BRADLEY

BRADLEY, RICHARD (d. Cambridge, England,5 November 1732), botany.

Bradley's main scientific contributions were hisstudies on the movement of sap and on the sexualreproduction of plants . His experiments, particularlyon trees, led him to consider sap as circulating in someway; from his work on tulips and hazel he drewanalogies with animal reproduction, and emphasizedthe significance of pollination and the importance ofinsects in fertilization . He then went on to discuss thenovel idea of cross-fertilization and the productionof different strains . This work was published in hisNew Improvements of Planting and Gardening (1717)and A General Treatise on Husbandry and Gardening(1724) .

Bradley was a prolific science writer, producingmore than twenty botanical works, as well as writingon the plague at Marseilles in 1720, advocating clean-liness and a "wholesome diet" as prophylactics . Hisstyle was clear and readable, and his reputationimmense ; indeed, his publications did much to en-courage a scientific approach to gardening and hus-bandry. Bradley claimed to have invented the kalei-doscope, which he used for preparing symmetricaldesigns for formal gardens, thus anticipating theclaims of Sir David Brewster by some ninety years .He also strongly advocated the use of steam to powerthe irrigation of gardens and farmland .

On I December 1712 Bradley was elected a fellowof the Royal Society of London, and on 10 November1724 was appointed to the chair of botany at Cam-bridge University . It is said he obtained the latter byclaiming a verbal recommendation from the botanistWilliam Sherard (1659-1728) and promising to pro-vide a botanic garden at his own expense. He pro-vided no garden and was unfamiliar with Latin andGreek, and, because of some supposed scandal, therewas a petition to remove him . It proved of no avail,and he died in office . Sir Joseph Banks and otherbotanists named genera to commemorate him .

BIBLIOGRAPHY

Bradley's books include The Gentleman and Farmer'sKalendar, Directing What Is Necessary to Be Done EveryMonth (London, 1718): New Improvements of Planting andGardening, Both Philosophical and Practical; Explainingthe Motion of the Sap and Generation of Plants (London,1718) : The Plague at Marseilles Consider'd :: With RemarksUpon the Plague in General (London, 1721) : PrecautionsAgainst Infection : Containing Many Observations Necessaryto Be Considered, at This Time, on Account of the DreadfulPlague in France (London, 1722) ; A General Treatise onHusbandry and Gardening, 2 vols . (London, 1724): Philo-

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sophical Account of the Works of Nature (London, 1725) :A Survey of Ancient Husbandry and Gardening (London,1725): The Country Gentleman and Farmer's Monthly Direc-tor (London, 1726); A Complete Body of Husbandry (Lon-don-Dublin, 1727) ; Dictionarium botanicum, 2 vols . (Lon-don, 1728); The Riches ofa Hop-Garden Explain 'd (London,1729); A Course of Lectures Upon the Materia Medica(London, 1730) ; and Collected Writings on SucculentPlants, with an introduction by G . D. Rowley, a facsimileed. (London, 1946) .

His scientific papers include "Motion of Sap in Vegeta-bles," in Philosophical Transactions of the Royal Society,29 (1716),486-490 ; and "Some Microscopical Observationsand Curious Remarks on the Vegetation and ExceedingQuick Propagation of Moldiness of the Substance of aMelon," ibid., 490-492 .A work containing information on Bradley is Richard

Pulteney, Historical and Biographical Sketches of the Prog-ress of Botany in England, II (London, 1790), 129-133 .

COLIN A . RONAN

BRADWARDINE, THOMAS (b . England, ca.1290-1300 ; d. Lambeth, England, 26 August 1349),mathematics, natural philosophy, theology .

Both the date and place of Bradwardine's birth areuncertain, although record of his early connectionwith Hartfield, Sussex, has often been taken as sugges-tive. His own reference (De causa Dei, p . 559) to hisfather's present residence in Chichester is too late tobe relevant .

Our knowledge of Bradwardine's academic careerbegins with the notice of his inscription as fellow ofBalliol College in August 1321 . Two years later wefind him as fellow of Merton College, a position hepresumably held until 1335 . We also have evidenceof a number of other university positions during thisperiod . The succession of his Oxford degrees wouldseem to be the following : B.A. by August 1321 ; M .A .by about 1323 ; B.Th. by 1333 ; D .Th. b) 1348 .

Bradwardine's ecclesiastical involvement appears tohave begun with his papal appointment as canon ofLincoln in September 1333, although his less officialentry about 1335 into the coterie of Richard de Bury,then bishop of Durham, was probably of greaterimportance in determining the remainder of hiscareer. For not only did this latter move placeBradwardine in more intimate contact with some ofthe more engaging theologians in England, but it alsomay well have proved to be of some effect in introduc-ing him to the court of Edward III . Indeed, shortlyafter his appointment as chancellor of St . Paul's,London (19 September 1337), we find him aschaplain, and perhaps confessor, to the king (about1338/1339). We know that he accompanied Edward's

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retinue, perhaps to Flanders, but certainly to Franceduring the campaign of 1346 . In point of fact, it waslate in that year, in France, that Bradwardine de-livered his (still extant) Sermo epinicius in the pres-ence of the king, the occasion being the commemora-tion of the battles of Crecy and Neville's Cross . Thecloseness of his ties with Edward might also be in-ferred from the fact that the king annulled Brad-wardine's first election to the archbishopric of Canter-bury (31 August 1348) . He was, however, elected asecond time (4 June 1349), apparently withoutEdward's opposition, and consecrated at Avignonapproximately a month later (10 July 1349) .Bradwardine immediately returned to England,where, after scarcely more than a month as arch-bishop, he fell before the then raging plague and diedat the residence of the bishop of Rochester inLambeth, 26 August 1349 .

