Ok. So, in light of the last post on understanding the full semantic punch of a Biblical Hebrew (BH) preposition, in this post we'll talk about 1) the primary sense of עִם and אֵת's gamut of meaningful units,
2) one way to properly grasp the development between a lexeme's network of senses, and 3) the primary theory which this "developmental scheme" rests upon (i.e. grammaticalization). [Written after: Well, I had high hopes—as you can see—but only 1) made the cut, so we'll save 2) and 3) for the next round.]
As a reminder, the primary sense of a lexeme's range of meaning is that meaningful use that drives all the others (in some way or another) and rests as the core of the full semantic spectrum in play. It's like the seed which sprouts into a tree. There were several guidelines we discussed that can help us identify this primary sense, but rather than go through the whole spiel on how I decided what's the semantic seed of עִם and אֵת I'm just gonna spoil the joy of the journey and tell you the happy ending.
The primary sense for, interestingly enough, both עִם and אֵת is what I've called Shared Presence. I'll give you a brief "definition" with some examples after I explain two components that are present in any given prepositional relation: a trajector (TR) and a landmark (LM). The trajector is the thing that is understood in relation to the landmark. It follows a trajectory. So, a trajector might be something that moves and is typically smaller than the landmark. It can be much more than this (and even less), but these English examples will help you understand what I mean. It's really quite simple.
I hit the ball so hard it (TR) went over the fence (LM)
Sam crushed his toe (TR) against the chair (LM)
My dog peed (TR) on the rug (LM) the other day
So you see, the TR is the entity that is related to the LM by the preposition. These relations always vary and are determined, largely, by the relator (i.e. preposition). With that said, let me tell you what Shared Presence—the primary sense of עִם and אֵת—relates between a TR and a LM.
עִם and אֵת describe a trajector that is located in general spatial proximity to a landmark
Here are two examples:
וַתִּקַּח מִפִּרְיוֹ וַתֹּאכַל וַתִּתֵּן גַּם־לְאִישָׁהּ עִמָּהּ וַיֹּאכַל
And so she took some of its fruit and she ate, and she also gave some to her husband, who (TR) was with her (LM), and he ate (Gen 3.6)
וַיִּזְכֹּר אֱלֹהִים אֶת־נֹחַ וְאֵת כָּל־הַחַיָּה וְאֶת־כָּל־הַבְּהֵמָה אֲשֶׁר אִתּוֹ בַּתֵּבָה
But God remembered Noah and all of the beasts and all of the cattle (TR) that were with him (LM) in the ark (Gen 8.1)
I hope that gives you a pretty good picture of this most foundational sense of עִם and אֵת. Something that's important to remember when discussing semantic primacy is that just because a particular sense is the so-called seed of a network does not entail that this meaning will always be present in every sense. In other words, just because a semantic network starts somewhere doesn't mean it has to carry forth these originating traits. With a little bit of digging, of course, you'll probably be able to trace the semantic extensions, but they will certainly not always be present.
Nonetheless, in the next post on this series we'll talk about how to identify (what I'll call) 'distinct senses', and from here, it might be fun for you to try and "connect the dots", before I do in the following following post. ;) I hope you enjoyed this brief overview. Please let me know if you've got any questions.
 For a more technical description, but still highly helpful, see J. Taylor (2003) Linguistic Categorization, Oxford: Oxford University Press, 113.