The fox and the wolf, and even the coyote, can readily distinguish whether a herd of sheep or cattle is guarded by three or four dogs, and whether there is one herdsman or two. They cannot tell the exact number of sheep, however; neither could a man without first counting them. Their knowledge of geometry is remarkable. They can orient themselves to the surrounding woods, measure distances, figure out the safest way of escape, and the power of the enemy even better than savage man. Yet in most of these problems, definite notions of number or figures have little part. A dog, when hunting, for example, on a prairie where he has to leap over ditches or quickly turn around a large tree, is able by a second's thought to do so without danger. He clears the wire fence, leaps the ditch, dashes through a closing gate, or escapes an infuriated enemy at a moment's notice. This natural wisdom is exercised spontaneously in him, it is the result of inborn theorems of which he may not even be aware, but which he uses with a sureness that defies the book-learning of all our teachers of mathematics. He uses speed, force, space, mass, and time with so small an effort, and by the quickest and shortest routes.
Suppose a wolf or a wild hog could not tell how many dogs were attacking it? There would be no way for it to defend itself. If four dogs attack it, they are counted and the tactics used that would be useless in other cases. If four dogs attack, two on each side, it retreats, with face toward the enemy. If a dozen dogs are in the attacking force, the hog becomes confused, loses all idea of number, and wildly bites at any enemy that comes nearest. Man in a similar condition would use practically the same tactics.
Cats undeniably count their kittens. If the mother loses one of three or four, she searches for it immediately. When dogs are chasing a hare, if they raise another, they become very confused, as if they did not know which to follow. Many shepherd dogs know if a sheep is missing from the flock and go to hunt it.
The efforts of scientific investigators, who work with so many learned theories, have been less successful in discovering the real facts about animals than of laymen, largely because the scientists have not yet learned that arithmetical notions are more difficult than geometrical ones. Our industrial civilisation has caused us to lose the idea of the insignificance that number has in animal life compared to the idea of size. Most animals have a remarkable sense of size; they measure time and distance better than civilised man. A hyena, for example, knows just how near he dare approach an unarmed man.
A sense of time is common among animals that daily eat at fixed hours. A donkey was accustomed to being fed at six o'clock in the morning, and when on one occasion his master did not appear on time, he deliberately kicked in the door to the barn and proceeded to feed himself.
Animals are capable of measuring lapses of time in which they are particularly interested. Houzeau claims that a female crocodile remains away from her eggs in the sand for twelve to twenty days, according to the species, but returns to the place exactly on the day they hatch.
Although we should hesitate to affirm that all animals have an extensive knowledge of figures and numbers, yet it can hardly be denied that the elephant, donkey, horse, dog, and cat, if given the proper training, become good mathematicians. It is undeniable that they have a love of mental acquisition, and it seems that the Creator has given to every animal, as a reward for its limitations in other respects, a definite innate knowledge and desire to advance educationally. There is in the breast of every animal an irresistible impulse which urges it to advance in the scale of knowledge. Where the animal is blessed with other mental powers, there is found a perfect harmony—of tact, intuition, insight, and genius—all that man himself possesses.