introduction, it would be a more valuable bit of evidence as to his

attitude toward astrology than anything that we have at present. His surely genuine mathematical works, as edited by Bubnov, consist solely of a short geometry and a few of his letters in which mathematical topics, mainly the abacus, are touched upon. His contemporary and disciple, the historian Richer, tells in the well-known passage[2808] how Borellus, “the duke of Hither Spain,” took Gerbert as a youth from the monastery at Aurillac in Auvergne back with him across the Pyrenees and entrusted his education to Hatto, bishop of Vich, in the north-eastern part of the peninsula. Whether Gerbert studied Arabic or not Richer does not state. Since he is still described as _adolescens_ when the duke and bishop take him with them to Italy and leave him there with the pope, one would infer that he probably had not engaged in the work of translation from the Arabic. Another almost contemporary writer, alluding very briefly to Gerbert, makes him visit Cordova, but is perhaps mistaken.[2809] Richer does, however, state that Berbert especially studied _mathesis_, a word which, as various medieval writers inform us, may mean either mathematics or divination. Apparently Richer uses it in the former sense, for later he mentions only Gerbert’s achievements in arithmetic, geometry, music, and astronomy.[2810] But Robert, king of France, 987-1031, whose teacher Gerbert had been, seems to refer to him as “that master Neptanebus” in some verses,[2811] a name which certainly suggests an astrologer, as well as an instructor of royalty, if not also a magician. [Sidenote: His posthumous reputation as a magician.] But Gerbert’s reputation for magic seems to start with William of Malmesbury in the first half of the twelfth century, who makes him flee by night from his monastery to Spain to study “astrology” and other arts with the Saracens, until he came to surpass Julius Firmicus in his knowledge of fate. There too, according to William of Malmesbury, “he learned what the song and flight of birds portend, to summon ghostly figures from the lower world, and whatever human curiosity has encompassed whether harmful or salutary.” William then adds some more sober facts concerning Gerbert’s mathematical achievements and associates.[2812] Michael Scot in his _Introduction to Astrology_ in the early thirteenth century speaks of a master _Gilbertus_ who was the best nigromancer in France and whom the demons obeyed in all that he required of them day and night because of the great sacrifices which he offered and his prayers and fastings and magic books and great diversity of rings and candles. Having succeeded in borrowing an astrolabe for a short time he made the demons explain its purpose, how to operate it, and how to make another one. Later he reformed and became bishop of Ravenna and pope.[2813] In a manuscript early in the thirteenth century is a statement that Gerbert became archbishop and pope by demon aid and had a spirit enclosed in a golden head whom he consulted as to knotty problems in composing his commentary on arithmetic. When the demon expounded a certain very difficult place badly, Gerbert skipped it, and hence that unexplained passage is called the _Saltus Gilberti_.[2814] [Sidenote: An anonymous astronomical treatise; its possible relation to Gerbert.] In a manuscript in the Bodleian library which seems to have been written early in the twelfth century[2815] is an astronomical treatise in four books which Macray suggested might be the _Liber de planetis et mundi climatibus_ which Ethelwold, bishop of Winchester from 963 to 984, is said to have composed.[2816] The present treatise indeed embodies a _Letter of Ethelwold to Pope Gerbert_ on squaring the circle.[2817] It seems, however, that this letter on squaring the circle was really written by Adelbold, bishop of Utrecht from 1010 to 1027.[2818] Adelbold speaks of himself in the letter as a young man[2819] and of course wrote it before Gerbert’s death in 1003, and very probably before Gerbert became Pope Silvester II in 999. But he could scarcely have written the letter early enough to have it included in a work written by Ethelwold who died in 984. Our astronomical treatise in four books is therefore not by Ethelwold, unless the letter be a later interpolation, but it is possibly by Adelbold or by Gerbert.[2820] Its opening words, “Quicumque mundane spere rationem et astrorum legem ...,” are similar to those of the treatise on the uses of the astrolabe which has often been ascribed to Gerbert, “Quicumque astronomice peritiam discipline....”[2821] [Sidenote: Contents of its first two books.] Our treatise then may be by Gerbert or it may be a specimen of the astronomy of the eleventh or early twelfth century. As it appears to be little known and never to have been published, it may be well to give a brief summary of its contents. An introductory paragraph outlines some of the chief points with which the treatise will be concerned, such as the twelve signs of the zodiac, their positions, “most varied qualities,” the reasons for their names, and the diverse opinions of gentile philosophers and Catholics as to their significations; the four elements; and the seven planets. In the text which follows, these topics are considered in rather the reverse order to that in which they were named in the preface. After some discussion of “the founders of astronomy and the doctors of astrology,” the first book is occupied with a description of the sphere or heavens. The second book is largely geographical, beginning with the question of the size of the earth, the zones, the ocean, and how to draw a T map. This geographical digression the author justifies in the prologue to his third book by the statement that often the position of the stars can be determined from the location of countries, and that if the habitat of peoples is known one can more easily arrive at the effect of the stars.