An emerald - Rather the garnet, which when cut with a convex face is termed the carbuncle.
A sapphire - Not the stone now called the sapphire; the lapis-lazuli is most probably meant.
A diamond - There is no trace of evidence that the ancients ever acquired the skill to engrave on the diamond, or even that they were acquainted with the stone. The "diamond" here may possibly be some variety of chalcedony, or (perhaps) rock crystal.

And the second row shall be an emerald, a sapphire, and a diamond. The first of these stones is by both the Targums of Onkelos and Jonathan rendered an "emerald", as by us; and which is described by Pliny (k) as of a green colour, exceeding delightful and pleasant, and to which he gives the third place among precious stones; though by many the stone here called Nophec is thought to be the carbuncle, and is so rendered by the Septuagint; the carbuncle of the ancients is no other than what we call the, "ruby"; and which Braunius (l) thinks is here meant, and so Abarbinel, which is just making an exchange of the last stone of the first row for this; and De Dieu observes, that if any chooses to render the preceding stone an emerald, as Braunius does, he must render this a carbuncle or ruby; and if he renders that a carbuncle, then he must this for an emerald. The next stone is "the sapphire", of which one would think there could be no doubt, it is the very Hebrew word itself that is here used; which Ruaeus (m) says is of a sky colour, and sparkles with golden spots or specks, with which agrees Job 28:6. The third stone of this row is the "diamond" or adamant; and that this stone is meant seems clear from its name Jahalom, which comes from a word which signifies to break; and from hence a hammer has its name, because this stone pierces, cuts, and breaks other stones, but cannot be broken itself. On these three stones were engraved, according to the Jerusalem Targum, the names of the three tribes of Judah, Issachar and Zebulun; but more truly, according to the Targum of Jonathan, the names of the tribes of Judah, Daniel and Naphtali, and so Jarchi; for the names here, as on the onyx stones, were according to the order of their birth.
(k) Ut supra, (Nat. Hist. l. 37.) c. 5. (l) Ut supra, (De Vestitu Sacerd. Hebrews. l. 2.) c. 11. sect. 2, 7. p. 661, 667. (m) De Gemmis, l. 2. c. 2.