At first glance, one may be discouraged because the article is written in Latin. However, we may take heart, remembering that Euler took great pains in writing to be understood. In particular, Latin was nobody's first language in the 18th century. Conscientious of this fact, Euler used straightforward vocabulary and linguistic constructions in his writing; the prose is not flowery. Three pieces of advice for the Latin amateur (from a fellow Latin amateur) are:
Most sections will require multiple readings. During a first reading, aim for a general idea of what is being discussed by looking at the roots of words. (Many of the words we use today have their roots in Latin; many examples will follow throughout the remainder of this article.) Do not get hung up on the tenses of words, on the distinction between adjectives and adverbs, and so forth. If the goal was to write a translation, then these details would be important, but we shall assume that the goal is to gain an understanding of the results and an appreciation of the style in which they were originally presented.
Do not be misled by the typesetting. Often times our letter "s" appears to be an "f", while the letter "i" is used in place of a "j", or a "u" in place of a "v". This is not always the case, but it does happen sometimes. If a word looks unfamiliar, trying a letter swap will sometimes reveal a word or root of a word that is familiar to us, allow us to glean an idea of what is being said. A few examples that appear repeatedly in this paper include the Latin words folidi (read as solidi), maior (read as major), and conuexium (read as convexium). Often, looking at the context will help.
Always keep a Latin-to-English dictionary handy. The author prefers the on-line dictionary, William Whitaker's "Words" (http://lysy2.archives.nd.edu/cgi-bin/words.exe) for efficiency and ease of use. Remember when looking up words, we only need to find the root of a word -- dictionaries will often not contain all conjugations, declensions, etc. Also, it will help to maintain a separate list of Latin words previously looked up and their definitions, in order to save time looking up similar definitions.
At first glance, this appears to be a lot of extra work, but you may find that it can also be quite an enjoyable activity, especially for students. Deciphering 18th century Latin is like cracking a code, uncovering a hidden message, or putting together pieces of a complex puzzle. In fact, these are many of the reasons why people choose to study mathematics professionally. Through this activity, we can convey some of this excitement to our students, while staying on a level where the mathematical complexity is not so great as to intimidate them. In addition, you may find that the self-proclaimed "literary" and/or "non-mathematical" students will enjoy this chance to shine. Here, the student has the opportunity to use the "other half of the brain" in a mathematics class.
Lee Stemkoski (Adelphi University), "Investigating Euler's Polyhedral Formula Using Original Sources - Understanding Latin," Loci (April 2010), DOI:10.4169/loci003297