# CHAPTER 11PRINCIPLES OF LEAST SQUARES

## 11.1 INTRODUCTION

In surveying, observations must often satisfy established numerical relationships known as *geometric constraints*. As examples, in a closed-polygon traverse, horizontal angle and distance observations should conform to the geometric constraints given in Section 8.4, and in a differential leveling loop, the elevation differences should sum to given a quantity. However, because the geometric constraints rarely meet perfectly, an adjustment of the data is performed.

As discussed in earlier chapters, errors in observations conform to the laws of probability; that is, they follow normal distribution theory. Thus, they should be adjusted in a manner that follows these mathematical laws. While the mean has been used extensively throughout history, the earliest works on least squares started in the late eighteenth century. Its earliest application was primarily for adjusting celestial observations. Laplace first investigated the subject and laid its foundation in 1774. The first published article on the subject, titled “Méthode des Moindres Quarrés” (Method of Least Squares), was written in 1805 by Legendre. However, it is well known that although Gauss did not publish until 1809, he developed and used the method extensively as a student at the University of Göttingen beginning in 1794, and thus is given credit for the development ...

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