Jesuits Mathematical Science

PETER REAR *

JESUIT MATHEMATICAL SCIENCE AND THE RECONSTITUTION OF EXPERIENCE IN THE

EARLY SEVENTEENTH CENTURY

I

AN ‘EXPERIMENT’ in modern science is often contrasted with simple ‘experience’ by claiming that the former involves the posing of a specific question about nature which its outcome is to answer, whereas the Iatter does nothing more than supply items of fact regarding phenomena, and is not designed to judge matters of theory or interpretation. Thus it has been pointed out that pre-modern, scholastic uses of ‘experience’ in natural philosophy tend to take the form of selective presentation of instances which illustrate conclusions generated by abstract philosophizing, and not the employment of such material as a basis for testing these conclusions. ‘Experiment’ became a characteristic feature of natural philosophy only in the seventeenth century.’

In its broadest terms this picture must be accepted, but enough is left out in the analysis of the nature of ‘experiment’ to obscure understanding of its historical emergence. The science-textbook definition of experiment fails to capture the reality of the new conceptions of the seventeenth century: Robert Hooke’s term ‘experimentum crucis’, so signally adopted by Newton, was certainly intended to pick out an aspect of Bacon’s teaching suitable to the notion of ‘experiment’ as a test of hypotheses, but Boyle’s ‘experimental histories’, also indebted to Bacon, had no immediate purpose beyond the mere collection of facts.’ The ‘experiments’ of the Accademia de1 Cimento were frequently designed to test hypotheses or decide between alternatives,3 but the empirical work of the Accademia’s Florentine forebear, Galileo, seems at

*Department of History, Corndl University, McGraw Hall, Ithaca. NY 14853-4601. U.S.A. Received 12 January 1986. ‘See, for instance, Charles B. Schmitt, ‘Experience and Experiment: A Comparison of

Zabarella’s View with Galileo’s in De mo0Z, Shrdies in fire Renaissance 16 (1969), 80- 138; Paolo Rossi, ‘The Aristotelian6 and the Moderns: Hypothesis and Nature’, Annali dell’lstifutu e Muse0 di St&a delta Sciema di Firenze 7 (1982). fasc.1. 3-27, both of which look particularly at Zabarella as a representative Aristotelian. .

*Robert Hooke, Microwaphia (London, 1665: facsimile edn. New York: J3over, 1%2), p. 54; the term seems to derive %rn Elafon’s ‘i&a& cruds’.

‘See W. E. Knowles Middleton, The Experimenters: A Study of the Accademia del Cimento (Baltimore: Johns Hopkins University Press, 1971).

S&d. H&t. Phil. Sci,, Vol. 18. No. 2, pp. 133 - 175, 1987. Printed in Great Britain.

133

0039- 3681/87 $3.00 f 0.00 Pergamon Journals Ltd.

134 Studies in History and Philosophy of Science

least partly to have been directed towards establishing premises for formal scientific demonstrations, a quite different function.’ As a novel feature of the Scientific Revolution, ‘experiment’ is not easily characterized.

The problem is really one of determining how the seventeenth century reconstructed the pre-existing concept of ‘experience’ to fit new ends. I have argued elsewhere’ that ‘experience’ as an element of scholastic natural philosophical discourse took the form of generalized statements about how things usually occur; as an element of characteristically seventeenth-century, non-scholastic natural philosophical discourse it increasingly took the form of statements describing specific events. The former were associated with the commentary as the typical literary genre, the latter with the research report exemplified by countless contributions to the Philosophical Tramactions by the early Fellows of the Royal Society. For the scholastic natural philosopher, writing his commentaries on Aristotle, the grounding in experience of the physical facts debated in his discussions was guaranteed by their generality as experiential statements - ‘heavy bodies fall’ is a statement to which all could assent, through common experience embodied in authoritative texts. If experiential statements referred to singular events, however, and if evidential weight attached to that singularity, as in the case of contrived experiences using special apparatus, this kind of common assent by which to establish their truth could not be anticipated. The new ‘experience’ of the seventeenth century, therefore, established its legitimacy in historical reports of events, often citing witnesses. The singular experience could not be evident, but it could provide evidence.

The development of the idea of ‘experiment’ as an arbiter of hypotheses, therefore, formed only one aspect of the change in the nature and function of experience in the Scientific Revolution. More fundamental was the emergence of the discrete experience as the primary empirical component of natural philosophy. Although no simple answer can be expected as to how this came about, aspects of the process can be found within the pre-established traditions of the mathematical sciences as practiced by members of what has long been regarded as a bastion of reaction - the Jesuit order.” The shift to a more modern concept of experience among Jesuit mathematicians may not typify

‘See notes 39 and 40 below. ‘Peter Dear, ‘Totius in v&o. Rhetoric and Authority in the Early Royal Society’, Isis 76

(1985), 145 - 161. ‘The assumption that the Jesuits played the part of reactionary Aristotelians seems largely to

have stemmed from their role in Galileo’s condemnation. Recent reassessments arc beginning to change that picture: see e.g. Gabriele Baroncini, ‘L’Insegnamento della filosofia naturale nei collegi italiani dei Gesuiti (1610- 1670): un esempio di nuovo aristotelismo’, La ‘rutio st~diontm’: Modelli culturali e pratiche educative dei Gesuiti in Italia tra Cinque e Seicento, Gian Paolo Brizzi (cd.) (Roma: Bulzoni Editore, 1981), pp. 163 - 215; William A. Wallace, Galileo and his Sowwes. The Heritage of the ColIegio Ramono in Galileo’s Science (Princeton: Princeton University Press, 1984); and subsequent references below.

Mafhematical Science and fhe Reconsfitution of Experience 135

this seventeenth-century phenomenon as a whoIe, but it impinges directly on the implications of moving from a scholastic to a characteristically early- modern natural philosophical framework. As such, it encapsulates an important aspect of the Scientific Revolution.

Ii

The importance of the mathematicaI disciplines in the Jesuit college curriculum of this period is weIl-known.7 In effect, the Jesuits had transplanted the medieval quadrivium, comprising the headings of arithmetic, geometry, astronomy and music, from the propaedeutic place (after the trivium) which it occupied in the medieva1 university to the second or third year of the three-year philosophy course, to be taught alongside either physics or metaphysics.’ The pattern varied from college to college, and over time, but by the early seventeenth century the mathematical disciplines held a prominent position in the courses of study offered by the Jesuits at the larger coheges, and in the ideal curricuIum enshrined in the 1599 Ratio studiorum The main constraint appears to have been an insufficient number of competent teachers to go around,” but this was not a problem at the major colleges, and

‘The fundamental work of Francois de Dainville on this issue includes La nu&rnce de I’humanisme moderne and La gdographie des humanktes (both Paris: Beauchesne, 1940), and articles collected in Dainville, L ‘l?ducation des J&uites XVI’- XWIc si&les (Paris: Editions de Minuit, 1978).

‘For a general account of Jesuit mathematical education see, in addition to the previous note, John L. Heilbron, Electricity in the I7th and I8th Centuries. A Study in Early Modern Physics (Berkeley etc.: University of California Press, I979), pp. IOI- 114. See aIso Camille de Rochemonteix, Un coffage de J&.&s aux XVII’ et XVIII’ s&k Le coflpge Henri IV de La FWhe, 4 vols (Le Mans: Leguicheux, 1889), especially vol. 4, pp. 27 and 32. Essays on the quad&al disciplines in the Middle Ages may be found in The Seven Liberal Arts in the Middle Ages, David L.-Wagner (ed.) (Bloomington: Indiana University Press, 1983).

‘Francois de Dainville. ‘L’Enseignment des mathtmatisues darts Ies colI&aes J&suites de Prance du XVI’au XVIII’ sit&‘, Revue-d’histoire des sciences 7 (1954), 6-21,109- 123; Giuseppe Cosentino, ‘Le matematiche nella ‘Ratio Studiorum’ della Compagnia di Gesu’, M&el/atreu Sforica Ligure II.2 (1970), 171-213; idem, ‘L’lnsegnamento de& matematiche nei collegi GcsuiticI neil’ltaha settentrionale: Nota introduttiva’, P&s& 13 (1971), 205-217; for comprehensive listings and biographical entries of mathematicians in the colleges see Karl Adolf Franz Fischer, ‘Jesuiten Mathematiker in da deutschen Assistenz bis 1773’, Archivum Historicurn Societutis Iesa 47 (1978), 159- 22~3; idem, ‘Jesuiten Mathematiker in der franziisischen und italienischen Assistenz bis 1762, bzw. 1773’, ibid. 52 (1983, 52-92; idem, ‘Die Jesuiten- mathematiker des nordostdeutschen Kulturgebietes’. Archives internationales d’histoire des sciences No.112 (1984). i24- 162.

“‘Cf. Heilbron, Electricity, pp. 102 - 103.


altogether the Jesuits were quite successful in producing practitioners and writers in the mathematical sciences during the seventeenth century.” The prime mover in establishing these subjects in the curriculum, Christopher Clavius, had a great influence on the style and attitudes manifest in subsequent Jesuit mathematical writing.‘*

Clavius’s textbooks formed the standard introduction to the quadrivial disciplines for pupils in the colleges, and the approach has been described as ‘utilitarian’. I3 The emphasis is certainly on the mathematical sciences as suppliers of techniques, but the purposes for which those techniques were to be applied were wide-ranging. Practical utility was at least matched by the usefulness of mathematics for other disciplines; in an educational policy document of the 1580s Clavius had argued for the importance of mathematics by claiming that physics could not be understood without it (and he mentions particularly the relevance of astronomy to cosmology).‘4 Furthermore, there were ‘an infinity of examples in Aristotle, Plato and their most illustrious interpreters which can in no way be understood without some knowledge of the mathematical sciences’ ,I5 This latter point was more than a mere afterthought; its value as a way of increasing the esteem of philosophers for their mathematical colleagues was apparently great enough to induce Clavius’s pupil Blancanus to publish, in 1615, a 283 page work entitled Arisfotelis locu mathematics ex universis ipsius Operibus collecta, & explicuta, which proceeds

” The more mathematically prominent colleges would include La H&he in Normandy, the alma ma&r of Descartes and Mersenne; Ingolstadt, where the astronomer Christopher Scheiner spent most of his career; W&burg, ornamented by the presence at one time or another by Athanasius Kircher and Gaspar Schott; and, above all, the Collegio Romano.

“On Clavius’s work and career see Charles Naux, ‘Le p&e Christophore Clavius (1537 - 1612). sa vie et son oeuvre’, Revue des questions scienf~fiques 154 (1983), 55 -67, 181 - 193, 325 - 347; Ugo Baldini, ‘Christoph Clavius and the Scientific Scene in Rome’, Gregoriun Reform of the Calendar. Proceedings of the Vatican Conference fo Commemorate ifs 400th Anniversary 1582-1982, G. V. Coyne, S. J., M. A. Hoskin and 0. Pedersen (eds) (Vatican City: Specola Vaticana. 1983), pp. 137 - 169. The latter is particularly well-documented.

“For example by Corn& de Waard in his ‘Vie de Mersenne’, Correspondunce du P. Marin Mersenne Vol.1 (1932). p. xxi. A document written by Clavius for his superiors in about 1579 or 1580, detaiting a projected mathematics course with subjecls and authors, is reproduced in Ugo Baldini, ‘La nova de1 1604 e i matematici e filosofi del Collegio Romano: Nota su un testo inedito’, Annaii dell’lstituto e Muse0 di Storia deiia Scierrza di Firenze 6 (198 I), fasc.2.63 - 98, on pp. 89-95.

“Christoph Clavius, ‘Modus quo disciptinae mathematicae in scholis Societatis possent promoveri’, Monumenta Paedagogicu SocietutiF Jest quae Primam Rationem Studiorum anno 1586 pruecessere (Matriti, 1901). pp. 471-474, on p. 472. Nicholas Jardine notes an apparent increase in the perceived relevance of mathematical astronomy for natural philosophy towards the end of the sixteenth century in his The Birth of History and Phibsophy of Science, Kepler’s A Defense of Tycho against Ursus, with Essays on its Provenance and Significance (Cambridge: Cambridge University Press, 1984). p. 246.

“Clavius, ‘Modus’, p. 472.

Mathematical Science and the Reconstitution of Experience 137

through Aristotle’s works in turn, picking out passages which admit of mathematical elucidation.16 While praising its general applicability in prefatory remarks,17 Clavius restricted his own writings to the more practical aspects of mathematics; his endeavours in the devising of the Gregorian calendar were very much in keeping with the tenor of his texts on geometry, astronomy, arithmetic and algebra - the latter in its sixteenth-century guise as a department of arithmetic providing analytical computational techniques.” Nonetheless, he argued strongly for the importance of mathematical knowledge to philosophers and characterized mathematics as a worthy, indeed noble, intellectual pursuit, quite apart from its practical applications. ‘Truly, if the nobility and pre-eminence of a science be judged by the certainty of the demonstrations which it uses, there is no doubt that the mathematical discipIines will have the first place among all the rest.‘lg Astronomy was the noblest of all, he said, since it fulfilled Aristotle’s criteria of exceIlence better than any other, using the most certain demonstrations (namely, geometrical), and dealing with the most noble subject, the heavens.20

Clavius played a genuinely seminal role in the formation of a Jesuit tradition, running throughout the seventeenth century, of work in the mathematical disciplines. His promotion of their intellectual status was of paramount importance because it provided a basis for a treatment of aspects of the natural world which would be on equal terms with Aristotelian natural philosophy, or physics. The most important point to establish was that the mathematical disciplines were true sciences, and hence on an equal methodological footing with physics. A number of Italian philosophers in the sixteenth century, starting with Alessandro Piccolomini and including the

‘*This work is a separatdy paginated treatise bound with, and following, another piece entitled: De mathemaiicarum natura dissertafio. Una cum clarorum mathematicorum chronologia. lath works bear the details: Aurhore [eodeml Iosepho BIancano e Societate IESU, Mathemaricarum in Parmensi Academia professore. Bononiae MDCXV. Apud Earfholomaeum Cochium. Sumptibus Hieronymi Tamburini.

“See, e.g. Christopher Clavius, Geometria practica, in Clavius, Qperum mathematicorum tomus secundus (Moguntiae. sumptibus Antonii Hierat, excudebat Reinhardus Eltz, MDCXI), ‘Praefatio’, p. 3.

“See, for Clavius’s calendrical work, Baldini, ‘Christoph Clavius’; on his algebra see Naux, ‘Clavius’, pp. 336-338. See also Frederic A. Homann S. J., ‘Christopher Clavius and the Renaissance of Euclidean Geometry’, Archivum Historicurn Societatis Iesu 52 (1983), 233 - 246. A full listing of Clavius’s works is contained in BibtiothPque de la Compugnie de J&us, 1 I vols, Carlos Sommervogel et at. (eds) (Brussels: Alphonse Picard, 1890- 1932), Vol.2, cols.1212- 1224. An excellent treatment of sixteenth-century cossist algebra is JoAnn S. Morse, ‘The Reception of Diophantus’ Arithmetic in the Renaissance’ (Ph.D. dissertation, Princeton University, 1981).

