Is there any ancient or modern connection between the word "sin" and the sport of archery?

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Idioms are notoriously difficult to translate, since they are specific to the language of origin and may cause confusion when translated literally. One example is the expression “to miss the mark” (from the meaning of the Greek word hamartia in Luke 1:77; John 1:29; 1 John 3:4). It is worth noting what 1 John 3:4 says about sin: “Everyone who sins breaks the law; in fact, sin is lawlessness” (NIV). True, the Greek word ‘hamartia’ has been linked to the expression “to miss the mark”:

Strong's Concordance: hamartia: a sin, failure: Original Word: ἁμαρτία, ας, ἡ Definition: a sin, failure Usage: prop: missing the mark; hence: (a) guilt, sin, (b) a fault, failure (in an ethical sense), sinful deed.

Thayer's Greek Lexicon: STRONGS NT 266: ἁμαρτία: ἁμαρτία, (ας, ἡ (from 2 aorist ἁμαρτεῖν, as ἀποτυχία from ἀποτύχειν), a failing to hit the mark (see ἁμαρτάνω. In Greek writings (from Aeschylus and Thucydides down). 1st, an error of the understanding (cf. Ackermann, Das Christl. im Plato, p. 59 Anm. 3 (English translation (S. R. Asbury, 1861), p. 57 n. 99)). 2nd, a bad action, evil deed. In the N. T. always in an ethical sense https://biblehub.com/greek/266.htm

However, we do well to go further back in time and take a quick look at the Hebrew meaning of sin. During the Greek Mycenean Period Moses was attributed as being the author/compiler of the Pentateuch, which includes the book of Genesis where the first mention of sin as a noun is recorded:

The first mention of sin as a noun is a zoomorphism, with sin (hattath) crouching at Cain's door (Genesis 4:7). The first as a verb is Abimelech being prevented from sinning (khata) against God in a dream.

Hebrew has several other words for sin beyond hata, each with its own specific meaning. The word pesha, or "trespass", means a sin done out of rebelliousness. The word aveira means "transgression". And the word avone, or "iniquity", means a sin done out of moral failing. The word most commonly translated simply as "sin", hata, literally means "to go astray." Just as Jewish law, halakha, provides the proper "way" (or path) to live, sin involves straying from that path. Source: https://en.wikipedia.org/wiki/Jewish_views_on_sin

The generic Hebrew word for any kind of sin is ‘avera’ which literally means “transgression.” The article goes on to describe the three levels of sin and is well worth reading.

I mention this to emphasise the point that biblical sin means a transgression against the law of God. The Greek idiom not only “misses the mark” but it doesn’t even get close to the target! As a description of biblical sin, it is wholly inept.

Back to your question asking if the Greek word ‘hamartia’ can be linked to archery. You have discovered that “Hamartia as it pertains to dramatic literature was first used by Aristotle in his Poetics.” I have to agree with your conclusion that “it seems that sin is not actually an archery term.” The Greek word may lend itself to “missing the mark” but not so with the Hebrew. Interesting word search, though.

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The Strong's definition of the word says the following:

ἁμαρτία, (ας, ἡ (from 2 aorist ἁμαρτεῖν, as ἀποτυχία from ἀποτύχειν), a failing to hit the mark (see ἁμαρτάνω. In Greek writings (from Aeschylus and Thucydides down). 1st, an error of the understanding (cf. Ackermann, Das Christl. im Plato, p. 59 Anm. 3 (English translation (S. R. Asbury, 1861), p. 57 n. 99)). 2nd, a bad action, evil deed. In the N. T. always in an ethical sense, and

Here, the "failing to hit the mark" meaning is attributed to Aeschylus (a playwright) and Thucydides (a historian and a general).

Given the fact that Thucydides was a general, it is possible that the word did literally refer to archery, but that is the biggest possible link that I found.

I did find a source that did seem to say it wasn't related to archery:

In reality though, outside of the Odyssey and the Illiad archery isn't mentioned a ton. That's simply because they didn't have it. Hoplite warfare was the theme of the day. Anyone here seen the movie "alexander" a very accurate depiction of Greek warfare was during the battle of Gaugamela when Parmenio's left flank was impailing the Persias on the scarisus (I honestly forget the spelling). The 15' spear with a backweight. That part when they started mowing people down was very much a reason why they didn't need archers.

There is a very telling part in the Odyssey thought that I think gives hint to why there wasn't archery. In the book, there is mention to Odysseus' father still being alive. However, he is relatively ineffective and is more content to water his flowers. The relationship between the Basileus and the Boule is such that titles WERE NOT hereditary. It was based on how well you were as a ruler and a warrior... and when you could no longer fullfil your warrior obligations, you would stand down and let someone else go for you.

If you want to be a good warrior, you want to be known as macho. That being said, Greeks prized competition and the willingness of the warrior to look their opponent eye to eye while killing them. It was a sign of manliness and therefore killing them at a distance with an arrow is not a macho thing to do.

If you look at the Odyssey, Odysseus is painted as a crafty and a man full of tricks. I think that says something there.

If you read Pindar's Olympic Odes, archery is not even mentioned in there (although there is a sport there call "pankritation" which its only rules were that it prohibited biting and eye gouging... whatever that's all about). Pindar was an honorable enough soul that when Alexander sacked Corinth in 336BC (I think), Pindar's house was the only thing left standing. If archery were to be mentioned... he'd have had to have done it.

But take that with a grain of salt.

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