Adjustment theory — identifying the principle behind most probable values In a set of observations of equal precision, the most probable values are those that minimize the sum of the squares of residual errors. This is the defining principle of which method?

Introduction / Context:
Survey adjustments reconcile redundant measurements to produce a consistent set of estimates and realistic precision measures. The foundational criterion used worldwide is to minimize the sum of squared residuals, leading to statistically optimal estimates under common assumptions. Recognizing this principle and its method name is essential for network adjustment and error analysis.

Given Data / Assumptions:

• Observations are of equal precision (same variance).
• Residuals are small, independent, and centered.
• Linearized observation equations are applicable.

Concept / Approach:
The least squares method states: choose parameter estimates that minimize Σ v^2, where v are residuals (observed minus computed). Under Gaussian error models, these are also maximum likelihood estimates. When observations have different precisions, a weighted least squares is used (minimize Σ w v^2). In equal-precision cases, weights are equal and the unweighted sum of squares is minimized.

Step-by-Step Solution:

Verification / Alternative check:
Compare with other criteria (e.g., least absolute deviations). Least squares yields closed-form solutions and optimal properties under Gaussian errors.

Why Other Options Are Wrong:

• Gauss’ mid-latitude formula: a navigation/astrometry computation, not an adjustment criterion.
• Delambre’s and Legendre’s named methods refer to historical contributions; the criterion described is universally known as least squares.
• “Method of equal parts” is unrelated to residual minimization.

Common Pitfalls:
Assuming equal precision when weights differ; forgetting to test residuals for blunders before adjustment.

Final Answer:
Least squares method