Chapter 10.  John Dee and Astrological Physics


          1.  A relatively late exemplar of a person who united astronomy/astrology with mathematics and recommended observation of nature was John Dee.  Dee was born in London in 1527 and died in Mortlake, Surrey, in 1608.  The following details of his life are based on the work by Nicholas H. Clulee, John Dee's Natural Philosophy (1988).  Dee took an MA at Cambridge in 1548.  In 1551 he became connected with the court of Edward VI of England, by way of family and university connections.  He was an adviser on matters of navigation in connection with the early English voyage of Richard Chancellor and Hugh Willoughby in 1553 to attempt to find a Northwest Passage to Cathay (the Indies, China).  Dee's patrons at court, which included the Duke of Northumberland (John Dudley), were executed or otherwise deprived of power during the reign of Mary.  In 1555 Dee was imprisoned, ostensibly for calculating and conjuring and witchcraft.  "Calculating" probably referred to casting nativity horoscopes.  According to Dee's own account, he was arrested because of his service to Princess Elizabeth, on the charge that he "endeavored by enchantements to destroy Queene Mary" (quoted by Clulee, p. 33; Clulee says this explanation is "substantially correct".)  He was acquitted of charges of treason, but remained for some time in the custody of Bishop Bonner of London, a prominent Catholic defender.  According to Clulee, speaking of Bishop Bonner: "Not only did he become a persecutor of Protestants, he was very unsympathetic to any form of magic and not likely to look kindsly on those associated with the governments of Somerset and Northumberland.  Yet Dee does not seem to have had a hard time of it.  ...  Either Dee was a very good dissembler [in matters of religion] capably of playing a very convincing role over the course of several months or he may in fact have had nothing to worry about."  (Clulee, p. 34).

          2.  Subsequently Dee found favor at the court of Queen Elizabeth (who acceded in 1558) and to some extent with the queen herself, especially by way of Robert Dudley, Earl of Leicester and son of John Dudley, and William Cecil, Elizabeth's principal secretary.  (Clulee, p. 121-123).  In 1558, Dee published a work commonly called the Propaedumata aphoristica (Aphoristic Introduction), and a second edition of the work in 1568..  The second edition has been translated into English by Wayne Shumaker in John Dee on Astronomy (1978), which also contains an essay on Dee's mathematics and physics by J. L. Heilbron.  Shumaker's translation of the full title is:  An Aphoristic Introduction by John Dee of London Concerning Certain Outstanding Virtues of Nature.  An earlier work by Dee called the Monas hieroglyphica had been defended by Elizabeth herself.  Clulee says (p. 189):  "He [Dee] was ... well considered at court.  Elizabeth defended his Monas  ...  and read the book with him ...  She rewarded his gift of a copy of the 1568 edition of the Propaedeumata with £20, and when her travels took her past Mortlake, she occasionally stopped to talk with him."

          3.  Evaluations of John Dee have varied widely, from his own day to the present.  In her biography of Dee, John Dee (1527-1608), published in 1909, Charlotte Fell Smith says (Chapter 1):  "There is perhaps no learned author in history who has been so persistently misjudged, nay, even slandered, by his posterity, and not a voice in all the three centuries uplifted even to claim for him a fair hearing.  Surely it is time that the cause of all this universal condemnation should be examined in the light of reason and science; and perhaps it will be found to exist mainly in the fact that he was too far advanced in speculative thought for his own age to understand.  For more than fifty years out of the eighty-one of his life, Dee was famous, even if suspected and looked askance at as clever beyond human interpretation."  And further (Chapter 23):  " ... Dee's memory may be entrusted to the kinder judges of to-day, who will be more charitable because more enlightened and less impregnated with superstition.  They may see in him a vain, presumptuous and much deluded person, but at any rate they must acknowledge his sincere and good intentions; his personal piety; his uncommon purity of thought and mind.  If in his thirst for knowledge of the infinite unknowable, he pushed back the curtain farther than was wise or justifiable, did he harm any one's reputation beside his own?  Did he not suffer all the penalty in his own miserable failure, so far as comfort and prosperity in material things were concerned?  In all the vague hopes held out by him to Queen, Princes and Emperors, of enriching them through his alchemical skill, he was no conscious charlatan, playing a part to lure them on, but a devout believer in man's power and purpose to wrest scientific secrets from the womb of the future.  Can we look back upon the discoveries of three hundred years and feel his certainty was vain?  The powers of electricity, the training to our uses that marvelous and long concealed agency and light; the healing virtues of radium, should be worth more to us than much manufactured gold."

          4.  In his biography of Dee, John Dee: The World of an Elizabethan Magus (1972), Peter J. French says (p. 4):  "John Aubrey's brief estimation [in 1718] of John Dee as 'one of the ornaments of his Age' may be as fair as any that has so far been made.  Although Dee was a major intellectual force in Elizabethan England, many of his contemporaries -- the 'Ignorant' Aubrey termed them -- branded him a conjurer.  Posterity has not been any kinder than his less learned contemporaries.  Because of Dee's interest in occult philosophy and because of the controversy surrounding his rather remarkable life, many erroneous notions developed about him and his activities, and these have frequently been embellished to the point of absurdity in successive centuries.  Opinions about Dee varied during his own lifetime.  Most erudite scholars on the Continent and in England respected him as a learned man and a dependable source of information.  So, also, did the English mechanicians, those self-educated and middle-class craftsmen and technologists who flourished in Elizabethan London.  In court circles Dee enjoyed almost universal esteem, though, as Aubrey suggests, the commonalty feared him as a sorcerer and a necromancer, a black magician left over from the medieval past."  French says further (p. 8-9):  "John Foxe, in his early editions of the Actes and Monuments, probably did more than anyone else to brand Dee as a conjurer; among other uncomplimentary references in the 1563 edition is the phrase, 'Doctor Dee the great Conjurer'.  The intensely Protestant Actes and Monuments enjoyed extraordinary popularity throught the Elizabethan period; in 1571, Convocation ordered a copy placed in every cathedral church, and the book was also to be found within most ordinary parish churches throughout the kingdom.  Finally, Dee could stand Foxe's 'damnable slaunder' no longer and, in 1576, he issued a plea that Foxe be refrained from describing him as a 'Caller of Divels', and the 'Arche Conjurer' of England.  Dee's plea was successful for all references to him by name were suppressed in the 1576 editioin of the Actes and Monuments.  The silencing of Foxe did not, however, end the vicious rumours about Dee's activities.  And during the final decades of his life there was, in fact, good reason for the continuing suspicions:  John Dee had spent from 21 September 1583 until 2 December 1589 on the Continent where he had quite openly practised cabalist angel-magic with the disreputable Edward Kelley acting as his skryer, or medium."  In his conclusion (p. 208-209), French says:  "His [Dee's] science and magic, his art and even his antiquarianism, all form part of a universal vision of the world as a continuous and harmonious unity.  Dee did not gain his European reputation, as one of his nineteenth-century biographers claimed, for writing 'sheer nonsense'; rather, he gained it because he was a brilliant representative of a philosophy that had inundated Renaissance Europe -- Hermeticism".  Hermeticism was, and still is, a multifaceted set of beliefs and practices founded on an ancient collection of works known as the Corpus Hermeticum, once thought to have been composed, or revealed, at about the time of Moses, but now considered to have been written sometime during the 1st to 3rd centuries A.D.

