DIOGO HOMEM ATLAS MS 501 INVESTIGATED MORGAN LIBRARY AND MUSEUM, NEW YORKINTRODUCTION
This atlas is held in the Morgan Library and Museum, reference MS M501 and was purchased by J Pierpont Morgan from Leo S Olschki, a publisher in Florence Italy in 1912. It is a bound atlas comprising 7 folios of which 6 folios are charts and the first of the seven is a cosmological wheel dated to 1560. This is the date that the library has stated for the atlas but M. Destombes correctly states that the Leningrad atlas of Diogo Homem also has a cosmological wheel dated 1560, and as with other atlases by Diogo Homem these Wheels are reused the date is more likely to be 1565. It is this date that I will be using in the text.
BIOGRAPHICAL INFORMATION
In 1946 the library produced a document describing the atlas and its provenance. There was a slight perturbation regarding the attribution as at first the reading of the Initials within the atlas was O.F.D.H, translated as Opus Fecit Diogo Homem, when it is actually C.F.D.H, Carta Fecit Diogo Homem and as my illustration shows there is a serif clearly visible on the letter C and thus this is the correct reading. There is a handwritten note on the front inner cover regarding a text by C Melzi d’Eril which is discussed later.
Similarly the extraordinary text to be found on the inside rear cover regarding planetary distances is discussed in the appendix text.
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ChJPM/1/D02
ChJPM/1/D03
THE ATLAS FOLIOS IN DETAIL
FOLIO 1, COSMOGRAPHICAL WHEEL
Dated 1560 on the section between December and January which effectively removes 8 days from the calendar it is reconciled as follows. In ancient Greece the year was based upon 12 lunar months of 29 ½ days giving 354 days when the actual year is 365 ¼ days. Thus this calendar is for 357 days and it illustrates the great problems early astronomers had in reconciling the lunations with the solar year. However in Babylonia c432BCE their mathematicians calculated that 7 years of 13 lunar months followed by 12 years of 12 lunar months would be almost exactly 19 solar years and this later became known as the Metonic Cycle. Hence we see in these wheels and tables the 19 lunar set against the 12 solar.
This wheel was gradually phased out as the Calendar became “Gregorian” with three equal years and one having an extra day to realign the year to the solar cycle. The fact that the Cosmological wheel was phased out can be shown by the change of 11 days for the Spring Equinox as this wheel indicates March 10th for the sun entering Aries and in 1582 that Equinox was moved to 21st march as we acknowledge today.
Curiously the two rectangular tables included have on the second with the lunar years 11 to 19 the Star signs indicated by their “figurative” drawings and a meteorological notation repeating three times with the following; “Hot and Dry”; “Cold and Dry”; “Hot and Humid” and “Cold and Humid”, bearing no reason as they are nonsense.
However as it is dated 1560 it is quite probable that the table actually refers to 1558/59 and is thus usable for 12 years.
ChJPM/1/D04
FOLIO 2, THE AEGEAN SEA
This chart is one of two larger scaled charts and illustrates the whole Aegean Sea from Crete in the south to the Dardanelles in the north. The visible scale bar in the south gives 28 divisions for 4 degrees, 23E to 27E and thus each division may be considered 10 miliaria or 70 miliaria per degree. From 25N to 40N it is 46 ½ units, which equates to 93 miliaria per degree, a reasonable unit and normal latitudinal measure here for a portolan chart. The perfect measures at 36N would be latitudes 90 miliaria and longitudes 72 miliaria, hence that is probably the intention.
ChJPM/1/D05
FOLIO 3, THE ADRIATIC SEA
Drawn from the Gulf of Venice to the Strait of Otranto and the 40N latitude, it has a scale bar equal to the previous chart but unfortunately they do not link together. The actual geographical graticule is distorted, whereas the Aegean graticule was acceptably rectangular. Thus the units per degree here are widely different and I suspect the Gulf of Venice has been deliberately over represented as it is a Venetian Atlas! However it is a beautiful chart.