Although our evidence is not absolutely conclusive,it seems highly probable that Bradwardine composedall of his philosophical and mathematical works be-tween the onset of his regency in arts at Oxford andapproximately 1335 .

Early Logical Works . In spite of the lack of anydirect testimony, it is nevertheless a reasonable as-sumption that the early logical works were the resultof a youthful Bradwardine first trying his hand at akind of activity common, even expected, among re-cent arts graduates in the earlier fourteenth century .A number of these logical treatises ascribed to Brad-wardine are undoubtedly spurious, but at least twoseem, to judge in terms of present evidence, to bemost probably genuine : De insolubilibus and Deincipit et desinit . Neither of these works has beenedited or studied, yet the likelihood is not great thatthey will eventually reveal themselves to be muchmore than expositions of the opinio communis con-cerning their subjects. Both treatises are of courserelevant to the history of medieval logic, but the Deincipit et desinit, like the many other fourteenth-century tracts dealing with the same topic, had adirect bearing upon current problems in naturalphilosophy as well. For the medieval works groupedunder this title (or under the alternative De primoet ultimo instanti) address themselves to the problemof ascribing what we would call intrinsic or extrinsicboundaries to physical changes or processes occurringwithin the continuum of time . Thus, to cite the funda-mental assumption of Bradwardine's De incipit (anassumption shared by almost all his contemporaries),the duration of the existence of a permanent entity(res permanens) that lasts through some temporalinterval is marked by the fact that it possesses a firstinstant of being (primum instans in esse) but no last

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instant of being (ultimum instans in esse) ; its termina-tion is signified, rather, by an extrinsic boundary, afirst instant of nonbeing (primum instans in non esse) .

Tractatus de proportionibus velocitatum in motibus .It is this work, composed in 1328, that has firmlyestablished Bradwardine's position within the historyof science . As its title indicates, it treats of the "ratiosof speeds in motions," a description of the contentsof the Tractatus that becomes more properly revealingonce one identifies the basic problem Bradwardineset out to resolve: How can one correctly relate avariation in the speeds of a mobile (expressed, as inthe work's title, as a "ratio of speeds") to a variationin the causes, which is to say the forces and resis-tances, determining these speeds? The proper answerto this question is, without doubt, the fundamentalconcern of the Tractatus de proportionibus . InBradwardine's own words, to find the correct solutionis to come upon the vera sententia de proportionevelocitatum in motibus, in comparatione ad moventiumet motorum potentias (Crosby ed., p . 64) .

Answers to this question were, Bradwardine pointsout, already at hand. Yet they failed, he argues,to resolve the problem satisfactorily . Basically, theirfailure lay in that they would generate results whichwere inconsistent with the "postulate" of Scholastic-Aristotelian natural philosophy which stipulated thatmotion could ensue only when the motive powerexceeded the power of resistance : when, to usemodern symbols, F > R. Thus, for example, one un-satisfactory answer was that implied by Aristotle. For,although Aristotle was certainly not conscious ofBradwardine's problem as such and, it can be argued,never had as a goal the firm establishment of anymathematical relation as obtaining for the variablesinvolved, the medieval natural philosopher took muchof what he had to say in the De caelo and the Physics(especially in bk . VII, ch . 5) to entail what is nowmost frequently represented by Vac F/R . But this willnot do. For, if one begins with a given F1>R1 , andif one continually doubles the resistance (i .e .,Rz = 2R 1, R3 = 2R2 , etc.), then F1 as a randomlygiven mover will be of infinite capacity (quelibetpotentia motiva localiter esset infinita [ibid., p . 98]) .In our terms, what Bradwardine intends by his argu-ment is that, under the continual doubling of theresistance, if we hold F 1 constant, then at some pointone will reach R, >F 1 , which, on grounds of thesuggested resolution represented by Vcc F/R, impliesthat some "value" would still obtain for V; this inturn violates the "motion only when F > R" postulate .Therefore, VccF/R is an unacceptable answer to hisproblem. Bradwardine also sets forth related argu-ments against other possible answers, which are usu-

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ally symbolized by VccF- R and Vcc(F- R)/R.The correct solution, in Bradwardine's estimation,

is that "the ratio of the speeds of motions follows theratio of the motive powers to the resistive powers andvice versa . or, to put the same thing in other words :the ratios of the moving powers to the resistive powersare respectively proportional to the speeds of the mo-tions, and vice versa . And," he concludes, "geometricproportionality is that meant here" (Proportio veloci-tatum in motibus sequitur proportionem potentiarummoventium ad potentias resistivas, et etiam econtrario .Vel sic sub aliis verbis, eadem sententia remanente : Pro-portiones potentiaruni moventium ad potentias resis-tivas, et velocitates in motibus, eodem ordine propor-tionales existunt, et similiter econtrario . El hoc degeometrica proportionalitate intelligas [ibid., p . 112]) .

