The Oration on the Dignity and the Usefulness of the Mathematical Sciences of Martinus Hortensius (Amsterdam, 1634): Text, Translation and Commentary, in: History of Universities 21(2006), 71-150

00-Feingold-Prelims.qxd 28/3/06 02:55 PM Page iii History of Universities VO L U M E X X I / 1 2006 1 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 71 The Oration on the Dignity and the Usefulness of the Mathematical Sciences of Martinus Hortensius (Amsterdam, 1634): Text, Translation and Commentary Annette Imhausen and Volker R. Remmert Introduction While discussing mathematics and philosophy in Proclus’s Commentary on Book I of Euclid’s Elements Ian Mueller observed that ‘Plato lived at a time when mathematical knowledge was expanding rapidly, and technical advance mingled with philosophical speculation to create a sense of unlimited possibility. Not until the early modern period, when mathematics again enters a period of rapid expansion, do we find as convincing a proclamation of the broad powers of mathematical science as we find in Plato’s Republic’.1 Although Mueller’s claim may be too strong, it is indeed striking that during the early modern period many proclamations were made, more or less convincingly, of the power of the mathematical sciences. An excellent example is Martinus Hortensius’s Oration on the Dignitiy and the Usefulness of the Mathematical Sciences (Oratio de dignitate et utilitate Matheseos). It provides an overview of an elaborate array of arguments for the power of the mathematical sciences, and its references range from the classical Greek tradition to contemporaneous developments. As such, it is indicative of a discipline, or rather a group of related disciplines, in search of a new position and an enhanced status within university systems and the hierarchy of the sciences. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 72 72 History of Universities The Mathematical Sciences in the Sixteenth and Seventeenth Centuries It had long been taken for granted that the mathematical sciences of the seventeenth century were at the core of what is commonly called the Scientific Revolution. They are no longer, however, thought of as the paramount factor in that period, as recent historiography, in step with current trends in the area of the sciences, has somewhat shifted focus. Much light has been shed of late on the important roles that other dis- ciplines, such as natural history and biology, played in the great upheaval which, in a historiography preoccupied with European pre-eminence, has long stood uncontested as the founding myth of a world characterised by ongoing and accelerating processes of scientification. Nonetheless, the mathematical sciences are of particular interest if the historical development of the system of scientific disciplines that dominated much of nineteenth- and twentieth-century science and society, during which the mathematical approach prevailed, is to be understood. Physics, having taken the role of leader in the hierarchy of scientific disciplines (a ‘Leitwissenschaft’ as Norbert Elias described it), stands as an emblem of this process. But in the early seventeenth century the struggle for supremacy in the realm of knowledge was wide open. During the Middle Ages and up to the late sixteenth century, the mathematical sciences were subordinate to theology, philosophy and, particularly, natural philosophy. Even though the mathematical sciences then began to challenge the primacy of philosophy and theology, the regal insignia in the realm of academic disciplines had not yet been passed over to the mathematical sciences. During the seventeenth century, however, the picture changed: Cinderella became mathesis Regia, the Royal Mathematical Sciences, as the Jesuit Claude François Milliet Dechales proudly declared in the dedicatory letter of his Cursus seu Mundus Mathematicus (Lyon, 1676).2 The mathematical sciences started to play a leading role in the hierarchy of scientific disciplines, and modes of explanation informed by them increasingly dominated many branches of the sciences and segments of society. In early modern Europe, the term mathematical sciences was used to describe those fields of knowledge that depended on measure, number and weight, reflecting the much quoted passage from The Wisdom of Solomon 11, 20: ‘but thou hast ordered all things in measure and number 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 73 Hortensius’ Oration on the Dignity of Mathematics 73 and weight’. This included astrology and architecture, as well as arithmetic and astronomy. The scientiae, or disciplinae mathematicae, were gen- erally subdivided into mathematicae purae, dealing with quantity, continuous and discrete as in geometry and arithmetic, and mathematicae mixtae or mediae, which dealt not only with quantity but also with quality: for example, astronomy, geography, optics, music, cosmography and architecture. The Jesuit Gaspar Schott even enumerated more than twenty fields among the mathematicae mixtae in his Cursus mathematicus of 1661 (Schott (1661)). It has been suggested that the term mathematicae mixtae came into use around 1600,3 but in fact, it was commonly used during the whole sixteenth century; Marsilio Ficino (1433–1499) had already distinguished between two grades (gradus) of mathematics, puri (arithmetic and geometry) and mixti (music, astronomy and stereometry) in his commentary on book VII of Plato’s Republic.4 The frequent analogy between mixed mathematics and modern applied mathematics is a misconception, because applied mathematics, like pure mathematics, is a subdivision of the modern scientific discipline of ‘mathematics’, which did not exist in its own right as a discipline around 1600. The mathematical sciences then, consisted of various fields of knowledge, often with a strong bent toward practical applications, and these only became independent as disciplines between the late seventeenth and early nineteenth centuries. One of the important preconditions of this process of the formation of scientific disciplines, and of the Scientific Revolution itself, was the rapidly changing social and epistemological status of the mathematical sciences as a whole from the mid-sixteenth through to the seventeenth century. The foundations of the social and epistemological legitimiza- tion of the mathematical sciences began to be laid by the work of mathematicians and other scientists from the beginning of this period. Justification of their activities was bipolar: since the late sixteenth- century debate about the certainty of mathematics, the quaestio de certitudine mathematicarum,5 the mathematicae purae were taken to guarantee the absolute certainty, and therefore the intrinsic worth, of knowledge produced in all the mathematical sciences, pure and mixed, while, on the other hand, the mathematicae mixtae proved the utility of this unerring knowledge.6 In this context it is important to keep in mind the conceptual inconsistencies in the use of the terms—(1) mathematicus, signifying either the activities of a (pure) mathematician or those of a practitioner of the mathematical sciences performing 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 74 74 History of Universities (mixed) mathematics; (2) mathematica, normally used as an adjective and only rarely, but confusingly, employed as a noun meaning pure mathematics; and (3) mathesis or mathematicae, instead of scientiae or disciplinae mathematicae, denoting the whole ensemble of the mathe- matical sciences. This inconsistency often makes it, and made it, difficult to distinguish between the two branches of the mathematical sciences under discussion (mathematicae purae or mixtae), and this was readily exploited at the time to illuminate or promote both their intrinsic value and the advantages of their practical utility.7 Praising the Mathematical Sciences in the Sixteenth and Seventeenth Centuries In the seventeenth century, efforts to legitimize the mathematical sciences were being actively driven forward by mathematicians who tried to move the mathematical sciences out of their seclusion through the use of various deliberate strategies (not all of which have yet been researched and understood). These strategies usually involved the use of print media in one way or another—mathematical textbooks, practical manuals, books of mathematical entertainments, editions of the classics, encyclopaedic works, and also inaugural speeches or other orations on the mathematical sciences.8 Of these, inaugural speeches were particularly important, as they were presented publicly, usually in universities (or comparable teaching institutions). They were there- fore addressed to mixed audiences of academic and non-academic, wealthy and noble, and young and mature listeners. Their goal was evident: to propagate and establish the relevance of the mathematical sciences. From the mid-sixteenth century on, they developed into a genre of their own, and by the early seventeenth century it had become common practice to praise and promote the mathematical sciences in inaugural lectures, quite a few of which were sent to the press by their authors.9 It seems that in the early seventeenth century, there was a real market for orations in praise of the mathematical sciences, and some publishers even looked to printing speeches from the sixteenth century. Thus, Tycho Brahe’s inaugural Copenhagen lecture of 1574 (De disciplinis mathematicis Oratio), first printed in 1610, was reprinted in 1621. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 75 Hortensius’ Oration on the Dignity of Mathematics 75 During the fifteenth century, three aspects of the mathematical sciences were usually singled out as praiseworthy: their propaedeutic value for the study of philosophy, their practical advantage for the community and—in humanist vein—their antiquity.10 It has been shown that during the sixteenth century, the arguments used in orations and prefaces became fairly standardized, and drew on a common basis of argumentation (the practical and propaedeutic role of the mathematical sciences) and examples (Archimedes’s burning mirrors being among the most popular). The educational value of the mathematical sciences, to which their epistemological status was closely related, was usually seen to be in their importance for training the mind and in their recreational potential, but not often in their necessity or worth for other disciplines, such as philosophy, medicine, law or theology.11 This situation gradually changed until, in the first half of the seventeenth century, mathemat- icians, emphasizing the absolute certainty of mathematical knowledge which had been so hotly debated in quaestio de certitudine mathemati- carum, boldly declared that the mathematical sciences deserved a new position in the modified hierarchy of scientific disciplines.12 Hortensius’s Oration on the Dignitiy and the Usefulness of the Mathematical Sciences reflects most of the strategies that were usually employed in the process of legitimization sketched above. Addressing a broad gamut of listeners—young students as well as mature merchants— Hortensius employed the whole range of standard arguments in praise of the mathematical sciences, covering biblical times and Greek antiquity, as well as the then most recent developments in astronomy, such as Galileo’s astronomical observations. Maarten van den Hove/Martinus Hortensius (1605–1639) Martinus Hortensius was born Maarten van den Hove in Delft in 1605.13 He was a student in the Latin school at Rotterdam, where he probably came under the influence of the natural philosopher Isaac Beeckmann. In 1625, he went to Leiden, but it was only in March 1628 that he registered as a student in the prestigious University of Leiden, where the well-known mathematician Willebrord Snel (1580–1626) taught from 1613 until his early death. It was probably under Snel’s guidance that Hortensius turned to the mathematical sciences and made astronomical observations in 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 76 76 History of Universities Leiden. After Snel died, Hortensius completed one of Snel’s books and translated it into Latin (Snel (1627)). Hortensius then came into contact with the reformed minister, physician, astronomer, and ardent propagator of the Copernican system, Philipp Lansbergen (1561–1632), with whom he closely collaborated, editing and translating some of Lansbergen’s works (Lansbergen (1630); Hortensius (1630), (1632)). In 1633, Hortensius published a small tract on the transit of Mercury of 1631 (Hortensius (1633)), which drew the attention of prestigious astronomers around Europe and which Pierre Gassendi treated in his Mercurius in Sole Visus et Venus Invisa (Gassendi (1632)). Hortensius endorsed Gassendi’s presentation of the measurements of the planets and fixed stars. On the basis of these and his own observations, he put forward his own table of apparent and actual planetary sizes, which was the first such table based on telescopic observations and which remained the only one of its type for almost twenty years.14 In the same year, Hortensius moved from Leiden to Amsterdam, hoping to get a position at the city’s recently established Athenaeum illustre. Several of these ‘illustrious schools’ had been founded all over the Dutch Republic in the 1630s in order to prepare students for the universities or even to compete with them (Deventer, Amsterdam, and Utrecht). Of these, only the Amsterdam Athenaeum illustre rose to a prominent position because the founding fathers used the immense wealth of the city of Amsterdam to lure professors away from Leiden with the promise of high pay. From 1632, Caspar Barlaeus (1584–1648) and Gerard Joannes Vossius (1577–1649) taught at the Athenaeum illustre, the former delivering an inaugural lecture on The Wise Merchant (Barlaeus (1632)), flattering the city fathers for their decision to establish the illustrious school.15 Barlaeus took a hand in recommending Hortensius to the authorities to teach mathematical sciences, and in particular navigation and astronomy, at the Athenaeum illustre. Hortensius began teaching there in May 1634, after first delivering his inaugural lecture, the Oration on the Dignitiy and the Usefulness of the Mathematical Sciences (Hortensius (1634)). If we are to believe his own testimony, his daily lecture courses were a success, attracting quite a number of listeners.16 The university authorities hired him as full professor in early 1635, and that summer he lectured on optics, again after delivering a formal inaugural lecture in July (Hortensius (1635)). But in this same year of 1635 he was complaining about a lack of students, which Vossius blamed on Hortensius’s frequent periods of absence travelling to Delft, The Hague, and Leiden. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 77 Hortensius’ Oration on the Dignity of Mathematics 77 In the years to follow, Hortensius’s reputation continued to grow. He was known to be an able astronomer and a convinced Copernican, as well as an admirer of Galileo. In the summer of 1634, he had already secured for himself a copy of Galileo’s 1632 Dialogue Concerning the Two Chief World Systems.17 Among his correspondents were distinguished scholars and gentlemen, such as Fabri de Pereisc, Galileo, Gassendi, Grotius, Constantin Huygens, Mersenne, and Schickard. Much of his energy between 1635 and 1639 was absorbed by an unfulfilled plan to bring his hero Galileo to the Dutch Republic. This project was not only intended as a humanitarian gesture, but also entailed high hopes of obtaining Galileo’s method of determining longitude at sea by means of the moons of Jupiter for the Dutch Republic. Apparently, Hortensius received a considerable sum of money to go to Italy and negotiate the arrangement, but he never went and later was accused of having embezzled the money.18 At the height of his fame, Hortensius received a professorship in Leiden, but he died shortly after moving there in August 1639. Although he did not count amongst the great luminaries of seventeenth-century science, and Descartes even considered him ‘very ignorant’,19 his appointment at Leiden shows that he was highly esteemed in the Dutch republic of letters. In his Oration on the Dignitiy and the Usefulness of the Mathematical Sciences, as well as in his other writings (particularly in the Canto on the Origin and Progress of Astronomy (Hortensius (1632)), Hortensius showed himself well-versed not only in astronomy and the mathematical sciences, but also in classical writings and traditions, an important achievement in an academic world still driven, to a considerable extent, by humanistic impulses.20 The Oration on the Dignitiy and the Usefulness of the Mathematical Sciences is imbued with allusions to and quotations from the classical authors, so that it demonstrates not only the dignity and the pract- ical advantages of the mathematical sciences, but also their antiquity. These aspects together made a convincing case for both the mathematical sciences and their representative, Hortensius (seeking a permanent academic position), in the prosperous city of Amsterdam in the Dutch Golden Age. The sources used by Hortensius As mentioned above, the Oration on the Dignitiy and the Usefulness of the Mathematical Sciences abounds with allusions to, quotations from, 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 78 78 History of Universities and paraphrases of, classical texts. While, in conformity with contem- poraneous practice, Hortensius rarely divulges his sources, it seems clear that he frequently alludes to Proclus’s Commentary on the First Book of Euclid’s Elements (Barozzi (1560); Proclus (1992)) and to the writings of Christopher Clavius, which Hortensius highly recommends in his Dissertatio de studio mathematico recte instituendo (Hortensius (1637)). Where we have been able to identify the exact sources Hortensius used or copied, we have supplied that information in the footnotes (as, for example, in the cases of Polybius (XVI.8f), Martianus Capella (XIX.5–9) or the Bible). In the many cases where he paraphrases texts or draws on then well-known stories, our references are to the probable ancient sources, for example, Diogenes Laertius or Plutarch (cf. Bibliography II). Note on text and translation The original text is divided into 27 paragraphs, which we have identi- fied with Roman numerals for easy reference. Within these paragraphs, the Latin sentences have been itemized with Arabic numeral suffixes, so that, for example, XV.(2) refers to the second sentence in paragraph XV. For cross-reference with the original of 1634, its page numbers have been included in square brackets, for example, thus: [13]. Every translator of an ancient language has to face the decision whether to translate literally, close to the original text, or, less literally, in a way that will be more accessible to the modern reader. As historians of mathematics, we have chosen the latter, but we are well aware of the pitfalls of dealing with the subtleties of the original—traduttore, traditore. Although neither of us is a native English speaker, our aim has been to render Hortensius’s speech in readable English and in that we are indebted to Jackie Stedall for her help in making the language of the translation flow. Acknowledgements Work on this project began at the Dibner Institute for the History of Science and Technology where we held fellowships in the academic year 2001/2002. We are grateful to the institute and its staff, especially Judith 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 79 Hortensius’ Oration on the Dignity of Mathematics 79 Nelson, for their support. The first draft of the translation has been gone through and corrected several times since, and it hugely benefited from the observations and suggestions of various colleagues and friends. We would like to thank Jochen Althoff, Colin Austin, Klaas van Berkel, Renate Emerenziani, Mordechai Feingold, Ben Kern, Eleanor Robson, David E. Rowe, Christine Salazar, Jackie Stedall, Anja Wolkenhauer, Liesbeth de Wreede and an anonymous referee for their kind support and comments. AG Geschichte der Mathematik und der Naturwissenschaften Institut für Mathematik FB08—Physik, Mathematik und Informatik Mainz University D-55099 Mainz Germany Department of History and Philosophy of Science Cambridge University Cambridge CB2 3RH United Kingdom REFERENCES 1. Cit. Mueller (1987), 307f. 2. Milliet Dechales (1676): ‘Plebeiae sunt ceterae disciplinae, mathesis Regia’. 3. See Brown (1991), 81. 4. See Ficino (1561), ii., 1411; cf. Remmert (1998), 79–83 and the discussion of scientia media and mathematica media in the Middle Ages in Gagné (1969), 984; Mandosio (1994); Olivieri (1995), 66–71; on the arts in general, see Kristeller 1951–2. 5. We do not want to give an account of this rather extensive and highly important debate, but just to state the main result accepted by most mathematical practitioners by the beginning of the seventeenth century: if mathematical proofs were not the most powerful within the ideal scientific hierarchy of early modern Aristotelians (demonstrationes potissimae), they still guaranteed the highest degree of certainty attainable by humans (demonstrationes certissimae). This perception was central to the revalua- tion of epistemological categories and deliberately ignored and undermined the Aristotelian hierarchy of the scientific disciplines. Cf. Dear (1995), 34–42; Feldhay (1998), 83–100; Jardine (1988); Mancosu (1992, 1996); Remmert (1998), 83–90; Romano (1999), 153–162. 6. See, e.g., Bennett (1991). 7. On this, see Remmert (1998), 79–90; cf. the remark of Peter Damerow that up to the early eighteenth century ‘mathematics as a discipline did exist, the mathematician specialized in mathematics did not’: Damerow (1996), 128. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 80 80 History of Universities 8. Dear (1995); Mancosu (1996); Remmert (1998); cf. also Biagioli (1993); on visual strategies of legitimization, see Remmert (2006). 9. For a selection of these and related pieces see our bibliography IV; cf. the discussion in Remmert (1998), 152–165; Schüling (1969), 37–9; Swerdlow (1993). 10. On the praise of the mathematical sciences by Alberti, Pacioli, Regiomontanus, and others see Høyrup (1992), 87–90; Swerdlow (1993); on the importance of humanism the locus classicus is Rose (1975). 11. On this see Hooykaas (1958), 82–4; Jardine (1984), 263f; Keller (1985), 354–61; Rose (1975). 12. See Remmert (1998), 152–4. 13. On Hortensius and his publications, see our bibliography, I.1, I.2 and III.1. On the history of Dutch science in this period, see Berkel, Helden, and Palm (1999), 13–67; Davids (1986), (2001); Hooykaas (1976); Struik (1981); Vermij (1993); Vermij (2002), 126–9; for a more general discussion cf. the chapter Intellectual life, 1572–1650 in Israel (1995), 565–91; North (1997). 14. On these achievements of Hortensius, see Helden (1985), 101–104 and 120f. 15. A French translation of Barlaeus speech, a model of its kind, is given in Secretan (2002); on Barlaeus, see Secretan (2002); on Vossius Blok (2000), 13–17; Rademaker (1981); cf. Burke (1994), 91f; Israel (1995), 773f. 16. He mentioned this in a letter to Pierre Gassendi in June 1634. Gassendi (1658), vi., 422f: ‘Nunc quotidie doceo elementa Astronomica in satis magno Auditorum numero’. Cf. Berkel (1997), 209. 17. Galilei (1890–1909), XX, 579f. 18. On this see Blok (2000), 161–3; Rademaker (1981), 247–50; Waard (1911), 1163. 19. Descartes to Mersenne, March 31, 1638: ‘il est tres ignorant’, quoted from Berkel (1997), 219. 20. The literature on this is vast; for the case of the mathematical sciences, see Rose (1975); Swerdlow (1993); de Wreede (2002); (2006). On the context of Hortensius’s Canto that he dedicated to Lansbergen, see Remmert (2006), chapter 6.2. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 81 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 82 82 History of Universities 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 83 Hortensius’ Oration on the Dignity of Mathematics 83 [1] Oration of Martinus Hortensius on the dignity and the usefulness of the mathematical sciences, delivered in the famous Gymnasium1 of the Senate and the people of Amsterdam, when, by the authority of the honourable Councillors and Senators of this City, he began to lecture on the mathematical sciences, on May 8 1634 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 84 84 History of Universities [2] Magnificis, Amplissimis, Prudentissimisque V.V. ac D.D. IOHANNI GROTENHVIS I.C. Inclytae civitatis Amstelodamensis Praetori, ANDREAE BICKER I.V.D. THEODORO BAS Equiti IOHANNI GEELVINCK, IACOBO BACKER, Augustae ejus Vrbis CONSVLIBVS, Nec non ejusdem Reip. SCABINIS, SENATORIBVS, & illustr. Scholae CVRATORIBUS, O R AT I O N E M hanc officiosè dedico, humillimè offero, MARTINVS HORTENSIVS. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 85 Hortensius’ Oration on the Dignity of Mathematics 85 [2] To the sublime, most eminent and most prudent gentlemen Jan ten Grootenhuys, lawyer.2 The burgomasters of the widely famous Town of Amsterdam, Andries Bicker, doctor of canon and civil law, Dirk Bas, knight, Jan Geelvinck, Jacob Backer.3 Councillors of this venerable Town, And also to the Jurors, Senators and the Curators of this famous School, I obligingly dedicate and humbly offer this speech, Martinus Hortenius 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 86 86 History of Universities [3] M A RT I N I H O RT E N S I I O R AT I O De dignitate & utilitate Matheseos. Amplme D. PRAETOR, Magnifici spectatissque CONSVLES, prudentmi SCABINI, SENATORES gravmi. Curatores dignmi, Clarissmi Professores, Pastores Ecclesiarum vigilantmi, Doctores, Magistri, Scholarum Rectores, Mercatores humanmi, ornatissima studiosorum Iuvenum corona. I. (1) Quod maximis aliquando viris contigisse nostis, ut nempe quum in publico pararent dicere, timerent: id si & mihi hodie evenire affirmavero, non utique credo mirum vobis videbitur aut novum. (2) Quoties enim oculos conjicio in frequentissimum hunc Procerum atque eruditorum virorum confessum; quoties animum intendo in augustae Vrbis famam, cujus magnitudinem ipsa terra jam non capit: toties splendor vester & propriae tenuitatis conscientia me terret, ne aut vobis injurius sim, exspectationi vestrae non satisfaciendo; aut mihi, nec pro dignitate hujus loci, nec pro argumenti amplitudine accuratè satis disserendo. (3) M. Tullium illum Romanae eloquentiae patrem, virum luci & publico assuetum, nunquam sine metu ad dicendum venisse accepimus: quid mihi futurum censeam, cui privato hactenus & publicarum actionum insueto, derepente in tam illustri Auditorum corona verba facienda sunt, ex ea cathedra, quam viri summi & bina eruditionis lumina sic illustrant, ut tenuis nostrae lampadis lucula ad eorum radios facilè evanescat? (4) Accedit aetas, & quam lubenter agnosco curta doctrinae supellex: quae vel sola potuisset ab incepto deterrere; nisi & de aequitate vestra fuissem quodammodo certus, & parendum habuissem imperio Majorum, quibus refragari neque licitum duxi, neque honestum. (5) Istâ factum est ut audacior, hoc ut securior ad dicendum accesserim: quippe faciliùs sciebam audiri quod cum animis audientium conspirat, & tutiùs dici quod publicâ autoritate communitur. (6) Non diu est quod Magnifici DD. CONSVLES, consensu Amplissimi SENATUS inclytae hujus Reipublicae, inter medias civitatis turbas & operosa mercantium negotia, Phoebo Musisque excitarunt ac consecrarunt hanc quam videtis Palladis arcem & Palaestram bonae mentis. (7) In qua civium liberi intra paterna moenia, doctrinae ac sapientiae praeceptis instruerentur; & ipsi docti 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 87 Hortensius’ Oration on the Dignity of Mathematics 87 [3] Oration of Martinus Hortensius on the Dignity and the usefulness of the mathematical sciences Eminent Mayor, honourable and most respected members of the city council, prudent Jurors, venerable Senators, most worthy curators, most famous professors, most watchful ministers of the churches, Doctors, Masters, schoolmasters, most learned merchants, most distinguished band of young students. I. (1) What you know to have at times happened to the greatest men, namely, that they are afraid when they are preparing to speak in public, if I shall attest that this is happening today to me also, it will not, I think, seem strange or new to you. (2) For as often as I gaze at this audience, so crowded with noble and learned men, as often as I turn my mind to the fame of the venerable city [Amsterdam], whose greatness the earth itself does no longer hold, so often your splendour and the consciousness of my own unimportance make me fear that I shall be unjust, either to you by not satisfying your expectation, or to myself by not discoursing carefully enough with respect to the dignity of this place or the richness of the subject. (3) We know that M. Tullius [Cicero], that father of Roman eloquence, a man accustomed to public scrutiny and publicity, never came forward to speak without apprehension;4 what might I think will happen to me, a private citizen up to this time and unaccustomed to public deeds, now that I must deliver an oration, without preparation, to so noble a band of listeners, from that very seat which very great men and two luminaries of erudition so adorn5 that the faint glimmer of my lantern easily fades out in their rays? (4) Add my age, and, as I willingly acknowledge, my inadequate stock of learning; this indeed by itself might frighten me away at the start, if I were not somehow sure of your fairness; and also I had to obey the order of the elders, which I considered neither lawful nor honourable to disregard. (5) It happens that by the former I come to speak more boldly, by the latter more securely; since I knew that what accords with the minds of the audience would more easily be heard, and what is strengthened by public authority would more safely be spoken. (6) It is not long since the members of the city council, with the assent of the eminent Senate of this glorious Republic, in the middle of the city’s turmoil and laborious business dealings, raised up and consecrated to Phoebus Apollo6 and the Muses this stronghold of Pallas Athena7 and gymnasium of sound thinking that you see, and consecrated it to them. (7) May the children of the citizens, within the city walls of their fathers, learn in it 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 88 88 History of Universities invenirent, quo animum toedio [sic] negotiorum fractum subinde reficerent. Laudabili profectò ac sapientissimo instituto. (8) Quo postquam Vrbem suam magnitudine, opibus, potentiâ, Mercatorum [4] frequentiâ, aedium publicarum privatarumque splendore, maximis Europae urbibus aut parem esse viderunt, aut superiorem: mentis quoque culturâ, & literarum ac doctrinae mercatu, non tulerunt eam ab aliis superari; aut ullatenus esse inferiorem. (9) Qua quidem in re quantum sibi gloriae, literatis ac studiosae juventuti utilitatis ac delectationis paraverint; indigenarum publicae loquuntur voces, exterorum ostendunt judicia, neque à me pluribus opus est confirmari. (10) Illud potius dicendum propter quod praecipuè hanc sedem conscendi. (11) Nimirum iidem Amplissimi ac Spectatissimi DD. CONSVLES ac SENATORES uti & nunc sunt heroicâ prudentiâ & generoso ad promovendas bonas artes animo conspicui; Mathematicas quoque Scientias doceri hîc voluerunt: cùm ut juventus earum cognitionem hauriat juxta studium Philosophiae ac literarum; tum quoque ut satisfiat non paucis Vrbis incolis, qui assidua sua & jam penè improba vota dudum ad hunc eventum direxere. (12) Quam quidem provinciam nobis demandandam, & his humeris quodcunque id est oneris imponendum censuerunt; non ex quadam singularis nostrae scientiae persuasione, sed proprio gratiosi affectus impulsu: qui etiam, ut verum fatear, ad ingrediendum hoc iter haud minimos mihi addidit stimulos. (13) Quoniam verò ex usitato Scholarum more nonnulla dicenda video, quibus instituti mei reddam rationem; decrevi inpraesentiarum non aliud pertractare argumentum, quam quod ipsam vobis depingat Mathesin. (14) Ab hujus objecto & denominatione incipiam: inde per varias partes decurrens, ostendam dignitate eam inter alias scientias eminere, & summam cum dignitate habere utilitatem. (15) Favete modo animis, & conatus nostros benignis votis prosequimini; ut quae à me exspectare vos sentio, iis excipiantur auribus, quales adfuturas ob eximiam vestram benevolentiam totus mihi persuadeo. II. (1) Philosophiae ea pars quae Contemplativa dicitur, sic comparata est, ut circà res necessarias occupata, non alium sibi praefixum habeat scopum, quàm ipsarum rerum veritatem. (2) Vbi eam novit ac comprehendit, mentem humanam ulterius non perducit; sed finem propositum assecuta, subsistit solius scientiae terminis contenta. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 89 Hortensius’ Oration on the Dignity of Mathematics 89 the precepts of knowledge and wisdom, and may they themselves, when taught, find in it a place where they may, from time to time, refresh their minds worn out by the tedium of business affairs. Certainly, a praiseworthy and most wise purpose. (8) By this they have since seen their own city to be equal or superior in size, wealth, power, large number of merchants [4], splendour of public and private buildings to the greatest cities of Europe; also in cultivation of mind, and commerce of letters, and learning they have not allowed her to be surpassed by others, or to be inferior in any way. (9) In this matter, indeed, the voices of inhabitants tell, the judgments of foreigners show how much glory for themselves, and utility and entertainment for the educated and studious youth they shall have provided, such as there is no need for me to confirm it with many words. (10) Rather, I ought to speak about the matter for which I have come up to this seat. (11) It is no wonder that these same most reverend and glorious members of the city council and Senate practiced even now a heroic wisdom and noble intent to promote the fine arts; they wanted the mathematical sciences to be taught here also, not only so that the youth might soak up this branch of knowledge along with the study of philos- ophy and letters, but also that they might satisfy not a few inhabitants of the city, who have directed their assiduous and now hardly presumptuous wishes for a long time to this end. (12) They have thought this duty ought indeed be demanded of me and whatever labour it entails imposed on these shoulders, not from some conviction of my outstanding learning but from the effect of their own gratiousness, which indeed, to be quite honest, was not the least stimulus to go along this road. (13) Since indeed, in the usual custom of scholars, I see a few things that ought to be said, in which I may give an account of my undertaking, I have decided at the present moment to treat at length for you no other subject than that which repres- ents the mathematical sciences themselves. (14) I shall begin with their objective and name; passing from there through their various parts, I shall show that they surpass other sciences in dignity, and that, along with dignity, they hold the greatest usefulness. (15) Please be well disposed in your minds and follow upon my attempts with your good wishes, so that what I feel you expect of me may be received by such ears, as I wholly persuade myself will be present [now], on account of your exceeding goodwill. II. (1) The part of philosophy that is called contemplative is so disposed that, so far as it is concerned with necessary matters, it has no other aim (set up) for itself than the truth of those matters. (2) Where it renews and under- stands the truth, it does not lead the human mind any further; but, having attained its proposed end, it rests content with the boundaries of science only. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 90 90 History of Universities (3) Rerum genera varia sunt, & sub triplici considerationis ordine, tanquam objecta cuique propria mentis oculis aprehensa, tres constituunt contemplativae Philosophiae partes, Metaphysicam, Physicam, & Mathematicam. (4) Objectum Metaphysicae sunt res seu entia, tam re quàm ratione abstracta à materia & omni ejus motu. (5) Objectum Physicae & re & ratione conjunctum est cum materia & ejus motu, utpote corpus naturale. (6) Objectum verò Mathematicae re innititur materiae & ejus conditionibus, sed ratione ab omni materia abstrahitur: estque Quantitas, quae mente concipitur ac definitur, etsi nunquam citra aliquod subjectum subsistat, aut substantiae non inhaereat. (7) Quamobrem Mathematica media habenda est inter Metaphysicam, quae mentem à sensibilibus rebus in summa simplicitate abstrahit; & Physicam, quae materiales qualitates considerat, & res sensibus ut plurimum subjectas. (8) Metaphysicae enim vicina est, cum nudam quantitatem ejusque affectiones varias contemplatur; Physicae, quando exercetur in rebus materiatis quantitati subjectis. III. (1) Vnde autem haec Scientia dicta sit  , hoc est, disciplina, invenio [5] inter autores non convenire. (2) Proclus Geometra solertissimus, commentariis in primum librum Euclidis censet à Pythagoraeis nomen Matheseos exortum, argumento ’  d , recordationis, quod omnis quae dicitur disciplina, recordatio sit, sed praecipuè ea quae Mathesis appellatur, quod sit aeternarum cogitationum in animo recordatio, mentemque dirigat ad impressas quasi à Deo rerum formas recolendas. (3) Alii è Philosophorum arbitrio profectum putant: sive quod illis seculis Mathematicae ante alias pueris tradi solebant, & sic primae quasi erant disciplinae, quibus perceptis transibant ad altiores Physicam & Ethicam: sive ob subtilitatem & acumen rerum quas tractant, quo prae caeteris diligentiam & laborem in addiscendo exigunt, & vix absque praeceptoris opera percipiuntur. Neque id sine ratione. (4) Nam etsi per Md omnes disciplinae intelligantur, credibile tamen est has solas hoc nomine dignas aestimatas, eò quod certitudine singulari & invicto demonstrationis ordine discentium animos confirment. (5) Et hanc ob causam quicunque se huic studio penitus dederant, prisco aevo soli Mathematici dicti sunt, aestimatique prae caeteris Philosophis certiorem elegisse Philosophiae partem; cùm quae de causa prima & Deo inter eos 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 91 Hortensius’ Oration on the Dignity of Mathematics 91 (3) There are various kinds of things, and under a three-fold order of consideration, just as to him who having properly apprehended objects in the mind’s eye there are three parts of contemplative philosophy, meta- physics, physics and mathematics. (4) The subjects of metaphysics are things or beings, both by their nature and by reason abstracted from matter and all its motions. (5) The subjects of physics are by their nature and by reason joined to matter and its motion, namely the natural body. (6) Now, the subject of mathematics rests upon the nature of matter and its conditions, but by reason it is abstracted from all matter; and it is quantity, which is mentally conceived and defined, although it may never exist apart from some subject, or does not inhere to substance. (7) Therefore mathematics ought to be considered the mean between metaphysics, which draws the mind from perceptible things to the greatest simplicity, and physics, which considers material qualities and things more subjected to perception. (8) For mathematics is close to metaphysics in that it examines pure quantity and its various effects and influences; close to physics when it is occupied with material things subject to quantity. III. (1) Why indeed this science comes to be called Mathesis, that is, a discipline [disciplina], I do not find [5] authors to agree upon. (2) Proclus, a very skilled geometer, in his Commentary on the First Book of Euclid’s Elements, thinks the name Mathesis arose from the Pythagoreans, proving it by anamnesis, recollection, because everything that is called a discip- line becomes recollection, but especially that which is called Mathesis, because it becomes the recollection of eternal thoughts in the human mind, and directs the mind to recollecting the forms of things, impressed [as they were] by God.8 (3) Others of the philosophers think it arose arbitrarily, either because in those times the mathematical sciences were usually taught to boys before everything else, and thus they were the first instances of disciplines, and when they had been absorbed, they crossed over to the higher studies, physics and ethics; or because of the precision and sharpness of the objects which it treats, so that it demands diligence and labour in learning beyond others, and the work can scarcely be seized without a teacher. And this is not without reason. (4) For although through Mathemata all disciplines are known, nevertheless it is credible that mathematics alone is considered worthy of this name, on this account, that it strengthens the minds of students with a singular certainty and unconquerable order of demonstration. (5) And for this reason those who in former times have given themselves wholeheartedly to this study were alone called mathematicians, and were highly esteemed compared to the other philosophers as they had chosen the more secure 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 92 92 History of Universities erat, valde incerta esset & conjecturis plenissima; quae verò de rebus naturalibus, ob insignem naturae obscuritatem etiam quoad minimas suas partes infinitis hallucinationibus obnoxia: haec contra, immotis nixa principiis, nihil concluderet, quod ex ante notis & concessis non esset confirmatissimum: & quod scientiae maximè est proprium, semper sine confusione eodem modo se haberet ac percipi posset. (6) Magnum quoque & venerabile inter Philosophos Mathematicorum fuit nomen. (7) Quippe viri summi de gravissimis Philosophiae controversiis disputaturi ad illorum confugiebant demonstrationes; iisque assertionum suarum fundamenta studebant stabilire; non ignari, eas & solas & firmas esse Philosophiae ansas, quod olim dixit Xenocrates. IV. (1) Sunt autem Disciplinae Mathematicae aliae purae & propriè sic dictae, abstractae ab omni materia; aliae mixtae & aliquatenus Physicae, conjunctae cum materia & ejus motu. (2) Purae duae sunt Arithmetica & Geometria, pro duplici specie quantitatis, discretae & continuae, numeri & magnitudinis. (3) Mixtae veteribus totidem, nempe Musica, quae quasi Arithmetica quaedam est in sonis; & Astronomia quae Geometria est in materia mobili, puta caelo & sideribus eo contentis. (4) Ad has quatuor, omnes alias partes existimarunt illi posse reduci: quales sunt Geodaesia, Optica, Geographia, Mechanica. (5) Sed recentiores Mathematici, partes Matheseos mixtas constituunt sex; Musicam, Logisticam, Geodaesiam, Opticam, Mechanicam & Astronomiam. (6) quarum prior versatur circà harmonicas concentuum rationes & sensus adminiculo utitur in distinguendis sonorum intervallis: altera exercet praxin numerorum: tertia metitur agrorum superficies & solida quaevis corpora: quarta considerat proprietates lucis & umbrae, variasque radiorum in speculis & corporibus pellucidis reflexiones, refractionesque: quinta machinarum & organorum rationes, quibus stupendi eduntur effectus, describit & explicat: sexta & ultima caelestium corporum scrutatur motus, eorumque magnitudines tradit ac distantias. (7) Sic in universum octo essent Mathesis [6] partes: quanquam alii sex duntaxat admittere velint, Logisticam subjicientes Arithmeticae, & Geodaesiam Geometriae: à quorum sententia minimè essem alienus, nisi Staticam quae ponderum momenta explicat, & Architecturam militarem quam barbarè Fortificationem dicunt, quae munimentis & vallis exstruendis incumbit, censerem adjungendas. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 93 Hortensius’ Oration on the Dignity of Mathematics 93 part of philosophy; since amongst them [the philosophers] what concerned the first cause and God was surely uncertain and most full of conjecture; what concerned nature, liable to infinite delusions on account of the notable obscurity of nature as to its smallest parts; but mathematics, resting on unmovable principles, comes to no conclusion that is not fully confirmed by what has been noted and allowed before, and, what is espe- cially characteristic of science, can always be perceived by the same method without confusion. (6) Great, also, and venerable was the name of the mathematicians among the philosophers. (7) Indeed, the greatest men, when disputing about the weightiest controversies of philosophy, took refuge in mathematicians’ demonstrations, and were eager to ground the principles of their assertions on them, not unaware that as Xenocrates once said, these are the only firm handles of philosophy.9 IV. (1) Now, some mathematical sciences are called pure, and rightly so, if abstracted from all matter; others are called mixed and up to a point physical, if connected with matter and its motion. (2) The two pure mathematical sciences are arithmetic and geometry, according to the two types of quantity, discrete and continuous, number and size. (3) The ancients had the same number of mixed mathematical sciences, that is, music, which is almost an arithmetic of sound, and astronomy, which is a geometry of moving matter, namely the sky and the stars contained in it.10 (4) The ancients thought that all other parts can be reduced to these four: such are geodesy, optics, geography, and mechanics. (5) But more recent mathematicians brought together six mixed parts of the mathematical sciences: music, practical arithmetic,11 geodesy, optics, mechanics and astronomy.12 (6) The first of these is engaged in harmonic theory and uses perception as a tool to distinguish intervals of sound; the second exercises the practice of numbers in its procedures; the third measures the area of fields and any solid bodies; the fourth considers the properties of light and shadow, and various reflections and refractions of rays in mirrors and clear bodies; the fifth describes and explains the theory of machines and tools by which marvellous effects are produced; the sixth and last explores the motions of the heavenly bodies and devotes itself to their size and distance. (7) Thus there would be eight parts of the mathematical sciences in all [6], although others might wish to allow only six, subordinating practical arithmetic to arithmetic and geodesy to geometry. I would differ very little from their opinion, if I did not think that statics, which explains the movements of weights, and military architecture (vulgarly called fortification), which concentrates on ramparts and walls, ought to be added. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 94 94 History of Universities (8) etsi & hanc ad Geodaesiam aliquatenus revocari posse, illam verò etiam ad Mechanicam, non ignorem. (9) Interim cuilibet suum relinquentes judicium distinguemus inter partes Mathesis, dicemusque Theoreticas esse duas, Arithmeticam & Geometriam, Practicas verò pro subjectorum varietate in quibus occupantur decem, nempe Logisticam, Geodaesiam, Architecturam militarem, Mechanicam, Staticam, Musicam, Opticam, Astronomiam, Geographiam & Nauticam: quarum usum infra latiùs prosequemur. V. (1) Diximus ante Metaphysicam, Physicam, & Mathematicam partes esse Theoreticae Philosophiae; atque inter illas certitudine eminere Mathematicam. (2) Eam principiis niti firmissimis, & demonstrationum vi ita occupare discentium animos, ut in media luce fateantur se esse constitutos. (3) Quod sanè eximiam ei dignitatem conferre nemo potest diffiteri. (4) Scientiae dignitati convenientius nihil est, quàm ea sibi principia assumere, quae non tantum nota & intellectu priora sunt, sed & ne micam quidem ambiguitatis in se continent, aut ullis disputationibus queunt convelli. (5) Talia autem sunt principia Mathematicae: nobiscum nata, animis nostris ingenita, clara & aperta, ab ipsa natura expressa, & quae semel accepta cogant traditis assentiri absque ulla tergiversatione. (6) Atque hinc ea inter socias Philosophiae partes dignitatem suam tuetur ac servat illibatam. (7) Cui si adjungere lubeat veritatis splendorem qui ubique elucet, cum nihil probabile aut dubium admittit, sed ex certis & concessis omnia deducit; majorem etiam autoritatem sibi comparare deprehendetur. (8) Illa, illa Diva, mentis actionumque rectrix, cui quidquid meditamur, quidquid animis concipimus, inniti ac dicari debet; nusquam non augustae suae majestatis fulgorem per Mathematicum palatium diffundit. (9) Huic serviunt, huic litant, & quam dogmatibus suis quaerunt, ex semetipsis quoque depromunt. (10) Sic incedunt regiâ ad cognitionem rerum viâ, quâ nec planior ulla nec certior. (11) Cumque aliae scientiae quod incertitudine & conjecturis plenae sint, neque veritatem per se assequi valeant, neque falsorum quae continent medicinam ex se depromere; Mathesis sibi sufficit nullius indiga; solo naturae ductu contenta ipsam veritatem venatur & capit. (12) Digna ergo quae tantum super alias eminere scientias censeatur, quantò latiùs se diffundit, & objecta quaevis altiùs penetrat. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 95 Hortensius’ Oration on the Dignity of Mathematics 95 (8) I am well aware that the latter could in some way be brought under geodesy, the former, also under mechanics. (9) Meanwhile, leaving each to his own judgment, let us distinguish among the parts of the math- ematical sciences, and let us say that two are theoretical, arithmetic and geometry, and the practical are, by the variety of subjects with which they are occupied, ten: that is, practical arithmetic, geodesy, military architecture, mechanics, statics, music, optics, astronomy, geography and naval science. We will follow up on their practice more fully below. V. (1) We said earlier that metaphysics, physics and mathematics are parts of theoretical philosophy; and among these mathematics excels by its certainty;13 (2) and that it is grounded on the firmest principles and that by the strength of demonstrations it takes hold of the minds of the students that they are acknowledged to have been established in broad daylight. (3) No one can deny that mathematics is, indeed, of extraor- dinary dignity. (4) Nothing is more suited to the sublimity of a science than to take up those principles which are not only through intellect well-known and superior, but also contain in themselves not even the least bit of ambiguity and cannot be destroyed by any disputations. (5) Such indeed are the principles of mathematics, born with us, implanted in our minds, clear and easily understood, imprinted by nature herself and, once accepted, they compel agreement with the traditions without any hesitation. (6) And hence mathematics guards and preserves its dignity among the allied parts of philosophy. (7) If you wish, add the splendour of truth shining everywhere, as it allows nothing probable or doubtful but deduces everything from what is certain and conceded.14 You find that thereby it acquires an even greater authority for itself. (8) That Goddess [mathematics], guide of the mind and actions, whom we ought to rely on and obey, whatever we have in mind, whatever we conceive in our minds, never does she fail to diffuse the gleam of her noble majesty through the palace of mathematics. (9) Her they serve, her they pray to, and how much do they inquire into her doctrines, bringing them forth also from themselves. (10) Thus they walk on the royal road to the knowledge of things, the road more smooth and certain than any other. (11) Where other sciences, being full of uncertainty and conjecture, can neither reach the truth by themselves, nor produce a remedy for the falsities they contain, mathematics, lacking nothing, suffices unto itself; content with the guidance of nature only, it hunts and captures truth itself.15 (12) It is worthy, therefore, to be thought to predominate over other sciences insofar as it spreads itself more widely, and penetrates further into any subject. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 96 96 History of Universities (13) Ipsa nimirum animum contemplantis dulcissimo veritatis gustu satiat, judicium excitat, ratiocinationem quammaximè [sic] confirmat. (14) Ad splendorem ejus oculi mentis conversi omnia vident aperta & in clara luce posita. (15) Quod cernens Plato non dubitativit Mathesin αn  ` , viam ad eruditionem appellare: quod qui eam calluerit, non tantum sine ulla difficultate reliquas artes superare ac perdiscere possit; sed praecipuè argumentorum necessitati assuetas, nihil admittat quod vero non sit consentaneum; nullas praetendat autoritates cùm rationibus pugnandum est in rei demonstratione: quod verae eruditionis unicum est & proprium fundamentum. (16) Pythagoras [7] quoque discipulos suos ad Physicam & Politicam non ante admittebat, quàm doctrinae hoc genus α ` , continens institutiones Mathematicas, probè percepissent. (17) ineptos esse judicans ad rerum naturae contemplationem, aut civitatum & rerumpublicarum administrationem; qui non ante in pulvere Mathematico strenuè se exercuissent, mentemque haberent ejus Scientiae usu subactam ac confirmatam. (18) Talium virorum judicio, vim Mathematicarum disciplinarum quam in inquirenda veritate exerunt, tanquam in tabella habemus depictam. (19) Addamus ipsi turbam praestantissimorum Philosophorum, quibus ab omni aevo cordi fuerunt, & valorem sui probarunt: inveniemus inde à nata Philosophia nobilissima quaeque ingenia studiosè incubuisse in earum notitiam. (20) Mathemata ab Aegyptiis ad Graecos transtulit Thales Milesius. (21) Auxerunt Pythagoras, Plato, Eudoxus, Archytas, Xenocrates, Aristoteles, Euclides, Eratosthenes, Pappus, Theon, Proclus, viri ingentes & primarii humanae sapientiae antistites. (22) Partes singulares subtilissimis inventis ornarunt Apollonius, Hipparchus, Ptolemaeus, Geminus, Posidonius, Menelaus, Diophantus, divini artifices. (23) Apicem Scientiae attigit tot Scriptorum monumentis celebratus Archimedes Syracusanus, ubique mirandus. (24) Mitto alios minorum gentium Philosophos, quorum nomina duntaxat memorantur, aut pauciora fuere inventa, quàm ut inter aequales duxerint familiam: qui non minimo occurrunt numero, quibusque Mathesis digna semper visa est, in qua seriò exercerentur & bonam aetatis partem consumerent. (25) Quinimò si à splendore discentium quidquam accedere disciplinis autoritatis dicendum est, major adhuc dignitatis Mathematicae nota erit super purpuram conspici, & Principum ac Regum munificentiâ foveri; quod nec rarum est, nec novum. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 97 Hortensius’ Oration on the Dignity of Mathematics 97 (13) No wonder that it satisfies the mind of the observer with the sweetest taste of truth, that it inspires judgment, that it especially strengthens reasoning. (14) The eyes of a mind turned to its splendour see all things open and positioned in clear light. (15) Seeing this, Plato did not hesitate to call mathematics kata paideusin hodon, the way to learning;16 because one who is versed in her can not only conquer and learn thoroughly without any difficulty the rest of the arts, but especially, accustomed to the necessity for proofs, will admit nothing that does not agree with the truth and offer no authority as an excuse when the battle in the demonstration of a thing must be fought using reason; and this is the only proper foundation of true learning. [7] (16) Also Pythagoras did not admit his disciples to physics and politics before they had well under- stood how much doctrines of this educational ( paideutic) type of teaching depended on mathematical instructions. (17) He judged those to be unsuited for the contemplation of the nature of things, or the administration of the state and republic, who had not previously strenu- ously exercised themselves in the field of the mathematical sciences, and had their minds disciplined and strengthened in the use of these sciences. (18) In judging such men we have portrayed the force of the mathematical sciences, which they reveal in investigating the truth, as in a painting. (19) Let us add the gathering of most pre-eminent philosophers, who took these disciplines to heart in every age and proved its value; we shall find from the birth of philosophy that the most noble talents were carefully incubated in the knowledge of these disciplines. (20) Thales of Miletus brought mathematics from Egypt to Greece.17 (21) Pythagoras, Plato, Eudoxus, Archytas, Xenocrates, Aristotle, Euclid, Eratosthenes, Pappus, Theon, Proclus, great men and the first experts in human wisdom, added to the mathematical sciences. (22) Apollonius, Hipparchus, Ptolemy, Geminus, Posidonius, Menelaus, Diophantus, the divine masters, enriched special parts by most ingenious inventions. (23) The height of science was attained by Archimedes of Syracuse, everywhere admired, celebrated in so many monuments of writings. (24) I pass over other lesser philosophers, whose names are merely remembered or whose inventions were too few to consider them equal to the former. They occur in no small number and with them the worth of mathemat- ical sciences, in which they were trained seriously and which used up a good part of their lives, is always visible. (25) Why, indeed, if from the glory of their pupils anything must be said to add to the authority of their teachings, an even greater sign of the dignity of the mathemat- ical sciences is to shine above the emperorship, and to be cherished by the munificence of princes and kings; which is neither rare nor new. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 98 98 History of Universities (26) Euclides Geometra à Ptolemaeo Lagi filio primo Aegyptiorum rege, in Aegyptum evocatus, in honore habitus, & Rege familiariter fuit usus. (27) Eratosthenes Mathematum laude celeberrimus, & ob eruditionem minor Plato dictus, Regi item Aegyptio Ptolemaeo tertio charissimus fuit, & regiae bibliothecae ab eo praefectus. (28) Archimedes miraculosis machinationibus apud reges Siciliae Hieronem & Gelonem tantum sibi paravit gratiae & existimationis, ut de quacunque re dicenti credendum publicè jusserint. (29) Iulius Caesar rerum potitus & redactâ in provinciam Aegypto, studia Mathematica impensè coluit, & Sosigenem Astronomum assiduè in consortio secum habuit. (30) Et ne vetera tantum respiciam, invenerunt Mathemata fautores suos Carolum Magnum & Fridericum II Imperatores, Boëthium Romanorum Consulem, Alphonsum Castiliae regem, & Matthiam Hungariae; avorumque nostrorum memoriâ Imperatores item potentissimos Maximilianum & Carolum quintum; nostrâ, Fridericum II Daniae regem, & Mauritium Vraniae principem, qui tum foverunt Mathemata, tum & manibus suis tractarunt. (31) At nunquam ad tantum eminentiae gradum ascendere ea potuissent, nisi verè à Regibus & Principibus judicatum fuisset, dignissima esse quibus sublimes animae delectentur; & in quorum jugi tractatione curis politicis fatigatam mentem anxiâ sollicitudine quandoque resolvant. VI. (1) Accedat demum eximia voluptas quam secum adferunt, & vel ob hanc solam aestimari digna esse illicò patebit. (2) Quicquid expetimus, aut utilitatis aut [8] honestatis, aut denique jucunditatis gratiâ à nobis expeti, certum est. (3) Vtilitate Mathemata non carere mox ostendemus. (4) Inter jucunditates autem num major esse potest, quam Mathematica quae ipsam mentem afficit & intimos animi sensus plenissimo gaudio perfundit? (5) Historiarum cognitio & fabularum lectio occasionem praebet delectationis. (6) Politicae, Ethicae, Logicae studia, suas habent delitias. (7) Mathematicae verò voluptates tam sunt validae, tam acres, ut velut illicibus quibusdam ad se trahant, & summam in animis discentium excitent alacritatem. (8) Quam ob rem Mathemata Plato dixit Q   n α in nempe quae alliciant, quae impellant mentem hominis ad abstrusarum rerum inventionem, & inventarum jucunditate ad ulteriora semper perducant. (9) Talis fuit voluptas, cujus sensu affectus Thales cùm inscriptionem trianguli 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 99 Hortensius’ Oration on the Dignity of Mathematics 99 (26) The geometer Euclid was called into Egypt by Ptolemy I, son of Lagos, king of Egypt, was held in honour and was treated on a level of familiarity with the king. (27) Eratosthenes, the most celebrated of mathematicians, called from his learning a lesser Plato, was likewise most dear to the Egyptian king Ptolemy III, and was put in charge of the royal library by him. (28) Archimedes by his miraculous machines gained for himself so much favour and reputation among the kings of Sicily, Hieron and Gelon, that they ordered the public to believe whatever he said on any subject.18 (29) Julius Caesar, after he was in possession of power and Egypt had been made a province, greatly cultivated the study of mathematics, and kept Sosigenes the astronomer continually in company with him. (30) And lest I look back to the ancients only, the mathematical sciences have been cherished by the emperors Charles the Great and Frederick II, by Boethius, consul of the Romans, Alphonse, king of Castile, and Matthew of Hungary; in the memory of our grandfathers, likewise by the most powerful emperors Maximilian and Charles V; in our memory, by Frederick II of Denmark and Maurice Prince of Orange, who not only fostered the mathematical sciences but also practised them with their own hands.19 (31) But never could they have ascended to such a level of eminence, if they had not indeed been judged by kings and princes to be most worthy to delight noble minds, and now and then to refresh their mind worn out by anxious care under the yoke of politics. VI. (1) Finally there may be added the exceptional pleasure that the mathematical sciences bring with them, and indeed on account of this alone it will be obvious that in that matter they are to be esteemed worthy.20 (2) Whatever we seek from them is certain, [8] whether it is sought by us for practical advantage and usefulness or merit or finally pleasure. (3) Soon, we will show that the mathematical sciences do not lack practical advantage and usefulness. (4) Among pleasures, can any be greater than the mathematical sciences stimulating the mind itself and flooding the inmost feelings of the spirit with fullest joy? (5) The knowledge of histories and the reading of tales offer occasions of delight. (6) The study of politics, ethics, logic, all have their pleasures. (7) But the joys of the mathematical sciences are so strong, so keen, that they attract as though by something seductive and excite the highest rapture in the minds of their students. (8) For which reason, Plato said mathematics was helktika kai agoga,21 that is, that which draws, which impels the mind of man to the discovery of hidden things, and always, by the pleasure of discovery, leads further on. (9) Such was the pleasure that Thales felt when he discovered the inscription of an equilateral 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 100 100 History of Universities aequilateri in circulo invenisset, Musis bovem, Pythagoras verò repertâ ratione laterum trianguli rectanguli multò liberalior, hecatomben immolavit. (10) Talis laetitia quâ perfusus Archimedes cùm in balneo rationem deprehendendi furti in corona regis aurea commissi invenisset, exiliens & nudus domum properans, identidem per plateas ingeminavit ;  , ;  , inveni, inveni. (11) Tanta fuit animi securitas, quâ idem captâ patriâ, schematibus geometricis quasi ignarus malorum intentus, ab imperito milite caesus, finem vitae simul & studiorum fecit. (12) Quâ Claudius Ptolemaeus Astronomorum Princeps, etsi mortalem se agnosceret, quoties sidera mente sequebatur, non jam pedibus terram se tangere, sed apud Iovem nectare & ambrosiâ frui gloriabatur. (13) Hoc est illud generosum sciendi desiderium, quo accensus Eudoxus Cnidius, industrius imprimis caelestium siderum contemplator, Phaethontis modo comburi voluit, eâ lege, ut sibi ante liceret ad Solem adstanti, figuram, magnitudinem, formamque astri perdiscere. (14) Tam solidis voluptatibus Mathemata cultores suos sibi devinciunt. & quanquam aspera initio ac dura videantur, gratâ mox dulcedine molestiam laboris attemperant. (15) Scilicet ut inter medios spinarum aculeos fragrantissima enascitur rosa; & nux pinea duritiem corticis dulcissimis redimit nucleis; sic & Mathesis, quicquid habet arduum ac difficile incredibili voluptate compensat. VII. (1) Honesta quoque esse exercitia Mathematica, & eo nomine expeti dignissima, non credo quenquam esse qui ambigat. (2) Quid enim honestius esse queat, quàm mentem tot ac tam variarum rerum scientiâ instruere? (3) quid liberali ingenio dignius, quàm ardua quaeque penetrare? (4) caeli terrarumque plagas emetiri? (5) siderum determinare magnitudines? (5) siderum determinare magnitudines? (6) regiones exteras ac toto orbe divisas intra paternas aedes pervagari? (7) Innocuae artes sunt, & quae mentem è terrenis hisce faecibus extollentes, hominem homini reddunt, imò super aethera evehunt, unde ortum habet haec aurae divinae particula. (8) Quod sciens Anaxagoras Philosophus generis gloriâ & opibus clarissimus, quum universum patrimonium suis concessisset, & ad contemplandam rerum naturam se conferens, rem & publicam & privatam omninò negligeret; cuidam ita 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 101 Hortensius’ Oration on the Dignity of Mathematics 101 triangle in a circle, that he sacrificed an ox to the Muses; and Pythagoras, much more generous, sacrificed a hecatomb when he found the pro- portion of the sides of a right-angled triangle.22 (10) Such was the plea- sure which filled Archimedes when in his bath he discovered a method of discerning the fraud committed with the royal golden crown, that he leapt up and rushed home naked, repeating over and over again through the streets heureka, heureka, I have found it, I have found it.23 (11) So great was the peace of mind, with which, when his country was con- quered, he was intent on geometrical schemes, as if ignorant of evils, that he was wounded by an ignorant soldier, putting an end to his life and his studies at the same time. (12) With this peace of mind Claudius Ptolemy, the foremost of astronomers, although he perceived himself as mortal, boasted that when his mind followed the stars, his feet no longer rested on earth but that he supped on nectar and ambrosia standing by Jove himself.24 (13) This is that noble desire for learning, to which arose Eudoxus of Knidos, an especially energetic watcher of the stars in the sky, when he was willing to be burned in the manner of Phaethon, with this proviso, that he first be allowed to stand by the sun and thoroughly learn its shape, size and configuration.25 (14) So genuine are the plea- sures with which the mathematical sciences bind to themselves their devotees; and however hard and harsh they seem at first, soon they tem- per the trouble of the labour with pleasing sweetness. (15) Just as the most fragrant rose is born among the points of spines, and the nut redeems the hardness of its shell with its sweetest core, so also are the mathematical sciences; whatever laboriousness and difficulty they hold, they compensate with unbelievable pleasure. VII. (1) That mathematical exercises26 are also honourable and therefore most worthy to be sought, I do not believe anyone doubts. (2) For what can be more honourable than to instruct the mind in the knowledge of so many and such varied things? (3) What more worthy of the free spirit than to penetrate these difficulties? (4) To measure the spaces of heaven and regions of earth? (5) To determine the sizes of the stars? (6) To wan- der over foreign lands, divided over the whole globe, in your own home? (7) These arts are innocuous, which raise the mind from these earthly dregs, give back man to himself, even bring him up to the heavenly things, whence he rises, this small particle of the divine gleam.27 (8) In the knowledge of this, after Anaxagoras, a philosopher most famous for the glory of his family and his wealth, had given over his whole inheritance to his family and wholly neglected matters both public and private, taking himself to the contemplation of the nature of things, when 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 102 102 History of Universities compellanti, nullane tibi patriae cura? respondit, mihi verò patria cura & quidem summa est, digitum in caelum extendens. (9) Idem rogatus cujus rei causa natus esset? inspiciendi inquit, caeli, & Solis, & Lune. (10) Eodem sensu si & ego Mathemata nobis excolenda asseruero, ut per ea ad caelestium siderum [9] cognitionem adspirantes, & acutiùs illum naturae librum inspiciamus, & legamus attentiùs, non multum à vero abiturus sum; cùm & Plato oculos quidem homini ad contemplanda sidera datos esse dixerit, sed & Arithmeticam ac Geometriam tanquam alas additas, quibus in altissima Mundi subvolet spatia. (11) Quod ipsum nobile profectò & honestum exercitium reputandum est; quia per id ad primam omnium rerum causam, Deum, perducimur; & immensam Mundi molem, infallibilem durationem, ordinem admirabilem, edocti; humanae fragilitatis memores, spiritus ac fastum continere cogimur, & spes magnas mortales geniti abjicere. VIII. (1) Addam & antiquitatem Mathematicae, neque supremam dignitatis notam substraham principi scientiarum. (2) à qua si ulla omnino ars aut doctrina aestimari meretur, haec palmam caeteris facilè praeripiet. (3) Arithmeticae ortum ad Phoenices, Geometriae ad Aegyptios, Astronomiae ad Babylonios aut Assyrios referunt, sed immeritò: nisi fortè de usu harum artium intelligant. (4) Alioqui longè vetustiores censendae sunt, &ab initio Mundi earum arcessenda origo. (5) Non citiùs homines nati fuere, quàm numerare noverint; & sublatis in caelum oculis astrorum lucidos ignes & mirandas conversiones observarint. (6) Quin & primi illi ac sanctioris vitae Patriarchae ante Diluvium, concessâ Divino beneficio vitae diuturnitate, cùm bonitatem & sapientiam Dei ex operum ejus inspectione venarentur; Mathematicas adiere scientias, & caelestium siderum universum ornatum ac periodos etiam posteritati enarrare conati sunt, monumentis inventorum suorum in aeternam memoriam lapideis columnis, Iosepho teste, insculptis. (7) Geometria verò etiam aeterna fuit in mente Dei, & in ipsis mundi corporibus cùm esse inciperent expressa. (8) Solem adspicite, & Lunam, & Terram; rotundo sunt corpore. (9) Ipse Mundus cujus circumflexu teguntur omnia, sphaericus est, figurae Mathematicae inter omnes alias perfectissimae, 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 103 Hortensius’ Oration on the Dignity of Mathematics 103 he was asked, ‘Have you no concern for your fatherland?’ he replied, ‘Indeed I do care for my fatherland and in truth it is in the heights’, and pointed to the sky. (9) Likewise, when asked why was he born? he said, to watch the sky and sun and moon.28 (10) In this same frame of mind, if I also assert that the mathematical sciences ought to be cultivated and honoured by us and their reputation enhanced,29 so that through them, aspiring to the knowledge of the stars in the sky [9], we may watch more carefully that book of nature30 and we may read it more attentively, I will not go very far from the truth; since Plato also said that eyes were given to men to watch the stars, but also arithmetic and geometry were given as added wings, by which he might fly into the highest spaces of the world.31 (11) This ought to be considered in this assuredly noble and honourable exercise: because through it we are led to the first cause of all things, God, and are instructed in the immense structure/machine32 of the world, its infallible duration and admirable order, we are compelled, mindful of human fragility, to moderate our spirit and our arrogance, and, born mortal, to throw away great hopes. VIII. (1) Let me add also the antiquity of the mathematical sciences, nor let me take away the supreme mark of merit of the first of the sciences. (2) If any art or science at all deserves to be highly estimated, this one easily snatches the winner’s prize away from the rest. (3) By tradition they attribute the origin of arithmetic to the Phoenicians, of geometry to the Egyptians, of astronomy to the Babylonians or Assyrians, but undeservedly, unless by chance they are referring to the use of these arts. (4) If not, they are to be thought far older, and their origin ought to be sought from the beginning of the world. (5) No sooner was man born than he learned how to count, and with eyes uplifted to heaven observed the bright fires and wondrous movements of the stars. (6) Yes, and the first men and in particular the patriarchs of holy life before the flood, having been granted great length of life by divine blessing, when they were trying to aspire to the goodness and wisdom of God by inspection of his works, turned to the mathematical sciences and sought to report for posterity the universe adorned with heavenly bodies and their periods, by written monuments of their inventions carved on stone columns in eternal memory, as [Flavius] Josephus testifies.33 (7) Geometry has indeed been continually in the mind of God, and expressed in the bodies of the world themselves, since they began to be. (8) Look at the sun and the moon and the earth, they are of round body. (9) The world itself, by whose circumference everything is covered, is spherical, the mathematical figure which among all others is most 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 104 104 History of Universities excellentissimae, & capacissimae. (10) Vidimus Mathesi non parvam accessisse dignitatem, quod à Principibus & Optimatibus semper fuerit exculta, quibus placuisse non ultima apud Poetam laus est. (11) At quantae dignitatis esse putabimus, autorem habere ipsum Deum, conditorem universi & directorem; qui Mathematicis rationibus totum Mundi ornatum disposuit, & per eas actiones naturae quotidie dirigit ac conservat? (12) Oraculum Sapientis est, Deum omnia condidisse in Numero, Pondere, & Mensura, quibus Mathesis partes nobilissimae denotantur, Arithmetica, Statica & Geometria; ad quarum actionem, vim, & proportionem, totam mundi machinam & singula ejus membra disposuit Deus ac sapientissimè contemperavit. (13) Nec minus verum est Platonis ’ ´  quo judicavit ` ` ’` ~ , Deum semper Geometriam exercere. (14) Quod etsi alii aliter interpretentur, ego sic intelligendum existimo, quod Deus O.M. non tantùm materiam Mundi indeterminatam & confusam in principio rerum definiverit, terminis ac figuris Mathematicis circumscripserit, numerorum ac ponderum proportionibus constrinxerit; sed & eandem indies ob innatam mobilitatem nullis non mutationibus, ortibus & occasibus obnoxiam, tanquam pater & opifex peritissimus iisdem mediis tueatur, & in optima compositione conservet. (15) Itaque ad Deum ipsum si Mathesis originem