Monalosphaerium; Cosmotheoriae & de Proportionibus J F Fernel, Measurements; Accuracy & Fantasy Abound

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Jean Francois Fernel, 1497-1558, was born at Montdidier; then the family moved a short distance to Clermont. He entered the College of Sainte-Barbe, Paris, where he first studied mathematics and astronomy before changing his subject to graduate in Medicine. Ultimately he became the court physician to Catherine de Medici and Henry II of France.
J F Fernel entitled all three texts commencing “Ioannis Fernelli Ambianatis” indicating his origins in the Somme valley and that he was therefore stating he is of the “Ambiani”, the ancient tribe who occupied that valley. Many later historians have inferred from the appellation that he was of Amiens, named originally “Samarobriva” from the “valley of the Samara” (Somme) and latterly “civitas Ambianensium”, and thus confusion set into their works. J F Fernel uses “Samarobriga quae & Ambiani” in his texts.
In the 1520’s J F Fernel wrote his three texts concerning early studies, entitled, Monalosphaerium, Cosmotheoriae and de Proportionibus, published from 1526 to 1528 in Paris. The first discusses longitude and latitude and the variable measurements involved, but it then becomes in the fourth part “Praxis Geometrica”, basically a treatise on surveying using Geometry, much of which is impractical. The second book discusses the size of the Earth via a practical methodology to determine the actual length of a degree of latitude using the Sun elevation and declination tables to ascertain the degree marks for measurement. It then changes suddenly into an astronomical treatise on the planets and stars.
Unfortunately there are many foibles within the texts, let alone wrong figures, either by the author or the copyist/compositor which have not been noted by the author in the errata. This negates some of the data as published and arguments used which will be explained as the texts are analysed. Thus it is first necessary to set down the parameters of the research.

In studying the three texts, written in Medieval Latin and thus obtuse at times, it became very apparent that the data contained within them was a sampling of a whole corpus of research and not three singular books with singular data. Thus I concluded that the research was probably dated to c1520/1525 with the texts written and published in 1526/1528. That factor negated research into the astronomical texts as not knowing which year any reading was taken it is hard to evaluate the findings in the texts. Not impossible, but just not worth the effort, particularly as he notes phenomena which will occur years later. This decision was also amplified by the fact that the data within the texts of J F Fernel was obviously badly copied from his notes, and thus at times utterly unacceptable.
What also became very apparent was the lack of an exact determination of the basic measurement J F Fernel was using. Yes, there is an example of the Digitus Geometricus drawn on one page, but that measurement had to be determined prior to any meaningful research being concluded. Thus the following section moves across the three texts.

Within Monalosphaerium folio 25v/25r headed Quarta Pars, “corporum magnitudines, in primis secundum unam tantum dimensionem, mox secundum duas, ac tandem tribus (quae omnes sunt) aperte manisestam”, we read firstly “Scalaru partes (quae 12 sunt coequals) punct aut digitos saepiuscule vocitari dignovimus: earuquae subdivisions ( si quae sint) digitorum minuta” which is followed by the following table as Diagram MsJFF/1/D01;

  • Granum, omnium mensurarum minima
  • Digitus = grana 4
  • Palmus = digitos 4Complectitur.
  • Pes = palmus 4Haec nostro instituto sufficient
  • Passus = pedes 5

To the left of this table is “Fuguratio pedis geometrici”, which is a physical representation of Granu, Digitus, Palmus and Pes Geometrici drawn as a line diagram set vertically on the page, as the insert indicates. But nowhere does he refer to the 12 coequals obviously uncia in the normal Pes measure. Basic research into this line length was carried out previously and I will quote the findings from the Penny Cyclopaedia, volume 27, as set down therein.


“In the first work he gives his foot or “figuration pedis geometrici” which he says is to be chosen with great care, on account of the great diversity of measures. This paper-foot is now within a sixtieth of an inch of nine inches and two thirds (English), which, increased in the proportion of 41 to 42 is nine inches and nine tenths.” The increase is to cover any shrinkage which may or may not have occurred in the printing process.
Here I must comment that this argument as seen in 1841/42 completely misses the ramifications of the most important section of data J F Fernel gives in his second book which I believe settles the measurement argument completely; they did not recognise it!
Thus let us analyse the actual statements by J F Fernel; “ Nam a Regio Palatio ad aedem sacram divi Dionisii passus 5950 dinumeravi; et inter ambas civitates passus median 4450”