[2822] [Sidenote: Attitude towards astrology.] This suggests that the author believes in astrological influence, and in the two following books he states a number of astrological doctrines, not, however, as his own convictions but as the opinions of the _genethliaci_ or astrologers, or “those who will have it that prosperity and adversity in human life are due to these stars.”[2823] On the other hand, he seldom subjects the astrologers to any adverse criticism. Indeed, early in the third book, he states that the belief of the _genethliaci_ that human wealth and honors, poverty and obscurity, depend upon the stars, pertains to another subject than that which he is at present discussing; namely, prognostication, concerning which he will treat fully in later chapters. But I cannot see that he fulfills this promise in the present manuscript, which seems to end rather abruptly,[2824] so that possibly there is something missing. In the previous passage, however, he immediately proceeded to admit that the sun and moon greatly affect our life and to tell further how it is connected with the other five planets. In the star of Saturn the soul is said to busy itself especially with reasoning and intelligence, logic and theory. Jupiter is practical and represents the power of action. Mars signifies animosity; Venus, desire; Mercury, interpretation. Men have proved the moon’s moist influence by sleeping out-of-doors and finding that more humor collected in their heads when they slept in the moonlight than when they did not.[2825] After mentioning the twelve signs, “through which the aforesaid planets revolving exert varied influences, and even, according to the _genethliaci_, make a good man in some nativities and a bad man in others,”[2826] the author goes on to tell which signs are masculine and which are feminine, to relate them to the four cardinal points and to the four elements, to define the twenty-eight mansions and their distribution among the twelve signs and seven planets,[2827] and to tell how the planets differ in quality.[2828] All this is providing at least the basis for astrological prediction. [Sidenote: The fourth book.] The fourth book of the treatise is mainly taken up with descriptions and figures of the constellations, concerning which the author often repeats the fables of antiquity. After discussing the six ages of the world, the author intended to insert a figure on what is the next to last page of the present text to show “the harmony of the elements, climates of the sky, times of the year, and humors of the human body,” for, as he goes on to say, man is called a microcosm by the philosophers. This missing figure or figures would have been analogous to those which Wickersheimer investigated in the early medieval manuscripts in the libraries of France. [Sidenote: Citations: Arabic names.] Our author does not make many citations, but among them are Eratosthenes,[2829] Aratus, Ptolemy, Macrobius, and Martianus Capella. Some of these authors are perhaps known to him only indirectly, and he seems to make use of Isidore and Pliny without mentioning them. He shows, however, an acquaintance with foreign languages, listing the seven heavens as “oleth, lothen, ethat, edim, eliyd, hachim, atarpha,” and giving Greek, Hebrew, and “Saracen” names for the seven planets, as well as a “Similitudo,” or corresponding metal, and “Interpretatio,” or quality such as “Obscurus, Clarus, Igneus.”[2830] He also gives the Arabic names for the twenty-eight mansions into which the circle of the zodiac subdivides.[2831] We now turn to another treatise, found in tenth and eleventh century manuscripts, in which Arabian influence is apparent. [Sidenote: The _Mathematica_ of Alchandrus or Alhandreus.] William of Malmesbury, writing in the first half of the twelfth century concerning Gerbert’s studies in Spain, says, probably with a great deal of exaggeration, that Gerbert surpassed Ptolemy in his knowledge of the astrolabe, Alandraeus in his knowledge of the distances between the stars, and Julius Firmicus in his knowledge of fate.[2832] It is rather remarkable that a work ascribed to Alhandreus or Alcandrus, “supreme astrologer,” should be found in two manuscripts of the eleventh century[2833] in both of which occurs also the work on the astrolabe which is perhaps by Gerbert, while in one is found also the _Mathesis_ of Julius Firmicus Maternus. Alchadrinus or Archandrinus is cited in Michael Scot’s long _Introduction to Astrology_ as the author of a “book of fortune making mention of the three _facies_ of the signs and the planets ruling in them,” and Michael adds that a similar method of divination is employed in general among the Arabs and Indians as can be seen in the streets and alleys of Messina where “learned women” answer the questions of merchants.