          5.  In a University of London dissertation of 1952 entitled "John Dee Studied as an English Neoplatonist", I. R. F. Calder writes (Chapter 1):  "Throughout the latter half of the sixteenth century John Dee enjoyed a thoroughly European reputation for profound scholarship:  his opinions were widely consulted, his authority invoked in many diverse fields of speculation and research.  Yet, without minimising the value value of his personal influence and attainments, the justification for a detailed study of these must depend less on the limited value of the accompanying attempt to assess Dee's own claims as an original thinker or direct contributor to scientific discovery, than on the fact that he may be significantly considered as the representative -- and in some respects the spokesman -- of an age.  Dee in his life and writings championed a certain vigorous "new philosophy" which flourished in the late Renaissance, and though this philosophy, or rather the particular form which it then assumed, fell later into barren obsolescence, yet some of its offshoots of that time were to bear rich, and unexpected fruit in succeeding centuries.  Dee's surviving works are perhaps only fragmentary illustrations of certain aspects of the general body of doctrine he maintained, yet an examination of them is illuminating since, however limited or idiosyncratic their subject matter, they exemplify a typical approach to various problems, and they also occasionally give clear expression to broad statements of principle, which should, Dee believed, provide a foundation for a multitude of particular applications.  In these respects, they throw some light, if only indirectly, on much contemporary endeavour and achievement, even in fields discussed not at all, or only incidentally by Dee, since these may often properly be regarded as related and comparable effects arising from a common intellectual tradition."

          6.  E. G. R. Taylor, in her Tudor Geography, 1485-1583 (1930) speaks of John Dee as holding "an important place in the history of sixteenth-century English Geography" (p. 76).  She says (p. 8):  "The fact that John Dee was a practitioner of Judicial Astrology has, however, created such prejudice against him, and has led to such a one-sided estimate of his place in history, that it is here necessary to state emphatically that a close examination of the evidence leaves no doubt of his intellectual honesty and genuine patriotism.  His fame as an astrologer lent Dee prestige among the vast number of his contemporaries who believed with him, that there was a legitimate as well as an illegitimate exercise of that art; while his preoccupation with the search for the Philosopher's Stone and the Elixir of Life lent urgency to his desire for a discovery of the way to Cathay, since it has been a constant tradition that Initiates and Adepts are to be found among the learned of the Far East.  That such was the case, nevertheless, does not detract from the value of his geographical studies or his geographical teaching." 

          7.  However, Taylor's extensive description of Dee's contributions to geography and exploration may not convince everyone of his importance in this regard.  For example, there is the matter of an instrument to aid navigation which Dee claimed to have invented.  Speaking of the era from 1547-1570 in England, she says (p. 95):  "The two great problems, from the theoretical standpoint, which faced the navigator in the far north, were in the first instance the rapid convergence of the meridians, with the resultant spiral curvature of the rhumb lines, and in the second instance the very great variation of the compass from point to point.  The current carta marina, with parallel meridians, was, of course, quite useless under such circumstances, and some fresh technique for setting and plotting a course was necessary.  To this end Dee invented an instrument or device, about which, although he several times refers to it, he was extremely reticent: his Paradoxal compass.  A very careful study of all the inventor's own references, together with those [references] made by the last of his pilot-pupils, leads to the conclusion that this Paradoxal Compass enabled the master to lay a course along a succession of rhumbs which would make an approximation to great circle sailing.  It was, in fact, a practical development of the teaching of Pedro Nuñez on this subject, and its invention belongs to a period when Dee is known to have been in personal touch with the great Portuguese."  One may note that Taylor says that Dee was "extremely reticent" about his compass, and that the word "invention" seems to have been applied by Dee to a supposed construction based on ideas or perhaps designs developed by Nuñez.  Taylor goes on to observe that dates given by Dee for the invention of this instrument, stated in at least one case 20 years or so after the event, were different in different places.  The dates were given by Dee as 1556, 1557 and 1559 in different places.  Taylor conjectures this may have been due to Dee's confusing "when Dee first considered the device" and "when it actually took practical shape in a form suitable for use".  One might also conjecture that the discrepancy in dating was due to Dee's faulty memory.  However, later in her book (p. 109), Taylor observes that Dee introduced the voyagers Martin Frobisher and Christopher Hall to the compass, and that "Frobisher told him [Dee] that its invention had been claimed twenty years since by someone else:  Frobisher further suggested that Dee should establish his claim by publishing an account of the Instrument.  The impudent plagiarist was probably James Alday, who had been trying to get employment with Frobisher, and who, twenty years earlier, must (as the assistant of the official Chief Pilot Sebastian Cabot) have been very jealous of the part of instructor played by John Dee.  As regards publication, it may be observed, Dee never got further than naming the Instrument in the title-page of his General and Rare Memorials, where he puts back its discovery to an earlier date than he claims for it anywhere else."  Taylor gives no reference to support what seems to be a gratuitous assumption about the behavior of Alday in this connection, and begs the question by referring to Alday as an "impudent plagiarist".   Taylor notes further (p. 108):  "Dee's instructions, given to Frobisher and Hall 'in the Rules of Geometry and Cosmography', for their better instruction in 'the Use of Instruments of Navigation', were not so successful as might have been hoped, for as the mariners hinted in a letter of thanks sent from the Scottish coast after they had started, their grounding in mathematics was too scanty for them to profit by his teaching."  When this letter of thanks, as described by Taylor, is compared with the page and a half of notables and nobles which Taylor lists when first establishing what an important person Dee must have been (p. 76-77), one may wonder whether or not Frobisher and Hall were diplomatically thanking a person who had many connections with influential and moneyed  people, and using their alleged lack of mathematical prowess as an excuse for rejecting the instrument -- if there was indeed an instrument, and not just a set of instructions.

          8.  Another conjecture, which I base only on statements made by Taylor herself, is that in fact Dee may have been something of a plagiarist, or at least a dissembler, in connection with his Paradoxal Compass.  Readers of biographies of Dee may well conclude that this behavior on Dee's part agrees well with numerous other estimations of his character and actions.  In this connection, one may also contemplate statements made by Taylor in her later book Late Tudor and Early Stuart Geography, 1583-1650 (1934):  "It would be not untrue to say that from about 1583 onwards, English geography entered upon a distinctly mathematical phase.  The connecting link between the two studies was, of course, astronomy, which formed part of every normal course of mathematical studies at the Universities.  The seamen's problems of position-finding by compass direction, latitude, longitude, and the use of the sea-chart, were at that time problems which taxed the highest skill of the astronomer-mathematician.  Each difficulty, it is true, had been attacked by John Dee, during the period of his active work for England's advancement by discovery between 1550 and 1583, but he did not offer his solutions to the public, and hence he advanced knowledge only among those who sought him personally." (p. 68)

