Adrain founded the short-lived journal Analyst or Mathematical Museum in 1808. In this he published an article in which for the first time is deduced the normal law of errors. Cleveland Abbe credits him with the independent invention of the method of least squares. Merriman suggests that Nathaniel Bowditch's solution to a question may have prompted Adrain to write his 1808 article.

• [1808]  "Research concerning the probabilities of the errors which happen in making observations, &." The Analyst or Math. Museum 1. pp. 93-109. This paper is difficult to locate, but was partially reprinted by Cleveland Abbe in the American Journal of Science (1871), Vol. 101, pp. 411-415 and in Mansfield Merriman A List of Writings Relating to the Method of Least Squares (1877),  pp. 163-165.
  This latter is an extract from The Transactions of the Connecticut Academy, Vol. IV, 1877.
• [1818]  "Investigation of the Figure of the Earth and of the Gravity in different Latitudes," Trans. Amer. Phil. Soc., Vol. 1, pp. 119-135. This includes an application of the method of least squares to find a formula for the length of the seconds pendulum.
  
• [1818] "Research concerning the mean diameter of the Earth," Trans. Amer. Phil. Soc., Vol. 1, pp. 353-336.