Thus we are informed of (but not how it was measured) that from the Ile de la Citie, Kings Palace to Saint Denis Basilica is 5950 Passus. Study a map of medieval Paris and this is seen as a direct route, nearly a straight line bearing slightly east of north via the Pont aux Meuniers, Rue St Denis, Porte St Denis and thence to the City of St Denis and the Basilica church as Diagram MsJFF/1/D02 illustrates. The Kings Palace is at 48° 51” 21’N & 2° 20’ 40”E; St Denis is at 48° 56’ 08”N and 2° 21’ 35”E, and thus we have an excellent set of figures to ascertain the direct distance and thus the length of a single Passus.
Using “Form of the Earth” measurements for a minute of latitude and longitude at 48° 53’ N, we have; 1 min lat = 1.853565Km and 1 min long = 1.222424Km. By the simple usage of the Pythagorean A2 = B2 + C2, the latitudinal shift is 4’ 47” or 4.7833’ x 1.853565km = 8.86616km or 5.9943 Roman Miles. The long shift is 0’ 55” or 0.91667 minutes and thus a distance of 1.12056km or 0.75759 Roman Miles. Hence the direct distance is 6.042 Roman Miles or 6042 Passus, but does not follow a road precisely.
J F Fernel quotes his measurement as 5950 Passus, a mere 92 Passus difference and hence we may thus state plainly that J F Fernel is probably using the Roman Pes of 11.64706 statute inches/ 295.835mm, with a Passus of 58.2353 statute inches or 1479.1766mm. But to state the measurement is precisely the Roman Pes and not within a few millimetres of that length would be rather stupid given the circumstances, but it can be shown to be acceptable.
However we must remember that J F Fernel states 5 Pes, a Passus is equal to 6 pedes geometricus and his diagram measures c9.667 statute inches. The answer is staggeringly simple; all the time J F Fernel discusses the “Digitus”, the sixteenth part of the Pes when in fact the real division is the Uncia, of which 12 make a Pes (“Quae sunt 12 coequals”). Thus the Pes Geometricus is equal to 10 Uncia; 295.835 divided by 12 = 24.65292mm and thus 10 uncia are 246.5292mm or 9.706 statute inches; I requote, ”this paper foot is now within a sixtieth of an inch of 9 inches and two thirds, i.e. 9.667 statute inches plus a sixtieth” Is this not the Pes Geometricus, and thus confirmation that the original measure is the PES?
But of course the Pes Geometricus is 5/6ths of the Pes and it is the 5:6 ratio which pervades the whole measurement system of the medieval period. I have oft times quoted the following facts; “L’Aune est une mesure de longueur ancienne instauree par l’Edit Royal de Francois 1er. Elle se devisa par seize. Mais l’Aune de Paris se voulait de quatre pieds Romains exactement.” The ancient Aune of Paris France, is 526 5/6ths lignes, or 46.79 inches totalling 1188.45mm. But 4 Roman Pedes would be 1183.34mm and there is possibly the minor difference between the actual Pes measurement of 295.835mm and a quarter of the Aune at 297.11mm, a mere 1.2775mm difference! This difference may be ignored as we do not have sufficient detail to be so very specific, but it points directly to the real measure.
I therefore find that all of the experts from the 17th to the 19th century who have examined these texts to ascertain the measurements, such luminaries as Picard and Delambre and many more, are seriously awry in their work, and have not researched all texts.
Armed with the measurement of the Roman Pes of 295.835 or 297.11mm I can return to a normal evaluation of the three texts as written, but, inform readers that many points are still to be questioned and re-evaluated. These are particularly the discrepancies in the texts with its “numbers,” which in various forms have been severely mis-copied by J F Fernel. He has not realised his errors, no doubt having carried out the research years before, trusted his notes, misread his notes and thus carried out spurious calculations as will be shown. However there is a second part of the sentence which must be evaluated as it impinges upon the basic measurement of the Roman Pes and 12 Uncia. The part-sentence follows the Royal Palace to St Denis distance given as 5950 Passus and is as follows, illustrated by Diagram MsJFF/1/D03;


“Inter ambas civitates passus median 4450.”, which translates as “between the two cities the middle distance is 4450 passus” Thus we have three sections of measurement to consider. The first from the Palace to Porte St Denis, the city boundary; the second is from there to the South or Paris gate of the City of St Denis, the middle distance, and finally from there to the Basilica. The coordinates are as follows, and I am only using the latitudes for this example;

Royal Palace 48° 51’ 21”N 2° 20’ 40”E
Porte St Denis (4th/wall) 48° 52’ 08.66”N 2° 21’ 08.2”E
St Denis South gate 48.929553°N 2.356392°E
Basilica 48.935607°N 2.359110°E

Thus the Royal Palace to Porte St Denis is 1.479145Km or 1 Roman Mile, 1000 Passus

Porte St Denis to South gate is 6.7232Km, 4.5435 Roman Miles or 4544 Passus (4450 JFF)
South gate to Basilica is 0.6733Km, or 0.4552 Roman Miles or 455 Passus.
The total latitudinal measure is 6000 Passus or 6 Roman Miles, and thus we may consider the 5950/6042 measurement previously calculated as accurate as can be. However did J F Fernel actually mean that the distance of 5950 passus was by latitude? But look at the measurements as they unfold, read the information about the Roman City of Paris and the Medieval City. The major roads are 300 Roman Feet apart and this grid has been found everywhere in the central section of Roman Lutetia. The experts have determined that the probable setting out of the Cardo Maximus, that is north/south, was from the Saint-Genevieve hill and that is where the College of Sainte-Barbe is situated.
I therefore consider that the foregoing calculations are proof positive of the basic measurement used by J F Fernel. It is a Roman Pes/Passus/Mile and may have been in fact 1.28mm longer than the actual Roman Pes Measurement. That expansion is 4 thousandths of the Pes, and therefore the difference in 5000 Pes or 1000 Passus is frankly irrelevant given the problems within the text of numbers, and the actuality of the measurement points.
Within this section which is trying to establish the correct measurements prior to the main analysis of the three texts, there is one last check that can be made to confirm that the Pes Geometricus is in fact 5/6ths Pes of 12 uncia or 16 digitus. J F Fernel states his measuring wheel for the latitudinal measure was 20 Pedum or 4 Passus circumference and its diameter was; “6 pedum sextquae paulo magis digitorum geometricus”. Thus the diameter is 6 normal Pes plus 6 and a little or somewhat more digits geometricus.
The digitus geometricus is 1/16th of 5/6ths Pes or 5/96ths Pes, and thus 6 digitus geometricus = 5/16ths Pes. Therefore the diameter is approximately 6+ 5/16 Pes. Thus the circumference is 6+ 5/16 x 22/7 = 2222/112 or 19.8393 Pedes, with J F Fernel stating 20 Pedes. Reverse the calculation and we have 20 divided by 22/7 = 6.36364 Pedes, that is 6 5.82/16 pedes. That is 6 + 5.82 digitus or 6 + 6.984 digitus geometricus, because 5.82 digitus = 5.82 x 6/5 = 6.984 digitus geometricus.
Thus not knowing the precise calculation by J F Fernel made the 6.984 Dig Geom, may well be “a little or somewhat more”. The problem is the possible 1.28mm discrepancy!
The last point which requires to be mentioned prior to the full evaluation is the measurement of the Sun Altitude via the homemade measuring device pictured on Diagram MsJFF/1/D04. The angular arm which provides the direct reading of altitude has appended two viewing holes set slightly above the arm. If these are aligned to the Sun centre and a reading taken
then there is a discrepancy in the altitude reading unless the scale bar has been set to the holes/bar distance north to compensate and thus ensure the alignment via the holes is correct.


