We recently purchased the machine you see here in the condition in which it was found, and received.  It is yet another Harris Visible No. 4, with serial number 107931.  There are two odd things about it, though.

First, it is the second known example displaying the front frame decal seen below.  The machine is labeled clearly as a Harris Visible, with this name on the front and on the paper table (and the name Harris above the keyboard as well) but the front frame clearly states that the machine was produced by Rex Typewriter Mfg. Co. 

The other odd thing is that the machine has three keys with fig-shifts that are fractions; this is the first known of this keyboard variation.
The machine shown here matches that above in all of its labeling details, although it does not have the fractions.

It is owned by Chris Amato, and is serial number 105462.

This might lead one to believe that the HARRIS name continued in use by Sears Roebuck for catalog sales, and that REX simply modified the decals when it supplied machines to Sears.
This is, seemingly, thrown out of consideration when we examine my first-found Harris Visible No. 4, seen at right.  It is serial number 106723, which places it between the other two seen above.  It does NOT carry the Rex name on the front frame, but the Harris Typewriter Mfg. Co. name.  Like the other two, though, it doesn't have the elaborate decorative pinstriping and so forth that many Harris machines have. 

This discovery, coupled with further findings in known serial numbers has led to the beginning of a complete examination of the Harris and Rex machines chronologically by serial numbers and advertising to determine just what the variants are.  We will attempt to briefly discuss these below as we see them here.
Ads from 1914 and 1916 show the elaborate pinstriping, and platen knobs that consist of a pulley-like wheel with a rubber ring around it (usually missing as it is on the actual example seen here from the David B. Davis collection with serial number 22925.  Machines from quite low in the Harris serial number range (ie down into the 14,000's) match these and as we see this overall format runs into nearly the 23,000 range.  Some very early ads show solid keys, like on an Oliver but none have been found in surviving machines.
HARRIS machines are found in two serial number ranges.  These run from around 11,000 to around perhaps 25,000, and there is also a "high range" of serial numbers that run from over 103,000 to over 111,000. 

REX machines run in two serial number ranges as well; machines are known to exist with serial numbers in the 8,000 and 9,000 range, and then the main range picks up above the lower Harris range around 25,000 and runs continuously through just over 81,000 although it is suspected that many numbers were not used. 

-Why are there REX machines with serial numbers BELOW HARRIS machines when we know for certain that Rex postdates Harris?

-Why are there two separate serial ranges for HARRIS machines so far apart?

Let's look through what we have and see if anything can be determined.
An advertisement from the introduction of the REX VISIBLE in February 1916 shows a marked reduction in pinstriping, and new solid one-piece platen knobs.

Remember that knobs get replaced over time, and in fact the first machine shown on this page has been fitted with what we might call re-engineered ersatz knobs.  However, we can see that the solid knobs have appeared in this early 1916 ad with no question.  We also see the ribbon selector buttons on the sides of the machine.
Here is a 1918 illustration of the Rex Visible No. 4.  Note that the ribbon selector is now a lever, mounted on the front of the machine.  There is a bead line (decorative) stamped into the front plate which has to be broken in the plate to allow the ribbon selector lever shaft hole to be punched. 

If we look at the first three machines shown on this page, we see a curious feature.  All of these have the break in the bead line on the front panel but NO hole and NO ribbon selector.  This makes one think that the design for the ribbon selector had been developed during Harris production, but not implemented. 

The two known Harris Visible No. 5 machines, in the 20,000 to 21,000 serial number range DO have ribbon selectors but they are side-mounted like that seen in the 1916 Rex ad above.
Rex Visible No. 4 in our collection, matching the 1918 illustration seen above with serial number 48935.

American Can Company bought, and then sold Rex Typewriter so that some Rex machines carry the labeling of Rex Typewriter Company but most carry the label "Sold and Guaranteed by American Can Co." on the front frame.  This American Can Co. range runs from serial numbers in the 33,000 range to near the highest Rex numbers, with one sole exception being the only known Rex Visible No. 10 labeled Rex Typewriter Company.
Having seen ALL this information, what are we left to assume?  Well, below are MY assumptions about how the production actually progressed chronologically with serial number blocks referenced.