“Christopher Clavius, Operum mathematicorum tomus primus (Mogundae, MDCXI). ‘In disciphnas mathematicas prolegomena’, p. 5.

mClavius, In sphaeram Ioannis de Sacro Bosco commenrurius, in Operum maihemaricorum tomus terrius (Moguntiae, MDCXI), p. 3.


Jesuit Per&a, had maintained that pure mathematics, geometry and arithmetic, were not true sciences in Aristotle’s sense because they did not demonstrate their conclusions through causes. 2’ Clavius avoided this problem by relying on Aristotle’s own characterization of mathematics, which built on a more general model of the structure of an ideal science. Sciences should be founded on their own unique, proper principles, which provided the major premises for deductive, syllogistic demonstration. Subject-matters were therefore strictly segregated among their appropriate sciences, a logical necessity arising from the principle of homogeneity, which required that the principles of a science be of the same genus as the objects of that science so as to ensure the possibility of a deductive link between them. Leaving aside the status of geometry and arithmetic themselves, clearly disciplines like astronomy and music which drew on the results of pure mathematics violated the principle of homogeneity for that very reason. Accordingly, Aristotle made a special accommodation for such subjects by classifying them as sciences subordinate to higher disciplines.22 The applied mathematical disciplines were later to be represented in the quadrivium by astronomy and music, although these two also stood for a host of others such as geography or mechanics, and became known variously as ‘subordinate’, ‘middle’ or ‘mixed’ sciences.23 Aristotle’s was in some ways an ad hoc solution to the classificatory problem, which provoked later scholastic discussions, particularly among the Jesuits, on whether true scientific knowledge was produced by demonstrations in something like optics if the presupposed theorems of geometry were not proved at the same time.” It served Clavius’s purpose perfectly well, however, because Aristotle’s attempt to fit the applied disciplines into his general model for a science itself showed his acceptance of the scientific status of all the mathematical disciplines. Clavius could therefore

l’ See Paolo Galluzzi, ‘II “Platonismo” del tardo Cinquecento e la filosofia di Galileo’, Ricerche sullu cul~ura dell’Ituliu moderna, Paola Zambelli (ed.) (Bari: Editore Laterza, 1973), pp. 37 - 79; see also Wallace, Galileo, p. 136, for further references, and Adrian0 Carugo, ‘Giuseppe Moleto: Mathematics and the Aristotelian Theory of Science at Padua in the Second Haif of the 16th Century’, A&ore/&no veneto e scienza moderna, 2 vols, Luigi Olivieri (ed.) (Padova: Editrice Antenore, 1983), Vol.1, pp. 509-517. Proponents of this view usually attributed the certainty of mathematical demonstration to the nature of its subject-matter. A summary of the scholastic - Aristotelian definition of a ‘science’ may be found in Wallace, Galileo, pp. 99 - 101.

=A good treatment of this is Richard D. McKirahan, ‘Aristotle’s Subordinate Sciences’, British Journal for the History of Science 11 (1978), 197 - 220.

23 Jesuit discussions include Clavius, ‘Prolegomena’ (note 19). pp. 3-4; Blancanus, De mathemuticarum natwu dimertatio, pp. 29 - 3 1.

=See Wallace, Galileo, p, 134.


present the latter as sciences on the strength of Aristotle’s generaIly accepted characterization of them without even touching on the tricky question of causes. 25 Blancanus later tackled that question head-on, arguing that demonstrations in geometry utilized formal and material causes, and that geometry dealt with the essences of geometrical objects.2” Blancanus even made the argument that geometrical optics could provide final causes in the study of the physiology of the eye, in that it explained why the eye needed to be more-or-less spherical.” This served to reinforce Clavius’s orthodox Aristotelian description of the logical structure of the mathematical disciplines, and it highlights the importance which the Jesuit mathematical tradition stemming from Clavius attached to maintaining the scientific status of mathematical knowledge.

Turning to the subject-matter of mathematics, Clavius adopted the conventional classification found in Aristotle and stemming from Plato, a classification frequently used in the West since the early Middle Ages.

Because the mathematical disciplines discuss things which are considered apart from any sensible matter, ahhough they are immersed in material things, it is evident that they hold a place intermediate between metaphysics and natural science, if we consider their subject, as is rightly shown by Pro&s, for the subject of metaphysics is separated from all matter, both in the thing and in reason: the subject of physics is in truth conjoined to sensible matter, both in the thing and in reason: whence since the subject of the mathematica1 disciplines is considered free from all matter,

*Although he does refer to it in ‘Modus’ (note 14), p. 473. In ‘Prolegomena’, p, 3, Qavius describes the mathematical disciplines as follows: ‘they alone retain the way and method [modurn twionemquel of a science. For they proceed always from certain Lquibusdaml foreknown principles to demonstrated conclusions, which is the proper duty and office of a doctrine or discipline, as even Aristotle, PaSrerior Anm’ytics I, testifies.. .’ (Clavius had commenced by showing that the etymological derivation of ‘mathematics’ linked it to the meanings of ‘discipline’ or *doctrine’).

LBlancanus, DemaH&emuticurum natwa dissertafio, pp. 7 - 10. A convenient summary may W found in Wallace, Galileo, pp. 142- 143. A somewhat later, very useful systematic discussion of the same issues, drawing on but not merely recapitulating Blancanus’s defence of mathematics BS a science, is in the following work: Hugonis Sempilii Craigbaitaei Scoti e Societate Iew de Mathemaroticis Disci~inir libri duodecim , . , An tverpiae, ex officina Planfiniana Bakhasari MO&i. MDCXXXV, pp. 7-u). Such defences of mathematics seem to have become almost commonplace in seventeenth-century Jesuit treatises on mathematics, and upholding scientificity was clearly seen as extremely important,

nBlancan~, De mathematicarum nature dkwtafio, Q. 30.


although it [i.e. matter] is found in the thing itself, clearly it is established intermediate between the other two.‘”

The advantage of this classification was that it placed mathematics as an integral part of philosophy, no less than physics or metaphysics. Thus Blancanus could reply to denials of mathematics’ philosophical status by saying that ‘among Aristotle and all the peripatetics nothing occurs more frequently than that there are three parts of philosophy, physics, mathematics, and metaphysics’.2g Indeed, Clavius used the scheme as a way of reiterating his theme of the pre-eminence of mathematics by denigrating the other two parts of philosophy, on Ptolemy’s authority: ‘For he says that natural philosophy and metaphysics, if we consider their mode of demonstrating, are rather to be called conjectures than sciences, on account of the multitude and discrepancy of opinions.‘3o

The Aristotelian definition of a scientific demonstration as one which demonstrates its conclusions from immediate and necessary causes represented the highest ideal of human knowledge for a 3esuit scholastic philosopher (which is why Clavius tried to argue for its applicability to mathematics). The

“Clavius, ‘Prolegomena’, p. 5. The career of this classification may be traced in James A. Weisheipl, ‘Classification of the Sciences in Medieval Thought’, Mediuevul Studies 27 (1%5), 54 - 90; idem, ‘The Nature, Scope and Classification of the Sciences’, Science in the Middle Ages, David C. Lindberg (ed.) (Chicago: University of Chicago Press, 1978), pp. 461- 482. See also A. C. Crombie, ‘Mathematics and Platonism in the Sixteenth-Century Italian Universities and in Jesuit Educational Poticy’, Prismata: Nattuwissenschaftsgeschichtliche Studien (Festschrift ftir W//y Hurtner), Y. Maeyama and W. G. Saltzer (eds) (Wiesbaden: Franz Steiner, 1977), pp. 63 - 94.

HBlancanus, De mathemuticarum natura dissertatio, p. 21. ‘°Clavius, ‘De sphaera’ (note 20). p. 4, citing Ptolemy, Almagest 1.1; see theEnglish translation

by G. J. Toomer, Ptolemy’s Aimagest (London: Duckworth, 1984), p. 36. In his De math. nat., Blancanus presents what looks like a very Platonic view of the nature of mathematical entities. Denying the charge that mathematical entities exist only in the intellect, he says that although geometrical figures are not found perfectly realized in the physical world, nonetheless they are realized to some degree, ‘for it is well-known [constutl that nature and art chiefly intend mathematical figures, although on account of the crassness and imperfection of sensible matter, which is unable entirely to receive perfect figures, they are frustrated in their aim.. .’ (p. 6). Although mathematical entities do not exist perfectly in the material world, ‘because, however, the Ideas of them exist, as much in the mind of the Author of nature as in the human [mind], as the most exact types of things, and also as exact mathematical entities, the mathematician therefore discusses these Ideas of them, which are primarily intended per se and which are true entities’ (p. 7). Despite the Platonic overtones, this somewhat un-Aristotelian picture in fact employs the standard scholastic interpretation of the ‘forms’ of things as corresponding to ‘ideas in the mind of God’; Blancanus simply tries to justify its extension to mathematical entities. His account (p. 6) of the subject of pure mathematics, the intelligible matter resulting from the abstraction of quantity (specifically, terminated quantity) from sensible matter, is thoroughly orthodox and Aristotelian.

Secondary accounts of Blancanus’s discussions in De moth. nut. can be found in Galluzzi, ‘Platonismo’, especially pp. 56-65; 0. C. Giacobbe, ‘Epigone nel Seicento della “Quaestio de certitudine mathematicarum”: Giuseppe Biancani’, Physir 18 (lY76), 5 - 40; Wallace, Gulileo, pp. 141- 144. For additional information on Blancanus, see Ugo Baldini, ‘Additamentu Gulilueanu. I. Galileo, la nuova astronomia e la critiea all’Aristotelismo nel dialog0 epistolare tra Giuseppc Biancani e i revisori romani della Compagnia di Gesu’. Annuli dell’lstituto e Muse0 di Storia de//u Scienzu di Firenze 9 (1984), fasc.2, 13-43, especially pp. 14- 17.


certainty, as well as the necessity, of such a demonstration was rooted in its premises; it was easy to construct a formally valid syllogism, but inventing the appropriate premises to yield scientific knowledge of a conclusion was not. CIavius outlined this problem, for which Aristotle’s Posterior Analytics was the Iocus classicus, in the ‘Prolegomena’ to his edition of Euclid:

While every doctrine, and every discipline, is produced from pre-existing knowledge, as Aristotle says, and demonstrates its conclusions from certain assumed and conceded principles, no science, however, according to the opinion of Aristotle and other philosophers, demonstrates its iown principles; the mathematical disciplines too will have their principles, from which, posited and conceded, they confirm their problems and theorems.”

But if a science cannot confirm its own principles, how are those principles, on which the certainty of the science depends, to be established?

Ideally, the principles or premises would be evident, and therefore immediately conceded by all. In the case of geometry, Euclid’s ‘common opinions’ reflect Aristotle’s intention, statements such as ‘the whole is greater than its proper part’, or ‘equals added to equals are equal’.” Extending this notion of ‘evidentness’ to sciences dealing with the physical world required something other than an intuitive grasp of a statement’s truth, however, because according to Aristotle knowledge here rested on the apprehension of empirical data - a doctrine encapsulated in the scholastic dictum ‘nihil in intellectu quod non prius in sensu’.33 Nonetheless, in practice, empirical principles could be made evident in a similar way to geometrical axioms: they would be evident if everyone agreed on their truth and judged argument to be unnecessary in the establishment of such agreement. This is implied in the very form of experiential statements used in scholastic natural philosophy. The term ‘experience’ designated a universal statement of fact, supposedly constructed from the memory of many singular instances, and its universality expressed its intended status as an evident truth which might form a premise in a scientific demonstration.34 Formal unive rsality did not in itself establish an

“Clavius, ‘Prolegomena’, p. 9. It might be nxed that this passage serves also to place the mixed mathematical sciences on a par with other sciences, despite their assumption of results from other disciptines, by stressing that u/I principles of all sciences are finally just conceded.

“Aristotle called these ‘axioms’. See G. E. R. Lloyd, Magic, Reasm undExperiemx. Studies in the Origin and Deve/opnenl of Greek Science (Cambridge: Cambridge University Press 1978), p. 1 I1 and, in general, chap. 2.

“On Aquinas, natural philosophy as scienriu. and the requirement for evident premises, see William A. Wallace, ‘The Philosophical Setting of Medieval Science’, Science in rhehfiddle Ages, Lindberg (ed.), pp. 91- 119, on pp. 98 - 99.

y Dear, ‘Tot& in verba’, pp. 148 - 149; see also Matthias Schramm, ‘Aristotelianism: Basis and Obstacle to Scientific Progress in the Middle Ages’, History 01 Science 2 (1963). 91 - 113, especially pp. 104- 105; idem. ‘Steps Towards the Idea of Function: A Comparison Between Eastern and Western Science of the Middle Ages’, ibid. 4 (1963, 70 - 103.


experience as ‘evident’, of course; it had to express, and derive from, the perennial lessons of the senses. 3s But experiential statements could not play a role in scientific discourse unless they were universal. If they were not, they could never be evident. The classical mixed mathematical sciences aspired to just such a grounding in evident empirical principles. The basic empirical premise of the mathematical science of optics, for example, that light rays (or visual rays) travel in straight lines, represented a perfect model; it was directly evident because everyone knows, from common experience, that you can’t see around corners..” The optical works of Euclid and Ptolemy represent attempts at developing a science on this kind of basis, and involve ways of presenting the simple law of reflection so as to make it appear a necessary corollary of everyday visual experience.37 Similarly, Archimedes’ mechanical works were geometrical elaborations of physical postulates chosen for their immediate acceptability by all reasonable people: Postulate 1 of On the Equilibrium of Planes states that ‘equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance’.38 This ideal form of a mathematical science continued to hold sway in the early seventeenth century. Winifred Wisan39 has argued that Galileo’s use of experiment in the

“A celebrated example is Aristotle’s observation that bees appeared to reproduce parthenogenetically: Aristotle held his conclusion open to correction by subsequent experience. Aristotle, De animalibus historiae 1X.42.

