo. (Mercator, 1678) [Elements V, 18]
 R43:
If four magnitudes are in proportion, any equimultiples of the first and the third are both greater, or both equal, or both lesser than any equimultiples of the second and the fourth. (Rohault, 1682) [cf. Elements V, def. 5]
 R44:
If four magnitudes are such that any equimultiples of the first and the third are both greater, or both equal, or both lesser than any equimultiples of the second and the fourth, they are proportional. (Rohault, 1682) [Elements V, def. 5]
 R45:
If four magnitudes are such that for some equimultiples of the first and the third and some equimultiples of the second and the fourth it may happen that the equimultiple of the first is greater than the equimultiple of the second but the equimultiple of the third is equal or lesser than the equimultiple of the fourth, the four magnitudes are not proportional, and the first ratio is greater than the second. (Rohault, 1682) [Elements V, def. 7]
 R46:
All ratios of a magnitude to a magnitude equal to it are equal to one another. (Arnauld, 1683)
 R47:
The ratio of a magnitude to another magnitude is equal to the sum of the ratio of a magnitude smaller than the first to the second and the ratio of their difference to the second. (Arnauld, 1683)
 R48:
The ratio of a magnitude to the part of another magnitude is greater than the ratio of the first magnitude to the whole second magnitude. (Arnauld, 1683)
 R49:
Ratios which have a common antecedent, are to one another as is the inverse ratio of the consequents. (Arnauld, 1683)
 R50:
Given four magnitudes in two ratios, three equalities are possible: the equality of the antecedents, the equality of the consequents, and the equality of the ratios. If any two of these equalities are given, the third is also given. (Arnauld, 1683)
 R51:
Given four ratios in two proportions, the first ratio of the first proportion is to the second ratio of the first proportion as the second ratio of the first proportion is to the second ratio of the second proportion. (Arnauld, 1683)
 R52:
A proportion does not change if the ratios are exchanged among them. (Arnauld, 1683)
 R53:
Lines and surfaces the ratio of which is equal to a ratio between numbers are commensurable; otherwise they are incommensurable. (Lamy, 1685)
 R54:
If a whole is to another whole as a part of the former is to a part of the latter, the remaining part of the former has the same ratio to the remaining part of the latter. (Newton, 1706)
 R55:
Given four magnitudes, if the first is the same equimultiple, or the same part, or the same parts of the second, as the third is to the fourth, the ratio between the first and the second will be equal to the ratio between the third and the fourth. (Marchetti, 1709) [Elements VII, def. 20]
 R56:
If equal ratios are added to unequal ratios, the composed ratios will be unequal. (Marchetti, 1709)
 R57:
If equal ratios are subtracted from unequal ratios, the remaining ratios will be unequal. (Marchetti, 1709)
 R58:
Ratios that are double, triple,&c., a half, a third,&c. of equal ratios (or of the same ratio) are equal. (Marchetti, 1709)
 R59:
Given four magnitudes in proportion, the first and the third magnitudes will both be greater, or will both be lesser, or will both be equal, to the second and the fourth. (Marchetti, 1709)
 R60:
Quantities having the same ratio to another quantity are equal to one another. (Simpson, 1747) [together with R62, a reformulation of R4]
 R61:
Equal quantities have the same ratio with the same quantity. (Simpson, 1747) [together with R63, a reformulation of R35]
 R62:
Quantities to which another quantity has the same ratio are equal to one another. (Simpson, 1747) [together with R60, a reformulation of R4]
 R63:
The same quantity has the same ratio with equal quantities. (Simpson, 1760) [together with R61, a reformulation of R35]
 R64:
If two magnitudes are divided into equal parts, the first is to the second as the number of equal parts of the first is to the number of equal parts of the second. (Simpson, 1760).
2.6 The Axiom of Archimedes and the theory of indivisibles
 AA1:
Any magnitude may be multiplied as many times as to exceed any given homogeneous magnitude. (Commandino, 1572)
 AA2:
Every solid is formed by an infinite number of parallel planes which have an insensible thickness. (Lamy, 1685)
 AA3:
If two solids have the same height and the planes that compose them have the same thickness, the two solids are formed by the same number of planes. (Lamy, 1685)
 AA4:
If a straight line arbitrarily small, and a straight or curved line arbitrarily long (but finite), are given, it is possible to take the first line a number of times such that it will exceed the second line. (Roberval, 1675)
 AA5:
If two or more quantities differ by a quantity less than any assignable quantity, these quantities will be perfectly equal to one another. (Deidier, 1739)
 AA6:
It is possible to take a straight line so small that its square is less than any assigned surface. (Simpson, 1760)
2.7 Principles of arithmetic and number theory
 N1:
A number that divides a whole and the part which is subtracted from it also divides the difference. (John of Tynemouth, twelfth century)
 N2:
Any number that divides two numbers also divides their sum. (John of Tynemouth, twelfth century)
 N3:
Any number that divides something also divides everything that is divided by it. (John of Tynemouth, twelfth century)
 N4:
To take any quantity of numbers equal to any given number. (Jordanus de Nemore, thirteenth century)
 N5:
To take a number however greater than any given number. (Jordanus de Nemore, thirteenth century)
 N6:
The series of numbers can be extended to infinity. (Jordanus de Nemore, thirteenth century)
 N7:
That part is lesser which has a greater denomination; and that part is greater which has a lesser denomination. (Jordanus de Nemore, thirteenth century)
 N8:
The unity is a part of any number, and denominated by it. (Jordanus de Nemore, thirteenth century)
 N9:
Every number is a multiple of the unity as many times as the unity is part of it. (Jordanus de Nemore, thirteenth century)
 N10:
Every number multiplied by the unity results in itself. The unity multiplied by any number results in this number. (Jordanus de Nemore, thirteenth century)
 N11:
The difference of the extremes is composed of the differences of the same extremes to the mean term. (Jordanus de Nemore, thirteenth century)
 N12:
To take any quantity of numbers equal to, or multiples of, any given number. (Campano da Novara, thirteenth century) [cf. N4]
 N13:
No number can be diminished to infinity. (Campano da Novara, thirteenth century)
 N14:
If a number is greater than another, and their difference is added to the lesser, or subtracted from the greater, the resulting numbers will be equal. (Lefèvre d’Étaples, 1496)
 N15:
Two numbers that have the same ratio to the same number are equal. (Lefèvre d’Étaples, 1496) [cf. Elements V, 9]
 N16:
A number may be subtracted from another number as many times as the former divides the latter. (Lefèvre d’Étaples, 1496)
 N17:
The greater number does not divide the lesser. (Lefèvre d’Étaples, 1496)
 N18:
If a given number is multiplied by another number and then is divided by it, the result is again the given number. If a given number is divided by another number and then is multiplied by it, the result is again the given number. (Lefèvre d’Étaples, 1496)
 N19:
If a number measures a part [i.e., a submultiple], it also measures the whole. (Candale, 1566)
 N20:
The sum of a given number with a number that is greater than another one is greater than its sum with the lesser number. (Pedro de Monzón, 1569)
 N21:
Numbers which are made by the same quantity of unities are equal. (Pedro de Monzón, 1569)
 N22:
Two number are equal if their parts with equal denomination are in equal quantity. (Pedro de Monzón, 1569)
 N23:
If two number are multiplied, the product is to the first as the second is to the unity. (Pedro de Monzón, 1569)
 N24:
If a number divides another number, the divided number is to the divisor as the quotient is to the unity. (Pedro de Monzón, 1569)
 N25:
If two numbers are added together which are greater than two other numbers, the sum of the greater ones is greater than the sum of the lesser ones. (Pedro de Monzón, 1569)
 N26:
The ratio of a number with another number is the same as the ratio of the parts of the two numbers with the same denomination. (Pedro de Monzón, 1569)
 N27:
If a unity is added to an odd number, the result is an even number. (Pedro de Monzón, 1569)
 N28:
If a unity is added to an even number, the result is an odd number. (Pedro de Monzón, 1569)
 N29:
If several ratios among numbers are all equal to the same ratio, they will also be equal to one another. (Billingsley, 1570) [cf. Elements V, 11; and R10]
 N30:
Those numbers which are the same part (or the same parts) of the same number (or of equal numbers) are equal to one another. (Commandino, 1572)
 N31:
Those numbers, which have the same (or equal) number of the same part (or parts) are equal to one another. (Commandino, 1572)
 N32:
The unity divides any number. (Commandino, 1572)
 N33:
Every number divides itself. (Commandino, 1572)
 N34:
If two numbers are multiplied, the one measures the product by the other and the latter by the former. (Commandino, 1572)
 N35:
If a number measures another number by a third one, the third also measures the latter by the former. (Commandino, 1572)
 N36:
If a number measures another number by a third one, the product of the first by the third, or of the third by the first, results in the second. (Commandino, 1572) [cf. N18]
 N37:
It is possible to make additions, subtractions, multiplications, divisions, extractions of roots, and squares and cubes of numbers. (Herigone, 1634)
 N38:
Every property of a number is also a property of a number equal to it. (Herigone, 1634)
 N39:
It is the same to multiply a whole with a whole, or a whole with each of the parts, or each of the parts with a whole. (Arnauld, 1667) [cf. Elements V, 1]
 N40:
No magnitude has a divisor greater than itself. (Prestet, 1675)
 N41:
The greatest common divisor of two parts composing a whole is also the greatest common divisor of one part and the whole. (Prestet, 1675)
 N42:
The greatest common divisor of a whole and one of its two parts is also the greatest common divisor of the whole and the other part. (Prestet, 1675)
 N43:
If a magnitude with the sign + is joined to the same magnitude with the sign –, the result is equal to nothing. (Lamy, 1685)
 N44:
Two numbers are always commensurable the one another. (Lamy, 1685)
 N45:
If a number is subtracted from another number, the result is a number. (Lamy, 1685)
 N46:
It is the same thing to multiply 12 times 8, or 8 times 12. (Malézieu, 1705) [cf. Elements VII, 16]
 N47:
It is the same thing to multiply 12 times 8, or 12 times many parts which, taken together, are equal to 8. (Malézieu, 1705) [cf. N39]
 N48:
If equal quantities are multiplied by equal quantities, their products will be equal. (Malézieu, 1705)
 N49:
If equal quantities are divided by equal quantities, their quotients will be equal. (Malézieu, 1705)
 N50:
Given a number, to add a unity to it. (Newton, 1706)
 N51:
Given a number, to subtract a unity from it. (Newton, 1706)
 N52:
If the roots of any two quantities are equal, their squares are also equal. (Hill, 1726)
 N53:
If the squares of any two quantities are equal, their roots are also equal. (Hill, 1726)
 N54:
If unequal quantities are multiplied by equal quantities, their products will be unequal. (Deidier, 1739)
 N55:
If unequal quantities are divided by equal quantities, their quotients will be unequal. (Deidier, 1739)
2.8 General principles of space, situation, and givenness
 S1:
Mathematical sciences are not abstracted from natural things; nor are they in the imagination, nor are they in the intellect; but rather, space is their subject. (Patrizi, 1587)
 S2:
Geometry takes into consideration the point, the line, the surface, and the body, in this natural order. (Patrizi, 1587)
 S3:
Space is extension, and extension is space. (Patrizi, 1587)
 S4:
Every space is a minimum, a maximum, or in the middle of these. (Patrizi, 1587)