ChJPM/1/D06
FOLIO 4, THE BLACK SEA & EASTERN MEDITERRANEAN
This is the first of the four folios which are the same scale and will if joined ( see later) form a portolan chart. The geographic graticule is rectangular and although varies, that is in accord with a standard portolan chart in this area, produces an overall c84 x c70 Sbu’s graticule. The actual graticule should be 90 x 72, but as often stated the scale bars accuracy is not always perceived.
ChJPM/1/D07
FOLIO 5, CENTRAL MEDITERRANEAN SEA
It is here on a standard portolan chart that we normally note the change from 75 Sbu’s latitude in Iberia to a gradually increasing number latitudinally, sometimes as high as 100 Sbu’s per degree. This chart is more restrained in its change which increases from 78 in the west to 83 1/3rd in the east with a longitudinal measure of 71 Sbu’s. The scale bar is however adequate in length to be utilised.
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FOLIO 6, BRITISH ISLES, FRANCE, IBERIA & N AFRICA
From 9W to 3E the average is 67 Sbu’s and from 36N to 60N it is 68 Sbu’s. But latitudinally from 36N to 58 ½ N, a distance geographically of 22 ½ degrees via the scale bar each degree is 72 Sbu’s. The isles of Britannia and Hibernia are drawn disproportionately large and the Atlantic coast of Europe above 48N longitudinally awry from 3W to 10E, the Jutland peninsula. The latitudinal scale bar reflects the compression of France and the elongation of the British Isles.
Thus the Iberian Peninsula is drawn square as is France which is the main reason for the 3W to 10E compression. However, it is obvious that the chart is drawn to ensure maximum geographical coverage and thus has been manipulated to produce the required chart format.
ChJPM/1/D09
FOLIO 7, IBERIA, N AFRICA AND THE ATLANTIC ISLES
This chart has an identical Iberia to folio 6, but, thence both the latitudes and longitudes are awry. The longitudes have been compressed thus the chart can encompass the Azores which stretch from 25W to 36W and are at 39N at their northerly point. But, the latitudinal scale bar is generally correct apropos the geography.
As an illustration of the West Atlantic from Iberia it is acceptable as it ignores any of the imaginary islands that plague other charts and concentrates on the geography.
The SW wind rose has on its eastern pointer the initials CFDH as previously indicated and being the last chart in the atlas no doubt has been used to record the cartographer.
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THE 1560 CHART BY DIOGO HOMEM AND THE ATLAS PAGES FOLIOS 4 TO 7.
Four folios have been joined to form an overall chart of the Mediterranean Sea basin and the Atlantic Coast plus islands. Latitudinally it is from 27N to 60N with the alignments as per the folios discussed. Longitudinally it is from 36W, the Azores to 42E and thus encompasses the Black sea.
With the exception of the Azores being included it probably represents the 1560 chart by Diogo Homem held in the Biblioteca Marciana, Venice, produced immediately prior to the atlas.
ChJPM/1/D11 & 12 & 13
ChJPM/1/D14
COMPARISON OF THE 1560 CHART TO THE COMPOSITE
ChJPM/1/D15 & 16 & 17
The three diagrams which form a complete chart indicate that the latitudinal scales are meant to be the same and when the actual coastlines are superimposed they show a similarity which clearly indicates that the atlas was drawn from a pattern/template matching the portolan chart. That should be expected as an atlas cannot be drawn without such a reference and the minor misplacements show the difficulty of its production piece by piece.
APPENDIX PAPER
MS7; CURIOUS ADDITIONS TO TWO ATLASES DATED 1550 & 1565, NOTES AND PLANETARY CIRCUITS, DISTANCES AND DIAMETERS
ABSTRACT
The front cover lining has pencil notes appended regarding the previous ownership and attribution
The rear cover lining is an “Astronomical Text” which can be traced to an Italian Cosmographer, Companus of Novara (1220-1296) and Arab astronomers of the 9th century. The examination of those texts led finally to Roger Bacon and his Opus Majus.
