It's actually less Galileo's failure than mine. After reviewing his pioneering work on solid mechanics and strength of materials (if you rule out Leonardo), I became convinced that he had used the theatrum machinarum for inspiration. I felt that his findings and approach were a perfect bookend for Errard's depictions of bridges.
I was wrong. It seems that Galileo did not own any of the TM. Although he may have owned Trasportatione dell'obelisco vaticano, the library list entry is marked with an asterisk and the note:
"This copy of the work is currently possessed by B.N.F. and carries a bookplate marked: 'B.R.9.1.' It contains some small anotations attributed to Galileo." [L'esemplare di quest'opera attualmente posseduto dalla B.N.F. ed ivi contrasegnato colla notazione: "B.R.9.1." contiene alcune poche postille attribuite a Galileo.]
Galileo was unfamiliar with the TM. Favaro's index of authors indicates that Galileo owned none of the works. There still may be a strength of materials angle. While Galileo's work was good, it was also wrong. In my first few years of civil engineering we didn't study Galileo. We did, however, study the next two big names to enter the engineering game: Euler and Bernoulli.
Euler was a famous and prolific mathematician. He also did some nifty work on the compressive strength of solids with the aid of one of his colleagues, Daniel Bernoulli. As a family, the Bernoullis were prolific but Daniel has probably had the greatest impact on modern sophmore engineering students. They study both the Euler/Bernoulli theories for solid mechanics and Bernoulli's theorem for incompressible fluid flow... exciting stuff!
Bernoulli and Euler were (obviously) colleagues. What's interesting is that they were both employed by the St. Petersburg Academy of Sciences, then under the direction of Schumacher. They certainly would have come into contact with Nartov. Indeed, there is evidence of letter writing between Nartov and Euler.
Nartov and Euler corresponded in 1743 with three letters, two of which were written by Euler. Euler also maintained communication with a great variety of other dignitaries. There were 39 letters with d'Alembert (1746-1773), 100 letters with Daniel Bernoulli (1726-1768), six letters with Condorcet (1775-1776), ten letters with Lomonosov (1748-1765) one letter from Kant (1749), and 307 letters with Schumacher (1730-1757). Given the acrimonious relationship between Schumacher and both Nartov and Euler, it's possible that these letters were related directly to the usurping of the directory.
There's also the possiblity that they were conferring about some of the works that were familiar to Nartov and were contained in Peter the Great's library: Besson, de Caus, and Ramelli, Bockler, and Leupold. Perhaps Nartov was the vector between Euler, Bernoulli, strength of materials, and the theatrum machinarum.
Euler's great work Methodus inveniendi lineas curvas appeared in 1744, the year after his correspondence with Nartov. At the time he was living in Berlin but still publishing in the Russian Academy's Commentarii Academiae Petropolitanae.
More letters between Nartov and Euler may exist. Volume 2 of Die Berliner und die Petersburger Akademie der Wissenschaften im Briefwechsel Leonhard Eulers, for example, is all about the correspondence of Euler wiith Nartov, Razumovskij, Schumacher, Teplov, and the Petersburg Academy, from 1730-1763. Perhaps I'll find some more hints; perhaps I'm smoking crack.
Favaro, A. "La Libreria di Galileo Galilei," in Bulletino Bibliografico di Storia Scientifica, Matematica & Fisica, 19 (1886), 219-90
Euler's Correspondents -- Alphabetical Listing
Post Script (December 29, 2005)
Die Berliner und die Petersburger Akademie was dissapointing. There are three letters between Euler and Nartov. Unfortunately, only the letter from Nartov is written in German. It seems that Euler's letters were written--and reproduced by the editors--in Russian. German is bad enough. I can't even decipher cyrilic letters!
A more promising avenue for pursuing the Nartov/Euler relationship is contained in Danilevskii's biography. The copy I had secured through ILL was a very poorly executed reproduction based on a microfiched archive copy. Luckily, I've been able to purchase a copy that was withdrawn from the University of Iowa library system... if only it had a better index! Thank goodness for abebooks.com.