parshat Tetzaveh: Identifying the Stones: לֶשֶׁם - lešem

The next stone, which is the first stone on the third row, is the stone of Dan. The Torah calls it לֶשֶׁם - lešem.

The pesukim describing the stones on the choshen (Shemot 28:17-21):

יז וּמִלֵּאתָ בוֹ מִלֻּאַת אֶבֶן, אַרְבָּעָה טוּרִים אָבֶן: טוּר, אֹדֶם פִּטְדָה וּבָרֶקֶת--הַטּוּר, הָאֶחָד.	17 And thou shalt set in it settings of stones, four rows of stones: a row of carnelian, topaz, and smaragd shall be the first row;
יח וְהַטּוּר, הַשֵּׁנִי--נֹפֶךְ סַפִּיר, וְיָהֲלֹם.	18 and the second row a carbuncle, a sapphire, and an emerald;
יט וְהַטּוּר, הַשְּׁלִישִׁי--לֶשֶׁם שְׁבוֹ, וְאַחְלָמָה.	19 and the third row a jacinth, an agate, and an amethyst;
כ וְהַטּוּר, הָרְבִיעִי--תַּרְשִׁישׁ וְשֹׁהַם, וְיָשְׁפֵה; מְשֻׁבָּצִים זָהָב יִהְיוּ, בְּמִלּוּאֹתָם.	20 and the fourth row a beryl, and an onyx, and a jasper; they shall be inclosed in gold in their settings.
כא וְהָאֲבָנִים תִּהְיֶיןָ עַל-שְׁמֹת בְּנֵי-יִשְׂרָאֵל, שְׁתֵּים עֶשְׂרֵה--עַל-שְׁמֹתָם; פִּתּוּחֵי חוֹתָם, אִישׁ עַל-שְׁמוֹ, תִּהְיֶיןָ, לִשְׁנֵי עָשָׂר שָׁבֶט.	21 And the stones shall be according to the names of the children of Israel, twelve, according to their names; like the engravings of a signet, every one according to his name, they shall be for the twelve tribes.

What is this lešem? JPS translates it jacinth. What is jacinth?

Wikipedia:

Jacinth is a red transparent variety of zircon used as a gemstone. Jacinth is also a flower of a reddish blue or deep purple (hyacinth), and hence a precious stone of that colour (Revelation 21:20). It has been supposed to designate the same stone as the ligure (Hebrew leshem) mentioned in Exodus 28:19 as the first stone of the third row in the high priest's breast-plate, the Hoshen. In Revelation 9:17 the word is simply descriptive of colour.

I am uncertain in what sense JPS intends it - whether they mean the modern sense of red zircon. Jacinth is the same as hyacinth, which in ancient times might have meant blue sapphire, and in the Middle Ages meant yellow gems (including yellow zircon) from East India. I have already discussed this by bareqet, since Midrash Rabba identifies bareqet as hyacinth. Read up on it there. I won't put a picture of it here, since there are many possibilities of what JPS meant.

Midrash Rabba states: דן כוחלין.

Jastrow (pg 618) defines this as "carbuncle." We have already discussed carbuncle, since it was JPS's definition of nop̄eḵ (and apparently other people's definition of bareqet.) Copying what I wrote there:

There, they write:

The word carbuncle occurs in three places in most translations of the Bible. Each use originates from the same Hebrew word בָּרְקַת בָּרְקַת or bâreqath bâreqath (baw-reh'-keth, baw-rek-ath'). In this sense, a carbuncle is usually taken to mean a gem, particularly a deep-red garnet, unfaceted and convex; however, the Hebrew definition is less definite and the precise color of the gems is not known.

Assuming the same definition will apply to nop̄eḵ, perhaps it is the "deep-red garnet, unfaceted and convex," mentioned above. For scientific identification of garnet, see the Wikipedia article on garnet:

The garnet group of minerals show crystals with a habit of rhombic dodecahedrons and trapezohedrons. They are nesosilicates with the same general formula, A₃B₂(SiO₄)₃. The chemical elements in garnet include calcium, magnesium, aluminium, iron²⁺, iron³⁺, chromium, manganese, and titanium.

Here is possibly an image of the carbuncle being described, from probertencyclopedia:

Though it matches the aforementioned description, I cannot be sure this is in fact the carbuncle described, because this encyclopedia writes to the side of it:

Carbuncle is a beautiful gem of a deep red colour (with a mixture of scarlet) . It was called by the Greeks anthrax and is found in the East Indies. When held up to the sun, it loses its deep tinge, and becomes of the colour of burning coal. The name belongs for the most part to ruby sapphire, though it has been also given to red spinel and garnet.

Interestingly, so far, all stones in the first column seem to be red, based on Midrash Rabba's account, assuming I have reconstructed it accurately.