Given just this much, it is not at all immediatelyclear what Bradwardine had in mind . His intentionsreveal themselves only when one begins to examinehis succeeding conclusions and, especially, theexamples he uses to support them . If we generalizewhat we then discover, we can, in modern terms, saythat his solution to his problem of the corresponding"ratios" of speeds, forces, and resistances is thatspeeds vary arithmetically while the ratios of forcesto resistances determining these speeds vary geo-metrically . That is, to use symbols, for the seriesV/n, . - . V/3, V/2 . V, 2 V, 3 V, . .- n V, we have thecorresponding series (F/R)'/", . . . (F/R)'", (F/R) 112 ,F/R, (FIR) 2 , ( F/R)3 , . . . (FIR)" . Or, straying an evengreater distance from Bradwardine himself, we canarrive at the now fairly traditional formulations ofhis so-called "dynamical law" :

(Fl/R 1 ) 1 -" 1 1= F.2/R2 or V = log„ F/R,

where a = F,/R, .

Furthermore, if we continue our modern way ofputting Bradwardine's solution to his problem, we canmore easily express the advantage it had over themedieval alternatives cited above . In essence, thisadvantage lay in the fact that Bradwardine's "func-tion" allowed one to continue deriving "values" forV, since such values-the repeated halving of V, forexample-never correspond to a case of R>F(as wasthe case with VccF/R) ; they correspond, rather, tothe repeated taking of roots of F/R, and if the initialF1 >R 1 (as is always assumed), then for any such rootF„/R„ - (F1/R 1 ) 11 ", F„ is always greater than R,rWith this in view, it would seem that Bradwardine'smost notable accomplishment lay in discovering amathematical relation governing speeds, forces, andresistances that fit more adequately than others theAristotelian-Scholastic postulates of motion involvedin the problem he set out to resolve .

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It is of the utmost importance to note, however,that almost all we have said in expoundingBradwardine's function goes well beyond what onefinds in the text of the Tractatus de proportionibusitself. Notions of arithmetic versus geometric increaseor of the exponential character of his "function" maywell translate his intentions into our way of thinking,but they also simultaneously tend to mislead . Thus,to speak of exponents at once implies or suggests amathematical sophistication that is not in Brad-wardine, and also obscures the relative simplicityof his manner of expressing (by example. t o be sure)his "function ." This simplicity derives from the sym-metrical use of the relevant terminology : If onedoubles a speed, he would say, then one doubles theratio of force to resistance, and if one halves the speed,then one halves the ratio . Although we feel con-strained to note that doubling or halving a ratioamounts, in our terms, to squaring or taking a squareroot, such an addendum was unnecessary forBradwardine, since for him the effect of applying suchoperations as doubling or halving to ratios was un-ambiguous. To double A/B always gave-again in ourterns-(A/B) 2 , and never 2(A/B) . What is more, theexamples Bradwardine utilizes to express his functiondeal merely with doubling and halving, a factor whichmakes it evident that he was still at a considerableremove from the general exponential function sooften invoked in explaining the crux of the Tractatusde proportionibus .

This limitation not only derived from the fact thatthe relevant material in Aristotle so often spoke ofdoubles and halves, but may also be related to thepossible origin of Bradwardine's function itself. Thelocale of this origin is medieval pharmacology, wherewe find discussion of a problem similar toBradwardine's ; in place of investigating the cor-responding variations between the variables of mo-tion, we have instead to do with an inquiry into theconnection of variables within a compound medicineand its effects . Given any such medicine, how is avariation in the strength (gradus) of its effect relatedto the variation of the relative strengths (virtutes) ofthe opposing qualities (such as hot-cold or bitter-sweet) within the medicine which determine thateffect? As early as the ninth century, the Arab philoso-pher al-Kindi replied to this question by stipulatingthat while the gradus of the effect increases arith-metically, the ratio of the opposing virtutes increasesgeometrically (where this geometric increase followsthe progression of successive "doubling," that is,squaring). Now, not only was the pertinent text ofal-Kindi translated into Latin, but the essence of hisanswer to this pharmacological puzzle was analyzed,

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developed, and, in a way, even popularized by thelate thirteenth-century physician and alchemistArnald of Villanova . From Arnald's work al-Kindi's"function" found its way into early fourteenth-centurypharmacological works, even into the Trifolium ofSimon Bredon, fellow Mertonian of Bradwardine inthe 1330's .