Does it matter? It can make a difference depending upon the accuracy required in the reading. The Sun has a nominal diameter as seen from Earth of 31’ 31” to 32’ 33”, although this is normally taken as 30’or half a degree. Thus if the difference in the viewing holes to bar surface measurement is ¼ degree then the top edge of the Sun should be used in order for the actual reading to be taken on the centre line of the Sun Disc. Thus there could be a ¼ degree error in any reading. But as this is a theoretical discussion with no methodology of ascertaining the answer, it unfortunately must be ignored and the figures given in the text accepted unless an obvious error is noted. But it should be investigated as J F Fernel is quite cavalier in his use of measurements; a problem thus!

Following the index there is an 18 line verse headed, “Ioannis le Lievr, AD lectorem Carmen”, which is basically, “Jean the Hare, to the reader a verse”, and is worth quoting.

Plaudat Americae gens aspera: plaudat & Indus,
Plaudat & ardenti soenore diues Arabs.
Plaudant regna quibus cura est monstrosa videre,
Plaudat qui celeres scandere gliscit aquas.
Ecce patent Asiea fines, iam regna propinquant
Europes: Libyae nausraga saxa patent.
Indomito agnoscat zonam sudore perustam,
Noscat fluctivagus nauta charybdis aquas.
Hic eclicen gnomon vobis signabit, & horas
praetenui umbella, ventus & unde ruat.
Hic facile serie geometrica signa recludens,
vos docet omnipari portio quanta soli est.
Hic docet humanos quae sidera flectere possint,
et quae stellifero sidera in axe micant.
Quisquis ades, animo gratanti hec scripta revolue,
quae tibi parturient non decus exiguum.
Fernlio gratare igitur gratare labori:
ingratorum animos torua megara premit.

The text, as far as this research is concerned commences with J F Fernel Pars “Tertia, Ex primo mobile utilitates quarta proposito,” where he discusses the eclipse which will occur on 17th October 1529 at 7;54 in the morning. He states that the “Parisiorum Longitudinem esse graduum 24 & minutorum 30.”
Here we meet the first real problem within the work of J F Fernel. According to the text of” Geographike Hyphegesis” by Claudius Ptolemy, written in section 2:7 is, “Below these are the Parisi and the town Lucotecia, 23° 30’ longitude east and 48° 45’ latitude north. Thus in 1526, J F Fernel using his notes from 1520/25 has miscopied the actual Ptolemaic longitude or is using a copy of the text which is itself in error. This can be shown later to be mis-copy by J F Fernel, but at this stage in his text it has significant repercussions for many of his later calculations in all of his texts. I have included a section of the text from the fourth proposition as it also sets down details such as the fact that 1 hour is equal to 15 degrees longitude and then gives examples of how to calculate via the following longitude distance table between cities, which I illustrate later.


Within the “Quinta Proposito”, folio 15, Pars Tertia, headed; “Quanta quorum cunquae locorum terrestris sit distant dimetiri” we read that he considers each degree of the 360 to be “Miliarii 60 Italicis, concessis; ac singulo cuiquae minute, miliaria una”. J F Fernel includes a “Tabula coversionis graduum in Italica miliaria, extra aequatorem”, which is herein included with the corrected longitudinal measures per degree of latitude. J F Fernel has obviously restricted his figures to whole numbers, but in places is actually 1 ½ miliaria adrift.
The table from folio 15 is as Diagram MsJFF/1/D06.


Quinta proposito, quanta quorumcunquae locorum terrestris sit distantia, dimetiri.
Tabula coversionis graduum in italic miliaria, extra aequatorem.

GradusLatitudinis Italicamiliaria Actualcosine
1 60 60
12 59 58.69
19 58 56.73
23 57 55.23
26 56 53.93
28 55 52.98
30 54 51.96Cos 30 = √3/2
32 53 50.88
34 52 49.74
36 51 48.54 Ptolemy 4/5
37 50 47.92
39 49 46.63
40 48 45.96
41 47 45.28
43 46 43.88
44 45 43.16
45 44 42.43√2/2
47 43 40.92
49 42 39.36
50 41 38.57
51 40 37.76
52 39 36.94
53 38 36.11
54 37 35.27
55 36 34.41
56 35 33.55
57 34 32.68
58 33 31.79
59 32 30.90
60 31 30.00Cosine 60 = 1/2
61 30 29.09
62 29 28.17
63 28 27.24
64 27 26.30
65 26 25.34
66 25 24.40
67 24 23.44
68 23 22.48
69 22 21.50
70 21 20.52
71 20 19.5370.53 = Cos 1/3
72 19 18.54
73 18 17.54
74 17 16.54
75 16 15.53
76 15 14.51
77 14 13.50
78 13 12,47
79 12 11.45
80 11 10.42

Thus J F Fernel has chosen a number for the latitudinal degree which has no substance in metrology except in the minds of mathematicians who have not explored the actuality of the measure on the earth surface. Thus his 60 miliaria are in fact each 1.25 Roman Miles.
J F Fernel then produces an “Exemplium” for the utilisation of the tabula, but it must be noted the longitudinal error continues here-in. Diagram MsJFF/1/D07 illustrates this.
“Clariora sient haec exemplo. Luteciae, teste Ptolemaeo, longitude est graduum 24, minutorum 30. At Brondentie Germaniae civitates logitudo, graduum 33, minutorum 30. Longitudinum itaque differetia est 9 graduum. Et quum utriusquae latitude sit prope graduu, 48, ingress tabellae coperio cuiquae gradui differentiae, 43 italica miliaria deberi: quibus per 9 ductis, 387 nascutur, quae sunt caru civitatu breuissima absistentia.”


