The quantitative axiom of mediate relation—“Magnitudes equal to the same magnitude are equal”—is, of course, never expressed by savages; but it is practically understood and applied by them whenever they use a common measure—the fingers for counting, the pace, or hand or arm-stretch to measure distance. These devices are gesture language; and the truth which the gestures assume has been forced upon the observation of primitive men whenever they saw three or more nearly equal things together—men in a row, birds in a flock, eggs in a nest. We need not suppose that conscious analysis is necessary to determine the relations between such things; the brain has its own method of analysis; and some day we learn the results. It was late in the day that the results became known; not, apparently, till the Greeks gave scientific form to the rudiments of mathematics. For then, for the first time, articulate axioms were wanted—to satisfy the form of science. The innumerable exact calculations of Egypt and Assyria could go on very well without them. Their discovery required the specific purpose of a scientist in search of them, the state of profound meditation and abstraction, excluding all irrelevant ideas, when in the emptiness and darkness of the mind their light became visible, like a faint sound in a silent room. Thus an idea, whose functioning has for ages controlled thought without being recognised, suddenly takes its place in the organisation of knowledge. § 8. THE WEAKNESS OF IMAGINATION-BELIEFS With immature minds their superstitions seem to rest on good evidence. Some of those who pray to Neptune are saved from shipwreck, and the drowned are forgotten: all confirmatory coincidences are deeply impressive, and failures are overlooked or excused. By suggestion the sick are often healed, and the hale are struck down. Curses and incantations, if known to the intended victim, fulfil themselves. So do good omens that give confidence, and bad omens that weaken endeavour. If a magician has the astuteness to operate for rain only when the wet season approaches, the event is likely to confirm his reputation. Sleight of hand, ventriloquism and the advantages of a dark séance are not unknown to sorcerers tutored in an old tradition of deceit, and their clients take it for demonstration. The constant practice by a whole village of both magic and industry for the same end, makes it impossible for ordinary mortals to see which of them is the real agent of success. For the failures of magic or of sacrifice the practitioners always have plausible explanations. Hence between imagination-beliefs and perception-beliefs, as to their causes, there may be, for the believers, no apparent difference.