Now it is certainly possible, indeed even probable,that Bradwardine may have appropriated his functionfrom the al-Kindian tradition (a borrowing that mayalso have occurred in the case of the use of"exponential" relations within certain fourteenth-century alchemical tracts as well) . But even admittingthis, Bradwardine did a good deal more than simplytransfer the function from the realm of compoundmedicines to the context of his problem of motion .For, quite unlike his pharmacological forerunners, hedeveloped the mathematics behind his function byaxiomatically connecting it with the whole medievalmathematics of ratios as he knew it . Thus, the entirefirst chapter (there are but four) of his Tractatus isdevoted to setting forth the mathematical frameworkrequired for his function . A beginning exposition ofthe standard Boethian division of particular numeri-cal ratios (e .g ., sesquiallera, superpartiens, etc .)furnishes him with the terminology with which he wasto operate . Second, and of far greater insight andimportance, he axiomatically tabulated the substanceof the medieval notion of composed ratios. That is,to use our terms, A/C is composed of (componiturex) A/B • B/C. Furthermore (and here lies the speci-fic connection with Bradwardine's function), whenA/B = B/C, then A/C = (A/B) 2 . Or, as Bradwardinestated in general, if a,/a, = a 9/a3 = . . . a,,,/a, ,,then a1/a„ = a l/a 2 • a2/a3. • . . a,,_ 1/a„ anda,/a„ = (al/a2)"-_1 . In point of fact, the insertion ofgeometric means and the addition of continuousproportionals that are here manipulated wereBradwardine's, and the standard medieval, way ofdealing with what are (for us), respectively, the rootsand powers involved in his function .

The fact that Bradwardine was thus able to statein its general form the medieval mathematics behindhis function suggests that, although his expression ofthe function itself in mathematical terms was nevergeneral, this was due to his inability to formulate sucha general mathematical statement . The best he coulddo was, perhaps, to give his function in the ratheropaque, and certainly mathematically ambiguous,form we have quoted in extenso above, and thenmerely to express the mathematics of it all by wayof example .

Proper generalization of Bradwardine's functionhad to await, it seems, his successors . Hence, John

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Dumbleton, like Bradwardine a Mertonian, gave amore general interpretation of the function througha more systematic investigation ofits connections withthe composition of ratios (see his Summa de logiciset naturalibus, pt. III, chs . 6-7) . He also extendedBradwardine by translating him, as it were, into thethen current language of the latitude of forms (thatis, equal latitudes of motion [ V] always correspondto equal latitudes of ratio [F/R], where the corre-sponding "scales" of such latitudes are, respectively,what we would term arithmetic and geometric) .

A further development of Bradwardine, in manyways the most brilliant, occurred in the Liber calcula-tionum of Richard Swineshead, yet anotherMertonian successor. In Tractatus XIV (entitled Demotu locali) of this work, Swineshead elaborates hispredecessor's function by setting forth some fifty-oddrules that, assuming Bradwardine to be correct, spec-ify which different kinds of change (uniform, difform,uniformly difform, and so on) in F/R obtain relativeto corresponding variations in V. Swineshead alsoextended Bradwardine in Tractatus XI (De loco ele-menti) of his Liber calculationum, where, in what issomething of a fourteenth-century mathematical tourde force, he applies his function to the problem ofthe motion of a long, thin, heavy body (corpuscolumare) near the center of the universe .

Another significant medieval development ofBradwardine's function was effected by NicoleOresme in his De proportionibus proportionum . Hereone observes an extension of the mathematics implicitin the function into a whole new "calculus of ratios"in which rules are prescribed for dealing with whatare, for us, rational and irrational exponents . More-over, Oresme then applies this calculus to the problemof the possible incommensurability of heavenly mo-tions and the consequences of such a possibility forastrological prediction.Many other Scholastic legatees of the

Bradwardinian tradition could be cited as well, but,unlike the three we have mentioned above, mostappear to have concerned themselves chiefly withrather belabored expositions of what Bradwardinemeant, although a few, such as Blasius of Parma andGiovanni Marliani, produced somewhat unimpressivedissents from his opinion .

One should not close an account of the Tractatusde proportionibus without some mention of its finalchapter. Here, in effect, Bradwardine attacks thequestion of the appropriate measure of a body inuniform motion, a matter that becomes problematicwhen rotational movement is considered. Again inves-tigating, and rejecting, proposed alternative solutionsto his question, he argues that the proper measure

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must be determined by the fastest-moving point ofthe mobile at issue . Once more his decision bore fruit,especially in his English successors. The resulting"fastest moving point rule" gave birth to an extensiveliterature treating of the sophisms that arise when oneattempts to apply the rule to bodies undergoingcondensation and rarefaction or generation and cor-ruption. (The work of the Mertonian WilliamHeytesbury furnishes the best example of this litera-ture .)