J F Fernel follows that with a second example where in that example they are not on the same latitude, (and so he manipulates the figures, check Ptolemy!).
“Exempli favour horum nihil te prorsus sugier. Lutecia, artiu omnium aluna, 24 gradus, & minuta 30 habet in longitudine: at in latitudine gradus 48, & minuta 30. Rome urbis longitude, est 36 graduum, & 30 minutoru: eiu squae latitudo41 graduum, & minutorum 30. Latitudinu idcirco differetia, est graduum 7: longitudinum vero, 12 graduum. Illam per 60 multiplico,& nascuntur milaria 420. His rursus in sequadrate ductis, 176400 prosiliunt. Ab his, tabulam ingredior cum latitudine 44 graduum, quae media est inter ambas civitates: ac 45 miliaria cuiquae gradui differetiae longitudinis in ea latitudine deberi, animaduerto. Proinde 12 gradibus per 45 multiplicatis,540 consurgent: quae in se ducta quadrate, 291600 coponunt. Hunc numeru iampridem innvento, adijcio. Sitquae; hic numerous 468000 cuius invenio hanc radicem quadrata 684; haec sunt ergo miliaria absistentiae haru civitatum.” { Please note this is an erroneous calculation!}
This example can be expressed simply as; take the longitudinal distance at the mid-points of the latitudes; multiply it by the longitudinal difference and then take the latitudinal measure and multiply by 60 and then use those two overall distances as the sides of a right angle triangle. Thus A2 = B2 + C2 and the hypotenuse is the direct distance between Paris and Rome in Miliaria at 60 per degree. The Diagram MsJFF/1/D07 illustrates the text solution and includes a calculation using the Cosine Rule for a Great Circle distance between two cities. That difference is not worth considering given the idea J F Fernel promulgates has already been amply illustrated in my texts, where it has been shown that for calculation on Portolan Charts and Maps of the 14th to 17th centuries the centre latitude longitudinal measure will suffice for calculating the overall distance with little inaccuracy.
However it must be clearly stated, as the diagram illustrates that J F Fernel has misread his figures and in fact inverted the 44/45 such that his calculation is spurious.
Moving to Folio 23, Trigisimasexta Proposito, we find the length of the year defined as, “Diebus 365, Horis 5, Minutis 49 & Secundis 15. Unfortunately it is possible to have up to 6 measurements for a “year” and is therefore not worth commenting upon.
The major portion of the text, “Monalasphaerium” for our purposes commences on Folio 25, “Quarta Pars Corporum” etc as already discussed with the “Figuratio Pedis Geometrici”. The problem identified already that the “Pedis Geometrici” has been subdivided into 16 parts to produce “Digitus Geometrici” seems to have convinced J F Fernel that the Barleycorn can be the arbiter of the length. The grain referred to in antiquity has always been the Barleycorn or Graine d’Orge and has differing measurements;
Angleterre; Le Barleycorn ou Graine d’Orge est la 36th partie du pied.
Arabie Ancienne; Le Graine d’OIrge des anciens Arabes faisait la 6th partie du goigt ou assbaa et se divisiat en 6 crins de cheval.
Portugal; Le Grao ou Graine d’Orge 72nd partie du pied se divise en 2 linhas ou 24 points.
The text “Ms2, Follow the Barleycorn”, page 4 explains the basis for the original grain measurements, as well as the ramifications of cross title for different measures.
The pages following Folio 25 are propositions using angles to determine survey measurements whereby J F Fernel likens each Tower or Tree to a Gnomon and thus its shadow length. From a surveying standpoint many are simply unworkable in the field and thus this is purely a paper exercise which interested parties may research themselves. However it is pertinent to these discussions that J F Fernel never mentions the Euclidean proposition 47 where the simple proof that A2 = B2 + C2 is given, and this would have aided his texts considerably, and shortened them as he mentions Euclid often there-in.