referamus, nullam erroris incurremus suspicionem; sed veritati [10] congrua dixisse, universa Mundi compages, indissolubilis rerum ordo, & vestigia Mathematum in corporibus mundanis expressa planissimè convincent. IX. (1) Tantum de Dignitate Mathesis, sequitur Vtilitas ad cujus adumbrationem nunc me confero. (2) Laudatus olim, & rectè, fuit Socrates, quod Philosophiam à contemplatione rerum naturalium primus ad vitam communem traduxerit & conformationem morum. (3) Vt enim qui ingentem thesaurum possidet, non habetur verè dives, quod repositam in loculis pecuniam contempletur, & solo nomine sibi gaudeat, nisi eâ commodè ad vitae utatur institutum; ita existimabat vir eximius Scientias omnes non tantum sui aut veritatis cognoscendae gratiâ excolendas, quod alioqui praeclarum quoque sit; sed ad usum praecipuè referendas, verum eruditionis nostrae scopum. (4) Idem nobis propositum in Mathesi. (5) Quam velut in contemplatione rerum alias scientias certitudine, subjecti nobilitate, & jucunditate, antecedere 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 105 Hortensius’ Oration on the Dignity of Mathematics 105 perfect, most excellent, and most capacious. (10) We have seen no small dignity accruing to the mathematical sciences, which have always been cultivated and honoured among princes and aristocrats, whom to have pleased is not the least praise in the opinion of the poet. (11) And how much dignity shall we attribute to them that have God himself as their founder, the creator and director of the universe, who has arranged the whole equipment of the world by mathematical principles and through them guides and conserves the course of nature day by day? (12) It is the saying of a wise man, that God has founded all things in number, weight and measure,34 by which the most noble parts of the mathemat- ical sciences are known, arithmetic, statics and geometry; by whose action, force and proportion God has disposed and most wisely harmonized the whole machine of the world and each of its members. (13) No less true is the passage of Plato, where he judged ton Theon aei geometrein, God is always doing Geometry.35 (14) Although others may interpret this in other ways, I think it ought to be understood thus, that God Almighty not only defined the indeterminate and confused matter of the world in the beginning of things, circumscribed it with mathematical boundaries and shapes, constrained it with proportions of number and weight, but also constantly as most skilled father and creator [of the world] he guards it by those means, liable on account of innate mobility to every change, rise and fall, and keeps it in the best composition. (15) And so if we refer the origin of the mathematical sciences to God himself, we incur no suspicion of error; but to have said things [10] congruent with the truth, that demonstrate the universal framework of the world, through the indissoluble order of things and the traces of mathematics expressed most obviously in the bodies of the world. IX. (1) So much for the dignity of the mathematical sciences; there follow the usefulness and practical advantage, to whose description I now turn. (2) Socrates was once praised, and rightly, because he was the first to transfer philosophy from the contemplation of natural things over to communal life and the strengthening of morals. (3) Just as the one who possesses a huge treasury is not considered truly rich, because he contemplates his money stored up in secret places and rejoices in it by name only, unless he should make use of it fittingly for his living expenses, so an outstanding man did not think of cultivating all sciences for himself or for the sake of knowing the truth, which anyway is notable as well, but of referring them especially for use, the true aim of our learning. (4) The same is for us the aim of the mathematical sciences. (5) As we have shown, in the contemplation of things they surpass the 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 106 106 History of Universities ostendimus; ita usus quoque nobilissimos hominibus conferre reddemus manifestum. (6) Non ignoro equidem Matheseos sacra per se constare, neque opus habere ut ullis rebus materialibus immisceantur, sed ex sensu Platonis sedem posuisse in solâ actione mentis: qui cùm coaetaneos suos Archytam Tarentinum & Eudoxum Cnidium Mathemata ad usum popularem transferre vidisset; iratus dignitatem Philosophiae vulgo prostitui, utrumque ab instituto deterruit: verùm cum jam ante Theoreticas Mathesis partes à Practicis distinguendas esse monuerimus; in iis quidem judicio Platonis locum concedimus, in his minimè. (7) Illae ut lubet purae & abstractae considerentur; nos harum omnivario usu vitam humanam carere minimè posse liquidò demonstrabimus. X. (1) Consideranda autem est Mathesis utilitas in genere, quatenus se diffundit per omnes Disciplinarum ac Facultatum ordines; & in specie, prout cujuslibet partis est propria. (2) Inter Facultates prima sit Philosophia, cujus Mathesis & ipsa partem constituit haud postremam. (3) Vsum ejus hic insignem esse, tum pro se in contemplatione sui objecti per quod inter naturalem & primam Philosophiam media est; tum ad perscrutandum alias Philosophiae partes, scriptaque praestantissimorum Philosophorum intelligenda, adeo clarum est, ut vix ulla indigeat probatione. (4) Sectae Philosophorum praecipuae hodie duae sunt, Platonica & Peripatetica. (5) Si ad Platonicam te conferas, foribus Gymnasii inscriptum invenies ’N o ’  ’ g, nullus Geometria expers intrato. (6) Scilicet Mathematicis rationibus libros Philosophiae suae implevit Plato, eique quidquid in illis mirabile ac splendidum est, tanquam fundamentum substernens, occultam esse voluit Geometriae ignaris. (7) Ita in Menone Socratem dissertantem cum puero insert de quadrato quadrati duplo; in Theaeteto de numero aequaliter aut inaequaliter aequali. (8) Ita in Timaeo Deum mundi animam rationibus arithmeticis & geometricis componere statuit, ac deinde corpus geometricis figuris fabricari. (9) Multa quoque disserit de elementorum creatione & proportione juxta figuras varías triangulorum & corporum regularium. (10) In Peripatetica Philosophia & libris Aristotelis, infinita sunt è quibus nemo absque Mathematum peritia se extricaverit. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 107 Hortensius’ Oration on the Dignity of Mathematics 107 other sciences in certainty, nobility of subject and delight; so we will make clear that they also confer the most noble benefits upon men. (6) In fact I know very well that the sacred matters of the mathematical sciences exist through themselves and there is no need for them to be mixed up with any material things, but in Plato’s sense they take their seat in the mind’s action only. When Plato’s contemporaries, Archytas of Tarentum and Eudoxus of Knidos seemed to be transferring the mathematical sciences to the use of the people, he frightened them both off the under- taking, angry that the dignity of Philosophy was being prostituted to the crowd.36 Indeed, since we have warned already before that the theoret- ical parts of the mathematical sciences ought to be distinguished from the practical, we yield to the judgment of Plato as to the former, but as to the latter, we yield not at all. (7) The former are to be considered pure and abstract as you wish; we will show clearly that by the multifarious benefits of the latter, human life can lack little. X. (1) Now, the advantage of the mathematical sciences ought to be considered in general, to what extent it spreads itself through all orders of disciplines and faculties, and in particular cases, according to what belongs to each part. (2) Among faculties, the first is philosophy, of whom the mathematical sciences constitute a part by no means the last. (3) That its advantage here is extraordinary is so clear that it exacts hardly any proof, both in itself in the contemplation of its own subject, through which it is the medium between natural and pure philosophy, and also for exam- ining the other parts of philosophy, and in understanding the writings of the most outstanding philosophers. (4) Today there are two principle philosophical sects, Platonists and Peripatetics [Aristotelians]. (5) If you turn to Platonism, you will find written on the doorway of the gymna- sium, ageometretos oudeis eisito, let no one ignorant of geometry enter.37 (6) Surely Plato filled the books of his own philosophy with mathemat- ical reasoning, and whatever in them is wonderful and splendid, as it was the underlying foundation, he wished to be hidden from those ignorant of geometry. (7) So, in Meno, he brings in Socrates discussing how to double a square with a slave-boy, and in Theaetetus, about numbers equally or unequally equal.38 (8) So, in Timaeus, he declared that God put together the soul of the world with arithmetic and geometric proportions, and then its body was created with geometric figures.39 (9) He also said many things about the creation and proportion of elements according to various figures of triangles and regular bodies. (10) In Peripatetic philosophy and the books of Aristotle, there are infinite matters from which no one can extricate himself without skill in the mathematical sciences. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 108 108 History of Universities (11) Physicam enim si consideremus, tota non solum [11] exemplis, sed & fundamentis Mathematica est. (12) Liber primus quadraturam habet circuli; secundus duos angulos rectos in triangulo plano; tertius gnomones numerorum è doctrina Pythagorica; alii alia. (13) Libri de Caelo, de infinitate magnitudinis agunt, de figura aquae, de compositione sphaerae ex pyramidibus, de figuris locum implentibus. (14) In Analyticis habetur tetragonismus circuli, lineae commensurabiles & incommensurabiles, parallelismus rectarum, anguli exteriores in figuris, aliaque complura. (15) In Meteoris, quot sunt loca Mathematicis rationibus explicanda? (16) De Cometis, de Galaxia, altitudine montium, de proprietatibus Iridum & Pareliorum. (17) Metaphysici quoque & Ethici libri, Geometricis aut Arithmeticis demonstra- tionibus scatent, & Peripateticam Philosphiam totam è Mathematicis rationibus constitutam arguunt & exstructam. XI. (1) Ad Theologiam Mathematum notitia tantum praebet utilitatis, ut nullâ ratione à cordato Theologo negligi debeant aut praeteriri. (2) Summum Theologiae scopum esse agnitionem Dei fatentur omnes. (3) Ad eam verò duplici viâ pervenitur, per intuitum nempe operum Dei, aut per lectionem S. Scripturae. (4) Vtrique Mathesis summè necessaria est, quia & manifestat mirabilia Dei in operibus ejus, & multorum Scripturae locorum faciliorem parit intellectum. (5) Quis enim Mundum & universum ejus ornatum rectè examinet sine auxilio Mathesis? (6) Aut quis potentiam Dei ac bonitatem erga filios hominum dignè suspiciat ac veneretur, nisi qui cùm Davide inspexerit Caelos opera digitorum ejus, & Solem ac Lunam quos praeparavit? (7) Immo quis haec corpora inspiciens, occasionem invenerit cum eodem vate exclamandi, Domine Deus noster, quam admirabile est nomen tuum in universa terra! nisi qui à Mathematicis acceperit motus, ordines, & vastas eorundem magnitudines: atque ab his mente ascenderit ad infinitam Dei potentiam? (8) Mathesis usu discimus, quanam ratione Caeli enarrent gloriam Dei, & opus manuum ipsius annunciet firmamentum: aut quo modo ex visibilibus hujus Mundi agnoscantur ejus invisibilia. (9) Eadem solidas subministrat rationes, quibus Mundum hunc non temerè sed certo ordine conditum esse, & sapientissimum habuisse Architectum evincitur. (10) Quibus finitum quidem eum sed infinito similem esse, & infinitudinis Divinae 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 109 Hortensius’ Oration on the Dignity of Mathematics 109 (11) Now if we consider Aristotle’s Physica, it is wholly mathematics, not only in examples, [11] but also in foundations. (12) The first book includes the squaring of a circle, the second two right angles in a plane triangle, the third has gnomons of numbers from Pythagoras’s doctrine, and the other books have other examples. (13) The books on the heavens, De Caelo, discuss the infinity of magnitude, the shape of water, the composition of the sphere from pyramids, the figures that fill a prescribed space. (14) In the Analytica priora et posteriora we find the squaring of a circle, commensurable and incommensurable lines, the parallelism of straight lines, the external angles of figures, and many others. (15) In his Meteorologica how many places are there explicable by mathematical reasons? (16) The sections on comets, on galaxies, on the height of mountains, on the properties of rainbows and parhelia. (17) The books of metaphysics, Metaphysica, and ethics, Ethica Nicomachea, also teem with geometric and arithmetic proofs, and reveal that the whole peripatetic philosophy is founded and built on mathematical reasoning. XI. (1) To theology, acquaintance with the mathematical sciences offers so much advantage that it in no way ought to be neglected or passed over by a wise theologian. (2) All agree that the highest aim of theology is the knowledge of God. (3) One may indeed come to that in two ways, through examination of the works of God, of course, or through reading of the Holy Scripture.40 (4) The mathematical sciences are highly necessary for both, because they make plain the wonders of God in his works, and provide an easier comprehension of many passages of scripture. (5) For who rightly examines the world and its whole ornament without the aid of the mathematical sciences? (6) Or who worthily admires and reveres the power of God and his kindness towards the sons of men, unless with David he has looked upon ‘the heavens, the works of his fingers, and the sun and moon that he has created’?41 (7) Indeed, who looking at these bodies shall have found occasion with the same bard to exclaim ‘O Lord our sovereign, how glorious is thy name in all the earth!’42 unless he has received from the mathematical sciences the motions, orders and vast size of these bodies, and ascended from these in his mind to the infinite power of God? (8) By the use of the mathematical sciences we learn how ‘the heavens tell out the glory of God’, and how ‘the vault of heaven reveals his handiwork’,43 or in what way from the visible things of this world the invisibles of it may be known. (9) They furnish the solid reasons with which this world was founded, in not rash but secure order, and prove that the Architect acted most wisely. (10) By these reasons, they show that the world is finite, but similar to infinite and that it bears 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 110 110 History of Universities elegantissimum ostenditur gerere typum. (11) Nec minor est ejus utilitas in explicandis quamplurimis S. Scripturae locis, è quibus pauca modò attingam. (12) Locum de creatione Mundi & luminarium caeli; dissertationem ipsius Dei de via lucis, Plejadibus, Orione, apud Iobum; eclipsin Solis miraculosam tempore passionis Domini; rationem anni Iudaici & festi Paschatis, ritè non explicabit Theologus sine cognitione Astronomiae ut nec sine Geographiae peritia, exitum Israelitarum ex Aegypto; distributionem Terrae sanctae; & peregrinationem Pauli. (13) Geometriae vestigia sunt in Arca Noei; Templo Salomonis; civitate Ezechieli per visionem ostensa; & nova Hierosolyma Apostolo Iohanni visa. (14) Arithmeticae in hebdomadibus Danielis; & numero electorum ex tribubus Israelis in Apocalypsi; ac passim alibi. XII. (1) Iurisprudentiam nunc adeamus, visuri an non & gravis illa ac severa humanae sapientiae vindex, Mathematicae operâ indigeat. (2) Romanae leges multis in locis Arithmeticas ac Geometricas requirunt demonstrationes, sine quibus [12] intelligi & explicari non possunt. (3) In quotidiana praxi absque iis nec judicium exerceri potest, nec lites dirimi, neque furta & infinitae inter mortales injuriae ac confusiones evitari. (4) Quacunque ortâ controversiâ, spatia temporum quibus quaeque res acta, pacta, aut locata est, petenda sunt ex Arithmetica & Astronomia. (5) Bello aut inundatione agrorum limitibus confusis aut injustè occupatis, geometrica dimensio suam cuique mensuram aequissimâ ratione restituit. (6) Haereditates si dividendae sunt, aut aestimanda noxa, dissolvenda vorsura [sic], foenus expendendum, distribuendum lucrum, ad Arithmeticam itur: si latifundia separanda, ducendae cloacae, paries inclinatus in alienum solum erigendus, arboris in confinio stantis partiendi fructus, insulae in alveo fluvii enatae adjudicandae, ad Geodaesiam. (7) Addo aridorum & liquidorum varias mensuras, staterae & librae momenta; quarum aequitas in republica servari nequit, nisi è Mathematicis fundamentis aestimentur, & publico examini subjiciantur. (8) Denique Astraea ipsa pro tribunali sedens, nihil efficit sine proportione Arithmetica & Geometrica, ad quas praemia & poenas distribuendo, aequae examine lancis Iustitiae pondera expendit; ostenditque Mathesin in ipsa curia, summo civitatis loco, aequi bonique arbitram esse, & unicam juris vindicem ac magistram. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 111 Hortensius’ Oration on the Dignity of Mathematics 111 the stamp of divine infinity. (11) No less is their advantage in explicating very many passages of Holy Scripture, out of which I will only touch on a few. (12) The passage on the creation of the world and the lights of the sky, the discussion of God himself on the way of light, the Pleiades, Orion, in Job, the miraculous eclipse of the sun at the time of the Passion of the Lord, the calculation of the Hebrew year and the feast of Easter— a theologian will not rightly explain these without knowledge of astronomy. As without skill in geography he cannot explain the exodus of the Israelites from Egypt, the distribution of the Holy Land and the journeys of Paul. (13) There are traces of geometry in Noah’s ark, the Temple of Solomon, the city shown to Ezekiel in a vision,44 and the New Jerusalem seen by the apostle John.45 (14) There are traces of arithmetic in the hebdomads of Daniel46 and in the number of the chosen from the tribes of Israel in the Apocalypse,47 and everywhere else.48 XII. (1) Let us now turn to jurisprudence, and we shall see whether or not that grave and severe protector of human wisdom needs the mathematical sciences in its work. (2) The Roman laws in many places require arithmetical and geometrical proofs, without [12] which they cannot be understood and explained. (3) In daily practice, judgment cannot be exercised without them, nor can disputes be brought to an end, nor can thefts and infinitely many injustices and confusions between mortals be avoided. (4) Wherever controversies arise, the space of time within which each action occurred, was agreed upon, or located, are to be sought by arithmetic and astronomy. (5) When boundaries of fields are confused or unjustly taken by war or by flooding, geometrical measuring restores its size by most fair calculation. (6) If inheritances are to be divided, or a damage to be estimated, a debt to be paid by a loan, interest to be paid, gain to be distributed, we go to arithmetic; if real estate is to be divided, sewers to be built, a wall slanting into another’s soil to be straightened, the fruits of a tree standing on a boundary to be divided, islands born out of the riverbed to be arbitrated, we turn to surveying. (7) I add the various dry and liquid measures, the weights of scale and balance, whose fairness cannot be kept in the state unless they are estimated through mathematical foundations and submitted to public scrutiny. (8) Finally, Astraea49 herself sitting in judgment does nothing without arithmetic and geometric proportion, according to which she has to distribute rewards and punishments, paying out the weights on the Scale of Justice impartially; and she demonstrates that in the court itself, the highest place of state, the mathematical sciences are an arbiter of what is fair and good, and the only protector and master of the law. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 112 112 History of Universities XIII. (1) Ad Medicinam quod attinet, certum est Mathematum cognitionem eruditum Medicum non modo egregiè ornare, verum etiam multis modis felicitatem praxis adjuvare, quapropter à principibus Medicorum & exculta semper fuere, & honestissimo elogio commendata. (2) Magnus Hippocrates Thessalo filio Geometriam & Arithmeticam ediscendas praecipit, non solùm ad splendorem vitae, sed & ad artis Medicae usum. (3) Et Geometriam quidem ad ossium situm, luxationem, repositionem, exemptionem & omnimodam curationem: Arithmeticam ad intensiones, periodos, & mutationes morborum rectè dijudicandas. (4) Galenus Medicos culpat qui cum Hippocratem laudent, ipsi tamen omnes aliud potiùs agunt quàm ut ei quem praedicant similes efficiantur; cum ille Geometriam & Astronomiam Medico necessariam esse dixerit, hi verò ab utriusque studio usque adeo abhorreant, ut alios etiam id conantes coarguant. (5) Neque iis tantum summorum in arte medica virorum judiciis Mathesis in Medicina stat utilitas, sed & experientiâ ipsâ. (6) Morborum periodos & intricatas crisium rationes, nunquam feliciùs expediet medicus, quàm si praeceptis astronomicis instructus, praeter naturae impetum in agitatione materiae morbificae, etiam motum Lunae consideraverit, à cujus influxu ordo dierum criticorum dependet, & majores minoresve morbi mutationes procedunt, quemadmodum multis docet Galenus libro 3 de Diebus Decretoriis. (7) Epidemicos morbos nunquam rectiùs judicabit, quam si ex sideribus anni statum examinet, cum quo & ventriculi hominum mutantur. (8) exortus item & occasus siderum notet, quo mutationes & excessus ciborum ac potuum, & ventorum & totius mundi, ex quibus morbi hominibus oriuntur, sciat observare. quod studiosè praecipit Hippocr. in lib. de Aere, Aquis & Locis, & 1 de Diaeta. (9) Neque à Geographia minus habebit praesidii quàm ab Astronomia. (10) Vt enim illa servit Medicinae ad causas caelestes mutationum anni inveniendas; ita haec valet in discernendis particularibus regionis cujusque morbis, & contemperandis remediis [13] pro occasione & natura loci. (11) Plurimum facit ad praenoscendas climatum qualitates & familiares locorum ventos, pro quorum varietate aliae atque aliae occurrunt aegritudines: aut ad praecavendum morborum incursum qui à certis mundi partibus ad alias solent transmigrare: quod Hippocratem olim egisse legimus, cùm peste à barbaris ad graecos pervadente, dimissis 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 113 Hortensius’ Oration on the Dignity of Mathematics 113 XIII. (1) As far as Medicine is concerned, it is certain that knowledge of the mathematical sciences not only adorns a learned doctor very highly, but also aids the success of practice in many ways, wherefore it was always honoured by the first doctors and commended in most honourable praise. (2) The great Hippocrates advised his son Thessalus to learn geometry and arithmetic, not only for brilliance of life but also for use in the arts of medicine. (3) And geometry indeed ought to be studied for setting of bones, dislocation, repositioning, amputation and every kind of cure, and arithmetic for rightly judging the intensities, periods and changes of diseases. (4) Galen50 blames doctors who praise Hippocrates but themselves nevertheless all do something other than bring about results similar to those of him they praise. When he said geometry and astronomy were necessary for a doctor, they in truth shy away from the study of both to such an extent that they refute others who even attempt this study. (5) Nor is the advantage of the mathematical sciences in medicine obvious only from the judgment of these men who are greatest in the medical art, but also from experience itself. (6) A doctor will never set right the periods of diseases and intricate calculations of crises more fortunately than if, instructed by the precepts of astronomy, beyond the impetus of nature in the agitation of the sickness, he shall have taken into consideration also the motion of the moon, from whose arrival the order of critical days depends and the greater and lesser changes of illness proceed, just as Galen teaches in book three of his De diebus decretoriis.51 (7) He will never judge epidemics more rightly, than if he should examine the state of the year from the stars, with which also the ventricles of men change. (8) He likewise notes the risings and settings of the stars, by which there are changes and excess of food and drink, and he knows how to observe the risings and settings of winds and of the whole world, from which the diseases of men arise, which Hippocrates expressly advises in his book De aere, aquis, locis, and in the first book of De victu. (9) Nor will he find less support in geography than in astronomy. (10) For as the latter serves medicine in finding the celestial causes of the changes of the year, so the former has value in discerning the particular regions and their diseases, and preparing [13] the remedies according to the occasion and nature of the place. (11) This does a great deal for predicting the qualities of climates and local winds, on behalf of whose variety this or that disease occurs. It also does much for warding off the onset of illnesses that are accustomed to migrate from certain parts of the world to others, which we read that Hippocrates once did, when, as a plague was coming from foreign lands to the Greeks, he helped 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 114 114 History of Universities per loca discipulis patriae succurrens, medelas indicavit, quibus qui uterentur instantem pestem securè effugere possent. (12) Quapropter etiam in magnis mysteriis apud Athenienses non secus ac Hercules Iovis filius publicè fuit initiatus, & coronâ aureâ mille aureorum donatus, ac toto vitae tempore in Prytaneo victu & jure civitatis donatus. XIV. (1) Hactenus vidimus quantum Mathemata praebeant caeteris facultatibus; supersunt partes, quarum singulae singulares quoque usus continent, silentio haudquaquam praetereundos. XV. (1) Logisticae seu Arithmeticae practicae tanta est necessitas ut verbis satis nequeat describi. (2) Hac consistit humana societas, & vita hominum mutuâ rerum permutatione faciliùs toleratur. (3) Sine hac nec respublica regitur, nec familia administratur: non bellum geritur, non pacis fructus metuntur. (4) Haec hominem acuit & attentum facit ad rem, nec facilè alterius fraude patitur circumveniri. (5) Intuemini quaeso Auditores hanc vestram Vrbem, & utilitatis Logisticae vivum habebitis exemplar. (6) Civium maxima pars cum Italis, Gallis, Anglis, Germanis, Afris & Indis, commercia exercet; in summa varietate ponderum, nummorum, & mensurarum. (7) Si quis roget quâ arte freti rerum suarum reddantur securi? (8) Respondebunt Logisticam esse quâ in commutationibus & comparationibus mercium, difficultatem atque obscuritatem omnem superant: & servata accepti atque expensi ratione, facultates suas integro statu servant aut explicant. (9) Si quis de usu artis quaerat, fatebuntur tantas eâ commoditates comprehendi, ut carere illâ nequeant nisi cum manifesto rerum suarum dispendio & familiae detrimento. (10) A Mercatura ad militiam vos convertite; cernetis Logisticam ad distribuendas & ordinandas acies prorsus esse necessariam. (11) Ordinum ratione & commodâ subsidiorum emissione ingens saepe stetit victoria. (12) Macedonica Phalanx & triplex Romanorum acies, aliquoties innumeram barbarorum multitudinem sustinuit, profligavit. (13) Adeo ut Logistica pacis ac belli fida administra dicenda sit, & ingentia utrobique hominibus conferre subsidia. XVI. (1) Geodaesiae multiplex quoque est usus ac necessitas. (2) Haec illa est quae superficies corporum, longitudines, latitudines, & profunditates quaslibet metitur; montium & turrium inaccessas prodit altitudines; quae insularum ambitus explorat & fluviorum latitudines; quae tormenta bellica dirigit, & scalarum mensuras praebet ad invadendas stratagemate civitates. