Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and Dresden, he entered the Polytechnische Schule, now Technische Universität, in Dresden to study engineering science in 1859. Finding him especially enthusiastic about geodesy, one of his teachers, August Nagel, hired him while still a student to work on the triangulation of the Erzgebirge and the drafting of the trigonometric network for Saxony. In 1863 Helmert became Nagel's assistant on the Central European Arc Measurement. After a year's study of mathematics and astronomy Helmert obtained his doctor's degree from the University of Leipzig in 1867 for a thesis based on his work for Nagel.
In 1870 Helmert became instructor and in 1872 professor at RWTH Aachen, the new Technical University in Aachen. At Aachen he wrote Die mathematischen und physikalischen Theorien der höheren Geodäsie (Part I was published in 1880 and Part II in 1884). This work laid the foundations of modern geodesy. See history of geodesy.
The method of least squares had been introduced into geodesy by Gauss and Helmert wrote a fine book on least squares (1872, with a second edition in 1907) in this tradition. Hald (p. 633) gives this assessment: "[It] is a pedagogical masterpiece; it became a standard text until it was superseded by expositions using matrix algebra." In 1876 Helmert published an article deriving the distribution of the sample variance for a normal population. The work was described in German textbooks, including his own, but the English statisticians 'Student' and Fisher did not know of it and re-derived the distribution.
Helmert received many honours. He was president of the global geodetic association of "Internationale Erdmessung", member of the Prussian Academy of Sciences in Berlin, was elected a member of the Royal Swedish Academy of Sciences in 1905, and recipient of some 25 German and foreign decorations.
There is an obituary at
There is a photograph of Helmert at
and three more at
The first edition of Helmert's textbook on least squares is available at the GDZ site
A partial scan of Die mathematischen und physikalischen Theorien der höheren Geodäsie (Part I) is available on the site
There is an account of Helmert's work on the theory of errors in section 10.6 of
For eponymous terms in Statistics see