Tractatus de continuo . In book VI of the Physics,Aristotle had formulated a battery of argumentsdesigned to refute, once and for all, the possiblecomposition of any continuum out of indivisibles .Like all Aristotelian positions, this received ampleconfirmation and elaboration in the works of hisScholastic commentators. Yet two features of themedieval involvement with this particular segment ofPhysics VI are especially important as backgroundto Bradwardine's entry, with his Tractatus de continuo,into what was soon to become a heated controversyamong natural philosophers . To begin with, from theend of the thirteenth century on, Scholastic supportand refortification of Aristotle's anti-indivisibilistposition almost always included a series of mathe-matical arguments that did not appear in the Physicsitself or in the standard commentary on it affordedby Averroes . Considerable impetus and authoritywere given to the inclusion of such arguments by thefact that Duns Scotus had seen fit to feature them,as it were, in his own pro-Aristotelian treatment ofthe "continuum composition" problem in book II ofhis Commentary on the Sentences. The second impor-tant medieval move in the history of this problemoccurred in the early years of the fourteenth century,when we witness the eruption of anti-Aristotelian,proindivisibilist sentiments . These two factors alonedo much to explain the nature and purpose ofBradwardine's treatise, for he wrote it (sometime after1328, since it refers to his Tractatus de proportionibus)to combat the rising tide of atomism, or indivisibilism,as personified by its two earliest adherents : Henryof Harclay (chancellor of Oxford in 1312) and WalterChatton (an English Franciscan, fl. ca. 1323). Further-more, in attacking the atomistic views of his twoadversaries, Bradwardine used as his most lethalammunition the appeal to mathematical argumentsthat, as we have noted, were by now standard Scho-lastic fare. But he developed this application of mathe-matics to the problem at issue far beyond that of hispredecessors .The Tractatus de continuo was, first of all, mathe-

matical in form as well as content, for it was modeledon the axiomatic pattern of Euclid's Elements, begin-

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ning with twenty-four "Definitions" and ten "Sup-positions," and continuing with 151 "Conclusions" or"Propositions," each of them directly critical of theatomist position . These "Conclusions" purport to re-veal the absurdity of atomism in all branches ofknowledge (to wit : arithmetic, geometry, music,astronomy, optics, medicine, natural philosophy, meta-physics, logic, grammar, rhetoric, and ethics), but thenucleus of it all lies in the geometrical argumentsBradwardine brought to bear upon his opponents .

To understand, even in outline, the substance andsuccess of what Bradwardine here accomplished, oneshould note at the outset that the atomism he wascombating was, at bottom, mathematical . The posi-tion of the fourteenth-century atomistic thinker con-sisted in maintaining that extended continua werecomposed of nonextended indivisibles, of points .Given this, Bradwardine astutely saw fit to expandthe mathematical arguments that were already popu-lar weapons in opposing those of atomist persuasion .Such arguments can be characterized as attempts toreveal contradictions between geometry and atomism,in which the revelation takes place when assortedtechniques of radial and parallel projection are ap-plied to the most rudimentary of geometrical figures .For example, parallels drawn between all the "point-atoms" in opposite sides of a square will destroy theincommensurability of the diagonal, while the con-struction of all the radii of two concentric circles will,if both are composed of extensionless indivisibles,entail the absurdity that they have equal circum-ferences . In applying these and related arguments,Bradwardine effectively demolished the atomist con-tentions of his opponents, at least when they main-tained that the atoms composing geometrical lines,surfaces, and solids were finite in number or, if notthat, were in immediate contact with one another . Hissuccess (and that of others who employed similararguments against atomism) was not as notable,however, in the case of an opponent who held thatcontinua were composed of an infinity of indivisiblesbetween any two of which there is always another .Here the argument by geometrical projection faltereddue to a failure-which Bradwardine shared tocomprehend the one-to-one correspondence amonginfinite sets and their proper subsets (although thisproperty was properly appreciated, it seems, byGregory of Rimini in the 1340's) .The major accomplishment of Bradwardine's

Tractatus de continuo lay, however, in yet anothermathematical refutation of his atomist antagonist . Torealize the substance of what he here intended, weshould initially note that Aristotle's own argumentsin Physics VI against indivisibilism made it abun-

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dantly clear that the major problem for any prospectivemathematical atomist was to account for the connec-tion or contact of the indivisibles he maintained couldcompose continua. As if to grant his opponent allbenefit of the doubt, Bradwardine suggests that thisproblematic contact of point-atoms might appro-priately be interpreted in terms of the eminentlyrespectable geometrical notion of superposition(superpositio), a respectability guaranteed for themedieval geometer in the application of this notionwithin Euclid's proof of his fundamental theorems ofcongruence (Elements 1, 4 and 8; III, 24) . However,immediately after this concession to the opposingview, Bradwardine strikes back and, in a sequenceof propositions, conclusively reveals that the super-position of any two geometrical entities systematicallyexcludes their forming a single continuum . Conse-quently, the urgently needed contact of atoms isgeometrically inadmissible .

Finally, as if to reveal his awareness of the mathe-matical basis of his whole Tractatus, toward its con-clusion Bradwardine puts himself the question ofwhether, in using geometry as the base of his refuta-tion of atomism, he had perhaps not begged the veryquestion at issue ; does not geometry assume thedenial of atomism from the outset? He replies bycarefully pointing out that while some kinds ofatomism are, by assumption, denied in geometry,others are not. And he explains why and how . In ourterms, he has attempted to point out just whichcontinuity assumptions are independent of the axiomsand postulates, both expressed and tacit, of Euclid'sElements and which are not. That he realized thepertinence of such an issue to the substance of themedieval continuum controversy is certainly much tohis credit .