%The object was to employ principles which were perse nota; on the rectilinear propagation of light in Antiquity see David C. Lindberg, Theories of Vision from AI-Kindi to Kepler (Chicago: Chicago University Press, 1976), pp. 12, 220n.79; p. 20 describes a demonstration of this by Al- Kindi involving shadows. The one glaring scientific deficiency of such empirical principles, that they fell short of the strict Aristotelian ideal by lacking any obvious necessity, seems to have been ignored by the Jesuit mathematical apologists, no doubt because it could not in its own tcrma be remedied. Necessity could accrue to empirical statements in physics, on the other hand, owing to the alleged possibility of grasping essences: propositions like ‘man is a rational animal’, once established, were necessarily true by definition. Actually establishing them was another matter, and generated methodological discussions in the sixteenth century concerning the demonstrative regress; the mathematicians kept clear of such niceties. See especially Nicholas Jardine, ‘Galileo’s Road to Truth and the Demonstrative Regress’. Studies in History and Philosophy of Science 7 (1976), 277 - 318.

“Schramm, ‘Steps’, pp. 71- 73; note, however, also the important differences in approach between Euclid and Ptolemy - Ptolemy is more self-conscious in establishing his suppositions. See also Saleh Beshara Omar, Ibn uI-Haytham’s Optics. A Study ofthe Origins ofExperimental Science (Minneapolis/Chicago: Bibliotheca Islamica, 1977), chap. 1, especially pp, 17 - 36.

‘“Archimedes, ‘On the Equilibrium of Planes’, Book I, in The Works of Archimedes, T. L. Heath (ed.) (New York: Dover, 1953), on p. 189.

39 Winifred L. Wisan. ‘Galileo and the Emergence of a New Scientific Style’, Theory Change, Ancient Axiumotics, and Galileo’s Methodology I= Proceedings of the 1978 Coderence on the History and Philosophy ofScience1, Jaako Hintikka, David Gruender and Evandro Agazzi (eds) (Dordrccht: D. Reidel, 1981), Vol. 1, pp. 311- 339. The tension between Galileo’s methodological predilection for this kind of demonstrative science and the constraints of his actual practice is examined in Ernan McMullin, ‘The Conception of Science in Galileo’s Work’, New Perspectives on Gulileo, Robert E. Butts and Joseph C. Pitt (eds) (Dordrccht: D. Reidel, 1978). pp. 209-257.


establishment of his ‘new science of motion’ was aimed at finding empirical principles governing falling bodies which would imitate the Archimedean model of a mathematical science by commanding immediate assent; it was not aimed at uncovering recondite facts accessible only through elaborate experimental procedures. William Wallacea has considered this aspect of Galileo as a development of scholastic regressus theory, concerned with the establishment of principles from phenomena and the subsequent confirmation of those principIes by verification of their consequences. But whether Galileo is portrayed as an Archimedean or as a methodologist drawing on scholastic sources, he certainly saw the construction of a mathematical science of nature in terms of classical mixed mathematics, Aristotle’s subordinate sciences.4’ His attempts to render as evident as possible the principles, or suppositions, of the ‘new science of motion’ illustrate very neatly the scholastic-Aristotelian concept of experience reflected in the traditional mathematical sciences.42 When Galileo was obliged to cite experimental tests in support of his arguments, it was tantamount to an admission of defeat.43 To be fully adequate, empirical premises needed to command assent because they were evident, not because of particular events adduced in their support.

At one level, then, ‘experience’ in schoIastic natural philosophy and mathematics typically took the form of universal statements because singular statements, statements of particular events, were not evident and indubitable, but relied on fallible historical reports. In a sense, the Aristotelian model of a science adopted by the Jesuits impIied that scientific knowledge must bepublic - the conclusions of scientific demonstration would in principle be truths perfectly graspable by all, because they were derived from necessary logical connections between terms formulated in premises commanding universal assent. Singular experiences were not public, but known only to a privileged few; consequently, they were not suitable elements of scientific discussion.

40Wallace, Guiileo, pp. 343 - 347; also Wallace, ‘The Problem of Causality in Galileo’s Science’, Review ofMetaphysics 36 (1983), 607 - 632. Wallace’s general suggestion is followed up in David Hemmendinger, ‘Galileo and the Phenomena: On Making the Evidence Visible’, Physical Sciences and History of Physics [ = Boston Stud& in the Philosophy of Science, Vo1.821, R. S. Cohen and M. W. Wartofsky (eds) (Dordrecht: D. Reidel, 1984), pp. IlS- 143. Wallace’s claims for Galileo’s use of regressus theory must be qualified by the discussion of the mathematical tradition of analysis and synthesis in Jardine, ‘Galileo’s Road to Truth’.

“This point is made by Peter Machamer, ‘Galileo and the Causes’, New Pempectives on Gulileo, Butts and Pitt (eds), pp. 161- 180.

“There are clear connections here to the question of Galileo’s use of ‘thought-experiments’: see Alexandre KoyrC, ‘Galileo’s Treatise De rnotu gruvium: The Use and Abuse of Imaginary Experiment’, in Koyre, Metaphysics and Measurement (London: Chapman & Hall, I%@, pp. 44-88; Thomas S. Kuhn, *A Function for Thought Experiments’, in Kuhn, The Essentirrl Temion. Selected Studies in Scientific Tradition and Change (Chicago: Chicago University Press, 1977). pp. 240-265.

” Wisan, ‘Galileo’, p, 321,


At another level, however, the universality of scientific experience was an aspect of the scholastic- Aristotelian view of nature.44 Nature was the principle of motion, that is, change in general, and change was a process striving towards a goal. Thus final causes gave a rational ordering to the flux of generation and corruption. To give a scientific explanation was to give an account of a thing’s particular nature or form, and a thing’s nature was revealed by discovering the end towards which it strove. The paradigm originally employed by Aristotle seems to have been biological? thus, one learns the nature of an acorn by noting its development into an oak-tree. This view of nature as an object of scientific knowledge possessed implications for the role of experience, because natural processes do not always achieve their

ends - acorns do not always develop into oak-trees. Processes might fail to fulfill themselves for a variety of reasons: a concatenation of various accidental causes, each in itself ‘natural’, could pervert the course of a given process, as could man-made or artificial causes; most radically, for Christianized Aristotelianism, God could always intervene directly to change things,46 This was a potential source of difficulty in building a science of the physical world, because scientific demonstrations had to express necessary connections between cause and effect. If natural processes lacked absolute uniformity, that necessity would seem to be lacking. A way around this problem was developed by Albertus Magnus and Thomas Aquinas, the technique of reasoning ex supposifione.47 Its central feature was that the empirical fact, the process, to be explained - acorns developing into oak- trees, for example - was taken, or supposed, to have actually occurred without impediment. Given this initial supposition, a scientific explanation could then in principle be given. The essential properties of an acorn explain why it grows into an oak-tree, given that it actually does so; if it does not, accidental impediments must have prevented it. These may or may not be identifiable, but the demonstration ex suppositione itself remains properly

WA useful recent discussion is James A. Weisheipl, ‘Aristotle’s Concept of Nature: Avicenna and Aquinas’, Approaches to Nature in the Middle Ages. Papers of the Tenth Annual Conference of the Center for Medieval and Early Renaissance Studies (Medieval and Renaissance Texis and Studies Vol. 16), Lawrence D. Roberts (ed.) (Binghamton NY: Center for Medieval and Early Renaissance Studies, 1982), pp. 137- 160. For a valuable survey of early seventeenth-century scholastic conceptions fitting this same general description see Sister Mary Richard Reif, ‘Natural Philosophy in some Early Seventeenth Century Scholastic Textbooks’ (Ph.D. dissertation, St. Louis University, 1%2), pp. 191-210.

“A perspective adopted in the interesting study by Marjorie G. Grene, A Portrait of A&tot/e (London: Faber & Faber, 1963).

16 Weisheipl. ‘Aristotle’s Concept of Nature’, especially p. 152; William A. Wallace, ‘Albertus Magnus on Suppositional Necessity in the Natural Sciences’, Albertus Magnrcs and the Sciences. Commemora?ivefisays 1980, James A. Weisheipl (ed.) (Toronto: Pontifical Institute of Medieval Studies, 1980). pp. 103- 128, especially pp. LlO-111.

“Wallace, ‘Albertus Magnus’; idem, ‘Galileo and Reasoning ex strpposirione’, in Wallace. Prelude lo Galileo. Essays on Medieval and Sixteenth-Century Sources of Galileo’s Thought (Dordrecht: D. Reidel, 1981). pp. 129- 159.


scientific.U The rationale behind this technique serves to underline the significance of the universality of experiential statements: singular instances might not represent the ordinary course of nature, and so could not establish the nature of a particular process. An acorn failing to develop into an oak-tree would shed no light on the nature of acorns. Only experience formed from instances supplied habitualIy by the senses couId have scientific relevance; there could be no ‘crucial experiments’.

Deviations from the ordinary course of nature, if sufficiently spectacular, would be dubbed ‘monsters’, and far from being regarded, as they were by Francis Bacon, as providing privileged insights, they were often taken to be portents or omens, literally supernatural occurrences due to God’s intervention.49 But whether portents or not, monsters were by definition contrary to nature, and hence not illuminating of the naturaI order. Although typically applied to monstrous births, the term was used quite generally: Clavius, when arguing against sceptics doubtful of the reality of astronomers’ mathematical devices, extended its connotations in a revealing way, by concluding that ‘From all these things, therefore, I judge it to be established that eccentrics and epicycles are not so monstrous [monstrososl and absurd as they are feigned [to be1 by adversaries, and that they are not introduced by astronomers without great cause.“’ Epicycles and eccentrics were not contrary to nature, monstrous, but seemed to be an integral part of it. More intriguingly, in a Ietter of 1611, Clavius confirms Galileo’s recent telescopic discoveries and predicts that other ‘monstrosities’ (monstnmsita) will be found in the planets. ‘I The idea seems to be that the new features of the heavens are contrary to what would be expected on the received view of their nature. The

g Wallace has claimed an important role for ex suppositione reasoning in Galileo’s thought: ‘Gaiileo and Reasoning ex suppositione’; ‘Aristotle and Galileo: The Uses of Nvpothesir (Supposirio) in Scientific Reasoning’, Sfudies in Aristotle, Dominic J. O’Meara (ed.) (Washington DC: Catholic University of America Press, 1981), pp. 47-77; Galileo, pp. 340- 343. The claims in the latter seem rather toned-down compared with Wallace’s previous pieces on the subject.

“See Lorraine J. Daston and Katherine Park, ‘Unnatural Conceptions: The Study of Monsters in Sixteenth- and Seventeenth-Century France and England’, Past and Presenl No. 92 (1981), 20-54. Aiso relevant is Keith Hutchison, ‘Supernaturalism and the Mechanical Philosophy’, ffirtory of Science 21 (1983), 297 - 333.

“‘Clavius, ‘De sphaera’, p. 304. On the context of Clavius’s remark, see Nicholas Jardine, ‘The Forging of Modern Realism: Clavius and Kepler Against the Sceptics’, Studies in History and Philosophy of Science 10 (1979), 141- 173; this acts as a partial corrective to the treatment in Pierre Duhem, To Suve the Phenomena. AR Essay on the Idea of Physicai Theory from Plato to Guliko, trans. by Edmund Doland and Chaninah Maschler, introductory essay by Stanley L. Jaki (Chicago: Chicago University Press, 1%9), pp. 92 - 96. Duhem’s discussion, first published in book-form in 1908, is largely followed by Ralph M. Blake in Theorim of Scientific Method: The Renaissance Through the Nineteenrh Century, Edward H. Madden (ed.) (Seattle: University of Washington Press, 1960), chap. 2, ‘Theory of Hypothesis among Renaissance Astronomers’, especially pp. 32 - 35.

“Cited by Naux, ‘Clavius’, p. 193, from a letter to Mark Welser, 22 June 1611, printed in Galileo, Opere.


fact that these were constant features previously unobserved, and not occasional anomalies such as new stars, meant that they were not truly monsters; they were instead indications that the ordinary course of nature might be different from what had hitherto been supposed. But the primary sense of ‘monster’ exploited here by Clavius illustrates well the scholastic understanding of the use of experience in sciences dealing with the physical world. Experience taught how nature usually behaved; it did not consist of knowIedge of discrete events, because such events might be anomalous, ‘monstrous’.

The scholastic conception of experience inherited by the Jesuits, then, formed an integral part of their (more-or-less Thomistic) conception of the nature of a science. In the specific case of the mathematical disciplines, the insistence of the Jesuit school upon their scientific status implied that empirical input should be of the evident and manifestly universal kind appropriate to a science. It might appear that in categorizing the mathematical disciplines as scientific according to Aristotelian canons, the Jesuits had strait- jacketed themselves by excluding the possibility of using discrete experiences, that is, observations and experiments, in their development. Such was not the case, however, because elements of the mathematical sciences themselves as they had been practiced since antiquity demanded that special accommodations be made in the strict Aristotelian framework.

III

The peculiarities of observational data in astronomy did not intrude themselves into considerations of the role of experience in natural philosophy so long as astronomy was cordoned off from philosophy as a specialized discipline concerned with models for the computation of tables.52 However, by emphasizing that mathematics was a part of philosophy, and by going SO far as to promote astronomy itself to the position of a pre-eminent science, Clavius drew direct attention to its methodological form. He did not himself take up the implicit challenge; he restricted his remarks to such commonplaces as that astronomy was a mixed mathematical science subordinate to

‘2 Robert S. Westman has drawn attention to the importance of the disciplinary division between astronomy and natural philosophy in the sixteenth century. See especially his ‘The Melanchthon Circle. Rheticus, and the Wittenberg Interpretation of the Copernican Theory’, Is& 66 (1975), 165 - 193; ‘The Astronomer’s Role in the Sixteenth Century: A Preliminary Study’, Hslory of Science 18 (1980), 105 - 147. See also the important essay by Nicholas Jardine, ‘The Status of Astronomy’, chap. 7 of his The Birth offfistorv undPhilosophy ofScience. Paul Oskar Kristeller makes some extremely prescient remarks on the subject in an essay originally published in 1950, ‘Humanism and Scholasticism in the Italian Renaissance’, in Kristeller, Renaissance Thought. The Clnssic, Schotostic, and Wum~?nist Strains (Harper Torchbooks, 1961), pp. 92- 119, on pp. ll8- Li9.

Mathematical Science and the Reconsfitrrtion of Experience 147

geometry,53 allowing elsewhere that it employed in addition arithmetical demonstrations.” As with the problem of causality in mathematical demonstrations, an explicit analysis of the role played by empirical data in astronomy was left to Blancanus, in his Sphaera mundi of 1620.55

The work is an introduction to the elements of astronomy and cosmography, and covers much the same ground as Clavius’s commentary on De sphaeru. Blancanus justified this apparent duplication in part by pointing out that the final edition of Clavius’s work still failed to take into account the new telescopic discoveries, an omission he would rectify.56 His regard for the importance of telescopic observations indicates how the speciaI features of astronomical practice could intrude into questions of scientific methodology: telescopic observations were produced by the use of special instrumentation not readily available to all. These ‘experiences’ could certainly be expressed as universal statements - ‘Venus displays phases’; ‘the moon’s surface appears rough’ - but they lacked ‘evidentness’ precisely because they were only possessed by telescopic observers. As far as everyone else was concerned, they were not part of the ordinary course of nature, and this may be what Clavius had in mind in calling them ‘monsters’. Astronomy had in fact always provided potential anomalies for an orthodox schoIastic - Aristotelian view of the place of experience in a science, because its techniques employed data consisting of discrete observations made at particular times and places, as well as with the aid of instruments.