The text is 12, A4 pages and has 3, A3 diagrams,
INTRODUCTION
As discussed above this atlas is adorned with notations on the inner covers, the front inner cover being pencil notes to aid the librarians and the rear inner cover a very strange item to be found adorning an atlas. That text is solely concerned with planetary lore.
THE BINDINGS AND THEIR TEXTS
Within the front binding we read of the text by C Melzi d’Eril which was published by Leo S Olschki in his house magazine called La Bibliofolia. There is also a note concerning M Destombes letter of Mars 15, 1959 where the error of the attribution is made.
The rear binding has the extraordinary text concerning the planets and their sizes and can be traced back to c1260 written in Venice. Rather than duplicate the text a separate appendix text is now included.
FRONT INNER
ChJPM/1/D02
The front inner contains a brief description of the provenance and then a note to see “Melzi d’Eril, Camillo, Di un altro portolano del sec XVI, la Bibliofilia, XIV (1912). This text commences by explaining that the “portolani” is part of the Collezione Olschki and that “La Bibliofilia Rivista dell’arte antica” is actually the house magazine of the Leo S Olschki publishing house which is extant today. The author provides a detailed description of the Cosmographical Wheel and tables; a short set of notes on the actual charts there-in and then a single page discussing the inner rear binding “Astronomical Text”. That text he compares to the Cosmography of Francesco Barozzi, printed in Venice by Gratioso Perchacino in 1607.
Camillo Melzi d’Eril (1851-1929) and was born in Pisa. He was a priest of the Barnabite Order from 1873 and a Professor of Mathematics and Natural Sciences. Francesco Barozzi (1537-1604) was an Italian Mathematician, Astronomer and Humanist. Born on Crete, educated at Padua, he was an independent scholar and his “Cosmography” is extant today in original form. However, as will be clearly shown this Additional Text is not his work.
The second note in pencil states; “Your MS M501 is most probably the work of Diogo Homem, as testify the writing and the letters OFDH (opus fecit Diogo Homem)”. M. Destombes at Marseilles-Corniche, in letter to Miss Greene, Mar 15 1959. I have dealt with this text in the main paper.
REAR INNER COVER
ChJPM/1/D18
ChJPM/1/D19
This is however a very different text and tabulation and having traced the origins, its provenance, but when it was added to the atlas is unknown and thus its raison d’être unexplained.
However I knew that on the circular chart drawn by Fra Mauro c1457 in the top left or south east corner there was a table of the distances and diameters of the planets taken from Companus of Novara. Fra Mauro has entitled it, “On the distance of the Heavens. Rubric.” It does contain several errors by the copyist, the one being that a mile measures 400 cubits and not 4000 cubits, which clearly indicates that the copyist did not know his errors.
Whilst researching this text I was informed of another copy which could be found in an atlas by Battista Agnese, dated 1550. It is actually a folio; the penultimate of 22 folios bound in a leather binding and is a beautifully written text. Compare it to the MS M 501 copy and it is obvious that it is a copy of the Agnese text and the errors, numerical, can be clearly seen.
The Planetary Text commences with three paragraphs which set down the basic medieval knowledge concerning the size of the earth and are now explained in detail.
PARAGRAPH ONE
Antiqui cognoverunt g centrum rotuditatis terre est centrum omnia spheras celesti et g unus gradus sphere terre conspondet uni gradui celi T habet proportionem ad ipsum T cum gradum terre mensuraverunt diligenter et invenerunt cum este miliaria 56 et duas tercias unius miliaris quod milisare costat ex 4000 cubits.