Geometria speculativa and Arithmetica speculativa .These two mathematical works, about which we lackinformation concerning the date of composition, areboth elementary compendia of their subjects and wereintended, it seems plausible to claim, for arts studentswho may have wished to learn something of thequadrivium, but with a minimal exposure to mathe-matical niceties . The Arithmetica is the briefer of thetwo and appears to be little more than the extractionof the barest essentials of Boethian arithmetic. Moreinteresting, both to us and to the medievals them-selves, to judge from the far greater number of extantmanuscripts, is the Geometria speculativa. From themathematical point of view, it contains little of start-ling interest, although it does include elementarymaterials not developed in Euclid's Elements (e .g .,stellar polygons, isoperimetry, the filling of space bytouching polyhedra [impletio loci], and so on). Of

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greater significance would seem to be Bradwardine'sconcern with relating the mathematics being ex-pounded to philosophy, even to selecting his mathe-matical material on the basis of its potential philosoph-ical relevance. Such a guiding principle was surelyin Bradwardine's mind when he saw fit to have hiscompendium treat of such philosophically pregnantmatters as the horn angle, the incommensurability ofthe diagonal of a square, and the puzzle of the possi-ble inequality of infinites . Indeed, it is precisely topassages of the Geometria dealing with such questionsthat we find reference in numerous later authors, suchas Luis Coronel and John Major. Such authors werefundamentally philosophers-philosophers, more-over, with little mathematical expertise-and it wouldseem fair to conclude that Bradwardine had just thistype of audience in view when he composed hisGeometria .

Theological Works . Bradwardine's earliest ventureinto theology is perhaps represented by his treatmentof the problem of predestination, extant in a questioentitled De futuris contingentibus . His major theo-logical work, indeed the magnum opus of his wholecareer, is the massive De causa Dei contra Pelagiumet de virtute causarum ad suos Mertonenses, completedabout 1344. Its primary burden was to overturn thecontemporary emphasis upon free will, found in thewritings of those with marked nominalist tendencies(the "Pelagians" of the title), and to reestablish theprimacy of the Divine Will . Although this reaffirma-tion of a determinist solution to the problem of freewill is not of much direct concern to the history ofscience, brief excursions into sections of the De causaDei have revealed that it is not without interest forthe development of late medieval natural philosophy .Thus, to cite but two instances, Bradwardine discussesthe problem of an extramundane void space and,within the context of rejecting the possible eternityof the world, again struggles with the issue of unequalinfinites. One is tempted to suggest that closer studyof the De causa Dei will reveal that Bradwardine'stheological efforts contain yet other matters of impor-tance for the history of science .

BIBLIOGRAPHY

1. Liri: . The fundamental point of departure is A . B .Emden, A Bibliographical Register of the University ofOxford to A .D. 1500, 1 (Oxford, 1957), 244-246 . One mayalso profitably consult H . A. Obermann, ArchbishopThomas Bradwardine, a Fourteenth Century Augustinian(Utrecht, 1958), pp. 10-22 . The Sermo epinicius has beenedited with a brief introduction by H . A. Obermann andJ . A . Weisheipl, in Archives d'histoire doctrinale et litterairedu mover age, 25 (1958), 295-329 .

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11 . WRITINGS AND DOCTRINE . The mostcomplete bibliog-raphy of the editions and MSS of Bradwardine's works isto be found in the unpublished thesis of J . A . Weisheipl,"Early Fourteenth-Century Physics and the Merton'School"' (Oxford, 1957), Bodl. Libr . MS D . Phil . d .1776 .

Logical Works . The unedited De insoluhilibus is extantin at least twelve MSS, including Erfurt, Amplon. 8© 76,6r-21v and Vat . lat . 2154, 13r-24r . For the equally uneditedDe incipit et desinit : Vat. lat . 3066, 49v-52r and Vat . lat .2154, 24r-29v . Although Bradwardine's treatises are notconsidered, the kinds of problems they bear upon are dealtwith (for the De insolubilibus) in 1 . M. Bochenski, A Historyof Formal Logic (Notre Dame, Ind ., 1961), pp . 237-251 ;and (for the De incipit) in Curtis Wilson, WilliamHevtesburv. Medieval Logic and the Rise o_f MathematicalPhysics (Madison, Wis ., 1956), pp . 29-56 . A variety of otherlogical writings, although often ascribed to Bradwardinein MSS, are most likely spurious ; they are too numerousto mention here .

