Line. First Column. 1 1 Second Column. 2 1 1 Third Column. 3 1 2 1 Fourth Column. 4 1 3 3 1 Fifth Column. 5 1 4 6 4 1 Sixth Column. 6 1 5 10 10 5 1 Seventh Column. 7 1 6 15 20 15 6 1 Eighth Column. 8 1 7 21 35 35 21 7 1 Ninth Column. 9 1 8 28 56 70 56 28 8 1 Tenth Column. 10 1 9 36 84 126 126 84 36 9 1 Eleventh Column. 11 1 10 45 120 210 252 210 120 45 10 1 Twelfth Column. 12 1 11 55 165 330 462 462 330 165 55 11 1 Thirteenth Column. 13 1 12 66 220 495 792 924 792 495 220 66 12 1 Fourteenth Column. 14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Fifteenth Column. 15 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 Sixteenth Column. 16 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 Seventeenth Col. 17 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
Examining these numbers, we find that they are connected by an unlimited series of relations, a few of the more simple of which may be noticed. Each vertical column of numbers exactly corresponds with an oblique series descending from left to right, so that the triangle is perfectly symmetrical in its contents. The first column contains only *units*; the second column contains the *natural numbers*, 1, 2, 3, &c.; the third column contains a remarkable series of numbers, 1, 3, 6, 10, 15, &c., which have long been called *the triangular numbers*, because they correspond with the numbers of balls which may be arranged in a triangular form, thus--
[Illustration]
The fourth column contains the *pyramidal numbers*, so called because they correspond to the numbers of equal balls which can be piled in regular triangular pyramids. Their differences are the triangular numbers. The numbers of the fifth column have the pyramidal numbers for their differences, but as there is no regular figure of which they express the contents, they have been arbitrarily called the *trianguli-triangular numbers*. The succeeding columns have, in a similar manner, been said to contain the *trianguli-pyramidal*, the *pyramidi-pyramidal* numbers, and so on.[102]
[102] Wallis’s *Algebra*, Discourse of Combinations, &c., p. 109.
From the mode of formation of the table, it follows that the differences of the numbers in each column will be found in the preceding column to the left. Hence the *second differences*, or the *differences of differences*, will be in the second column to the left of any given column, the third differences in the third column, and so on. Thus we may say that unity which appears in the first column is the *first difference* of the numbers in the second column; the *second difference* of those in the third column; the *third difference* of those in the fourth, and so on. The triangle is seen to be a complete classification of all numbers according as they have unity for any of their differences.
Since each line is formed by adding the previous line to itself, it is evident that the sum of the numbers in each horizontal line must be double the sum of the numbers in the line next above. Hence we know, without making the additions, that the successive sums must be 1, 2, 4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the Logical Alphabet. Speaking generally, the sum of the numbers in the *n*th line will be 2^{*n* - 1}.
Again, if the whole of the numbers down to any line be added together, we shall obtain a number less by unity than some power of 2; thus, the first line gives 1 or 2^{1} - 1; the first two lines give 3 or 2^{2} - 1; the first three lines 7 or 2^{3} - 1; the first six lines give 63 or 2^{6} - 1; or, speaking in general language, the sum of the first *n* lines is 2^{*n*} - 1. It follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines increased by a unit. For the sum of the *n*th line is, as already shown, 2^{*n* - 1}, and the sum of the first *n* - 1 lines is 2^{*n* - 1} - 1, or less by a unit.
This account of the properties of the figurate numbers does not approach completeness; a considerable, probably an unlimited, number of less simple and obvious relations might be traced out. Pascal, after giving many of the properties, exclaims[103]: “Mais j’en laisse bien plus que je n’en donne; c’est une chose étrange combien il est fertile en propriétés! Chacun peut s’y exercer.” The arithmetical triangle may be considered a natural classification of numbers, exhibiting, in the most complete manner, their evolution and relations in a certain point of view. It is obvious that in an unlimited extension of the triangle, each number, with the single exception of the number *two*, has at least two places.
[103] *Œuvres Complètes*, vol. iii. p. 251.