The two key words in the medieval and early-modern astronomical vocabulary were ‘phenomenon’ and ‘observatio’, both corresponding to Greek terms found pre-eminently in Ptolemy’s Alrnagest.57 Ptolemy’s usage is fairly straightforward: a phenomenon was any kind of appearance in the heavens, whether the path of a planet or an eclipse of the moon, and an observation was an act whereby a phenomenon became known through the senses .” Neither Ptolemy nor subsequent astronomical writers seem to have felt any methodological discomfiture with this terminology; it required an acute self-consciousness about the scientific status of astronomy to create it.

S3CIavius, ‘Prolegomena’, p. 3. “Clavius, ‘De sphacra’, p. 3. This follows Ptolemy. “Sphaera mundi, seu cosmographia. demonstrativa, ac facili method0 tradita: in qua totius

mundi fabrica, una cum novis, Tychonis, Kepleri, Galilaei, aliorumque astronomorum adinventis contittetw. . . . Authore losepho Blancano Bononiensi e Societate IESU. _ . Bononiae, Typis Sebastiani Eorromij Sumptibus Xieronymi Tamburini. 1620.

zblbid.. ‘Praefatio’, 2nd p. -On astronomy in the Middle Ages see, for a general survey, Olaf Pedersen, ‘Astronomy’,

Science in the Middle Ages, Lindberg (ed.), pp. 303 -337. The literature tends to concentrate either on planetary models themselves or on instruments themselves, rather than on the interaction of instruments, data and the process of modelling.

uSee, e.g. his usage in Almagest 1.3, 4 (Toomer trans., pp. 38 and 40).


Blancanus, however, fulfilled this criterion, and his concern with methodology led him to an apparently quite novel, and acute, redefinition of the two central terms. Clavius, typically, left the matter alone, but his use of the word ‘experience’ during discussion of astronomy, whilst it retains its usual scholastic sense, also hints at the complications in its function created by the role of singular experiences in astronomical practice. In discussing questions arising from the rotation of the heavens and precession, for example, Clavius concludes:

Wherefore faith is to be had in the experiences [experientiisl of astronomers, until something else is brought forward to the contrary by which it be demonstrated that what is propounded by astronomers concerning the motion of the stars from the west towards the east above the poles of the zodiac is not trueV5’

The ‘experiences of astronomers’ refers to their general accumulated experience in this matter rather than to a body of discrete observations, but the acknowledged possibility that ‘something else’ could be ‘brought forward to the contrary’ admits the practical dependence of astronomical doctrine on such observations.

In terms of the formal construction of a science, the phenomena were what provided the empirical premises on which astronomical demonstrations were based (either directly or through the intermediary of geometrical models). Properly speaking, as experiences they needed to be general statements of how things are or behave in the heavens. In fact, however, they were not all known through common experience; they were not all, in that sense, evident:

precession of the equinoxes, for example, was a phenomenon which could only be constructed from discrete data collected over long periods of time. Blancanus gave this point explicit recognition in Sphaera mundi. Following elementary material on the celestial sphere, the treatise gets underway with a section called ‘Sphaerae materialis et mundanae simul explicatio’.60 The opening chapter is headed ‘Suppositiones’, the technical term for those things which must be accepted at the outset before a science can be constructed - definitions, statements of existence and undemonstrated (though not unjustified) principles generally.6’ Blancanus explains what suppositions are involved in astronomy:

Apart from those things which astronomy supposes that have been received from outside, as much from geometry as from arithmetic, as has been said in the apparatus at the beginning, it supposes as well other things internal to it [intrinsecul, and principles proper to it, and even as it were foundations, which are indeed of two kinds, for astronomers call some ‘Phaenomena’, or appearances, for the reason that

‘pClavius, ‘Ue sphaera’, p. 33. MBlancanus, Sphaera mundi, pp. 15ff. ” See Wallace, Galileo, pp. 112- 113; and, more fully, idem, ‘Aristotle and Galileo’.


they appear and are manifest even to all the common people [omnibus etiam vulgol, such as: the rising and setting of the stars [stellar], moon and sun; that aI1 stars byderal move from the east to the west; that the sun moves lower in winter, and higher in summer; that the sun does not always ascend from the same place on the horizon; and many other things of that kind which we suppose as very well-known to alLU

Phenomena, then, are defined not simply as appearances in the heavens; they

are also characterized by their being known to all - they are evident, a part of

common experience. That is what one would expect of the empirical suppositions of a true science; that is how they are justified. But Blancanus must recognize other, less evident principles in astronomy.

Astronomy has from itself another kind of principles not sought from elsewhere. which are called ~Q~~ELC, that is, observations: they are, moreover, certain items of knowIedge [cugniciorpesl provided from experiences Lab experiment&l which do not become known by everyone as appearances do, but only by those who, skilfully labouring hard at it with diligent work, and instruments, apply themseIves zealously to the science of the stars Melloruml . . . .6’

This privileged knowIedge includes such things as the apparent diameters of the sun and moon, and the retrogradations and speeds of the planets.64

Blancanus’s distinction between ‘phenomena and ‘observations’ appears to be quite novel, resting as it does on a distinction between their respective cognitive statuses. Phenomena are evident, whilst observations are recondite; furthermore, observations are constructed from discrete experiences, often acquired using instruments (recall Blancanus’s interest in telescopic discoveries), whereas phenomena are more-or-less given in ordinary experience.

9 Blancanus. Sphuera mundi. p. 15. bllbid., pp. I5 - 16. ul&id., p. 16.


Blancanus’s description of the way in which ‘observations’ are produced from discrete experiences ultimately derives from Ptolemy’s account of his procedures in the Almugesf.65 The Greek word Blancanus gives as the equivalent of ‘observatio’ comes from the same source, but not from the same context - his use of the two, procedure and word, in conjunction, is certainly not Ptolemaic. At one point, Ptolemy describes how he had confirmed Hipparchus’s belief that the stars within the zodiac maintained their positions relative to those outside it, so that precession was common to all the stars. Hipparchus had reached his conclusions by comparing his own positional data with those from earlier astronomers; Ptolemy used Hipparchus’s data in a similar fashion and, owing to its superiority to that available for comparison to Hipparchus himself, he had been able to test the belief with some reliability.% What Ptolemy describes, then, is the production, or construction, of a piece of astronomical knowledge from various sets of data. It accords exactly with Blancanus’s description of the nature of an ‘observation’. But Ptolemy used that term to refer to the gathering of the discrete items of data, not to the knowledge manufactured from them. It should be noted that the contrast between ‘phenomena’ and ‘observations’ on Blancanus’s account is not one between universal and discrete experiences: both are universal. Observations, however, are universal experiences (experiences expressed as universal statements) which are explicitly constructed, using appropriate computational techniques, from discrete experiences. That gives them a special status compared with phenomena, which are universal experiences which do not need to be explicitly constructed, and which are therefore identical to the Aristotelian definition of experiences. It is the process of construction, as well as the specialized nature of the data which itself must be ‘manufactured’ using instrumental techniques, that sets ‘observations’ apart from ‘phenomena’.

Blancanus’s distinction between phenomena and observations was important for astronomy’s scientific status because, as suppositions (suppositiones), observations were not evident in the way that phenomena were; they were suppositions relying on the specialized work of astronomers, who to that extent had privileged access to them. Those who were not astronomers had to accept them on faith. Only Blancanus’s sensitivity to the

6sSee A. I. Sabra, ‘The Astronomical Origin of Ibn al-Haytham’s Concept of Experiment’, Ack du XIF CongrPs Intematfional d’tiirtoire des Sciencq Paris 196% Tome 111 A (Paris: Albert Blanchard, 1971), pp. 133 - 136. The passage is Ptolemy, Ahugest VU.1 (Toomer trans., pp. 321- 322).

-Sabra, ‘Astronomical Origin’, p. 134. See also on Ptolemy’s use of observational data Lloyd, Magic, Reason and Experience, pp. 183 - 200.

Mathematical Science and the Reconstitution of Experience IS1

requirements of a genuine science impelled him to recognize these two distinct cIasses of astronomical suppositions; no one before him had formulated the matter in such a way as to raise the issue at all.

The goaI of a science to provide evident, public knowledge, demonstrated and necessary knowledge which could be possessed by all, therefore ran into problems in the case of astronomy.67 Those problems wouId demand a reformulation of the criteria by which empirical suppositions, in Blancanus’s astronomical terms ‘observations’, could be judged ‘evident’, and therefore

“The character of these problems is illustrated by considering Tycho Brahe’s quite different conception of astronomy. For Tycho, astronomy was not an Aristotelian science (he always referred to it as an ‘art’); instead, it was akin to alchemy. The same secrecy, the same attitude towards the private character of knowledge acquired through personal experience and endeavour, applied for Tycho to astronomy, astrology, and alchemy, the latter referred to as ‘terrestrial astronomy’ because of the astrological linkages between planetary influences and the generation of metals and other minerals on earth [see, e.g. Tychonis Broke Duni Opera Omniu, 1. L. E. Dreyer el rrl. (eds) (reprint Amsterdam: Swets & Zeitlinger, 1972), V, p. 118; VI, p. 145; VII, p. 2381. In respect of his work in astrology, Tycho said that ‘we are not inclined fo communicate this kind of astrological knowledge to others’; the reason was that ‘it is not given to everyone to know how to use it with caution, without the superstition or excessive confidence which shouId not be shown towards created things’ [Omro V, p. 117; translation adapted from @Z/IO Brahe’s Descripfion of his Instruments and Scientific Work as given in Astronomiue Instauratas Mechanica {Warrdesbtugi, 1X&?), trans. and cd. by Hans Raeder, Elis Stromgren and Bengt Stromgren (Kobenhavn: I Kommission hos ejnar Munksgaard, 1946). pp. 117 - 1181. In the case of aIchemy, ‘it serves no useful purpose, and is unreasonable, to make such things generally known. For although many people profess these things, it is not given to everybody to treat these mysteries properly according to the demands of nature, and in an innocent and useful way’ (@era V, p. 118, adapted from Description, p. 118). Nonetheless, ‘I shall be willing fo discuss these questions frankly with distinguished and princely persons. . . and I shall on occasion communicate something to them, as long as I feel sure of their good intentions towards me and that they will keep it secret’ (ibid.). Tycho’s possessiveness towards his astronomical data is well-known; his attitude towards the instruments producing those data was of a piece with that towards the allied fields of astrology and alchemy. Regarding projected new instruments, he said: ‘I shall hardly publish anything about these and similar matters that I have invented recently. . nor about those that I shall invent in future. . But to distinguished and princely persons. . . shall I be willing to reveal and explain these matters when convinced of their gracious benevolence, but even then only on condition that they will not give them away’ (Opera V, p. 101, adapted from Description, p. 101). Tycho’s perception of his work as generating p&ate knowledge is tempered only by his need for patronage, and the three passages just quoted date from 1598, when his position was somewhat shaky prior to his move to Prague: see J. R. Christianson, ‘Tycho Brahe’s German Treatise on the Comet of 1577: A Study in Scientific Patronage’, Isis 70 (1979). 110-140. When Tycho approvingly quoted Paraceisus on alchemy, he expressed equally his own view of astronomy: ‘For Paracelsus says most rightly that no one knows more in this art than he who knows by experience through the fire [per ignem expertus sit]’ (Opera VI, p. 146). Tycho’s conception of knowledge in the art of astronomy was that of experience through the quadrant. This went along with a general dislike of scholastic philosophy - ‘the academies of the peripatetics. demented and encrusted with sophisms and pretenm [have neither learnt nor teach] to examine the reality and very kernel itself of truth’ hidden in alchemy and astrology (O~ru VI, p. 145). That astronomical practice lent itself so readily to such a view of knowledge indicates starkly the problems faced by Blancanus. See on these aspects of Tycho: Charles Webster, From Parace&u.s to Newton. Mugic and the Muking of Modern Science (Cambridge: Cambridge University Press, 1982), pp. 29 - 30; Victor E. Tboren, ‘Tycho Brahe as the Dean of a Renaissance Research Institute’, Religion, Science, and Worldview. &~UJU in Hcwor of Richard S. West&K Margaret J. Osler and Paul Lawrence Farber (eds) (Cambridge: Cambridge University Press, 1985), pp. 275 -295, especially pp. 286- 290; and especially Owen Harmaway, ‘Laboratory Design and the Aim of Science: Andreas Libavius versus Tycho Brahe’, Is& 77 (1986), 585 - 610.


acceptable. The new criteria would provide a different way of employing experience, a way approaching a characteristically seventeenth-century notion of experiment. Blancanus’s distinction between ‘phenomena’ and ‘observations’ shows his awareness of the difficulty, and Jesuit work in other areas of the mixed mathematical sciences shows a concern with establishing and elucidating their methodological structure leading to just the same kind of results as those of Blancanus in astronomy.

Geometrical optics was at least as well-established a discipline as astronomy, with an extant textual tradition stretching from Euclid through Ptolemy to Alhazen. The work of the latter remained, together with Witelo’s contributions in the thirteenth century, the major text on the subject at least until Kepler’s Paralipomena ad Vitellionem of 1604.68 In the second decade of the seventeenth century there appeared two optical works by Jesuits, the Opticorum libri sex of Franciscus Aguilonius, in 1613, and Christopher Schemer’s Oculus, in 1619. 6g Both display a sensitivity to what might be described as experimental procedures, something which, in one form or another, had been part of optics from Ptolemy onwards, being particularly prominent in Alhazen.7D Whereas in astronomical treatises the relationship between positional data and derived generalizations could be characterized, as it was by Ptolemy, in terms of the comparison of data, in optics there was a greater emphasis on the singular experiences, or experiments, themselves. As with astronomical data, these singular experiences were constructed using special apparatus, and were often quantitative. The best-known example in Ptolemy’s optical work, namely his measurement of refraction, bears an apparent similarity to astronomical measurements, but it is, as has been indicated earlier, untypical of his usual approach;” the data given, furthermore, appear to have been generated with the aid of a computational paradigm of second differences rather than by the bald measuring technique described by Ptolemy.” In fact, as A. I. Sabra and S. B. Omar have

argued,73 it is Alhazen who for the first time incorporates experiment, in the

tiSee Lindberg, Theories of Vision, for a thorough survey. wFruncisci Aguilonii e societate Iesu Opticorum libri sex. Philosophis iuxta ac Mathematicis

utiles. Antverpiae, ex officina Plantiniana, Apud Viduam et FiIios IO. Moreti. MDCXIII; Christophorus Scheiner, Oculus. Hoc est: fundamentum opticum (Oenoponti, spud Danielem Agricolam 1619). August Ziggelaar, Fraqois de Aguildn S. J. (1567-1617). Scientist and Architect I = Bibliotheca Insfituti Historici S.I., Vol. 441 (Roma: lnstitutum Historicum %I., 1983), devotes chaps I and 2 of Part I1 to a study of the Opticorum libri. Further references to material on Scheiner may be found in William R. Shea, ‘Galileo, Schemer, and the Interpretation of Sunspots’, Isis 61 (1970), 498 -519.