This text commences with a dedication as follows; “Praepotenti ac Serenissimo Lusitaniae Regi Ioanni Tertio, Ioannes Fernalius Ambianas Saluten.
In 1484 King John II of Portugal (1455-1495) formed a commission to solve the problem of altitude determination in the southern hemisphere. Thus a table of Solar Declination named, “The Regiment of the Sun”, was drawn up and tested off N W Guinea in 1485. Following that introduction, the routine taking of solar sights & thus all instruments were graduated in degrees rather than using place names etc. There was also the manual, “Regimento do Astrolabio e do Quadrante” which provided rules for the use of astronomical instruments in navigation. Thus we can assume J F Fernel was well aware of these texts.
{These texts and the ramifications of their contents are covered in my Ch/Charts texts.}
There are in Book 1, folio 2, a series of statements which will be simply listed and then as appropriate commented upon.
1)Eratostheni siquide philosopho (cuius placita apud Strabonem videre licet) visum est gradui cuique circuli maioris 700 stadia deberi, quae Italica miliaria 87 cum semille efficient.
This is a completely erroneous measurement concocted by medieval scholars who took as “gospel” the statements of ancient philosophers. In the early medieval period the degree of latitude was known as 75 Roman Miles but the cartographers and sailors thought there was a problem as the inter-connectivity of the stadia numbers was awry at 5:6 ratio. Thus 75 x 8 = 600 Stadia and that morphed into 90 Miliaria by virtue of the 5:6 ratio. Then the medieval scholars were introduced to Eratosthenes 700 stadia degree latitude. But 90 miliaria is 90 x 8 stadia= 720 (of undisclosed but calculable length) and thus it was reduced to 700 stadia which of course became 87.5 miliaria, a nonsense degree.
2)Ab hac sentential Ioannes de monte region parum deficiens, parte quamlibet stadia 640 continere, posteritati passim scriptis suis demandauit, quibus Italica miliaria 80 debentur.
A completely ‘off the wall’ idea with no foundation in measurements known.
3)Ptole. Vero (ut ad minors quantitates seriatim deducar) 500 stadiorum, seu 62 Italicorum milliariorum cum semille, partem unamquanquae stabiliuit.
A direct comparison to the degree of 75 Roman Miles without foundation as the stadia length has not been regularised.
4)Campanus, Thaebitius, Almaeon & Alphreganus, qui post Ptole. Astronomicum, cosmographicumque principatus rexerunt, 56 milliaria cum biffe seu tertijs duabus, partibus singulis tribueda dixere.
In Iraq, Al Ma-mun instructed his experts to measure the degree of latitude. The text ChMEA/1 details the four trials and illustrates quite clearly that Alfraganus chose the wrong length. It should be 56 1/3 Arab Miles of 4000 Black cubits.
5)Ob id ergo causae, idipsum experiment comprobans, deprehend accurate supputatione, cuiquae gradui circuli maioris tam in terrae quam in maris convexo 68 Italica miliaria, passus 95 cum una quarta respondere.
J F Fernel carried out his experiment to measure a degree of latitude and arrived at the distance of 68 Italian Miliaria and 95 ¼ passus. The exercise is fully discussed in the following section where J F Fernel explains his actions and readings for the degree measurement. It is however not really explained in his text, merely a passing comment.
6)Haec autem stadia Romana 544, passus 45 (sic = 95) cum una quarta vel exactus cum septemdecim septuagesimissecundis efficient.
The first part is merely a quantifying as Stadia. The second part is quite frankly hard to determine as there are three possible translations of the number sequence as written. It is discussed separately as it indicates perhaps a different and original reading for the degree.
7)Cuius enim gradui 56 milliaria cum duabus tertijs tribuentes, aiunt miliaria quodquae 4000 cubitis, seu passibus 1200 constare.
Mathematically correct at 1 passus = 3.333 cubits, but awry in actual length.
8)Milliarioru itaque 56 cum duabus tertijs quos aiut unico gradui respondere, passus erut 68000, qui per 1000 distributi, plane declarat cuilibet gradui 68 milliaria Italica ad amussim deberi; eritquae differentia passuum prope 95.
If J F Fernel had measured the degree correctly then this statement would have been correct with the two measurements of a degree being different miliaria in length.
9)Ducto nempe circunferentiae numero per 7, ac product per 22 distributo, diametric quantitas numeru squae succrescit; inuersa quae operatione, diametric inquam numero per 22 multiplicto, si consurgens per 7 fecetur, ambitus circunferentia quae prosiliet.
The PI figure is confirmed as 22/7, the well known and used ratio.
10)Dignosces igitur harumsce regularum ope, subiectae figurae quantitates terrae in unguem deberi.
Cuius gradui ambitus terrae;68 milliaria + 95 ¼ passus
Totus terrae & maris ambitus; 24514 milliaria + 285 5/7 passus
Terrae diameter7800 milliaria
Eiusdem semidiameter, sequentiu basis3900 milliaria
Quo facile sit cuiquae arithmetices officio, cunctas quantitates milliarijs & passibus expressas, in caeteras mensuras resoluere, tabella supposuimus mensurum varietate refertam.
Granum hordei mensurarum omnium minima.
Digitus grana habet4 {digitus geometricus}
Palmus digitus constat4 {digitus simplex}
Pes palmos habet4 {Pes simplex}
Cubitus sesquipes est palmus habens6 {cubitus simplex}
Passus simplex palmos habet10 {5 x4 = 20}
Passus Geometricus pedes habet5 { obviously passus simplex}
Pertica est pedum10
Stadium Italicum passus habet125 {625 pedes}
Milliarium Italicum stadia habet8
seu passus1000
Milliarium Germanicum habet passus4000
Milliarium Sueuicum habet passus5000.
Where-as figures may be correct in some instances the measurement is awry.
Cosmotheoriae, Book 1, folio 3, has 26 lines describing the known world by degrees and Miliaria, which is too complicated for this text and will be the subject of a second, thus enabling the usage of the work by Claudius Ptolemy to be explored.
However the second sheet of folio 3 contains the main text regarding the degree of latitude measurement and this will be investigated fully, with a copy of the Folio appended for general use, as Diagram MsJFF/1/D03 in the previous section.
That page is number 9 in the text section of “Cosmotheoria”, and the lines there-in pertinent to this investigation are as follows; 3-11 discuss the diagram of a gallows frame to read off the sun altitude; lines 11-34, the solar readings for the degree change in latitude and lines 34-45 discuss the measurement method used.
Commencing with the diagram text, the “semidiametros” is 8 feet and thus the CD arm (the scale bar) is 8√2 or 11.314 feet. This is to be subdivided by 90 degree sections each having 60 minute sections for correct reading of the sun’s altitude on any given day. Thus each degree is c1.5 inches and each minute division c0.025 inches or 1/40th inch. Even manufactured to the greatest accuracy in modern terms that is a line every 0.635mm, and when we consider the Ligne of Paris the standard of the age is the 144th part of the Pied du Roi of 324.8mm it is 2.255mm per Ligne, thus requiring 3.5 divisions for the minute distance.
But if the actual size of the device is considered, it must be at least 12 feet tall and as such not portable from viewing site to viewing site as some historians suggest when they state J F Fernel walked from Paris to Amiens taking readings over 4 days. If you read the first three lines of the section following, on the 25th August he is in Paris and on the 26th August he is one degree latitude north; that is 111Km travel in a single day. This is not only highly improbable as it means travelling at some 10KPH for the 12 hours of daylight. Thus I question the text as written and concluded it is a conflation of work carried out in the previous 5 years. The readings given are the clue to the travel or lack of it.
If we now look at the solar information contained in the text, I will freely translate it into common everyday parlance; please note g = gradus/degree and m = minute.
“On the 25th August the sun at the meridian was at g49 m13 elevation and as that day was the 11th day in Virgo the declination was g7 m51 which calculates for latitude g48 m38. On the 26th August at latitude g49 m38 the sun was elevated g47 m51. On the 27th August it was g47 m26; on the 28th August it was g47 m05 and on the 29th August it was g46 m41. He then goes on to say that he had ascertained from local inhabitants he was 25 leagues from Paris. The return journey was used to measure the distance by the revolutions of the carriage wheel which is 20 feet or 4 passuum circumference and it took 17024 revolutions to return to my start point of the 25th August. The total distance for the journey was therefore 68096 passus, equivalent to 68 Italian miles and 96 passus. I have reduced this to 68 Italian Miles and 95 ¼ passus for simplicity of calculation.”
If we now look at the details of the Sun’s declination we can determine the units of declination used and thus confirm the daily position of Jean Fernel, north of Paris. He states that on the 25th August, the 11th day of Virgo the declination is g7 m51 or 7.85 degrees. The sun is at 49.2167 and thus we have 90 – 49.2167 = 41.3667 to which we add the declination of 7.85 and the latitude is therefore 48.6333, that is, g48 m38. But we must note that on the following day, 26th August the latitude is given as g49 m38 and he has travelled 25 leagues north, which in our terms is 111Km or 75 Roman Miles and thus each league is 3 Roman Miles, not two miliaria. The Roman Degree of latitude is 75RM, and thus it would appear there is a discrepancy of 6 Roman Miles + 904.75 passus, or 6904.75 passus from his 68 + 95 1/4. Diagram MsJFF/1/D08 illustrates the physical positions of the latitudinal readings.