Though the properties above explained are highly curious, the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula (p. 182), for the numbers of combinations of *m* things out of *n*, for all possible values of *m* and *n*. Out of seven things one may be chosen in seven ways, and seven occurs in the eighth line of the second column. The combinations of two things chosen out of seven are (7 × 6)/(1 × 2) or 21, which is the third number in the eighth line. The combinations of three things out of seven are (7 × 6 × 5)/(1 × 2 × 3) or 35, which appears fourth in the eighth line. In a similar manner, in the fifth, sixth, seventh, and eighth columns of the eighth line I find it stated in how many ways I can select combinations of 4, 5, 6, and 7 things out of 7. Proceeding to the ninth line, I find in succession the number of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out of 8 things. In general language, if I wish to know in how many ways *m* things can be selected in combinations out of *n* things, I must look in the *n* + 1^{th} line, and take the *m* + 1^{th} number, as the answer. In how many ways, for instance, can a subcommittee of five be chosen out of a committee of nine. The answer is 126, and is the sixth number in the tenth line; it will be found equal to (9 . 8 . 7 . 6 . 5)/(1 . 2 . 3 . 4 . 5), which our formula (p. 182) gives.
The full utility of the figurate numbers will be more apparent when we reach the subject of probabilities, but I may give an illustration or two in this place. In how many ways can we arrange four pennies as regards head and tail? The question amounts to asking in how many ways we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the *fifth* line of the triangle gives us the complete answer, thus--
We can select No head and 4 tails in 1 way. " 1 head and 3 tails in 4 ways. " 2 heads and 2 tails in 6 ways. " 3 heads and 1 tail in 4 ways. " 4 heads and 0 tail in 1 way.
The total number of different cases is 16, or 2^{4}, and when we come to the next chapter, it will be found that these numbers give us the respective probabilities of all throws with four pennies.
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.*
There exists a close connection between the arithmetical triangle described in the last section, and the series of combinations of letters called the Logical Alphabet. The one is to mathematical science what the other is to logical science. In fact the figurate numbers, or those exhibited in the triangle, are obtained by summing up the logical combinations. Accordingly, just as the total of the numbers in each line of the triangle is twice as great as that for the preceding line (p. 186), so each column of the Alphabet (p. 94) contains twice as many combinations as the preceding one. The like correspondence also exists between the sums of all the lines of figures down to any particular line, and of the combinations down to any particular column.
By examining any column of the Logical Alphabet we find that the combinations naturally group themselves according to the figurate numbers. Take the combinations of the letters A, B, C, D; they consist of all the ways in which I can choose four, three, two, one, or none of the four letters, filling up the vacant spaces with negative terms.
There is one combination, ABCD, in which all the positive letters are present; there are four combinations in each of which three positive letters are present; six in which two are present; four in which only one is present; and, finally, there is the single case, *abcd*, in which all positive letters are absent. These numbers, 1, 4, 6, 4, 1, are those of the fifth line of the arithmetical triangle, and a like correspondence will be found to exist in each column of the Logical Alphabet.
Numerical abstraction, it has been asserted, consists in overlooking the kind of difference, and retaining only a consciousness of its existence (p. 158). While in logic, then, we have to deal with each combination as a separate kind of thing, in arithmetic we distinguish only the classes which depend upon more or less positive terms being present, and the numbers of these classes immediately produce the numbers of the arithmetical triangle.
It may here be pointed out that there are two modes in which we can calculate the whole number of combinations of certain things. Either we may take the whole number at once as shown in the Logical Alphabet, in which case the number will be some power of two, or else we may calculate successively, by aid of permutations, the number of combinations of none, one, two, three things, and so on. Hence we arrive at a necessary identity between two series of numbers. In the case of four things we shall have
2 = 1 + 4/1 + (4 . 3)/(1 . 2) + (4 . 3 . 2)/(1 . 2 . 3) + (4 . 3 . 2 . 1)/(1 . 2 . 3 . 4).
In a general form of expression we shall have
2 = 1 + *n*/1 + (*n* . (*n* - 1))/(1 . 2) + (*n* (*n* - 1)(*n* - 2))/(1 . 2 . 3) + &c.,
the terms being continued until they cease to have any value. Thus we arrive at a proof of simple cases of the Binomial Theorem, of which each column of the Logical Alphabet is an exemplification. It may be shown that all other mathematical expansions likewise arise out of simple processes of combination, but the more complete consideration of this subject must be deferred to another work.