          9.  Francis R. Johnson, in his Astronomical Thought in Renaissance England (1937) continues with  praise for Dee, although his evaluation of Dee's scientific work seems to be based to some fair extent on what E. G. R. Taylor had published a few years before his own work.  He says (p. 135-136):  "Throughout the two decades following [Robert] Recorde's death in 1558, Dee was recognized as the supreme scientific authority in England, and was special adviser to the principal English voyages of the period, beginning in 1553 with the first Muscovy venture, to which Recorde also gave scientific counsel.  Dee's later career, during which this unrestrained optimism concerning the possibilities of natural science made him the dupe of the charlatan Edward Kelley and caused him to turn his energies to alchemy and crystal gazing, has tended to obscure his real merit as a scientist and his very great services to his country.  Even Dee's modern biographer [Charlotte Fell Smith, see above] has emphasized his reputation as an astrologer, alchemist, and dabbler in spiritualism, at the expense of his significant work in legitimate science.  One recent scholar, however, has done much to restore the proper balance and give a truer picture of the man.  Professor E. G. R. Taylor, in her Tudor Geography, devotes three chapters to Dee's labors in geography and the sciences directly related thereto.  She gives the following brief summary of his position and contemporary influence."  Johnson here inserts a long part of Taylor's list of influential or noted people with whom Dee is said to have connections with, of one sort and another.  Johnson then continues (p. 137):  "As the passage quoted indicates, Dee's great significance in English science is due to his work as teacher, adviser, and friend to most of the English mathematicians, astronomers, and geographers of his day."  It isn't clear how Taylor or Johnson arrive at the conclusion that Dee was teacher and adviser to most of the people in the list given by Taylor, nor who are the sources, other than Dee himself, for reports about his nature of his contacts with these people.  Johnson goes on:  "His [Dee's] published works were very few, though he lists many manuscript treatises which were never printed."  And, one may add, have been lost -- or, perhaps, never existed except as titles which Dee hoped to write on some day?  "Whereas Recorde's influence." says Johnson, throughout the century, was transmitted by his excellent textbooks in the vernacular, Dee's was exerted through his personal advice and teaching, and passed on by the pupils he had trained."  Of course, we have no records today of his personal advice and teaching.  There were contemporaries who praised Dee's great knowledge of mathematics (a term which was broader in scope than it is today), but none, it seems, who go into any detail about it.  Johnson, as do all the writers about Dee I have referred to so far, discusses Dee's magnificent library of several thousand volumes, including many medieval manuscripts, which was often consulted by others.  Johnson says (p. 138, 139):  "During the third quarter of the [16th] century, John Dee and his friends and pupils constituted the scientific academy of England. ... This great library was always at the disposal of Dee's fellow scientists among his friends and pupils.  If one believes that the first essential and the true center of any university is its library, Dee's circle might truly be termed the scientific university of England during the period from about 1560 to 1583. ... The position of Recorde and Dee as the outstanding and most influential English scientists at the beginning of the second half of the sixteenth century was universally conceded by their contemporaries among their countrymen."  Of course, strictly speaking, there were no people called "scientists" in this era, but rather "natural philosophers", "mathematicians", and "astrologer/astronomers", etc.  We will see shortly that some more recent writers about Dee have not concurred with these evaluations of Dee by Taylor and Johnson.

          10.  First, however, there is another person to consider who adheres to an evaluation of Dee of the sort presented by Taylor and Johnson.  In her book Theatre of the World (1969), Frances Yates writes from the standpoint of history of architecture during the Renaissance, especially in connection with the Renaissance theater.  She says (p. 7-8), referring to the works of E. Taylor and Johnson, and after quoting a passage from Taylor's Tudor Geography:  "Thus modern historians of science have rehabilitated Dee, have drawn aside the veil of the ridiculous and deluded conjuror of nineteenth-century legend to show behind it the practical scientist fully abreast of the latest scientific thought, translating it into practical use for the service of his countrymen.  Dee comes out now as in the van of Elizabethan movements, the maritime expansion, the scientific activity of all kinds, and moreover as particularly Elizabethan in spirit in his appeal to the rising artisan classes.  His was a new and modern kind of learning which included technology as well as abstract speculation, which made an appeal to new social classes, as well as to the queen and to courtiers.  In these most important respects, no more complete mirror of the Elizabethan age could be found than John Dee."   On p. 14, Yates says:  "It is surely time that Dee should be judged objectively and without prejudice, and that critical historical enquiry should be made, not only into the nature of his science and its place in the history of thought, but also into the nature of his religion and its place in the history of religion."  Yates quotes from a letter of 1592 from Dee to the Archbishop of Canterbury, in which Dee speaks of his religious beliefs, and she concludes:  "It was the religion of a mathematician who believed that the divine creation was held together by magical forces.  If we substitute mechanics for magic as the operative force used by the Creator, Dee's religion was perhaps not altogether unlike that of Isaac Newton."  This comparison is open to question, and I will return to it later.  On p. 17 of her Theatre of the World, Yates says:  "Dee comes straight out of a main Renaissance Hermetic stream which has reached him at rather a late date and bearing with it accretions gathered in the sixteenth century during which it developed, both in more extremely occult directions and in more precisely scientific directions."  A major thesis in Yates writing was that a revival of the magic of Hermeticism (or Hermetism), based on works ascribed to a person known as Hermes Trismegistus, was a dominant influence in many areas during the Renaissance, and in particular that this revival was an important source of the new sciences which arose in the 16th and 17th centuries.  In her earlier book Giordano Bruno and the Hermetic Tradition (1964), she says (p. 449-450):  "Taking a very long view down the avenues of time a beautiful and coherent line of development suggests itself -- perhaps too beautiful and coherent to be quite true.  The late antique world, unable to carry Greek science forward any further, turned to the religious cult of the world and its accompanying occultisms and magics of which the writings of 'Hermes Trismegistus' are an expression.  The appearance of the Magus [Wise One] as an ideal at this time was, as Festugière has said, a retreat from reason into the occult.  The same writer compares the appearance of the Magus idea in the Renaissance as similarly a retreat from the intense rationalism of mediaeval scholasticism.  In the long mediaeval centuries, both in the West and in the Arabic world, the traditions of rational Greek science had made progress.  Hence, it is now suggested, when 'Hermes Trismegistus' and all that he stood for is rediscovered in the Renaissance, the return to the occult this time stimulates the genuine science.  The emerging modern science is still clothed in what might be described as a Hermetic atmosphere.  Francis Bacon's New Atlantis is perhaps not a very good example to take since Bacon's former position as Father of Experimental Science is now weakened.  Nevertheless, the New Atlantis is a scientist's Paradise where every kind of discovery and invention is put to the service of the happy people. ...  Whether or not there is any real connection between the New Atlantis and the City of the Sun [a work by Tommaso Campanella, placed by Yates in the Hermetic tradition, along with Giordano Bruno], those two Utopias come out of the same stream, and the stream is Hermetic, or Hermetic-Cabalist."

          11.  In a turn toward studying the writings of Dee which bear on natural philosophy, or as we now say, the sciences, J. L. Heilbron and Wayne Shumaker published in 1978 a book which contains a translation by Shumaker of Dee's first publication devoted to this topic, the Propaedeumata aphoristica (Aphoristic Introduction) of 1568, a long introductory essay by Heilbron, and notes by Shumaker, assisted by Heilbron, on each of the 120 aphorisms.   "John Dee (1527-1608)," says Heilbron (p. 1), "was a geometer, physicist, astrologer, antiquarian, hermetist, and conjurer, a mixture of mathematician and magician, of scholar and enthusiast, of schemer and dupe."  Heilbron observes that " 'mathematics' in Dee's time included many subjects that have since become branches of physics or engineering, or entirely separate disciplines:  optics, architecture, surveying, fortification, cartography, astronomy, navigation." (p. 4).  As to Dee's capabilities as a mathematician in this sense, Heilbron quotes a statement made by Dee late in life in a work called the Compendious Rehearsal (1592), addressed to Queen Elizabeth in the hope of refuting charges of conjuring made against him and of obtaining a pension from a grateful nation, which runs as follows:

"At the request of some English gentlemen, made unto to me to doe somewhat there [Paris] for the honour of my country, I did undertake to read freely and publiquely Euclide's Elements Geometrical, Mathematicé, Physicé, et Pythagoricé; a thing never done publiquely in any University of Christendome.  My auditory in Rhemes College [the insignificant Collège de Reims -- note by Heilbron] was so great, and the most part elder than my selfe, that the mathematicall schooles could not hold them; for many were faine, without the schooles at the windowers, to be auditors and spectators, as they best could help themselves thereto.  I did also dictate upon every proposition, beside the first exposition."  (quoted by Heilbron, p. 6)

