I gave in p. 181 a calculation of the number of ways in which eight planets can meet in conjunction; the reader will find all the numbers detailed in the ninth line of the arithmetical triangle. The sum of the whole line is 2^{8} or 256; but we must subtract a unit for the case where no planet appears, and 8 for the 8 cases in which only one planet appears; so that the total number of conjunctions is 2^{8} -1 - 8 or 247. If an organ has eleven stops we find in the twelfth line the numbers of ways in which we can draw them, 1, 2, 3, or more at a time. Thus there are 462 ways of drawing five stops at once, and as many of drawing six stops. The total number of ways of varying the sound is 2048, including the single case in which no stop at all is drawn. One of the most important scientific uses of the arithmetical triangle consists in the information which it gives concerning the comparative frequency of divergencies from an average. Suppose, for the sake of argument, that all persons were naturally of the equal stature of five feet, but enjoyed during youth seven independent chances of growing one inch in addition. Of these seven chances, one, two, three, or more, may happen favourably to any individual; but, as it does not matter what the chances are, so that the inch is gained, the question really turns upon the number of combinations of 0, 1, 2, 3, &c., things out of seven. Hence the eighth line of the triangle gives us a complete answer to the question, as follows:-- Out of every 128 people-- Feet Inches. One person would have the stature of 5 0 7 persons " " 5 1 21 persons " " 5 2 35 persons " " 5 3 35 persons " " 5 4 21 persons " " 5 5 7 persons " " 5 6 1 person " " 5 7 By taking a proper line of the triangle, an answer may be had under any more natural supposition. This theory of comparative frequency of divergence from an average, was first adequately noticed by Quetelet, and has lately been employed in a very interesting and bold manner by Mr. Francis Galton,[104] in his remarkable work on “Hereditary Genius.” We shall afterwards find that the theory of error, to which is made the ultimate appeal in cases of quantitative investigation, is founded upon the comparative numbers of combinations as displayed in the triangle. [104] See also Galton’s Lecture at the Royal Institution, 27th February, 1874; Catalogue of the Special Loan Collection of Scientific Instruments, South Kensington, Nos. 48, 49; and Galton, *Philosophical Magazine*, January 1875. *Connection between the Arithmetical Triangle and the Logical Alphabet.*