Month: February 2018

More fun with Ptolemy’s Ὁ Καρπός

Our premodern reader didn’t just add Latin glosses to his copy of Ptolemy’s Ὁ Καρπός, now and then he emended the Greek. For example, on the second aphorism the copiest wrote “τὴν κρεῖττον”. The reader seems to have been sufficiently bothered by this mistake that he wrote the correct article, “τὸ,” above the incorrect “τὴν.”

The second and third aphorism from the copy of Ptolemy’s Ὁ Καρπός in BnF gr. 2180.

Here’s a transcription of second and third aphorisms (including sigma chaos):

ὅτε ἐπιζητήσει τὴν κρεῖττον ὁ βουλόμενοϲ, οὐκ ἔσται μεταξὺ αὐτοῦ καὶ τῆϲ ἰδέαϲ ἀυτοῦ τοῦ πράγματοϲ διαφορά τισ.

ὁ πρόϲ τι πρᾶγμα ἐπιτήδιοϲ ἕξει πάντωϲ καὶ τὸν δηλοῦντα [ἀστέρα] τοιοῦτον ἐνδύναμον ἐν τῷ ἰδίῳ γενεθλίῳ.

In other manuscripts Ptolemy’s second aphorism two seems displays a bit of drift toward the end, where “τοῦ πράγματος διαφορά” sometimes becomes the grammatically problematic “ἀυτοῦ πράγματων διαφορά” (as in Biblioteca Angelica, gr. 029 below) and sometimes the grammatically ok “τῶν πραγμάτων διαφορά.” The wording in the third aphorism likewise varies a bit toward the end, where “ἰδίῳ γενεθλίῳ” becomes “ὀικείῳ γενεθλίῳ.” Two 14th-century manuscripts illustrate these variations. Biblioteca Angelica, gr. 029 shows the variation in both aphorisms; Biblioteca Vallicelliana, fonds principal F 086 illustrates the variation in just the third aphorism.

Variations of Aphorisms β’ and γ’ from 14th-cent. Mss.
Aphorism Biblioteca Angelica, gr. 029 Biblioteca Vallicelliana, fonds principal F 086
β’ ὅτε ἐπιζητήσει τὸ κρεῖττον ὁ βουλόμενος, οὐκ ἔσται μηταξὺ αὐτοῦ καὶ τῆς ἰδέασ ἀυτοῦ πράγματων διαφορά τίσ. ὅτε ἐπιζητήσει τὸ κρεῖττον ὁ βουλόμενος, οὐκ ἔσται μεταξὺ αὐτοῦ καὶ τῆς ἰδέασ τοῦ τοῦ πράγματος διαφορά τις.
γ’ ὁ πρόσ τι πρᾶγμα ἐπιτήδειοσ, ἕξει πάντωσ καὶ τὸν δηλοῦντα [ἀστέρα] τὸ τοιοῦτον ἐνδύναμον ἐν τῷ ὀικείῳ γενεθλίῳ. ὁ πρός τι πρᾶγμα ἐπιτήδειος, ἕξει πάντωσ καὶ τὸν δηλοῦντα [missing ἀστέρα] τὸ τοιοῦτον ἐνδύναμον ἐν τῷ ὀικείῳ γενεθλίῳ.

This particular variation carries over into some fifteenth-century manuscripts, e.g., Vatican Barb.gr.127 or BnF gr. 2027, and even sixteenth-century copies, e.g., Harley ms 5597.

The third aphorism from Ptolemy’s Ὁ Καρπός, with the common variant highlighted (BL Harley ms. 5597).

As promised in the previous post, Struggling with Ptolemy’s Ὁ Καρπός, here are initial translations of the second and third aphorisms (with the same caveat that I reserve the right to admit this translation is horrible and to change it).

Translations of Ptolemy’s Aphorisms β’ and γ’ from BnF gr. 2180
Aphorism Greek English
β’ ὅτε ἐπιζητήσει τὴν (corrected above to τὸ) κρεῖττον ὁ βουλόμενοϲ, οὐκ ἔσται μηταξὺ αὐτοῦ καὶ τῆς ἰδέαϲ ἀυτοῦ τοῦ πράγματοϲ διαφορά τισ. When anyone seeks the superior thing, there will not be a particular difference between it and the form of the thing itself.
γ’ ὁ πρόϲ τι πρᾶγμα ἐπιτήδιοϲ ἕξει πάντωϲ καὶ τὸν δηλοῦντα [ἀστέρα] τοιοῦτον ἐνδύναμον ἐν τῷ ἰδίῳ γενεθλίῳ. He who is suited to a particular activity will certainly have the star signifying such an influence in his own nativity.

