107 



grams shown in Figures !), 10, or 11. Every triangle may 

 thus be regarded as half of a possible parallelogram. In every 

 parallelogram thus formed of two triangles, it is evident that 

 each triangle has the same base and vertical height as the 

 parallelogram, and half its area ; and equally evident that, if 

 multiplying its base by its whole height will give the area of 

 the whole parallelogram, multiplying its base by half its height 

 will give half its area, that is, the area of the triangle. Where- 

 fore the area of a triangle is equal to its base multiplied by 

 half its height. 



As triangles can be thus measured, all plane surfaces 

 bounded by straight lines can be measured, for they can all be 

 divided into triangles. Thus a field of the shape of Figure 12 

 can be divided as shown, and then the area of each triangle 

 being found, the sum of these areas will be the area of the 

 field. 



The power we thus possess of measuring triangles, is the aim 

 and goal of all geometry. The angles and the areas of other 

 figures cannot be ascertained by measuring their sides. Four 

 or any greater number of lines of any given lengths may de- 

 scribe figures of infinitely varied shapes and areas. Thus these 

 four lines, each of four inches in length, may describe a square 

 of 16 square inches in area, as shown in Figure 13, or a 

 parallelogram of any smaller area, one of which is shown by 

 Figure 14. But these three lines, each four inches long, can 

 only describe the one shape and area shown in Figure 15. To 

 alter its shape or area, the length of one or all of its sides 

 must be altered, consequently the lengths of the sides of a 

 triangle fix its shape — that is, determine its angles also ; but 

 the lengths of the sides of a polygon do not. So the dividing 

 of a polygon into triangles ties the figure into a fixed shape, 

 and therefore, in surveying, the diagonal lines that do so arc 

 often called tie lines. 



By these concrete, visible demonstrations, 

 proved the truth of rules that enable us to 



we have 

 measure 



now 

 any 

 accessible plane bounded by straight lines. But very often 

 there are parts of planes that are so inaccessible that we cannot 

 apply our measures to them. How, then, are we to measure 

 them, and how are we to be sure that our measurements are 

 exact ? 



1 cannot occupy your time, nor pretend within the limits 

 of this paper to give the complete course of Geometry 

 that is involved in the answer to this question. But to show 

 how much this course can be facilitated by concrete methods, 

 let me visibly demonstrate the 47th proposition of the 1st 

 Book of Euclid's Elements without having to previously 

 demonstrate all the preceding propositions that lead up to it. 



