In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given on page 243 shows how in early days a simple device was used for this purpose. Two similar triangles are formed in this way, and we have only to measure the height of the eye above the ground, and the distances of the mirror from the tower and the observer, to have three terms of a proportion. All of these instruments are easily made. The mirror is always at hand, and a paper protractor on a piece of board, with a plumb line attached, serves as a quadrant. For a few cents, and by the expenditure of an hour or so, a school can have almost as good instruments as the ordinary surveyor had before the nineteenth century. [Illustration: THE SPECULUM Finaeus's "De re et praxi geometrica," Paris, 1556] A well-known method of measuring the distance across a stream is illustrated in the figure below, where the distance from _A_ to some point _P_ is required. [Illustration] Run a line from _A_ to _C_ by standing at _C_ in line with _A_ and _P_. Then run two perpendiculars from _A_ and _C_ by any of the methods already given,--sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point _D_, on _CD_, to _P_, putting a stake at _B_. Then run the perpendicular _BE_. Since _DE_ : _EB_ = _BA_ : _AP_, and since we can measure _DE_, _EB_, and _BA_ with the tape, we can compute the distance _AP_. There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. Other figures are suggested on page 244, and the triangles need not be confined to those having a right angle. A very simple illustration of the use of similar triangles is found in one of the stories told of Thales. It is related that he found the height of the pyramids by measuring their shadow at the instant when his own shadow just equaled his height. He thus had the case of two similar isosceles triangles. This is an interesting exercise which may be tried about the time that pupils are leaving school in the afternoon. [Illustration] Another application of the same principle is seen in a method often taken for measuring the height of a tree. [Illustration] The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in which _AB_ = _BC_. He holds _AB_ level and walks toward the tree until he just sees the top along _AC_. Then because _AB_ = _BC_, and _AB_ : _BC_ = _AD_ : _DE_, the height above _D_ will equal the distance _AD_. Questions like the following may be given to the class: 1. What is the height of the tree in the picture if the triangle is 5 ft. 4 in. from the ground, and _AD_ is 23 ft. 8 in.?