Study of the variations and other scholarly contributions will have to wait. For the moment, I’m just having fun working through the copy of Ptolemy’s Ὁ Καρπός in BnF gr. 2180. I confess, I’m also enjoying the amazing botanical illustrations, such as this one:[1]

One of the many amazing botanical illustrations from BnF 2180.

  1. If you’re interested in Byzantine herbals and botanical illustrations, this ms is loaded with amazing images of plants. Unfortunately, many of the illustrations were never added to the manuscript, just large blank spaces awaiting a skilled illustrator.  ↩

Doritos™ Alchemy

I recently stumbled across this spec Doritos™ commercial David Ward made a six years ago. Really well done adaptations of the alchemist motif.

You gotta love the list of ingredients (also, not listed, Unicorn tears):

Staff of Anubis
Philosopher Stone
Rubber Hammer
Rubber Nails
Moon Rock Salt
Archimedes Screw
Harpsichord
Parachute
Blank
Bag of Holding
Cloud Mist
Elven joy
Lucky Penny
Love Song
Erlenmeyer Flask
Marcoscope
Sense of Wonder
Blankety Blanks
Temporal Glitch
Haiku
Nods
Sweeps
Beeps
Deeps
Sneeps
Reeps
Winks
Memories
Fireballs
Congratulations
Laughter
Lightening
Star Dust
Rings of a Tree
Mother’s Approval
Mountain Air
Cheesiness
Inspiring Footage
Smiles
Secret Ingredient
Smell of Morning
Love
Salt

Nice job guys—David was joined by Jack Dreesen, John Ramsey, and Byron Brown. I enjoy creative work like this.

Struggling with Ptolemy’s Ὁ Καρπός

It is perversely reassuring to see that other people have had to labor to understand Ptolemy’s aphorisms.[1] Consequently, this 15th-century copy of Ptolemy’s Ὁ Καρπός (more widely known by its Latin title, Centiloquium) makes my day.

Copied sometime in the latter half of the fifteenth century by a certain George Mediates, this manuscript was later owned by Jean Hurault de Boistaillé, who amassed an impressive collection of Greek manuscripts, see, e.g., this list.[2] Philippe Hurault de Cheverny inherited Boistaillé’s manuscripts. Shortly after his death in 1620 the collection was purchased for the Bibliothèque royale de France.

A page from a manuscript copy of Ptolemy’s Ὁ Καρπός with numerous Latin interlineations.

Some premodern reader worked through Ptolemy’s text adding Latin translations above most of the Greek words. He worked diligently through the first 50 or so aphorisms, adding such interlinear glosses throughout. Then he suddenly stopped.

An enlargement showing more clearly the interlinear, Latin glosses.

Here is a transcription of the preface and the first aphorism (most of what you see in the image immediately above):

βιβλίον κλαυδίου πτολομαίου ὁ λεγόμενοϲ καρπόϲ κεφαλεα ρ’
Προεκθέμενοι, ὦ σῦρε, τὰϲ ἐνεργείαϲ τῶν ἀστέρων τὰϲ ἐν τῷ ϲυνθέτῳ διενεργουμένας κόϲμῳ κατὰ πολλὺ λυσιτελεισ, ουσας πρὸϲ τὴν πρόγνωσιν, καὶ τὸ παρὸν ἐξεθέμεθα πόνημα, ὅπερ καρπόϲ ἐστι τῶν βιβλίων ἐκείνων, γυμνασθὲν διὰ τῆσ πείρασ τὴ ἀληθεία σύστοιχον

δεῖ οὖν τὸν μέλλοντα τοῦτω απιέναι προότερον διελθεῖν τὰϲ τῆσ ἐπιστήμης ἁπάσας μεθόδους, εἶτα πρὸϲ τὴν ἀνάγνωσιν τουτῶν χωρῆσαι.

