2+8 THE POPULAR SCIENCE MONTHLY. 



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 calcu- 

 lated, 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 measure- 

 ment of a geodetical base leaves all ordinary methods of which sur- 

 veyors dispose altogether out of the question as too incorrect ; a sys- 

 tem of measurement has therefore to be applied which permits of the 

 measurement being executed in a line scrupulously straight, of all vari- 

 ations 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 ex- 

 pansion 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, mi- 

 croscopes, etc. "When the rods have been carefully placed in line, the 

 distance between the end of one rod and the following has to be ascer- 

 tained, and also, for getting at the actual length of each rod, the dif- 

 ference 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. 



