figure.

(c) Prolong AB to any point C.

(d) Draw any line from C as CE.

(e) Draw DE || BC.

(f) Sight from E to A.

=

(This may be done by making / CDE C.) Mark the intersection F. (g) What are similar? Why?

(h) Measure BC, CF, FD, and DE. Find AC and then AB.

(i) NOTE: This problem is taken from a book written in Latin and printed in 1645. It is the same book from which the other drawings were taken.

8. By using one of these methods, find the width of a river, creek, or street in your vicinity.

9. (a) In the measurements thus far, only one end of the line has been inaccessible. There is a very easy method for measuring distances, both ends of

which are inaccessible. It is by means of an

old instrument called a baculus, meaning rod. (b) The baculus is really made of two rods, one very much longer than the other and marked off in segments of equal length. The shorter rod is equal in length to one of the segments and is made to slide over the longer one easily, but always remains perpendicular to it.

A BACULUS

10. The following picture and explanation showing the use of the baculus are also taken from the book by Bettinus.

(a) FG is the distance required.

(b) Let CD be at a certain mark on the baculus AB. (c) Sight from A so that points F and G are just seen along C and D.

(d) Then move the shorter rod nearer A, if you approach FG; or nearer B, if you move away from FG for the next observation.

(e) Find the point V, so that F and G may still be just visible past the ends of the shorter rod at O and P.

(f) By measuring the distance AV, between the two stations, the desired length FG will be had.

NOTE: The above explanation is translated from the Latin. The proof is by proportions derived from several sets of similar triangles, but it is too difficult to be given here. The construction and use of the baculus, however, are very simple.

11. Make estimates of distances between objects on the opposite bank of a river, then measure with a baculus and tape.

12. Another way to measure inaccessible distances.

Let AB be the required distance. Standing at C let observer sight A through D on a rod placed at B. Do likewise at F, some other convenient point.

Draw BC, CF, and EF.

Draw BH || EF.

It is proved in geometry that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.

Which three lines can be measured to find AB?

13. To measure an inaccessible distance by a quadrant with a plumb line, by drawing a small similar triangle.

Let AB be the required distance.
What kind of a triangle is ABC?

With the quadrant measure angle DCE.

Draw a small right triangle with ZF C.

=

Which three lines can be measured to find the distance AB?

The name of this book by Marius Bettinus is,

Apiaria Universae Philosophicae

Mathematicae

Progymnasma Primum

The first problem is called Proposition I. An exact translation of it is given below.

"PROPOSITION I. To measure an inaccessible distance by the twenty-sixth proposition of Book I of Euclid.

at sea.

"Let A be the position of a ship at sea and let Thales be on the shore at B. How shall he find the distance AB?

"Let him withdraw in a straight line AB to any desired point, as C. At B, with the aid of a norma, make a perpendicular and mark off any length, as BD.

"Then let the angle BDA be noted. On the other side, let the angle BDC be marked off equal to the angle BDA.