Experienced Monotype folks seem to think that identifying this equipment is trivial. I find this not to be so. The central problem is that the basic Monotype machine is a protean device capable, with the correct addition and subtraction of Attachments, of doing nearly anything. One might think that the consequent need for clarity in naming and describing the machine would lead, in fact, to clarity. One might think.
Like most machines, the Monotype had a prehistory. I won't cover that here, but will recommend strongly that you acquire Richard L. Hopkins' book Tolbert Lanston and the Monotype . It covers the entire history of (American) Monotype equipment (including early and pre-production equipment) better than any other source.
For those who prefer a quick summary, here is a table which lists possible combinations of machine capabilities/attachments, together with the names for these machines, for all variations of the basic Monotype caster (but not for later machines such as the Giant Caster). (You won't be able to read the image below; click on it to get a larger, scalable PDF version):
1. If you have a Monotype casting machine which can cast composition (and which is not a Monomatic), then you have a "Composition Caster." This is true regardless of any additional (optional) capabilities that this machine may have for casting types for the cases or for casting leads/slugs/rule. If a Composition Caster has Attachments for casting both types for the cases and leads/slugs/rule then it may be termed a "Combination Machine."
2. If you have a machine with the same basic structure as a Composition Caster which cannot cast composition, then you have a "Type-&-Rule Caster." This is true even if the machine is not equipped to cast leads/slugs/rule. (This is of course a bit illogical, but this usage is supported by the Lanston Monotype parts lists.) The Type-&-Rule Caster in its base state was equipped to cast type for the cases in display sizes (12-36 point) from Lanston Monotype display matrices. It could be further equipped to cast type for the cases in text sizes from Lanston Monotype composition (cellular) matrices.
3. As an exception to the above, if you happen to have a machine with the same basic structure as the Composition Caster which can only cast leads/slugs/rule (but not type), then you have a "Lead, Slug-&-Rule Caster." At present this machine is attested only in a very few references in the Monotype literature. None are known to exist.
4. Otherwise you have a machine which was assigned its own name (Giant Caster, Supercaster, Monotype-Thompson Type-Caster, Material Making Machine, Junior Material Making Machine, Monomatic, Monomatic II.) These are much easier to identify.
[See also the CircuitousRoot Notebooks of ../ Monotype Composition Caster Literature, both sales and technical.]
By 1900 a machine was placed on the market which is recognizably the Monotype "Composition Caster" (with its associated, separate, keyboard). This machine embodied Tolbert Lanston's basic ideas but was to a great extent redesigned by John Sellers Bancroft. If someone says just "a Monotype," this is probably what they mean.
Here's a illustration showing two views of a Composition Caster. It is shown here in a 1903 advertisment, but later machines would be difficult to distinguish from it. Note the paper tape reading apparatus (the "Paper Tower") at the left side of the view of the machine on the left.
Here's an early style of Monotype Keyboard (1903). Later keyboards differed only in detail, not in principle. Note that the operation of the Keyboard requires a three-piece suit, while the operation of the caster requires a smock and cap. Class distinctions run deeper in typography than one might hope.
(Both images are from the same advertisement in The Inland Printer. Vol. 32, No. 2 (November, 1903) . See also the ad from which this came.)
The Composition Caster was manufactured by both the US-based Lanston Monotype Machine Company and the UK-based Monotype Corporation Limited. When necessary, they may be distinguished by calling them "American Comp. Casters" and "English Comp. Casters." The two machines differ in detail; I don't know them well enough to articulate the differences. However (I believe) each kind of machine may be equipped so as to be able to use the matrices of the other. 
This casting machine is perhaps best referred to as the Composition Caster (or Comp. Caster), and indeed it is most often so named by enthusiasts today. Until the advent of its successor, the much less common "Monomatic" casters, it was the only caster from the Monotype companies capable of casting composed matter.