In antiquity it was understood that the centre of the earth was the rotation point for the celestial sphere. That is Claudius Ptolemy’s earth centric format as per his Almagest. The earth was measured diligently to establish 1° latitude as stated. In fact this is the Alfraganus quoted measurement carried out at the bidding of Caliph Al Ma’mum as the text ChMEA/1 fully describes. Alfraganus chose to accept this spurious measurement as noted in the text “Differantie in quibusdam collctis scientie astrorum”, section 8, “In fractionibus Mensure Terre et in divisione septum climatum eius habitabilum” as follows;
“—- inveniemus quoque post hoc quod portio unius gradus circuli ex rotunditate terre sit 56 miliariorum et 2/3 unius miliaria per miliarium quod est 4000 cubitorum per gradus equals secundun quod solicite probatum est in diebus Allmenon et convenerent super probationen eius sapientes plures numero”.
Alfraganus then states, 360 degrees will give the circle of the earth as 20400 miliaria with a diameter of 6500 miliaria. A rather unfortunate rounding off as is discussed later.
PARAGRAPH TWO
Cubitum autem sic invenerum posuerunt 6 grana ordei contigua in uno ordine et spatium quod dicta 6 grana ordei occupant apellaverunt polis et de 4 polis fecerunt palma et de 4 palmis fecerunt cubitum qui a quibusdam vocatur pes aliprandi et de cubitus 4000 fecerunt miliare cum co mensuranerunt omnia”.
This seeks to explain how the cubit length was arrived at by placing 6 grains of Barleycorn in one line and that will be a finger width. Then 4 fingers (24 barleycorn) equals a Palm and four Palms equal a Cubit; i.e. 6 x 4 =24 x 4 = 96 Barleycorns per cubit. The Mesopotamian methodology is; 6 hairs of a camel or horse = 1 barleycorn; 6 barley corn = 1 Asbaa or finger and 4 Asbaa = 1 Cabda or palm and 8 cabda = 1 coudee or cubit. That is 6x4x8 = 24×8 or 192 Barleycorn, and in this calculation the Barleycorn is used widthwise not lengthwise. Text MS2,”Follow the barleycorn; the beginnings of metrology” fully explains.
PARAGRAPH THREE
Cum ergo quilibet gradus terre sit miliaria 56 2/3 T sphere terre sit 360 gradus in circumferential maioris circuli eius si multiplica verimus 360 per 56 miliaria et duas tercius miliaris pervenient miliaria 20400 et sic patebit quod ambitus totius terre sive rotunditas eius esr 20400 miliaria et si hec miliaria diviserimus per 3 1/7 pervenient miliaria 6490 10/11 que sunt diametre sphere terre et si hanc diameter p diviserimus per 2 pervenient miliaria 3245 5/11 que sunt se midiametrum sphere terre id est distantia a centro terre ad superficium terre.”
It can be seen immediately that these figures are not those of Alfraganus who writes 6500 diameter and thus the above is an accurate mathematical summation. Unfortunately he chose the wrong measurement for a degree of latitude and we must look to Ahmad ibn Abdallah, known as Habash for the correct measure of 56 Arab miles of 4000 black cubits.
We can also read Al-Biruni’s comments upon these measurements as follows;
On the other hand it is related of Al-Farghani that he reported two thirds of a mile in addition to the 56 miles mentioned above. Similarly I found all the records confirming these additional 2/3rds and I may not attribute that to their having dropped out of the manuscript of the Kitab al-Abád wa-‘l-ajram because Habash derived from that value the circumference and diameter of the earth and all of the planetary distances. When I investigated these I found that they result from fifty-six miles only (that is without the 2/3rds). It is preferable to imagine that these different results for the length of one degree derive from two accounts by two teams.
In fact four teams were involved measuring, 56; 56 1/4; 56 2/3 and 57 Arab Miles. A full explanation of this must wait until the tables on the folio have been explained and commented upon.
THE TABULATED MEASUREMENTS
The tables, actually two only as the first is split across the centre of the folio are taken from the text, “Theorica Planetarum” by Companus of Novara (English translation and comments by F S Benjamin Jr. and G J Toomer) and have been compiled by a later astronomer for this text to be written. That is obviously so as the extraction of the data is a laborious exercise and they do not accord with the figures as originally written. They are also perhaps incomplete as they do not contain the “semi diameter corporis” and hence the diameters.
