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Field observations refer to the vertical of the geoid, because the geodetic instruments are put in the working position in respect to the vertical. It is not appropriate for the calculations on the ellipsoid surface, which request measured elements to be referred to the normal of the reference ellipsoid. This means that the field observations must be reduced from the vertical to the normal, i.e. they must be somehow reduced from the Earth's physical surface to the surface of the reference ellipsoid. There are two methods to do this: Pizzetti's method and Helmert's method. Pizzeti's method projects the measured elements with the aid of the vertical onto the geoid surface and then, with the aid of the normal to the ellipsoid, projects them further onto the surface of the reference ellipsoid. Helmert's (or Bruns-Helmert's) method projects the measured elements directly onto the surface of the reference ellipsoid with the aid of the normal to the ellipsoid. But, until the middle of the 20th century, none of them were used. This means that the field observations were not properly reduced onto the ellipsoid's surface; they were reduced only to the geoid's surface and then used as if they were reduced to the ellipsoid. The reasons fot that are simple. On the one hand, it was almost impossible to determine the relation between the geoid and the ellipsoid at each desired point on the Earth's physical surface (first of all the geoid undulations N and the components of the deflection of the vertical î and ç), so it was very difficult to calculate the reductions. On the other hand, the consequences of neglecting the reduction from the geoid to the ellipsoid were too small, compared with the accuracy of the observations in that time

Reduction of geodetic observations to the reference ellipsoid