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Investigating Euler's Polyhedral Formula Using Original Sources - Euler's Proposition 1

Lee Stemkoski (Adelphi University)


Next we shall consider some of Euler's results -- propositions and corollaries -- in the original Latin! In particular, we hope to convey to the reader that the language is not an obstacle to be overcome but a part of the experience, and something to be enjoyed along the way.

Proposition 1 (paragraph 21) says (in part):

In quouis solido numeris omnium acierum est semissis numeri omnium angulorum planorum. . . .

Figure 4. Euler's Proposition 1.

Again, we emphasize that the reader knows more Latin than they suspect. Some of these words are already familiar: solido means "solid", acierum means "edges", angulorum planorum means "plane edges. It is a fair guess that numeris means "number". The word omnium is a bit harder, but the root, omni, appears in many words: omnipotent (all-powerful), omniscient (all-knowing), etc. Thus we will guess (at least for the purposes of a first pass) that omnium means "all". The word semissis also appears intimidating, but the root, semi, is also common in English, and we will translate it as "half". The word quouis we check in a Latin-to-English dictionary and find that it refers to an indefinite object (whoever/whichever/whatever); based on context, we'll try "whatever". Finally, we try the smaller words. A glance at our dictionary reveals that in is a preposition which could have various meanings; here we will try using "in" as our translation. The word est will be recognized by those familiar with French as a conjugation of that all-important verb "to be"; we will translate est as "is". Thus our amateur translation of Proposition 1 is:

In whatever solid number all edges is half number all plane angles. . . .

Making a second pass and inserting a few articles from English will increase the readability:

In whatever solid the number of all edges is half the number of all plane angles. . . .

Mathematically speaking, this is the formula:

\(E = \frac{1}{2}P\)

We have now uncovered the first formula in the paper! Of course, the next natural question to ask is: Why is this true? Depending on one's goals, one could use guided discovery ("Since E is an integer, what must be true about P? It must be—it is necessary that it is—divisible by two, or in other words, P is even!"), students could reason this out without reading further, students could translate Euler's explanation (as Euler calls it, the Demonstratio), or groups could work on these tasks in parallel.

Continuing on, let us examine Corollary 2 (paragraph 23) of this Proposition:

Si igitur omnes hedrae ambitum solidi cuiuspiam constituentes fuerint triangula, earum numerus necessario erit par. . . .

Figure 5. Euler's Corollary 2 to Proposition 1.

There are some familiar words here, and a few new unfamiliar ones. Euler keeps the total amount of vocabulary required for reading his papers relatively low, so the translations will eventually become easier as we progress through the paper.

Looking up the word si in a dictionary, we translate it as "if"; we translate igitur as "therefore". The word omnes appears similar in root to omnium from the previous Proposition, so we translate it as "all" once again. Hedrae will still mean "faces". The word ambitum appears to have many possible translations, so we will choose one possible definition that seems general for now and try to find a better choice of meaning later once we have a better idea of the context. We will choose "around" for ambitum. Solidi will still mean "solid". Cuiuspiam appears to mean "any". Constituentes appears similar in root to the English word "constitute". Fuerint is another form of the verb "to be", which we loosely translate as "is". We can guess that triangula means "triangle(s)" or "triangular"; again, the form we need will be more clear once we have an idea of the context. Looking up earum, we find the meaning "they". Numerus we will guess as "number" again, and necessario looks a lot like the word "necessary", so that will be our guess for now. Erit is yet another form of "to be", and again we will translate it as "is" until it is clear what conjugation we require. Finally, par is an interesting word with many definitions, such as "mate", "spouse", or "partner". Reading down the list of possibilities in our dictionary, we find a particularly mathematical translation: "even" (as in "even number"), which we will use as our guess for now. Upon reflection, we note that par is the root of "parity".

Thus, our very rough first translation is:

If therefore all faces around solid any constitute is triangle(s), they number necessary is even. . . .

This translation isn't perfect, but remember, our main goal is to locate and understand the mathematical content. Euler appears to be saying that:

If all faces around any solid are made from triangles, their (total) number is necessarily even. . . .

The word "therefore" occurs at the beginning of Euler's sentence since this is a Corollary; that this result appears as a Corollary indicates that the main part of the argument will follow from the truth of the Proposition previously mentioned. "The Proposition is true, therefore the Corollary is true" is a standard and well-known presentation format to today's mathematicians; however, this dependence is not obvious to students and hence appropriately emphasized by inclusion of the word "therefore".

Now, assuming that we are comfortable with the mathematical content of this paragraph, let us investigate it in more detail. The proof is another good topic for discussion or guided discovery... why is this result true? Since this is a Corollary, as mentioned above, it should depend on the Proposition, which was

\(E = \frac{1}{2}P\)

If the faces of a solid are all triangles, what do we know about the value of P? What do we know about the value of E when the number of triangles is odd? Try some examples. What about the value of E when the number of triangles is even? Again, Euler has a nice, gentle progression of ideas suitable for exploration in the classroom.

Lee Stemkoski (Adelphi University), "Investigating Euler's Polyhedral Formula Using Original Sources - Euler's Proposition 1," Convergence (April 2010), DOI:10.4169/loci003297