INTERLUDE
At this juncture I must make reference to the last line of the folio
“Miliare est divisum in 660 partes adeo quod totum est 660 peci” This is a complete mis-reading of the figures and is even misunderstood on page 26 of the English translation. That states; “The solemn calculation of all the planetary distances to 660ths of a mile is only the most blatant example of a ludicrous pedantry—–“
It is patently obvious that the figures for the first five items at the beginning of the table and then the first 3 items in the second schedule have been manipulated because Companus does not write 107936 400/660 but 107936 20/33. Thus Companus has realised that the multitude of actual fractions in his computations require to be reduced to the “Lowest common denominator” which is 660. Thus it is nothing to do with parts of a mile it is merely a mathematical construct to ensure all fractions appear co-ordinated. Each of the 7 simple fractions can be changed to a 660 base except for the 2/7ths division for Mercury.
WHERE DID COMPANUS OF NOVARA OBTAIN HIS INFORMATION FROM?
Mentioned already is the work of Alfraganus and Habash, but the text by Habash, “Book of Bodies and Distances” described in the text by Y Tzvi Langermann Centaurus 28, pp 108-128 (1985) where the author has collated the computations of Habash concerning the Earth, Moon and Sun into a table as follows;
Below are the results of Habash’s calculations of the sizes and distances on the basis of 1° = 56 Arab Miles. Where the result is expressed sexagesimally, we exhibit as it stands.
| circumference of Earth | 20160 |
| diameter of Earth | 6414.54 |
| radius of Earth | 3207;16,30 |
| diameter of “aether” (= closest distance of moon) | 215,208;9,9 |
| circumference of “aether” | 676,368;28,45,25,43 |
| radius of furthest distance of moon | 205,800;8,45 |
| diameter of furthest distance of moon | 411,600.216 |
| circumference of furthest distance of moon | 1,293,600.916 |
| diameter of orbit of sun | 7,761,605.5 |
| circumference of orbit of sun | 24,392,571.38 |
| diameter of sun | 35,280;1.30 |
| circumference of sun | 110,880;4,43 |
| Diameter of moon | 1886.8 |
| circumference of moon | 5927.025 |
| one degree along orbit of sun | 67,700.05 |
| one minute along orbit of sun | 1129.283 |
E S Kennedy: Survey of Astronomical Tables has the following note under Habash;
P. Miscellaneous.
There is a table (f.150) of the apparent diameters of the seven planets, to two places, for each six degrees of the anomaly, apparently from Hindu sources. (Cf. Surya-Siddhanta, p196). There is an interpolation table for lunar and solar distances, computed to two places, for each two degrees of the anomaly.
In the Surya-Siddhanta we read on page 195, 13) The diameters upon the moon’s orbit of Mars, Saturn, Mercury and Jupiter are declared to be thirty, increased successively by half the half; that of Venus is sixty.
On page 196 there is a short tabulation giving the planets in Yoyanas as 30, 37 ½ 45, 52 ½, 60; Then planetary distances in Miles
| modern | Surya-Siddhanta | error | |
| Mercury | 3032 | 3008 | -1 |
| Venus | 7523 | 4011 | -47 |
| Mars | 4218 | 3772 | -11 |
| Jupiter | 88748 | 41624 | -53 |
| Saturn | 74580 | 73882 | -1 |
Geocentric Orbital Circumferences
| planet | S-S in Yoyanas | S-s orbit radius in miles |
| Moon | 324000 | 258000 |
| Mercury | 4332000 | 3447000 |
| Venus | 4332000 | 3447000 |
| Sun | 4331500 | 3447000 |
| Mars | 8147000 | 6483000 |
| Jupiter | 51376000 | 40884000 |
| Saturn | 127668000 | 101595000 |
With the Yoyana being 5 miles, in verse 1.59 the diameter of the earth is given as 1600Y/8000M and the Sun 6500Yand 32500M and the Moon 480Y and 2400M with the Earth Moon distance as 258000M.
