The visual display of planetary latitudes.

Although reprinted in black-and-white, a planetary graph of this type appears on page 28 of Edward Tufte’s Visual Display of Quantitative Information. Yesterday evening, while the first few snowflakes fell, I thought of writing a quick post on the eleventh-century Vatican Pal. lat. 1577 (fol.82v) to provide some context for this image, and a partial correction. Together with the Commentariorum in Somnium Scipionis of Macrobius, Plato’s Timaeus (through Calcidius) and Martianus Capella, certain passages in Pliny’s Historia Naturalis were the principal sources of Carolingian planetary theory. It is far from unique: there are at least 26 other graphs like this, charted on the vellum of manuscripts which lie scattered on shelves from the shadow of the Radcliffe Camera to the streets of Geneva. Each visually models the data for planetary latitudes derived from Pliny, HN II. The manuscript at hand opens with: cur aut[em] magnitudines suas (et) colores mutent (et) ead[em] ad septe[n]triones accedant abeant[que] ad austru[m] latitudo signiferi (et) obliquitas facit. That is to say: As to why they [the planets] change in color and magnitude, approach from the north and recede to the south, this is from the latitude and [zodiacal] obliquity. Pliny proceeds to give the following values for the inclinations of the planets to the plane of the ecliptic: Venus 14°, the Moon, 12°, Mars 4°, Jupiter 3° and Saturn 2°. Thus the y-axis of the diagram indicates the 12° zodiac, centered on the solar ecliptic; the x-axis depicts the path of the planet within its limits. Pliny’s confusing descriptions often began from the heliacal rising of the respective planets, but the extant diagrams seem to have been devised for didactic purposes, and not precise calculation.

Due to the extensive scholarship of Bruce Eastwood and Gerd Graßhoff, the 1936 Osiris article that Tufte used in 1983 has been superseded, but mistaken descriptions (which follow from Tufte) have accumulated across the internet like fallen snow. Nevertheless, the clarity of the visual model remains. Yesterday, I almost posted this note, but found that somebody else (who shares my enthusiasm for stemmatic diagrams) had already raised this point. Like the galerie des Glaces on the north-west side of Versailles, the web offers an apparently endless progression of imperfect reflections.

Doppelmayr and the illustration of early modern physics.

During his long tenure at professor of mathematics in Nürnberg, Johann Gabriel Doppelmayr (1677-1750) translated the Astronomia Carolina of Thomas Streete into Latin, composed a book on sundials, wrote on spherical trigonometry, and rendered the sectors, calipers and scales of Nicolas Biot into German. He wast most famous for his celestial atlas, but also compiled an influential textbook on contemporary electrical theory, the Neu-entdeckte Phaenomena von bewunderns-würdigen Würckungen der Natur (1744). In this, he compiled the experiments of Hauksbee, Dufay, Pieter van Musschenbroek, and Willem ‘sGravesande into a single volume of accessible gebrochene Schrift, complete with folded engravings which have since become quite famous.

The image above depicts several of the experiments conducted by Stephen Gray (1666-1736), a dyer from Canterbury and amateur natural philosopher, who began the first sustained experiments in electrical conduction while living as an impoverished pensioner in Charterhouse in Smithfield, London. Doppelmayr extracted these primarily from Gray’s descriptions, which were published in Philosophical Transactions as a letter to Cromwell Mortimer (vol.37, 1731); in this, Gray used a cork-capped flint-glass tube and bit of eiderdown for his first experiment (fig.1), before moving on to suspend guineas, shillings, brick, tiles and chalk from the line; he then used flakes of leaf-brass and lengths of cane and silken thread to observe electrical conduction at significant lengths, and in various directions. After successfully charging a paper world-map, umbrella and soap-bubble, Gray attempted an experiment using a volunteer from Charterhouse, in sessions from 8-21 April 1730, by suspending him from the ceiling with clothesline, and placing leaf-brass on paper plates. The lower diagrams are from Gray’s collaborator, J. Desaguliers, and suggest a preliminary question. How does one visually depict force in an engraving, and capture the passage of time in the amber of a printed page?