 S5:
Every space is long; or long and wide; or long and wide and deep. (Patrizi, 1587)
 S6:
Given a thing, to take in it a point or a straight line. (Ricci, 1651)
 S7:
The object of pure geometry is space, which is considered as extended in three dimensions. (Pascal, 1655) [cf. S1 and S5]
 S8:
Space is infinite in each dimension. (Pascal, 1655)
 S9:
Space is immovable as a whole and its parts are also immovable. (Pascal, 1655)
 S10:
Points may only differ in situation. (Pascal, 1655)
 S11:
Lines may differ in situation, magnitude and direction. (Pascal, 1655)
 S12:
Surfaces may differ in situation, length, breadth, content, and direction. (Pascal, 1655)
 S13:
Equal straight lines only differ in situation. (Pascal, 1655)
 S14:
Equal circles only differ in situation. (Pascal, 1655)
 S15:
Equal arcs of the same circle only differ in situation. (Pascal, 1655)
 S16:
Cords of equal arcs of equal circles only differ in situation. (Pascal, 1655)
 S17:
If an end of a straight line is given in position, and the straight line is given in magnitude, the other end of the straight line will fall on the circumference of the circle with that given center and radius. (Arnauld, 1667)
 S18:
To assume a figure, whose properties are looked for in a theorem, even though it is not constructed. (Fabri, 1669)
 S19:
There exists a solid body. (Roberval, 1675)
 S20:
A straight line is given in position and magnitude when its endpoints are given. (Kästner, 1758) [Euclid, Data 26; cf. also I30].
2.9 Principles of intersection, incidence and connection
 I1:
A surface cuts a surface in a line. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century) [cf. EX3]
 I2:
If two surfaces which cut one another are planes, they cut one another in a straight line. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century)
 I3:
A line cuts a line in a point. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century) [cf. EX3]
 I4:
To indefinitely extend any plane. (AnNayrīzī, transl. by Gerardo da Cremona, twelfth century)
 I5:
Two straight lines do not have a common segment. (Clavius, 1574) [from Proclus, In Euclidis 214; cf. I3]
 I6:
Two straight lines converging into a point will necessarily intersect one another at that point when produced. (Clavius, 1589)
 I7:
If a point lies on two straight lines, it will be at their intersection or meeting. (Herigone, 1634)
 I8:
If two points are on the same plane, the straight line joining them will be on the same plane. (Herigone, 1634)
 I9:
Two concurrent and intersecting straight lines have no common part. (Grienberger, 1636) [another version of I6]
 I10:
From any point of a circular arc that is not one of its two ends, two straight lines may be drawn to the ends of the arc, forming a triangle with the chord between the ends. (Richard, 1645)
 I11:
Given two points, to draw a circle having one of the points as its center and passing through the other point. (Ricci, 1651)
 I12:
Through any line, to draw a plane. (Borelli, 1658)
 I13:
Two planes do not have a common [plane]segment. (Borelli, 1658)
 I14:
Two planes do not enclose a solid figure. (Borelli, 1658)
 I15:
The common section of two planes is a line, and if a line lies on two planes it is their common section. (Borelli, 1658) [cf. I2]
 I16:
Two intersecting straight lines lie on the same plane. (Borelli, 1658) [Elements XI, 2]
 I17:
Two intersecting straight lines may have at most one point in common. (Arnauld, 1667) [cf. P6*]
 I18:
A bounded solid body can be divided into two parts in any direction. Taking together all the divisions, it will be divided into an innumerable number of parts. (Roberval, 1675)
 I19:
The division of a body may be physical, thus disconnecting the parts of the body, or mathematical, thus preserving its continuity. (Roberval, 1675)
 I20:
Besides the surfaces that bound a solid body, innumerable other surfaces may be conceived of which bound its parts. (Roberval, 1675)
 I21:
A bounded surface can be divided into an innumerable number of lines, and a bounded line into an innumerable number of points. (Roberval, 1675)
 I22:
The existence of a bounded solid body implies the existence of surfaces, the existence of bounded surfaces that of lines, the existence of bounded lines that of points. (Roberval, 1675)
 I23:
However big a bounded solid body is posited, a bigger one may be conceived of that contains the former as a proper part. The included body may be conceived of as separated from the bigger one by some space. (Roberval, 1675)
 I24:
A solid body revolved around two fixed points has an entire line fixed. (Roberval, 1675) [cf. T2]
 I25:
If a line is perpendicular to another line, its extension is also perpendicular to it. (Lamy, 1685)
 I26:
If a line is perpendicular to another line, the latter is also perpendicular to the former. (Lamy, 1685)
 I27:
If a straight line perpendicular to a plane moves uniformly in a straight line on the plane, its flow will produce a plane perpendicular to the first plane. (Lamy, 1685)
 I28:
If a straight line is perpendicular to a plane, a plane passing through the straight line is also perpendicular to the first plane. (Lamy, 1685)
 I29:
The extremities of a line are points. (Lamy, 1685) [Elements I, def. 3]
 I30:
A straight line is determined by two points only. (Lamy, 1685)
 I31:
Only one straight line may be drawn between two points. (Lamy, 1685) [cf. I17]
 I32:
Two straight lines that have two points in common coincide. (Lamy, 1685) [cf. I17]
 I33:
A straight line cannot be partly on one plane and partly on another plane. (Lamy, 1685) [Elements XI, 1]
 I34:
Every triangle lies on a plane. (Lamy, 1685) [Elements XI, 2]
 I35:
If a part of a plane is congruent to a part of another plane, the two planes are congruent to one another. (Marchetti, 1709) [cf. I13]
 I36:
A straight line [i.e., a segment] may only be bisected in one point. (Rondelli, 1719)
 I37:
To draw a straight line perpendicular to any given straight line in any given point. (Simpson, 1747) [Elements I, 11 and 12].
 I38:
An infinite straight line divides the infinite plane in two parts. (Kästner, 1758).
 I39:
It is possible to extend a plane through two intersecting lines. (Simpson, 1760)
2.10 Principles of continuity
 C1:
If a straight or curved line is drawn from a point which is within a figure to another point in the same plane which is outside the figure, it will intersect the sides or boundary of the figure. (Fine, 1532)
 C2:
If a straight line drawn from any angle of a rectilinear figure meets the opposite side or angle of the figure, it also intersects this latter side or angle. (Fine, 1532)
 C3:
If a circumference with a given center and a radius of a given length is produced, and the radius is extended beyond the center to infinity, it will intersect the circumference. (Richard, 1645)
 C4:
Only two points on a circumference are on one straight line. That is: if a straight line intersects a circle entering in its area and leaving it, it will intersect the circle at two points only, the one when it enters it and the other when it leaves it. (Richard, 1645)
 C5:
If a circumference is described which has an end of a given segment as its center and passes through the other end of the segment, it will also pass through the ends of all segments equal to the given one that are produced from the given center. (Richard, 1645)
 C6:
If a circumference is described which has its center on the side or boundary of a given figure, and whose radius falls inside the figure (not on its boundary), it will intersect the figure at least once. (Richard, 1645)
 C7:
If a circumference is described which has its center outside a given figure, and whose radius falls inside the figure (not on its boundary), it will intersect the figure at least once. (Richard, 1645)
 C8:
If a straight line drawn inside a figure is produced to infinity (if needed) in both directions, it will intersect the boundary of the figure. (Richard, 1645)
 C9:
If a straight line divides a rectilinear angle and is produced to infinity, it will intersect a given straight line which is applied [with its ends] to the straight lines forming the angle. And a straight line drawn from a rectilinear angle to a point inside its base will divide the angle. And a straight line drawn from a point of the base of a triangle to the opposite angle will divide the angle. (Richard, 1645)
 C10:
If a point is taken inside a space which is everywhere bounded by lines, and an infinite straight line passes through the point, it will cut the boundary of the space at at least two points. (Pascal, 1655)
 C11:
If two points are taken on the different sides of a straight line, a straight line going from one point to the other will intersect the first line at exactly one point. (Pascal, 1655)
 C12:
An infinite straight line passing through a point inside a circle intersects the circumference at exactly two points. (Pascal, 1655)
 C13:
A circumference passing through a point inside a circle and a point outside it intersects the circumference of the circle at exactly two points. (Pascal, 1655)
 C14:
If two circumferences both have some points inside one another, they will intersect one another at exactly two points. (Pascal, 1655)
 C15:
If a circumference has a point on one side of an infinite straight line and the center on the other side, it intersects the straight line at exactly two points. (Pascal, 1655)
 C16:
If the same straight line is inside two figures, the two figures will have a common part and will intersect one another. (Borelli, 1658)
 C17:
If two equal circles have their centers on the ends of a segment taken as their radius, they meet. (Schott, 1661)
 C18:
If a point F is taken in the base BD of a triangle BCD and a straight line FG is drawn inside the triangle and extended enough, it will intersect one of the other sides BC or CD at a point G, or both of them at their common point C (and C is then the same point as G). (Roberval, 1675)
 C19:
If an infinite straight line passes inside a plane or a solid figure, it will intersect the boundaries of this figure. (Roberval, 1675)
 C20:
If a segment has one end inside a circle and one end outside it, it intersects the circumference at one point. (Kästner, 1758)
2.11 Principles of congruence, motion, and transportation
 T1:
Quantities and their properties may be superposed by an intellectual act. (Richard, 1645)
 T2:
Quantities may be conceived to move completely or around something fixed. (Richard, 1645)
 T3:
Quantities the divisible boundaries of which are congruent are themselves congruent and equal to one another. Equal straight lines are congruent with one another. Equal rectilinear angles are congruent with one another. Equal circles are congruent with one another. [Equal] arcs of equal circles are congruent with one another. Equilateral and equiangular rectilinear figures are congruent with one another, and equal. (Richard, 1645)
 T4:
If the ends of two straight lines are congruent with one another, the straight lines are themselves congruent and equal. (Richard, 1645)
 T5:
The mere change of position of a triangle or any other plane and rectilinear figure does not change anything in its sides or angles. (Richard, 1645)
 T6:
Given a thing, to reproduce it. (Ricci, 1651)
 T7:
Given two things, to superpose them. (Ricci, 1651)
 T8:
If the boundaries of two plane figures are congruent, the two figures are themselves congruent. (Ricci, 1651)
 T9:
If two plane figures are congruent, their boundaries are congruent as well. (Ricci, 1651)
 T10:
Equal straight lines are congruent with one another. (Tacquet, 1654) [cf. T3]
 T11:
Equal circles are congruent with one another. (Tacquet, 1654) [cf. T3]
 T12:
Equal angles are congruent with one another. (Barrow, 1655) [cf. T3, and E4]
 T13:
To draw a straight line equal to a given straight line. (Arnauld, 1667) [Elements I, 2]
 T14:
Things with equal motions traverse equal spaces in equal times. (Fabri, 1669)
 T15:
To move the compass keeping its opening. (Fabri, 1669)
 T16:
To bend a straight line through the movement of the sides. (Fabri, 1669)
 T17:
If a rectilinear angle is congruent with another rectilinear angle and it is reflected without changing its magnitude or width, it will be still congruent with the second angle. (Roberval, 1675)