“See note 73 below. “See note 37 above. “Schramm, ‘Steps’, especially pp. 74-75; an attempt to explicate the nature of Ptolemy’s

enterprise here is A. Mark Smith, ‘Ptolemy’s Search for a Law of Refraction: A Case-Study in the Classical Methodology of “Saving the Appearances” and its Limitations’, Archive for History of Exact Sciences 26 (1982), 221- 240.

“Sabra, ‘Astronomical Origin’; Omar, Ibn al-Haytham’s Optics.

Muthematical Science and the Reconstitution of Experience 153

sense of deliberate construction of empirical fact, into optics and explicit optical methodology. The term Alhazen employs is not the standard Arabic translation of the peripatetic (and Galenic) E~IT.&Z, ‘experience’, but one drawn from Arabic astronomica usage, corresponding to Ptolemy’s word for a test or proof involving the comparison of distinct sets of data.74 This term, in its various forms, was rendered into Latin as CK-erimentum or experimentatio, experimentare, and experimentutor.” AIhazen appears deliberately to have used a word distinguished in meaning from the usual ‘experience’; it is, however, questionable to what extent he had in mind the notion of ‘experiment’ as a test of a pre-conceived hypothesis. Sabra maintains that Alhazen’s use of the astronomical term retains the technical specificity of Ptolemy’s concept of ‘test by comparison’, and that he therefore intended just this notion.76 Omar, disagreeing with Sabra, goes to the further extreme of arguing that Alhazen employed ‘experiment’ as a way of generating hypotheses, not merely testing them, and cites in his support the way in which Alhazen’s treatise develops its ideas in tandem with its described experiments.77 One may perhaps doubt this latter point on the twin grounds that the presentation of materia1 in a systematic treatise need bear little relation to any methodological ‘logic of discovery’, and that in any case such a procedure by Alhazen wouId be egregiously out of keeping with pre-modern ideals of science. For our present purposes, however, it is enough to note that Alhazen’s ‘experiments’ seldom actually correspond even to the astronomical usage to which Sabra etymologically traces them. Alhazen usually employs an ‘experiment’ in order to demonstrate something to be the case, not to test a hypothesis or to derive sets of comparable data. When discussing refraction in dense media (to choose a typicai example), Alhazen provides instructions for experiencing the behaviour he describes, not a report of an attempted test of his claims.

Since therefore the experimenter would wish to make tria1 of [.CTXJJW+~] the passage of light Mninisl in water through this instrument: let him take a straight-sided vessel, such as a copper urn, or an earthenware jar, or something similar. . . .”

Alhazen concIudes this discussion by saying: ‘By this method [vial, therefore, the passage of light [luck] through a body of water will be experienced

“Sabra, ‘Astronomical Origin’, and see above. “ibid., p. 133, referring to Opticue thesaurus. Alhuzeni arabis libri septem, nuncprimutn editi.

Eiusdem liber De cremwxdis & Nubium ascensionibus. Item Vitellionis Thuringopoloni iibri X. Omnes instaurati, figuris illusfrati 8r aucti, adiectk etiam in Alhuzenum commen?a&s, a Federico Risnero. Basileae, per episcopios. ML&XXII.

“Sabra, ‘Astronomical Origin’, pp. 134- 135. “Omar, Ibn al-Haytham’s Optics, chap. 3, especially p. 68. “Alhazen, Upticue thesaurus, p. 233. This is, of course, the edition familiar to early

seventeenth-century optical writers.


[experimentabiturl . . r79 The sense clearly indicates that the translation ‘experienced’ rather than ‘tested’ or ‘tried’ is appropriate here; Alhazen has given a recipe allowing his reader to see what Aihazen himself already knows, and not a protocol to provide the reader with an opportunity of checking his claims. The same intention can be seen more straightforwardly elsewhere, as for example when, in introducing some theorems in catoptrics, Alhazen remarks: ‘What we have said will be evident [patebit] in spherical mirrors polished on the outside.“’

The development of these ideas in optics by the Jesuits involved explicit recognition of the special cognitive status of constructed experiences. Although Alhazen used a special technical term, apparently drawn from astronomy, to designate an active production or construction of phenomena, he did not worry about the problems of casting optics as an Aristotelian science. Methodologically, he saw himself as being within a properly optical tradition stemming from Euclid, who is his main source of references on such matters,” and as such did not concern himself with confronting wider issues of scientific argument. In consequence, he could employ ‘experiment’ without being required to consider it methodologically because he was writing in a delimited genre of ‘optics’, not a broader genre of ‘mathematical sciences’. For Aihazen, ‘experiment’ was a procedure, not a cognitive category. Both Aguilonius and Scheiner, by contrast, display a concern with establishing optics as a science fulfilling Aristotelian canons. Scheiner is much more conscious of the implications of such a characterization than Aguilonius, who gives no sign of seeing any problems whatever in the use of specially constructed experiences in scientific demonstration.” Book IV of his treatise, devoted to optical illusions, presents as the first of its ‘Hypotheses’ a consideration of the fallibility of the senses.*’ Aguilonius denies that either the Epicurean doctrine that the senses never deceive, or the sceptical doctrine that they can never be relied upon, are valid, and presents an orthodox account of the role of sensory experience in scientific knowledge.w The particular context of his discussion provides an additional reason, apart from those

“Ibid., p. 235. mIbid., p. 126; see also, for a similar formulation, p. 134. “See e.g material in ibid., pp. 30- 32. Roshdi Rashed, ‘Optique geometrique et doctrine I .

optique chez Ibn al Haytham’, Archive~or History ofExad Sciences 6 (1970). 271-298, makes the point that Alhazen uses traditional optical terminology even while departing from Euclidean/ Ptolemaic conceptions of the nature of optics.

B2A g uilonius seems often to follow Alhazen’s conception of optics, for example in his stress on vision as an aspect of cognition: compare, e.g. Aguilonius, Upricomm libri, preface ‘Lectoris’, 1st p. , and Book III pas&, with Alhazen, Opticae thesaurus, Book 11, chap. 1.

“Aguilonius, Opticorum libri, ‘Visum subinde & falli & fallere’, pp. 213 ff. u Ibid., pp. 2 I3 - 215. On sceptical doctrine in this period see Richard H. Popkin, The Nisrory

oj tiepricism from Erasmus to Spinoza (Berkeley: University of California Press, 1979); C. B. Schmitt, ‘The Rediscovery of Ancient Skepticism in Modern Times’, The Skeptical Tradition, Myles Burnyeat (ed.) (Berkeley: University of California Press, 1983), pp. 225 -251.

Muthematical Science Qnd the Reconstitution of Experience 155

considered earlier, for the necessity of repeated sensory acts to constitute a single valid experience.

If the senses always mocked the acuteness of the mind, there couId be no science, which [is what1 the Academics strive to assert; and if they never failed, the most certain experience would be had by a single act. Now, the matter is, however, in the middle [of these extremes]: for although the senses are sometimes deceived, usually however they do not err. Hence it is, that whenever experience is certain the first time, it is confirmed by the repetition of many acts agreeing with it. For a single act does not greatIy aid in the estabIishment of sciences [ad scientiarum primordial and the settlement of common notions, since error can exist which lies hidden for a single act, but having been repeated time and again LFnepe ac suepius it strengthens the judgement of truth, until finally it passes into common assent; whence afterwards they are put together through reasoning as if from the first principles of a science.”

Aguilonius thus uses the standard Aristotelian conception of experience, and regards such experience as adequate to establish the empirical ‘common notions’ (a term equivalent to Aristotle’s ‘axioms’s6) as the basis of a science. AII this accords with what we have considered earlier. Aguilonius, however, nowhere seems to regard features peculiar to optics as requiring special methodological elucidation in the way that Blancanus was to do for astronomy. Scheiner, by contrast, follows the latter’s approach very closely.

The preface to the 0cuZu.s opens by categorizing optics in standard Aristotelian fashion.

Optics, truly and properly called a science [here a marginal reference to Aristotle, Posterior AnQ&tiCs 1.30”1, has much distinct from, and much in common with, physics. Common and foreknown Lpruecognifal are its objects. For both, as much physicists as opticians, are concerned with visible things and the organ of sight; however, in different ways. For geometry, as the Philosopher declares, Physics 11.20, considers the physical line, but not insofar as it is the physical line of the physicist: optics @rspectivu, here used as a synonym for optical, however, indeed considers the mathematical line, but not insofar as it is physical. They both investigate the truth of the same thing, therefore, but by different methods [ViiSl. . . .II

Both physics and optics, Scheiner continues, deal with those things which enter the senses:

‘IAguilonius, Opticotwm Ii&H, pp. 215 - 216. “Cf. note 32, above. “The reference is actually An. pmt. 1.3. In addition, Scheiner here gives references to Pereira.

Villapandus, and to Blancanus’s De nzulh. nat. w!3chciner, Oc&s, ‘Praefatio’, 1st p. The reference is Phys. IJ.2.


of those things, some which so occur as happen by nature and are evident [obvial to everyone, and require only the observation Ianimadversionem] of the sedulous investigator, are called ‘Phaenomena’ or appearances: others, which either don’t occur or don’t become evident [non patescuntl without the industry of special empirics [peculiari Empirici] are called ‘Experientiae’ . . . .89

This unusual dichotomy exactly matches Blancanus’s for astronomy, with the slight difference that what Blancanus calls ‘observations’ Scheiner calls ‘experiences’. Scheiner’s ‘experiences’ need to be constructed by ‘special empirics’, just as Blancanus’s ‘observations’ required the specialized work of astronomers. Since they would be couched in the form of universal statements (if only as conditionals), they were still close enough to Aristotle’s general concept of ‘experience’ to allow Scheiner to invoke Aristotle’s authority, as in his description of the contents of Book I, Part II of the Oculus: ‘In the second part we will bring forward experiences produced for the purpose [pro re natal, so that from them we establish truth and rebut errors. For one true experience,

the Philosopher declares, is worth more than a thousand deceitful subtleties of underhand reasons.‘gO

A particularly noteworthy feature of Scheiner’s ‘experiences’ is that, as he had said, they were such that without the ‘industry of special empirics’ they would not only fail to become evident, but might not even occur at all. Indeed, he planned in his treatise to ‘bring forward experiences produced for the purpose’ to establish his arguments. This last point is elaborated in Part II of Book I, entitled ‘Experientiae variae.’

This other part of the first book is occupied with putting forward and explaining various abstruse [reconditis] and well-tried [probutisque] experiments, most diligently investigated with singular industry, tenacious labour, much exertion, and faithfully brought into the light from the hidden treasury of nature, so that from them, just as from foreknown, undoubted [principles], it may be permitted finally to reach into the true throne of the visual power without obscurities.”

Scheiner is here concerned with establishing ‘undoubted’ principles as premises for scientific demonstration. He speaks in that connection of ‘abstruse and well-tried experiments’, not ‘experiences’, and the distinction is a functional one embodied in the structure of the presentation. Thus the subsequent text commences with Chapter I, ‘Experientia Prima. Pupillae variatio.‘92 This

O9 Ibid. “Ibid., p. 1. ” fbid.. p. 29. “Ibid., pp. 29- 32.


chapter contains three sub-sections headed ‘Experimentum’, each consisting of a set of instructions whereby an aspect of the variation in size of the pupil can be evinced. Most of the other chapters in this part of Oculus are also ‘experientiae’, as its title indicates, and they deal with a number of matters apart from pupil size, such as the effect of different degrees of illumination on the apparent size of objects. Although only the first chapter is broken down into sections explicitly 1abeIled ‘experimentum’, the other ‘experiential chapters consist of exactly the same kinds of things - ‘experiments’, similar in nature to those of Alhazen93 (Scheiner is, of course, covering quite similar material), presented variously in the form of instructions or in theorem-form (typically commencing ‘Sit. . . ‘).w Scheiner’s terminology is, therefore, quite clear: an ‘experience’ is sensory knowledge about an aspect of the world which needs to be deliberately brought into being, or constructed, and it is expressed as a universal statement, thereby corresponding to Aristotle’s definition and so fit for use in scientific demonstration. An ‘experiment’, by contrast, is a particular procedure whereby the experience may be instantiated. Thus the ‘experiments’ are the means by which an ‘experience’ is constructed. One can come into possession of the ‘experience’ that the pupil contracts when exposed to bright light by performing a number of different ‘experiments’ which contribute to its formation. The contrast with ‘phenomena’ is that the individual instances constituting the latter are not codified or made explicit, because they are presented routinely by nature.95

At the beginning of Book I, Part I, Scheiner considers ‘the necessity of anatomical inspection concerning the eye’,% and he expresses that necessity as arising from the requirement for firm premises on which to build optical science. As Clavius had said, every science demonstrates its conclusions from ‘certain assumed and conceded principles’, but no science ‘demonstrates its own principles’. Scheiner has therefore to justify his principles through experience, and that justification will, strictly speaking, be outside the science itself.

Foreknown [Praecognita, a word echoing ‘praecognitio’, a term designating everything which a science must take as given”1 for our purpose are not so much

“In subsequent parts of the work, esp. Book III, Part I, Scheiner deals with pinhole images, lenses and the telescope, though with less reference to Kepler (who is mentioned now and agein throughout OCU~IJS) than one might expect.

‘All this to ibid., p. 52. R There is again a contrast with Aguilonius here, who in dealing with this same question adduces

the way in which the pupil is known to dilate habitually in the doom so as to admit more light, and vice versa. He considers it necessary only to refer to common experience, not to formulate discrete experiments. Aguiionius, Op~icorum l&i, pp. 19 - 20.

