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This new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.
Bulletin of the American Mathematical Society, vol. 23, 1916.
Investigating Euler's Polyhedral Formula Using Original Sources
Euler's Proposition 2
As we progress through the paper, our vocabulary becomes larger and the propositions become easier to comprehend. Euler's second Proposition (paragraph 26) is:
Figure 6: Euler's Proposition 2
Vel numerus angulorum planorum nunquam minor esse potest, quam triplum numeri hedrarum ambitum solidi cuiusquam.
Many of these words are familiar by now; we discuss those which are new. The word vel means "or"; the construction "vel ... vel" means "either ... or". We may guess that aequalis means "equal(s)". The word maior looks unfamiliar, but we may try swapping the "i" with a "j" as described earlier. Major we will translate as "greater". The word ter is the root of words such as "tertiary"; we will thus translate ter as "three". Sumto appears to have sum as a root; this might mean "sum" or "addition". In parallel with our translation of major above, we will translate minor as "lesser". Using our Latin-to-English dictionary, we guess that esse will mean "is", potest will mean "possible", quam could mean "who", "what", or "which", and nunquam could mean "at no time", "never", or "in no circumstances". It is a reasonable guess that triplum means "triple". And so our rough first translation becomes:
Or number plane angles never lesser is possible, which triple number faces around solid any.
Gently rearranging words and using articles, our attempt at translation yields:
The number of plane angles can never possibly be less than triple the number of faces around any solid.
Mathematically speaking, this proposition states (in modern notation) that:
It is never the case that P < 3F.
This is slightly more complicated than the last proposition. Students could find the proof using guided discovery or by translating Euler's line of reasoning -- whatever method works best for the course and students at hand. Again, it might be helpful for students to see examples of solids for which the equality holds and for which the inequality holds. Teachers could prepare a batch of examples for use in the classroom, or see what Euler has to say in Corollary 1 to this proposition:
Figure 7: Euler's Corollary 1 to Proposition 2
The only new word here is ergo, which we may recognize as part of the famous quote "cogito ergo sum" -- "I think, therefore I am". Thus we translate ergo as "therefore", and our rough first translation is:
A slight revision would yield:
Again, the word "therefore" arises because a corollary follows from the proposition appearing before it. Euler has given us an example of when equality holds in Proposition 2. Why not ask students to check it for the tetrahedron, octahedron, and dodecahedron?
Let us examine the second corollary of Proposition 2.
Figure 8: Euler's Corollary 2 to Proposition 2
quia numeris angulorum planorum = 2A, erit vel 2A = 3H vel 2A > 3H.
Impossibile ergo est, ut sit 2A < 3H.
The further we progress, the easier the translations become! New words appearing here include ponatur, which could mean "put" or "set" (we will use the latter), quia, which we look up and find means "because", and ut, which we will translate as "while". The rough translation is:
because number plane angles = 2A, is either 2A = 3H or 2A > 3H.
Impossible therefore is, while is 2A < 3H.
Cleaning up the translation (and using modern notation), we have:
because the number of plane angles equals 2E, it is either [the case that] 2E = 3F or 2E > 3F.
While therefore it is impossible that 2E < 3F.
The proof of this is easy: we are doing a simple substitution, combining Proposition 1 and Proposition 2. This would be such a small step that most authors would omit it, much less set it apart as a corollary by itself. Euler, however, is not most authors. He sets a good pace for students of mathematics, from any century, seeing this for the first time.
Proposition 3 is similar to Proposition 2, and therefore we omit a discussion of it here, although it is still important and should be discussed as it will be important later. Proposition 3, in modern notation, states that
The second corollary to this proposition states that:
Stemkoski, Lee, "Investigating Euler's Polyhedral Formula Using Original Sources," Loci (April 2009), DOI: 10.4169/loci003297
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