Heilbron comments:    "Now Dee had lectured on only the first two of Euclid's thirteen books, not upon 'every proposition," and his treatment was by no means unprecedented. ... It was already a commonplace to refer to Pythagoras, Plato, and the sublime when recommending the study of the Elements. ... One can hardly avoid the conclusion that Dee's 'Pythagorean' elucidation of 1550 was but a reworking of the traditional uplifting prolegomena to the study of Euclid. ... The fact that no contemporary reference to Dee's lectures has been found, not even in the lengthy preface on the dignity of mathematics published by his friend Montaureus in 11551, provides further evidence that they contained little unfamiliar to the Parisians.  It appears that the aging Dee, dissatisfied with the reception of his life's work and pressed to defend it, misrepresented the commendable but unexceptional Euclidean lectures of his youth as a spectacular and unprecedented achievement." (p. 6, 7-8).  During the years 1551-1553, Dee served the Duke of Northumberland, "a man for whom no one has ever had a kind word."  (Heilbron quotes here G. R. Elton, England under the Tudors, 3rd ed., 1969, p. 209).  Dee tutored the Duke's children, including Robert Dudley, later Earl of Leicester and a favorite of Queen Elizabeth.  Also, Dee helped the Duke "to promote a search for northern routes to to the riches of the orient.  Dee's confidence, enthusiasm, and mathematics, not to mention his training by Gemma and Mercator [two noted geographers and cartographers], were just what was wanted to reassure uneasy investors in expensive voyages through unknown seas." (p. 9)  

          12.  Of Dee's prowess as a mathematician, Heilbron says (p. 17):  "The fact of Dee's contemporary reputation is easier to ascertain than its basis.  We can dismiss the suggestion that he was admired for 'profundity'.  He quite rightly does not figure on van Roomen's list of the chief mathematicians of the later sixteenth century.  Dee's contributions were promotional and pedagogical; he advertised the uses and beauties of mathematics, collected books and manuscripts, and assisted in saving and circulating ancient texts; he attempted to interest and instruct artisans, mechanics, and navigators, and strove to ease the beginner's entry into arithmetic and geometry.  It is in this last role, as pedagogue, that Dee displayed his competence, and made his occasional small contributions (which he classed as great and original discoveries) to the study of mathematics."  As a sample of the sort of thing Dee added to Euclid, Heilbron notes (p. 25) that Dee shows how to find lines x, y, z such that x/y = y/z  = a/b and xyz = c3 , where a, b and c are given.  (I have used anachronistic notation.  Heilbron states the proportions as x:y::y:z::a:b.)  What Dee does, in effect, though with techniques adapted to the use of proportions prevalent in his time, is set z = c, x = (a/b)/c and y = c/(a/b).  With the advantage of algebra as we know it nowadays, one sees by multiplying x, y and z, that indeed xyz = c3 .  The technique then available to Dee for handling proportions was a little more involved, and he it was necessary for him to work in terms of line segments directly, and not with lengths of line segments.  However, the procedure involved were elementary and commonplace according to the standards of the time.  What is interesting, though, in connection with evaluating Dee's status as a mathematician, as Heilbron observes, is that Dee connects this to the ancient problem of duplicating a cube, i.e. constructing a cube with volume equal to twice the volume of a given cube with only a straight edge and an ordinary compass, or more abstractly, using only the axioms and postulates given by Euclid in his Elements.  Heilbron quotes Dee (p. 25):  "Listen to this and devise, you couragious Mathematicians:  consider how nere this creepeth to the famous Probleme of doubling the Cube."  In fact, as Heilbron notes, the problem solved by Dee is useless in this regard.  This can be seen by noting that presumably Dee has in mind taking a cube with length of edge equal c, and using his technique to find x such that x3 = 2c3 .   But the 2, necessary to accomplish the doubling, is nowhere involved in Dee's construction.  Again, this wouldn't have been so apparent to Dee, since algebra was in his time in the process of being developed into the system, or systems, we know today, and Dee was evidently working with traditional Euclidean geometrical constructions of line segments.  In this regard, however, Heilbron notes that "although he [Dee] usually sets up his problems and manipulates his proportions geometrically, his treatment is strongly algebraic in spirit.  The examples so far given show his tendency to set up equations (as proportions) and to juggle them until a solution emerges in the form of a constructible line."

          13.  Heilbron gives a detailed analysis of the contents of Dee's Propaedeumata aphoristica as related to astrology of Dee's time in Part II of his introductory essay.  He says (p. 51):  "Dee's approach to astrology, although not entirely original with him, deviated in several ways from that of the ordinary literature.  There is no trace of the hermetic planetary souls, for example:  stars are not intelligences to be cajoled but unalterable sources of exploitable power; they are not like people but like radiators; their influences may be concentrated by optical instruments, not by songs, prayers and incense.  Again, Dee takes literally the demands of optical theory; and since the intensity of radiation diminishes with distance and increases with the size of the luminous source, he insists that an exact astronomy, one that yields precise values of the sizes and distances of all planetary configurations, is a prerequisite to a competent astrology."  Dee appears to have adopted the idea of considering radiations, modeled chiefly after the example of light, as fundamental to astrology from works of Roger Bacon (1214-1294), and possibly from works of Robert Grosseteste (c1170-1253).   This point of view led Dee to recommend that geometrical optics, as explained by the Arab astrological write Al-Kindi, who flourished about 850 A.D., and some others, be applied to the study of astrology.  Of the physics, astronomy, and astrology used by or alluded to by Dee in his aphorisms, Heilbron says (p. 60):  "Nothing exceeds the grasp of a sixteenth-century undergraduate, even of Oxbridge."  Dee spends some time discussing from the point of view of elementary geometry matters of the intensity and scope of radiations from the seven known planets (counting the sun and moon as planets, but not counting the earth).  The diagrams and constructions involved are explained by both Heilbron and Shumaker.  As an example of what Dee recommends as a basis for astrology is his calculation of 25,341 possible conjunctions (angular separation of zero degrees as seen from the earth) of the planets (2 at a time, 3 at a time, and so on, up to all 7 at once), taking into account the various possibilities of relative differences in their powers or intensities of radiations as to which planet has greater or lesser power than another.  (Dee says 25,335, but he made an error in an intermediate number, putting 120 for 126.)   To deal with each of these conjunctions in a manner recommended by Dee would involve a knowledge the distances of the planets from the earth, of the relative powers of the planets, such matters as the time the planet spends above the horizon on different occasions (the morae) and differences due to retrograde motion of planets with respect to the stars, etc. Traditional astrologers are concerned also with aspects, which are (approximately) angles between planets of 60 or 90 or 120 or 180 degrees, and taking into account the various possibilities of different powers in this case would add many more cases.  Heilbron says (p. 91-92):  "How is this vast mathematical apparatus -- these morae directions, distances and periods, not to mention the 25,341 conjunctions -- to be worked by the astrologer?  How, if he lives long enough to make the calculations, can he compute celestial influences upon the "'entire and unchangeable order of every generation (XXI); upon events both immediate and proximate (LVII); upon the 'chief and truly physical causes of the procreation and preservation of all things which are born and live' (CVII); upon the occasion of the death of the body (CIX).  [The Roman numerals refer to Dee's numbering of his aphorisms.]  Dee's counsel in these matters is not very helpful."  The Propaedeumata aphoristica was written relatively early in Dee's career.  As he grew older, he began to despair more and more of attaining the kind of knowledge and wisdom he longed for.  Eventually he turned, with the help of an associate, Edward Kelley, to trying techniques for conversing with angels.  Heilbron says (p. 15):  "As he [Dee] told an emissary of the Hapsburg Emperor in 1584, he had by degrees passed through 'all manners of studies ..., as many as were commonly known and more than are commonly heard of,' and all ultimately unsatisfactory.  'At length I perceived that onely God (and by his good Angels) could satisfy my desire, which was to understand the natures of all his creatures."