COMPANUS OF NOVARA; TEXT DISTANCE MEASURES
Page 146-147; Theory of the Sun
Lines 30 to 44 deal with the size of the earth, being 360 degrees of 56 2/3rds miliaria per degree and thus 20400 circumference. He then calculates the diameter = 6490 10/11 and the radius before mentioning the Sun.
“Therefore from the earth surface to the centre of the Suns body will be 3923754 6/11ths miliaria and the Suns body will be 35700 miliaria, while the circumference of its rounded surface will be 112,200 miliaria. And if we subtract 17850 which is the radius of the Suns body in Miliaria from 3923754 6/11ths which is the distance in miles of the centre of the suns body from the surface of the Earth, the remainder will be the distance of the surface of the Suns body from the surface of the Earth, and that is 3905904 6/11ths miliaria.
The body of the Sun contains the body of the Earth 166 times plus 3/8ths.
Page 190-191; Theory of the Moon
Commencing at line 430, but omitting extraneous data, we have; The radius of the moons body will be 948 13/33rds miliaria. The whole diameter will be double that, namely 1896 26/33rds miliaria. The circumference of the moons body will be 5958 22/33rds miliaria. ——- distance of the convex surface of the moons sphere from the centre of the earth; that is 209,198 13/33rds miliaria. ——Spissitudo/thickness spere lune 101261 26/33rds.
The size of the moons body is about 140 part the size of the earth, 40 + quintam deciman.
Page 242-243; Theory of Mercury
Commencing at line 450. But, the radius of Mercury’s body is 115 13/33rds miliaria and the distance of the surface of the concave sphere of Mercury from the centre of Earth; this is 209198 13/33rds miliaria. Thickness/Spissitudo = 370122 5/11ths miliaria. The circumference of Mercury’s body is 725 11/33rds miliaria. The body of the Earth contains the body of Mercury 22,247 161/512ths times.
Pages 326-327; Theory of Venus
Commencing at line 442; the radius of Venus’s body is 1442 280/600ths miliaria. The circumference of its body will be 9095 63/660ths miliaria. The concave surface of Venus to the centre of the Erath is 579320 560/660ths miliaria. Thickness or Spissitudo of Venus’s sphere will be 3313545 650/660ths miliaria. The body of the Earth contains the body of Venus 11 25/64ths of it.
Pages 336-337; The Theory of Mars
Commencing at line 602; the radius of Mars body is 3786 240/660ths miliaria and the circumference is 23800 miliaria. The distance of the concave surface of its sphere from the centre of Earth is 4268629 110/660ths miliaria. The thickness/spissitudo of its sphere will be 28083446 310/660ths miliaria. The body of Mars contains the body of the Earth once plus 127/216 of it.
Pages 340-341; The Theory of Jupiter
Commencing at line 653; the radius of Jupiter’s body will be 1482 500/660ths and the circumference of Jupiter’s body will be 93100 miliaria. The distance of the concave surface of the sphere from the centre of Earth will be 32352075 420/660ths miliaria. The thickness/spissitudo of its sphere is 20192626 320/660ths miliaria. The body of Jupiter contains the body of the earth 95 times + 6353/27000 of it.
Pages 342-343; The Theory of Saturn
Commencing at line 700; the radius of Saturn’s body will be 14604 360/660ths miliaria. The circumference of Saturn’s body will be 91800 miliaria. The distance of the concave surface of its sphere from the centre of earth is 52544702 280/660ths miliaria. Thus the thickness/spissitudo of its sphere will be 20843044 480/660ths miliaria. The body of Saturn contains the body of the Earth 91 times + 1/8th of it.