ἀπὸ σοῦ καὶ ἀπὸ τῆϲ ἐπιστήμηϲ οὐ γάρ ἐστι δυνατὸν τὸν ἐπιστήμοναν τὰϲ μερικὰσ ἰδέαϲ τῶν πραγμάτων ἀναγκεῖλαι, ὥσπερ οὐδὲ ἡ αἴσθησισ δέχεται τὴν μερικὴν ἰδέαν τοῦ αἰσθητοῦ ἀλλά τιναν γενικήν. καὶ δεῖ τὸν μετιόντα καταστοχάζεσθαι τῶν πραγμάτων μόνοι γὰρ οἱ ἐνθουσιοντες προλέγουσι τὰ μερικά.[3]

A number of things about this copy interest me, starting with the Greek itself. Perhaps the least significant: this scribe ranges freely across the three forms of sigma with no apparent rhyme or reason: the typical internal form, σ, occurs frequently at the end of words; the typical Byzantine form, ϲ, appears at the beginning and end of words; and the terminal form, ς, appears only occasionally at the end of a word. Further, the scribe either misspells a number of words or, as seems possible in some cases, spells them to capture pronunciation. Then there are the places where the wording itself varies from other copies.

For those who care, here’s a comparison of the preface and first aphorism in BNF gr. 2180 and those in the now quite old critical edition (I added line breaks to the critical edition text to make it easier to compare to BNF gr. 2180).

A Comparison of BNF gr. 2180 and the critical edition
BNF gr. 2180 Critical edition
Preface Προεκθέμενοι, ὦ σῦρε, τὰϲ ἐνεργείαϲ τῶν ἀστέρων τὰϲ ἐν τῷ ϲυνθέτῳ διενεργουμένας κόϲμῳ κατὰ πολλὺ λυσιτελεισ, ουσας πρὸϲ τὴν πρόγνωσιν, καὶ τὸ παρὸν ἐξεθέμεθα πόνημα, ὅπερ καρπόϲ ἐστι τῶν βιβλίων ἐκείνων, γυμνασθὲν διὰ τῆσ πείρασ τὴ ἀληθεία σύστοιχον.

δεῖ οὖν τὸν μέλλοντα τοῦτω απιέναι προότερον διελθεῖν τὰϲ τῆσ ἐπιστήμης ἁπάσας μεθόδους, εἶτα πρὸϲ τὴν ἀνάγνωσιν τουτῶν χωρῆσαι.

Προεκθέμενοι, ὦ Σῦρε, τὰς ἐνεργείας τῶν ἀστέρων τὰς ἐν τῷ συνθέτῳ διενεργουμένας κόσμῳ καὶ πολὺ λυσιτελούσας πρὸς τὴν πρόγνωσιν, καὶ τὸ παρὸν ἐξεθέμεθα πόνημα, ὅπερ καρπός ἐστι τῶν βιβλίων ἐκείνων, γυμνασθὲν διὰ πείρας.

δεῖ οὖν τὸν μέλλοντα τοῦτο μετιέναι πρότερον διελθεῖν τὰς τῆς ἐπιστήμης ἁπάσας μεθόδους, εἶτα πρὸς τὴν ἀνάγνωσιν τούτου χωρῆσαι.

First aphorism ἀπὸ σοῦ καὶ ἀπὸ τῆϲ ἐπιστήμηϲ οὐ γάρ ἐστι δυνατὸν τὸν ἐπιστήμοναν τὰϲ μερικὰσ ἰδέαϲ τῶν πραγμάτων ἀναγκεῖλαι, ὥσπερ οὐδὲ ἡ αἴσθησισ δέχεται τὴν μερικὴν ἰδέαν τοῦ αἰσθητοῦ ἀλλά τιναν γενικήν. καὶ δεῖ τὸν μετιόντα καταστοχάζεσθαι τῶν πραγμάτων μόνοι γὰρ οἱ ἐνθουσιοντες προλέγουσι τὰ μερικά. Ἀπὸ σοῦ καὶ ἀπὸ τῆς ἐπιστήμης οὐ γάρ ἐστι δυνατὸν τῷ ἐπιστήμονι τὰς μερικὰς ἰδέας τῶν πραγμάτων ἀναγγεῖλαι, ὥσπερ οὐδὲ ἡ αἴσθησις δέχεται τὴν μερικὴν ἰδέαν τοῦ αἰσθητοῦ ἀλλά τινα γενικήν. καὶ δεῖ τὸν μετιόντα καταστοχάζεσθαι τῶν πραγμάτων· μόνοι γὰρ οἱ ἐνθουσιῶντες προλέγουσι καὶ τὰ μερικά.

I have nothing profound to add to this post, no insight to give. The goal of this post was merely to draw attention to this Byzantine copy of Ptolemy’s Ὁ Καρπός because, well, I find it interesting. Now back to work on this text.