 T18:
To put two straight lines directly one to the other. (Scarburgh, 1705)
 T19:
Any extended entity you please may move with any given law whatever. (Newton, 1693–1706)
 T20:
A moving body may halt in any given position whatever. (Newton, 1693–1706)
 T21:
Extended entities may describe the paths marked out through the courses of moving things or by the position of stationary ones. (Newton, 1693–1706)
 T22:
To draw through two given points a line congruent to a given line at a distance no less than that of the points, an ordained point of which shall coincide with one or other given point. (Newton, 1693–1706)
 T23:
To draw any line on which there shall always fall a point which is given according to a precise rule by drawing from points through points lines congruent to given ones. (Newton, 1693–1706)
 T24:
To extend from any given point a straight line equal to a given straight line. (Marchetti, 1709) [Elements I, 2; cf. T13]
 T25:
To cut from any given straight line a portion equal to a shorter given straight line. (Marchetti, 1709) [Elements I, 3]
 T26:
To reproduce any straight line or rectilinear angle. (Simpson, 1747) [Elements I, 2–3 and Elements I, 23].
 T27:
The magnitudes that are congruent with one another are similar and equal, and the magnitudes that are similar and equal are congruent with one another. (König, 1758)
2.12 Principles of the equality and the comparison of measure
 E1:
The shortest line between two points is the straight line. (Bradwardine, fourteenth century) [Archim. De sphaera et cylindro I, Ass. 1]
 E2:
If two more sides are drawn on the base of a triangle, they will be greater or lesser than the previous sides. (Vögelin, 1528) [Elements I, 7]
 E3:
If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal sides equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively. (Pelletier, 1557) [Elements I, 4]
 E4:
If one of several equal angles is a right angle, all angles will be right angles. (Herigone, 1634) [cf. Proclus, In Euclidis 189]
 E5:
The distance between two points is the straight line. (Pascal, 1655) [cf. Archim. De sphaera et cylindro I, Ass. 1; cf. E1]
 E6:
The diameter bisects the circle. (Pascal, 1655) [Elements I, def. 18].
 E7:
All lines drawn from the center to the circumference of a circle are equal to one another. (Borelli, 1658)
 E8:
Two circles that have equal radii are equal. (Borelli, 1658)
 E9:
Two equal circles have equal radii. (Borelli, 1658)
 E10:
If, from the two ends of a straight line, there is drawn an arc of a circle, as well as two straight lines meeting at a point outside of the circle and not intersecting the circle, and a curve entirely outside the circle, the latter curve and the two straight lines will both be longer than the circle, and the circle will be longer that the initial straight line. (Borelli, 1658)
 E11:
If two lines on the same plane have their ends in common and are both convex toward the same part, the one which is contained by the other is shorter than it. (Arnauld, 1667) [Archim. De sphaera et cylindro, Ass. 2]
 E12:
The degrees of equal circumferences are equal. (Arnauld, 1667)
 E13:
Among all straight lines drawn from the center of a circle, those smaller than the radius have their other end inside the circle, those equal to the radius have their other end on the circumference, and those longer than the radius have their other end outside of the circle. (Arnauld, 1667)
 E14:
If two straight lines intersect, and two points of the cutting line are singularly equidistant from two points of the other line, then all other points in the cutting line will also be equidistant from those two points of the other line. (Arnauld, 1667)
 E15:
Squares with equal sides are equal to one another. (Arnauld, 1667)
 E16:
Rectangles with equal bases and heights are equal to one another. (Arnauld, 1667)
 E17:
Similar segments of circles on equal straight lines are equal to one another. (Arnauld, 1667) [Elements III, 24].
 E18:
A line between two points that deviates toward one part of the other from the straight line between the same points is longer than the straight line. (Lamy, 1685)
 E19:
A polygon is larger than the circle inscribed in it. (Lamy, 1685)
 E20:
A polygon is smaller than the circle in which it is inscribed. (Lamy, 1685)
 E21:
If one end of a straight line lies on a plane and is conceived as the center of a circle in the plane, the straight line is perpendicular to the plane if and only if the other end is equidistant from the circumference of the circle. (Lamy, 1685)
 E22:
A solid figure is bigger than the one to which it is circumscribed and smaller than the one to which it is inscribed. (Lamy, 1685)
 E23:
Among prisms with equal height, the one with the smaller base is smaller. (Lamy, 1685)
 E24:
In equal circles, arcs of equal chords are equal and chords of equal arcs are equal. (Lamy, 1685) [Elements III, 23 and 24; cf. E17]
 E25:
The space ABC formed by the straight lines running from a point A to all the points of a line BC is the smallest space bounded by AB, BC, and AC. (Prestet, 1689)
 E26:
Equal circles have equal circumferences. (Marchetti, 1709)
 E27:
If two circles have equal circumferences, they are equal. (Marchetti, 1709)
 E28:
A greater circle has a greater circumference and greater radii. (Marchetti, 1709)
 E29:
If a circle has a greater circumference or radius than another circle, it is greater than the latter. (Marchetti, 1709)
 E30:
In a triangle two sides taken together in any manner are greater than the remaining one. (Marchetti, 1709) [Elements I, 20]
 E31:
If two straight lines forming an angle have the same length as two other straight lines forming an equal angle, the segment joining the ends of the former lines will be equal to the segment joining the ends of the latter lines. (Simpson, 1747) [Elements I, 4; cf. E3].
 E32:
If two solids of the same height are cut by planes parallel to their bases and the sections at equal distance from the bases are either equal or in constant ratio, the solids themselves are equal or in the same constant ratio. (Simpson, 1747)
 E33:
Arcs of a circle that contain equal angles are similar to one another. (König, 1758)
 E34:
Upright prisms with equal bases and heights are equal to one another. (Simpson, 1760) [cf. Elements XI, 29]
 E35:
The circumference of a circle is greater than the perimeter of any inscribed polygon, and lesser than the perimeter of any circumscribed polygon. (Simpson, 1760) [cf. E19 and E20]
2.13 The parallel postulate
 PP1:
If two straight lines lying on the same plane and extended to infinity in both directions do not meet, they are equidistant. (Fine, 1532)
 PP2:
If a straight line meets one parallel, it will also meet the other parallels. (Ricci, 1651)
 PP3:
Parallel lines have a common perpendicular. (Tacquet, 1654)
 PP4:
Two perpendiculars cut off equal segments from each of two parallel lines. (Tacquet, 1654)
 PP5:
If a straight line perpendicular to a given straight line is translated into a plane remaining with one of its ends on the given straight line, the other end of it will draw with its flow another straight line. (Borelli, 1658)
 PP6:
If two straight lines extended toward one side come closer and closer, they will eventually meet. (Arnauld, 1667)
 PP7:
If two lines are parallel, all the perpendiculars from one to the other are equal to one another. (Dechales, 1672)
 PP8:
If two straight lines diverge more and more when they are extended in one direction, they will converge and meet in the other direction. (Roberval, 1675)
 PP9:
Two straight lines do not incline toward one another if one of them does not incline toward a third, unless the other also inclines toward it. (Mercator, 1678)
 PP10:
Two nonequidistant straight lines lying on the same plane, if produced far enough, will meet on one side. (Rondelli, 1719) [cf. PP6]
 PP11:
If two points on a straight line have different distances to another straight line, the two straight lines, infinitely produced, will meet. (Simpson, 1747) [cf. PP10]
 PP12:
A straight line passing through a point that is inside an angle cuts at least one of the sides of the angle. (Lorenz, 1773)
 PP13:
Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another. (Playfair, 1795)
 PP14:
If two straight lines forming an angle are infinitely produced from a point, their distance will exceed any finite magnitude. (Commandino, 1572)
 PP15:
If one assumes that two straight lines which form internal angles equal to two right angles when intersected by a third straight line meet in that direction, one should also grant that the two lines always meet when they form the same angles with a third line. (Richard, 1645)
2.14 Hilbert’s Axioms of geometry
2.14.1 Axioms of incidence
 I, 1:
For every two points A, B there exists a line a that contains each of the points A, B.
 I, 2:
For every two points A, B there exists no more than one line that contains each of the points A, B.
 I, 3:
There exist at least two points on a line. There exist at least three points that do not lie on a line.
 I, 4:
For any three points A, B, C that do not lie on the same line there exists a plane \(\alpha \) that contains each of the points A, B, C. For every plane there exists a point which it contains.
 I, 5:
For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C.
 I, 6:
If two points A, B of a line a lie in a plane \(\alpha \) then every point of a lies in the plane \(\alpha \).
 I, 7:
If two planes \(\alpha , \beta \) have a point A in common then they have at least one more point B in common.
 I, 8:
There exist at least four points which do not lie in a plane.
2.14.2 Axioms of order (or Betweenness)
 II, 1:
If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B then also lies between C and A.
 II, 2:
For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B.
 II, 3:
Of any three points on a line there exists no more than one that lies between the other two.
 II, 4:
Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC. (Pasch’s Axiom)
2.14.3 Axioms of congruence
 III, 1:
If A, B are two points on a line a, and A \(^{\prime }\) is a point on the same or another line a\(^{\prime }\) , then it is always possible to find a point B \(^{\prime }\) on a given side of the line a\(^{\prime }\) through A \(^{\prime }\) such that the segment AB is congruent or equal to the segment \(A^{\prime }B^{\prime }\). In symbols \(AB \cong A'B'\).
 III, 2:
If a segment A \('B'\) and a segment \(A''B''\), are congruent to the same segment AB, then the segment A \('B'\) is also congruent to the segment \(A''B''\), or briefly, if two segments are congruent to a third one they are congruent to each other.
 III, 3:
On the line a let AB and BC be two segments which except for B have no point in common. Furthermore, on the same or on another line a\('\) let A \('B'\) and B \('C'\) be two segments which except for B \('\) also have no point in common. In that case, if \(AB \cong A^{\prime }B^{\prime }\) and \(BC \cong B^{\prime }C^{\prime }\), then \(AC \cong A^{\prime }C^{\prime }\).
 III, 4:
Let \(\angle \) (h,k) be an angle in a plane \(\alpha \) and a\('\) a line in a plane \(\alpha '\) and let a definite side of a\('\) in \(\alpha '\) be given. Let h \('\) be a ray on the line a\('\) that emanates from the point O \('\). Then there exists in the plane \(\alpha '\) one and only one ray k \('\) such that the angle \(\angle \) (h,k) is congruent or equal to the angle \(\angle \) (h \('\), \(k'\) ) and at the same time all interior points of the angle \(\angle \) (h \('\), \(k'\) ) lie on the given side of a\('\). Symbolically \(\angle \) (h,k) \(\cong \angle \) (h \('\), k’). Every angle is congruent to itself, i.e., \(\angle \) (h,k) \(\cong \angle \) (h,k) is always true.
2.14.4 Axiom of parallels
 IV.1:
Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a. (Playfair’s Axiom)
2.14.5 Axioms of continuity
 V, 1:
If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B. (Axiom of Archimedes)
 V, 2:
An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follow from Axioms I–III and from V1 is impossible. (Axiom of Linear Completeness)
3 The editions of the Elements
3.1 Greek manuscripts of the Elements used by Heiberg

