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A Crucible is a container used to melt metal in. As such, it has to withstand high temperatures. More specifically, it must not melt or weaken at the temperatures used to melt the metal it contains. In addition, it must be able to withstand rapid, large changes in temperature without weakening.

In this article, I will start by assembling what I can find in primary sources on the construction or design of crucibles.

This is what Theophilus wrote in his treatise, On Divers Arts, about making a crucible:

Chapter 22. Crucibles for Melting Gold and Silver

With all these things at hand, take white clay and grind it very fine. Then take old pots in which gold or silver has previously been melted and crush them up separately. If you do not have these, take pieces of a white earthenware pot and put them on the fire until they are red-hot, and if they do not crack, allow them to cool, and grind them up separately. Then take two parts of ground clay and a third part of the burned pots and mix them with warm water. Knead it well and make crucibles out of it, both large and small ones, in which you will melt gold and silver.
Hawthorne and Smith, On Divers Arts, Dover Press, 1979, p. 96.

Theophilus assumes the reader knows the steps for firing a clay object. He also has another chapter on creating crucibles for use with brass (and I suspect for bronze or copper):

Chapter 65: Making the Crucibles

Take fragments of old crucibles that have previously been used for the melting of copper or brass and crush them into tiny pieces on a stone. Then take clay from which earthenware pots are made - there are two kinds of this, one white, the other gray: the white is good for coloring gold, the other for making these crucibles.  Whenyou have ground it for a very long time, mix the raw clay in with the other (i.e., the burnt clay that you first ground) in proportion like this. Take any small part and fill it twice with the raw clay and three times with the burnt clay so that there are two parts of raw and three of burnt. Put them together into a large pot and pour warm water over them; then knead them vigorously with hammers and with your hands until the mixture is completely tenacious. Then take a round piece of wood and cut it to the size that ou want the crucible to have [inside], according to the capacity of the furnace. Shape a crucible upon this and, after it has been shaped, coat it at once with dry ashes and so put it close to the fire until it is dried. Make as many crucibles as you wish to in this way. When they have been carefully dried, put three, four, or five of them into the furnace, up to the number it can hold, and heap charcoal around them.
Hawthorne and Smith, On Divers Arts, Dover Press, 1979, p. 142-143.

Vannoccio Birunguccio, in his work, The Pirotechnia, has this to say about making crucibles:

The Thirteenth Chapter. How Very Good Small Crucibles Are Made, and Small Shells for Melting All Kinds of Metal

Since I promised you above when speaking of the melting of metals to teach you to make small crucibles and shells for melting, I do not wish to fail in my promise. For, in truth, since they are instruments that are much used for this procedure, they increase the work if they are not good, and often cause great extra expense as well as necessitating repetition of the work. If they are good they save everything.
Therefore, if you have to make them, do it so that they are greatly aided by art. It is first necessary to have the clay of a good nature, that is, resistant to the force of the fire by its own natural virtue, either when it is lean or when its viscousness is mixed with much talc, and whether it is yellow or white. This should be well freed from small stones and beaten well with an iron, and then thoroughly mixed by hand. With this should be mixed an eighth part of finely ground and sifted iron scale, and also some young ram's-horn ashes. These things should be well mixed together by hand and by beating them. If this clay should not be strong enough by itself, it is mixed with another lean clay, with crushed peperino, flintstone, or some other stone that in your judgement seems to be arid and resistant.
Crucibles are worked on a low potter's wheel turned by hand, or on a high one turned by foot, just as plates are made. Both large and small ones are made. Most of them are given a triangular shape at the mouth, and the shells have a kind of spout for ease in pouring out the metal. This done they are will dried, and baked in furnaces like pots or other vessels. At the end, when they have been thus made they are used for melts.
Smith and Gnudi, The Pirotechnica of Vannoccio Biringuccio, Dover Press, 2005, p. 391.

Smith and Gnudi have a note that the inclusion of the iron scale would ruin the refractoriness of the clay.  I have not yet verified that statement.

Allan, in Persian Metal Technology: 700-1300 AD, quotes al-Birūnī's chapter on iron. The full section describes casting steel ingots in crucibles. From the context, it appears the ingots were cooled in the crucible:

...iron was mixed with a small amount of manganese and various bits of plant or vegetable matter, heated in a sealed crucible, allowed to cool, and the resulting ingot (baida) used as the basis for a sword. A similar method is described in al-Birūnī's chapter on iron, but the author gives al-Kindī as his authority (Bir p. 256). In fact al-Birūnī's own description of fūlādh (Bir p. 252) suggests a different process was used in his experience. "The mixture of narm-āhan and its water, which is the substance which flows when the narm-āhan is purified, is fūlādh. The area of Herāt is especially noted for it and it is called baidāt ("eggs") on account of its shape. The eggs are long and round-bottomed, following the shape of the crucibles, and from them Indian swords and others are fashioned. ..."
Allan, Persian Metal Technology: 700-1300 AD, Ithaca Press, London, 1979, p. 75.

This is an interesting passage. Fūlādh is steel. Narm-āhan is iron with a low carbon content. The "water of narm-āhan" is not water. "Water of" is a chemical term of the time, referring to a substance that flows out of another substance when the first is purified. This is believed by the translaters to be cast iron. (FYI, the "d" in baida and baidāt should have a dot underneath it, I just haven't located the correct unicode character number for that letter yet.)

The crucibles have lids, presumably to keep oxygen or other pollutants out of the metal. Second, the crucibles are also molds for the ingots, they are not using a separate mold. Third, the crucible interior produces a long "egg-like" shape. I do not take that too literally, but an elongated shape with wider diameter at the top and a narrower diameter at the bottom might (along with a smart marketing mindset) account for the description. To make it "really" egg shaped, the top of the ingot would have to be domed, and the quoted description does not mention that.

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Gingery, Vince, Making Crucibles, 2003
 
 
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This book covers making crucibles for metal casting. Crucibles are fairly expensive and this provides an inexpensive way to make crucibles.
Hawthorne, John, Smith, Cyril Stanley, On Divers Arts, 1979
90
1120
Yes
Yes
Describes the method of making clay crucibles to use for casting. If you are interested in period medieval metal working, glass or painting processes, this is a must-have work. Theophilus covers a wide variety of techniques and processes, including recipes for various materials.
Smith, Cyril Stanley, Gnudi, Martha Teach, The Pirotechnia of Vannoccio Biringuccio, 2005
90
1530
Yes
Yes
Describes a recipe and the process of making one. This was originally published circa 1530. It is a must-have book for any medieval metalworker. Biringuccio was a very pragmatic man and his work reflects that.