Pages 356-363; Theorica Planetarum; Summary tables for the Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn
COMMENTS ON THE FOREGOING TEXT
The text affords a comparison to the folios in the two atlases and clearly indicates the usage of “Theorica Planetarum” of “Companus of Novara”. However, there is no raison d’être for that inclusion. What it does indicate is that c1550 the Medieval Mind was exploring the Cosmos as the plethora of astronomers in this period clearly shows. These are sure to have encouraged educated persons who would purchase an atlas for their own studies, not actual usage, to illustrate to others their wide range of interests, to be thought to be highly intelligent and have an understanding of the latest sciences and of course wishing to show off their supposed abilities. Thus in many atlases the cosmological wheel and tables are outdated and not applicable to the calendar and are thus included for show!
A SECOND APPENDIX
This appendix to an appendix is required to perhaps indicate the cross-fertilisation of knowledge from the infant universities of England, France and Italy.
ROGER BACON AND THE OPUS MAJUS
One of the greatest minds of the 13th century was Roger bacon and his juxtaposition to Campanus of Novara, 1220-1296 is rather strange. Firstly, Companus is noted as being Chaplain to four Popes, namely Urban IV 1261-1264; Adrian V, 1276; Nicholas IV 1288-1292 and Boniface XIII, 1294-1303. It is obvious that Pope Urban IV was the sponsor of Companus as there are several copies of the Theorica Planetarum with a dedicatory letter to Urban IV. Thus the Companus text was composed before 1259, the date of the earliest known manuscript and before to Pope was elected.
We then read that Pope Clement IV, 1265-1268, who followed Urban IV asked in June 1266 Roger Bacon to send his texts regarding the whole subject of the cosmos, and this was the Opus Majus sent in 1267, and probably written in 1265/66 given its length and spread of subject matter, and of course copying it! In fact Roger bacon composed not only the Opus Majus, but also the Opus Minus and Opus Tertiae, to fully explain his theories, and the knowledge available in Universities at the time. But, would Pope Clement IV ask Roger Bacon to write a text if his preceding Pope Urban IV already possessed the text of Companus of Novara. One possibility is that both the Pope and Bacon were in the same University in Paris prior to Clements Ordination and he preferred to trust him.
In 1910, John Henry Bridges produced a copy of Volume 1, edited with an introduction and analytical table. “Operis Majoris Pars Quarta”, “Mathematicae in Divinus Utilitas”, Bridges has provided a resume of pages 219-236, “The Sixth Head” which deals with the same subjects as Companus, although with widely differing calculations. The introduction pages CXI and CXII are here-in copied followed by Roger Bacon’s actual measurements and his use of the Lowest Common Denominator of 630.
Roger Bacon explicitly quotes Alfraganus as the originator of the length of a degree as 56 2/3rds miliaria and each miliaria contains 4000 cubits. “Oportet igitur supponere, quod cubitus acqualis et geometricus continent pedem et dimidium, et milliare continent 4000 cubitorum et sic accipit Alfraganus in sua consideration, (p224) But on page 225 we read, “Semi diamtria quantitatem scilcet 3250 (miliaria) et veram diameter scilicet 6500.”
Bacon continues, “Ergo oportet quod supponat radicem veram et completam, quae set in quantitate arcus terrae respect gradus coeli, licet non exprimat eam perfecte. Quapropter ipse supponit quod sit 56 miliaria et duo tertiae miliaria et 27 nonagesime, et una sexcentisima tricesimas, vel 56 milliaria et 2984 cubiti et quinque septimae unius cubiti.”
That resolved becomes 56+2/3+27/90+1/630 or 56 2/3 + 19/63 and obviously the 27 is a spurious number, an error and on Page 226 we read, “Quae est quantitas arcus terrae respect gradus coelo, 50 sexcentesimus tricemas unius miliaria, sive quod idem est 317 cubitus et tertiam cubiti, 28 sexagesimas terittias unius cubiti, quoniam non exprimit ibi nisi quod arcus iste terrae 56 milliaria et duae tertiae unius miliaria.”