But where was he from 27th August to 29th August; the only methodology we have available is to calculate his latitude from declination tables of the age. I am using those produced in 1543, being the nearest I could find in terms of date.
25th August; 49.2167 – 7.4167 = 41.8 and 90 – 41.8= 48.2 latitude
26th August; 47.85 – 7.05 = 40.8 and 90 – 40.8= 49.2 latitude
27th August; 47.433 – 6.6667 = 40.7666and 90 – 40.7666 = 49.23 latitude
28th August; 47.083 – 6.30 = 40.7833 and 90 – 40.7833 = 49.2167 latitude
29th August; 46.6833 – 5.933 = 40.75 and 90 – 40.75 = 49.25 latitude
Thus it is a simple matter of stating the following scenario; on the 25th August Jean Fernel measured the latitude of his start point in Paris and then appears to have moved one degree north such that on the 26th August he was 25 leagues/75 RM north and confirmed the reading. He then sojourned there until 29th August checking his readings and calculating his latitude accurately. After the reading at noon on the 29th August he prepared to depart and probably on the 30th August he returned by carriage to Paris carefully counting the revolutions. But again, I question that possibility. A Horse Drawn Coach would in all probability travel at 5KPH over 111km thus ensuring the count is possible, the coach wheels are steady for the differing surfaces, and obviously the Horses/driver requires rest-stops. Thus the journey would in all probability take 3 days, or is this the 4 days in the text which have been mis-copied?
At this point it is necessary to indicate that J F Fernel is using the Julian calendar and hence 25th August is his 11th day of Virgo, which in our terms and the Gregorian calendar it would be the 4th September. The rather fraught 11 days lost through the calendar changes brought about by the Papal Bull of 4th October 1582, which although readily accepted by Italy, Portugal and Spain was roundly rejected in many countries of Europe.
Thus we arrive at the distance measure given of 17024 wheel revolutions, which J Fernel equates to 68096 passus or 272384 pedes. There are arguments in the “London, Edinburgh and Dublin Philosophical Magazine and Journal of Science” of 1841, 1842 and 1843 written by Professor De Morgan and Thomas Galloway. A.M., F.R.S, where they explain fully the measurements and arguments put forward by many illustrious persons of the age, including changing the actual measurements given to the Toise.
I here-in state quite clearly that I do not believe the illustrious persons actually investigated the two texts to determine what J F Fernel had actually written!
I repeat a single statement made by J F Fernel on folio 2; “Haec autem stadia Romana 544, passus 45(sic) cum una quarta vel exactius cum septemdecim septuagesimis secundis efficient”. It appears nobody has been willing in the past to consider that sentence as a whole statement and thus perhaps the true measurement is staring at us in the text.
Consider this first; Paris is given as 48° 38’N and the next point given is 49° 38’N, which is in fact Breteuil. The 48° 38’N point is south of Paris and could in fact have been the Tour de Montlhery at 48.6349°N, close to Ste-Genevieve des Bois, the same appellation for the place of the College Ste- Barbe. But even curiouser is the fact that Montdidier, Somme, has the latitude of 49° 38’ 55”. Diagram MsJFF/1/D08 amply illustrates these facts.
My translation of the text is as follows; “These nevertheless 544 Roman Stadia plus 95 ¼ passus are exactly seventeen seventy two”. Yes the Medieval Latin could possibly also be, a figure 1772, or, 17/72nds or 17-70-2 as a final measurement, but not explained. However 17702 x 4 = 70808 passus and is in all probability the real measurement which has been reduced because of the roads meandering and the fact that the measurement of a degree must be due north/south for any meaningful usage. The difference is 2713 passus or 2.713 Roman Miles. If the road north from Paris via St Denis through Chantilly and Creil was used it would meet the known Roman Road at the N31 to the east of Clermont and thence it is northwards through St Just en Chausee to the D930 slightly East of Breteuil. If the actual route as described is measured then the difference is 111km latitude and 117km route. Thus this is 4 Roman Miles or 4000 passus deviation. The 2713 passus is therefore quite possible, even probable as the difference J F Fernel observed given the possible dimension of a Pes.
The expansion from 68095 to 70808 is 1.0398, where-as 111km to 117km is in fact a 1.054 expansion and thus we can understand that the 17702 passus is probably correct.
This of course may be just “pie in the sky” extrapolation, but why has J F Fernel included the figures 17, 70, 2 as the exact figure if it is not a pointer to the whole story and illustrates the unfortunate usage of notes taken over 5 years and not properly recorded or remembered for usage in preparing these texts.

