CHAPTER IX.

THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.

Nature may be said to be evolved from the monotony of non-existence by the creation of diversity. It is plausibly asserted that we are conscious only so far as we experience difference. Life is change, and perfectly uniform existence would be no better than non-existence. Certain it is that life demands incessant novelty, and that nature, though it probably never fails to obey the same fixed laws, yet presents to us an apparently unlimited series of varied combinations of events. It is the work of science to observe and record the kinds and comparative numbers of such combinations of phenomena, occurring spontaneously or produced by our interference. Patient and skilful examination of the records may then disclose the laws imposed on matter at its creation, and enable us more or less successfully to predict, or even to regulate, the future occurrence of any particular combination.

The Laws of Thought are the first and most important of all the laws which govern the combinations of phenomena, and, though they be binding on the mind, they may also be regarded as verified in the external world. The Logical Alphabet develops the utmost variety of things and events which may occur, and it is evident that as each new quality is introduced, the number of combinations is doubled. Thus four qualities may occur in 16 combinations; five qualities in 32; six qualities in 64; and so on. In general language, if n be the number of qualities, 2^{n} is the number of varieties of things which may be formed from them, if there be no conditions but those of logic. This number, it need hardly be said, increases after the first few terms, in an extraordinary manner, so that it would require 302 figures to express the number of combinations in which 1,000 qualities might conceivably present themselves.

If all the combinations allowed by the Laws of Thought occurred indifferently in nature, then science would begin and end with those laws. To observe nature would give us no additional knowledge, because no two qualities would in the long run be oftener associated than any other two. We could never predict events with more certainty than we now predict the throws of dice, and experience would be without use. But the universe, as actually created, presents a far different and much more interesting problem. The most superficial observation shows that some things are constantly associated with other things. The more mature our examination, the more we become convinced that each event depends upon the prior occurrence of some other series of events. Action and reaction are gradually discovered to underlie the whole scene, and an independent or casual occurrence does not exist except in appearance. Even dice as they fall are surely determined in their course by prior conditions and fixed laws. Thus the combinations of events which can really occur are found to be comparatively restricted, and it is the work of science to detect these restricting conditions.

In the English alphabet, for instance, we have twenty-six letters. Were the combinations of such letters perfectly free, so that any letter could be indifferently sounded with any other, the number of words which could be formed without any repetition would be 2^{26} - 1, or 67,108,863, equal in number to the combinations of the twenty-seventh column of the Logical Alphabet, excluding one for the case in which all the letters would be absent. But the formation of our vocal organs prevents us from using the far greater part of these conjunctions of letters. At least one vowel must be present in each word; more than two consonants cannot usually be brought together; and to produce words capable of smooth utterance a number of other rules must be observed. To determine exactly how many words might exist in the English language under these circumstances, would be an exceedingly complex problem, the solution of which has never been attempted. The number of existing English words may perhaps be said not to exceed one hundred thousand, and it is only by investigating the combinations presented in the dictionary, that we can learn the Laws of Euphony or calculate the possible number of words. In this example we have an epitome of the work and method of science. The combinations of natural phenomena are limited by a great number of conditions which are in no way brought to our knowledge except so far as they are disclosed in the examination of nature.

It is often a very difficult matter to determine the numbers of permutations or combinations which may exist under various restrictions. Many learned men puzzled themselves in former centuries over what were called Protean verses, or verses admitting many variations in accordance with the Laws of Metre. The most celebrated of these verses was that invented by Bernard Bauhusius, as follows:[94]--

“Tot tibi sunt dotes, Virgo, quot sidera cœlo.”

[94] Montucla, *Histoire*, &c., vol. iii. p. 388.

One author, Ericius Puteanus, filled forty-eight pages of a work in reckoning up its possible transpositions, making them only 1022. Other calculators gave 2196, 3276, 2580 as their results. Wallis assigned 3096, but without much confidence in the accuracy of his result.[95] It required the skill of James Bernoulli to decide that the number of transpositions was 3312, under the condition that the sense and metre of the verse shall be perfectly preserved.

[95] Wallis, *Of Combinations*, &c., p. 119.

In approaching the consideration of the great Inductive problem, it is very necessary that we should acquire correct notions as to the comparative numbers of combinations which may exist under different circumstances. The doctrine of combinations is that part of mathematical science which applies numerical calculation to determine the numbers of combinations under various conditions. It is a part of the science which really lies at the base not only of other sciences, but of other branches of mathematics. The forms of algebraical expressions are determined by the principles of combination, and Hindenburg recognised this fact in his Combinatorial Analysis. The greatest mathematicians have, during the last three centuries, given their best powers to the treatment of this subject; it was the favourite study of Pascal; it early attracted the attention of Leibnitz, who wrote his curious essay, *De Arte Combinatoria*, at twenty years of age; James Bernoulli, one of the very profoundest mathematicians, devoted no small part of his life to the investigation of the subject, as connected with that of Probability; and in his celebrated work, *De Arte Conjectandi*, he has so finely described the importance of the doctrine of combinations, that I need offer no excuse for quoting his remarks at full length.