The half-redacted sphere of Pythagoras.

Over the passage of decades, layers of meaning and controversy can gather in manuscripts like accumulated seasons of fallen autumn leaves. At some point in the ninth century, a scribe in the Abbaye de Saint-Amand in the Foret de Vicoigne in far northern France copied a creed ascribed to St. Ambrose in tiny Carolingian minuscule on what is now fol.162v (which has parallels in the Codex Muratorianus in Milan, fol.73v), but left a wide expanse of empty parchment below. At some point, the manuscript passed to the Abbey and Altenmünster at Lorsch, in the Bergstraße district of Hesse, and an opportunistic scribe used a later, thorny, script to complete a Sphaera Pythagoras in the vacant space.

There are many such iatromathematical devices in medieval manuscripts, and were used for simple prognostication. If a fellow monk fell ill, the idea was to assign the Roman numerals in the column to the left with the respective letters of his name, add them together with the day of the synodic month on which the illness first occurred and the day of the week, and divide by 30. If the remainder is in the upper half of the crossbones, the patient will live; if below, they will either perish quickly or linger in sickness until consumed by death. Really, the thing reminds me of nothing more than the origami paper ‘cootie-catchers’ we used to make in elementary school. But apparently, it meant something much more serious to a third monk, who wrote anathema above, and quickly crossed out the column, x-y axis of the disc, and instructions with quick strokes of red ink. I assume the heresy was not quite enough to deserve more than a hasty redaction, for despite the warning, the sphere can be easily read.

Johannes Trithemius and the orthographies of invention

Soon after Johannes Trithemius (1462-1516) arrived at the Schottenkloster in Würzburg, he was again expressly invited to join the company of the Holy Roman Emperor Maximilian I (1459-1519) in Mainz and Cologne; his Polygraphiae was still in process. In his efforts to establish Trojan origins for the Hapsburgs, Trithemius managed to fabricate the existence of manuscript which contained the chronicles of Hunibald, supposedly a contemporary of Clovis (c.466-511); like Russian nesting dolls or black lacquered Japanese concentric boxes, the invention of Trithemius claimed to be based on older sources: the Scythian Wasthald, Heligast and Doracus, who traced the exploits of the ancient Franks to the wellsprings of history and the Fall of Troy. After inquiries, Trithemius claimed the manuscript was in Sponheim, or possibly the cloister library at Hirsau Abbey, but after repeated efforts, no manuscript of the sort was ever found. Until his death, Trithemius was derided as a forger by Konrad Peutinger and Johannes Stabius. All of this has been well treated by Tony Grafton in his most recent collection of essays.

What I recently found remarkable is the alphabet above, which I found in the 1550 edition of the Polygraphiae (fol. gir). Trithemius supposedly transcribed this from Wasthald, and it seems to visually link Gothic to Greek. Of course, Trithemius claimed the codex was difficult to read, but certain of the characters are genuinely rare (the doubled H for m can be found in a manuscript in St. Germain des Prés), and my access to an obvious source, the very real Frankish Benedictine Hrabanus Maurus (c.780-856), is distorted by the poorly printed entry in Migne’s Patrologia Latina. In any case, these letters were passed along by Henricus Cornelius von Agrippa to the cursory attention of Gerard Johannes Vossius in his Aristarchus, sive de arte Grammatica (vol. I.p.38), thence to the criticism of George Hickes and various Enlightenment-era French paleographers. In time, individuals like Johannes Goropius Becanus would take arguments drawn from etymology and inscriptions to entirely new levels of effort.

Terminus: a copy and imperfect mirror.