9( Scheiner, Oculrcs, p. 1. “See Wallace, Gu/iko, pp. 101- 110.


phenomena as singular experiments derived from study [studio], and they are of two kinds; one from inspection of the eye, the other selected from consideration of appearances extended out [specierum. . .diffusaruml from observable things [a rebus uspectabifibusl into the eye. We will gather something from b0th.l

Once the premises based on the resultant experiences (to do both with the eye and with reflection and refraction) were established, as they were in the two parts of Book I, they could be used in properly scientific demonstrations. In Book II especially, propositions are supported by a variety of demonstrations founded on the ‘experiences’ previously constructed. For example, in a proposition concerning the refraction of light at the surface of the cornea, the form is of this kind: ‘Demonstratio II. Per Experientiam citati Capitis 8. . . ‘;

‘Demonstratio III. Sumitur ex Experientia 4.c.4.p. 2.1.1. . _‘; ‘Demonstratio IV. Obvia est ex Experientia 8. Capite II allata. . . .‘*

Scheiner’s approach to optics, therefore, differs from Alhazen’s precisely to the extent that it is similar to the view of astronomy found in Blancanus, and both reflect the renewed stress on establishing the scientific credentials of the mathematical disciplines inaugurated by Clavius. Alhazen had been less concerned than Ptolemy with the axiomatic ideal of science whereby the truth of the premises should be immediately evident. On that basis, he had gone a long way in developing deliberately conceived experiments which would refine ordinary experience by creating conditions not usually met with in the ordinary course of events. These experiments were ‘potential experiences’ which could be actualized by the reader so as to reproduce Alhazen’s own optical experience (at least in principle; see below). The Jesuit opticians, Aguilonius and Scheiner, differed from Alhazen in their characteristic concern to maintain the status of their discipline as an Aristotelian science - their scientific ideal in optics was therefore closer to Ptolemy’s than to Alhazen’s. In Scheiner’s case, the constructed experiences figuring in optical science since Ptolemy, and essential to the treatment of, particularly, refraction and vision, acquired a special and problematical status which they had not possessed for Alhazen and which Aguilonius had not confronted.‘00 Scheiner distinguished constructed ‘experiences’ which required the work of ‘special empirics’ from

* Scheincr. Ocks. p. 1. psIbid., p. 19. ‘mAguilonius’s use of specific experiences is almost casual; he typically treats them as

unproblematic and supplementary to the dlscussion rather than central. Thus, For example. in arguing that cold bodies don’t radiate their coldness, Aguilonius says: ‘only by touching is any cold thing felt, which we have often demonstrated [osfendimusl by instruction with an entertaining experiment fiqticiun~ibibus Iudicro qrimenlol’ @_ 359). Another ‘ludicrum experimenturn’. to do with the impossibility of judging distances with just one eye. Aguiloniur said he heard from some boys, ‘but we have judged it worthy of a philosopher’ @. 154). ‘Experiences’ arc primarily illustrative for Aguilonius.

Mathemntical Science and the Reconstitution of Experience 159

the evident ‘phenomena’ which would, ideally, have been adequate but in practice were not. This mirrored Blancanus’s distinction, in astronomy, between constructed ‘observations’ and evident ‘phenomena’. And just as Blancanus’s ‘observations’ were ‘cognitiones quaedam ab experimentis comparatae’,“’ that is, prepared or provided from ‘experiments’, the instrumental manufacture of individual pieces of data, so also Scheiner’s ‘experiences’ were constructed from contrived ‘experiments’.

To say that the Jesuit mathematicians recognized, and in effect tried to codify, the methodological problems involved in making the mixed mathematical disciplines into genuine Aristotelian sciences is not to say that they succeeded in solving them. Indeed, the challenge they posed required in practice a good deal of compromise. The central problem, that of establishing the premises, or suppositions, from which scientific demonstrations proceeded, did have the advantage of not being part of scientific methodology at all. It was a sine quo non of a true science that the premises should be evident and acceptable, but the means by which the premises were established was not itself a scientific matter. There were some restrictions, in that premises should be primitive and indemonstrable (with the special provisions for subordinate sciences providing a partial exception), but, in general, if suppositions could be made evident, they could be used in scientific demonstration. The classifications used by Blancanus and Scheiner, however, allowed that certain kinds of supposition were only really evident to specialized investigators operating with tools designed for the purpose. The Jesuit mathematicians could not avoid this problem, especially in the case of astronomy, and although in the purely pedagogical context such suppositions might be treated as petitiones, premises voluntarily granted on the word of the teacher,lM this recourse was inadequate to establish scientific status. A partial solution was to present material prior to the formulation of the ‘experience’ itself, which for astronomy might consist of observational data on the positions of celestial bodies. Astronomical tables, derived from predictive models themselves resting on observational data, and hence equivalent to Blancanus’s ‘observations’, served a similar purpose: the gradual realization of the predictions (assuming this occurred) itself provided a continuing opportunity for experiencing the planetary motions embodied in the models. In optics, the detailed description of experiments represented potential

‘*‘Blancanus, Sphueru mundi, p. 15 (see above). There is an interesting parallel between the two kinds of empirical supposition recognized by Blancanus and Scheiner. and Descartes’ explicit distinction between ‘common experience’ and particular, contrived experiments (both usually referred to as ‘experiences’): see Desmond M. Clarke, I%sc&e~ Philosophy of Science (Manchester: Manchester University Press, 1982), pp. 22- 23.

?Gx Wallace, Galileo, p. 113.


experiences, being, not historical events witnessed and reported by the author, but accounts of what happens when certain situations are created.

The inadequacy of this solution lay in the extent to which the generation of certain kinds of empirical knowledge depended on the ‘diligence’ and expertise of observers and experimenters, with their specialized instruments and skills. It

was not a straightforward matter for anyone to reproduce for himself the experiences claimed by astronomers or opticians. This problem only became crucial when there was dispute, or the likelihood of dispute, over such claims. But in that event, the need for new norms of evidence became pressing. How could ‘experiences’ be established as common property if most people lacked direct access to them?

There was also another methodological difficulty. It was suggested above that Blancanus’s concern to include telescopic observations in his Sphuera mundi indicates his awareness that much astronomical knowledge was instrumentally constructed. Scheiner treated the telescope (the tubus oculu.s) in Book III, Part I of his optical treatise,“’ along with lenses and pinhole images (he appears to be greatly indebted to Kepler), and it is probable that ‘experiences’ constructed using lenses were what he had in mind when referring to things which ‘don’t come into being’ without the ‘industry of special empirics’. These ‘experiences’, apparently fully acceptable in scientific demonstration, could only be constructed by ‘empirics’. Similarly, Aguilonius described how, in his optical work, ‘I consider things intentionally altered by me [Consulto. . . consilio a me mutara red’, as well as those provided by occasion. ‘04 Now, such experiences were, by definition, artificial, not natural; they did not represent the ‘ordinary course of nature’. The particular case of experiences constructed using lenses, it will be recalled, prompted Clavius to refer to Galileo’s discoveries as ‘monsters’. The employment of constructed experiences in the mathematical sciences threatened to violate not only the requirement that scientific premises be evident, but also the strict artificial/natural distinction at the heart of the Aristotelian world-view.

There were pragmatic ways around both these difficulties. In the next section we will consider how the Jesuit mathematicians partially circumvented the artificial/natural distinction by means of a repeated emphasis on the disciplinary and philosophical division between mathematics and physics. In the section following, consideration will be given to the development of new evidential norms surrounding singubr experiences.

‘mBlancanus, Sphaera mmdi (in the appendix entitled ‘Apparatus ad mathematicas addiscendas, et promovendas’), p. 413, refers to Scheiner’s book and to the great potential value of the telescope and its relation to optics. He also refers to the many experiments in reflection and refraction brought forward by Scheiner.

‘“Aguilonius, Opriconrm l&i, preface ‘Lectoris’, 9th p.


IV

Because the natures of things were revealed by undisturbed natural processes, the artificial/natural distinction performed an important function in physics.‘” However, the Aristotelian stress on the difference between the subject-matters of mathematics and physics, which underpinned Clavius’s arguments for the superiority of mathematical over physical demonstrations, rendered the distinction irrelevant to mathematics. Even though the mixed mathematical disciplines concerned physical phenomena, they only considered the quantitative aspects: physical causes fell inside the domain of physics, and outside that of mathematics. The mathematician did not seek to reveal essential natures, and so he could interfere with natural processes quite legitimately. But to do so required a constant policing of the disciphnary boundary between mathematics and physics. The integrity of the methodological structure of the mathematical sciences, conceived within an Aristotelian framework, was maintained by making it clear whenever necessary that certain questions were beyond the purview of the mathematician. A typical example is Blancanus’s remark in Sphaera mundi, a treatise of astronomy and cosmography, not of cosmology, when he discusses the earth’s motion. His subject, he says, is not ‘terrae motu’, the motion of the earth as an eIement, but ‘motu terrae’, the motion of the earth itself, ‘for the former has nothing astronomical, and accordingly is to be left entirely to physics ~physicisl’.‘M

This concern can be seen especially clearly in cases where the mathematician finds himself tempted to discuss properly physical questions, and therefore makes a point of underlining the fact that he is, for the moment, stepping beyond his strict jurisdiction. Blancanus’s Aristoteiis iota mathematics

contains a fairly full paraphrase and commentary on the pseudo-Aristotelian Mechanical Questions,‘07 and at the end, where the peripatetic author touches on vortices and their tendency to push heavy objects inwards towards the centre of rotation, Blancanus excuses himself from treating the matter on the grounds that it is physical, not mathematical. Having provided that caveat, he then proceeds to add that Aristotle is wrong anyway, because bodies get thrown outwards, not inwards, ‘for experience teaches [itl’.lM Again, in

“‘On early seventeenth-century discussions of the art/nature distinction, and of its applicability to mechanical contrivances, see Reif, ‘Natural Philosophy’, pp. 228 - 241, and Wallace, Galileo. pp. 209-211.

‘*Blancanus, Sphaero mundi, p. 74. The whole issue of the demarcation between physics and mathematics, and the importance to Jesuits of maintaining it, is dealt with from a different perspective, stressing its ideological rather than intellectual functions, in extremely valuable forthcoming work by Dr Rivka Feldhay.

‘“Blancanus, Arisfoteiis km (note la), ‘In mechanicas quaestiones’, pp. 148- 195. ‘“[bid., p. 195.


Sphaera mundi, Blancanus gives an account of the new stars seen in the late sixteenth and early seventeenth centuries (up to 1604),‘09 and asks what they might be, and whether they constitute generation in the heavens. That, he says, is a matter for physicists and not appropriate for discussion here.“’ But, he continues, perhaps they are like comets, approaching and receding so that they are not always visible.“’ Blancanus had signalled his brief foray across disciplinary boundaries precisely because those boundaries were too important to be blurred.

The same concern to maintain the autonomy of the mathematical sciences even while briefly considering physical questions under their auspices can be found in Orazio Grassi’s Libra astronomica of 1619.1’2 Grassi held the chair of mathematics at the Collegio Romano, previously occupied by Clavius until his death in 1612, during the years 1616-1624, and again between 1626- 1628.‘13 This work, published under the pseudonym of Lothario Sarsi, formed part of the celebrated exchange with Galileo over the comets of 1618 which produced the latter’s Assayer. The core of the controversy was the question of whether comets were bodies traversing the heavens or, as Galileo claimed, refractive phenomena in vaporous exhalations rising up from the earth. The central issues were optical and astronomical, but owing to the wide- ranging character of Galileo’s arguments Grassi, the mathematician, found himseIf obliged on occasion to venture outside his proper territory. Thus, at one point’14 Grassi takes on Galileo’s suggestion that comets and sun-spots were composed of the same matter, and, accepting the suggestion, uses the concept of natural motions to cast doubt on Galileo’s claims for cometary

paths - because sun-spots travel in circles around the sun. Grassi concludes the digression abruptly with the words: ‘But these things are physical rather than mathematical.‘“’ The next paragraph begins: ‘Now 1 come to the optical reasons by which it is proved far more effectively that the comet never was a vain apparition, nor did it ever wander as a spectre among the nocturnal

‘OPBlancanus, Sphaera mundi, pp. 344 - 35 I. “‘Ibid., p. 350. “‘Ibid., p. 351. “‘Lothario Sarsii Sigensani [Horatii Grassi Salonensisl, Libru ustronomica ac phikwophica

116191, in Le opere di Galileo Galilei Vo1.W (Nuova ristampa della Edizione Nazionale; Firenze: G. Barbera, 1%5), pp. 109- 179, trans. by C. D. O’Malley as Horatio Grassi, TheAs#onomicul Balance, in The Controversy on the Comers of 1618, Stillman Drake and C. D. O’Malley (eds and trans.) (Philadelphia: University of Pennsylvania Press, l%O), pp. 67 - 132.

“‘Drake, ‘Introduction’ to ibid., p. xv. Grassi plays a leading role in Pietro Redondi, Galileo eretico (Torino: Einaudi, 1983); see especially pp. 151- 156, and chap. 7. See also Claudio Costantini, Baliani e i Gesuiti. Annotazione in margine alla corrkpondenta del Baliani con Gio Luigi ConJulonieri e Orazio Grossi (Istituto Italian0 per la storia della tecnica, Sezionc IV, Volume 3) (Milano: Giunti. 1%9).

““Grassi, Libru, pp. l&- 141 (O’Malley, pp. 90 - 91). “‘Ibid.. p. 141 (O’Malley, p. 91). In Sphaera mundi. Blancanus remarks similarly that the

nature of comets is a physical question, not an astronomical (p. 304).

Muthetnuticul Science and the Reconstitution of Experience 163

shadows, but that it displayed itseIf to the view of all in one place and that it was always one and the same in appearance.“‘6 For Grassi, not only were mathematical arguments to be strictly segregated from physical, but they were also far more effective and conclusive, a position echoing Clavius.

The disciplinary boundary also acted, therefore, as a convenient way of preserving the solidity of demonstrations in the mixed mathematical sciences. Blancanus’s Sphaeru mundi contains a chapter headed ‘On the parts of the world, and first on the elementary part’.‘17 This begins by carefully delimiting its territory: ‘This inferior part of the world, which is composed from the elements (which whether they be three or four we leave to the disputation of the physicists [Physiologis disputanduml). . . .“” The implication that he will discuss firm knowledge, not the conjectures of the physicists, is dear. The same attitude appears in Scheiner’s Rosa ursina of 1630, which defines the subject-matter of astronomy as follows:

The astronomer. according to Ptolemy, Ahugest Book I, chap. I, considers the quantity of celestial bodies, and indeed Aristotle himself decIares that [also] in the Cutegories, and teaches in [his] physics that quantity, and those things which are connected with it, is considered as much by the physicist as by the mathematician, but in different ways.“’

These differences were important. As Scheiner said further on, ‘Many things are not known in physical matters about what concerns the heavens; few things are known for certain; there are many doubtful things, many false things are asserted; many true things denied.‘lz4 The mathematical approach, Scheiner seems to suggest, is surer than the physical.

