Hello! Welcome to this lesson dedicated to measuring distances. Measuring distances is one of the fundamental problems of surveying. In this lesson, we will see first a short introduction, posing some questions with the notion of distances, which gives some historical background, then we describe the electronic method for measuring distances, and finally, we will address the fundamental problem with trigonometric leveling. In this introduction, we will first ask some basic questions. Everyone has already measured a short distance with a yardstick or a measuring tape. When it comes to measuring longer distances, other methods should be used, in particular electronic methods. Then we will ask what are the factors that influence the measurements of distance: the physical factors, but also geometric. And a fundamental question we must ask, is "what type of distance?" If I take my example here on the right, I can consider a horizontal distance between the two lake sides, but I can also consider from this summit of the mountain to the lake side, a slant distance. I have already two notions of distances. Then, if I take a small step back, I can start asking myself the question: "is a distance necessarily straight or do we have to take into account the curvature of the earth?" We will try to answer these questions in this lesson dedicated to measuring distances. Before addressing the electronic measurement of distances, we give a brief history of some principles of optical measurements or direct measurement of distances. We have in this example here, the measurements with a steel tape or with an Invar tape. These measurements are possible when we are near the objects. In general, we make the measurements within a limited framework, so it can be a small site, it can be inside, and in general we take horizontal measurements. To determine the distances, surveyors used the optical concept of their measuring devices, namely, the theodolite or the level. In this figure, one can place an observer with his theodolite, who look at a sight. I can measure an angle and determine a portion of the distance on the sight. I have a simple relationship, between the tangent of the angle <i>α/2</i> that is equal to <i>(b/2)/S</i>, <i>S</i> is the distance I am trying to determine. So <i>S</i> will be equal to <i>b/2</i> times the <i>cotangent of (α/2)</i>. In the optical principle there are two approaches: when <i>b</i> is a constant, and <i>alpha</i> is measured, or when <i>alpha</i> is a constant, and it is <i>b</i> that is measured. The first method is the method called "stadia". We have here the objective and what the operator can observe, namely stadia lines. We have here, a gap between two fixed lines, that means that <i>alpha</i> is a constant in this case. So we will determine the portion intercepted by these two stadia lines to determine the distance. The other method is called "parallax", in which we measure the angle <i>alpha</i> between the two sides of an Invar sight, the angle of 2 meters. For more details on these methods, you can refer to the handout that describes these methods, which are practically no longer used in surveying today, but that is good to review for historical purpose.