  1. Modern scholarship has shown to its own satisfaction that the collection of aphorisms attributed to Ptolemy, well known in Latin as the Centiloquium, were not, in fact, composed by Ptolemy. For the moment, I don’t care if the text was or was not written by Ptolemy. The copiest and the pre-modern owners of this manuscript thought Ptolemy had composed the aphorisms—for my purposes now, that’s more important than insisting on a ps- prefix for Ptolemy.  ↩

  2. See also D. Jackson, “The Greek Manuscripts of Jean Hurault de Boistaillé,” Studi italiani di filologia classica 2(2004): 209–252.  ↩

  3. Translations from the Latin versions of the Centiloquium are easy to find. I know of only one translation from the Greek. A second, it seems to me, could be useful. So I will slowly add translations from the Greek. Here, then, is a really rough translation of the first aphorism (I reserve the right to admit I totally messed up this translation and to improve it when I realize that):

    For it is not possible that the wise man from himself or from knowledge reports the particular forms of events, just as perception cannot grasp the particular form of the thing perceived but a certain general form. And so it is necessary to infer the course of events, for only those inspired by a god can predict the particulars.  ↩

Jon Pace & the Latest Prime Number

In case you missed it, the world has a new largest prime number called M77232917:

46733318335923109998833558556111552125132110281771449579858233859356792348052117720748431109974020884962136809003804931724836 … and on and on and on for another 23,249,300 digits.

That number is unimaginably large. It is so large that even analogies meant to help us wrap our minds around it are nearly unimaginable. Who can imagine a number that takes up 9,000 pages? Or imagine a number that stretches 73 miles?[1] Or imagine writing 5 digits per second for 54 days? Such are the ways mersenne.org tries to describe the number. (I might add, parenthetically, mersenne.org has the nicest website 1996 can buy).

The Great Internet Mersenne Prime Search page is Internet Retro at its finest.

The history of prime numbers in general is fascinating as is the history of Mersenne primes in particular. But I will save those for another post. Here I just want to draw attention to how the newly recognized prime status of this number has been noted in the press.[2] In particular, I’m intrigued by how different reports apportion credit for the “discovery” and how they describe Jon Pace, whose computer first identified the number as prime.

We might as well start with the post at mersenne.org that announces: GIMPS Project Discovers Largest Known Prime Number: 277,232,917–1. In case there was any doubt that the GIMPS project wanted to take credit for the discovery, the post continues:

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 277,232,917–1, having 23,249,425 digits. A computer volunteered by Jonathan Pace made the find on December 26, 2017. Jonathan is one of thousands of volunteers using free GIMPS software available at www.mersenne.org/download/ … GIMPS, founded in 1996, has discovered the last 16 Mersenne primes.

Make no mistake, GIMPS discovered this latest prime, along with the previous 16 Mersenne primes. And although Jon Pace installed and ran the software on his computer, the result had to be checked by four other people who also deserve credit. Oh, and credit goes to all the people who didn’t find a prime. Credit for the discovery, in this version, echoes the distributed nature of the labor but reserves pride of place for the GIMPS software and its developer:

GIMPS Prime95 client software was developed by founder George Woltman. Scott Kurowski wrote the PrimeNet system software that coordinates GIMPS’ computers. Aaron Blosser is now the system administrator, upgrading and maintaining PrimeNet as needed. …
Credit for this prime goes not only to Jonathan Pace for running the Prime95 software, Woltman for writing the software, Kurowski and Blosser for their work on the Primenet server, but also the thousands of GIMPS volunteers that sifted through millions of non-prime candidates. In recognition of all the above people, official credit for this discovery goes to “J. Pace, G. Woltman, S. Kurowski, A. Blosser, et al.”

Thanks for the CPU cycles and paying for the electricity to run your computer, Jon. But don’t feel slighted, GIMPS tends to celebrate its software over the people donating the computer time.

Who is this Jon Pace, well GIMPS say simply “Jonathan Pace is a 51-year old Electrical Engineer living in Germantown, Tennessee.” A nice, reasonable, scientific sounding person.

Science Daily just reposts the GIMPS report, changing the title slightly: “Largest known prime number discovered 50th known Mersenne prime ever found, on computer volunteered in collaborative project.”

Things begin to drift in the NPR story, “A Tenn. Man Recently Discovered The Largest Prime Number Known To Humankind.” Perhaps we shouldn’t be surprised that NPR with its preference for the human-interest story would highlight Jon Pace:

This past week, a FedEx employee from Germantown, Tenn., made a massive discovery — and it wasn’t in any packages. John Pace found the largest prime number known to humankind. …
Pace found his prime as part of an online collective called the Great Internet Mersenne Prime Search, or GIMPS. Pace and thousands of volunteers ran software on their personal computers crunching numbers day-in and day-out.