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 115 Hortensius’ Oration on the Dignity of Mathematics 115 his country by sending his pupils all over and pointing out the cures to use for securely escaping the disease. (12) Wherefore also he was ini- tiated into the Great Mysteries among the Athenians no less than Hercules the son of Jove,52 and given a golden crown of a thousand gold coins, and also dinner in the State House, Prytaneum,53 for the rest of his life, and the right of citizenship.54 XIV. (1) Up to this point, we have seen how much the mathematical sciences have to offer to the other disciplines; there remain the particular branches, containing particular benefits, not at all to be passed over in silence. XV. (1) So great is the necessity for logistica or practical arithmetic that it can hardly be described in words. (2) Human society stands on her [logistica], and the life of men is borne more easily by mutual exchange of goods. (3) Without her neither is a state governed, nor a family ordered; no war is waged, no fruits of peace gathered. (4) She has trained men and has made them attentive to affairs, and is not easily liable to be defrauded by another. (5) I ask my listeners, gaze upon your city and you will have a living example of the value of practical arithmetic. (6) The greater part of the citizens engage in trade with Italy, France, England, Germany, Africa and India, with the greatest variety of weights, coinage and measures. (7) If anyone should ask them, trusting in which art their goods return safely? (8) They will answer that it is logistica, by which in exchanges and comparisons of merchandise, they overcome every problem and obscurity, and, having kept a calculation of what is received and spent, they preserve their wealth in an impeccable state, or enlarge it. (9) If anyone should enquire about the profit of the art, they will confess that so many conveniences are comprehended in it, that they could do without it only with clear loss of their possessions and harm to their families. (10) From trade now look to war; you will see that practical arithmetic is necessary for the deploying and correct ranking of battle-lines. (11) A great victory often depends on the calculation of ranks and convenient flow of supplies. (12) The Macedonian phalanx and the Roman triple battle-line, as many times as they withstood the uncounted multitude of barbarians, it has overwhelmed them. (13) So it should be said that logistica is the trustworthy servant of peace and war and that it brings great support to men in both. XVI. (1) There is also a multiple advantage in and need for geodesy. (2) This is the art that measures the surface of bodies, longitudes, latitudes and depths, of all kinds; it gives away the inaccessible heights of mountains and towers, it explores the circuits of islands and the width of rivers, it directs catapults and offers the measure of ladders for invading cities by stratagem. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 116 116 History of Universities (3) Ejus usu ut magna hominibus cedunt commoda, ita ignoratione gravissima eveniunt damna & errores periculosissimi. (4) Vulgaris opinio est, agrorum qui in ambitu eandem mensuram colligunt, contenta spatia esse aequalia; at Geodaesia docet, si dentur duo agri quorum ambitus sit decempedarum 160, unus verò sit figurae quadratae alter triangularis, illius aream esse decempedarum 1600, hujus tantum 1200, quartâ parte istâ minorem. (5) Ac tantum damni emptoribus accedit ex hac aut simili pseudographia, quando ex ambitu agrorum [14] putant se easdem emere areas, nisi contrarium edocti noverint esse cautiores. (6) Idem apud Historicos usu venit, cum insulas aut urbes pares esse tradunt quae eodem navigationis aut circuitionis ambitu continentur; quod falsum esse Geodaesia apertè evincit. (7) Ejus rei elegans locus est apud Polybium, quem non pigebit referre. (8) Megalopolis (inquit) ambitu fuit quinquaginta stadiorum, Lacedaemon quadraginta octo: & tamen Lacedaemon duplo major Megalopoli. (10) Hoc ignaris Mathematum incredibile videatur. (10) Quod si dixero, fieri posse ut civitas ambitu quadraginta octo stadiorum sit dupla civitatis centum stadiorum ambitu; insanum (ait) atque amens videatur; attamen utrumque verum est & geometrica neceßitate demonstratum. XVII. (1) Geodaesiae sociam demus Architecturam militarem, quae muniendis, defendendis & oppugnandis civitatibus inservit. (2) Hujus notitia regibus ac principibus maximè convenit, & militae ducibus est quasi propria. (3) A parvis initiis exorta eò necessitatis ac praestantiae ascendit, ut sine illa nec bellum gerere queant Principes, neque civitates proprias tueri, aut hostiles in suam redigere potestatem. (4) Quod notius est, & inter Belgas ubi ante omnes Europae tractus meliùs excolitur, frequentius; quàm ut multis à me postulet describi. (5) Hac enim arte, post benedictionem Dei, curas Patrum, & Principum Auriacorum vigilem industriam, respublica nostra ad illud columen quod hodie cernimus, evecta est: eâdemque contra omnimodas hostium technas, salvam tuemur & inconcussam. (6) Magna olim gessere machinationibus & castrametationibus suis, Pirrhus Epirotarum rex, Demetrius Poliorcetes, Cajus Caesar; sed ea, si cum Belgicis victoriis conferantur, vix in prima acie consistere possunt, quod civitates invictissimae brevi tempore expugnatae; aliae extremis hostium viribus tentatae, & felicissimè defensae; & quod amplius, usque in viscera soli hostilis prolata reipublicae pomoeria; irrefragabili testimonio evincunt. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 117 Hortensius’ Oration on the Dignity of Mathematics 117 (3) As by its use great benefits come to men, so the weightiest harms and most dangerous errors result from the ignorance of it. (4) The common opinion is that the spaces of fields with the same measurement in the circuit are equal; but geodesy teaches that given two fields whose circuit is 160 yards, one in the shape of a square, the other of a triangle, the area of the former is 1600 yards, of the latter 1200, a fourth less. (5) And so much harm comes to buyers from this or similar false contracts, whenever from the circuit [14] of fields they think they are buying the same areas, unless having been taught the contrary they know to be more cautious. (6) It likewise comes in handy among the historians, when they say that islands or cities that have the same circumference by means of navigation or walking are equal; geodesy has plainly proved that this is false. (7) There is an elegant passage on this matter in Polybius, which it will not be annoying to quote. (8) ‘Megalopolis,’ he says, ‘was fifty stades in circumference and Sparta forty-eight; and nevertheless Sparta was twice as large as Megalopolis. (9) This may seem incredible to those ignorant of mathematics. (10) What if I say that it is possible for a state with a border of forty-eight stades to be twice the area of a state with a border of one hundred stades? It may seem astounding,’ he says, ‘and witless, but nevertheless both are true, and shown by geometrical necessity.’55 XVII. (1) Let us make military architecture, which serves to fortify, defend and assault cities and states, the ally of geodesy. (2) Knowledge of it is greatly suited to kings and princes and as appropriate to military leaders. (3) Arising from small beginnings, it rose to be so necessary and prestigi- ous that without it, princes could not wage war, nor could states be defended or bring the enemy under their control. (4) This is better known and more common among the Dutch, where before all European regions it is better cultivated and perfected, than that it needed to be described in depth by me. (5) For by this art, after the blessing of God, the cares of fathers and watchful industry of the princes of Orange, our republic has been lifted up to that height that we see today; and by that same art, we keep it safe and unharmed against every type of enemy craft. (6) Pyrrhus, king of the Epireans,56 Demetrius Poliorates,57 Gaius Caesar accomplished great things once upon a time by war machines and camp planning; but these things, if they are to be compared with Dutch victories, can hardly stand in the first rank. For invincible states were conquered within a short time, others attacked by greatest strength of the enemy, and most fortunately defended, and what is more, the frontier of the republic was moved forward incessantly even into the inward parts of enemy territory, and they endure according to irrefutable testimony. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 118 118 History of Universities XVIII. (1) Ad Mechanicam venio & Staticam, illas admirandorum operum effectrices, & insignia humanae solertiae documenta; in quibus praecipuè elucet quantum valeant Mathemata ubi rebus materiatis applicantur. (2) Facultas Mechanica non modò omnes alias amplitudine superat, sed & antiquissima est inter homines, & à mundi primordiis usurpata. (3) Ea principium dedit culturae agrorum, domorum aut tuguriorum structurae, confectioni vestimentorum, & innumeris deinde instrumentis ad opificum usum necessariis. (4) Ea est qua fabri & architecti vastissimas lapidum trabiumve moles attollunt, ac sine ullo fere labore quo volunt dirigunt: qua lapicidae durissima marmorum frusta dividunt: qua statuarii, sartores, aurifabri, typographi, quilibet materiam sui opificii incidunt, secant, cudunt, premunt: quâ nautae exiguo clavo ingentes naves pro lubitu regunt. (5) Staticis rationibus constant organa hydraulica & pneumatica: libra item & omne quod vehitur in humido. Ea est cujus usu Patriam nostram salvi incolimus, quando aut redundantes & terram obruituras machinis exhaurimus aquas; aut redituras ex mari catarractis aggeribusque cum stupore exterorum arcemus & excludimus. (6) quando ponderis majoris aedificia non extruimus locis uliginosis, ut in hac ipsa Vrbe, nisi fundamento palis sublicisque bene praemunito. (7) Mechanica & Statica in rebus ad voluptatem aut dolum comparatis, varias efficiunt praestigias: cùm modò [15] statuas ambulantes, modo vocem instar oraculi edentes machinantur, & automata construunt quibus tempora distinguimus aut rerum gestarum exhibemus historiam. (8) Tales fuere tripodes Vulcani apud Homerum qui sponte praeliabantur, Ctesibii merulae vocem humanam imitantes, & columba lignea volans Archytae Tarentini. (9) Vtriusque potentiam unus nobis ostendere potest Archimedes, qui sphaeram vitream fabricatus est, quae Solis, Lunae, & Planetarum motus caelestibus analogos perpetuò exhibuit; qui organis suis quinquies mille modiorum pondus solus attraxit; & navem regiam quam totius Siciliae vires movere non poterant, deduxit in mare; qui item Syracusas adversus Romanorum oppugnationes aliquandiu defendit; & artis suae fiduciâ jactare ausus fuit, si haberet ubi consisteret totam se moturum Terram. XIX. (1) Musica varios habet usus & delectationem non contemnendam. (2) Nam ut omittam omnis generis instrumenta quae singulari voluptate audientium animos afficiunt, facit ad contemperandos hominum affectus: generosas mentes excitat ad eminentiores actiones: morum ferociam 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 119 Hortensius’ Oration on the Dignity of Mathematics 119 XVIII. (1) I come to mechanics and statics, those producers of wondrous works, and famous examples of human skill, in which it is especially clear how much value the mathematical sciences have when they are applied to material affairs. (2) The applicability of mechanics not only overcomes all others in its extent, but also it is the most ancient among men, and employed from the beginning of the world. (3) It gives us the origin of agriculture, of the building of homes and huts, of the production of garments, and in addition to that of countless tools necessary for craftwork. (4) It is the one by which builders and architects raise the hugest masses of stones and beams, and direct them where they wish almost without any labour. It is the one by which stone cutters divide the hardest chunks of marble, by which sculptors, tailors, gold-workers, printers carve, cut, beat, print whatever material belongs to their craft, by which sailors direct large ships at their will by a small rudder. (5) Hydraulic and pneumatic instruments58 exist through static methods and procedures; the scales likewise, and everything that is carried in water. (6) By its use we inhabit our country in safety, because either we drain by machines waters that overflow and ruin the land, or we barricade and force out water about to return from the sea by sluices and dykes that are the wonder of foreigners, and because we have not constructed buildings of greater weight in damp locations, as in this city, unless on a pre-fortified foundation with stakes and piles. (7) Mechanics and statics create various wonders in matters designed for enjoyment or trickery, when [15] they contrive statues now walking, now emitting a voice like an oracle, and they construct automata59 by which we determine and measure the time or display the history of deeds. (8) Such were the tripods of Vulcan, according to Homer, which competed of their own accord,60 and the blackbirds of Ctesibios that imitated human voices,61 and the flying wooden dove of Archytas of Tarentum.62 (9) One man, Archimedes, can show us the power of both, in the glass sphere that he made, which continuously exhibited the motions of the sun, moon, and planets, analogous to the heavenly bodies; he alone drew a weight of 5.000 pounds by his own machines; and he led into the sea the royal ship, which the whole force of Sicily could not move; and he likewise defended for some time the Sicilians against the assaults of the Romans; and he dared to boast, in his faith in his skill, that if he had a place to stand, he could move the whole earth.63 XIX. (1) Music has various benefits, and a charm not to be despised. (2) For (I shall here pass over instruments of every kind that touch the minds of listeners with singular pleasure), it facilitates the tempering of men’s emotions. It excites noble minds to very great actions; it softens 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 120 120 History of Universities emollit & ad aequalitatem reducit. (3) Vnde Orpheus apud Poetas, feras, leones, tigres, sono testudinis placasse fingitur; & Amphion Thebarum conditor etiam saxa permovisse. (4) Magnas quoque vires habet in curandis morbis; quae quanquam hodie fere ignorentur, veteribus non fuere inexploratae. (5) Illi enim si Martiano Capellae credimus, febres & vulnera cantione curabant. (6) Asclepiades item tubâ surdissimis medebatur. (7) Theophrastus ad animi affectiones adhibebat tibias. (8) Thales Cretensis cytharae suavitate fugavit morbos ac pestilentiam. (9) Xenocrates organicis modulis liberavit lymphaticos. (10) cujus rei exemplum in sacris quoque literis est, ubi David cytharae cantu demulcet furibundum Saulem. XX. (1) Optica extendit se per universam Philosophiam, magistra ac directrix scientiae nostrae meritò dicenda. (2) De abstrusioribus enim Naturae miraculis philosophari non licet; nisi mentem adhibeamus opticis rationibus imbutam, quibus tuta, cum oculis simul in errorem non trahatur. (3) Sola optica est quâ instructus Philosophus discit nihil admirari, quae res & beatum facere potest & servare. (4) Quid dicam de portentosis effectis quos profert, cùm per speculorum composi- tiones pro una imagine refert centum; hominem capite deorsum verso ambulare facit; colorem faciei ad lubitum variat; radiis Solis ad cer- tum punctum collectis plumbum liquefacit, lignum ac stipulas accen- dit? quod Archimedem fecisse tradunt in obsidione Syracusanâ quando ad teli jactum naves Marcelli velut fulmine ictas combussit & in cineres redegit: cùm manes ab inferis revocat, & per machinas catoptricas Hectorem in conspectum sistit, aut Achillem, aut Helenam? (5) Ejus beneficio pictores in tabulà planâ eminentes colles, protuberantes arbores, & atria (quod mirum) introrsum ducentia repraesentant. (6) Senes oculos aetate debilitatos adhibitis perspicillis emendant. (7) Haec est quae scalas mundo injecit, & distantiam ac magnitudinem Solis, Lunae, Planetarum, astronomos edocuit. (8) Quae plura nos- tro seculo in lucem protulit, quàm toti Philosophorum scholae ante nos datum fuit cognoscere. (9) Ad instrumentum illud respicio, nuper inventum, quod Tubum Dioptricum vocant, quo res longè dissitas intuemur tanquam propinquas. [16] (10) Hoc enim clausa mundi atria reseravimus, & mirandum eruimus naturae thesaurum. (11) Maculas in Sole, illo lucis fonte, oriri; Lunae superficiem inaequalem & montibus ac vallibus obsitam esse; viam lacteam & stellas nebulosas multarum stellularum conglomeratione constare, didicimus. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 121 Hortensius’ Oration on the Dignity of Mathematics 121 the ferocity of behaviour and makes it smooth. (3) Wherefore it is said among the poets that Orpheus calmed wild animals, lions, tigers, by the sound of his lyre; and that Amphion the founder of Thebes even moved stones.64 (4) It also has great power to cure disease, which, although this is almost unknown today, was not unexplored by the ancients.65 (5) For they, if we are to believe Martianus Capella, cured fevers and wounds by incantation. (6) Asclepiades healed the deaf with the trumpet. (7) Theophrastus used the flute with mentally disturbed patients. (8) Thales of Crete dispelled diseases and pestilence by the sweetness of his cithara playing. (9) Xenocrates cured insane patients by playing on musical instruments.66 (10) There is an example of this in the Bible, where David soothed the maddened Saul by singing to the lyre.67 XX. (1) Optics extends itself through the whole of philosophy, worthy to be spoken of as teacher and director of our science. (2) For it is not allowed to philosophize on the more hidden wonders of nature, unless we use the mind imbued with optical theorems, by which it is kept safe, and not drawn into error along with the eyes. (3) It is only optics by which the learned philosopher learns to ‘marvel at nothing’, and this ‘can make a man happy and keep him so’.68 (4) What shall I say about the miraculous effects which it confers, when by the combination of mirrors it returns a hundred images for one; it makes a man walk upside-down; it varies the colour of a surface at will; it melts lead by the rays of the sun brought together at a certain point, and sets wood and straw on fire (which they say Archimedes did in the siege of Syracuse when by fling- ing a sunray he burned up the ships of Marcellus as if they had been struck by lightning and reduced to ashes69); when it calls back the souls of the dead from the underworld and by the use of catoptric machines puts Hector on view, or Achilles or Helen? (5) By its benefits, painters represent on a flat board high hills, bulging trees, and (what wonder!) halls that open inwards. (6) Old men improve their age-weakened vision by lenses. (7) This is the science that has put a ladder on the world and informed astronomers of the distance and size of the sun, moon and plan- ets. (8) This has brought more light to our century than was given to all the schools of philosophy before us to know. (9) I look back to that instrument, recently invented, which they call a dioptric tube [telescope], by which we see things sited far off as if they were [16] close up. (10) By this means, we have unlocked the closed halls of the world, and we have discovered the miraculous treasury of nature. (11) We have learned that spots arise on the sun, that fount of light; that the unequal surface of the moon is covered with mountains and valleys; that the milky way and mists of stars consist of a conglomeration of many small stars.70 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 122 122 History of Universities (12) Mundum in mundo deteximus, Iovem nempe quatuor comitatum Planetis certis intervallis & periodis eum circummeantibus. (13) Eodem instrumento, Venerem Planetarum lucidissimum Lunae instar in cornua abire; Saturni globum tergeminum esse; Mercurium corpore opaco cum caeteris Planetis lucem omnem à Sole recipere deprehendimus. (14) Quorum omnium apud veteres nec certa ulla mentio exstat, nec observationis vestigium XXI. (1) Astronomia, regina Mathematum, caelestis palatii supellectilem nobis recludit, & velut inde contemplantibus, aeternas nobilissimorum corporum periodos, immensas caelorum moles, & stupendum ordinem ob oculos ponit: cujus usum insignem esse nemo ferè est qui ignorat. (2) Temporum, annorum, dierum distinctio nulla esset, nisi Astronomi conversionem Solis Lunaeque observarent. (3) At verò quàm utile sit certam exstare temporum rationem in vita communi, rem paulò attentiùs consideranti nequit esse obscurum. (4) Sine ea nec agi quicquam inter homines nec dirigi potest, sed vita vivitur inordinata ac confusa, velut inter bruta animantia. (5) Lapsu seculorum tempestates anni confunduntur, aestas transit in hyemem, hyems in aestatem, quod ex neglectu Sacerdotum Romanorum propemodum contigerat post mortem Iulii Caesaris, ni Augustus emendatione anni Romani maturè occurrisset. (6) Historiarum fides incerta est & suspecta, nisi ab Astronomia robur suum accipiat & firmamentum. (7) Quis enim in tanta varietate annorum, Aegyptiorum, Atticorum, Arabicorum, Iudaicorum, Romanorum, sine observationibus & canonibus Astronomorum, non facilè se confundat, & Aerarum intervalla malè constituat aut connectat? (8) Eclipsium Solis ac Lunae consignatio, sola intricatissimas Chronologorum rixas dissolvit, quando annus & anni dies quâ res gesta est, à caelesti charactere omnis dubitationis experte, confirmatur. (9) Eaedem eclipses anni lunaris modulum prodidere, ut Aequinoctiorum observationes anni solaris; quibus omnis constat numeratio temporis; & Paschatis legitimè celebrandi ratio, quae universam Ecclesiam superioribus seculis exercuit, determinatur. (10) Aequinoctia & Eclipses habentur ab observationibus, observationes perficiuntur organis