In his 1848 Gromatici Veteres (a title which makes me think of Nick Park’s stop-motion animation), Lachmann relied on the famous (and obviously ancient) Codex Acerianus. But he also used another manuscript from the former ducal library at Wolfenbüttel (Cod. Guelf.105 Gudianus Lat.) as a principal source, and relegated the collations of Vatican Pal. lat 1564 to the variant readings at the bottom of the page. Lachmann’s favored manuscript was from the collection of the German philologist Marquard Gude (1635-1689), and was acquired for the Herzog August around 25 October 1709 (I think), by Gottfried Wilhelm Leibniz. Quite simply, Codex Gud.105 is a copy of Pal. lat. 1564 from late in the ninth century, but the correct order of descent was not decisively established until the Swedish scholar Carl Olof Thulin (1871-1921) published his Die Handschriften des Corpus Agrimensorum Romanorum in 1911 (ibid.58-69).

Of course, he used methods pioneered by Lachmann decades before. Although the paths the two codices took through the passage of centuries were quite distinct, we can witness the imperfect mirror formed from an image and its copy. Unfortunately, few current digitization projects provide us with tools to explore these paths of transmission, circulation and reconstruction.  On this point, I will not yield.

Terminus: concedo nulli

The image above is from a lavish Carolingian manuscript ( fol.50v) made at Aachen c.825. The manuscript incorporates fragments of various Roman Agrimensores, or gromatic writers, including Frontinus and Siculus Flaccus, who wrote technical treatises on the rather complicated practice of surveying land in Rome. As a whole, the Latin of the fragments is quite corrupt, and was first systematically analyzed by the superb classical scholar Karl Lachmann in 1848. For this particular passage, ascribed to a certain Latinus v.p. Togatus, the text reads: Terminus sive petra naturalis si branca lupi habuerit facta, arborem peregrinam significat. Terminus sive petra naturalis si branca ursi habuerit lucum significat. So, among buried shards of pottery, Greek letters gamma and coins, we have a boundary-stone with a wolf’s claw signifying an exotic tree (if we are to remember Pliny, the peach tree comes to mind, as does a scribal error for a third-declension plural). We also have a stone with a bear’s claw (branca ursi) that signifies a sacred grove.

Interestingly, small illuminations of blocks of stone are inserted directly into the text; they are nothing like the Roman adaptation of Greek herma to which Erasmus applied his motto. But one remembers the stories in Plutarch and Appian of wolves clawing at the boundary-stones of the Gracchi. Here, the images refuse to yield their place to text, and both are set in the vellum to provide a single, lucid explanation, and it’s strange to imagine this manuscript crossing the alps on the back of a mule after the fall of Heidelberg in 1622.

Galileo, the diagram, and the physical object

Around 1598-1600, Galileo Galilei composed a compact and elegant treatise on mechanics for his pupils at Padua; from fol.16r from his manuscript at the BCN Firenze, Ms.Gal.72, we have the simplest example I could think of to illustrate the contemporary tension between his diagrams and drawings. At the top right, we have a crosshatched cube of shadowed stone atop a triangular fulcrum and plank of wood, which sits in stasis, and seems to occupy space. Below, the lever is abstracted into a geometric diagram, in two sequential states. As Galileo explains: Impero che, ripigliando la med(esima) [linea] BCD, della quale sia C il sostegno, e la distanza CD pogasi…per essempio quintupla alla distanza CB, e mossa la lieva sin che pervenga al sito ICG. That is, for again assuming the…[lever] BCD, with fulcrum C, let the distance CD be taken to be five times the distance CB, and move the lever to position ICG. In that time the the force will have moved through space DI, and the weight will have moved from B to G. Because the distance DC was five times that of CB, it is clear that the weight placed in B may be five times greater than the force applied in D. In the pages of this manuscript, Galileo would fuse mathematical and physical arguments in the same way he linked geometrical diagrams (which would begin to depict spatial and temporal changes) to drawings of physical objects. But at this point, the tension is still evident.

Of course, one should refer to Michael Mahoney’s lucid critique of Samuel Edgerton et al. (Diagrams and Dynamics) before reading a lasting revolution into the marginal strokes of Galileo’s illustrations alone: the emergence of analysis cannot be ignored.

