For this reason, the success of the whole quantitative study of migration depends upon our ability to make directional analyses of primary data. As I have already pointed out, the flight directions of birds may be recorded with convenience and a fair degree of objectivity by noting the slant of their apparent pathways across the disc of the moon. But these apparent pathways are seldom the real pathways. Usually they involve the transfer of the flight line from a horizontal plane of flight to a tilted plane represented by the face of the moon, and so take on the nature of a projection. They are clues to directions, but they are not the directions themselves. For each compass direction of birds flying horizontally above the earth, there is one, and only one, slant of the pathway across the moon at a given time. It is possible, therefore, knowing the path of a bird in relation to the lunar disc and the time of the observation, to compute the direction of its path in relation to the earth. The formula employed is not a complicated one, but, since the meaning of the lunar coördinates in terms of their corresponding flight paths parallel to the earth is constantly changing with the position of the moon, the calculation of each bird's flight separately would require a tremendous amount of time and effort. [Illustration: FIG. 7. The sampling effect of a square. In Diagram A eight evenly distributed birds are flying from south to north, and another four are proceeding from east to west. Three appear in each of the smaller squares. Thus, if we were to treat any of these smaller sections as a directly proportionate sample of the whole, we would be assuming that 3 × 16, or 48, birds had traversed the square mile--four times the real total of 12. If we consider the paths separately as in Diagram B, we see quite clearly what is wrong. Every bird crosses four plots the size of the sample and is being computed into the total over and over a corresponding number of times. Patently, just as many south-north birds cross the bottom tier of squares as cross the four tiers comprising the whole area. Just as many west-east birds traverse one side of the large square as cross the whole square. In other words, the inclusion of additional sections _athwart_ the direction of flight involves the inclusion of additional birds proceeding in that direction, while the inclusion of additional sections _along_ the direction does not. The correct ratio of the sample to the whole would seem to be the ratio of their perimeters, in this case the ratio of one to four. When this factor of four is applied to the problem it proves correct: 4 × 3 (the number of birds that have been seen in the sample square) equals 12 (the exact number of birds that could be seen in the square mile).]