The next step is a figure adjustment like that for a chain of triangles, except that the sum of the interior angles of a quadrilateral is (4 - 2) 180, or 360.

Next, for a quadrilateral, comes another figure adjustment, based on the four overlapping triangles within the quadrilateral. To understand this figure adjustment, study the quadrilateral shown in figure 15-28. The diagonals in this quadrilateral intersect to

form vertically opposite angles 9-10 and 11-12. From your knowledge of geometry, you know that when two straight lines intersect the vertically opposite angles thus formed are equal. From the fact that the sum of the angles in any triangle is 180, it follows that for any pair of vertically opposite angles in figure 15-28, the sums of the other two angles in each of the corresponding triangles must be equal.

For example: In figure 15-28, angles 11 and 12 are equal vertically opposite angles. Angle 11 lies in a triangle in which the other two angles are angles 1 and 8; angle 12 lies in a triangle in which the other two angles are angles 4 and 5. It follows, then, that the sum of angle 1 plus angle 8 must equal the sum of angle 5 plus angle 4. By similar reasoning, the sum of angle 2 plus angle 3 must equal the sum of angle 6 plus angle 7.

Suppose now, that the values of angles 2, 3, 6, and 7, after adjustment for the sum of interior angles, areas follows:

The difference between the two sums is 8 seconds. This means that, to make the sums equal, 4 seconds should be subtracted from the 2-3 sum and added to the 6-7 sum. To subtract 4 seconds from the 2-3 sum, you subtract 2 seconds from each angle; to add 4 seconds to the 6-7 sum, you add 2 seconds to each angle.

The final step in quadrilateral adjustment is related to the fact that you can compute the length of a side in a quadrilateral by more than one route. The final step in adjustment is to ensure that, for a given side, you will get the same result, to the desired number of significant figures, regardless of the route your computations take.

By the same law,

Again by the law of sines we have

By the same law,

It follows that

By the law of proportions, this can be expressed as

You know that in logarithms, instead of multiplying you just add logarithms; also, instead of dividing one number by another, you just subtract the logarithm of the second from the logarithm of the first. Note that the logarithm of 1 is 0.000000. Therefore, the above equation can be expressed as follows:

(log sin 1 + log sin 3 + log sin 5 + log sin7) - (log sin 2 + log sin 4 + log sin 6 + log sin 8) = 0

Suppose, now, that after the second figure adjustment, the values of the angles shown in figure 15-28 are as follows:

A table of logarithmic functions shows the log sines of these angles to be as follows:

By subtracting the two sums, you get the following:

Therefore, the difference in the sums of the log sines is 0.000047. Since there are eight angles, this means the average difference for each angle is 0.0000059.

The next question is how to convert this log sine difference per angle into terms of angular measurement To do this, you first determine, by reference to the table of log functions, the average difference in log sine, per second of arc, for the eight angles involved. This is determined from the D values given in the table. For each of the angles shown in figure 15-28, the D value is as follows:

The average difference in log sine per 1 second of arc, then, is 20.01/8, or 2.5. The average difference in log sine is 5.9; therefore, the average adjustment for each angle is 5.9 +2.5, or about 2 seconds. The sum of the log sines of angles 2, 4, 6, and 8 isS greater than that of angles 1, 3, 5, and 7. There for, you add 2 seconds each to angles 1, 3, 5, and 7 and subtract 2 seconds each from angles 2, 4, 6, and 8.