Tractatus de proportionibus velocitatum in motibus . Thishas been edited and translated, together with an introduc-tion, by H. Lamar Crosby as Thomas of Bradwardine . HisTractatus de Proportionibus . Its Significance for the Develop-ment of Mathematical Physics (Madison, Wis ., 1955) . Cor-rections to some of Crosby's views can be found in EdwardGrant, ed . . Nicole Oresme. De proportionibus proportionumand Ad pauca respicientes (Madison, Wis ., 1966) . pp . 14-24,a volume that also contains a text, translation, and analysisof Oresme's extension of the mathematics of Bradwardine's"function." For the problem and doctrine of Bradwardine'sTractatus one should also note Marshall Clagett, TheScience of Mechanics in the Middle Ages (Madison, Wis .,1959), pp . 215-216, 220-222,421-503 : and Anneliese Maier,Die Vorldufer Galileis im 14. Jahrhundert, 2nd ed. (Rome,1966), pp. 81-110. Discussion of some of the factors ofthe above-cited application of Bradwardine's function inthe Liber calculationum of Richard Swineshead can befound in John E . Murdoch, "Mathesis in philosophiamscholastican: introducta : The Rise and Development of theApplication of Mathematics in Fourteenth Century Philos-ophy and Theology," in Acts of the I Vth InternationalCongress of Medieval Philosophy, Montreal, 1967 (in press) ;and M . A. Hoskin and A. G . Molland, "Swineshead onFalling Bodies: An Example of Fourteenth CenturyPhysics," in British Journal for the History of Science, 3(1966), 150-182, which contains an edition of the text ofSwineshead's De loco elementi (Tractatus XI of his Libercalculationum) . For a new interpretation of howBradwardine's function should be understood . see A. G .Molland. "The Geometrical Background to the 'MertonSchool': An Exploration Into the Application of Mathe-matics to Natural Philosophy in the Fourteenth Century,"in British Journal for the History of Science, 4 (1968), 108-125 . A brief discussion and citation of the relevant texts inDumbleton that treat of Bradwardine will appear in an articleby John Murdoch in a forthcoming volume of the BostonUniversity Studies in the Philosophy of Science . Finally, theissue of the probable pharmacological origin ofBradwardine's function is treated in Michael McVaugh .

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"Arnald of Villanova and Bradwardine's Law," in Isis, 58(1967), 56-64 .

Tractatus de continuo. The as yet unpublished text ofthis treatise was first indicated, and extracts given, inMaximilian Curtze, "Uber die Handschrift R . 4© 2 :Problematum Euclidis Explicatio, des Konigl . GymnasialBibliothek zu Thorn ." in Zeitsch rift fur Mathematik andPhysik, 13 (1868), Hist .-lit . Abt., 85-91 . A second articlegiving a partial analysis of the contents of the Tractatusis Edward Stamm, "Tractatus de Continuo von ThomasBradwardina," in Isis, 26 (1936), 13-32, while V. P. Zoubovgives a transcription of the enunciations of the definitions,suppositions, and propositions of the Tractatus, with anaccompanying analysis of the whole, in "TraktatBradvardina '0 Kontinuume,"' in Istoriko-mate-maticheskiie Issledovaniia, 13 (1960), 385-440 . A criticaledition of the text has been made from the two extantMSS (Torun, Gymn . Bibl . R. 4© 2, pp. 153-192; Erfurt,Amplon. 4© 385, 17r-48r) by John Murdoch and willappear in a forthcoming volume on mathematics and thecontinuum problem in the later Middle Ages . Some indica-tion of the issues dealt with in the De continuo can befound in Anneliese Maier, Die Vorldufer Galileis im 14.Jahrhundert, 2nd ed. (Rome, 1966), pp. 155-179; and JohnMurdoch, "Rationes mathematice." Un aspect du rapportdes mathematiques et de la philosophic au moyen age,Conference, Palais de la Decouverte (Paris, 1961), pp .22-35; "Superposition, Congruence and Continuity in theMiddle Ages," in Melanges Kovre, I (Paris, 1964), 416-441 ;and "Two Questions on the Continuum : Walter Chatton(?), O .F.M . and Adam Wodeham, O .F.M .," in FranciscanStudies, 26 (1966), 212-288, written with E . A. Syrian.

Mathematical Compendia . The Arithmetica speculativawas first printed in Paris, 1495, and reprinted many timesduring the fifteenth and sixteenth centuries . 'The Geometriaspeculativa (Paris, 1495) was also republished, and hasrecently been edited by A. G . Molland in his unpublisheddoctoral thesis, "Geometria speculativa of ThomasBradwardine : Text with Critical Discussion" (Cambridge .1967) ; cf. Molland's "The Geometrical Background to the'Merton School,"' cited above . Brief consideration of theGeometria can also be found in Moritz Cantor, VorlesungenUber Geschichte der Mathematik, 2nd ed., II (Leipzig, 1913),114-118 . One might note that a good deal of Bradwardine'sGeometria was repeated in a fifteenth-century Geometriaby one Wigandus Durnheimer (MS Vienna, Nat . Bib . 5257,lr-89v) . The Rithmomachia ascribed to Bradwardine (MSSErfurt, Amplon . 4© 2, 38r-63r ; Vat. Pal. lat . 1380.189r-230v) is most probably spurious .The Questio de futuris contingentibus was edited by

B . Xiberta as "Fragments d'una questio inedita de TomasBradwardine," in Beit •dge zur Geschichte der Philosophicdes Mittelalters, Supp . 3, 1169-1180. The editio princeps ofthe De causa Dei at the hand of Henry Savile (London,1618) has recently been reprinted (Frankfurt, 1964) . Thebasic works dealing with Bradwardine's theology areGordon Leff, Bradwardine and the Pelagians (Cambridge,1957), and H . A . Obermann, Archbishop ThomasBradwardine. A Fourteenth Century Augustinian (Utrecht,

BRAGG

1958), whose bibliographies give almost all other relevantliterature. For the discussion of void space and infinity inthe De causa Dei, see Alexandre Koyre, "Le vide et l'espaceinfini au XTVe siecle," in Archives d'histoire doctrinale etlitteraire du moyen age, 17 (1949), 45-91 ; and JohnMurdoch, "Rationes mathematice" (see above), pp . 15-22 .Also of value are A . Combes and F . Ruello, "Jean de Ripa1 Sent. Dist. XXXVII : De modo inexistendi divine essentiein omnibus creaturis," in Traditio, 23 (1967), 191-267 ; andEdward Grant, "Medieval and Seventeenth-Century Con-ceptions of an Infinite Void Space Beyond the Cosmos,"in Isis (in press) .