To complicate matters, however, the Lanston company was not consistent in naming this machine. For example, The Monotype System (1912) they call the Composition Caster the "Monotype Composing Machine and Type Caster," and call its display casting variation (see below) the "Type Caster (Convertible)" (pp. 172-173). In a booklet of "Monotype Matrix Information" from the 1930s or 1940s, their terminology varies considerably. They do not call this machine the Composition Caster, but instead refer to it as either the "Typesetting Machine" (on its own) or the "Composition-and-Display Caster" (when display casting capabilities are added but composition capabilities not subtracted). In a 1930 Parts PriceList, they call it the "Monotype Casting Machine." In a 1952 Parts List they call it (on the cover, as its primary name) the "Monotype Typesetting Machine" but subtitle it "The Composition Type-Caster." You would think that a company could come up with a consistent name for its primary product, but Lanston Monotype did not.
Of course, just because the Composition Caster can cast composed text doesn't mean that the text it casts has to say anything. In particular, it is perfectly possible to punch up a tape on the Keyboard which will cause a Comp. Caster to cast sorts, or indeed entire fonts, as desired. This isn't a particularly efficient use of tape, however, and the size of the sorts cast is limited to the size of the faces available for composition (originally 12 point; for exceptions see below). Moreover, the alloy used in the Composition Caster is a slightly softer alloy appropriate for the higher speeds used in casting composed matter. For sorts casting it would be desirable to run at a slower speed which would allow the use of a harder typemetal alloy.
[See also the CircuitousRoot Notebook of Type-&-Rule Caster Literature.]
In a December 1903 advertisement, Lanston introduced a attachments which allowed the Composition Caster to cast single types (not composed matter) in body sizes from 5 point up through 36 point.
These were provided by "Attachment 9CU: Display Type Attachment, Including Speed Regulating Attachment." So equipped, the basic caster could cast either as a regular composition caster or (with equipment changeover) a caster for display type for the cases in 14 to 36 point.
If in addition you wished the ability to cast as individual types for the cases text-size type of up to 12 point (using ordinary composition matrices) you could add "Attachment 19CU: Composition Matrix Sorts Casting (Type-&-Rule Caster)"
Note that at this point in time (1903-1914) Amos L. Knight had not yet invented the fusion-casting technology which allowed Monotype equipment to cast strip material. In this period you couldn't have called this machine a "Type-&-Rule Caster" because Monotype rule casting hadn't been invented yet.
In the 1912 first edition of The Monotype System (which postdates the introduction of the display equipment but predates the introduction of the lead and rule casting equipment) the Monotype caster equipped for display casting was called the "Type Caster [Convertible]". (This same source calls the Composition Caster the "Monotype Composing Machine and Type Caster.")
(A note on matrices. The basic patent by Bancroft and Indahl which extended the original Monotype caster to the casting of types for the cases in larger sizes was US patent 883,378. This patent was filed 1904-12-01 (a date more or less consistent with the announcement of the display type capability), but was not in fact issued until 1908-03-31. Most Lanston Monotype display matrices have stamped on their backs a patent date of Nov. 24, 1908. This is the date of US patent 904,510 (filed 1907-01-12 and issued to Chalfant on 1908-11-24). It is not clear to me what kind of matrix Lanston Monotype was supplying for display sizes (14 - 36 point) in the period 1903-1907. For a further discussion and reprints of these patents, see the CircuitousRoot Notebook on Lanston Monotype Type-&-Rule Caster Matrices.)
On 1914-10-03, Amos L. Knight filed the basic patent for the "fusion casting" technology which was used for all stripcasting on all Monotype casting machines. (This patent application was later divided in three parts and three patents were issued: 1,222,415 (1917-04-10), 1,237,058 (1917-08-14), and 1,256,807 (1918-02-19). For a further discussion of these, including reprints of these patents, see the CircuitousRoot Notebook on thea Lanston vs. Elrod Dispute.)
The initial use of this fusion-casting technology was as an attachment to the display-caster version of the basic Monotype caster (that is, to what had been known in 1912 as the "Type Caster [Convertible]"). This was "Attachment 11CU: Lead and Rule Mold Operating Attachment" (at least it was referred to by that symbol and name in the 1930s).