“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged or combined together, it often happens that men, even of the best understandings and greatest circumspection, are guilty of that fault in reasoning which the writers on logic call *the insufficient or imperfect enumeration of parts or cases*: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions.

“It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our understandings, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of *the art or doctrine of combinations*. Nor is this art or doctrine to be considered merely as a branch of the mathematical sciences. For it has a relation to almost every species of useful knowledge that the mind of man can be employed upon. It proceeds indeed upon mathematical principles, in calculating the number of the combinations of the things proposed: but by the conclusions that are obtained by it, the sagacity of the natural philosopher, the exactness of the historian, the skill and judgment of the physician, and the prudence and foresight of the politician may be assisted; because the business of all these important professions is but *to form reasonable conjectures* concerning the several objects which engage their attention, and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes that are capable of producing them.”[96]

[96] James Bernoulli, *De Arte Conjectandi*, translated by Baron Maseres. London, 1795, pp. 35, 36.

*Distinction of Combinations and Permutations.*

We must first consider the deep difference which exists between Combinations and Permutations, a difference involving important logical principles, and influencing the form of mathematical expressions. In *permutation* we recognise varieties of order, treating AB as a different group from BA. In *combination* we take notice only of the presence or absence of a certain thing, and pay no regard to its place in order of time or space. Thus the four letters *a*, *e*, *m*, *n* can form but one combination, but they occur in language in several permutations, as *name*, *amen*, *mean*, *mane*.

We have hitherto been dealing with purely logical questions, involving only combination of qualities. I have fully pointed out in more than one place that, though our symbols could not but be written in order of place and read in order of time, the relations expressed had no regard to place or time (pp. 33, 114). The Law of Commutativeness, in fact, expresses the condition that in logic we deal with combinations, and the same law is true of all the processes of algebra. In some cases, order may be a matter of indifference; it makes no difference, for instance, whether gunpowder is a mixture of sulphur, carbon, and nitre, or carbon, nitre, and sulphur, or nitre, sulphur, and carbon, provided that the substances are present in proper proportions and well mixed. But this indifference of order does not usually extend to the events of physical science or the operations of art. The change of mechanical energy into heat is not exactly the same as the change from heat into mechanical energy; thunder does not indifferently precede and follow lightning; it is a matter of some importance that we load, cap, present, and fire a rifle in this precise order. Time is the condition of all our thoughts, space of all our actions, and therefore both in art and science we are to a great extent concerned with permutations. Language, for instance, treats different permutations of letters as having different meanings.

Permutations of things are far more numerous than combinations of those things, for the obvious reason that each distinct thing is regarded differently according to its place. Thus the letters A, B, C, will make different permutations according as A stands first, second, or third; having decided the place of A, there are two places between which we may choose for B; and then there remains but one place for C. Accordingly the permutations of these letters will be altogether 3 × 2 × 1 or 6 in number. With four things or letters, A, B, C, D, we shall have four choices of place for the first letter, three for the second, two for the third, and one for the fourth, so that there will be altogether, 4 × 3 × 2 × 1, or 24 permutations. The same simple rule applies in all cases; beginning with the whole number of things we multiply at each step by a number decreased by a unit. In general language, if *n* be the number of things in a combination, the number of permutations is

*n* (*n* - 1)(*n* - 2) .... 4 . 3 . 2 . 1.

If we were to re-arrange the names of the days of the week, the possible arrangements out of which we should have to choose the new order, would be no less than 7 . 6 . 5 . 4 . 3 . 2 . 1, or 5040, or, excluding the existing order, 5039.

The reader will see that the numbers which we reach in questions of permutation, increase in a more extraordinary manner even than in combination. Each new object or term doubles the number of combinations, but increases the permutations by a factor continually growing. Instead of 2 × 2 × 2 × 2 × .... we have 2 × 3 × 4 × 5 × .... and the products of the latter expression immensely exceed those of the former. These products of increasing factors are frequently employed, as we shall see, in questions both of permutation and combination. They are technically called *factorials*, that is to say, the product of all integer numbers, from unity up to any number *n* is the *factorial* of *n*, and is often indicated symbolically by *n*!. I give below the factorials up to that of twelve:--

24 = 1 . 2 . 3 . 4 120 = 1 . 2 ... 5 720 = 1 . 2 ... 6 5,040 = 7! 40,320 = 8! 362,880 = 9! 3,628,800 = 10! 39,916,800 = 11! 479,001,600 = 12!

The factorials up to 36! are given in Rees’s ‘Cyclopædia,’ art. *Cipher*, and the logarithms of factorials up to 265! are to be found at the end of the table of logarithms published under the superintendence of the Society for the Diffusion of Useful Knowledge (p. 215). To express the factorial 265! would require 529 places of figures.