          14.  In his John Dee's Natural Philosophy (1988), Nicholas H. Clulee says (p. 1):  "Until the present century [20th], much of the biographical literature on Dee has been concerned with him as a personality and with the drama of his life; it has devoted little attention to the study of his writings and has been less than scholarly."  Clulee comments on some of the writers I have referred to above.  He says, for example, of works about Dee in the 20th century:  "Initially, the approach adopted typically ignored the more questionable dimensions of Dee's biography and focused upon his position within the evolution of some narrow scientific development that aimed to indicate the importance of his role in that evolution.  E. G. R. Taylor took the lead in rehabilitating Dee's reputation, emphasizing his work and his role as a teacher of practitioners in mathematics and navigation. ... Likewise, Frances R. Johnson considered Dee largely in terms of his accomplishments as an astronomer, his attitude toward Copernicus, and his formulation of an idea of experimental method.  This approach, while not entirely claiming for Dee a position of the first rank, did serve to earn for him some right to consideration in sixteenth-century intellectual history, but at the expense of divorcing aspects of Dee's activity from the rest of his work and biography."  Clulee speaks of a number of writers on Dee since 1950 as adhering to what he calls the Warburg interpretation of Dee's position in intellectual history of the Renaissance, referring to the Warburg Institute of the University of London, an organization which, as one of their World Wide Web pages states, "exists principally to further the study of the classical tradition, that is of those elements of European thought, literature, art and institutions which derive from the ancient world."  Frances Yates was a prominent participant in the Warburg Institute, and I. R. F. Calder was a student of hers there.  Peter French, Clulee says, "drew the foundation of his interpretation from her [Yates'] works and acknowledges her guidance." (p. 2)  Clulee also mentions a dissertation by Graham Yewbrey which "takes the framework developed by Yates and French as the starting point for interpreting Dee's ideas."  Of Frances Yates herself, Clulee says that she "herself contributed significantly to this view in a series of writings in which Dee assumed an increasingly central place." These various authors have been mainly concerned, according to Clulee, with placing Dee's works into an intellectual tradition of some prominent kind.  Calder examined Dee's works in the light of the thesis put forth by Edwin Burtt in his The Metaphysical Foundations of Modern Science (1924) to the effect that "the mathematical aspects of 17th century science were an outgrowth of the conception of mathematics implicit in the Platonism that was revived and developed in the Renaissance." (Clulee, p. 3; Calder says in a Prefatory Note "The basic assumptions of this study (set out in Ch. I.) are similar to those of Burtt's Metaphysical Foundations of Modern Physical Science".)  As I indicated above, Yates places Dee in the tradition of Hermetic occultism and kabbalism (Cabbalism).  Clulee says (p. 8):  "The culmination of Yates's progressive unravelling of the English Renaissance is The Occult Philosophy in the Elizabethan Age.  Here she finds the key component of Elizabethan culture to be a kabbalistic variety of the Hermetic occult philosophy built from Lull, Ficino, Pico, Agrippa, and Francesco Giorgi, given a particularly English and Rosicrucian expression by Dee with his addition of alchemy.  In the context of this philosophy, she interprets English literature as a battle ground between competing ideologies, with Sidney, Spenser, Shakespeare, and Chapman defending the occult philosophy of Dee, and Marlow, the proponent of reaction, attacking the occult philosophy and Dee."

          15.  Clulee studies in detail the main three surviving works of Dee related to natural philosophy and mathematics.  These are the Propaedeumata aphoristica, whose presentation by Shumaker and Heilbron was discussed above; the Monas heiroglyphica, a baffling piece evidently related to alchemy and a proposed project for recovering the original language of God (cf.. the book of Genesis) of which the languages of people are corrupt descendants; and the Mathematicall Praeface which appeared as a preface to a translation of Euclid's Elements  attributed to Henry Billingsley, which, as Clulee shows in his Chapter 6 owes much to the commentary of Proclus on Euclid's Elements which dates from the 5th century A.D.  Passages from the Mathematicall Praeface have often been cited, here and there, as showing that Dee was an early advocate of the emphasis on experiment which many (often enough influenced directly or indirectly by the late 16th and early 17th century jurist, and publicist and theorist about science, Francis Bacon) think was characteristic of what they take to be a Scientific Revolution in the later 16th and 17th centuries.  These latter two works are not much concerned with astrology, as is the case with the Propaedeumata, and since my concern here is with history of astrology, won't be discussed here.  They are, however, major documents for anyone who wants to study Dee's beliefs about magic.

          16.  As to the Propaedeumata,  Clulee gives a succinct but detailed analysis of it.  He says about this work, after presenting his analysis of individual aphorisms (p. 52):  "What has emerged in the Propaedeumata is an astrological physics in which Dee has developed and elaborated his earlier ideas with much greater detail and precision concerning the mechanism of astrological influence and has also developed a method for quantifying the strength of astral virtues and mathematically determining the nature and strength of astrological effects, thereby fusing his interest in astrology with his early interest in mathematics.  Much of Dee's general theory can be traced to standard works on astrology and to Renaissance formulations of astrological theory, despite the fact that Dee rejects or minimizes the usual components of astrological prediction:  the constellations, the aspects, the houses, ascendants, and so forth.  These sources could have provided Dee with a vague notion of rays of force analogous to light as the mechanism of astrological influence, which is present in Dee's astrological notes from the beginning.  These works do not, however, account for all the details of Dee's theory.  Most important, they do not apply the mathematical treatment, which is characteristic of Dee's mature astrology, to the diffusion of virtues.  It is in this regard that Dee's study of optics in the mid-1550s provides the essential foundation for the astrology of the Propaedeumata and the development of Dee's early natural philosophy in general."  Clulee studies at some length the influences of Al-Kindi, the 9th century Arab astrologer, Robert Grosseteste and Roger Bacon on Dee, of whom Clulee says Bacon's influence was  preeminent (p. 64).  Of these influences, Clulee says (p. 57):  "The theory of astrology in the Propaedeumata thus reflects an effort of intellectual archeology through which Dee recovered a tradition of optically based natural philosophy that had continued only in attenuated form after its culmination in Roger Bacon. ... Yet al-Kindi did not suggest that the mathematical definition and study of radiation was relevant in an astrological context, and although Grosseteste and Bacon implied that astrological influences were subject to study by the philosophy of multiplication of species [transmission of or transformation by radiation], they never systematically developed the full implications of their ideas for astrology.  It was left to Dee, who in his early years was most intensively interested in both mathematics and a causal mechanism for astrological physics, to take these ideas out of the contexts in which he found them and weld them into a theory specifically focused on the problem of astrology."  In his conclusions (p. 70-73) about this work by Dee, Clulee argues that "Dee's natural philosophy and astrology was an eclectic individual creation that does not have very much specifically in common with any distinctly Renaissance philosophy whether Aristotelian, Neoplatonic, or Hermetic. ... Dee was working very explicitly within the context of an Aristotelian model of natural philosophy and science. ... The fact that Dee's idea of natural magic [feats performed by human artifice, rather than by spirits or demons] was derived in close association with his study of Roger Bacon also should warn us that an interest in magic is not a specific indicator of a particular philosophic position in the Renaissance. ... It would be vain to argue that Dee's work on astrology represents any contribution to the progressive development of science.  His most fundamental insights were adaptations of medieval ideas whose only novelty was their resurrection after centuries of obscurity and relative neglect.  His concept of science is Aristotelian, his natural philosophy is still limited to a qualitative view of nature, and his ideas on method and 'experimental science; come from Aristotle with the admixture of [Roger] Bacon.  The significance of Dee's astrology lies more in its attempt to deal with contemporary criticisms and doubts concerning the validity and precision of astrology, than with its role in any long-term revolution in natural science.  In these terms Dee's astrology was a significant and novel attempt to develop a physical theory and mathematical method for astrology."