The Parts Price List, Monotype Casting Machine and Type-&-Rule Caster (Philadelphia, PA: Lanston Monotype Machine Company, 1930) provides some information on what you would call machines with various sets of equipments.
To further complicate matters, enthusiasts today will colloquially call this machine an "Orphan Annie." This name is said to come from the prefix "OA" on the serial numbers of the machine. I'm not sure of this, though, since I cannot even find the serial number on my machine (it was put on a plate on the Paper Tower, which component is of course absent on an Orphan Annie!)
Here's a view of a Type-&-Rule Caster. It lacks the Paper Tower and, as shown, has the Lead and Rule Mold Operating Attachment and the Automatic Cutting Attachment (the bit sticking out on the right), but is clearly the same underlying machine as the Composition Caster.
From Adjustments of the Type-&-Rule Caster (1926)
Finally, of course, it is not necessary to remove the composition capabilities of the machine. In "Monotype Matrix Information", Lanston refers to a "Composition-and-Display Caster," but I'm not entirely sure what you would call such an omnipotent machine today.
I have come across several slightly cryptic (and un-illustrated) references to a Monotype machine called the "Lead, Slug-&-Rule Caster." Although these were the materials produced by the Material Making Machine (see below), these references seem to suggest that it was instead a Type-&-Rule caster with only the Lead-&-Rule Casting Attachment and no type casting equipment.
Note that the 1930 Parts Price List indicates that the 11CU "Lead and Rule Mold Operating Attachment" required the 9CU "Display Type Attachment, Including Speed Regulating Attachment." The Speed Regulating Attachment was not listed as available by itself. This would seem to contraindicate a "Lead, Slug-&-Rule Caster," yet such a machine was referenced by the Lanston Monotype literature.
For a further discussion, see the section on the Lead, Slug-&-Rule Caster in the CircuitousRoot Type-&-Rule Caster Notebook .
A December 1903 advertisement gives the body sizes for composed matter as five to twelve point.
A November 1903 advertisement notes that it casts quads and spaces "up to pica in size" (= 12 point, of course). No surprise to this.
In the description of this machine in The Monotype System (1912) it is said to cast composed type in body sizes from 5 to 12 1/2 points (according to tables given). The Directions for Care and Cleaning the Style 1E Composition Mold, however, indicate that the mold was capable of 5 to 12 point.
By the 1921 Pony Specimen Book, the lower limit of to the body size for both the Composition Caster and the Type-&-Rule Caster is given as 4 points (p. vi).
A December 1903 advertisement gives the body sizes for composed matter as five to twelve point.
Later, an attachment was introduced which allowed the casting of composed matter at fourteen and eighteen point. This required the use of special composition matrices. In the 1930 Parts PriceList, this is "Attachment 10CU: Eighteen Point Attachment."
Still later an attachment enabled the casting of 24 point composition. I have few details of this. On the English side, it is mentioned as the "Large-Type Composition" attachment in 'Monotype' Attachments, Accessories and Moulds. (Salfords, Surrey, UK: The Monotype Corporation Limited, [no date; post-WWII]). This, in turn refers to "Information Sheet No. 154. Rich Hopkins, in "Monotype Composition Matrices: A Preliminary Discussion" in the Newsletter of the American Typecasting Fellowship, No. 32 (August, 2008), pp. 8-15, indicates that "Large Composition" arrangements for sizes up to 24 point were made by both the American and English companies; he illustrates several Large Composition matrices.
A Type-&-Rule Caster can be equipped to cast sorts from composition matrices throughout their entire range of sizes (from 4 (or 4 1/2?) points up to 12 points normally, or 14 & 18 points optionally, or 24 points optionally).
Note that there is some overlap between the range potentially covered by composition matrices and that potentially covered by display matrices. In the 12 to 18 (and sometimes 24) point range one could in theory use either. However, both my own (very limited) experience and an examination of the lists of matrices available from Lanston would indicate that for sorts casting display mats were typically suppplied in the 14 to 36 point range. Handy Index of 'Monotype' Rental Matrices (1955) , which shows display matrices available for rental from Lanston in 14 to 36 point only.