*Possible Variety of Nature and Art.*
We cannot adequately understand the difficulties which beset us in certain branches of science, unless we have some clear idea of the vast numbers of combinations or permutations which may be possible under certain conditions. Thus only can we learn how hopeless it would be to attempt to treat nature in detail, and exhaust the whole number of events which might arise. It is instructive to consider, in the first place, how immensely great are the numbers of combinations with which we deal in many arts and amusements.
In dealing a pack of cards, the number of hands, of thirteen cards each, which can be produced is evidently 52 × 51 × 50 × ... × 40 divided by 1 × 2 × 3 ... × 13. or 635,013,559,600. But in whist four hands are simultaneously held, and the number of distinct deals becomes so vast that it would require twenty-eight figures to express it. If the whole population of the world, say one thousand millions of persons, were to deal cards day and night, for a hundred million of years, they would not in that time have exhausted one hundred-thousandth part of the possible deals. Even with the same hands of cards the play may be almost infinitely varied, so that the complete variety of games at whist which may exist is almost incalculably great. It is in the highest degree improbable that any one game of whist was ever exactly like another, except it were intentionally so.
The end of novelty in art might well be dreaded, did we not find that nature at least has placed no attainable limit, and that the deficiency will lie in our inventive faculties. It would be a cheerless time indeed when all possible varieties of melody were exhausted, but it is readily shown that if a peal of twenty-four bells had been rung continuously from the so-called beginning of the world to the present day, no approach could have been made to the completion of the possible changes. Nay, had every single minute been prolonged to 10,000 years, still the task would have been unaccomplished.[105] As regards ordinary melodies, the eight notes of a single octave give more than 40,000 permutations, and two octaves more than a million millions. If we were to take into account the semitones, it would become apparent that it is impossible to exhaust the variety of music. When the late Mr. J. S. Mill, in a depressed state of mind, feared the approaching exhaustion of musical melodies, he had certainly not bestowed sufficient study on the subject of permutations.
[105] Wallis, *Of Combinations*, p. 116, quoting Vossius.
Similar considerations apply to the possible number of natural substances, though we cannot always give precise numerical results. It was recommended by Hatchett[106] that a systematic examination of all alloys of metals should be carried out, proceeding from the binary ones to more complicated ternary or quaternary ones. He can hardly have been aware of the extent of his proposed inquiry. If we operate only upon thirty of the known metals, the number of binary alloys would be 435, of ternary alloys 4060, of quaternary 27,405, without paying regard to the varying proportions of the metals, and only regarding the kind of metal. If we varied all the ternary alloys by quantities not less than one per cent., the number of these alloys would be 11,445,060. An exhaustive investigation of the subject is therefore out of the question, and unless some laws connecting the properties of the alloy and its components can be discovered, it is not apparent how our knowledge of them can ever be more than fragmentary.
[106] *Philosophical Transactions* (1803), vol. xciii. p. 193.
The possible variety of definite chemical compounds, again, is enormously great. Chemists have already examined many thousands of inorganic substances, and a still greater number of organic compounds;[107] they have nevertheless made no appreciable impression on the number which may exist. Taking the number of elements at sixty-one, the number of compounds containing different selections of four elements each would be more than half a million (521,855). As the same elements often combine in many different proportions, and some of them, especially carbon, have the power of forming an almost endless number of compounds, it would hardly be possible to assign any limit to the number of chemical compounds which may be formed. There are branches of physical science, therefore, of which it is unlikely that scientific men, with all their industry, can ever obtain a knowledge in any appreciable degree approaching to completeness.
[107] Hofmann’s *Introduction to Chemistry*, p. 36.
*Higher Orders of Variety.*
The consideration of the facts already given in this chapter will not produce an adequate notion of the possible variety of existence, unless we consider the comparative numbers of combinations of different orders. By a combination of a higher order, I mean a combination of groups, which are themselves groups. The immense numbers of compounds of carbon, hydrogen, and oxygen, described in organic chemistry, are combinations of a second order, for the atoms are groups of groups. The wave of sound produced by a musical instrument may be regarded as a combination of motions; the body of sound proceeding from a large orchestra is therefore a complex aggregate of sounds, each in itself a complex combination of movements. All literature may be said to be developed out of the difference of white paper and black ink. From the unlimited number of marks which might be chosen we select twenty-six conventional letters. The pronounceable combinations of letters are probably some trillions in number. Now, as a sentence is a selection of words, the possible sentences must be inconceivably more numerous than the words of which it may be composed. A book is a combination of sentences, and a library is a combination of books. A library, therefore, may be regarded as a combination of the fifth order, and the powers of numerical expression would be severely tasked in attempting to express the number of distinct libraries which might be constructed. The calculation, of course, would not be possible, because the union of letters in words, of words in sentences, and of sentences in books, is governed by conditions so complex as to defy analysis. I wish only to point out that the infinite variety of literature, existing or possible, is all developed out of one fundamental difference. Galileo remarked that all truth is contained in the compass of the alphabet. He ought to have said that it is all contained in the difference of ink and paper.