P (Vatican, Early ninth century; partially belonging to a nonTheonine tradition)

Open image in new window : P1, P2, P3, P4, P5, P6*.

Open image in new window : CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5.

B (Oxford, 888 AD)

Open image in new window : P1, P2, P3, P4, P5.

Open image in new window : CN1, CN2, CN3, CN6*, CN8*, CN9*, CN4, CN5, P6*.

F (Firenze, tenth century)

Open image in new window : P1, P2, P3, P4, P5, P6*.

Open image in new window : CN1, CN2, CN3, CN6*, CN8*, CN9*, CN4, CN5.

b (Bologna, eleventh century)

Open image in new window : P1, P2, P3, P4, P5.

Open image in new window : CN1, CN2, CN3, CN6*, CN8*, CN9*, CN4, CN5, P6*.

V (Wien, twelfth century)

Open image in new window : P1, P2, P3, P4, P5, [P6*].

Open image in new window : CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

p (Paris, twelfth century)

Open image in new window : P1, P2, P3, P4, P5.

Open image in new window : CN1, CN2, CN3, CN6*, CN8*, CN9*, CN4, CN5, P6*.

q (Paris, twelfth century)

Open image in new window : P1, P2, P3, P4, P5, P6*.

Open image in new window : CN1, CN2, CN3, CN6*, CN7*, CN8*, CN4, CN5.
3.2 Medieval translations in Latin manuscripts

[Boethius], Geometria.

Book I

Petitiones, sive Postulata: P1, P2, P3, P4, P5.

Communes animi conceptiones: CN1, CN3, CN2, CN4.

Adelard of Bath, first half of twelfth century.

Book I

Petitiones quinque: P1–P2, P3, P4, P5, P6*.

Scientia universaliter communis: CN1, CN2, CN3, CN6*, A1, CN9*, CN4, CN5.

Hermann von Carinthia, first half of twelfth century.

Book I

Quinque modi investigandi: P1–P2, P3, P4, P5, P6*.

Sapientia: CN1, CN2, CN3, CN6*, CN9*, A1, CN4, CN5.

Robert of Chester, first half of twelfth century.

Book I

Peticiones sunt quinque: P1–P2, P3, P4, P5, P6*.

Communes animi concepciones: CN1, CN2, CN3, CN6*, A1, CN9*, CN4, CN5.

Gerardo da Cremona, Liber Euclidis philosophi, mid. twelfth century.

Book I

Ea in quibus necesse est convenire: P1, P2, P3, P4, P5.

Communes animi conceptiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Gerardo da Cremona, [Anaritii in decem libros priores elementorum Euclidis commentarii], mid. twelfth century.

Book I

Petitiones: P1, P2, P3, P4, P5, [P6*].