debitâ magnitudine in hunc finem praeparatis. (11) Ab his est omne quod ex Astronomia ad nos redit utilitatis. (12) Habuitque ea cura Reges ac Principes tantopere quondam sollicitos, ut Alexandriae publicis sumptibus Armillae ac Regulae exstructae sint, ad capienda Aequinoctia: & 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 123 Hortensius’ Oration on the Dignity of Mathematics 123 (12) We have uncovered a world in the world, indeed Jupiter, accompanied by four planets orbiting him at certain intervals and periods of time.71 (13) By this instrument, we perceive that Venus, brightest of the planets, goes away into horns like the moon72, that Saturn has a triple globe73, that Mercury with its obscure body receives, with the rest of the planets, all its light from the sun. (14) Among the ancients there is no mention whatsoever of all these matters nor any trace of their investigation. XXI. (1) Astronomy, the queen of the mathematical sciences, reveals the treasure of the heavenly palace to us, and, for those contemplating thence from that source, places before the eyes the eternal periods of the most noble bodies, the immense structures of the heavens, and their astonish- ing arrangement.74 Its conspicuous benefit almost no one is unaware of. (2) There would be no distinction of seasons, years, days, if astronomers did not observe the changes of the sun and moon. (3) But indeed how profitable it becomes, to have a secure calculation of time in communal life, cannot be unknown to one examining the matter a little more attentively. (4) Without this, nothing can be done or arranged among men, but life would be out of control and confused, as among mindless animals.75 (5) In the flow of the centuries, times of the year were confused, summer crosses over into winter, winter into summer, which would nearly have happened from the neglect of Roman priests after the death of Julius Caesar, if Augustus had not at the right time come to the rescue of the Roman year.76 (6) The reliability of historical knowledge is uncertain and suspect, unless from astronomy it receives its strength and foundation. (7) For who, in so great a variety of years [calendars]—Egyptian, Attic, Arab, Jewish and Roman—would not easily become confused without the observations and rules of astronomy, and construct and connect the intervals of the chronologies badly? (8) Only the documentation of eclipses of the sun and moon can dissolve the most intricate quarrels of chronology, whenever the year and the days of the year about which things are reported are confirmed by a celestial sign, free from all doubt. (9) These same eclipses produce a model of the lunar year, as observations of the equinoxes do for the solar year, on which the accounting of all times rest, and the calculation of when to celebrate Easter correctly, which has exercised the whole church in preceding centuries, is determined. (10) Equinoxes and eclipses are gained by observations; the observations are achieved by tools prepared in due size for this purpose. (11) From these everything of use in astronomy comes to us. (12) And in former times its diligence and care won over kings and princes, such that in Alexandria parallactic rulers and armillary spheres were constructed at public expense to capture the equinoxes;77 and 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 124 124 History of Universities Aristoteles ante omnia Orientis spolia ab Alexandro Magno petierit, ut captâ Babylone observationes Chaldaeorum mitterentur in Graeciam, quae erant prope bis mille annorum. (13) Noverat scilicet vir maximus, tantum ac tam necessarium esse observationum astronomicarum usum, ut absque iis nec scientia caelestis constitui valeat, nec certa ulla annorum quantitas obtineri. (14) Conducit quoque Astronomia lectioni veterum Poetarum & rei rusticae Scriptorum. (15) Poetae enim antequam Calendarium Romanum à Iulio Caesare ad normam motus Solaris foret restitutum, tempora arandi, serendi, navigandi, descripsere per ortus [17] & occasus certorum siderum, ut Plejadum, Sirii, Arcturi, & aliorum, prout multis in locis apud Hesiodum, Virgilim, Ovidium, Columellam, alios, videre est; quae difficulter ab ignaris hujus scientiae percipiuntur. (16) In Politicis & Militaribus non minorem dicenda est habere necessitatem. (17) Quippe causarum & caelestium eventuum ignoratio aut praenotio, magnos interdum exercitus aut pessumdedit aut servavit. (18) Nicias Atheniensium dux pavore eclipsis lunaris veritus classem portu educere, opes eorum afflixit. (19) Contra Dio Siciliae rex adversus Dionysium navigaturus, eclipsi Lunae cujus causae gnarus erat, nil territus, rem prosperè gessit. (20) Christophorus Columbus novi orbis inventor, in Iamaica insula commeatus penuriâ circumventus, praedictâ eclipsi quam futuram ex Astronomia noverat, barbaros quasi deorum iram incursuros, in metum egit; se suosque servavit. XXII. (1) Verum longè magis enitebit Astronomiae utilitas, si illi conjunxerimus Geographiam & Nauticam, quibus fundamenti vice subjicitur. (2) Geographia oculus historiarum est, sine qua non rectiùs versamur in narratione rerum quàm noctua ad Solem, aut vespertilio in luce diei. (3) Habet hoc rerum gestarum descriptio, ut nisi ad circumstantias locorum & regionis indolem contemperetur, vix moveat lectorem, aut veram referat historiae imaginem. (4) Quoties Carthaginem à Scipione excisam, aut miserabilem Crassi cladem ad Carras legimus, parum dignâ cognitione fruimur, nisi conterminas regiones & situm locorum ubi haec contigere simul habeamus perspectum. (5) Carthaginem nempe Tyriorum coloniam in littore Africae oportunissimo loco positam fuisse ex adverso Italiae, & ob hoc diu Romani imperii aemulam: Carras circà vastas, siccas, & arenosas Mesopotamiae solitudines 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 125 Hortensius’ Oration on the Dignity of Mathematics 125 Aristotle sought from Alexander the Great, beyond all the spoils of the ori- ent, that at the capture of Babylon the observations of the Chaldeans might be sent into Greece as they covered nearly 2000 years. (13) Surely that great man knew how important and how necessary the use of astronomi- cal observations was, that without them the science of the heavens could neither be established, nor any certain measurement of the year obtained. (14) Astronomy is also useful for the reading of old poets and agricultural writers. (15) For poets, before the Roman calendar was brought back to the regular movement of the sun by Julius Caesar, described the times of ploughing, sowing, and sailing through [17] the rising and setting of par- ticular stars, as the Pleiades, Sirius, Arcturus and others, as can be seen in many places in Hesiod, Virgil, Ovid, Columella and others; and this caused difficulties for those ignorant of this science. (16) In politics and military affairs astronomy must be said to be no less indispensable. (17) Why indeed, ignorance or advance notice of causes and celestial events has either destroyed or saved on occasion great armies. (18) Nicias the Athenian gen- eral brought ruin upon his fleet as in fear of an eclipse of the moon he was afraid to lead the fleet out of the harbour. (19) On the other hand, Dion king of Sicily, when about to sail against Dionysius, was not frightened by a lunar eclipse whose cause he understood, and waged his battle with suc- cess.78 (20) Christopher Columbus, the discoverer of the New World, when he was sailing around the island of Jamaica low on supplies, brought the barbarians into fear, as if they were visited by the wrath of the gods, and has saved himself and his men by predicting an eclipse which he knew from astronomy would happen.79 XXII. (1) Truly the usefulness of astronomy will shine much more, if we join geography and navigation, under which it is placed like a founda- tion. (2) Geography is the eye of history, without which we would be no more correctly engaged in the narrative of events than the owl with the sun or the bat with the light of day. (3) The description of deeds is such that unless it is tempered to the circumstances of the places and the native quality of the region, it would scarcely move the reader, or present a true image of history. (4) However many times we read of Carthage slashed by Scipio [146 B.C.] or the wretched defeat of Crassus against Carrhae [53 B.C.], we enjoy knowledge worth little, unless we have in mind the neighbouring areas and the sites of the places where the things happened. (5) Carthage, the Tyrian colony, was placed in a most advantageous location on the shore of Africa, opposite Italy, and on account of this a rival of the Roman Empire for a long time. Carrhae was sited along vast, dry, sandy Mesopotamian deserts, which were the 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 126 126 History of Universities sitas, quae Romanis militibus internecionis fuere causae. (6) Geographia totum orbem terrarum parvâ tabellâ comprehendit & exprimit; locorum situm & civitatum ordinem docet; mores hominum, soli caelique qualitatem tradit; climatum rationes ac proprietates describit; spectatorem denique domi remanentem & à peregrinantium periculis tutum, jucundissimo spectaculo per universum terrae marisque ambitum circumducit. (7) Sine hac nec Principum geruntur bella; nec Rerumpublicarum jura ac limites defenduntur; nec Mercatorum prosperè succedunt negotia. (8) Imperitia locorum multas perdidit militum copias, ducesque aliàs prudentissimos parvâ tabellâ comprehendit & exprimit; locorum situm & civitatum ordinem docet; mores hominum, soli caelique qualitatem tradit; climatum rationes ac proprietates describit; spectatorem denique domi remanentem & à peregrinantium periculis tutum, jucundissimo spectaculo per universum terrae marisque ambitum circumducit. (7) Sine hac nec Principum geruntur bella; nec Rerumpublicarum jura ac limites defenduntur; nec Mercatorum prosperè ac fortissimos in ruinam egit. (9) Eadem Mercatorum fortunas aliquoties subvertit: ut contra, situs & genius regionum locorumque certò exploratus, & mercium inibi nascentium nota conditio, plurimas iis contulit divitias. XXIII. (1) Alter haud postremus Astronomiae foetus est Nautica; quae à Phoenicibus ante multa secula exculta & per Thaletem Milesium in formam artis redacta, tandem se diffudit ad omnes mundi incolas. (2) Quippe Phoenices oportunitate maris ad navigandum allecti, primi ad Cynosuram & ejus conversiones circumpolares respicientes, vitae suae securitatem astronomicis fulcierunt praeceptis: quae deinde repertâ Pyxide nauticâ, & notatâ conversione acus Magneticae ad Septentrionem, universaliora evasere, & pluribus gentibus communia. (3) Haec ea est quae regiones toto mari divisas navibus adire docet, & peregrinos populos quaquaversum latè dispersos frequentare. (4) Cujus fiduciâ mortales inter monstra marina & saevas tempestates; inter horridas syrtes & mille [18] mortis discrimina, ingentes auri & argenti gazas, instabili Oceano committunt; tenui ligno opes Indorum & exoticas Afrorum merces domum comportant. (5) Navigatione non privatorum tantùm res, sed & civitatum ac regnorum aut stetere aut cecidere fortunae. (6) Tyrii ac Sidonii ob crebras navigationes in tantum divitiis ac potentiâ successu temporis accreverunt, ut quatuor nobiles deduxerint colonias, Leptim, Vticam, Carthaginem, & Mediterranei maris imperio navigando sibi vindicato, ad extremum Solis occasum Gades. (7) Alexandrini Tyro excisâ orientalium & occidentalium populorum commercia ad se traxere, ac diu soli possedere; donec paulatim in potentiam Venetorum abierint & Genuensium. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 127 Hortensius’ Oration on the Dignity of Mathematics 127 cause of the complete demise of the Roman soldiers. (6) Geography comprehends and expresses the whole world on a small map; it teaches the sites of places and arrangements of states; it hands down the customs of men and the quality of soil and sky; it describes the reasons and qualities of climates; finally, it conducts the spectator sitting at home and safe from the perils of travel, on a most pleasing spectacle through the whole circuit of land and sea. (7) Without this, princes would neither wage war, nor would the rights and limits of states be defended, nor would the business of merchants succeed prosperously. (8) Lack of experience of places has destroyed military power, and led the most prudent (in other respects) and brave leaders into ruin. (9) The same thing has repeatedly overturned the fortunes of merchants, as, on the other hand, exploring securely the site and attribute of regions and places and knowing the condition of the merchandise there has brought them great riches. XXIII. (1) Another, but not the least child of astronomy is naviga- tion, which was studied by the Phoenicians many centuries ago, and through Thales of Miletus was brought into the form of an art and finally it has spread to all the inhabitants of the world. (2) Indeed, the Phoenicians, drawn to navigation through the favourable position by the sea, were the first to refer to the Polar Star/Ursa minor and its circumpolar revolutions and guarded the safety of their lives by astronomical precepts. These astronomical precepts spread out more universally and were common to many peoples after the naval compass had been discovered and the turning of the magnetic needle to the north was known. (3) It is this that teaches to travel by ship to regions separated by a whole sea and to visit foreign peoples widely dispersed in all directions. (4) Trusting to this art, mortals, among sea monsters and savage storms, among rough straits and a [18] thousand dangers of death, commit huge treasuries of gold and silver to the unstable ocean, and convey home in a light piece of wood the wealth of India and exotic merchandise of Africa. (5) Not only individual affairs depend on navigation, but also both the continuation and the fall of the fates of kings and states. (6) The Tyrians and Sidonians, on account of frequent sea journeys, amassed such riches and power over time that they founded four magnificent colonies, Leptis, Utica, Carthage and, vindicating for themselves by navigation their rule over the Mediterranean, Cadiz, at farthest west. (7) When Tyre was destroyed, the Alexandrians drew to themselves trade with people east and west, and possessed it alone for a long time, until little by little they withdrew before the power of the Venetians and Genoese. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 128 128 History of Universities (8) Nempe opulentissima ac potentissima Venetorum Respublica ad tantum fastigium assurrexit studio & peritiâ navigandi: & superbus Genuensium splendor marinis in totum debetur negotiationibus. (9) Successere Hispani & Lusitani, qui longinquis tentatis regionibus, Orbe novo invento & occupato, veteri ad ultimos orientis fines lustrato, incredibile dictu quàm brevi tempore quantas collegerint opes, & quàm latè potentiam suam cum invidia Orbis Europaei extenderint. (10) Ac tandem nos Batavi (ne exterorum curiosus domestica praeteream) excusso Hispanorum jugo, ubi remotiores mundi oras adire incepimus, nullis istorum aut studio aut successu fuimus inferiores. (11) Quondam Oceanum Atlanticum vix ingressi vitam modicis navigationibus sustentavimus: ac tum qui Flandricas aut Canarias insulas viderant, tanquam ex alio orbe delati reduces cum admiratione conspiciebantur. (12) Sed postquam Matheseos notitia hîc accrevit, & ars Nautica uberius exerceri coepta est, universa maria navigationibus nostris implevimus: Indiae orientalis & occidentalis ditissima loca adivimus, vidimus, hosti extorsimus: Orbem circumnavigavimus; terras deteximus; freta nova invenimus: & ne quid maneret inexpoloratum, extra anni Solisque vias, inaudito exemplo, per mediam glaciem & plus quàm scythicas pruinas, aditum ad divites Cathajae & Sinarum regiones quaesivimus. (13) Ita omnis mercaturae forum intra Bataviam orbis angulum contraximus ac stabilivimus. (14) Quod faxit Deus ut tantum indies augmentum capiat, quantum emolumentum Mathesis attulit ad Nauticam, Nautica ad Mercaturam, & Mercatura ad Patriae nostrae solidam ac firmam prosperitatem. XXIV. (1) Sed tempus est ut vela contraham, & dum navigationis commoda prosequor, navem Orationis ulteriùs abripi non sinam. (2) Ad vos igitur me converto Magnifici atque Amplissimi D D. CONSVLES AC SENATORES; & gratias quàm maximas vobis ago, quod aures vestras tantisper Orationi nostrae commodare haud estis gravati, donec Matheseos Dignitatem & Vtilitatem pro viribus descripsi. (3) Eam nun porrò quâ possum reverentiâ vobis commendo. (4) Vrbem regitis toto orbe terrarum celeberrimam & potentissimam. (5) Ejus incrementum à studiis fuit Mathematicis, astronomicis imprimis & nauticis. (6) Ejus vigor, agite, ne iis unquam destituatur; sed quantum ipsa accrescit, tantum 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 129 Hortensius’ Oration on the Dignity of Mathematics 129 (8) Surely the very rich and powerful republic of Venice rose to so great a height by its study of and skill at navigation, and the proud splendour of Genoa is owed entirely to its sea trade. (9) The Spanish and the Portuguese came next, who, having attempted to reach the farthest regions, found and occupied the new world, illuminated the boundaries of the old east to the utmost—it is incredible to state in how short a time they collected so many riches and how widely they extended their power, to the envy of the European world. (10) Now, finally, we Dutch (lest I pay too much heed to foreign successes and pass over domestic ones), having struck off the Spanish yoke, when we began to approach the remotest shores of the world, were inferior in eagerness and success to none of the others.80 (11) At one time we hardly ever entered the Atlantic ocean, but sustained life on moderate voyages; and then those who saw the Flandric or Canary islands, they were looked at with admiration as though they were coming back from another world. (12) But after the knowledge of the mathematical sciences increased here, and the navigational art began to be practiced more intensively, we filled all the seas with our voyages; we came to the richest lands of the East and West Indies, saw them and snatched them away from the foreigners;81 we circumnavigated the globe; we discovered lands; we found new straits; and lest anything should be left unexplored, beyond the paths of the year and the sun, with no equal, through the middle of the icy and more than Scythian frosts, we searched for an approach to the rich regions of China.82 (13) So we have contracted the market of all merchandise within the angle of the world, Holland, and we have established it. (14) What God did so that Holland might daily expand so much, so much advantage have the mathematical sciences contributed to navigation, navigation to trade, and trade to the solid and firm prosperity of our country. XXIV. (1) But now it is time for me to strike my sails, and while I follow the advantages of navigation, not to haul the ship of my speech further away from its destination. (2) To you therefore I now turn, great and worthy members of the city council and Senators; and I give you very great thanks, that you were not at all unwilling to lend your ears to my speech for so long, while I described the dignity and advantage of the mathematical sciences to the best of my ability. (3) These I now further commend, as far as I can, in reverence to you. (4) You rule a city which is very famous and powerful in the whole world. (5) Its growth was from the study of the mathematical sciences, especially astronomy and navigation. (6) Use its [the city’s] energy so that they [the mathematical sciences] never lose their strength; but as much as it [the city] increases 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 130 130 History of Universities quoque favoris ac benevolentiae exhibete Mathemata profitentibus. (7) Vidistis à paucis retrò annis totius Belgii navigationes in Civitatem Vestram confluxisse; opes stupendas incolis cumulatas; Vrbis moenia ter aut ampliùs in immensum spatium prolata & extensa. (8) Magna haec sunt & quae [19] sic ipsam queant fatigare famam: sed majora erunt, si (ut laudabiliter incepistis) Artium ac Scientiarum promotionem adjiciatis, & Mathesin publicè doceri faciatis atque exerceri. (9) Iubete modo, nec deerit successus. (10) Exempla habetis nobilissimarum urbium, quae vobis hanc viam praeivere. (11) Tyrus illa, navigationum studio indefessa, omnis generis eruditione floruit, & inter alios Mathematicos Marinum fovit Geographum, Ptolemaeo toties memoratum. (12) Alexandria ut mercatoribus, sic & praestantissimis Mathematicis semper abundavit; & communis velut eorum fuit schola. (13) Haec nobis Timochares, Hipparchos, Ptolemaeos, Pappos, Theones dedit: haec instrumentis publicis Artem Astronomicam & hinc Nauticam indesinenter promovit. (14) Quid si Vos uti commercia Alexandrinorum, ita & Mathematum studia transferatis Amstelodamum? & organis erectis pro margaritis & gemmis, caducae fragilitatis thesauris; tot noctu lucentes gemmas, mundo coaevas, posteris annumerari mandetis; & nomina vestra ut magni olim heroes, Orion, Chiron, Hercules, quos sideribus adscripsit antiquitas, transmittatis in secula? (15) Vienna Austriae aluit suum Purbachium; Noriberga Regiomontanum, Waltherum, & Schoneros. (16) Veneti, Parisienses, Londinenses publicos Mathematum habent Professores. (17) Cur Amstelodamenses iis difficiliores audiant in promovendis Mathematis, aut boni publici minorem gerant curam? (18) Favoris ergo vestri radiis nascentem Mathematicae doctrinae segetem illustrate, fovete. (19) fructus videbitis insignes; & nos quicquid possumus, merita vestra celebrabimus & laudando per totum differemus mundum. XXV. (1) Vos quoque caeteri quotquot hic adestis Auditores, Theologi, Iurisconsulti, Medici; amate Mathesin & colite. (2) Audivistis ejus usum in omni disciplinarum genere esse permagnum: majora percipietis, si caeteris vestris studiis Mathematica velitis conjungere. (3) Eadem vos manet utilitas, quam toti humano generi Mathesin 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 131 Hortensius’ Oration on the Dignity of Mathematics 131 so much exhibit the favour and kindness to those publicly teaching the mathematical sciences. (7) You have seen a few years back the ships flowing into your city from all over Belgium; the inhabitants amassing marvellous wealth; the walls of the city extending and being brought forward three times or more into a huge space.83 (8) These things are great [19], and could wear out fame itself; but they will be greater if (as you have begun so laudably) you increase the promotion of the arts and the sciences, and bring it about that the mathematical sciences are publicly taught and practiced. (9) Only give the order, success will not fail. (10) You have examples from the noblest cities, which have paved the way for you. (11) Tyre itself, tireless in the study of navigation, flourished in every type of learning and, amongst other mathematicians, cherished the geographer Marinus, mentioned so many times by Ptolemy. (12) Alexandria abounded, as in merchants, so also in the most out- standing mathematicians, and there was as it were a communal school of them. (13) This [school] has given us Timochares, Hipparchus, Ptolemy, Pappus and Theon; it promoted by public instruments the art of astronomy and hence of navigation without limit. (14) What if you were to transfer to Amsterdam, as the commerce of Alexandria, so also the study of the mathematical sciences, and erect tools84 instead of pearls and gems, treasures of perishable fragility? What if you were to order to be counted for posterity so many jewels gleaming at night [stars], as old as the world, and transmit your names to them, like the former great heroes, Orion, Chiron, Hercules, whom antiquity has written up in the stars, for later generations? (15) Vienna in Austria fostered its citizen Peurbach, Nuremberg did the same for Regiomontanus, Walther and Schoener. (16) The Venetians, the Parisians, the Londoners, have public professors of the mathematical sciences. (17) Why should the citizens of Amsterdam find it harder than they to agree to promote the mathemat- ical sciences, or take less care for the public good? (18) Therefore shine with the beams of your favour on the crops of mathematical learning, now springing up; cherish them! (19) You will see remarkable fruit, and we can do something, we will celebrate your merits and spread them in your praise over the whole world. XXV. (1) You also, the other listeners, as many as are present here, theologians, judges, doctors: love the mathematical sciences and venerate them. (2) You have heard that their benefit is very great in every kind of discipline; you will learn greater things, if you wish to join the mathematical sciences to your other studies. (3) You will find the same advantage we have shown the mathematical sciences to confer to the 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 