FIRST HARRIS PRODUCTION for Sears began at the 10,000 range and moved to the 100,000 range after modifications..

REX took over in 1916 and converted to side button-type ribbon selectors; these began initially in the unused serial numbers below 10,000.  REX apparently continued to furnish some machines to Sears, still in the 100,000 plus serial number range, but with NO ribbon selector.  It may be that frames were simply manufactured, stamped with serials and stockpiled and then used (assembled) out of order.  Note that NO Harris-branded machines have ribbon selectors (except the few No. 5 machines) and ALL Harris-branded machines appear to have sound-deadening material inside the side and rear panels .. you can see two screws in the center of the side panels which affix the material.  (See the Davis Typewriter Works site for more on this feature.)  The sound-deadening may be at Sears-Roebuck specification, and was dispensed with in REX machines since these don't have it.  It should be noted that these are fairly early fully-enclosed typewriters -- most everything else "standard" was still open-sided and open-backed.

REX later restarted using serials above the last-built HARRIS machines in the five-digit range and eventually converted to using the front mounted lever type ribbon selector (roughly by 1918 and around serial 33,000.)
We can take this information and change it around to form blocks of serial numbers in serial order for Harris, Rex, Demountable and variants.

up to about 10000:   Rex Visible machines, made roughly 1916-1918

11,000 to about 25,000:  Harris Visible machines, made 1912-1913.* 

SPECIAL BLOCK:  19,965 through 20,509 contains all four known AUTOCRAT, REPORTER'S SPECIAL (2) and BETZ VISIBLE, and no others.

25,000 to about 62,000:  Rex Visible machines, 1916 and later (33,000 through 50,000 American Can; all others Rex Typewriter)**

100,000 to about 112,000:  Harris Visible No. 4 machines, built either by Harris 1913-1916 or Rex 1916 and later, for SEARS ROEBUCK***

125,000 to about 151,000:  Demountable No. 1, 2 and 5 machines (Demountable Typewriter Co.)

*:  Sole surviving Smith Visible No. 6 (serial 17019) is from this group.

** :  One machine labeled as Rex Visible No. 4 reported to have serial number in 81,000+ range but there are none others at all known between the given ranges in the listing above, ie none near this number so it seems suspect.  It may be misread or struck incorrectly.

***:  None in this group has a ribbon selector.  Made either by Harris or by Rex for Sears.  Last guaranteed date in a Sears-related publication is a date of 1920 (printing code) in the Office Machine Americana reprint of a Sears-labeled Harris Visible No. 4 manual.  One machine in this block discovered actually with its original Sears manual.  Two known machines in this block labeled as made by Rex, mixed serials randomly..  Frames / decks were probably stockpiled; this was well before "on time delivery" of parts in modern manufacturing, so they were used (painted and assembled) somewhat out of serial number order. All machines in this block have interior sound deadening material except for the earliest known example, serial 100,794 (owned by Craig Burnham.)
Harris Visible No. 4 serial number 107,931, labeled on the front as "Made by Rex Typewriter Mfg. Co." is in our shop for work.  We found inside the machine a boss, normally used on a Rex Visible No. 4 with the front-mounted ribbon selector lever.  The machine was made by Rex, for Sears-Roebuck who still called the machine the Harris Visible No. 4 the whole time, from 1912 through roughly 1920 and is concurrent with Rex Visible No. 4 machines in the high range with front mounted, lever-type ribbon selectors although it has none.
One further interesting assumption can be made using the serial information above and a privately circulated serial number list provided by owners.  We think that there were probably about 66,000 Harris Visible No. 4, No. 5 / Rex Visible No. 4, No. 10 machines made in complete total, including those for Sears-Roebuck, during the entire Harris/Rex period from 1912 until the conversion to Demountable Typewriter Company.  It would seem that there were only about 26,000 Demountables of all models.
June 2008:  Yet another HARRIS page!  Click the button.