changed by about 21%, or by about 3.7, or, say, 4. Then b lies between 144 and 152, and this makes A doubtful between 88° 20′ 10′′ and 88° 27′ 24′′, which is still a wider range than the sine gave, 88° 21′ 30′′ and 88° 25' 10".

The proper attitude toward all cases like this, where the field-measurements lead, primarily, to a function for which the tables reply uncertainly as to the angle, is to go again to the field and make other measurements. Any attempt to overcome the effect of injudicious field-work by high-art paper-work, involves generally, as here, multiplying a small quantity, subject to possible error, by a similar relatively large quantity. Thus the error in the small quantity is multiplied by the large quantity, vitiating the calculated quantities.

The proper direction to the engineering student is to avoid field-measurements leading to the determination of an angle from the sine, when the angle is near 90°.

The same textbook from which I have quoted goes on to say:

The same difficulty is encountered in oblique triangles when the law of sines is applied to calculate angles near 90°.

and B lies between 88° 56′ 10′′ and 89° 0' 30".

To determine B more accurately, drop the perpendicular CD on AB. Let CD=p and DB=t.

Here again is a small quantity, subject to rejection error, multiplied by a relatively large quantity, and the case is as before.

Of course, however, if the figures given for b, a, C (2, 1.06, 37°) represent actual measurements, the whole procedure is only an amusing refinement in calculation. The length, b=2, is subject to the rejection error, 0.5, which renders the five decimals in p meaningless. But even when the lengths represent sane (even though very sensitive) measurements, the difficulty remains ineradicable.

The following example, taken from another textbook, will illustrate this.

Let c=1.08261 and b = 1.08249, to solve the rightangled triangle. B is near 90° and the tables give an uncertain answer from sine B.

We may use

B sin =

c

Here the field measurements represent considerable refinement.

The fifth decimal place is cared for. If the figure, 1, preceding the decimal in b, c represents a foot or an inch, the accuracy indicated is on the borderland of all possible accuracy. Even if a mile is in mind, an accuracy of measurement of about 0.4 of an inch in a mile is indicated.

The indicated error in both b and c is the small quantity 0.000005.

Thus either c-b or c+b is liable to the error 0.00001. Thus

a2=0.00,02,59,81 is liable to the error

0.00,00,21,65.

Thus a2 is uncertain in the third doublet of decimals, and a2 lies between 0.00,02,81,46 and 0.00,02,38,16. Thus instead of a being exactly 0.016119, as the book gives it, it is certain only as far as 0.01; that is, to only one significant figure.

Thus is the determination of B by this process worthless.

To insure B correct to the nearest second we should

know a to five significant figures. To calculate a to this closeness requires that b, c be measured far more closely.

If a has the error x, a2 has the error 2ax, nearly. If a here has an error of 1 in the last significant figure, the error in a2 is 2 times 0.016119 times 0.000001. But by the process, the error in a2 is at most c + b times the error in cb, or b+c times twice the error in b or c.

.. allowable error in b or c=

0.016119 0.000001 2.1651

or less than half the numerator.

The allowable error in b and c is then less than eight billionths of a mile in a mile measurement, which is about five ten-thousandths of an inch in measuring a mile!

A NEGLECTED OPPORTUNITY TO

TEACH

CONSISTENT MEASUREMENT IN
TEACHING TRIGONOMETRY.

BY R. D. BOHANNAN,

Professor of Mathematics, Ohio State University.

When an engineer says a line is 27 units long, he rarely wishes to be understood as meaning exactly 27. If he is reading to the nearest unit, he calls any reading 27 which lies between 26.5 and 27.5, the lower limit included, the upper limit excluded.

When a textbook on trigonometry gives the side of a triangle as 27, it means, very generally, 27 exactly, just 27 and nothing else, and the calculation of ungiven parts is carried to a closeness having no bearing on the data and limited only by the tables at hand.

With an engineer 27, 27.0, 27.00, have very different meanings, when they indicate measurements. In the first the figure in tenths place is not known and the 7 may have come from 6.5. In the second, the figure in hundredths place is not known, and so on.

In a textbook on trigonometry, 27 and 27.0000000,zeros "world without end,"-mean one and the same thing.

These two points of view are very different. The one has to do with a real world, the other with triangles that never were, either on land or sea.

The student trained in the textbook point of view, will go into a physical or mechanical laboratory, or