If one disregards the various epitomes of the De causaDei, it would appear that the only remaining work whichmay well be genuine is a Tractatus de meditatione ascribedto Bradwardine (MSS Vienna, Nat . Bibl . 4487, 305r-315r ;Vienna, Schottenkloster 321, 122r-131v) . Of the numerousother works that are in all probability spurious, it willsuffice to mention the Sentence Commentary in MS Troyes505 and the Questiones physice in MS Vat . Pal . lat . 1049,which is not by Bradwardine but, apparently, by oneThomas of Prague .

JOHN E . MURDOCH

BRAGG, WILLIAM HENRY (b. Westward, Cumber-land, England, 2 July 1862 ; d. London, England, 12March 1942), physics.

Born on his father's farm near Wigton, Bragg wasthe eldest child of Robert John Bragg, former officerin the merchant marine, and Mary Wood, daughterof the vicar of the parish of Westward . His motherdied when he was seven . A bachelor uncle, WilliamBragg, a pharmacist and the dominant member ofthe family, then took his namesake to live with him .After six years Bragg's father removed his son fromthe uncle's house in Market Harborough (50 milesnortheast of Cambridge) and sent him to KingWilliam's College, a public school on the Isle of Man .Bragg continued, however, to return to Market Har-borough during vacations even after he had gone upto Cambridge, and to look forward to his uncle's pridein his accomplishments .

Bragg was always at the top of his school class,quiet and rather unsocial but tall, strong, and goodat competitive sports. Having outstripped his school-mates, he made little progress in his final year,1880-1881 . "But a much more effective cause for mystagnation was the wave of religious experience thatswept over the upper classes of the school during thatyear . . . . we were terribly frightened and absorbed ;we could think of little else ."' The mature Braggpreserved his composure by refusing to take literallythe biblical threat of eternal damnation, although heretained his faith and his abhorrence of atheism .

Bragg entered Trinity College on a minor scholar-

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ship, obtaining a major scholarship the following year .Beginning his work at Cambridge in the long vaca-tion, July and August 1881, he went up every "long"afterward. Under Routh's coaching he read mathe-matics, and only mathematics, "all the morning, fromabout five to seven in the afternoon, and an hour orso every evening" for three years, coming out thirdwrangler in Part I of the mathematical tripos in 1884 ."I never expected anything so high . . . . I was fairlylifted into a new world . I had new confidence : I wasextraordinarily happy ." 2 Bragg obtained first-classhonors in Part III of the mathematical tripos in 1885,and left Cambridge at the end of that year upon beingappointed to succeed Horace Lamb as professor ofmathematics and physics at the University ofAdelaide. Although in his last year at CambridgeBragg attended lectures by J . J. Thomson at theCavendish Laboratory, at the time of his appointmenthis physical studies had not included any electricity ;he subsequently attempted Maxwell's Treatise onlyafter reading more elementary texts .At Cambridge, Bragg published nothing ; in his first

eighteen years at Adelaide (1886-1904) he publishedthree minor papers on electrostatics and the energyof the electromagnetic field . Rather, his efforts wereinvested in the development of a marvelously, indeedbeguilingly, simple and comprehensible style of pub-lic and classroom exposition, in the affairs of hisuniversity, and in those of the Australasian Associa-tion for the Advancement of Science . One of theAustralian notables by virtue of his office, in 1889he married the daughter of the postmaster and govern-ment astronomer, Charles Todd, and fell in with theextensive but relaxed social life and out-of-doorsrecreations . His elder son, William Lawrence, caddiedfor his father, a fine golfer ; his daughter was a de-voted companion .

This is not the sort of life that brings election tothe Royal Society of London (1907), the Bakerianlectureship (1915), the Nobel Prize in physics (1915),the Rumford Medal of the Royal Society (1916),sixteen honorary doctorates (1914-1939), presidencyof the Royal Society (1935-1940), and membershipin numerous foreign academies, including those ofParis, Washington, Copenhagen, and Amsterdam .The new life began, at age forty-one, in 1903-1904 .

In 1903 Bragg was once again president of Sec-tion A (astronomy, mathematics, and physics) ofthe Australasian Association for the Advancementof Science . His presidential address, delivered atDunedin, New Zealand, on 7 January 1904, wasentitled "On Some Recent Advances in the Theoryof the Ionization of Gases ." 3 Conscious that he wasaddressing Rutherford's "friends and kindred," and