### 3. Summary

In surveying we have linear and non-linear mathematical relationships. Examples of both are:

• **Differential leveling**. Elevations are determined by addition and subtraction which are simple linear relationships.

• **Traversing**. Horizontal positions are determined by applying non-linear trigonometric functions to distances and angles.

If the number of variables is reasonably small and the mathematical relationships simple, “long-hand” solutions aren’t overly difficult nor time consuming. Three or more variables and non-linear mathematics increases computations along with error opportunity as demonstrated by the circle-circle intersection example. There are two ways to address this.

#### a. Systematic solution method

The *Substitution* and *Gaussian Elimination* methods require the user make decisions how to proceed step-by-step. Although each person may ultimately arrive at the correct solution(s), not all will follow the same solution process. A systematic method is efficient, repeatable, and applicable to multiple situations.

#### b. Replace the person with software

Almost any computation is quicker using software than paper and pencil. Software still needs a person who understands how to set up data and interpret results. Software can be simple, acting like a high speed calculator, or complex, providing greater flexibility. Because a computer can not “think” in the conventional sense, software must be based on a systematic solution method.

The former will be discussed in greater detail in later chapters along with general comments on software tools.