          17.  What are we to make of John Dee, and in particular of his beliefs about and contributions to astrology and astralism of other kinds?  I have shown here, mostly by direct quotation, a considerable variety of opinions about the originality and depth of Dee's works.  I expect what one thinks about Dee depends on who one is -- what one knows, what one believes, and what one wants to believe.  There is a work about Dee which I haven't mentioned up to now, called John Dee:  Scientist, Geographer, Astrologer and Secret Agent to Elizabeth I by Richard Deacon (1968).  Deacon presents a theory to the effect that Queen Elizabeth was a canny user of spies, and that Dee was one of her most trusted agents.  The evidence which Deacon presents seems to me to be slender, based on unsupported conjectures, and very trusting of Dee's statements about himself.  To get some idea, though, of how Dee appeared to Deacon, and what his motives for his judgments of Dee might have been (back in the 1960s), here is what Deacon says about Dee at the end of his book (p. 276-277):  "Here was a complete man in every respect of that overworked phrase, meticulously conventional at Court, yet with some of the qualities of the uncaring Bohemian.  Almost every aspect of his character could form the theme for an enlightening essay.  His religious outlook, his inquiring ecumenical spirit, his insistence on a religious approach to scrying and experiment mark him out as almost as fascinating a character as Sir Thomas More.  As an adviser on naval matters and an imperialist planner he deserves at least a chapter in Britain's naval history.  But it is Dee the dreamer, the seer of visions, the romantic Hermetist that causes one not only to look at Tudor England in a new light, but to ponder on what lessens Dee might have for the psychedelic experiments of today.  For if Hermetist is the right word to apply to him, there was no other Hermetist so practical, so sober and so far-seeing in his life-time.  ...  Thus, with and through Dee, one sees Tudor mysticism in the light of modern science and modern psychotical mysticism being submitted to the tests of a critical appraiser of scrying and the commonsense of Saint Teresa of Avila.  With Dee one sees much of the present and not a little of the distant future in his own past.  It is like looking at the whole of eternity through the shewstone of Elizabethan England."  I do not find shewstone (or showstone) in the Compact Oxford English Dictionary (1991), but I presume it refers to a crystal ball or other device used in scrying, which is the search for visions, of the future or of eternity, in crystal balls or other reflecting or semitransparent material.  One may compare this with a conclusion about Dee made by Nicholas Clulee (loc. cit., p. 240-241):  "The need to distinguish between different types of magic, both on the grounds of their content and theoretical presuppositions and on the grounds of the motivation behind the attention paid to each, also applies to the issue of the relation of magic to science in the Renaissance.  Dee's case indicates that the different traditions of magic in the Renaissance had different implications for science.  In his case the Florentine/Neoplatonic approach, in which magic had a predominantly religious function, was quite separate from his use of the medieval tradition of a natural magic, with most of his scientific work that can be related to magic being related to the latter.  Dee also suggests that the place of mathematics, usually as a mystical and symbolic view of numbers and figures as reflective of occult correspondences, in magical philosophies does not justify concluding that magic encouraged a mathematical approach to science preparatory to seventeenth-century science.  While he shows considerable interest in mystical mathematical correspondences, this interest was quite separate from his actual work involving the application of mathematics and and mathematical reasoning.  The sources that encouraged the expression of a concrete approach to nature through mathematics were Proclus and Cusanus, not any magical texts."  I leave it to any reader of this that might be to decide for yourself what you think about Dee and his works.  I don't feel the need to come to a conclusion about Dee.  I enjoy having met him, and I will no doubt muse about him and what he did as I muse about others I have met, in person or in books.  After all, I was born in the sign of Libra.  And I am a mathematician by profession, in the sense of mathematics prevalent nowadays (year 2000 A.D.) in universities of the world.

     18.  Since writing the 17 paragraphs above, I have come across a translation of Dee's Monas Hieroglyphica ("A Translation of John Dee's 'Monas Hieroglyphica' (Antwerp, 1564), with an Introduction and Annotations", C. H. Josten, Ambix, vol. XII, nos. 2 & 3, June and October, 1964, p. 84-221).  From my reading of the secondary sources I have discussed above, I was under the impression that the former work, the Monas, was not of much interest in connection with the history of astrology, being mainly about of alchemy and a kind of symbolism or language suitable for communicating secrets of nature, as indicated by the title, which can be read as The Hieroglyphic Monad.   However, the work contains astrological material as well.  The term monas is related to the term monad, which refers to an indivisible unit.  We know that Dee was acquainted with works of Proclus (5th century A.D.), such as Proclus's commentary on Euclid (translated into English by Glenn R. Morrow, A Commentary on the First Book of Euclid's Elements, 1970, 1990).  Here and elsewhere (such as in commentaries on Plato's works), Proclus discusses the One, used in a way still sometimes heard from philosophers, in such phrases as "the problem of the One and the Many", in connection with works of Plato, Plotinus and others.  For example, on p. 4 of Morrow's translation of Proclus' commentary by Glenn R. Morrow (1972, 1990), Proclus speaks of "the Limit and the Unlimited" as "the two highest principles after the indescribable and utterly incomprehensible causation of the One". We may conjecture that Dee's monad was a symbol for the One, and moreover that Dee was bent on describing and comprehending this One, regardless of what Proclus said about it.  Josten discusses the choice of the term monad (p. 106-108), and says:  "Essential oneness, or monas, -- the constituent of numbers, though not itself a number -- is the notion which links the alchemical contents of Dee's message with those many digressions on number symbolism, especially that of the Pythagorean tetraktys, and on the symbolism of geometry and of letters, with which his magical parable is enriched."  Presumably with the term hieroglyphica, Dee was referring to a method which he took to underlie such symbolism, and was wide-ranging, to be found in geometrical diagrams in the manner of Euclid, to the traditional symbols used by astrologer/astronomers to denote the planets, to the interpretation of letters and combinations of letters found in the doctrines of the Cabala (Cabbala, Kabala, Kabbala; "cabala" means something like "doctrine").  The Cabala is an elaborate system which has been popular with some from European medieval times up to the present.  Some of its adherents make use, among other things, of interpretations of Hebrew scriptures based on interpretations and associations of letters and combinations of letters with numbers (those we call positive integers, or whole numbers, or natural numbers).  Josten says (p. 84):  "Dee goes so far as to assert that, although he called the work hieroglyphic, it is endowed with a clarity and rigour almost mathematical; yet at the same time he leaves it to the reader even to guess that the subject of the elaborate display, which he is asked to view in such dim light, is the hermetic quest.  The semblance of clarity is achieved by discussing the dark subject under the guise of a symbolic sign invented by Dee, which is his monad.  This symbol indeed lends itself easily to digressive secondary interpretations of a numerological, cabbalistic, astrological, cosmological, or mathematical nature, all which, however, are without any doubt given so as to establish significant connexions with the all-embracing central theme, alchemy, which is barely mentioned."  Josten is referring here to a passage at the end of Dee's dedication of the Monas to  Maximilian of Habsburg, King of Bohemia, the Romans and Hungary, found in the translation at p. 121.  Dee continues here:  "Or is it not rare, I ask, that the common astronomical symbols of the planets (instead of being dead, dumb, or, up to the present hour at least, quasi-barbaric signs) should have become characters imbued with immortal life and should now be able to express their especial meanings most eloquently in any tongue and to any nation."   In this connection, Josten comments (p. 103-105):  "The symbol for Mercury [as used by Dee in his Monad] represents in alchemy the matter, the method, and the result of the alchemical process.  Accordingly the modified mercurial symbol which is that of Dee's monad may be assumed to stand for the subject, and the final achievement in Dee's notion of the hermetic discipline.  It represented to him in its broadest interpretation, therefore, the principle of transmutation itself, that principle of which Mercury is the universal agent and of which mercurial man, i.e. the true alchemist or magus, as a fit recipient of that influence, is the noblest subject. ... One would try in vain to go any further in the alchemical interpretation of Dee's symbol and to derive from the text any information on the practical, or psychological, application of Dee's hermetic doctrine. ... One practical application of the symbol of his doctrine is, however, indicated in the text.  Theorem XXIII contains eleaborate instructions for the mathematical construction of the symbol of the monad, each part of which has to be of a size that is in strict numerical proportion to the size of every other part. ... These instructions are intended for the benefit of readers who would like to bear the symbol of the monad "on rings or seals, or to use it in other ways".  They are addressed to the Mechanicus, i.e. the goldsmith or engraver, who manufactures such articles. ... It seems, therefore, reasonable to assume that Dee attributed to his symbol a power similar to that of other magic characters which were supposed to perpetuate stellar influences, beneficial or maleficent, when they were engraved, on metal or others materials, at astrologically suitable times."

