of which have been calculated, form with other imaginary lines other triangles, which may be designated as triangles numbers 2, 3, 4, etc. Each triangle has one side in common with the following triangle. Triangle number 1, for instance, may have one side in common with number 2, and one with number 3. If, therefore, the lengths of the three sides of triangle number 1 are known, these, together with the observations of the angles of number 2 and number 3, permit of the lengths of the sides of triangles number 2 and number 3 to be calculated, and so on.
The straight line that has to be actually measured in the field is known as a geodetical base. The accuracy necessary in the measurement of a geodetical base leaves all ordinary methods of which surveyors dispose altogether out of the question as too incorrect; a system of measurement has therefore to be applied which permits of the measurement being executed in a line scrupulously straight, of all variations in temperature which can affect the length of the measuring-rods being carefully noted and kept account of, and of the rods themselves being kept in a perfectly horizontal position. The measuring-rods are themselves very delicate and costly instruments. They consist of a prism of iron or steel, four metres long, on the upper surface of which another rod of metallic zinc rests, the zinc rod being somewhat shorter than the other, both being so placed on their supports as to prevent their bending and allowing them free expansion. The coefficient of expansion of zinc is much greater than that of iron, therefore the expansion or contraction of the zinc rod is much greater than that of the other. Changes in temperature in the two metals can thus be easily ascertained by actual measurement of the distance which separates the end of the zinc rod from a given point, marked on the iron rod. Four or six such rods are used for measuring a geodetical base. The rods are each in a long, wooden case, provided with micrometric arrangements for placing the rods in a straight line, raising or lowering the ends so as to have them perfectly horizontal; with spirit-levels, glasses, microscopes, etc. When the rods have been carefully placed in line, the distance between the end of one rod and the following has to be ascertained, and also, for getting at the actual length of each rod, the difference in the length of the iron and zinc rod in each of the cases. A small space is always left between two rods, which are not made to touch each other, in order to avoid sudden and too sharp contact. This intervening space, as well as the varying distance between the ends of the zinc and iron rods, is measured by means of small pieces of crystal a few inches long which have the form of half-prisms, being larger at one end and growing gradually smaller toward the other end. In fact, two of the four sides lengthwise have the form of a trapeze, while the other two are rectangles. One of these rectangular faces is divided to scale, and the observer has only to insert this piece of crystal between the two ends of the rods, without forcing it in, and to call the scale.