132 132 History of Universities conferrre multis ac variis in rebus ostendimus. (4) Eandem quoque ejus voluptatem experiemini, quam sensere Thales, Pythagoras, Archimedes, Ptolemaeus, aliique viri in erudito hoc pulvere non segniter versati. (5) Nec minus erit è re vestra, Mathematum notitiâ conspicuos esse, quàm aliarum Scientiarum, quarum non mediocrem vobis jam collegistis thesaurum. XXVI. (1) Vos Merctores, jucundum habebitis ea studia tractare, quorum beneficio merces vestrae vasto pelago commissae tutae eunt redeuntque. (2) Nolite objicere vitam vestram curis ac sollicitudine plenam, Mathematicas contemplationes non admittere: invenietis subinde horulam quâ tetricas negotiorum molestias amoenitate Matheseos diluatis. (3) Thales unus è septem Graeciae Sapientibus, & studiis Mathematicis vacavit, & Mercaturae. (4) Praevisâ enim olei ubertate omnia Milesiorum praela ac trapeta conduxit; iisque postea inenti pretio elocatis, ostendit amicis, non tantùm sapientem cùm velit ditescere posse, sed & contemplationes Philosophicas Mathematicasque, à Mercatura minimè esse alienas. (5) Plato quoque insignis fuit mathematicus, & Hippocrates Chius mercator industrius: at nihilominus ille in Aegypto olei mercatum exercuit; hic peritiâ Mathematum certavit cum Thalete, Pythagora, & ipso Euclide. XXVII. [20] (1) Vos denique doctissimi ac studiosissimi Iuvenes, qui aut literarum aut Philosophiae studiis incumbitis, & ad summam doctrinae arcem contenditis; Mathematum studia nolite negligere. (2) Plato & Aristoteles exemplis Mathematicis Philosophiam suam illustrarunt, quia ea aetate adolescentes jam tum perceperant Mathemata antequam ad Physicam aut Methaphysicam admitterentur. (3) Hoc & vos agite, si Platonem, si Aristotelem sequi vultis, & Philosphiam non perfunctoriè excolere. (4) Poetae item, Historici & Oratores, Mathemata tractare amant. (5) eorum doctrinam ut percipiatis, sit vobis à Mathesi studiorum initium, & brevi ingentes facietis progressus. (6) Quod si bene semel coeperitis, ego quantum muneris mei postulabit ratio, proposito vestro spondeo me haud defuturum. DIXI 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 133 Hortensius’ Oration on the Dignity of Mathematics 133 whole human race in many and varied affairs. (4) You will also experience their delight, which Thales, Pythagoras, Archimedes, Ptolemy and other men felt, who were very well versed in this learned arena. (5) Nor is it of less moment in your affairs to be conspicuous in the knowledge of the mathematical sciences, as in that of other sciences, whose treasury, not small, you have already put together. XXVI. (1) You merchants will have a pleasant time in employing these studies, by whose benefit your wares entrusted to the vast sea go out and return safely. (2) Do not object that your lives are full of cares and anxiety, and cannot admit mathematical contemplation; you will often find a small space of time in which you may dilute the worrisome troubles of business with the pleasure of the mathematical sciences. (3) Thales, one of the Seven Wise Men of Greece85, had time for both mathematical studies and trade. (4) For, having foreseen the richness of the olive crop, he hired every press and mill in Miletus; and afterwards when he leased them out at huge prices, he showed his friends not only that a wise man could be rich if he chose, but also that philosophical and mathematical studies are not at all foreign to trade.86 (5) Plato also was a famous mathematician, and Hippocrates of Chios a busy merchant; but nonetheless the former exercised trade in Egyptian olive oil, the latter contested in mathematical skill with Thales, Pythagoras, and Euclid himself.87 XXVII. [20] (1) Finally, you learned and eager youths, who are plunging into the studies of letters or philosophy, and are striving for the greatest height of learning: do not neglect the study of the mathematical sciences. (2) Plato and Aristotle illustrate their own philosophy with mathemat- ical examples, because in that time youths had already learned mathematics before they were admitted to physics and metaphysics. (3) Do likewise, if you wish to follow Plato, if you wish to follow Aristotle, and if you do not want to cultivate philosophy just superficially. (4) Poets likewise, historians and orators love to reflect on the mathematical sciences. (5) So that you may perceive their doctrine, let your studies begin with the mathematical sciences and in a short time you will make great progress. (6) Once you have made a good beginning, I solemnly promise that I will hardly fail your purpose, whatever duties it will demand of me. I have now finished my speech. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 134 134 History of Universities REFERENCES 1. i.e. the Athenaeum illustre. 2. Grootenhuys (1573–1646) had been appointed trustee of the Athenaeum illustre in 1632 along with Albert Coenraetsz: Elias (1963), i., 275; Rademaker (1981), 242. 3. The city affairs were conducted by the Town Council of thirty-six councillors, nine magistrates and four burgomasters or mayors: Burke (1994), 17. In 1634 the four burgomasters were Andries Bicker (1586–1652), Dirk Bas (1569–1637), Jan Geelvinck (1582–1666) and Jacob Backer (1572–1643); on them cf. Elias (1963), i., 346–8, 245f, 352f and 238 respectively. 4. Cicero, Pro Roscio Amerino 4, 9. Also referred to by Erasmus (1971), 97. 5. Reference to his colleagues Caspar Barlaeus who taught philosophy and Gerard Joannes Vossius who taught history; cf. our introduction. 6. Apollo, son of Zeus, had since classical times been considered as protector of the sciences in general, in particular of astronomy, mathematics and, naturally, music. 7. According to Greek myths Athene, whose residence was sometimes called arx Palladis, had early on taught the science of numbers. In classical times she became the Goddess of wisdom. Her symbol, the owl, still carries this connotation. 8. Reference to Proclus: Barozzi (1560), 27: ‘Haec itaque Mathesis est, sive disciplina, quae aeternarum in anima rationum reminiscentia est’. Cf. Proclus (1992), 38; on Proclus’s Commentary on the First Book of Euclid’s Elements see Mueller (1987). 9. As reported by Diogenes Laertius: Lives of Eminent Philosophers, IV, 10: Diogenes Laertius (1925), i., 384f; cf. the article on Xenocrates in Der Neue Pauly xii./2, 620–3. 10. Proclus comments on this: Barozzi (1560), 21; Proclus (1992), 29f. 11. Logistica (in other texts supputatio or arithmetica practica), which we render as practical arithmetic, is sometimes translated as computation or calculation; cf. Masi (1983), 148. 12. Clavius was among those who identified these six mixed parts: Clavius (1611–12) [1574], i., 3f; he in turn followed Barozzi: Barozzi (1560), 22f; Proclus: Proclus (1992), 31f; cf. Feldhay (1998), 96f. 13. In these self-confident words the whole might of the quaestio de certitudine mathematicarum reverberates; cf. introduction. 14. Clavius had stressed this aspect repeatedly: Clavius (1611–12) [1574], I, 5; (1611–12) [1581], iv., Praefatio, [1]. On the meanings and context of the term probabile see Daston (1998); Hacking (1993), 18–30. 15. Erasmus used the expression ‘solo naturae ductu’ in his Praise of Folly: ‘Thus the happier branches of knowledge are those which are more nearly related to folly, and by far the happiest men are those who have no traffic at all with any of the sciences and follow nature for their only guide’. Erasmus (1971), 111, No. 32. On the epistemology of science as a hunt see Eamon (1994). 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 135 Hortensius’ Oration on the Dignity of Mathematics 135 16. Plato, Timaeus 53c: Plato (1914–1927), ix, 126f: ‘. . . yet, inasmuch as you have some acquaintance with the technical method (kata paideusin hodon) which I must necessarily employ in my exposition, you will follow me’. Another general reference can be found in Plato, The Republic 522c–527c, especially 526b: Plato (1914–1927), vi., 166f: ‘Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were’. 17. This view had been widely spread by Diogenes Laertius, Lives of Eminent Philosophers I, 24f: Diogenes Laertius (1925), i., 26f and Proclus: Barozzi (1560), 36f; Proclus (1992), 52; cf. e.g. Clavius (1611–12) [1574], i., 4: ‘Hanc Thales Milesius ex Aegypto in Graeciam primus transtulisse fertur’. 18. Again, the story goes back to Proclus: Barozzi (1560), 37; Proclus (1992), 51. It is also told, e.g., in the collected works of Archimedes: Archimedes (1615): Archimedis vita, 6 and by Clavius: Clavius (1611–12) [1574], i., 7); cf. Gorman (2003), 40, 113. 19. The well-known tradition of patronage of the mathematical sciences by emperors, kings, and princes was regularly invoked by those seeking legitimization of the mathematical sciences; cf. Remmert (2006), ch. 6. On the close relationship between the Stadholder Maurice of Orange and Simon Stevin see van Berkel, Klaas, Stevin and the Mathematical Practitioners (Berkel/Helden/Palm, 1999), 13–36; Hopper (1982). 20. The ‘voluptas’ of the mathematical sciences is also a wide-spread topos, usually referred back to Plato as e.g. in Clavius, Christopher: In disciplinas mathematicas prolegomena: Clavius (1611–12), i., 3–9, here 6: ‘Testatur, magnam animi voluptatem ex his artibus percipi, Divinus Plato in 7. de Rep. [. . .]’. 21. Plato, The Republic VII, 523a–525a: Plato (1914–1927), vi., 152–161:    in 523a and  in 525a. 22. A hecatomb is a sacrifice of several oxen. Both stories have been related by Diogenes Laertius, Lives of Eminent Philosophers I, 24 and VIII, 12: Diogenes Laertius (1925), i., 24–27 and ii., 330f. 23. Hortensius’s account of this story may be based on Vitruvius: On Architecture IX, preface, 9–12: Vitruvius (1931), ii., 202–7. 24. Reference to an epigram, attributed to Ptolemy, preceding book I of his Almagest: ‘Well do I know that I am mortal, a creature of one day. But if my mind follows the winding paths of the stars then my feet no longer rest on earth, but standing by Zeus himself I take my fill of ambrosia, the divine dish’ (translated from the German in Ptolemy (1963), i.). Rarely had humans been granted the privilege of consuming ambrosia, the food of immortality. Apollo had been nursed on nectar and ambrosia by the nymph Themis and Athene had granted Achilles this privilege: Homer, Iliad XIX, 347–54). 25. As reported by Plutarch, Moralia 1094b: Plutarch (1927–69), xiv., 66f. 26. In the seventeenth century the Latin exercitia could carry religious undertones as, in particular, in the Ignatian spiritual exercises in the Society of Jesus, amongst whose members the mathematical sciences, too, were 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 136 136 History of Universities highly cultivated. Kepler and other seventeenth-century scholars saw their work and exercises in the mathematical sciences as a way to serve God; cf. Remmert (2005); for the case of Descartes see Jones (2001). 27. Possibly a reference to Horace who speaks about ‘a fragment of the divine spirit’ Saturae II, 2, 79: Horace (1929), 142f: ‘divinae particulam aurae’. Hortensius’s position brings to mind Galileo’s insinuation in the Dialogue Concerning the Two Chief World Systems of 1632 that pure mathematics was the only way open to the human intellect to gain knowledge equivalent in quality to divine knowledge: Remmert (2005). 28. As reported by Diogenes Laertius, Lives of Eminent Philosopher, II, 7: Diogenes Laertius (1925), i, 136f. 29. The Latin excolo has several undertones which are of importance in Hortensius’s campaign to gain patronage for the mathematical sciences: to cultivate, to honour, to enhance the reputation and to bring to perfection. 30. The juxtaposition of the book of revelation and the book of nature was standard in the seventeenth century, and their relation stood at the core of many debates, e.g. the Galileo affair; cf. Biagioli (2003); Blumenberg (1981); Bono (1995); Curtius (1984), 323ff; Harrison (1998), 193ff; Pedersen (1992); Scholz (1993); cf. below X.(3). 31. The image of arithmetic and geometry as wings of the mind/astronomy was wide-spread in the sixteenth and seventeenth centuries. Andrea Argoli for instance used it in a frontispiece: Argoli (1667). Philipp Melanchthon refers to it in connection with Plato’s Phaedrus: Melanchthon (1536), Av[r]: ‘Sunt igitur alae mentis humanae, Arithmetica et Geometria’. While the wings of the soul are important in the Phaedrus (246a–e), we have not been able to trace a precise reference to arithmetic and geometry as wings in Plato; cf. Jardine (1984), 186, fn 168. 32. The imagery of the world being a machine was wide-spread; cf. below VIII.(12): ‘mundi machinam’; cf. Ovid, Metamorphoses I, 257f: Ovid (1921–84), i., 20f): ‘quo mare, quo tellus correptaque regia caeli ardeat et mundi moles obsessa laboret’ (‘when sea and land, the unkindled palace of the sky and the beleaguered structure of the universe should be destroyed by fire’). 33. Flavius Josephus, Jewish Antiquities I, 69–71: Josephus (1930–1965), iv., 32f. 34. Wisdom of Solomon, 11, 20. 35. As reported by Plutarch: Moralia 718b–720c: Plutarch (1927–1969), ix., 118–31; cf. Ohly (1982), Mueller (2005). 36. Cf. the dedicatory letter of Federico Commandino to Cardinal Ranuccio Farnese: Archimedes (1558). Plutarch Moralia 718e: Plutarch (1927–69), ix., 121; Lives, ‘Marcellus’ XIV, 5f: Plutarch (1914–26), v., 471 mentions that Plato was incensed at their taking recourse to mechanical arrangements in order to tackle a geometrical problem. 37. The Byzantine commentator Johannes Tzetzes reported this in the twelfth century Chiliades/Book of Histories VIII, 972–3, quoted in Selections Illustrating the History of Greek Mathematics (1939–41), i., 386f. Cf. Elias 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 137 Hortensius’ Oration on the Dignity of Mathematics 137 Philosophus, In Aristoteles Categorias Commentaria, ed. A. Busse (Berlin, 1900), 118:18, and Joannes Philoponus, In Aristotelis de Anima Libros Commentaria, ed. Michael Hayduck (Berlin, 1897), 117: 29. Copernicus programmatically put this warning on the title-page of his De revolutionibus orbium coelestium in 1543: Copernicus (1543); cf. Mueller (2005). 38. Plato: Meno 82b–85e (Plato 1914–1927, II, 304–321); Plato: Theaetetus 147e–148a (Plato 1914–1927, VII, 26f). 39. Plato: Timaeus 34b–36d (Plato 1914–1927, IX, 64–73); also mentioned by Proclus (Proclus 1992, 14). 40. Again, Hortensius refers to the juxtaposition of the book of nature and Holy Scripture; cf. above VII.(10). 41. Hortensius refers to Psalm 8,4: ‘When I look up at thy heavens, the work of thy fingers, the moon and the stars set in their place by thee’ [quoniam videbo caelos tuos: opera digitorum tuorum lunam et stellas quae tu fundasti]. 42. Psalm 8,1. 43. Psalm 19,2 [Vulgata 18,2]: ‘Caeli enarrant gloriam Dei et opus manus eius adnuntiat firmamentum’. 44. Ezekiel 40–48; cf. the famous reconstruction of the Temple of Solomon by Juan Bautista Villalpando Prado and Villalpando (1596–1604). 45. Cf. Revelation 21, 9–21. 46. Cf. the groups of seven in Daniel 9, 24–27. 47. Cf. Revelation 7, 1–8. 48. On the mathematical sciences and biblical exegesis see Remmert (forthcoming). 49. Rivault tells about Astraea in similar words in his introduction ‘Nobilibus Gallis pro mathematicis’ to Archimedis (1615), 11: ‘Denique Astraea ipsa pro Tribunali sedens, & distributivae & commutativae iustitiae Mathesim nec- essariam adiucat, [. . .]’. On Astraea/Justitia see Ovid, Metamorphoses I, 127–31 and 149f: Ovid (1921–84), i., 10–13; Yates (1975), 29–87; on Astraea in early modern astronomy see Remmert (2003), 247–95, 281f, 286. 50. The works of Galen (129–199), the last famous physician and medical writer of antiquity, were still widely influential in early modern medicine. 51. For the importance of the mathematical sciences to early modern medicine see e.g. the exposition by Argoli (1639); cf. Sudhoff (1902). 52. Yearly feast in Athens, cf. Der Neue Pauly viii., 611–26. 53. This was one of the highest honours conferred in Athens mentioned, e.g., in Plato, Apology 36d: Plato (1914–1927), i., 128f; cf. Der Neue Pauly x., 493. 54. Cf. Hippocrates (1990), 106f, No. 25: ‘Decree of the Athenians’: ‘The Council and the People of Athens have decreed: Whereas Hippocrates of Cos, being a physician and descended from Asclepius, has shown great con- cern for the safety of the Greek people, And whereas on the occasion of a plague coming from the land of the barbarians towards Hellas, he sent out his pupils to different places to proclaim what therapies they had to use to keep themselves safe from the imminent plague, and, in order that 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 138 138 History of Universities medical science bequeathed to the Greeks would preserve safe those that were ill from it he generously published his writings on medical science because he wanted there to be many physicians who saved people, [. . .]. Therefore [. . . it] is decreed by the people to initiate him into the great mysteries at public expense as was done with Heracles, the son of Zeus, and to crown him with a gold crown worth one thousand gold pieces, and to proclaim the crown at the great Panathenaia at the athletic competition, [. . .] And that there be for Hippocrates citizenship, and sustenance in the Prytaneum for his lifetime’. On Hippocrates and the plague see also Hippocrates (1990), 116–19, No. 27.7 and Pinault (1992), 35–60). 55. Paraphrase of Polybius, The Histories IX, 26a: Polybius (1922–7), iv., 60–3. 56. Der Neue Pauly x., 645–8, on war see especially 647. 57. Conqueror of Athens, 307 B. C. 58. In particular, organum refers to a hydraulic or water organ as described by Kircher (1650) and many others in the seventeenth century; cf. Gouk (1999). 59. On automata see Bedini (1964); Hankins and Silverman (1995); Karafyllis (2004); Marr (2004); Mayr (1986); Wolfe (2004). 60. Reference to the tripods of Hephaestus/Vulcan in the Iliad, XVIII, 369ff: Homer (1999), ii., 314f. The tripods were a very prominent topic in mechanics and in particular in the literature on automata. Vulcan was considered as the founding father of the art of automata. A standard reference for the early modern period is de Caus (1615). 61. Ctesibios, who lived in Alexandria in the beginning of the third century B.C., was counted among the foremost engineers along with Heron and Archimedes. One of the main sources on his many inventions is Vitruvius, On Architecture, IX, 8, 2–7 and X, 7, 1–4: Vitruvius (1931), ii., 256–61, 310–13, who also mentions the blackbirds: De architectura, X, 7, 4. On Ctesibios see Drachmann (1948); cf. the article in Der Neue Pauly vi., 876–8. 62. Fragmente der Vorsokratiker, ‘Archytas Leben’, A.10a Fragmente der Vorsokratiker 1954, I, 424f; cf. the article in Der Neue Pauly i., 1029–31. 63. Cf. Plutarch, Lives, Marcellus 14, 7: Plutarch (1914–26), v., 473. 64. As reported by Martianus Capella whom Hortensius quotes below: Martianus Capella (1878), 908. 65. On the importance and traditions of musical healing see Horden (2000); Kümmel (1977). 66. Close paraphrase of a passage in book IX ‘De harmonia’, of Martianus Capella’s De nuptiis philologiae et mercurii, IX, 926; translation quoted from Martianus Capella (1977), 358; for the Latin see Martianus Capella (1878), 493. 67. Cf. 1 Samuel 16, 23. 68. Hortensius paraphrases Horace Epistulae I, 6, 1f: ‘Nihil admirari prope res est una, Numici, solaque quae possit facere et servare beatum’: Horace (1929), 286f. 69. This famous story had first been told by Lucian in the second century (cf. Selections Illustrating the History of Greek Mathematics (1939–41), ii., 20f. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 139 Hortensius’ Oration on the Dignity of Mathematics 139 See also Cassius Dio, Roman History: Cassius Dio (1914–27), ii., 171. On its credibility and tradition see Knorr (1983); Mills and Clift (1992); Simms (1977, 1994). It was of immense importance for the legitimization of the mathematical sciences in the seventeenth century: Remmert (1998), 201–205). 70. All due to the observations of Galileo in 1609–10 and published in his Sidereus Nuncius of 1610. Hortensius, too, was a Copernican. 71. To Galileo the four moons of Jupiter had been a precedent of the Copernican system as they clearly did not circulate around the earth. 72. Meaning that Venus has phases like the moon. 73. In his Letters on Sunspots of 1613 Galileo had described Saturn as being touched by two small stars, thus having the appearance of a triple globe: Galileo (1957), 101f. 74. Again, ‘immensas caelorum moles’ carries undertones of celestial machin- ery; cf. above VII.(11) and VIII.(12). 75. On this classical topos see Wolkenhauer (2005). 76. The Julian calendar was used from 45BC on. It consisted of 12 months of 30 or 31 days each, resulting in a year of 365 days. Once every four years an additional day was put in after February 24th (leap year). However after the death of Julius Caesar the rule was used incorrectly, resulting in leap years every three years. The first leap year was 45BC, then every year divisible by three until 9BC. These (false) additional leap years were corrected by Augustus by having the next leap year in 8AD and returning to the four year cycle. See Radke (1990), 67–8. 77. On parallactic or Ptolemy’s rulers, see Evans (1990), 241f; cf. the description of the parallactic ruler in Ptolemy’s Almagest: Toomer (1998), 244–7. 78. On Nicias and Dion see Plutarch, Lives, ‘Nicias and Crassus’, 23: Plutarch (1914–26), iii., 291. Nicias and his lack of astronomical knowledge are also discussed by Polybius, The Histories IX, 19: Polybius (1922–7), iv., 44f, whom Hortensius quotes earlier (XIV, 8–10). 79. Reference to Columbus’s prediction of an eclipse in February 1504. This story can also be found in the section on the usefulness of astronomy (‘De utilitate astronomiae’) in Clavius’s Sacrobosco commentary: Clavius (1611–12) [1570], iii., 4f. 80. On the Dutch colonial empire and the Dutch primacy in world trade in the early modern period see Israel, (1989). 81. Allusion to Caesar’s Veni, vidi, vici. 82. Reference to the search for a North-East passage to China and in particular the famous expedition of Willem Barents in 1596/7. Whale-hunting, too, brought the Dutch to the North, in particular to Spitzbergen and Greenland: Israel (1989), 111f. Kepler alluded to this context in his Elegia in obitum Tychonis Brahe: ‘Uranie Batavos saeva servavit ab Arcto | Quos fugit multo tempore clausa dies’: Kepler (1992), 24, lines 145f. 83. Allusion to the considerable territorial expansion of the city of Amsterdam in the early seventeenth century; on this see ’t Hart (2001), in particular 130–2. 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 140 140 History of Universities 84. He refers to the tools mentioned in XXI.(12). 85. The Seven Wise Men of Greece were first mentioned by Plato, Protagoras 343a: Plato (1914–1927), ii., 196f; cf. Diogenes Laertius, Lives of Eminent Philosophers i., 22–44 (Thales): Diogenes Laertius (1925), i., 22–47; Snell (1971). 86. This story is recounted by Aristotle: Politics 1259a9: Aristotle (1944), 54–7. 