=

1.7241379310 and a = 1.7320689655.

that q Computing roughly the value of the expressions in (4) we find that .00001815 <e < .000018199+, and, reducing a by the smaller number, we have 1.7320508 as correct figures of the root. This shows how we may use the expression to the left in (4) as a correction in order to obtain more correct figures, or, in other words, that h2

(5)

a

8a

is a better approximation to r than a. It may be seen from what follows that (5) gives really more correct figures than were indicated above, in fact, we may use ten decimals of the correction .0000181579, thus obtaining as correct figures of the root 1.7320508076.

It is possible to find a second correction, a third, and so on, but it is not so easy to do this graphically, and so we shall proceed algebraically. The error in (5) is

which may be factored by replacing (a−r) by its value in (3) and reduced further by the same substitution, thus

The expressions on the right and left of the inequality sign do not differ greatly and hence we may use the one to the left as a second correction. We obtain in this way the still better approximation

By subtracting r from this expression, and factoring and reducing as before in order to find the error, we shall find a third correction and the approximation,

The reductions become troublesome when we proceed much further in this way, but by the use of the binomial theorem for fractional exponents all of these terms and as many more as may

be desired may be easily obtained. For we have only to write the equation (3) in the form r2 = a2-h2/4 = a2(1-h2/4a2), or r = a(1 — h2/4a2)*, and expand the second factor in order to obtain the above development. This development by the binomial theorem will yield very easily any number of terms, but it depends upon the demonstration of the validity of the binomial theorem for fractional exponents, and this demonstration is quite difficult. But for any ordinary computation only one or two of the correction terms would ever be needed since they diminish very rapidly when h is small, as in the computation of 3 above, and in this case the first method of derivation has the advantage of using only simple algebraic reductions and of furnishing very convenient upper and lower limits for the error. Thus the original process of successive divisions and taking the average may be supplemented by the above explained method of subtracting from the last average found one or more corrections. However, for general use the unsupplemented method is by far preferable as a rule easily remembered and understood, and sufficiently rapid.

The simpler rule has in addition an important theoretical significance, for it gives a very simple and satisfactory definition of the square root of any positive number. Thus if N is any positive number we may define √N in the following way: Divide N by any convenient positive number d and denote the quotient by q and the average of q and d by a1. Now divide N by a, and find the corresponding quotient q1 and average a 2. Proceeding in this way we find two endless sequences of numbers:

such that the sequence A decreases while the sequence B increases, and the difference between a corresponding pair of terms of A and B, a;-qi, is positive and approaches zero as i increases. Also the square of any term in A is greater than N, while the square of any term in B is less than N. The sequences A and B approach therefore the same limit and this limit is √N.

These facts may be proved without assuming the existence of VN, as is desirable in such a definition. The proofs are not difficult and are left to the reader as an exercise. A similar definition may be formulated for any root of a positive number.

TWO METHODS OF LOCATING THE GERMAN SUPER GUN.
BY HARRIS F. MAC NEISH,
College of the City of New York.

METHOD BY MEANS OF CIRCLES.

Three observation stations, A, B, and C, are established near the front line, Station A being somewhat in advance of B and C. Station A is connected by wire with Stations B and C, and an instrument is set up at A so that the pushing of a button will start clocks going at B and C. When the discharge of the gun is heard at A the button is pushed, and when the discharge is heard at B and C the time is recorded by the observers.

Suppose the time recorded at B is two seconds, and at C three seconds. The velocity of sound is accurately determined in advance, but for simplicity assume the velocity to be 1,000 feet per second. It is evident then that A is 2,000 feet nearer to the gun than B, and 3,000 feet nearer than C.

A, B, and C are located accurately on a military map (see Figure 1), and two circles are drawn on the map to scale, one with B as center and 2,000 feet as radius, and the other with C as center and 3,000 feet as radius. Two circles are then drawn passing through A and tangent externally to the circles about B and C. Call the centers of these circles G1 and G2. G, and G, are evidently 2,000 feet farther from B than from A, and 3,000

1

feet farther from C than from A, and hence one of them is the required position of the gun. In practice one of the two positions, G1 or G2, may be excluded by direct observation as one of the two points usually lies back of the lines. Moreover, the determination is repeated many times, based upon successive discharges of the gun, and upon different positions of the stations, and the vicinity of the gun is located accurately within certain limits.

METHOD BY MEANS OF HYPERBOLAS.

Three stations, A, B, and C, are established as before. The stations are provided with very accurate clocks which are set at exactly the same time (clocks are available measuring time accurately to one one-hundredth of a second). When the discharge of the gun is heard, the time is recorded at the three stations. Suppose that as before the velocity of sound is taken as

Figure 2.

1,000 feet per second and that the gun is heard at A two seconds sooner than at B and three seconds sooner than at C.

The gun is therefore 2,000 feet nearer to A than to B. The locus of all points, the difference of whose distances from A and B is 2,000 feet, is a hyperbola drawn with A and B as foci and 2,000 feet as transverse axis. The branch adjacent to A is the locus of all the points 2,000 feet nearer to A than to B.

A, B, and C are located accurately on a military map (see

Figure 2), and the branch of the hyperbola drawn on the map to scale. The gun is situated somewhere in this curve. Similarly, a hyperbola is drawn with A and C as foci, and 3,000 feet as transverse axis. The gun will also be situated on the branch of that hyperbola adjacent to A. These two curves will intersect in two points, one of which will be the required position of the gun. In general, as before, no difficulty would be encountered in determining which of the two positions is the correct one. However, a third hyperbola may be drawn from the data with B and C as foci and 1,000 feet as transverse axis, and the gun will lie on the branch adjacent to B.

This branch will in general pass through only one of the two points previously determined, and the position of the gun will be determined uniquely.

AN APPLIED COURSE IN HIGH SCHOOL PHYSIOGRAPHY. BY O. W. FREEMAN,

Fergus County High School, Lewistown, Mont.

It is becoming a habit for science teachers to urge the merits of general science for high school students in preference to any particular science. Physiography especially has suffered as a result, and the subject is no longer taught in many schools. The writer admits that general science has a place in the curriculum and teaches a first-year class in the subject, but believes that physiography offers a splendid opportunity to direct the students along many practical lines. Physiography is elected by seventythree per cent more students than general science in the Fergus County High School at Lewistown, Mont. It is popular with the upper classes, and fifty-eight per cent of those taking physiography are from the three upper classes. Thirty-seven per cent of the class have had some other science course previously. Lewistown is in an exceedingly interesting part of Montana from a geologic, physiographic, and mining standpoint, and the subject of physiography fills a real need in an institution located there. The writer has adopted the suggestions of others along with his own for an applied course in physiography, and the course as given is the result of many years of teaching the subject and is modified from year to year.

The school year at the Fergus County High School is thirtyeight weeks long, of which two weeks are used for examinations. A textbook is followed so far as general topics are concerned, but recitations on the assignments are the least part of the course.