Thus his actual figure is 56 milliaria et duas tertias unius milliaris, 7 nonagesimas miliaria et unum sexcentesimam tricesimam; i.e. 56 + 2/3 + 7/90 + 1/630 which is 56 + 2/3 + 5/63rds and his 56 + 2984 5/7ths miliaria equals 56 + 0.7461781714 milliaria or 0.0795119 = 318 cubits longer than Alfraganus. Thus the foregoing text exposes the error by Alfraganus in his calculation which is as follows; 360 x 56 2/3 = 20400 miliaria and should then read using the PI ratio of 22/7 the diameter is 6490 10/11ths and the radius is 3245 5/11ths miliaria.
Alfraganus commits a cardinal sin in mathematics by resolving the actual figures to 6500 and 3250 miliaria in his text not realising that a reverse calculation will give a spurious result, just as Roger Bacon has exposed. The 6500 diameter gives a circle of 20428.57143 miliaria and thus when divided by 360 we have 56.74603 miliaria per degree.
What it clearly indicates is the mathematics of medieval texts should always be questioned when “round numbers” appear when the sum included either Root2, 99/70 or PI, 22/7 in the equation.
Returning to the Planets on page 228 we read;”Et quilbet potest haec experiri per computationem, et ideo omitto haec propter prolixitatem. Quoniam vero subtracta longitdine propiore a longiore remanet spissitudo orbis, ideo papet, quod spissitudo orbis lunae est 99504 miliaria et spissitudo Mercrii 334209 et Veneris 3097250 et Solis 325000 et Martis 24882000 et Jovis 17969250 et Saturni 18541250. Haec ex radicibus Alfregani et Ptolomaei in Almagest sumuntur.”
Thus Roger Bacon is clear on his authorities and it tells us just how many ancient texts were circulating and being studied in the fledgling Universities of Europe.
CONCLUDING REMARKS TO TEXT MS7
The intention of these appendices was merely to ask why such a folio was included in the Battista Agnese 1550 Atlas and copied at an unknown time or place into the inner rear lining of the Morgan Library MS M501 atlas by Diogo Homem. The only link is that both cartographers were working in Venice at the same period and undoubtedly would be aware of each other’s work. Not exactly a substantial reason for such a long appendix pairing but, once the data flowed it was important to explain it all as knowledge of the cosmological influences on cartography was up until then basically a cosmographical wheel.
Therefore, if Diogo Homem knew of the Agnese Folio, then why was it not properly included? Hence, it is reasonable to opine that the text in the rear of the atlas is copied well after it was bound in the leather covers and could well have been added after it was sold when the new owner learnt of the Agnese folio. The script is reasonable but it is obviously just copied out as it is not well aligned and errors are there-in.
That still leaves the simple question; “Why was such a very involved Planetary mathematical Table introduced into a Geographical Atlas in the first place?”
William of Oakham concluded that the simplest answer is always the best. Hence, it is very likely the client for this luxurious atlas by Battista Agnese, (just look at the latter folios-magnificent) and knew of or was told of the Companus of Novara text, “Theorica Planetarum” quite possibly by Francesco Barozzi (1537-1604), who was then commissioned to prepare the folio text. Barozzi was an independent scholar with no requirements for a clientele. Therefore the Atlas client dictated certain additions to the Agnese 1550 atlas which actually commences with two cosmological folios and perhaps the Planetary folio should have been the third as it now sits between highly decorated charts of Crete and Sicily and is as such badly positioned as the penultimate folio.
To view the atlas go to;
https://digi.ub.uni-muenchen.de/digitalisate/BV044370732/images/BV044370732_pdf/ubimu_0031450024.pdf
Munich University Library have digitized it along with many other charts and atlases.
Thus I conclude this investigation without establishing a raison d’être for its inclusion unless my hypothesis is acceptable.