Concerning Kalman's original paper published in 1960, "A New Approach to Linear Filtering and Prediction Problems", it seems the majority is to show the orthogonal projection is the optimal estimation under certain conditions and solutions to Wiener's problem.
But I did not manage to find the exact KF algorithm in this original paper as the ones in nowadays textbooks.
For example, is there an explanation of "Kalman gain" in his paper in 1960 ? Does Kalman's paper provide a mathematical derivation of Kalman Filter Algorithm ?