No rule can be given for discriminating between coincidences which are casual and those which are the effects of law. By a fortuitous or casual coincidence, we mean an agreement between events, which nevertheless arise from wholly independent and different causes or conditions, and which will not always so agree. It is a fortuitous coincidence, if a penny thrown up repeatedly in various ways always falls on the same side; but it would not be fortuitous if there were any similarity in the motions of the hand, and the height of the throw, so as to cause or tend to cause a uniform result. Now among the infinitely numerous events, objects, or relations in the universe, it is quite likely that we shall occasionally notice casual coincidences. There are seven intervals in the octave, and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals. It is hardly yet decided whether this apparent coincidence, with which Newton was much struck, is well founded or not,[175] but the question will probably be decided in the negative. [175] Newton’s *Opticks*, Bk. I., Part ii. Prop. 3; *Nature*, vol. i. p. 286. It is certainly a casual coincidence which the ancients noticed between the seven vowels, the seven strings of the lyre, the seven Pleiades, and the seven chiefs at Thebes.[176] The accidents connected with the number seven have misled the human intellect throughout the historical period. Pythagoras imagined a connection between the seven planets and the seven intervals of the monochord. The alchemists were never tired of drawing inferences from the coincidence in numbers of the seven planets and the seven metals, not to speak of the seven days of the week. [176] Aristotle’s *Metaphysics*, xiii. 6. 3. A singular circumstance was pointed out concerning the dimensions of the earth, sun, and moon; the sun’s diameter was almost exactly 110 times as great as the earth’s diameter, while in almost exactly the same ratio the mean distance of the earth was greater than the sun’s diameter, and the mean distance of the moon from the earth was greater than the moon’s diameter. The agreement was so close that it might have proved more than casual, but its fortuitous character is now sufficiently shown by the fact, that the coincidence ceases to be remarkable when we adopt the amended dimensions of the planetary system.