"How is that possible?" Amy asked. "Well, you see there are plenty of buds on all the young branches, and we can cut a branch just above the bud we wish to grow which will continue to grow in the direction in which it points. Thus we can shape each summer's growth in any direction we choose." "How can you be sure to find a bud just where you want it?" "I know we always do." "Of course we do," said Webb, "for buds are arranged spirally on trees in mathematical order. On most trees it is termed-the 'five-ranked arrangement,' and every bud is just two-fifths of the circumference of the stem from the next. This will bring every sixth bud or leaf over the first, or the one we start with. Thus in the length of stem occupied by five buds you have buds facing in five different directions--plenty of choice for all pruning purposes." "Oh, nonsense, Webb; you are too everlastingly scientific. Buds and leaves are scattered at haphazard all over the branches." "That shows you observe at haphazard. Wait, and I'll prove I'm right;" and he seized his hat and went out. Returning after a few minutes with long, slender shoots of peach, apple, and pear trees, he said: "Now put your finger on any bud, and count. See if the sixth bud does not stand invariably over the one you start from, and if the intervening buds do not wind spirally twice around the stem, each facing in a different direction." The result proved Webb to be right. He laughed, and said: "There, Len, you've seen buds and branches for over forty years, and never noticed this. Here, Alf, you begin right, and learn to see things just as they are. There's no telling how often accurate knowledge may be useful." "But, Webb, all plants have not the five-ranked arrangement, as you term it," his mother protested. "Oh, no. There is the two-ranked, in which the third leaf stands over the first; the three-ranked, in which the fourth leaf stands over the first. Then we also find the eighth and thirteenth ranked arrangements, according to the construction of various species of plants or trees. But having once observed an arrangement of buds or leaves in a species, you will find it maintained with absolute symmetry and accuracy, although the spaces between the buds lengthwise upon the stem may vary very much. Nature, with all her seeming carelessness and _abandon_, works on strict mathematical principles." "Well," said Alf, "I'm going to see if you are right tomorrow. I don't half believe you are." And on the following day he tried his best to prove Webb wrong, but failed.