Radical departures from the text of Apollonius

 The symbol Rx, which we see on prescription slips from the physician’s office or in the windows of pharmacies, was once most commonly a brevigraph for recipere, which the third conjunction allows us to identify as the present infinitive of recipio, and which leads us to the countless quoted passages of things being taken back, or simply taken, in the pages of the Lewis & Short. But Lucia Pacioli also used the Rx glyph to signify radix or radices in the margins of his Summa Arithmetica (Venice, 1494). From radix we derive our word radish, which is, after all, an edible root.

If you look closely, you might detect this symbol in the image above, of fol.3v of the first Latin translation of the Conics of Apollonius of Perga, which was printed in Venice in 1537; a variant of Pacioli’s m for meno (-) also occurs, clearly demonstrating Giovanni Battista Memo’s debt to his predecessor. Ultimately, Frederico Commandino was a much better scholar, and his two-volume 1566 Apollonii Pergaei Conicorum Libri Quattuor would not be surpassed until Edmond Halley’s edition, the late (1654) appearance of Francesco Maurolico notwithstanding. But digressions on the complex and fascinating textual history of the Conics must wait for future posts.

For now, the most interesting aspect of Memo’s Latin editio princeps is his curious use of arithmetic to explain the geometrical propositions, which reveals far more about how mathematics was read in sixteenth-century Venice than it does about about the propositions themselves.

Visualizing the order of the world in Isidore of Seville

On folio 138r of Boulogne-sur-Mer Bibliothèque Municipale Ms.0001 (another edition of the tremendously popular Etymologiae of Isidore of Seville), we have a small illustration of his description of the Earth (XIV.2), dovetailed above the decorated A. In the manuscript, the passage reads:

Orbis a rotunditae circuli dictus, quia sicut rotus est. Unde brevis ae(tiam) rotella, orbiculus appellat(ur). Undique enim occeanus circumfluens eius in circuitu ambit fines. Divisus e(st) autem trifaria. Ex quibus una pars asya, altera europa, ter(tia) affrica que libya nuncupatur…

This basically translates as: “the world (orbis) is named for the roundness of a circle, for it is like a wheel; thus a small wheel is called a little disc. In fact, the Ocean flows around it, and contains its boundaries in a circle. It is divided into three parts: Asia, then Europe, and the third is called Africa or Libya.” If taken literally, the characteristic T/O map results: our example is a circle of red ink within the green circumference of the encircling ocean, and divided into Europe, Africa and Asia, oriented toward the east. The square V represents the directions taken by Shem, Ham, and Japeth, the sons of Noah, after the waters of the flood receded, and the earth became fertile and dry.

As a simple aid to visualizing the ambiguous passage of Isidore, this example is much less complex than many other tripartite mappa mundi, and cannot quite qualify as an attempt at cartography. Nevertheless, the conceptual beauty of a tiny model of the world (whether a disc or drawn hemisphere) within the vellum darkness of a manuscript recalls the last lines of the Wallace Stevens poem The Planet on the Table:

It was not important that they survive.

What mattered was that they should bear

some lineament or character,

Some affluence, if only half-perceived,

In the poverty of their words,

Of the planet of which they were part.

Mapping the shadow of the earth.

In May, 1555, the Bohemian astronomer, mathematician and astrologer Cyprián Karásek Lvovický (c.1514-1574) dedicated this hybrid print/manuscript (Bayerische Staatsbibliothek Cod.icon 181) treatise on lunar eclipses through 1600 to Ferdinand I, King of Bohemia and Hungary, and future Holy Roman Emperor. It was based upon data established in the Prutenic Tables of Erasmus Reinhold. On fol. 36v the coordinates for a lunar eclipse on 31 January 1580 were entered into blank and printed tables by hand with red ink; folio 37r depicts an idealized schematic for the eclipse above an illustration of a night city like Prague or Augsburg, with a plum-colored moon above the central spire of the cathedral. Torches are held to the stars, and a single figure plays a lute. Incidentally, as we assume the Julian Calendar was used, there was a lunar eclipse from Saros 107 that occurred on 31 Jan. 1580; it would have been visible over Eritrea and western Ethiopa. With some adjustments, it seems the occurrence on a night 25 years in the future was predicted sufficiently enough.