The Imperial Russian battleship Orel (Eagle in English) should have capsized at the Battle of Tsushima. Her three Borodino-class sisters did, each turning keel-uppermost before sinking out of sight. Orel stayed upright because shipbuilder Vladimir Kostenko spent his voyage from the Baltic, around Africa and to the Pacific, studying Alexei N. Krylov's mathematics of unsinkability and prepared her for controlled flooding (to keep her upright).

When numbers decide who drowns, mathematics acquires a consequence that modern computing, with its rollbacks and do-overs, has largely forgotten.

Through the Eye of the Eagle follows that thread of consequence forward from Kostenko's battle report, which was heavily contested by those responsible for the disaster; through Krylov's 1906 lectures on approximate calculations; through Academician Viktor Glushkov and Candidate Ekaterina L. Yushchenko's Kiev computer (classified a State Secret) revealed in 1962; and into the reduced-precision arithmetic that powers today's machine learning.

The method throughout is the same: work directly from Soviet and pre-Soviet primary sources, reproduce the calculations ourselves, and lay the results side by side so you can see the through line for yourself, rather than taking it on trust.

Three ideas organize the journey:

• Economy of precision. Krylov's insight that a measurement only good to within 5% cannot honestly be reported to six decimal places, and his formula (derived from the Laplace theorem, direct ancestor of the Central Limit Theorem) showing how rounding error accumulates with the square root of the number of operations.
• Overcoming the barrier. The Kiev computer's group instructions, which fold a loop's increment-and-test into a single hardware instruction, turn 4n instructions into 3n+1 instructions for a scalar product of two vectors. That seems pointless, being the same number of operations, but in first generation computers, instruction fetch was an order of magnitude slower than register-to-register operations. The Kiev computer converted the instruction-fetch barrier into a sharp advantage.
• Economy of calculation. Academicians Gurii I. Marchuk and Andrei P. Ershov, and Dr. Mikhail R. Shura-Bura, exemplify what a computing culture builds when it invests in people rather than hardware. Ershov realized that multiplying by zero always produces zero, and multiplying by one always produces the operand, so his compiler completely removed instructions multiplying by zero or one before runtime.

Glushkov, in 1962, documented "the well-known rule of A.N. Krylov" to use an extra (decimal) digit of precision to encompass accumulating rounding errors. That rule can be derived from Krylov's published lectures, and Glushkov's phrasing suggests the rule was oral tradition.

This book's provocative example is that the West surfaced Krylov's rule twice in 2019: once via Higham and Mary's paper on probabilistic error analysis, and again with Sakr et al. writing about bit precision during deep-learning accumulation. Krylov's teaching on approximate calculations regained relevance because AI/ML floating-point arithmetic is not infinite precision, and therefore each calculation is an approximation.

When placing Krylov (1906), Glushkov (1962), Higham and Mary (2019), and Sakr et al. (2019) side by side, the difference is striking. All ask the same fundamental question concerning precision and validity of calculated results, state the same basic premise (rounding errors independent of each other, centered on zero, therefore tending to cancel each other out), with differing calculation models (manual calculations originating with shipbuilding; fixed-point digital arithmetic; floating-point mantissa; constrained coefficient/mantissa combinations), and differing conclusions. What, then, is the striking difference?

• Krylov and Glushkov remain anchored in physical reality and produce engineering results directly applicable to the task at hand.
• Higham and Mary begin with hardware-based premises and continue (section 3) "We now apply Theorem 2.4 within the error analysis of a variety of algorithms in numerical linear algebra." The opening sentence (section 1) addresses "the rise of large-scale, mixed-precision computations" but the ensuing discussion shifts from hardware to software, abstracting away from the hardware constraint in sharp contrast to Glushkov.
• Sakr et al. begin (abstract, first sentence) "Efforts to reduce the numerical precision of computations" based in hardware, measure and graph three specific benchmarks, state "these are very encouraging signs" (last statement before conclusion), conclude (section 6) "our theoretical concepts are application agnostic" and claim (last statement of abstract) "Overall this analysis enables precise tailoring of computation hardware to the application, yielding area- and power-optimal systems."

Both 2019 papers are well-respected, and rightly so. The striking difference is the shift away from remaining grounded in physical reality. This shift began in 1995 when we began hiding physical reality behind abstractions. We disconnected problem from solution, and both papers exemplify this shift. An entire generation has grown up on the assumption of infinite hardware availability. Not of infinite-precision hardware, but of the availability of more hardware to spin up as needed.

Unlike Soviet (and U.S.) supercomputing of the 1960s, modern AI/ML high-performance computing has become a frantic and supremely unnecessary buildout of countless new data centers: a multi-billion-dollar mistake in progress.

This century-wide comparison is not a priority dispute. Each mathematician was answering a slightly different question, demonstrating that the same invariant keeps surfacing. Technology changes, but the mathematics, and the physics, do not.

A board of stellar Academicians gathered in 1948 to issue the multi-volume Collected Works of Krylov. I caught two numeric errors in the 1911 Lectures on Approximate Calculations, Chapter V, Section 51 (page 189), so I checked the Collected Works. The Collected Works has both errors corrected to the values I expected (page 191).