Propositiones per se note: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN4, CN5, P6*, M1, M2, I1, I2, I3, P2–I4.

Anonymous, second half of twelfth century.

Book I

Unlabeled principles: P1, P2, P3, P4, P5; CN1, CN2, CN3, CN6*, CN8*, CN9*, CN4, CN5, P6*.

John of Tynemouth, late twelfth century or beginning of the thirteenth century.

Book I

Petitiones sunt quinque: P1–P2, P3, P4, P5, P6*.

Communes animi conceptiones: CN1, CN2, CN3, CN6*, A1, CN9*, CN4, CN5, A3, R1. (...)

Book VII

Communes animi conceptiones: N1, N2, N3. (...)

Jordanus de Nemore, De elementis arithmetice artis, first half of the thirteenth century.

Petitiones sunt tres: N4, N5, N6.

Communes animi conceptiones sunt octo: A2, N7, M3, M4, N8, N9, N10, N11.

Campano da Novara, around 1255–1259.

Book I

Petitiones sunt quinque: P1–P2, P3, P4, P5, P6*.

Communes animi conceptiones: CN1, CN2, CN3, CN6*, A1, CN9*, CN4, CN5; A3, R1. (...)

Book VII

Petitiones: N12, N5, N6, N13.

Communes animi conceptiones: A2, M3, M4, N8, N7, N9, N10, N2, N3, N1.

Thomas Bradwardine, Geometria speculativa, around 1320–1335.

Book I

Petitiones ab Euclyde ponuntur quinque: P1–E1, P3, P4, P5, P6*.

Communes scientie multe sunt, sed sufficiant novem: A4, CN5, CN1, A5, CN2, CN3, CN6*, CN7*, CN4. (...)
3.3 Printed editions

Campano da Novara, Preclarissimus liber elementorum Euclidis perspicacissimi, in artem geometrie incipit quamfoelicissime, Venezia, Ratdolt, 1482.

Book I

Petitiones sunt quinque: P1–P2, P3, P4, P5, P6*.

Communes animi conceptiones: CN1, CN2, CN3, CN7*, CN6*, A1, CN9*, CN4, CN5, A3, R1. (...)

Book VII

Petitiones: N12, N5, N6, N13.

Communes animi conceptiones: A2, M3, M4, N8, N7, N9, N10, N2, N3, N1.

Thomas Bradwardine, Breve compendium artis geometrie, ed. Pedro Sánchez Ciruelo, Paris, Guy Marchant, 1495.

Book I

Petitiones ab Euclyde ponuntur quinque: P1–E1, P3, P4, P5, P6*.

Communes scientie multe sunt, sed sufficiant novem: A4, CN5, CN1, A5, CN2, CN3, CN6*, CN7*, CN4. (...)

Jacques Lefèvre d’Étaples, Arithmetica decem libris demonstrata, Paris, Higman, 1496.

Dignitates: A2, N7, M3, M4, N8, N9, N10, N11, N14, CN1, CN3, M7, M8, CN2, M6, A4, N15, N16, R2, M5.

Petitiones: N4, N5, N6, N13, N17, N18.

Giorgio Valla, De expetendis et fugiendis rerum opus, Venezia, Manuzio, 1501.

Book I

Petitiones: P1, P2, P3, [P4], P5.

Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Bartolomeo Zamberti, Euclidis megarensis philosophi platonici Mathematicarum disciplinarum Janitoris: Habent in hoc volumine quicumque ad mathematicam substantiam aspirant: elementorum libros XIII. cum expositione Theonis, Venezia, Tacuino, 1505.

Book I

Postulata: P1, P2, P3, P4, P5.

Communes sententiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Luca Pacioli, Euclidis megarensis philosophi acutissimi mathematicorumque omnium sine controversia principis opera a Campano interprete fidissimo translata, Venezia, Paganino, 1509.

Book I

Petitiones: P1–P2, P3, P4, P5, P6*.

Communes animi conceptiones: CN1, CN2, CN3, CN7*, CN6*, A1, CN9*, CN4, CN5, A3, R1. (...)

Book VII

Petitiones: N12, N5, N6, N13.

Communes animi conceptiones: A2, M3, M4, N8, N7, N9, N10, N2, N3, N1.

Jacques Lefèvre d’Étaples, Euclidis Megarensis Geometricorum elementorum libri XV. Campani Galli transalpini in eosdem commentariorum libri XV. Theonis Alexandrini Bartholomaeo Zamberto Veneto interprete, in tredecim priores, commentariorum libri XIII, Paris, Henri Estienne, 1516.

Johannes V\(\ddot{\textsc {o}}\) gelin, Elementale geometricum ex Euclidis geometria, Wien, Singren, 1528.

Book I

Postulata: P4, P5, P6*, E2.

Communes sententiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, A4.

Oronce Fine, Protomathesis, Paris, 1532.

Book I

Postulata, seu Petitiones: P1, P2, P3, P4, PP1, C1, C2. (...)

Axiomata, vel Effata, seu Communes sententiae: CN4, CN1, M3–M4, CN2–CN3, CN6*–CN7*, P6*, CN5–A4.

Simon Grynaeus, Open image in new window , Basel, Herwagen, 1533.

Book I

Open image in new window : P1, P2, P3.

Open image in new window : CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Oronce Fine, In sex priores libros geometricorum elementorum Euclidis Megarensis demonstrationes, Paris, Simon de Colines, 1536.

Book I

Postulata: P1, P2, P3, P4, P5.

Communes sententiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Niccolò Tartaglia, Euclide Megarense philosopho, solo introduttore delle scientie mathematice, Venezia, Rossinelli, 1543.

Book I

Petitioni: P1, P2, P3, P4, P5, P6*.

Commune sententie: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5.

Book VII

Petitioni: N12, N5, N6, N13.

Commune conceptioni dell’animo: A2, M3, M4, N8, N7, N9, N10, N2, N3, N1.

Pierre de la Ramée, Euclides, Paris, Grandin, 1545.

Book I

Postulata: P1, P2, P3, P4, P5.

Communes sententiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Angelo Caiani, I quindici libri degli Elementi di Euclide, di greco tradotti in lingua thoscana, Roma, Blado, 1545.

Book I

Domande: P1, P2, P3.

Comuni pareri: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Joachim Camerarius, Euclidis elementorum geometricorum libri sex, conversi in Latinum sermonem, Leipzig, Valentin, 1549.

Book I

Petita: P1, P2, P3.

Communes notitiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Johann Scheybl, Euclidis Megarensis philosophi et mathematici excellentissimi, Sex libri priores de geometricis principijs, Basel, Herwagen, 1550.

Book I

Postulata: P1, P2, P3.

Communes notitiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Robert Recorde, The Pathway to Knowledg, containing the First Principles of Geometrie, London, Wolfe, 1551.

[Book II]

Grauntable requestes: P1, P2, P3, P4, P5, P6*.

Common sentences: CN1, CN2, CN3, CN7*, CN6*, CN8*, CN9*, CN4, CN5, A4.

Valentin Naboth, Euclidis Megarensis mathematici clarissimi elementorum geometricorum liber primus, Köln, Birkmann, 1556.

Book I

Petita: P1, P2, P3.

Communes notitiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Jacques Peletier du Mans, In Euclidis Elementa Geometrica Demonstrationum Libri sex, Lyon, Tornes, 1557.

Book I

Postulata: P1, P2, P3, P4, [P5].

Animi notiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, [P6*]; E3. (...)

[Jean Magnien, Pierre de Montdoré, St. Gracilis], Euclidis Elementorum libri XV, Graecè et Latinè, Paris, Cavellat, 1557.

Book I

Postulata: P1, P2, P3.

Communes notiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Xylander (Wilhelm Holtzmann), Die Sechs ersten Bücher Euclidis, vom Anfang oder Grund der Geometrj, Basel, Kündig, 1562.

Book I

Petitiones, seu Postulata: P1, P2, P3.

Communes notitiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Conrad Dasypodius, Euclidis quindecim Elementorum Geometriae primum, Strasbourg, Mylius, 1564.

Book I

Postulata: P1, P2, P3.

Communes notiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Pierre Forcadel, Les six premier livres des Elements d’Euclide traduicts et commentez par Pierre Forcadel de Bezies, Paris, Marnef, 1564.

Book I

Demandes: P1, P2, P3.

Communes sentences: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Christian Herlin, Conrad Dasypodius, Analyseis geometricae sex librorum Euclidis, Strasbourg, Rihel, 1566.

Book I

Postulata: P1, P2, P3.

Communes notiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, [P4], [P5], P6*.

François de FoixCandale, Euclidis Megarensis mathematici clarissimi elementa geometrica libri XV ad germanam geometriae intelligentiam è diversis lapsibus temporis iniuria contractis restituta, Paris, Royer, 1566.

Book I

Postulata: P1, P2, P3.

Communes sententiae: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Book VII

Communes sententiae: N19, N1.

Francesco Maurolico, Euclidis elementorum compendia, 1567 (unpublished).

Book I

Postulata: P1–P2, P3, P4, P5, P6*.

Communes sententiae: CN1, CN2–CN3–CN6*–CN7*, CN8*–CN9*, CN4, CN5–A6.