Any evaluation of medieval texts prior to c1600 requires the Julian calendar to be used unless the text dates have been updated by an editor and it is clearly stated as such.
But that means when J F Fernel gives sun elevations and calculations are made from them we also require using the correct declination tables. The “Manuel de Pilotage a l’usage des pilotes Bretons” par Guillaume Brouscon actually contains three differing declination tables and the first is used here-in to determine the J Fernel latitudes.

Aug 24th g7 m51 g7 m58 g8 m00
Aug 25th g7 m28 g7 m36 g7 m38
Aug 26th g7 m06 g7 m14 g7 m17
Aug 27th g6 m43 g6 m51 g6 m55
Aug 28th g6 m19 g6 m29 g6 m32
Aug 29th g5 m57 g6 m07 g6 m08
Aug 30th g5 m34 g5 m45 g5 m45

There is another slightly different version by G Brouscon in his 1543 Nautical Almanac, now held by the Huntington Library, USA. Thus if declinations were measured on a personal basis at the solstitial dates they could be used as an accurate figure and by interpolation to the day length the others resolved. The problem is of course the observer, is he accurate and how wedded to personal observation, i.e. first principles, and as printed lists were probably not generally available it was the only methodology, but it is playing with numbers and accuracy is thus at risk.

I have no doubt that many researchers have had reason to access the NOAA website for their device to determine the sun declination, altitude and many other facets of surveying data required to establish locations.
But please note that if you are researching any date prior to October 1582 the dates are given by the Gregorian calendar and must be adjusted to obtain correct declinations and thus latitudes for the Julian. Go to;

On folio 43 J F Fernel produces a list of places for which he appears to have taken the Ptolemaic longitude and recalculated those to a new zero point the longitude of Paris. I have already shown that J F Fernel has unfortunately used a spurious longitude for Paris, that is 24° 30’ instead of the Ptolemaic 23° 30’ and now illustrate the consequences of his rather lackadaisical attitude to accuracy. The tabula is included as per the Cosmotheoriae text and then subjected to a recalculation. Diagrams MsJFF/1/D09 to MsJFF/1/D13 which follow, illustrate the tables and the calculated points. There are 80 place names chosen by J F Fernel in the tabula, all have been recalculated and indicate that J F Fernel was well aware at some point of the correct longitude apropos Ptolemy but his lackadaisical and cavalier attitude to the accuracy of his text leaves a lot to be desired. Unfortunately he also uses Cities which are not within the “Geographike Hyphegesis” of Claudius Ptolemy and thus verification of his figures is rather a problem. I have chosen not to use the geographical co-ordinates in this investigation as the Ptolemaic longitude is bound up with his lengthening of the Mediterranean Sea to fit the coastlines into a 500 stadia world as my text CgMtCp/1 clearly illustrates. Basically from the Sacred Promontory of Iberia to Paris is 755Km, but Ptolemy has it as 1412Km and thus it is not feasible to estimate the alignment that 24 30’ or even the correct 23 30’ would represent. All we can assume is that perhaps the Cardo Maximus used by J F Fernel was that line.
I am sure that all are aware that the Paris Meridian through the position of the soon to be established Observatory was very close to this alignment.
At this point in the text I find it necessary to reiterate my comment regarding J F Fernel and his use of figures which are sadly generally awry and misleading.






IOANNIS FERNELLI Ambianatis de Proportionibus Libri duo Prior, qui de simplici proportione est & magnitudinum & numerorum tum simpliciumtum fractorum rationes edocet posterior ipsas proporttiones coparat; earumquae rationes colligit.
Parissis, exaedibus Simonis Colinaei, 1528
Ioannis Baptistae Lusitani ad Emanuelem de Tienes, elegiacum Carmen.
Then follows another of J F Fernel’s verses. However the final words of the preamble are; Parisiis apud clarissimum divae Barbarae gymnasiu ad calendas Novembris 1528.
NOTA; The divine Barbara is normally depicted as a Princess with palm and tower (or chalice and host, or, sword) She is one of the fourteen Holy Helpers, patroness of Architects, builders and firework-makers.
Genevieve of Paris is the Patroness of Paris and is normally shown with book and candle or torch Variations are; 1) devil tries to extinguish her candle while angel keeps it alight; 2) nun or maiden with sheep near her; 3) key in one hand, candle in the other; 4) devil at her feet with bellows; 5) restoring sight to her mother.
Thus the College Ste-Barbe is situated on the Mont Sainte-Genevieve, adjacent to the Pantheon. The college was founded in 1460.

The whole text is a mathematical ramble through his “ De Proportiones”, which we term as fractions. He starts with simple multiplication tables as if teaching a child that 1,2,3,4 are when doubled 2,4,6,8 or exampled as trebled and quadrupled. If a researcher is interested in medieval thinking about mathematics, albeit in a rather strangely presented methodology it will make for a long laborious study which will probably have no raison d’être at the end.