[2834] Peter of Abano in his _Lucidator astronomiae_,[2835] written in 1310, mentions Alchandrus as a successor of Hermes Trismegistus in the science of astronomy but as flourishing before the time of Nebuchadnezzar. Alchandrus was probably scarcely as ancient as that, but the treatise ascribed to him also exists in Latin in a manuscript of the tenth century,[2836] and seems to be a translation from the Arabic. In any case it is full of Arabic and Hebrew words, and professes to cite the opinions of Egyptians, Ishmaelites, and Chaldeans in general as well as those of Ascalu the Ishmaelite and Arfarfan or Argafalan or Argafalaus[2837] the Chaldean in particular. Since the name Alchandrus or Alhandreus is found so far as I know in no historian or bibliographer of Arabian literature or learning,[2838] we shall treat somewhat fully of the work and its author here. [Sidenote: An account of its contents.] The “Mathematic of Alhandreus, supreme astrologer,” as it is entitled in one manuscript, opens somewhat abruptly with a terse statement of the qualities of the planets. Two estimates of the number of years between creation and the birth of Christ are then given, one “according to the Hebrews,” the other “according to others.”[2839] There follow letters of the Greek alphabet with Roman numerals expressing their respective numerical values, perhaps for future reference in connection with some sphere of life or death. Next is considered the division of the zodiac into twelve signs for which Hebrew as well as Latin names are given. The movements of the planets through the signs are then discussed, and it is explained in the usual astrological style that Leo is the house of the sun, Cancer of the moon, while two signs are assigned to each of the other five planets. Every planet is erect in some one sign and falls in its opposite, and any planet is friendly to another in whose house it is erect and hostile to another in whose house it declines. Presently the author treats of “the order of the planets according to nature and their names according to the Hebrews,”[2840] and then of their sex and courses, which last leads to considerable digressions anent the solar and lunar calendars.[2841] Then the twelve signs are related to the four “climates” and elements. [Sidenote: Astrological doctrine.] All this implies a favorable attitude to astrology, and the author has already expressed his conviction more than once that human affairs are disposed by the seven planets according to the will of God.[2842] Since man like the world is composed of the four elements it is no false opinion which persuades us that under God’s government human affairs are principally regulated by the celestial bodies.[2843] To make this plainer the author proposes to insert an astrological figure “which Alexander of Macedon composed most diligently,” and which presumably would have been of the microcosmus or Melothesia type, but the space for it remains blank in the manuscript. Next comes a paragraph on the sex of the signs and their rising and setting, and then lists of the hours of the day and night governed by the signs and by each planet for all the days of the week.[2844] [Sidenote: Nativities and name-calculations.] Then we read, “These are the twenty-eight principal parts or stars (i.e. constellations) through which the fates of all are disposed and pronounced indubitably, future as well as present. Anyone may with diligence forecast goings and returnings, origins and endings, by the most agreeable aid of these horoscopes.”[2845] These twenty-eight parts are of course the sub-divisions of the zodiac into mansions of the sun or moon which we have already encountered, and Arabic names are given for them beginning with _Alnait_, the first part of the sign Aries. First, however, we are instructed how to determine under which one of them anyone was born by a numerical calculation of the value of his name and that of his natural mother similar to that of the spheres of life and death except that it is based upon Hebrew instead of Greek letters.[2846] Then follow statements of the sort of men who are born under each of the twenty-eight mansions, their physical, mental, and moral characteristics, and any especial marks upon the body,—either birth-marks or inflicted subsequently by such means as hot irons and dog-bite,—their health or sickness, term of life, and manner of death,—which in the case of Alnait, the first mansion, will be “by the machinations or imaginations of the magic arts.”[2847] Also the number of their children is roughly predicted. [Sidenote: Interrogations and more name-calculations.] Next is discussed the course of the planets through the signs, the houses of the planets, and their positions in the signs at creation.[2848] The author then turns to the influence of the planets upon men and gives another method of numerical calculation of a man’s name in order to determine which planet he is under.[2849] Under the heading “Excerpts from the books of Alexander, the astrologer king,”[2850] directions are given for the recovery of lost or stolen articles and descriptions of the thief are provided for the hour of each planet. The letter of Argafalaus to Alexander instructs how to read men’s secret thoughts as Plato the Philosopher used to do, and how to tell what is hidden in a person’s hand by means of the hours of the planets.[2851] After some further discussion of astrological interrogations the manuscript at the British Museum closes with the Breviary of Alhandreus, supreme astrologer[2852], for learning anything unknown by a method of computation from Hebrew and Arabic letters. [Sidenote: Alchandrus or Alhandreus not the same as Alexander.] Someone may wonder if the names Alhandreus and Alchandrus may not be mere corruptions of Alexander who is cited and quoted even more than has yet been indicated[2853], and if some careless head-line writer has not inserted the name _Alchandri_ or _Alhandrei_ instead of _Alexandri_ in the _Titulus_. But this would leave the statements of William of Malmesbury and of Peter of Abano to be explained away. Or, if it is argued that the name of Alhandreus should be attached only to the Breviary, it must be remembered that in the earliest manuscript, which does not contain the Breviary, the treatise is none the less called the Book of Alchandreus. As a matter of fact there is found also in the manuscripts a “Mathematica Alexandri summi astrologi,”[2854] but while the title is the same, the contents are different from the “Mathematica Alhandrei summi astrologi.” However, the treatise itself is found together with the _Mathematica Alhandrei_ in a tenth century manuscript.