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Adding to what Bill Morris provided, specifically on "How" the Wild company reached such precision- below is the mechanical and optical part- additionally,  in the field observers would take 16-32 repeated sets of observations over 4-10 nights to allow for repetition and improved measurement accuracy using statistical processing of the observational data.

The T-4 is primarily designed in part as a larger version of the Wild T-2 / T-3 theodolites- which by the time Wild designed the T-4 in the mid 1940's they had over 20+ years of making the T-2.      The Wild T-4 instrument can read directly to .1 second (that's 1 in  12.96 million divisions).  Key to this capabilty was a very large glass horizontal circle, an impersonal eyepiece micrometer to read the horizontal glass circle, and of course the incredibly precise optical and mechanical fabrication capability of the Wild company. 

- the large horizontal circle (240mm, or 24cm, or 9.5 inches)  provided a basic instrument capable of 2-second readings  by scaling the same design capability already in use with the Wild T-2 and T-3.    For comparison, the Wild T-2 can be read directly to .4 seconds, with a horizontal circle of 8.5cm; the T-4 design uses a glass circle 3X larger with the abilty for measuring about 3.3X greater divisions compared to the T-2 (.1 sec   vs .4 sec)

- the optical micrometer at the eyepice further divided the 2-second reading into 20 divisions-  so 2-seconds, divided into 1/20- became .1 second direct reading.

(think of using a microscope with 20-divisions/ intervals across the field-of-view  to read the horizontal circle markings)

The key technology that provided this capability is due to the use of photo-lithographic methods to mark the horizontal circle, not mechanical engraving- The T-4, with a 240mm (24cm/ 10inch) wide circle, had a lot more room for photo-lithographically etching the needed markings to allow 2-second graduations on the circle. 

This would be similar to the way that microprocessor designs are photo-lithographed and chemically photo-etched into silicon- 

I'm not sure of the exact process Wild used on the T-4 (timeframe was 1940's, so photo-chemical processes were well known by then), but I'm guessing that the precision template for the glass circle would have been designed at a much larger scale, (perhaps using a dividing engine for part of this design) - then the template design of the glass circle could have been photographed, and converted into a transparent negative.  The negative would then have been photographically projected via optical-reduction lenses onto the 240mm glass circle which was coated with a photo-resisitive coating.   The light of the exposed pattern of gradations would have changed the photo-resisitive coating, which when treated with another chemical would have selectively removed the coated graduations leaving bare glass- the bare glass would have been exposed to a etching chemical, which etched the lines into the glass- then, the etched lines in the glass circle could be treated with a dye or another chemical to darken the 2-second marks.      For computer chips this is done using a much smaller silicon wafer, using UV or X-ray light for the photo-lithographic reduction; the current silicon chip technology is limited by the wavelength of light used as to the minimum spacing between etched lines, currently around 14 nano-meters, or about 1/1000 the spacing on the T-4 circle. (physical distance between 2-second gradation marks on the T-4 horizontal circle was 2000 nano-meters (2 micrometers, or .002 mm)

The divisions on the 240mm circle to reach 2-second graduations needed to be spaced .002mm apart (or 2 micrometers)-  certainly not trivial, but Wild had the capabilityfor this by using their own high-quality optical lenses, of which Wild had lots of experience with from their aerial-camera lenses (which provide the same type of reduction capability- imaging the actual surface of the earth say 2x2 km square, and reducing the image to a film negative 20x20cm- factor of 10,000, very common in aerial photography.

The micrometer eyepiece was also similar to the Wild method of reading the circles of the T-2/T3;  it is basically a microscope used to read the exact difference between the set of 2-second markings,  with 20 intervals of the micrometer equal to a single 2-second gradation.   (2 seconds divided by 20 intervals  =  .1 second direct reading)

The T-4 main telescope had either a 60x or 80x telescope- which was useful for sighting the various stars used for astronomical observations-  but the telescope magnifcation by itself wasn't the key to the T-4's capability- (you can get a simple telescope to be 100x power quite easily, although the image won't be very bright)-  instead it was the ability to use a very bright, optical telescope with corrected optics for chroma and astigmatism to observe a star- and have that telescope precisely measure the incredibly small individual angles with the accompanying glass circle measurements in both the horizontal/vertical which allowed for it's accuracy and precision.   Wild was using coated glass optics at this point as well, which improved light transmission greatly, to over 95%- compared to uncoated optics of competing instruments, reducing light transmission to the 50-70% range.

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