Let us now work through a complete sentence from Euclid. Our example is the statement of Proposition 1, Book 4, shown in the box on the right. The cases of nouns are colour-coded so you can identify them at once. The core of the sentence is the last two words. The verb is an aorist active infinitive, meaning 'to fit,' and its object is 'a straight line,' in the accusative. This straight line is equal to something, as we see by the adjective in the same case just preceding it. What it is equal to is in the dative, and we see a string of datives: 'to the given straight line not greater being.' The comparative adjective requires something to be greater than, and this is put in the genitive: '(than) the diameter.' In English, the 'than' must be used explicitly; in Greek, it is part of the case of the article. Another short genitive phrase is sandwiched: 'of the circle,' which again requires more words in English, while the case does the job in Greek. We now know what the line to be fitted is to be equal to. Finally, where the line is to be fitted is in the accusative, helped by the preposition 'into.' One usually says that this preposition 'takes' the accusative, but more exactly it really helps the accusative and makes it more specific, which is a more general way of thinking about it. Notice that the definite article introduces each group of related words in the sentence, and the cases hold each group together. There is no nominative case in this sentence, since the verb is an infinitive. Any subject of an infinitive would be in the accusative, but here it has only an object, and no subject.
The vocabulary shows the nominative masculine singular forms of participles and adjectives, which are changed to suit the gender and number in the text. If we call anything that takes noun endings a noun, these are nouns. What we usually call a noun is really a noun-substantive, and the others noun-adjectives. This sentence consists almost entirely of nouns (it's 82% noun!), and shows the help the cases give in keeping things sorted out. There is an aorist participle and an aorist infinitive. These don't express so much a past tense as a single, completed action at any time. In fact, the action of the infinitive here is actually future. Since Euclid doesn't do much description of events, the aorist has this interpretation in the Elements, and is detached from the concept of tense. Note in the verb that the ζ becomes simply a σ when we would otherwise have the combination ζσ. You can remember the meaning of the verb by associating it with 'en-harmonize.'
Review the endings for each noun (in the broad sense) in the text, referring to the paradigms. Did you notice that th~j, obviously feminine, is used with dia/metrou, which looks masculine or neuter? Actually, the word grammh/s is understood, dia/metroj being an adjective with the same form for masculine and feminine (as happens frequently in Euclid). All the words in this example are frequently used. The participle o/)n, "being," is the model for all participles and adjectives of this type which end in ων in the masculine singular (like mei/zwn), and is worth knowing well. Carefully identify the word-groups in the sentence. When analyzing a sentence, always try to separate it into groups of related words, and do not take it doggedly word by word. Know the reason for the use of each case that appears. Incidentally, it is a help to learning to sound out the sentence, using whatever pronunciation scheme you prefer, while observing the accents and breathings.
The problem is solved, if you have forgotten, by choosing any point A on the circumference of the circle. Then draw a circle with radius equal to the given line segment, and connect A and either point of intersection.
Composed by J. B. Calvert
Created 15 August 2000