[2855] But no author is mentioned, and instead of _Mathematica_ the title reads “Incipiunt proportiones cppfcfntfs knkstrprx indxstrkb,” which may be deciphered as “Incipiunt proportiones competentes in astrorum industria.”[2856] Possibly therefore this treatise is a part of the work of Alchander, and the title _Mathematica Alexandri_ is an error for _Mathematica Alhandrei_. [Sidenote: Alkandrinus or Alchandrinus on nativities according to the mansions of the moon.] Moreover, in later manuscripts we encounter authors with names very similar to Alchandrus and works by them of the same sort as that we have just considered. In a fifteenth century manuscript at Oxford we find ascribed to Alkandrinus an account of the types of men born in each of the twenty-eight mansions of the moon[2857] such as we have seen formed a part of the _Mathematica Alhandrei_. And in a fifteenth century manuscript at Paris occurs under the name of Alchandrinus what seems to be a Christian revision of that same part of the _Mathematica Alhandrei_.[2858] What appears to be another revision and working over of this same discussion of nativities according to the twenty-eight mansions of the moon[2859] appeared in print a number of times in the sixteenth and seventeenth centuries and in French and English translations as well as Latin. The author’s name in these printed editions is usually given as Arcandam, but the English edition of 1626 adds “or Alchandrin.”[2860] [Sidenote: Albandinus.] Two other manuscripts at Paris[2861] contain under the name of Albandinus a “book of similitudes of the sons of Adam, fortunate and unfortunate, of life or death, according to nations, that is, their nativities according to the twelve signs.” The treatise opens with a method of calculating a person’s nativity from the letters in his own and his mother’s name similar to that which occurs in the course of the _Mathematica Alhandrei_, but then applies it directly to the twelve signs rather than to the twenty-eight mansions of the moon. It also does not bother with the Hebrew alphabet but gives numerical equivalents directly for the Latin letters. Some treatise by Albandinus on sickness and the signs in a manuscript at Munich[2862] is perhaps identical with the foregoing. [Sidenote: Geomancy of Alkardianus or Alchandiandus.] To an Alkardianus or Alchandiandus is ascribed a geomancy,[2863] and since it also is arranged according to the twenty-eight divisions of the zodiac with 28 judges and 28 chapters each consisting of 28 lines in answer to as many questions, it would seem almost certain that it is by the same author who treated of the influences of the 28 houses or _facies_ of the twelve signs upon those born under them. Moreover, this Alkardianus or Alchandiandus states in his preface that he has composed certain books on the dispositions of the signs and the courses of the planets and on prediction of the future from them. “But since moderns always rejoice in brevity,” he has added this handy and rapid geomantic means of answering questions and ascertaining the decrees of the stars. The 28 tables of 28 lines each of this Alkardianus or Alchandiandus are identical with one of the two such sets[2864] commonly included in the _Experimentarius_[2865] of Bernard Silvester, a work of geomancy which he is said to have translated from the Arabic.[2866] He lived in the twelfth century and will be the subject of one of our later chapters. [Sidenote: An anonymous treatise or fragment of the tenth century.] It still remains to speak of a portion of our tenth century manuscript at Paris which begins, after the book of Alchandrus seems to have concluded, with the words, “Quicunque nosse desiderat legem astrorum....”[2867] This _Incipit_ is so similar to that of the twenty-one chapters on the astrolabe, “Quicumque astronomiam peritiam disciplinae ...” and to that of the four books of astronomy, “Quicumque mundane spere rationem et astrorum,” that one is tempted to imply some relation between them, and, in view of the tenth century date of the one at present in question, to connect it like the others with the name of Gerbert. Our present treatise or fragment of a treatise is largely astrological in character, “following for the present the wisdom of the _mathematici_ who think that mundane affairs are carried on under the rule of the constellations.” This refusal to accept personal responsibility for astrological doctrine is similar to the attitude of the author of the four books of astronomy, so that perhaps the present text is the missing fragment required to fulfil his promise to treat of the subject of prognostication in later chapters. If so it indulges in some repetition, as it goes into the relations existing between signs, planets, and elements, and gives the “Saracen” names[2868] for the twenty-eight mansions of the moon. It includes a way to detect theft for each planet and a method of determining if a patient will recover by computation of the numerical value of the letters in his name. These features are suggestive of the _Mathematical_ of Alchandrus.