In this version, Pace discovered the prime and, moreover, its “his prime” (later the article refers to it as “Pace’s prime”). Not only is Pace the only one to get credit for the discovery, he is no longer an electrical engineer. Instead, he’s a FedEx employee. The two are by no means mutually exclusive. I am not suggesting they are. I am, rather, drawing attention to the fact that in the NPR version Pace’s primary marker of identity is his status as a FedEx employee whereas in the GIMPS post his primary identity is as an electrical engineer.

Popular Science adds a bit more to the story, “How a FedEx employee discovered the world’s largest prime number.” In this version Pace gets credit for the discovery, that he used the GIMPS software is just part of the process.

What does change in this version is Pace’s identity. Now he’s not only a FedEx employee—“a flight operations finance manager with the Memphis-based delivery behemoth”— but also a deacon at his church, the Germantown Church of Christ where he does, among other things, IT support. He installed the GIMPS software on one of the computers at the church.

Yes, Jon Pace is a Deacon at the Germantown Church of Christ, for whatever that’s worth.

The NY Times shifts the emphasis again: “How a Church Deacon Found the Biggest Prime Number Yet (It Wasn’t as Hard as You Think).” By this point, Pace is given sole credit for having discovered the prime number. Equally interesting, his identity has shifted to emphasize his role as a deacon at the Germantown Church of Christ:

A prime number discovery in December was made in the unlikeliest of places: on a church computer in a Memphis suburb.…
But for this behemoth to come to light, someone had to have installed free software used to search for Mersenne prime numbers, and that someone is Jon Pace, a deacon, FedEx finance manager and math aficionado who had spent 14 years hunting for such a number.

According to the article, Pace is prouder of “the 20 years [he has] served as deacon at Germantown” than of discovering the new prime number.

Discovery appears to be a rather slippery concept. What does it mean to discover a prime number? In what sense do numbers exist or not exist before we write them down? Surely nobody is suggesting the number 277,232,917–1 was discovered. But its primeness was discovered, or perhaps realized. In this case, discovery seems to be rather traditional, like discovering gold or the New World. Something pre-existed in some way before a human stumbled into it, but the identity, characteristics, and value of that something had to be recognized before we could talk about it and apportion credit. Here we see GIMPS asserting credit for the “discovery” whereas NPR or the NY Times are perfectly happy to give Pace the credit. Credit for the discovery is ambiguously related to labor—GIMPS thinks the intellectual labor that created the software more important than the volunteer labor (and donated electricity and other operating expenses); the NY Times thinks the human who owns the donated computer and pays to operate it more important.

And what about that human, in this case Jon Pace? GIMPS emphasizes Pace’s identity as an electrical engineer, though electrical engineering is only one part of his identity—along with his undergraduate degree in electrical engineering he has an M.B.A., is a flight operations finance manager,[3] and is a deacon at his church. And according to the NY Times, he is most proud of being a deacon. How is Pace’s reported identity related to the values different organizations hold? How does Pace’s identity as an electrical engineer reinforce the image that GIMPS wants to project? Why does Popular Science and the NY Times draw attention to Pace’s identity as a deacon in a church? I don’t believe that GIMPS didn’t know Pace was a deacon, though they chose not to acknowledge that aspect of his identity. The NY Times, by contrast and somewhat surprisingly to me, chose to draw attention to Pace’s work as a deacon at his church.

Discovery, it turns out, is pretty difficult to pin down.


  1. I suspect most people can identify cities that are roughly 73 miles apart but have no concept of what that distance actually is (FYI, it’s about 73 miles from Haverford College to the New Jersey shore, e.g., Ocean City). Now remove the bookending cities and challenge people to think about 73 miles of highway stretching through undulating grasslands or through the desert—suddenly the distance becomes meaningless. And how are we supposed to relate to 73 miles? A person walking will have a very different relationship to that distance than a person who thinks of driving it.  ↩

  2. And yes, that tortured expression, “newly recognized prime status” was intentional, an effort to avoid words like “discovered” and “found,” and an effort to draw attention to what strikes me as interestingly new here, i.e., our recognition that this number has a set of features that we call prime.  ↩

  3. I confess, I have no idea what a flight operations finance manager is. Flight operations manager I can imagine. Finance manager again makes sense. The two together, however, confuse me.  ↩