87. The latter is reported by Philoponus in his Commentary on Aristotle’s Physics (A 2 (185a16)); cf. Selections Illustrating the History of Greek Mathematics (1939–41), i., 234f. That Plato sold oil is reported by Plutarch, Lives, ‘Solon’, II, 4: Plutarch 1914–26, i., 408f, and had already been quoted by Barlaeus in his inaugural lecture on the Wise Merchant: Barlaeus (1632), 31. BIBLIOGRAPHY I. Works of Hortensius I.1 Publications of Hortensius Hortensius, Martin, ‘Ad Candidum ac Benevolum Lectorem Praefatio’, in: Lansbergen, Philipp: Commentationes in motum terrae Diurnum, & Annuum; et in verum adspectabilis caeli typum. In quibus    ostenditur, Diurnum, Anuumque Motum, qui apparet in Sole, & Caelo, non deberi Soli, aut Caelo, sed soli Terrae: simulque Adspectabilis Primi Coeli Typus, ad vivum exprimitur. Ex Belgico Sermone in Latinum versae, à Martino Hortensio Delfensi: unà cum ipsius Praefatione, in quâ Astronomiae Brahaeanae Fundamenta examinatur; & cum Lansbergianâ Astronomiae Restitutione conferuntur (Middelburg, 1630) [38 pages by Hortensius on recent developments in astronomy (Brahe, Kepler etc.)] Hortensius, Martin, Responsio ad additiunculam D. Ioannis Kepleri, Caesarei Mathematici, praefixam Ephemeridi eius in Annum 1624. In qua Cum de totius Astronomiae Restitutione, tum imprimis de observatione Diametri Solis, fide Tubi dioptrici, Eclipsibus utriusque Luminaris, luculenter agitur (Leiden, 1631) Hortensius, Martin, ‘In viri clarissimi Philippi Lansbergii Opus astronomicum tabulasque motuum caelestium dudum ab omnibus desideratas Carmen, quo ortus & progressus astronomiae ad nostra usque tempora ostenditur’ in Lansbergen, Philipp: Tabulae motuum coelestium perpetuae; Ex omnium temporum Observationibus constructae, temporumque omnium Observationibus consentientes. Item Novae & genuinae Motuum coelestium theoricae & Astronomicarum observationum Thesaurus (Middelburg, 1632), **1r–**4v [i.e. 18–24] Hortensius, Martin, Dissertatio de Mercurio in sole viso et venere invisa (Leiden, 1633) Hortensius, Martin, Oratio de dignitate et utilitate Matheseos. Habita in illustri Gymnasio Senatus Populique Amstelodamensis (Amsterdam, 1634) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 141 Hortensius’ Oration on the Dignity of Mathematics 141 Hortensius, Martin, Oratio de oculo eiusque praestantia. Habita in illustri Gymnasio Amstelodamensi (Amsterdam, 1635) Hortensius, Martin, ‘Dissertatio de studio mathematico recte instituendo’ in Grotius, Hugo (ed.), De Omni genere studiorum recte instituendo dissertationes (Leiden, 1637), 111–33 [reprint in: Grotius, Hugo (ed.), De Studiis instituendis (Amsterdam, 1645), 585–93] I.2 Translations into Latin by Hortensius Blaeu, Willem Janszoon, Institutio Astronomica De usu Globorum & Sphaerarum Caelestium ac Terrestrium: Duabus partibus adornata, una, secundum hypothesin Ptolemaei, altera, juxta mentem N. Copernici, per terram mobilem. Latinè reddita à M. Hortensio, in Ill. Amsterdamensium Schola, Matheseos Professore (Amsterdam, 1634) [various reprints] Lansbergen, Philipp, Commentattiones in motum terrae Diurnum, & Annuum; et in verum adspectabilis caeli typum. In quibus    ostenditur, Diurnum, Anuumque Motum, qui apparet in Sole, & Caelo, non deberi Soli, aut Caelo, sed soli Terrae: simulque Adspectabilis Primi Coeli Typus, ad vivum exprimitur. Ex Belgico Sermone in Latinum versae, à Martino Hortensio Delfensi: unà cum ipsius Praefatione, in quâ Astronomiae Brahaeanae Fundamenta examinatur; & cum Lansbergianâ Astronomiae Restitutione conferuntur (Middelburg, 1630) [reprinted: (Middelburg, 1653), and in Lansbergen’s Opera omnia (Middelburg, 1663)] Snel, Willebrord, Doctrinae triangulorum canonicae libri IV, quibus canonis sinum constructio, triangulorum tam planorum quam sphaericorum expedita dimensio breviter ac perspicuae traditur: post morte autoris in lucem editi a Martino Hortensio, qui istis problematum geodaeticorum & sphaericorum tractatus singulos adjunxit (Leiden, 1627) II. Sources Archimedis opera quae extant, ed. David Rivault (Paris, 1615) Archimedis opera, ed. Federico Commandino (Venice, 1558) Aristotle, Politics. With an English Translation by H. Rackham. (London/ Cambridge (Mass.), 1944) Barlaeus, Caspar, Mercator sapiens, sive Oratio de conjungendis Mercaturae & Philosophiae studiis (Amsterdam, 1632) Cassius Dio, Roman History. With an English Translation by Earnest Cary on the Basis of the Version of Herbert Baldwin Foster (9 vols, London/Cambridge (Mass.), 1914–27) Clavius, Christoph, In disciplinas mathematicas prolegomena in his Opera math- ematica (5 vols, Mainz 1611–12), i., 3–9 [first published with his Euclidis Elementorum Libri XV of 1574] Clavius, Christoph, Gnomonices libri octo in Opera mathematica (5 vols, Mainz, 1611/12), iv. [first published in 1581] 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 142 142 History of Universities Copernicus, Nicolaus, De revolutionibus orbium coelestium, libri VI (Nuremberg, 1543) De Caus, Salomon, La raison des forces mouvantes (Frankfurt, 1615) Diogenes Laertius, Lives of Eminent Philosophers (2 vols, London/Cambridge (Mass.), 1925) Erasmus, Desiderius, Praise of Folly and Letter to Martin Dorp. Translated by Betty Radice (Harmondsworth, 1971) Ficino, Marsilio, Opera (2 vols, Basel, 1561) Fragmente der Vorsokratiker, ed. H. Diels and W. Kranz (3 vols, Berlin, 1954) Galilei, Galileo, Le opere di Galileo Galilei. Edizione Nazionale diretta da Antonio Favaro (20 vols, Florence, 1890–1909) Galilei, Galileo, Discoveries and Opinions of Galileo. Translated with an Introduction and Notes by Stillman Drake (New York, 1957) Gassendi, Pierre, Mercurius in Sole Visus et Venus Invisa (Paris, 1632) Gassendi, Pierre, Opera omnia (6 vols, Lyons, 1658) Hippocrates, Pseudepigraphic Writings. Edited and Translated with an Introduction by Wesley D. Smith (Studies in Ancient Medicine 2, Leiden/New York, 1990) Homer, Iliad. With an English Translation by A.T. Murray. Revised by William F. Wyatt (2 vols, London/Cambridge (Mass.), 1999) Horace, Satires, Epistles, ars poetica. With an English Translation by H. Ruston Fairclough (London/Cambridge (Mass.), 1929) Josephus, Jewish Antiquities, with an English translation (7 vols, London/ Cambridge (Mass.), 1930–65) Kepler, Johannes, Keplers Elegie In obitum Tychonis Brahe. Übertragung und Kommentar von Hans Wieland (Munich, 1992) (Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Abhandlungen, Neue Folge, Heft 168 (Nova Kepleriana, Neue Folge—Heft 8)) Kircher, Athanasius, Musurgia Universalis (Rome, 1650) Martianus Capella, De nuptiis philologiae et mercurii, ed. Adolf Dick, (Stuttgart, 1878) Martianus Capella. The Marriage of Philology and Mercury. Translated by William Harris Stahl and Richard Johnson with E. L. Burge (Martianus Capella and the Seven Liberal Arts, Vol. II, New York, 1977) Milliet Dechales, Claude François, Cursus seu Mundus Mathematicus (Lyons, 1676) Ovid: Metamorphoses. With an English Translation by Frank Justus Miller (2 vols., London/Cambridge (Mass.), 1921–84) Plato, With an English Translation (12 vols., London/Cambridge (Mass.), 1914–1927) Plutarch, Moralia. With an English Translation (15 vols., London/Cambridge (Mass.), 1927–69) Plutarch, Lives. With an English Translation by Bernadotte Perrin (11 vols, London/Cambridge (Mass.) 1914–26) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 143 Hortensius’ Oration on the Dignity of Mathematics 143 Polybius, The Histories. With an English Translation by W. R. Paton (6 vols, London/Cambridge (Mass.), 1922–7). Proclus, Procli Diadochi Lycii philosophi platonici ac mathematici probatissimi in primum Euclidis elementorum librum Commentariorum ad universam mathematicam disciplinam principium eruditionis tradentium Libri IIII, ed. and translated by Francesco Barozzi (Padua. 1560) Proclus, A Commentary on the First Book of Euclid’s Elements. Translated, with Introduction and Notes, by Glenn R. Morrow (Princeton, 1992, 1st edn Princeton, 1970) Ptolemy, Handbuch der Astronomie. Deutsche Übersetzung und erläuternde Anmerkungen von K. Mantius. Vorwort und Berichtigungen von O. Neugebauer (2 vols, Leipzig, 1963) Melanchthon, Philipp, In arithmeticen praefatio (Wittenberg, 1536) Schott, Kaspar, Cursus mathematicus, (Würzburg, 1661) Selections Illustrating the History of Greek Mathematics. With an English Translation by Ivor Thomas (2 vols, London/Cambridge (Mass.), 1939–1941) Snell, Bruno (ed.), Leben und Meinungen der Sieben Weisen. Griechische und lateinische Quellen (Munich, 1971) Tacitus, Dialogus. Translated by Sir W. Peterson, revised by M. Winterbottom. In Tacitus in Five Volumes, (5 vols, London/Cambridge (Mass.), 1914), i. Vitruvius, On Architecture. Edited from the Harleian Manuscript 2767 and translated into English by Frank Granger (2 vols, London/Cambridge (Mass.), 1931) III. Secondary Literature III.1 On Hortensius Berkel, Klaas van, ‘Alexandrië aan de Amstel? De illusies van Martinus Hortensius (1605–1639), eerste hoogleraar in de wiskunde in Amsterdam’in E. O. G. Haitsma Mulier, C. L. Heesakkers, P. J. Knegtmans, A. J. Kox, and T. J. Veen (eds), Athenaeum Illustre. Elf studies over de Amsterdamse Doorluchtige School 1632–1877 (Amsterdam, 1997), 201–25. Revised version: ‘De illusies van Martinus Hortensius: Natuurwetenschap en patronage in de Republiek’ in Berkel, Klaas van, Citaten uit het boek der natuur: Opstellen over Nederlandse wetenschapsgeschiedenis (Amsterdam, 1998), 63–84 Minnaert, M. G. J., ‘Martinus Hortensius’ in Dictionary of Scientific Biography (16 vols, New York, 1970–80), vi. 520f Moes, E. W., Martinus Hortensius. De eerste Hoogleraar in de Mathematische Wetenschappen te Amsterdam in Oud-Holland 3 (1885), 209–16 and 18 (1900), 13 Remmert, Volker R., Ariadnefäden im Wissenschaftslabyrinth. Studien zu Galilei: Historiographie—Mathematik—Wirkung, (Berne:, 1998), 154–8 Remmert, Volker R., Widmung, Welterklärung und Wissenschaftslegitimierung: Titelbilder und ihre Funktionen in der Wissenschaftlichen Revolution (Wolfenbüttel/Wiesbaden, 2006), chapters 4.3 and 6.2 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 144 144 History of Universities Vermij, Rienk H., The Calvinist Copernicans: The Reception of the New Astronomy in the Dutch Republic, 1575–1750 (Amsterdam, 2002), 126–9 Waard, Cornelis de, ‘Martinus Hortensius’ in Nieuw Nederlandsch Biografisch Woordenboek (10 vols, Leiden, 1911–37), i. 1160–64 III.2 General Argoli, Andrea, De Diebus Criticis et de Aegrorum decubitu libri duo (Padua, 1639) Argoli, Andrea, Primi mobilis tabulae (Padua, 1667) Bedini, Silvio A., ‘The Role of Automata in the History of Technology’, Technology and Culture 5 (1964), 24–42 Bennett, James A., ‘The Challenge of Practical Mathematics’ in Stephen Pumfrey, Paolo Rossi and Maurice Slawinski, (eds), Science, Culture and Popular Belief in Renaissance Europe (Manchester/New York, 1991) 176–90 Berkel, Klaas van, Albert van Helden and Lodewijk Palm (eds), A History of Science in the Netherlands. Survey, Themes and Reference (Leiden, 1999) Biagioli, Mario, Galileo, Courtier: The Practice of Science in the Culture of Absolutism (Chicago/London, 1993) Biagioli, Mario, ‘Stress in the Book of Nature: The Supplemental Logic of Galileo’s Realism’, Modern Language Notes 118 (2003), 557–85 Blok, F. F., Isaac Vossius and His Circle: His Life Until His Farewell to Queen Christina of Sweden, 1618–1650 (Groningen, 2000) Blumenberg, Hans, Die Lesbarkeit der Welt (Frankfurt a. M., 1981) Bono, James, The Word of God and the Languages of Man (Madison, 1995) Brown, Gary I., ‘The Evolution of the Term “Mixed Mathematics” ’, Journal of the History of Ideas 52 (1991), 81–102 Burke, Peter, Venice and Amsterdam. A Study of Seventeenth Century Élites (Cambridge, 1994, 1st edn 1974) Curtius, Ernst Robert, Europäische Literatur und Lateinisches Mittelalter, (Bern/Munich 1984, 1st edn 1948) Damerow, Peter, ‘Mathematics Education and Society’ in his Abstraction and Representation. Essays on the Cultural Evolution of Thinking. With an Introduction by Wolfgang Edelstein and Wolfgang Lefèvre. Translated from the German by Renate Hanauer (Dordrecht/Boston/London, 1996), 111–48 Daston, Lorraine, ‘Probability and Evidence’ in Dan Garber and Michael Ayers (eds), The Cambridge History of Seventeenth-Century Philosophy (Cambridge, 1998), 1108–44 Davids, Karel, ‘Amsterdam as a Centre of Learning in the Dutch Golden Age, c. 1580–1700’, in O’Brien, Patrick et al. (eds), Urban Achievement in Early Modern Europe: Golden Ages in Antwerp, Amsterdam and London (Cambridge, 2001), 305–325 Davids, Karel, Zeewezen en wetenschap. De wetenschap van de navigatietechniek in Nederland tussen 1585 en 1815 (Amsterdam, 1986) (Dissertation) Dear, Peter, Discipline & Experience: The Mathematical Way in the Scientific Revolution (Chicago/London, 1995) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 145 Hortensius’ Oration on the Dignity of Mathematics 145 Drachmann, Aage Gerhardt, Ktesibios, Philon and Heron: a Study in Ancient Pneumatics (Copenhagen, 1948) Eamon, William, ‘Science as a Hunt’ in, Physis. Rivista internazionale di storia della scienza, n. s. 31 (1994), 393–432 Elias, Johan E., De Vroedschap van Amsterdam 1578–1795 (2 vols, Amsterdam, 1963, 1st edn Haarlem, 1903–05) Evans, James, ‘The Material Culture of Greek Astronomy’, Journal for the History of Astronomy 30 (1999), 237–307 Feldhay, Rivka, ‘The Use and Abuse of Mathematical Entities: Galileo and the Jesuits Revisited’ in Machamer, Peter (ed.), The Cambridge Companion to Galileo (Cambridge, 1998), 80–145 Gagné, Jean, ‘Du Quadrivium aux scientiae mediae’ in Arts libéraux et philosophie au Moyen Âge. Actes du quatrième congrès international de philosophie médiéval. Université de Montréal, 27 août—2 septembre 1967 (Montréal/Paris, 1969), 975–85 Gorman, Michael John, ‘Mathematics and Modesty in the Society of Jesus: The Problems of Christoph Grienberger’ in Feingold, Mordechai (ed.), The New Sciences and Jesuit Science: Seventeenth Century Perspectives (Dordrecht/ Boston/London, 2003), 1–120 Gouk, Penelope, Music, Science and Natural Magic in Seventeenth-Century England (New Haven/London, 1999) Hacking, Ian, The Emergence of Probability, (Cambridge et al, 1993, 1st edn, 1975) Hankins, Thomas L. and Silverman, Robert J., Instruments and the Imagination (Princeton, 1995) Harrison, Peter, The Bible, Protestantism, and the Rise of Natural Science (Cambridge, 1998) ’t Hart, Marjolein, ‘The Glorious City: Monumentalisation and Public Space in Seventeenth-Century Amsterdam’, in: O’Brien, Patrick et al. (eds), Urban Achievement in Early Modern Europe: Golden Ages in Antwerp, Amsterdam and London (Cambridge, 2001), 128–150 Helden, Albert van, Measuring the Universe. Cosmic Dimensions from Aristarchus to Halley (Chicago/London, 1985) Hooykaas, Reyer, Humanisme, science et réforme: Pierre de la Ramée (1515–1572) (Leiden, 1958) Hooykaas, Reyer, The Reception of Copernicanism in England and the Netherlands in Charles Wilson, Reyer Hooykaas, A. Rupert Hall, J.H. Waszink (eds): The Anglo-Dutch Contribution to the Civilization of Early Modern Society (Oxford, 1976), 34–44 Hopper, Florence, The Dutch Classical Garden and André Mollet, Journal of Garden History 2 (1982), 25–40 Horden, Peregrine (ed.), Music as Medicine: The History of Music Therapy since Antiquity, (Aldershot, 2000) Høyrup, Jens, ‘Archimedism, not Platonism: On a Malleable Ideology of Renaissance Mathematicians (1400 to 1600), and on Its Role in the Formation 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 146 146 History of Universities of Seventeenth-Century Philosophies of Science’, in Dollo, Corrado (ed.), Archimede: Mito Tradizione Scienza. Siracusa—Catania, 9–12 ottobre 1989 (Florence, 1992), 81–111 Israel, Jonathan I., Dutch Primacy in World Trade, 1585–1740 (Oxford, 1989) Israel, Jonathan I., The Dutch Republic: Its Rise, Greatness, and Fall 1477–1806, (Oxford, 1995) Jardine, Nicholas, The Birth of History and Philosophy of Science: Kepler’s ‘A Defence of Tycho against Ursus’ with Essays on Its Provenance and Significance (Cambridge, 1984) Jardine, Nicholas, ‘Epistemology of the Sciences’ in Schmitt, Charles B. (ed.): The Cambridge History of Renaissance Philosophy (Cambridge, 1988), 685–711 Jones, Matthew L., ‘Descartes’s Geometry as Spiritual Exercise’ in Critical Inquiry 28 (2001), 40–71 Karafyllis, Nicole C., ‘Bewegtes Leben in der Frühen Neuzeit: Automaten und ihre Antriebe als Medien des Lebens zwischen den Technikauffassungen von Aristoteles und Descartes’ in Engel, Gisela and Karafyllis, Nicole C. (eds), Technik in der Frühen Neuzeit—Schrittmacher der europäischen Moderne: special issue of Zeitsprünge. Forschungen zur Frühen Neuzeit 8 3/4) (2004)) Keller, Alexander, ‘Mathematics, Mechanics, and the Origins of the Culture of Mechanical Invention’, Minerva 23 (1985), 348–61 Knorr, Wilbur, ‘The Geometry of Burning-Mirrors in Antiquity’, Isis 74 (1983), 53–73 Kümmel, Werner Friedrich, Musik und Medizin: Ihre Wechselbeziehungen in Theorie und Praxis von 800 bis 1800 (Freiburg/München, 1977) Mancosu, Paolo, ‘Aristotelian Logic and Euclidean Mathematics: Seventeenth- Century Developments of the “Quaestio de certitudine mathematicarum” ’, Studies in History and Philosophy of Science 23 (1992), 241–65 Mancosu, Paolo, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (New York/Oxford, 1996) Mandosio, Jean-Marc, ‘Entre mathématiques et physique: note sur les “sciences intermédiaires” à la Renaissance’, in: Charpentier, Agnès (ed.): Comprendre et maîtriser la nature au Moyen Age: Mélanges d’histoire des sciences offerts à Guy Beaujouan (Geneva, 1994), 115–38 Marr, Alexander, ‘Understanding Automata in the Late Renaissance’, Journal de la Renaissance 2 (2004), 205–222 Masi, Michael, ‘Arithmetic’ in Wagner, David L. (ed.), The Seven Liberal Arts in the Middle Ages (Bloomington, 1983), 147–68 Mayr, Otto, Authority, Liberty & Automatic Machinery in Early Modern Europe (Baltimore/London, 1986) Mills, A. A. and Clift, R., ‘Reflections on the ‘Burning Mirrors of Archimedes’. With a Consideration of the Geometry and Intensity of Sunlight Reflected from Plane Mirrors’, European Journal of Physics 13 (1992), 268–79 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 147 Hortensius’ Oration on the Dignity of Mathematics 147 Mueller, Ian, ‘Mathematics and Philosophy in Proclus’ Commentary on Book I of Euclid’s Elements in Pépin, Jean and Saffrey, H. D. (eds), Proclus: lecteur et interprète des anciens (Paris, 1987) 305–18 Mueller, Ian, ‘Mathematics and the Divine in Plato’ in Bergmans, Luc and Koetsier, Teun (eds) Mathematics and the Divine. A Historical Study (Amsterdam, 2005) 99–121 North, Michael, Art and Commerce in the Dutch Golden Age (New Haven/ London, 1997) Ohly, Friedrich, ‘Deus Geometra: Skizzen zur Geschichte einer Vorstellung von Gott’, in Kamp, Norbert and Wollasch, Joachim (eds): Tradition als historische Kraft. Interdisziplinäre Forschungen zur Geschichte des frühen Mittelalters (Berlin/New York, 1982), 1–42 Olivieri, Luigi, ‘Dalle “scientiae mediae” alle “due nuove scienze”: linee di sviluppo dell’epistemologia galileiana in Baldo Coelin, Milla (ed.), Galileo e la scienza sperimentale (Padua, 1995), 65–86 Pedersen, Olaf, The Book of Nature (Vatican City, 1992) Pinault, Jody Rubin, Hippocratic Lives and Legends (Leiden/New York, 1992) Prado, Jéronimo de and Villalpando, Juan Bautista, In Ezechielem explanationes et apparatus urbis ac templi hierosolymitani commentariis et imaginibus illustratis (3 vols, Rome, 1596–1604) Rademaker, C. S. M., Life and Work of Gerardus Joannes Vossius (1577–1649) (Assen, 1981) Radke, Gerhard, Fasti Romani. Betrachtungen zur Frühgeschichte des römischen Kalenders (Münster, 1990) (Orbis Antiquus 31). Remmert, Volker R., Ariadnefäden im Wissenschaftslabyrinth. Studien zu Galilei: Historiographie—Mathematik—Wirkung (Berne, 1998) Remmert, Volker R., ‘Die Einheit von Theologie und Astronomie: zur visuellen Auseinandersetzung mit dem kopernikanischen System bei jesuitischen Autoren in der ersten Hälfte des 17. Jahrhunderts’, Archivum Historicum Societatis Iesu 72 (2003), 247–95 Remmert, Volker R., ‘Galileo, God, and Mathematics’ in Bergmans, Luc and Koetsier, Teun (eds), Mathematics and the Divine. A Historical Study (Amsterdam, 2005) 347–60 Remmert, Volker R., Widmung, Welterklärung und Wissenschaftslegitimierung: Titelbilder und ihre Funktionen in der Wissenschaftlichen Revolution (Wolfenbüttel/Wiesbaden, 2006) Remmert, Volker R., ‘Für diejenigen, die die heiligen Schriften auslegen, ist es notwendig, sich die mathematischen Disziplinen anzueignen’. Bibelexegese und mathematische Wissenschaften in der Gesellschaft Jesu um 1600’, in Zeitschrift für historische Forschung, forthcoming. Romano, Antonella, La contre-réforme mathématique: Constitution et dif- fusion d’une culture mathématique Jésuite à la Renaissance (1540–1640) (Rome, 1999) Rose, Paul Lawrence, The Italian Renaissance of Mathematics. Studies on Humanists and Mathematicians from Petrarch to Galileo (Geneva, 1975) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 148 148 History of Universities Schüling, Hermann, Die Geschichte der axiomatischen Methode im 16. und beginnenden 17. Jahrhundert, (Hildesheim/New York, 1969) Secretan, Catherine, Le ‹‹Marchand Philosophe›› de Caspar Barlaeus: Un éloge du commerce dans la Hollande du Siècle d’Or. Étude, texte et traduction du ‹‹Mercator sapiens›› (Paris, 2002) Simms, D. L., ‘Archimedes and the Burning Mirrors of Syracuse’, Technology and Culture 18 (1977), 1–24 Simms, D. L., ‘Galen on Archimedes: Burning Mirror or Burning Pitch?’, Technology and Culture 32 (1994), 91–6 Scholz, Bernhard F., ‘Marginalien bij het Boek der Natuur’, Feit en fictie. Tijdschrift voor de geschiedenis van de representatie 1 (1993), 51–74 Struik, Dirk J., The Land of Stevin and Huygens: A Sketch of Science and Technology in the Dutch Republic during the Golden Age (Dordrecht/Boston/ London, 1981) Sudhoff, Karl, Iatromathematiker vornehmlich im 15. und 16. Jahrhundert (Breslau, 1902) Toomer, Gerald J., Ptolemy’s Almagest (Princeton, 1998) Vermij, Rienk H., ‘Het copernicanisme in de Republiek: een verkenning’, Tijdschrift voor Geschiedenis 106 (1993), 349–367 Wolfe, Jessica, Humanism, Machinery, and Renaissance Literature (Cambridge, 2004) Wolkenhauer, Anja, ‘Ordo vitae: Die Entwicklung der Uhrenmetapher als Sinnbild guter Herrschaft in der spätantiken lateinischen Literatur’ in Susan Splinter, Sybille Gerstengarbe, Horst Remane, and Benno Parthier (eds), Physica et historia. Festschrift für Andreas Kleinert zum 65. Geburtstag (Acta Historica Leopoldina 45, Stuttgart, 2005), 43–50 Wreede, Liesbeth de, ‘Willebrord Snellius (1581–1626), humanist-mathematicus’, Nieuwsbrief Universiteitsgeschiedenis 8 (1) (2002), 12–18 Wreede, Liesbeth de, ‘Willebrord Snellius: a humanist mathematician’, Proceedings of the XII International Congress for Neo-Latin Studies (Bonn, 2003), to be published 2006 Yates, Frances A., Astraea. The Imperial Theme in the Sixteenth Century (London/Boston, 1975) Der Neue Pauly. Enzyklopädie der Antike (19 vols, Stuttgart/Weimar, 1996–2003) IV Praising the Mathematical Sciences Aggiunti, Niccolò, Oratio de mathematicae laudibus. Habita in florentissima Pisarum Academia cum ibidem publicam illius scientiae explicationem aggressurus foret (Rome, 1627) Barozzi, Francesco, Opusculum, in quo una Oratio, & duae Quaestiones: altera de certitudine, & altera de medietate Mathematicarum continentur (Padua, 1560) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 149 Hortensius’ Oration on the Dignity of Mathematics 149 Brahe, Tycho, De disciplinis mathematicis Oratio (1574) in Dreyer, John Louis Emil (ed.), Tychonis Brahe Opera Omnia (15 vols, Amsterdam, 1972), i., 143–173 (German translation by Karl Zeller: ‘Über die mathematischen Wissenschaften. Eine Rede Tycho Brahes’ in Die Sterne 11 (1931), 98–123) Cavalieri, Bonaventura, ‘Trattato delle Scienze Matematiche in generale’ in Sandra Giuntini, Enrico Giusti, and Elisabetta Ulivi (eds), Opere inedite di Bonaventura Cavalieri in Bollettino di Storia delle Scienze Matematiche 5 (1985), 1–350, 47–55 Chesnecopherus, Niels, Oratio de Matheseos laudibus, ejusque contra Epicureos & Aristippos defensione. Scripta et pronunciata publice in Philosophorum Collegio Marpurgi Cattorum, 32. (sic) Decem. anni 1593 (Marburg, 1594) Clavius, Christopher, In disciplinas mathematicas prolegomena in Clavius, Christoph, Opera mathematica (5 vols, Mainz, 1611–12) i. 3–9 Commandino, Federigo, De Scientiis Mathematicis Dissertatio in Michaelis Pselli Compendium Mathematicum aliaque Tractatus Eodem pertinentes (Leiden, 1647) Curtius, Joachim, Commentatio de certitudine Matheseos et Astronomiae (Hamburg, 1616) Dasypodius, Conrad, Protheoria Mathematica in qua non solum disciplinae Mathematicae omnes, ordine convenienti enumerantur: verum etiam universalia Mathematica praecepta, explicantur (Strasbourg, 1593) Dee, John, The Mathematical Preface to the Elements of Geometrie of Euclid of Megara (1570). With an Introduction by Allen G. Debus (New York, 1975) Gaurico, Luca, ‘Oratio de inventoribus, utilitate et laudibus astronomia, habita per Lucam Gauricum, vertente anno humanitati verbi MDVII dum in Ferrariensi Gymnasio mathematicas disciplinas publice profiteretur’ in Gaurico, Luca, Operum omnium (3 vols, Basel, 1575), i., 1–8 Grienberger, Christoph, ‘ “Praefatio” in Praise of the Mathematical Disciplines’ in Gorman ‘Mathematics and Modesty in the Society of Jesus: The Problems of Christoph Grienberger’ in Feingold, Mordechai (ed.), The New Sciences and Jesuit Science: Seventeenth Century Perspectives (Dordrecht/Boston/ London, 2003), 1–120, 33–40 Hood, Thomas, ‘The Speache by the Mathematicall Lecturer’ in Johnson, Francis R., ‘Thomas Hood’s Inaugural Address as Mathematical Lecturer of the City of London’, Journal of the History of Ideas 3 (1942), 94–106, 99–106 Jungius, Joachim, ‘Oratio de propaedia Philosophica sive Propaedeutico Mathematum usu.’. Edited and translated into German by Johannes Lemcke and Adolf Meyer in Beiträge zur Jungius-Forschung 1929, 94–120 Malapert, Charles, Oratio Habita Duaci dum lectionem Mathematicam auspicaretur. In qua De novis Belgici Telescopij phaenomenis non iniucunda quaedam Academice disputantur (Douay, 1620) 03-Feingold-Chap03.qxd 27/3/06 07:06 PM Page 150 150 History of Universities Monantheuil, Henri, Oratio pro mathematicis artis, Parisiis habita (Paris, 1574) Peletier, Jacques, ‘Oratio Pictavii habita, in praedelectiones Mathematicas’ (Poitiers, 1579), ed. Paul Laumonier, Revue de la Renaissance 5 (1904), 281–303 Pell, John, Oratio inauguralis Joannis Pellii in Inauguratio illustris scholae ac illustris collegii auriaci (Breda, 1647), 168–183 Velsin, Justin, De mathematicarum disciplinarum vario usu dignitateque (Strasbourg, 1544)