For blank I,
the elimination of speculation from the text,, isnt conclude that it was limited

The following sentence

The historian of ancient science Otto Neugebauer concluded that Babylonian astronomical texts are (i)_____ because everything has been eliminated from the astronomy except observations and the mathematical consequences of an initial hypothesis about the fundamental character of the astronomical movements.

The sentence above implies that by eliminating something from the rest of observations mean that that elimination follow a logic path. Not randomly

Therefore B is the answer

Blank (i): Neugebauer's Conclusion

- Context: Neugebauer's judgment is based on the fact that everything has been eliminated from the astronomy except observations and mathematical consequences. This suggests the texts are narrowly focused, rigorous, and empirical, not speculative or imaginative.
- Fit: B. scientific is the best fit. The elimination of everything except observations and mathematics means the texts adhere strictly to scientific (empirical and computational) principles.

Blank (ii): The Role of Mathematical Theory

- Context: The second sentence starts with "This judgment cohered with..." meaning the high level of mathematical theory agreed with the "scientific" conclusion. This theory is described as doing something to mathematical computation and empirical observation.
- Fit: The theory must affirm the importance of these elements. E. admitted works because it means the theory recognized and accepted computation and observation as valid components. (The word must be positive, fitting the "scientific" nature).

Blank (iii): The Nature of Computation and Observation

- Context: The theory recognized mathematical computation and empirical observations as the _______ of science, while simultaneously denying any role to speculative hypotheses. This establishes computation and observation as the essential elements.
- Fit: H. necessary characteristics precisely describes computation and observation as the essential, fundamental features of science, contrasting them with the rejected element: speculative hypotheses.