The Teaching of Geometry Part 22

N. Bion's "Traite de la construction ... des instrumens de mathematique," The Hague, 1723]

[Ill.u.s.tration: THE QUADRANT USED FOR ALt.i.tUDES

Finaeus's "De re et praxi geometrica," Paris, 1556]

The quadrant was practically used for all sorts of outdoor measuring.

For example, the ill.u.s.tration from Finaeus, on this page, shows how it was used for alt.i.tudes, and the one reproduced on page 240 shows how it was used for measuring depths.

A similar instrument from the work of Bettinus is given on page 241, the distance of a ship being found by constructing an isosceles triangle. A more elaborate form, with a pendulum attachment, is seen in the ill.u.s.tration from De Judaeis, which also appears on page 241.

[Ill.u.s.tration: THE QUADRANT USED FOR DEPTHS

Finaeus's "Protomathesis," Paris, 1532]

[Ill.u.s.tration: A QUADRANT OF THE SIXTEENTH CENTURY

De Judaeis's "De quadrante geometrico," Nurnberg, 1594]

[Ill.u.s.tration: THE QUADRANT USED FOR DISTANCES

Bettinus's "Apiaria universae philosophiae mathematicae," Bologna, 1645]

The quadrant finally developed into the octant, as shown in the following ill.u.s.tration from Hoffmann, and this in turn developed into the s.e.xtant, which is now used by all navigators.

[Ill.u.s.tration: THE OCTANT

Hoffmann's "De Octantis," Jena, 1612]

In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The ill.u.s.tration 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.

[Ill.u.s.tration: THE SPECULUM

Finaeus's "De re et praxi geometrica," Paris, 1556]

A well-known method of measuring the distance across a stream is ill.u.s.trated in the figure below, where the distance from _A_ to some point _P_ is required.

[Ill.u.s.tration]

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 ill.u.s.tration 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.

[Ill.u.s.tration]

Another application of the same principle is seen in a method often taken for measuring the height of a tree.

[Ill.u.s.tration]

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 cla.s.s:

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.?

2. Suppose a triangle is used which has _AB_ = twice _BC_. What is the height if _AD_ = 75 ft.?

There are many variations of this principle. One consists in measuring the shadows of a tree and a staff at the same time. The height of the staff being known, the height of the tree is found by proportion.

Another consists in sighting from the ground, across a mark on an upright staff, to the top of the tree. The height of the mark being known, and the distances from the eye to the staff and to the tree being measured, the height of the tree is found.

[Ill.u.s.tration]

An instrument sold by dealers for the measuring of heights is known as the hypsometer. It is made of bra.s.s, and is of the form here shown. The base is graduated in equal divisions, say 50, and the upright bar is similarly divided. At the ends of the hinged radius are two sights. If the observer stands 50 feet from a tree and sights at the top, so that the hinged radius cuts the upright bar at 27, then he knows at once that the tree is 27 feet high. It is easy for a cla.s.s to make a fairly good instrument of this kind out of stiff pasteboard.

An interesting application of the theorem relating to similar triangles is this: Extend your arm and point to a distant object, closing your left eye and sighting across your finger tip with your right eye. Now keep the finger in the same position and sight with your left eye. The finger will then seem to be pointing to an object some distance to the right of the one at which you were pointing. If you can estimate the distance between these two objects, which can often be done with a fair degree of accuracy when there are houses intervening, then you will be able to tell approximately your distance from the objects, for it will be ten times the estimated distance between them. The finding of the reason for this by measuring the distance between the pupils of the two eyes, and the distance from the eye to the finger tip, and then drawing the figure, is an interesting exercise.

Perhaps some pupil who has read Th.o.r.eau's descriptions of outdoor life may be interested in what he says of his crude mathematics. He writes, "I borrowed the plane and square, level and dividers, of a carpenter, and with a shingle contrived a rude sort of a quadrant, with pins for sights and pivots." With this he measured the heights of a cliff on the Ma.s.sachusetts coast, and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and distances in which he is interested.

THEOREM. _If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:_

1. _The triangles thus formed are similar to the given triangle, and are similar to each other._

2. _The perpendicular is the mean proportional between the segments of the hypotenuse._

3. _Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side._

To this important proposition there is one corollary of particular interest, namely, _The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter_.

By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please.

Thus we may make _AD_ = 2 in., _DB_ = 3 in., and erect _DC_ [perp] to _AB_. Then the length of _DC_ will be [sqrt]6 in., and we may find [sqrt]6 approximately by measuring _DC_.

[Ill.u.s.tration]

Furthermore, if we introduce negative magnitudes into geometry, and let _DB_ = +3 and _DA_ = -2, then _DC_ will equal [sqrt](-6).

In other words, we have a justification for representing imaginary quant.i.ties by lines perpendicular to the line on which we represent real quant.i.ties, as is done in the graphic treatment of imaginaries in algebra.

It is an interesting exercise to have a cla.s.s find, to one decimal place, by measuring as above, the value of [sqrt]2, [sqrt]3, [sqrt]5, and [sqrt]9, the last being integral. If, as is not usually the case, the cla.s.s has studied the complex number, the absolute value of [sqrt](-6), [sqrt](-7), ..., may be found in the same way.

A practical ill.u.s.tration of the value of the above theorem is seen in a method for finding distances that is frequently described in early printed books. It seems to have come from the Roman surveyors.