The sequence of restriction site of EcoRI - GAATTC was identified in the early 1970s, before Sanger Sequencing was invented.(1977)
How was the restriction site of EcoRI sequenced ?
The first determination of a recognition site for a restriction endonulease was reported in:
Kelly & Smith (1970) A restriction enzyme from Hemophilus influenzae II. Base sequence of the recognition site. J Mol. Biol. 51: 393-409
The enzyme was then called endonuclease R, but is now known as HindII (or HincII).
The method used was to cut DNA with the enzyme and then remove the exposed terminal phosphates with alkaline phosphatase. These ends could then be labelled with 32P using polynucleotide kinase and γ-labelled ATP.
The end-labelled DNA was then treated with nucleases in three different ways to release mononucleotides, dinucleotides, and a mixture enriched for trinucleotides. In each case the labelled products were identified by their behaviour in 2D thin-layer chromatography, and were further characterised for their total base composition after digestion to mononucleotides where appropriate.
The deduced site was/is
5'----GTY RAC----3' 3'----CAR YTG----5'
Thus the labelled mononucleotides were found to be A and G (purine=R), the dinucleotides were found to be GA and AA and so on.
The EcoRI sequence was determined by essentially the same methodology and was reported by
Hedgpeth et al (1972) DNA nucleotide sequence restricted by the RI endonuclease. Proc. Natl. Acad. Sci USA 69:3448 - 3452
Although Sanger sequencing (the plus-minus method) may have been developed through the 1970s, it really didn't catch on until the early 80s. So for example the first complete plasmid sequence to be determined was pBR322 in 1979:
Sutcliffe (1979) Complete nucleotide sequence of the Escherichia coli plasmid pBR322. CSH Symp. Quant. Biol. 43: 77 - 90
"I have determined the 4362-nucleotide-pair sequence of the plasmid cloning vector pBR322 using the DNA-sequencing technique of Maxam and Gilbert (1977). The DNA structure has several interesting features that lead to testable predictions."
The standard sequencing method in use at that time was chemical sequencing using the Maxam-Gilbert methodology. When I started a postdoc in the USA in 1979 M-G sequencing was the standard method. Sanger sequencing really only took off following the development of the M13 mp vectors (and later the pUC plasmids) by J Messing in the early 80s. And what a relief it was: M-G sequencing was extremely laborious, as anyone who ever did it will testify.