The disciplinary division between mathematics and physics could sometimes act, therefore, as a sort of cordon sanitaire to protect the claims of mathematics to possess a demonstrative pre-eminence. BIancanus employed it to remove what he appears to have seen as a tainting of astronomy by astrology. Clavius had divided astronomy into theoretical and practical branches, the first comprising alI the apparatus of positional astronomy, including use of instruments and drawing-up of tables, and the second, ‘which others call judicial, or prognostic, or divinatory’, consisting of astrology. Clavius sanctioned ody natural astrology, including the effect of celestial bodies on temperament, and condemned judicial astrology proper as

“‘Grassi. Libm, p. 141 (O’MalIcy, p. 91). “‘Blancanus, S&era mundi, p. 67. “‘l&f. “‘Christopharus Scheina, Rosa wsina sive Sot LX udmimndo facularum d mucularum suarum

Phuenomeno vurhs.. . (Bracciani, spud Andream Phacum, 1630). p. 602.11. ‘*Ibid., p. 606.II.


superstitious and theologically suspect’2’ (it was to be officially denounced by Urban VIII in 1631’2z). In his writings, however, he neglected treatment of astrology entirely, restricting himself to the ‘theoretical’ branch, Blancanus, too, failed to discuss astrology in the Sphueru mundi, but unlike Clavius he gave reasons, reasons denying the propriety of Clavius’s division of astronomy. The ‘most noble’ of all the mathematical sciences on account of its subject-matter, astronomy is subordinate to both geometry and arithmetic; astrology, Blancanus continues, does not really deserve its common assignment to astronomy. This is because, even if astrology were valid, celestial influences are natural causes, and demonstration through natural causes belongs to physics, not mathematics.‘23 In his subseqent listing of the divisions of practical mathematics, Blancanus includes the compilation of astronomical tables,lz4 again differing from Clavius, who had reserved the category of practical astronomy for astrology, and Blancanus’s attitude suggests that excluding the latter from the mathematical science of astronomy preserved astronomy’s integrity and ‘pre-eminence’ stemming from the sureness of its demonstrations.

The division between the mathematical and the physical continued to be of use to Jesuit mathematicians even after they had begun to assimilate the boundary-threatening work of Galileo, a fact which is not surprising given that their very identity was to a large degree determined by it.lz5 In Gaspar Schott’s Mugiu universalis, published in four volumes between 1657 and 1659,‘26 for instance, two distinct approaches to mechanics (the subject of Part III) are identified. One is mathematical, the other ‘physico-mechanical’, distinguished from the first by its search for physical causes.‘27 Among those taking the former approach, says Schott, are Guidobaldo, Stevin and Galileo, and the latter is taken by Aristotle (in the Mechanical Questions) and certain followers of his approach, including Honor4 FabrLLz8 At the end of this discussion, Schott concludes that although there must be some physical cause of the increase of force in simple machines beyond the mere disposition of weights and forces (because disposition itself is not ‘activa physice’), we don’t know what that cause is.“’ Accordingly, he proceeds to abandon

‘“‘Clavius, ‘De sphaera’, p. 3. InSee D. P. Walker, Spiritual and Demonic Magic from Ficino to Campanella (London:

Warbvrg Institute, 1958), pp. 205 - 206. There had been a previous d=enunciation in 1586. :z;%;ys, Sphuera mundi, p. 390.

‘“Cf. Heilbron, Electricity, p. 104, who holds that Jesuit mathematicians used the boundary as a way of avoiding the ‘new physics’.

“Gaspar Schott, Magia universalis naturae et artis, 4 vols (Herbipoli, Excudebat Henricus Pigrin, 1657 - 1659).

lnlbid., pp. 211-212. ‘“Ibid., p. 212. ‘“Ibid., p. 225. Cf. Lynn Thorndike, A History of Magic and Experimental Science VoLVtI

(New York: Columbia University Press, 1958), pp. 64U- 602.


consideration of the ‘physico-mechanical’ approach, and continues with a conventional treatment of statics, the simple machines, and hydrostatics falling under his ‘mathematica1’ category. The maintenance of disciplinary boundaries allowed the mathematician to avoid, on the one hand, committal to any particular controversiaI physical doctrine, and on the other, a confession of ignorance regarding his proper subject-matter.

Because the mathematical sciences did not employ natural causes, the tendency of the latter to be obscure highlighted the contrasting cIarity of mathematical demonstrations, and thereby supported Clavius’s arguments in support of mathematics’ status vis-ri-vis other disciplines. The prestige of mathematics therefore relied to a considerabIe extent on preserving a sharp distinction between it and physics, a philosophical distinction mirroring a disciplinary and curricular one. Eschewing natural causes also, as we have noted earlier, legitimated the use of empirica premises which were established artificiah’y: since these premises had to do only with quantity of one sort or another, and not with process, the Aristotelian dichotomy of natural and artificial did not apply. The enforcement of the disciplinary boundary between physics and the mathematical sciences was thus essential to both the status and the methodoIogica1 health of the latter. But although the mathematicians were content to leave natural causes to the physicists, that did not mean that they, in effect, deferred to them as if to the higher discipline. Clavius’s largely successful attempts to give mathematics a significant place in the Jesuit educational curriculum had created teachers and professors of mathematics in the colleges whose academic position in principle equalled that of the natural philosophers.‘30 Clavius, and above al1 Blancanus, were so concerned with estabIishing the mathematical disciplines as genuine sciences, and with arguing for a high estimation of their value, because they were attempting to achieve a genuine equality. Blancanus’s fulminations against Pereira and the Coimbra commentators for questioning the scientific character of mathematics shows that there was opposition; Clavius’s educational policy document of the 1580s suggested strongly that teachers should be prevented from criticising mathematics as being of no value to philosophy, and that teachers of mathematics should attend and participate in the regular formal disputations to show themselves on a level with the other philosophers.r3’ Since they wished to portray their subject as a partner of natural philosophy, therefore, the Jesuit mathematicians had no interest in maintaining the disciplinary boundary between them solely by disqualifying themselves as competent to treat physical questions. On the contrary, the physicists could learn things from the mathematical disciplines; indeed, mathematics was indispensable to

‘%ee Dainville, Nuismme, p. 60. “‘Blancanus, De malh. not., e.g. p. 13; Clavius, ‘Modus’ (note 14).


physics, as Clavius always insisted. Aguilonius’s Opticorum libri sex bears the advertisement that it is ‘useful as much to philosophers as to mathematicians’, and in the preface, Aguilonius, praising Archimedes over Plato for judging the union of mathematics with matter ‘in no way to derogate anything from mathematics, but indeed even to adorn and perfect it’, claims that ‘We, trusting in his authority, undertake the cultivation of this part of mathematics, which comprehends both genuses, namely philosophy and mathematics.“32 It was to the extent that mathematics considered the physical world that it became relevant to natural philosophy; the focus, that is, was on the mixed mathematical sciences. Anyone who has seriously considered the question, says Aguilonius, ‘will have approved very greatly the ways of explaining [of mathematics] to be appropriated for the manifold diversity of things.“33

Scheiner was particularly concerned to make this point in his Rosa ursina, a large work on astronomy which concentrated on his own sunspot observations. Following the passage quoted earlier setting out the difference in treatment of quantity on the parts of physics and astronomy, Scheiner lists some of the things which, as a mathematical science, astronomy was competent to treat. All kinds of change in the heavens, local motion, duration, augmentation, diminution and so forth, came under the auspices of astronomy, and so in the case of the Rosa ursina’s special subject-matter Scheiner could present results such as that sunspots are on the surface of the sun, that (from the observed motion of sunspots) the sun is a globe, and that the sun rotates around its own centre.“’ A subsequent section of the work is headed: ‘Many truths are disclosed to celestial physics [phrsiologium coelesfeml from solar phenomena.“” The advent of the telescope, Scheiner says, now allows access to things in the heavens hitherto inaccessible, things having relevance for physics. In particular, the new-found rotation of the sun certainly has important physical implications. I’6 Scheiner thus points out the indebtedness of physics to the new ‘mathematical’ discoveries and demonstrations. His marginal gloss avoids unwonted confusion, however, by observing that these physical questions emerging from the new knowledge of solar appearances are to be settled by physics.‘37

The disciplinary division which allowed the use of artificial empirical premises in mathematical sciences functioned precisely because physics was understood as a search for physical, or natural, causes, whereas mathematics sought demonstrations based on formal properties of magnitudes. The

‘J2Aguilonius, Opticomm libri, preface ‘Lectoris’, 10th p. “‘Ibid. %cheiner, Rosa ursina, pp. 602.11- 604.1. “slbid., p. 606.11. ‘%lbid., p. 607.1 -II. Tbid., gloss. to p. 607.iI.

Muthemotical Science and the Reconstitution of Experience 167

radically different kinds of premises represented an irreducible division insofar as they related to radicaily different kinds of questions about the world. Nonetheless, both physics and the mathematical sciences required the establishment of empirical premises, whether strictly artificial or not, to provide them with statements about how things behaved. Their job as sciences was then to use such statements to account for other events or states of affairs causally derived from them. But in the case of the mathematical sciences, as has been stressed, the use of experimental manipulation created difficuhies in rendering empirical principles evident: experimental knowledge was recondite, constituting private rather than public experience, and if it failed to achieve a public warrant it could not form part of a science. In order to legitimate experimental statements, therefore, the mathematician had to find ways of extending private experience to his audience through the medium of the mathematical treatise itseIf. Natural phiIosophers possessed special facihties for establishing experiences, in that the commentary-genre conventionally employed could rely on the authoritative text as the source for its empirical content; common experience was located in standard texts in the sense that its very presence in such a text rendered an experience common property. Mathematicians required different literary techniques to make their empirical statements acceptable in scientific discourse, and the use of experimental manipulation presented severe difficulties. In effect, the author’s private, indeed privileged, experience amounted to a historicaI report of events the veracity of which had to be made transparent to the scrutiny of the reader. The Jesuits began to find characteristically seventeenth-century techniques to accomplish this.

V

The most obvious way to establish a report’s veracity was to present it in the form of a constructional problem in geometry. This is exactly what is done, as we saw above, in optical treatises: the reader is either instructed to perform a series of operations and told what outcome will result, or (more revealingly) the series of operations is presented in the subjunctive mood of a geometrical problema so that the outcome emerges with a sober inevitabihty.‘38 The status afforded to experiential claims by this Iiterary device transcended what would appear to us to be the obvious difference between geometrical construction and experimental result: the former is transparent, and in following its steps the reader sees the outcome generated before his eyes; the latter, however, is not transparent to the extent that its outcome seems to be pulled from a hat. The reader does not need to have recourse to compass and ruler in order to

“*See Section III, above, especially text to note 94.


comprehend the geometrical construction, but he would need to become a ‘special empiric’ and manipulate specialized apparatus in order to make evident the experimental result. However, the adoption of a geometrical literary structure did in fact serve to accord a sort of transferred transparency to reported experimental procedures. This worked to the extent that procedure and outcome became formally inseparable - optical experiments detailed particular procedures for generating conditions under which specific effects would be manifest (the apparent position of a pointer having changed, for example, in refraction). Since the descriptions of experiments were really constructional techniques, and were therefore justified by the intended outcomes, there appeared to be no question of the experimental result differing from the one presented.

The use of a geometrical paradigm served to recreate experience by giving the illusion of generating it in the same way as conclusions or constructions in geometry were generated.“’ The mathematical sciences therefore provided a formal structure for legitimating constructed experiences, quite independent of their subject-matter. The power of this legitimatory device is indicated by its appropriation, for closely related purposes, by Niccolo Cabeo, in his Phiiosophia magnetica of 1629.L40 Cabeo insisted that he was doing physics, not mathematics, and he used the distinction on occasion to do work for him, much as his mathematical colleagues did.14’ Despite this, however, he quite deliberately borrowed the mathematicians’ rhetorical resources for his own ends. Book III of his treatise, on the directional property of magnets, opens with a chapter discussing ‘Ratio, & forma addendi hunt librum’,‘” noting that the proper way to investigate the nature of magnets is to use Aristotle’s demonstrative regress. 143 The previous books, we are told, had presented material ex signis, that is, from the effects or appearances, together with a posteriori arguments, touching the subject of this present book; now all this material would be brought together, and in a particular way.IU

This, furthermore, will be the method [ratio] of explication: I will put forward as a title the effect itself by the clearest words 1 can in their brevity; generally in that way whereby mathematicians are wont to present their propositions, which they then collect disposed for demonstration: next, I will begin to explicate the matter with additional figures where needed, and with the prominent appearances of things [expressis rerum formisl. By a clear method, and certified experience [probofa

“‘Once again the parallel with thought-experiments emerges. Cf. Kuhn, ‘Function’ (note 42). ‘“Nicolaus Cabeus. PhiIosophia magneticu, in qua magnetis natura penitus explicatur, et

omnium quae hoc iapide cernutttur, causae propriae afferuntur.. (Coloniae. apud loannem Kincldum, MDCXXIX).

“‘For example ibid., pp. 69 and 77. ‘“ibid., p. 197. “‘Ibid., p. 198. ‘“Ibid.


experiential, I will attempt to deliver the reason of the [effect] from the proper principles of the magnetic philosophy. , , .I45

This is a clear statement of the perceived value, even to a non-mathematician, of geometrical form as a way of mimicking the supposed clarity of geometrical content.

The efficacy of this technique in establishing empirical premises was enhanced if no controversial issue rested on the claimed results of experiments. If those results were neither especially surprising nor overly recondite, inherent plausibility reinforced the persuasive effect of the overall presentation. It is significant, for example, that Aguilonius did not even regard as necessary the detailed experimental protocols provided by Scheiner to support his empirical suppositions concerning vision.‘46 An experiment described by Blancanus for showing the rarefaction and condensation of the air (a sort of Dutch weather- glass arrangement) was eminently presentable preciseIy because it was not crucial: this behaviour of air ‘is established by many experiences,’ but ‘it is however agreeable to bring forward now a more beautiful as well as most evident [one]: let a glass flask be constructed, as you see in the figure. . . .‘I”

As long as not too much rested on any particular experiment, the geometrical form sufficed. Controversy, however, or the threat of controversy, demanded more radical measures, and at the same time placed greater emphasis on discrete events as justification for assertions.

Examples among the Jesuits in this period are rare, and serve to indicate a direction opened up to them rather than a general move unequivocally taken. Only controversy, the direct questioning of claims and assumptions, actualized the possibility. Appropriately, Grassi’s exchange with Galileo over comets produced perhaps the clearest example. The third part of Grassi’s Libra

astronomica is an ad hominem refutation of Galileo’s arguments against Aristotle - Grassi is not really concerned to maintain the Aristotelian positions, but rather to scorn Galileo’s objections to them. Galileo had identified an apparent contradiction between the dictum that the celestial spheres were perfectly smooth and the claim that air can be kindled to produce shooting stars and comets by the motion of the lunar sphere. Galileo had asserted that ‘neither air nor fire adheres to polished and smooth bodies,’ as the moon’s sphere is said to be, and hence cannot be kindled by the latter’s motion+ 14’ He supported the point by an appeal to experience:

“‘Ibid. Reif, ‘Natural Philosophy’, p. 309, notes an increasing tendency among her early seventeenth-century textbook writers to ‘pattern the format of their manuals on that of a geometrical treatise’.