De Proportionibus

The Diagram MsJFF/1/D14 here-in has two separate pages set on a single sheet and they illustrate the contents of the text. But significantly J F Fernel states quite clearly that a Miliaria has 8 stadia of 125 Passus, then that a passus has 4 cubitos, that is 6 pedes and must be the Pes Geometricus, but next he states that Cubitus, pedem cum semille; pes, 4 palmos; palmus, 4 digitos; digitus, 4 ordeacea grana; nec ultra procedit usus.
I have clearly shown that the passus is 5 standard pedes or 6 pedes geometricus and thus J F Fernel is yet again cross referencing his measurements and causing utter confusion there-by. Why did he not retain the normal Roman measurement with its simple rational?
The second section illustrated is equally confused in that J F Fernel has taken a square and used the diagonal, his diameter to form a second square. This is then subdivided into triangles by drawing the second diagonal. In fact splitting the square forms two equilateral triangles and is thus an excellent method of proving Euclid’s 47th proposition easily expressed as A2 = B2 + C2, and could have been written much simpler as an explanation.
Further on in the text J F Fernel constantly uses the Pedes as an example of division etc., rather than just plain numbers. He multiplies Pes/Digitus by others as illustration of quantifying fractions etc and uses multiple examples of similar calculations. Thus I do not consider the text to be capable of explaining the surveying and mathematics of that art.

Had the work by J F Fernel been as scrupulous as I am sure he would have wished in its correct presentation of the mathematics and the use of known facts then these texts would have given an even better insight into the medieval mathematics and surveying techniques?
The fact that J F Fernel took and used the method promulgated in the work of Vitruvius, Book 10, chapter 9, The Hodometer, to measure long distances, by a wheels rotation, is a first for this era, and no doubt should be celebrated. But, it is in fact doubtful that it took place as inferred from the text. Travel in those days was not fast and to infer he travelled 111Km in one day is a non-starter. The device he built to take the sun elevation readings if the arm was 8 feet long as described would be a 12 foot high unit, with a heavy stabilising weight on the base. It is hardly the easiest device to carry around and take elevation measurements, let alone move it 111Km in a day. We are not informed of the declination readings for four days altitude readings, only the dates on which they were taken. Thus in using tables available it is quite clear that the information is false or has been rewritten as a shortened event and thus impossible to clarify.
The measurement of a Roman Pes, be it within millimetres of the actual length or not, is actually irrelevant in the calculations of the degree of latitude. We do not know the two points from which the measures were taken and many historians have jumped upon the title Ambianatis to mean J F Fernel measured from Paris to Amiens, a point unproven. But the following text is a good indicator of the romance applied to the text of J F Fernel and of this measurement. I quote from the “Bulletin of the Societe archeologique et historique de Clermont de l’Oise, Tome XXXIV, p105 et suivantes” which is available from BNF gallica.
“L’historien Dechambre nous a laisse une traduction de l’experience decrite par Fernel lui-meme. En latin dans sa Cosmotheria (1528), rapporteee par Charles Ansart dans une publication de la SAHC. “je commencais a rassembler des regles mobiles en forme de triangle-rectangle, a la facon conseillee par Ptolemee; puis je me livrais a Paris, pendant trios jours de suite, a des calculs preparatoires pour preciser la hauteur du soleil a midi. Il s’agissait de determiner ensuite vers quell endroit, et quell jour, je trouverais la meme meridienne, en tenant compte de la declinaison et du changement de latitude. Je partis un 25 aout et m’avancais a pied le plus droit possible vers le nord. De mon pas ordinaire, don’t il faut compter 1000 pour faire 400 coudees, je marchais pendant quatre jours et me trouvais a 25 lieues de Paris (vers Breteuil) {aujourd’hui dans le departement de l’Oise}, quand je pus reliever la meme hauter du soleil qu’au depart. Mon degree de meridian faisait donc 25 lieues. Mais il me fallait verifier mon calcul pedestre; je montais alors sur une voiture qui rentrait directement a Paris, et je m’y tins tout le temps du trajet de maniere a compter les revolutions d’une des roués, lesquelles je fixais a 17024- deduction faite de ce qu’il fallait pour les montees et les descents. Le diameter de la roué etait de six pieds et un peu plus de six pouces geometriques et, par consequent sa circonference de 20 pieds ou 4 pas. En multipliant par quatre le nombre des revolutions, je trouvais 68096 pas, qui font 68 milles italiens(?), plus 96 pas que j’ai pu devoir convertir en 95 ¼ pour n’avoir pas de fraction a introduire dans le diameter de la Terre; j’en ai conclu qu’une meme mesure appartenait a tous les degres d’un grand cercle”
Claude Teillet of SAHC wrote in October 2012 a short paper containing this passage and then the comments of persons such as Cassini. It is interesting, but as with it all, flawed.
Thus the examination of the works by J F Fernel, although interesting and informative leave the distinct impression of a person for whom mathematical accuracy and the checking of data for a text was not of paramount importance. I cannot understand how such an inquisitive person was so lackadaisical with his text recording of his obvious notes from years bygone.

I conclude with a plan of the Medieval City of Paris which illustrates that the Roman Cardo Maximus was probably the route chosen (if he actually did) by J F Fernel as the Royal Palace to St Denis route, as Diagram MsJFF/1/D15 illustrates did not continue adequately south to the Mont Ste Genevieve and the College of Ste Barbe. Thus did J F Fernel measure the whole in sections such as the Royal Palace to St Denis and it is not a continuous measurement at all and thus suffers from the inaccuracy it portrays and from the way the texts are written.


The text is so lacking in real detail apropos these details that I contend it is an attempt to regularise his studies as exemplified by his annotation to the teacher at Ste- Barbe in his De Proportionibus accreditation. But why were they published, and who paid? Who were they written for, and who purchased the books? Were they in fact known widely in the 1530’s and is it a factor that these are just now interesting. Yes, they were possibly read by eminent persons a century later, but it is obvious from their comments they probably never read them all, but used other person’s analysis on which to base their comments.
If you just read the texts as written and forget the hyperbole, problems abound!

M J Ferrar June 2017.