Book V

Communes animi conceptiones: R3, R4, R5, R6, R7, R8, R9, R10, R11, R12, R13, R14, R15, R16.

Pedro Juan de Lastanosa (de Monzón), Elementa Arithmeticae ac Geometriae, Valencia, Huete, 1569.

Dignitates arithmeticae: CN1, N7, N8, N9, A4, N20, N21, N22, CN2, CN3, CN6*, CN7*, N10, N14, N23, N24, N15, N25, N26, N16.

Petitiones seu postulata [arithmeticae]: N6, N13, N27, N28, N4, N17.

Axiomata, seu dignitates [geometriae]: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5–A4.

Postulata [geometriae]: P1, P2, P3, P4, P5, C1, P6*. (...)

Henry Billingsley, John Dee, The elements of geometrie of the most ancient Philosopher Euclide of Megara, faithfully (now first) translated into the English tongue, London, Daye, 1570.

Book I

Peticions, or requestes: P1, P2, P3, P4, P5, P6*.

Common sentences: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5.

Book VII

Common sentences: N7, M3, M4, N1, N3, N2, N29.

Federico Commandino, Euclidis elementorum libri XV una cum Scholiis antiquis, Pesaro, Franceschini, 1572.

Italian version as De gli elementi d’Euclide libri quindici con gli scolii antichi, Urbino, Frisolino, 1575.

Book I

Postulata (postulati, overo dimande): P1, P2, P3, P4, [P5].

Axiomata, seu communes notiones (assiomi, overo communi notitie): CN1, CN2, CN3, CN6*, CN7*, [CN8*], CN9*, CN4, CN5, P6*; PP14.

Book V

Notiones communes (communi notitie): M3, M4.

Book VII

Petitiones (petitioni, overo dimande): N12, N5, N6–N13.

Notiones communes (communi notitie): M3, M4, N30, N31, N8, N32, N33, N35, N36, N34, N2, N3, N1.

Book X

Notiones communes (communi notitie): AA1, M9, M10, M11.

Christoph Clavius, Euclidis elementorum libri XV, Roma, Vincenzo Accolto, 1574.

Book I

Petitiones, sive Postulata: P1, P2, P3, A7.

Communes notiones, sive Axiomata, sive Pronunciata, sive Dignitates: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, [P5], P6*, I5, M1, M2, M12, M13, A4, M14.

Book V

Axioma: R17.

Book VII

Postulata, sive petitiones: N12, N5, [N6–N13].

Axiomata, sive pronuntiata: M3, M4, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Postulatum, sive petitio: AA1.

Axiomata, sive pronunciata: M11, M9, M10.

Rodrigo Zamorano, Los seis libros primeros dela geometria de Euclides, Sevilla, La Barrera, 1576.

Book I

Peticiones: P1, P2, P3, P4, P5.

Comunes sentencias: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Giovan Battista Benedetti, In quintum Euclidis librum, in Diversarum speculationum mathematicarum, & physicarum Liber, Torino, Belvilacqua, 1585.

Book V

Postulata: R11–R12, CN3–CN2, R3–R5, R4–R6, R8, R9, R10, R18, R19, R20, R21, R22.

Francesco Patrizi, Della nuova geometria, Ferrara, Baldini, 1587.

[Book I]

Supposizioni: S1, S2, S3, S4, S5.

Assiomi: A8, A9, A10, CN5, A11, A12.

Christoph Clavius, Euclidis elementorum libri XV, Roma, Bartolomeo Grassio, \(1589^{2}\).

Book I

Petitiones, sive Postulata: P1, P2, P3, A7.

Communes notiones, sive Axiomata, sive Pronunciata, sive Dignitates: CN1–A13–A14, CN2, CN3, CN6*–M15, CN7*–M16, CN8*–M17, CN9*–M5, CN4, CN5, I5, I6, P4, [P5], P6*, M1, M2, M12, M13, A4, M14.

Book V

Axioma: R17.

Book VII

Postulata, sive petitiones: N12, N5, [N6–N13].

Axiomata, sive pronuntiata: M3, M4, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Postulatum, sive petitio: AA1.

Axiomata, sive pronunciata: M11, M9, M10.

Nas ī r adDī n atTū s ī, Euclidis elementorum geometricorum libri tredecim ex traditione doctissimi Nasiridini Tusini, Roma, Typographia Medicea, 1594.

Book I

alusul almawdu’a: P6*, P2, P4, P5.

al’ulum almuta’arifa: CN1, CN2, CN3, CN6*–CN7*, CN8*, CN9*, CN4, CN5.

Jan Pietersz Dou, De ses eerste Boecken Euclidis, Leiden, Bouwensz, 1606.

Book I

Eenige stucken tot dese konst behoorende: P1, P2, P3.

Ghemeene bekantenissen: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Pietro Antonio Cataldi, I primi sei libri de gl’Elementi d’Euclide ridotti alla Prattica, Bologna, Cochi, 1613.

Book I

Petitioni, overo domande: P1, P2, P3, [P4], [P5], P6*.

Comuni concessioni, o comuni notitie: CN1, CN2, CN3, CN6*, CN7*, [CN8*], CN9*, CN4, CN5–A4.

Book V

Petitione: M14.

Comuni concessioni: CN1, CN2.

Denis Henrion, Les quinze livres des elements d’Euclide traduicts de Latin en François, Paris, Joallin, 1615.

Book I

Petitions, ou demandes: P1, P2, P3.

Communes sentences: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Book VII

Communes sentences: M3, M4, N30, N31, N8, N32, N34, N35, N36, N2, N3, N1.

Book X

Demande: AA1.

Communes sentences: M11, M9, M10.

Pierre Herigone, Cursus mathematicus. Cours mathematique, Paris, Le Gras, 1634.

Book I

Petitiones, sive Postulata (Petitions, ou Demandes): P1, P2, P3, A7.

Communes notiones, sive Axiomata, sive Pronunciata, sive Dignitates (Communes notions, ou communes sentences, ou maximes): CN1, A15, A13, A14, A16–A17, A18, A19, CN2, CN3, M18, CN6*, M6, M15, CN7*, M7, M16, CN8*, M19, M20, M21, CN9*, M22, M23, M24, CN4, CN5, A20, I5, I6, P4, E4, P5, P6*, I7, I8, M1, M2, M12, M13, A4, M25, M14, M26, A21.

Book V

Axioma: M27.

Book VII

Postulata: N12, N5, N37.

Axiomata: N38, A22, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Postulatum: AA1.

Axiomata: M11, M9, M10.

Christoph Grienberger, Euclidis sex primi elementorum geometricorum libri cum parte undecimi ex majoribus Clavij commentarijs in commodiorem forma contracti, Graz, Widmanstad, 1636.

Book I

Petitiones, seu Postulata: P1, P2, P3.

Axiomata, seu Pronunciata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, I9, P4, P5, P6*, A4, M14.

Marin Mersenne, Universae geometriae, mixtaeque mathematicae synopsis, Paris, Bertier, 1644.

Book I

Postulata: P1, P2, P3, P4, P5.

Axiomata, seu Communes notiones: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Book VII

Petitiones: N12, N5, N6–N13.

Communes notiones: M3, M4, N30, N31, N8, N32, N33, N35, N36, N34, N2, N3, N1.

Book X

Communes notiones: AA1, M9, M10, M11.

Claude Richard, Euclidis elementorum geometricorum libros tredecim, Antwerp, Verdus, 1645.

Book I

Petitiones, sive Postulata: P1, P2, P3, A7, T1, T2.

Axiomata, seu Pronunciata, seu Dignitates, seu Communes notiones: CN1–A13–A14–A15–A16–A17, CN2, CN3, CN6*–M15, CN7*–M16, CN8*–M17, CN9*–M5, CN5–A6, CN4–T3, I5, I6, P4, P5–PP15, P6*, M1, M2, M12, M13, A4, [M14], T4, C3, C4, I10, C5, C6, C7, C8, C9, T5.

Book V

Axiomata: R17, A23, M28; [A4, CN2, CN6*, CN3, CN1, CN7*, A13, M1].

Book VII

Petitiones: N12, N5.

Axiomata: M3, M4, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Petitio: AA1.

Axiomata, sive pronunciata: M11, M9, M10.

Evangelista Torricelli, De proportionibus liber, 1647 (unpublished).

Suppositiones et Axiomata: R10, R3–R4, R5–R6, R23, R24, R20.

John Wallis, In Elementa Euclidis Praelectiones, 1651 (unpublished).

Book I

Postulata: P1, P2, P3.

Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN9*, CN8*, CN4, CN5, P4, [P5], P6*.

Giovanni Ricci, De gli Elementi di Euclide li primi sei Libri, Bologna, Longhi, 1651.

Book I

Postulati, overo Dimande: P1, P2, I11, S6, T6, T7.

Assiomi, overo communi sentenze: CN1, A13, A14, CN2, CN3, CN6*, M15, CN7*, M16, CN8*, CN9*, CN4, CN5, P6*, A4, P4, I6, T8, T9, PP2, A21.

Book II

Assioma: A1.

Andreas Tacquet, Elementa geometriae planae et solidae, Antwerp, Meurs, 1654.