‘“For example in the matter of the dilation of the pupil: cj. note 95, above. ‘“Blartcanus, Sphuera mundi. p. 111, ‘“Grassi’s version of Galileo’s position, in Grassi. Libra, p. 152 (O’Malley, p. 106).


For if some hemispherical vessel, polished to smoothness, is revolved about its centre, the air enclosed in it will not be moved at its motion; this is proved by a lighted candle placed very close to the interna surface of a vessel; for if the air were carried along at the motion of the vessel it would also draw that flame with it.“’

Galileo thus seeks experiential support for a premise required in his demonstration of Aristotle’s fallacy. The experience is presented as a general unproblematic statement of how things behave, glossing over the specialized character of the conditions. Grassi, in deciding to call into question the truth of the experience, had to change the rules for constituting experiential statements. He could not merely counter-claim, since this would not have given him any argumentative edge over Galileo. So he introduces a quite different technique.lsO

Even smooth dishes when rotated about their axes carry the contents, whether water or air, around with them, says Grassi. He then proceeds with the literary construction of a discrete experience to justify this remark.

Lest anyone believe that we tested this negligently and carelessly, we obtained a brass hemispherical vessel skillfully hollowed out on a lathe, and we undertook that the axis would pass through its centre as a spherical axis - if it were prolonged. Furthermore, we constructed a firm and stable foot lest it be agitated by the motion of the vessel. . . .“I

And so forth, the whole accompanied by a drawing of the apparatus. What is noteworthy here is not the detailed description of apparatus (familiar from optics and astronomy), but the description of its construction as a historical event. Having reported the outcome (‘But not only water was carried around by the motion of the vessel, but the air itself, from which especially Galileo took his example”52), together with procedural details, Grassi concludes thus:

1 have no few witnesses to the fact that I say this not more surely than truly; first, many fathers of the Collegio Roman0 - however, many others were willing to recognize this on the authority of my teacher IGrassi is writing under the guise of one of his own pupils] - and many others as well. I ought not be silent about that one

‘“Ibid. ‘mRelevant to this discussion is Steven Shapin’s idea of ‘virtual witnessing’ as a means of

understanding the function of BoyIe’s detailed, circumstantial accounts of air-pump experiments. Steven Shapin, ‘Pump and Circumstance: Robert Boyle’s Literary Technology’, S&ial Studies of Science 14 (1984), 481- 519. See also Steven Shapin and Simon Schaffer, Leviathan and the Air Pump. Ho&es, &y/e and the Experimental Life (Princeton: Princeton University Press, 1985), chap. 2, especially pp. 55 - 69.

“‘Grassi, Libra, p. 156 (O’Malley, p. 110). lIzIbid.

Mathematical Science and the ReconstiCution of Experience 171

among them whose name is very well known to me not more by birth than by his singular erudition, and who can verify my activities and substantiate my remarks. I speak of Virginio Cesarini, who was astonished that a thing that many today hold certain could ever be proved false; and yet he saw what was done, which many denied could be done.lS3

Grassi, in order to substantiate his claims for a particular historical occurrence, a discrete experiment, cites witnesses. In effect, the requirement in scientific demonstration for empirical premises to be evident is here replaced by a surrogate technique of evidence in a quasi-Iegal sense. This step was never far away in conventional practice in the mathematical sciences, but equally it was seldom expIicitly made in the way Grassi does here. A clue to this is Grassi’s remark that ‘many others were willing to recognise’ the truth of the claimed results ‘on the authority of my teacher’, Grassi himself. This is no different from accepting astronomical data (but not, it is crucial to note, astronomical phenomena) on the authority of the observer who recorded it, and indeed Grassi appears to expect that his authority ought under most circumstances to be enough. But in making the issues of testimony and authority explicit, Grassi makes a virtue of necessity by multiplying witnesses and exploiting the inherent authority of one ‘whose name is very well known to me not more by birth than by his singular erudition’. (Cesarini had the added advantage, for Grassi, of being something of a disciple of Galileo.‘54)

‘These things are certain from experiment; yet if this experiment were Iacking, reason itself would also teach them’, Grassi continues.“’ Singular, constructed experiences remained an undesirable expedient, for all the literary techniques designed to accommodate them; Grassi attempts to justify his results with arguments drawn from physical rather than mathematical considerations, namely the humidity and hence adhesiveness of water and air.lS6 He soon returns, however, to more refined versions of the original experiment, presented as a report in the first person singular of his construction and manipulation of special apparatus.“’ The report ends by noting: ‘Those same ones who were mentioned by me earlier saw these final experiments, but I do not consider it necessary that they testify again,‘15* By this means, of course, they are already seen to have done so.

Extraordinary circumstances - a controversy with Galileo - thus called forth extraordinary resources in scientific argumentation. Those resources were potentialIy availabIe within the Jesuit tradition of mathematical sciences

“‘lb& p. 157 (O’Malley, p. 111). IsDrake ‘Introduction’ to Drake and O’Malley, Controversy, pp. xii-xiv. ‘%rass~ Libru, p. 157 (O’Malley, p. Ill). ‘yIbid. ‘571bid., pp. 157- 159 (O’Malley, pp. 113 - 115). ‘5eZbid.. p. 159 (O’Malley, p. 115).


because of awareness of the methodological issues involved in using experience in the formulation of a true science. Personal authority was always implicitly invoked when astronomers or opticians presented the results of their expertly- contrived manipulations; Grassi’s citing of witnesses was a way of going one better than Galileo when the latter employed a quite usual form of deploying experience.

Cabeo’s work on magnetism provides another example of an extraordinary circumstance drawing forth such a response. Although he claimed to be doing physics, we have seen how Cabeo’s approach to the presentation of experience drew on the mathematicians’ techniques. Indeed, as a physical treatise Cabeo’s work is somewhat heterodox in format, being a systematic topical treatise rather than a commentary.15g Cabeo sought properly physical explanations for effects, but he was not averse to mathematical reasoning, and he accepted that it was peculiarly conclusive - after discussing electrical attraction and presenting his effluvial explanation of it, for instance, he confessed its difficulties and said that he would willingly change his opinion if anyone came up with a better one, with the exception of ‘such things as I advance having been demonstrated mathematically’.‘60 Cabeo respected the solidity of mathematical demonstrations even though they failed to get at the more difficult matters of physics, and the mixed mathematical sciences provided him with a paradigm for the use of experiment in scientific demonstration. Like Grassi, he found himself in an extraordinary situation, but in this case what was extraordinary was the subject-matter itself.

Magnetism and magnetic effects (as also electrical) were by no means part of everyday experience (except in the crudest sense), and they required precise experimental contrivance, More obviously than in most cases, therefore, the authority of the experimenter was at issue and the evident character of the effects compromised. 16’ Cabeo tried to deal with this question in his

‘Praefatio ad Lectorem’:

I have delayed the reader in the very antechamber of the treatise, moreover, advising him that I am about to bring forward nothing in these magnetical disputations

“‘Cf. Heilbron’s observations in Electricity, pp. 109- 110. ‘@Cabeo, Philosophiu mugneticu, p. 195. On the effluvial theory, see Heilbron, Electricify, pp.

180- 183. lb’An interesting short treatise by Blancanus constitutes something of a halfway-house in this

regard. ‘Echometria, sive de natura echus geometria tractatio. publice habita a quodam Academico’, is appended to Sphuero mrrndi as pp. 415 -443, and in effect represents a treatment of sound along the same lines as the treatment of light in catoptrics. Although some of the ‘experiments’ are reported in the subjunctive (e.g. Theorem I, p. 419), others are in the first person singular, active voice (e.g. Theorem XVII, p. 434; Theorem XIX, pp. 436 - 437, reporting his own observations of echoes from rocks above a torrent), but with no citation of witnesses. The results are not especialy controversial, but they are presented as the foundation of the work, not as mere illustrations (see p. 420 for a statement of this). Blancanus is keen to claim that his results come ‘not from the books of others, but from experiments’ (p. 442).

Mathemakal Science and the Reconstitution of Experience 173

which, so far as I have been able, I did not confirm by experiments again and again, even with others gathered to watch [ad spectaculum convocafisl, so that not onIy would I have many witnesses, but I would remove myself from suspicion of error, as long as many of the most attentive k~riosi~~l observe the same experiment.‘”

Cabeo is here referring above all to effects which he wishes to explain through

appropriate physica principIes using the demonstrative regress, rather than to the establishment of the principles themseIves, but his attitude towards experience itself typifies the points we have been making. Firstly, Cabeo repeated his experiments many times, in effect creating an ‘experience’ from individua1 sense-impressions in the proper Aristotelian fashion; secondly, the private nature of that ‘experience’ was vitiated by using witnesses whose implied testimony could help to constitute it as shared, or public; thirdly, Aguilonius’s requirement that ‘experiences’ be based on many instances because the senses sometimes err was additionally met by the multiplication of sensory impressions due to a number of observers, The use of testimony relating to discrete experimental events is therefore an inseparabIe part of Cabeo’s methodological grounding for his ‘magnetical philosophy’. As with Grassi, the scientific imperative that experiential claims be evident has led to the development of new techniques of justification centred on the citing of Iegitimatory, semi-public events - witnessed experiments. But this development was predicated on the prior use of discrete experience

(‘experiments’ or astronomical data, for example) as a means of grounding the less-than-evident claims associated with constructed universal experiences.

Conclusion

Jesuit mathematicians in the early seventeenth century went a long way towards the formulation of techniques designed to incorporate singular experiences, discrete events, into properly accredited knowledge about the natural world. The norms of scientific procedure to which they adhered derived from the Aristotelian model employed by Jesuit natural philosophers and logicians, and the incentive to do so was the attempt, initiated by Clavius, to raise the status of the mathematical disciplines and their practitioners in the Jesuit academic system. The independent traditions of practice in the disciplines of astronomy and optics had evolved particular ways of generating and using experiential data which created problems for the characterization and presentation of these subjects as sciences according to Jesuit criteria, and this fact demanded the development of techniques for turning contrived and private experiences into evident empirical suppositions suitable for use in scientific demonstrations. Blancanus and Scheiner were especially aware of the

‘“Cabeo, Philosophia magnetica, ‘Praefatio’, 3rd p.


fact that much of the experiential basis of astronomy and of optics was manufactured by expert practitioners, and could not easily be transformed into the evident experience which would provide adequate principles for a true science. ‘63 To some degree this problem could not be solved, but it could be vitiated: accounts of procedure which were claimed to yield (that is, make evident) the relevant experience could be presented so as to turn a private actual experience into a public potential (or virtual) experience. Astronomical data and accounts of instrumental techniques, or experiments detailed as instructions or quasi-geometrical constructions, served to provide an underpinning of discrete experiences to the statements of universal experience informing scientific demonstrations, and to go some way towards making private knowledge-claims into public, self-evident truths. The closer that such accounts could come to appearing like thought-experiments - procedures which, although they have not actually been carried out for the reader, will with correct presentation create conviction of the truth of an untried outcome - the closer to proper scientific suppositions became their conclusions.

The deeper implications of this increasing, and paradoxical, reliance on

discrete experience as a means of scientizing the mathematical disciplines appeared when the possibility of challenge arose. The appropriate technique of response then became the historical report of a discrete empirical event, preferably witnessed. As later counterparts to Grassi’s experimental demonstrations at the Collegio Romano, the investigations of the Jesuits Arriaga and Riccioli into the behaviour of falling bodies make telling examples .‘a Each provides an account of experiments conducted, with the aid of assistants, in the most public of all locations - dropping weights from the tops of church and cathedral towers. The singular, historical experience was here supplanting the general experiential statement, a step not taken even by Galileo. 16’ The case of Niccolo Cabeo shows how the techniques developing within mathematics became available, through that legitimation,

‘%cheiner is, as has been shown, a long way from Tycho’s ‘alchemical’ approach to knowledge, but also some distance from Aguilonius, who in many ways shares Schtiner’s ideal of science but does not confront the methodological specificities of making an Aristotelian science out of the optics bequeathed by Alhazen. Perhaps this is because Aguilonius’s optics has no dimension of discovery; the foundations from which the science develops are fairiy tlxed. Scheiner’s approach, by contrast, leaves open the possibility of acquiring, and rendering ‘evident’, new empirical facts, not just new formulations or inferences.

‘“On Arriaga see Charles B. Schmitt, ‘Galilei and the Seventeenth-Century Text-Book Tradition’, Novitd celesti e crisi del sapere. Atti de1 Convegno Internazionaie di Studi Galileiuni (Supplemento agJi Annali dell’lsrituto e Muse0 di Storia della Scienza Anno 1983 - Fascicolo Z), Paolo Galluzzi (cd.). pp. 217 - 228, especially pp. 223 - 225. on the Cursus philosophicus. On Riccioli, see Alexandre KoyrC, ‘An Experiment in Measurement’, in KoyrC, Metaphysics and Mwswenumf, pp. 89- 117, especially pp. 102- 108.

‘“And note that while Riccioli wrote as an astronomer, Arriaga wrote as a natural philosopher, another indication (besides Cabeo) of a transfer of techniques from the mathematical sciences to physics.

Mathernaticai Science and the Reconstitution of Experience 175

for the use of natural philosophers, and suggests that the growing prestige of mathematical approaches to nature in the seventeenth century may have been coextensive with the move towards a new conception of experience, the ideal of the discrete event - summed up in John Wilkins’ term, ‘Physico- MathematicalI-Experimental1 Learning’.lti

Acknowledgements - Versions of this article were given at the Science Studies Unit. University of Edinburgh and at the joint meeting of the British Society for the History of Science and the British Society for the History of Philosophy at Churchill College, Cambridge, April 1986. I would like to thank all those who commented on those presentations and in particular Steven Shapin. For reading and commenting on drafts, I would like to thank the editors of this journal, Nicholas Jardine and Gerd Buchdahl, as well as Simon Schaffer and Pauline Dear. My thanks also to Rivka Feldhay for discussion and for providing me with material which, unfortunately, has come too late for incorporation but which would materially alter some of the arguments.

‘“See Barbara J. Shapiro, John Wilkins MI4- 1672. An Intellectual Biography (Ekrkeley: University of California Press, 1%9), p. 192.

Jesuits Mathematical Science

Documents

Transcript of Jesuits Mathematical Science