A particularly elaborate 1903 advertisement for the Monotype gives the maximum measure (line length) of the machine as 42 picas.
The Monotype System (1912) gives the normal measure (line length) of the casting machine as 42 picas, but observes that a "Sixty Pica Attachment" was available. (section 349, p. 168). The 1930 Parts PriceList calls this "Improvement No. 28: The Sixty Pica Device."
By the 1921 Pony Specimen Book, the maximum measure has been increased to 84 picas (p. vi).
I presume that these measures didn't change with the Type-&-Rule caster, but the concept of "measure" is not really meaningful for sorts casting. I presume that it merely gives the size of the galley mechanism. My own Type-&-Rule caster has had the galley mechanism removed entirely and a single curved channel lets type come out in a long line until it falls off the machine onto the floor.
The user (keyboard operator) of the Monotype may also vary the set width of of the type cast and thus do letterspacing without the casting of separate spaces between letters. This is discussed in detail in The Monotype System (1912) .
This of course raises issues for the modern letterpress printer in breaking up old forms of Monotype composition to salvage type. It is not necessarily the case that the set of the type for composed matter is the same as the set for type cast from the same matrix font as sorts. Neither is it necessarily the case that the set of type for composed matter is uniform throughout a form.
In both composition casting and sorts casting the caster may cast the type upon bodies of greater size than the default. Thus leading may be incorporated into the body of the type, and separate leading need not be used.
Note however that this involves attention to lining. From The Monotype System (1912) : "Faces cast on the leaded body line at the bottom with faces designed for that point size; for example, eight-point faces cast on ten-point body line with ten-point faces [and would not line with the same face cast on an eight-point body]." (283)
Both the potential presence of leading cast with the type and the issues of lining on type so cast introduce issues for the modern letterpress printer breaking up old forms of Monotype composition in search of type.
The original Monotype caster for casting composed matter could cast types in composition from 5 to 12 point in composition on a 42 pica maximum measure. Later the miniumum body size went down to [either 4 point or 4 1/2 point]. With special attachments, the maximum body size for composition went up to 14 & 18 point, and later 24 point. With special attachments, the maximum measure went first up to 60 picas and then to 84 picas.
To the Composition Caster you could add the 9CU "Display Type Attachment, Including Speed Regulating Attachment." The resulting machine was called in 1912 the "Type Caster (Convertible)," but this name seems not to have been used after that date. What you have is a Composition Caster with the additional capacity to cast display type for the cases.
It is not yet clear to me whether you could cast text size (5-12, 14/18, 24 point) types for the cases on this machine using the composition mold equipment, or whether you could add to it attachment 19CU "Composition Matrix Sorts Casting" (the 1930 Parts List indicates that the 19CU attachment was for the Type-&-Rule Caster only).
If you wanted to cast composition above 12 points, you were also required to add the 9CU "Display Type Attachment, Including Speed Regulating Attachment." I presume that this was really to add the Speed Regulating Attachment (necessary for slowing the machine for casting larger types), but it would end up giving you a machine capable of casting display types for the cases as well.
Finally, you could also add the 11CU "Lead and Rule Mold Operating Attachment" to extend the capabilities of this machine to the casting of leads, rule, and slugs. This attachment required the 9CU "Display Type Attachment, Including Speed Regulating Attachment" (but not the 19CU "Composition Matrix Sorts Casting" attachment.
The most elaborate machine, equipped for composition, for casting types for the cases in text and display sizes, and for casting leads, rule and slugs has been termed by Richard L. Hopkins a "Combination Machine." .
Note that the composition casting and type-for-the-cases casting capabilities, while possibly simultaneously present on the same machine, were mutually exclusive. You could cast composed matter, or you could cast types for the cases. You could not do both simultaneously, and going from one to the other required an equipment changeover.