     19.  "The symbol of the monad," says Josten, "as it appears on the title-page and in the illustrations of the text, is essentially the common alchemical and astronomical sign of Mercury (d) to which the common sign of the first division of the zodiac, Aries (q)has been added at the bottom.

The half-circle and the circle forming the upper part of the common Mercury symbol are represented as intersecting so as to convey the idea of a conjunction of Moon and Sun.  Besides, the lunar half-circle  has been enlarged into a crescent, and a central point has been added to the solar circle, in order to achieve complete identity of those upper elements of the monad symbol with the common signs of Moon (s)and Sun (a).  The central point of the solar circle symbolizes also the Earth around which the Sun, the Moon, and the other planets, revolve; it is what Dee calls the terrestrial centre of the monad.  The sign of Aries appended to the cross at the bottom of the Mercury symbol, thus modified to suit Dee's intentions, is the first of three signs in the zodiac which the astrologers assigned to the element of fire, the so-called fiery triplicity (Aries, Leo, Sagittarius).  Its addition to the symbol of the monad is intended, as Dee states expressly, to signify that in the work of  his monad ... the aid of fire is required.  If one leaves aside all refinements and all secondary interpretations with which Dee so confusingly invests his concept of the monad, the most general and obvious idea conveyed by its symbol is, therefore, that of the alchemical process:  Mercury, i.e. the philosopher's mercury, is seen as being activated by alchemical fire ... " (p. 102-103).  Further, Josten observes (p. 110):  "The signs of the Sun, of the Moon, of Aries, and the cross of the elements (i.e. the component parts of the monad symbol which had been derived from the straight lines and parts of the circumference aof  circle) are shown to be the component parts also of the five remaining planets.  Their shapes, inasmuch as they are at the same time the signs of metals (lead, tin, iron, copper, mercury) are subjected to a symbolical analysis bristling with alchemical allusions whose interpretation defeats the powers of the present writer."  On p. 100, Josten speaks of Dee's claim that "the power of a cosmic symbol invented by himself could seem to make the astronomers' work superfluous."  And indeed, in his dedication to King Maximilian, Dee says (p. 131):  "And will not the astronomer be very sorry for the cold he suffered under the open sky, for [all his] vigils and labours, when here, with no discomfort to be suffered from the air, he may most exactly observe with his eyes the orbits of the heavenly bodies under [his own] roof, with windows and doors shut on all sides, at any given time, and without any mechanical instruments made of wood or brass?"  So much for the astrologers' use of astronomical observations, of the sort that Dee had called for in his work about the reformation of astrology, the Propaedeumata aphoristica, involving a gathering of a huge amount of data with a very demanding precision.  I will indulge in a conjecture here.  Dee turned to what we see in his Monas hieroglyphica when he realized, or faced up, to the immense difficult, or indeed the impossibility, of carrying out this observational program.  He was searching for a method of finding out all about celestial influences on human affairs in, as he indicates, his own home, by contemplating what he took to be a powerful symbol which somehow contained all he wanted to know, which was commensurate in his mind, it seems, with all there is to know.

     20.  I have come across (9 Nov 2000) another translation of the Monas hieroglyphia, this one published in 1947 by J. W. Hamilton-Jones, under the title The Hieroglyphic Monad and with a commentary by him.  In the translation with analytical introduction published in 1964 by C. H. Josten, discussed above, Josten says (loc. cit. p. 85) of the translation by Hamilton-Jones and another translation into French:  "There are two translations of the Monas Hieroglyphica by modern occultists: [E. A.] Grillot de Givry, Jean Dee de Londres, La Monade Hiéroglyphique, Paris, 1925; it leaves out Dee's letter to the printer; J. W. Hamilton Jones, The Hieroglyphic Monad, London, 1947.  The second of these is partly based on the first, and leaves out the important letter of dedication to King Maximilian and the letter to the printer.  Neither translation may be regarded as accurate."  Josten gives no examples of such inaccuracies.  I will make some comparisons between parts of Josten's and Hamilton-Jones's translation.  To begin with, Dee wrote this work in the form of a sequence of numbered paragraphs, each of which he calls a "theorem".  However, Dee gives no proofs of these theorems, so the term "theorem" loses the force it has in such a work as Euclid's Elements.  The structure of Euclid's Elements, already described before Euclid in Aristotle's Prior Analytics, furnishes a pattern which has been followed since been followed by many mathematicians and many others who seriously want to present their work in the manner of Euclid.  In modern terminology, one often distinguishes between statements which are to be proved or disproved, called "propositions", and statements which have been proved according to principles of logic and axiomatic procedure, called "theorems".  A well-formed statement in a formal system of the sort Euclid gave becomes a "theorem" only after a proof has been furnished for it.  This already may lead one to wonder about the status of John Dee as a mathematician.  Certainly, mathematicians and logicians will wonder, and I conjecture that some such people wondered already during Dee's lifetime, whatever some of his influential acquaintances at Elizabeth's court may have thought.