Book I

Postulata: P1, P2, P3.

Axiomata: CN1–A13, CN2, CN3, CN6*, CN7*, CN8*–CN9*, CN4, T10–T11, CN5, P4, PP3, PP4, P6*, [I5].

Book V

Axioma: R25.

Isaac Barrow, Euclidis Elementorum Libri XV breviter demonstrati, Cambridge, Nealand, 1655.

Book I

Postulata: P1, P2, P3.

Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4 [T10, T12], CN5, I5, I6, P4, P5, P6*, M1, M2, M12, M13, A4, M14.

Book V

Axioma: M27.

Book VII

Postulata: N12, N5, N37.

Axiomata: N38, A22, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Postulatum: AA1.

Axiomata: M11, M9, M10.

Blaise Pascal, Introduction à la géométrie, around 1655 (unpublished).

[Book I]

Principes: S8, S9, S10, S11, E5, S12.

Théorèmes connues naturellement: S13, E8–S14, S15, S16, E6, I3, C10, C11, C12, C13, C14, C15.

Andreas Tacquet, Arithmeticae theoria et praxis, Leuven, Coenestenius, 1656.

Book VII

Axiomata: M3, M4, N30, N31, N8, N9, N34, N36, N5, N2, N3, N1.

Giovanni Alfonso Borelli, Euclides restitutus, sive prisca geometriae elementa, Pisa, F. Onofrio, 1658; Roma, Mascardi, \(1679^{2}\). Italian ed. Euclide rinnovato ovvero gli antichi elementi della geometria, Bologna, Ferroni, 1663.

[Book I]

Petitiones, seu Postulata: P1, P2, P3, A7.

Axiomata, seu Pronunciata: CN1–A13–A14, CN2, CN3, CN6*–CN7*, CN8*–CN9*, M29, CN5, A4, CN4, I9, P6*, E7, C16; PP5.

[Book II]

Axioma: E8–E9.

[Book III]

Axiomata: M9, R17, M30, R26, R27.

[Book V]

Axioma: E10.

[Book VI]

Postulata: I12, I4.

Axiomata: I13, I14, I15, I8, I16.

[Book IX]

Axiomata: M2–M13, M12.

Claude François Milliet Dechales, Huict Livres des Elements d’Euclide rendus plus faciles, Lyon, Coral, 1672.

Book I

Demandes: P1, P2, P3.

Axiomes, ou Principes: CN1–A13, CN2, CN3, CN6*, CN7*, CN8*–CN9*, CN4, T10–T12, CN5, P4, PP7, P6*, I9.

Book V

Axioma: R25.

Caspar Schott, Cursus mathematicus, sive Absoluta omnium mathematicarum disciplinarum encyclopaedia, Würzburg, Hertz, 1661.

Book I

Petitiones, sive Postulata: P1, P2, P3, A7.

Axiomata, sive Pronunciata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, [P5], P6*, I5, I6, I7, I8, M1, M2, M12, M13, A4, M14, M26, A24, E8, C17, A16.

Antoine Arnauld, Nouveaux Élémens de géométrie, Paris, Savreux, 1667.

[Book I]

Axiomes de l’égalité et inégalité: CN5, A4, CN1, CN2, CN3, CN7*, CN6*, M3, M4.

[Book II]

Axiomes: R20, R13, R29, R30, M31, R31.

[Book V]

Axiomes: E11, P1, P2, I5, I17, PP6; P3, T13, E8, E12, S16–E17, E13, S17; E14.

[Book XIII]

Axiomes: E15, E16, N39.

GillesFrançois de Gottignies, Elementa geometriae planae, Roma, Angelo Bernabò, 1669.

Postulata: P1, P2, P3.

Axiomata: CN1–A13–A14, CN2, CN3, CN6*, CN7*, CN8*–CN9*, CN4, T10–T11, CN5, P4, P6*, I5, R32, R28.

Honoré Fabri, Synopsis geometrica, Lyon, Molin, 1669.

Axiomata: CN1–CN8*–CN9*, CN2–CN3–CN6*–CN7*, CN5–A4–M14–A24, R33, I17–I5, T14, CN4, R34, R35, R36.

Postulata: P1, P3, P2–T15, A7, T2, S18, T16.

Vincenzo Viviani, Quinto Libro degli Elementi di Euclide, ovvero Scienza Universale delle Proporzioni spiegata colla dottrina del Galileo, Firenze, Condotta, 1674.

Assiomi, ovvero Comuni notizie: R38, R39, R35, R36, R16–R37, R10, R4, R40, R11.

Domanda: R17.

Gilles Personne de Roberval, Éléments de géométrie, around 1673–1675 (unpublished).

[Book I]

Axiomes: A4, CN5, CN1, CN2.

Postulats: S19, I18, I19, I20, I21, I22, I23, T2, I24, P1, A7, A25.

[Book II]

Axiomes: CN4, T17, E1.

Postulat: C18.

[Book III]

Postulats: P3, AA4, C19.

[Book IV]

Axiome: PP8.

[Book VIII]

Axiome: E19.

Jean Prestet, Elemens des mathematiques, ou principes generaux de toutes les sciences qui ont les grandeurs pour objet, Paris, Pralard, 1675.

Axiomes: A26, M32, A4, CN5, M33, CN1, CN2, CN3, CN6*, CN7*; M3, M4, N40, M11, M10, N41, N42.

Mercator (Nikolaus Kauffmann), Euclidis Elementa Geometrica Novo Ordine ac Methodo fere demonstrata, London, Martyn, 1678.

Axiomata: I5–P6*, P4, PP9, CN4–T10–T12, CN5–A4, CN1, CN8*–CN9*, CN2–CN3, CN6*–CN7*; R35, R8, R41, R10, R42.

Principia practica, sive Postulata: P1, P2, P3.

Vitale Giordano da Bitonto, Euclide Restituto, ovvero gli antichi elementi geometrici ristaurati e facilitati, Roma, Angelo Bernabò, 1680.

Book I

Postulati: P1, P2, P3, A7.

Assiomi: CN1–A13, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, [P5], [P6*], [I5].

Book VII

Postulati: N12, N5.

Assiomi: M3, M4, N30, N31, N32, N33, N34, N35, N36, N2, N3, N1.

Book X

Postulato: AA1.

Assiomi: M11, M9, M10.

Jacques Rohault, Les six premiers Livres des Elemens d’Euclide, in Ouvres posthumes, ed. C. Clerselier, Paris, Desprez, 1682.

Book I

Demandes: P1, P2, P3, A7.

Axiomes: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, A4, CN5, P4, P6*, I5, I6, M1, M2, M12, M13, M14.

Book V

Axiomes: R43, R44, R45.

Antoine Arnauld, Nouveaux Élémens de géométrie, Paris, Desprez, \(1683^{2}\).

[Book I]

Axiomes de l’égalité et inégalité: CN5, A4, CN1, CN2, CN3, CN7*, CN6*, M3, M4.

[Book II]

Raisons égales naturellement connues: R46, R20, R13, R29, R30.

Axiomes: R18, R47, R48, R28, R49, R10, R4, R50, R51, R52.

[Book V]

Axiomes: E11, P1, P2, I5, I17, PP6; P3, T13, E8, E12, S16–E17, E13, S17; E14.

[Book XIII]

Axiomes: E15, E16, N39.

Bernard Lamy, Les elemens de geometrie, Paris, Pralard, 1685.

Principes generaux, ou Axiomes: CN5, A4, CN1, CN2, CN3, CN7*, CN6*, CN4, CN8*, CN9*, N43.

[Book I]

Propositions, ou Demandes: I29, I3, E18, P1, P2, I31, I32, I30, I17; P3, E24, E6, E13, E8; E14, I25, I26; PP1.

[Book II]

Demandes: E19, E20.

[Book III]

Demandes: N44, N45, R53.

[Book IV]

Axiomes: I4, I33, I16, I34, I15, E21, I27, I28; E22, E23, AA2, AA3.

Jean Prestet, Nouveaux elemens des mathematiques, ou principes generaux de toutes les sciences qui ont les grandeurs pour objet, Paris, Pralard, 1689\(^{2}\).

Axiomes, ou notions communes, et principes generaux du corps entier des Mathematiques: CN5, A4, M33, CN2, CN3, CN6*, CN7*, M12, A13, A27, A28.

Axiomes, ou notions communes, et principes generaux du corps entier de la Geometrie: P1–P2–P6*, CN4, E25.

[John Keill], Euclidis elementorum libri priores sex, item undecimus et duodecimus ex versione latina Federici Commandini in usum juventutis academicae, Oxford, Theatrum Sheldonianum, 1701.

Book I

Postulata: P1, P2, P3.

Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*, P4, P5.

Book V

Axiomata: M3, M4.

David Gregory, Open image in new window , Euclidis quae supersunt omnia, Oxford, Theatrum Sheldonianum, 1703.

Book I

Open image in new window , Postulata: P1, P2, P3.

Open image in new window , Communes notiones, sive Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*.

Edmund Scarburgh, The English Euclide, being The First Six Elements of Geometry, translated out of the Greek, Oxford, Theatrum Sheldonianum, 1705.