One consequence of successive combination is that the simplest marks will suffice to express any information. Francis Bacon proposed for secret writing a biliteral cipher, which resolves all letters of the alphabet into permutations of the two letters *a* and *b*. Thus A was *aaaaa*, B *aaaab*, X *babab*, and so on.[108] In a similar way, as Bacon clearly saw, any one difference can be made the ground of a code of signals; we can express, as he says, *omnia per omnia*. The Morse alphabet uses only a succession of long and short marks, and other systems of telegraphic language employ right and left strokes. A single lamp obscured at various intervals, long or short, may be made to spell out any words, and with two lamps, distinguished by colour, position, or any other circumstance, we could at once represent Bacon’s biliteral alphabet. Babbage ingeniously suggested that every lighthouse in the world should be made to spell out its own name or number perpetually, by flashes or obscurations of various duration and succession. A system like that of Babbage is now being applied to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John Hopkinson.
[108] *Works*, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s *Encyclopædia*, art. *Cipher*.
Let us calculate the numbers of combinations of different orders which may arise out of the presence or absence of a single mark, say A. In these figures
+---+---+ +---+---+ +---+---+ +---+---+ | A | A | | A | | | | A | | | | +---+---+ +---+---+ +---+---+ +---+---+
we have four distinct varieties. Form them into a group of a higher order, and consider in how many ways we may vary that group by omitting one or more of the component parts. Now, as there are four parts, and any one may be present or absent, the possible varieties will be 2 × 2 × 2 × 2, or 16 in number. Form these into a new whole, and proceed again to create variety by omitting any one or more of the sixteen. The number of possible changes will now be 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2, or 2^{16}, and we can repeat the process again and again. We are imagining the creation of objects, whose numbers are represented by the successive orders of the powers of *two*.
At the first step we have 2; at the next 2^{2}, or 4; at the third (2^{2})^{2}, or 16, numbers of very moderate amount. Let the reader calculate the next term, ((2^{2})^{2})^{2}, and he will be surprised to find it leap up to 65,536. But at the next step he has to calculate the value of 65,536 *two*’s multiplied together, and it is so great that we could not possibly compute it, the mere expression of the result requiring 19,729 places of figures. But go one step more and we pass the bounds of all reason. The sixth order of the powers of *two* becomes so great, that we could not even express the number of figures required in writing it down, without using about 19,729 figures for the purpose. The successive orders of the powers of two have then the following values so far as we can succeed in describing them:--
First order 2 Second order 4 Third order 16 Fourth order 65,536 Fifth order, number expressed by 19,729 figures. Sixth order, number expressed by figures, to express the number of which figures would require about 19,729 figures.
It may give us some notion of infinity to remember that at this sixth step`, having long surpassed all bounds of intuitive conception, we make no approach to a limit. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity.
It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most remarkable writings of antiquity, the *Liber de Arcnæ Numero*, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arithmetical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes--those of the sixteenth magnitude--are supposed to have a distance of about 33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for supposing that there do not exist more than from 3 × 10^{24} to 10^{26} molecules in a cubic centimetre of a solid or liquid substance.[109] Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 68 × 10^{90} atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be immensely less than the fifth order of the powers of two.
[109] *Nature*, vol. i. p. 553.
In the variety of logical relations, which may exist between a certain number of logical terms, we also meet a case of higher combinations. We have seen (p. 142) that with only six terms the number of possible selections of combinations is 18,446,744,073,709,551,616. Considering that it is the most common thing in the world to use an argument involving six objects or terms, it may excite some surprise that the complete investigation of the relations in which six such terms may stand to each other, should involve an almost inconceivable number of cases. Yet these numbers of possible logical relations belong only to the second order of combinations.