     21.  Theorem I of the Monas reads in Josten's translation:  "The first and most simple manifestation and representation of things, non-existent as well as latent in the folds of Nature, happened by means of straight line and circle."  In Hamilton-Jones's translation, it reads:  "It is by the straight line and the circle that the first and most simple example and representation of all things may be demonstrated, whether such things be either non-existent or merely hidden under Nature's veils."  For those that read Latin, the original version of Dee's Theorem 1 is (as given by Josten in a reproduction):  "Per Lineam rectam, Circulumque, Prima, Simplicissimaque fuit Rerum, tum, non ex[s]istentium, tum in Naturae latentium Inuolucris, in Lucem Productio, representatioque" (Josten, p. 154).  I am indebted to Gareth Prosser and Richard Kay of the MEDIEV-L e-list on the Internet for the following translations, which seem to me to be better than those of Josten and Hamilton-Jones:  (Prosser) "The first and simplest way of producing and representing into the light things either not existing or hiding in the concealed parts of nature, is by a straight line and a circle"; Kay (taking account of tum ... tum) "The first and most simple way to lead forth into the light, and to represent, things that first did not exist and then were hidden under the veil of nature [or "were latent in nature's envelope"] was by means of the straight line and the circle."  One might want the definite article "the" instead of the indefinite "a", and put "the straight line and circle" in the last phrase, but that's debatable.  I like the more literal "representing into the light" or "lead forth into the light" better than the "demonstrated" of Hamilton-Jones and "happened" of Josten on account of what Dee thought about the nature of light,  as he learned about it, mainly, from works of Robert Grosseteste and Roger Bacon (see paragraphs 13 and 16 above).  Such are the perils of language translations.  In any case, one may ponder how "non-existent" things can be made to appear by means of straight lines and circles.  There seems to be an echo here of the first book of Euclid's Elements, an even cursory reading of which will reveal that Euclid bases his geometry on straight lines and circles.  One may be reminded also of Proclus's commentary on the first book of the Elements.  But one will not find in Euclid or Proclus any indication that the problem of generation or coming-to-be, to use English versions of Aristotle's terms, is solved by postulating that everything comes from straight lines and circles, even if all of Euclid's plane geometry is based on these geometrical figures, be they ideal or constructed mechanically.  So it seems that at the very beginning of Dee's Monas hieroglyphica we have a questionable "theorem", which would be more appropriately called an axiom.  Indeed, many of Dee's "theorems" are in the form of what have considered to be axioms or postulates, since at least the time of Aristotle.  Euclid famously makes very clear a distinction between axioms and postulates on the one hand, and theorems on the other hand.

     21.  Let's see what we can do with Dee's Theorem II.  In Josten's translation, this reads:  "Yet the circle cannot be artificially produced without the straight line, or the straight line without the point.  Hence, things first began to be by way of a point, and a monad.  And things related to the periphery (however big they may be) can in no way exist without the aid of the central point."  Hamilton-Jones has:  "Neither the circle without the line, nor the line without the point can be artificially produced.  It is, therefore, by virtue of the point and Monad that all things commence to emerge in principle.  That which is affected at the periphery, however large it may be, cannot in any way lack the support of the central point."  The original Latin is:  "At nec sine Recta, Circulus, mec sine Puncto, Recta artificiose fieri potest.  Puncti proinde, Monadisque ratione, Res, & esse coeperit primo; Et quae peripheria sunt affectae (quantaecumquae fuerint), nullo modo carere possunt Ministerio."  One notes first that Josten has the "things" being however big, whereas Hamilton-Jones has the periphery being however large.  However, let's not dwell on that.  More interesting for present purposes is the comment Hamilton-Jones makes on this passage.  He says (p. 54-55):  "Here the principle of a point within a circle is established, from which all parts of the circumference are equidistant.  Dee says the central point supports the periphery.  Let us consider a sphere, which equally depends upon the support of a central point, and we shall see immediately that we have produced the seventh Platonic body.  You will say that you were taught to be cautious and that you were informed that there are only five.  Think again, Brother!  The sphere is certainly one of them, because all the five popular shapes are perfectly regular when placed within the sphere, and you are going to ignore the point, particularly when you have been told to find it on the centre?  We suggest the five Plaonic solids, regular bodues, forms, etc. are in reality incomplete without the point and the sphere.  In this theorem, Dee shos that although the point and the Monad are identical, nevertheless in manifestation the Monad is an extension of the point in a peculiar occult fashion which he proceeds to develop."  

     22.  It's enough to make a mathematician cry, or at least cry uncle.  Euclid  proceeds by way of definitions, and there is no consistent way that a sphere can be regarded as a regular "solid", i.e. a polyhedron with place faces.  If we look at Plato's Timaeus, we might be willing to adjoin to Plato's theory that four of the regular solids (terahedron, cube, octahedron, icosahedron) are fundamental or primary figures in nature, corresponding to the four elements or "bodies" (fire, air, earth and water) a proposition that points and spheres are fundamental figures of the same basic status.  However, this poses problems if we are to adhere to Plato and Euclid.  In connection with the fifth regular solid (and there are only five regular solids, as Euclid shows in his Elements), F. M. Cornford says in his commentary on Plato's Timaeus (Plato's Cosmology, 1937, p. 218-219):  "The dodecahedron is not constructed [by Plato in the way he constructs the other four].  Plato knew that the pentagonal faces cannot be formed out of either of his two elementary triangles [picked as basic by Plato] ... Not requiring a dodecahedron with plane faces for any primary body [the four elements], the Demiurge [Plato's creator divinity] 'uses it for the whole', i.e. for the sphere, to whcih this figure approaches most nearly in volume, as Timaeus Locrus remarks."  Plato's exact words on this topic, as translated by Cornford, are brief:  "There still remained one construction, the fifth; and the god used it for the whole, making a pattern of animal figures thereon."  It is a matter of conjecture whether or not Plato intended to identify the dodadecahedron with the sphere in some way.  Cornford alludes to the proposal that a model of a dodecahedron made of leather could easily be deformed into (a model of) a sphere, although of course these would not be the abstract or ideal dodacahedron and sphere treated by Euclid.  If such an identification is intended by Plato, then Hamilton-Jones is not right to adjoined the sphere to the five regular solids as being (somehow) an additional "regular solid".  If such an identification is not intended by Plato, but rather Plato only intended to account for the fifth solid (for which there was no corresponding primary element available), then Hamilton-Jones is arbitrarily calling a sphere an additional "regular solid" by making a kind of pun on "regular":  a sphere exhibits certain "regularities", but not in the sense that the Platonic or Euclidean regular solids do, with their plane faces.  Thus Hamilton-Jones is, in effect, redefining "regular" so as to include spheres along with cubes, etc., in the same class.  To a mathematician, this brings to mind countless problems involving the relationship of curved figures, such as the surface of a sphere, to plane figures, such as the faces of a polyhedron.  As Archimedes well knew, and well dealt with (by approximating circles with polygonal figures), the difficult number pi, the ratio of a circumference of a circle to its diameter, is involved here.

     23.  There are 24 "theorems" in Dee's Monas hieroglyphica.  It seems useless to go on criticizing Dee's work from the point of view of mathematics and its history.   Even what I have done so far might be regarded as picayune pedantry if it weren't for the fact that a good many people have asserted and still assert that John Dee was a great mathematician of his time, partly on the basis of Dee's preface to an edition of Euclid's Elements (see paragraphs 12 and 15 above).  As I discussed earlier, J. L. Heilbron has called Dee's reputation as a mathematician into question.  Frances Yates and other members of or people influenced by the Warburg Institute of London, and earlier people such as Johnson and Taylor (see paragraphs 3 through 9 above), and others, particularly literary critics and theorists, and occultists, developed or propagated in numerous works theories to the effect that the sciences of the period from Copernicus and Galileo to that of Newton's time were much affected by neo-Platonic, Cabalistic and Hermetic theories of the Renaissance, and indeed that the sciences in question somehow grew out of the latter kinds of theories.  I have discussed this above; for example, in paragraph 10.  There, among other things, I quote Yates as suggesting that Dee's religion was perhaps not that different from Newton's religion.  This is questionable.  However, I think it is unquestionable that Dee's mathematics was very different from that of Newton.