Book I

Postulates: P1, P2–[T18], P3.

Axioms: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, [P5], P6*, PP1.

Nicholas de Mal \(\acute{\textsc {e}}\) zieu, Elemens de geometrie de Monseigneur le Duc de Bourgogne, Paris, Boudot 1705.

[Book I]

Axiomes, ou verités connues d’elles mêmes: CN5, A6, A4, CN1, CN2, CN3, N46, N47, N30, N48, N49.

Isaac Newton, [The Science of Axiomatc Deduction], around 1693–1706 (unpublished).

Geometriae libri duo

Postulata: P1, T22, T23.

[Arithmetic and Geometry]

Axiomata: CN1, CN2, CN3, R4, A4, R54.

Postulata: N50, N51, ... .

De constructione problematum geometricorum

Postulata: T19, T20, T21.

Angelo Marchetti, Euclides reformatus, Livorno, Celsi, 1709.

Book I

Petitiones, seu Postulata: P1, P2, T24, T25, P3, A7.

Axiomata, seu Pronunciata: CN1, A13, CN8*–CN9*, CN2, CN3, M6, M7, CN5, A4, M11, T4–T8, P4, I5, E6, E7, E8–E26, E9–E27, PP5, E28, E29, CN4, E30; PP1, AA1.

[Book III]

Axiomata: R35, R8, R55, R10, R16, R37, R11, R12, R14, R56, R15, R57, R58, R59, R17.

[Book VI]

Postulata: I12, I4.

Axiomata: I35, I14, I15, I8, I16.

Geminiano Rondelli, Sex priora Euclidis elementa, quibus accesserunt undecimum, & duodecimum, Bologna, Longhi, 1719.

Book I

Petitiones, seu Postulata: P1, P2, P3, S6, T7, A7.

Axiomata: CN1–A13–A14, CN2, CN3, CN6*–M15, CN7*–M16, CN8*–M17, CN9*–M5, CN4, CN5, I5, I6, P4, [P5], P6*, M2, M1, M13, M12, A4, M14, A16, A24, PP2, PP10, I36.

Book V

Axioma unicum: R25.

Henry Hill, The Six First, Together with the Eleventh and the Twelfth Books of Euclid’s Elements, Demonstrated after a New, Plain, and Easie Method, London, Pearson, 1726.

Book I

Postulates: P1, P2, P3.

Axioms: CN1, A13, A14, A29, A30, A31, CN8*, CN9*, N52, N53, CN2, CN3, N48, N49, M2, M13, CN5, A4, CN4, P6*, P4, P5.

Guido Grandi, Elementi geometrici piani e solidi di Euclide posti brevemente in volgare, Firenze, Tartini, 1731.

Book I

Dimande: P1, P2, P3.

Assiomi: CN1, CN2–CN3, CN6*–CN7*, CN8*–CN9*, CN4, CN5, P4, [P5], P6*, I5.

Manoel de Campos, Elementos de geometria plana, e solida, segundo a ordem de Euclides, Lisboa, Rita Cassiana, 1735.

Book I

Postulados: P1, P2, P3.

Axiomas: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5 [PP3], PP4, P6*, I5.

Antoine Deidier, La science des géomètres, Paris, Jombert, 1739.

Book I

Axiomes: CN5, A4, CN2, CN3, N48–N49, CN6*–CN7*–N54–N55, CN8*–CN9*, AA5, CN1.

Samuel Klingenstierna, Euclidis elementorum libri sex priores una cum undecimo et duodecimo, Upsala, Höjer, 1741.

Book I

Postulata: P1, P2, P3.

Axiomata: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*, P4, P5.

Book V

Axiomata: M3, M4.

Mårten Str \(\ddot{\textsc {o}}\) mer, Euclidis Elementa eller Grundeliga inledning til geometrien, til riksens ungdoms tienst på swenska språket utgifwen, Upsala, Kiesewetter, 1744.

Book I

Postulater: P1, P2, P3.

Axiomer: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*, P4, P5.

Thomas Simpson, Elements of Plane Geometry, London, Farrer & Turner, 1747.

Book I

Postulates, or Petitions: P1, P2, P3, I37–T26.

Axioms, or Selfevident Truths: CN1, CN5, A4, CN2, CN3, CN6*–CN7*, P4, PP10, E31.

[Book IV]

Axioms: R60, R61, R62, R8, R10.

[Treatise of Regular Solids]

Postulatum: E32.

Robert Simson, Euclidis Elementorum libri priores sex item undecimus et duodecimus ex versione latina Federici Commandini, Glasgow, Foulis, 1756.

English version as The Elements of Euclid, viz. the first Six Books together with the Eleventh and Twelfth. In this Edition, the Errors, by which Theon, or others, have long ago Vitiated these Books, are Corrected, and some of Euclid’s Demonstrations are Restored, Glasgow, Foulis, 1756.

Book I

Postulata (Postulates): P1, P2, P3.

Axiomata (Axioms): CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*, P4, [P5].

Book V

Axiomata (Axioms): M3, M4, M34, M35.

Samuel K\(\ddot{\textsc {o}}\) nig, Elemens de geometrie contenant les six premiers livres d’Euclide, Den Haag, Scheurleer, 1758.

Book I

Demandes: P1, P2, P3.

Axiomes ou notions communes: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN5, T27, P4, [P5–PP2], P6*.

Book II

Axiomes: A4, E16.

Book III

Axiomes: E8, E33.

Book V

Demandes: M36, A32, [N37].

Abraham Gotthelf K\(\ddot{\textsc {a}}\) stner, Anfangsgründe der Arithmetik, Geometrie, ebenen und sphärischen Trigonometrie und Perspectiv, Göttingen, Vandenhoeck, 1758.

[Book I]

Forderungen: P1–P2–S20, P3.

Grundsätze: I17–P6*, I5–I6, CN4, T10–T12, P4, E6, C20, I38; P5; I33.

Thomas Simpson, Elements of Geometry, London, Nourse, 1760.

Book I

Axioms, or Selfevident Truths: CN1, CN5, A4, CN2, CN3, CN6*–CN7*, P4, I17, PP11, E31.

Postulates, or Petitions: P1, P2, P3, [I37–T26].

[Book IV]

Axioms: R63, R61, R60, R62, R8, R41, R10, R64, M37.

[Book VII]

Axiom: E34.

Postulate: I39.

[Book VIII]

Postulates: AA1, AA6, E35.

Johann Friedrich Lorenz, Euklid’s Geometrie, oder die sechs ersten Bücher der Elemente nebst dem eilften und zwölften aus dem Griechichen übersetz, Halle, Weisenhaus, 1773.

Book I

Forderungen: P1, P2, P3.

Grundsätze: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, P5, P6*; [PP12].

John Bonnycastle, Elements of Geometry, London, Johnson, 1789.

Book I

Postulates: P1, P2, P3, PP2.

Axioms: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, A4, CN4.

AdrienMarie Legendre, Éléments de géométrie avec des notes, Paris, Didot, 1794.

Book I

Axiomes: CN1, CN2, CN3, N30, N31, CN5, A4, I31, CN4, [P5]. (...)

John Playfair, Elements of Geometry, containing the first six books of Euclid with two books on the geometry of solids, Edinburgh, Bell & Bradfute, 1795.

Book I

Postulates: P1, P2, P3.

Axioms: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P4, PP13.

Book V

Axioms: M3, M4, M34, M35.

AdrienMarie Legendre, Éléments de géométrie avec des notes, Paris, Didot, 1799\(^{2}\).

Book I

Axiomes: CN1, CN5, A4, I31, CN4, [P5]. (...)

Vincenzo Flauti, I primi sei libri degli elementi di Euclide, Napoli, Stamperia Reale, 1810.

Book I

Postulati: P1, P2, P3, P4, [P5].

Assiomi, o nozioni comuni: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5, P6*.

Book V

Postulato: R1.

François Peyrard, Open image in new window , Euclidis quae supersunt, Les oeuvres d’Euclide, Paris, Patris, 1814.

Book I

Open image in new window , Postulata, Demandes: P1, P2, P3, P4, P5, P6*.

Open image in new window , Notiones communes, Notions communes: CN1, CN2, CN3, CN6*, CN7*, CN8*, CN9*, CN4, CN5.
In the following years, several older editions were reprinted many times, but Peyrard’s text and system of principles gained a growing acceptance by the scholarly community. Another Latin edition based on Peyrard’s text was prepared by Niede in 1825, and in 1826 it was translated into English by Philips. In 1824, Camerer and Hauber published, in Germany, a very influential and heavily annotated edition of the Elements also based on Peyrard’s manuscript (and axioms). In 1826–1829, E.F. August published another edition of the Greek text, also following Peyrard’s manuscript and correcting the latter’s edition in a few places. Many years later (1883–1916), Heiberg and Menge built their edition of Euclid on the same Vatican manuscript found by Peyrard (even though they emended and integrated it in several places), and this edition is still the most authoritative reconstruction of the text of the Elements that we have.
Notes
Acknowledgments
Open access funding provided by Max Planck Society (Max Planck Institute for the History of Science).
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