If the caster did not have composition casting capabilities, then the most basic casting equipment it would have would be the 9CU "Display Type Attachment, Including Speed Regulating Attachment." This machine was in later Lanston Monotype literature termed a "Type-&-Rule Caster." An examination of the equipment configurations listed in the 1930 Parts Price List indicates that the machine was called a "Type-&-Rule Caster" even when it did not have the 11CU "Lead and Rule Mold Operating Attachment." Thus a "Type-&-Rule Caster," as officially defined by Lanston Monotype, need not have the capacity to cast rule. This same source also indicates that the 19CU "Composition Matrix Sorts Casting" attachment was applied to the Type-&-Rule Caster. So the display-size version of the machine is the most basic.
To the basic Type-&-Rule Caster you could add the 11CU "Lead and Rule Mold Operating Attachment" (which also required the 12CU "Automatic Cutter Mechanism." This would give you a Type-&-Rule Caster which could actually cast rule.
Richard L. Hopkins, in Tolbert Lanston and the Monotype, p. 205 , indicates that the colloquial name "Orphan Annie" has been applied to:
The Lanston Monotype Machine Company introduced the Giant caster (q.v.) in 1926. This was a noncomposing type caster of different basic engineering design capable of casting type and (fusion-cast) strip material from 14 to 72 point. (Type from 14-36 point was cast using display matrices and type from 42-72 point was cast using special Giant Caster matrices).
The (English) Monotype Corporation Ltd. made the Supercaster (or Super Caster) a noncomposing type caster which could cast type and (fusion-cast) strip material up to 72 point. A wide variety of mold and matrix equipment was available; this allowed casting from an equally wide range of different matrices. The (American) Lanston Monotype Machine Company never made the Supercaster.
Both the English and American Monotype companies made the Monotype-Thompson Type-Caster after the acquisition of the formerly independent Thompson Type Machine Company by Lanston in 1929. The Thompson (or Monotype-Thompson) is a noncomposing type caster casting up to 48 point type, quads, low quads, and spaces. A rule-casting attachment was once offered, but no example of this attachment is known to survive.
This image, from Monotype-Thompson Adjustments (1950) , isn't very good. Monotype never really did justice to the Thompson. I'm a bit obsessed by Thompsons, and am trying to collect as much information about them as I can. See the Thompson Notebook here on CircuitousRoot. See also Skyline Type Foundry, which casts exclusively on Thompsons.
The Lanston Monotype Machine Company introduced the Monotype Material Making Machine (commonly called the "Material Maker") in 1923.  This was a dedicated fusion-casting machine for producing strip material. It could cast "Rules, Leads, and Slugs, in continuous strips or cut to labor-saving sizes; Column Rules and Ornamental Borders; Single-Column Want-Ad Rules, Braces and Cut-Off and Decorative End Dashes ... in various sizes from 1 to 18 point" (from a 16 page brochure on the "Monotype Material Making Machine"). It could not cast type.
The Lanston company also produced, briefly, a Monotype Junior Material Making Machine. Little is known about it, and none are known to have survived.
At a relatively late point, the Monomatic and Monomatic II were introduced as successors to the Composition Caster. I know very little about them.
2. Although the size of the ordinary composition matrix up to 12 point has remained the same throughout the history of the Composition Caster (0.2 x 0.2 inch square, at the casting face), around 1910 Lanston redesigned the mechanical details of this matrix and reduced its depth of drive from the original 0.050 inch to the now familiar (well, familiar to Monotype folks) 0.030 inch. The resulting style of composition matrix is termed a "cellular" matrix. Very few Lanston pre-cellular matrices remain, as the company attempted to take them out of circulation. The English Monotype company, however, did not adopt this redesign, and retained a variation on the earlier form and also the 0.050 inch depth of drive. Rich Hopkins has written of this in his article "Monotype Composition Matrices: A Preliminary Discussion" in the Newsletter of the American Typecasting Fellowship, No. 32 (August, 2008), pp. 8-15. He also illustrates many of the variations in composition and display matrices.
3. The date of 1923 for the introduction of the Material Making Machine is that given by Hopkins. 
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