Many writers have from time to time remarked upon the extraordinary magnitude of the numbers with which we deal in this subject. Tacquet calculated[97] that the twenty-four [sic] letters of the alphabet may be arranged in more than 620 thousand trillions of orders; and Schott estimated[98] that if a thousand millions of men were employed for the same number of years in writing out these arrangements, and each man filled each day forty pages with forty arrangements in each, they would not have accomplished the task, as they would have written only 584 thousand trillions instead of 620 thousand trillions.

[97] *Arithmeticæ Theoria.* Ed. Amsterd. 1704. p. 517.

[98] Rees’s *Cyclopædia*, art. *Cipher*.

In some questions the number of permutations may be restricted and reduced by various conditions. Some things in a group may be undistinguishable from others, so that change of order will produce no difference. Thus if we were to permutate the letters of the name *Ann*, according to our previous rule, we should obtain 3 × 2 × 1, or 6 orders; but half of these arrangements would be identical with the other half, because the interchange of the two *n*’s has no effect. The really different orders will therefore be (3 . 2 . 1)/(1 . 2) or 3, namely *Ann*, *Nan*, *Nna*. In the word *utility* there are two *i*’s and two *t*’s, in respect of both of which pairs the numbers of permutations must be halved. Thus we obtain (7 . 6 . 5 . 4 . 3 . 2 . 1)/(1 . 2 . 1 . 2) or 1260, as the number of permutations. The simple rule evidently is--when some things or letters are undistinguished, proceed in the first place to calculate all the possible permutations as if all were different, and then divide by the numbers of possible permutations of those series of things which are not distinguished, and of which the permutations have therefore been counted in excess. Thus since the word *Utilitarianism* contains fourteen letters, of which four are *i*’s, two *a*’s, and two *t*’s, the number of distinct arrangements will be found by dividing the factorial of 14, by the factorials of 4, 2, and 2, the result being 908,107,200. From the letters of the word *Mississippi* we can get in like manner 11!/(4! × 4! × 2!) or 34,650 permutations, which is not the one-thousandth part of what we should obtain were all the letters different.

*Calculation of Number of Combinations.*

Although in many questions both of art and science we need to calculate the number of permutations on account of their own interest, it far more frequently happens in scientific subjects that they possess but an indirect interest. As I have already pointed out, we almost always deal in the logical and mathematical sciences with *combinations*, and variety of order enters only through the inherent imperfections of our symbols and modes of calculation. Signs must be used in some order, and we must withdraw our attention from this order before the signs correctly represent the relations of things which exist neither before nor after each other. Now, it often happens that we cannot choose all the combinations of things, without first choosing them subject to the accidental variety of order, and we must then divide by the number of possible variations of order, that we may get to the true number of pure combinations.

Suppose that we wish to determine the number of ways in which we can select a group of three letters out of the alphabet, without allowing the same letter to be repeated. At the first choice we can take any one of 26 letters; at the next step there remain 25 letters, any one of which may be joined with that already taken; at the third step there will be 24 choices, so that apparently the whole number of ways of choosing is 26 × 25 × 24. But the fact that one choice succeeded another has caused us to obtain the same combinations of letters in different orders; we should get, for instance, *a*, *p*, *r* at one time, and *p*, *r*, *a* at another, and every three distinct letters will appear six times over, because three things can be arranged in six permutations. To get the number of combinations, then, we must divide the whole number of ways of choosing, by six, the number of permutations of three things, obtaining (26 × 25 × 24)/(1 × 2 × 3) or 2,600.

It is apparent that we need the doctrine of combinations in order that we may in many questions counteract the exaggerating effect of successive selection. If out of a senate of 30 persons we have to choose a committee of 5, we may choose any of 30 first, any of 29 next, and so on, in fact there will be 30 × 29 × 28 × 27 × 26 selections; but as the actual character of the members of the committee will not be affected by the accidental order of their selection, we divide by 1 × 2 × 3 × 4 × 5, and the possible number of different committees will be 142,506. Similarly if we want to calculate the number of ways in which the eight major planets may come into conjunction, it is evident that they may meet either two at a time or three at a time, or four or more at a time, and as nothing is said as to the relative order or place in the conjunction, we require the number of combinations. Now a selection of 2 out of 8 is possible in (8 . 7)/(1 . 2) or 28 ways; of 3 out of 8 in (8 . 7 . 6)/(1 . 2 . 3) or 56 ways; of 4 out of 8 in (8 . 7 . 6 . 5)/(1 . 2 . 3 . 4) or 70 ways; and it may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers of ways are 56, 28, 8, and 1. Thus we have solved the whole question of the variety of conjunctions of eight planets; and adding all the numbers together, we find that 247 is the utmost possible number of modes of meeting.

In general algebraic language, we may say that a group of *m* things may be chosen out of a total number of *n* things, in a number of combinations denoted by the formula

(*n* . (*n*-1)(*n*-2)(*n*-3) .... (*n* - *m* + 1))/(1 . 2 . 3 . 4 .... *m*)

The extreme importance and significance of this formula seems to have been first adequately recognised by Pascal, although its discovery is attributed by him to a friend, M. de Ganières.[99] We shall find it perpetually recurring in questions both of combinations and probability, and throughout the formulæ of mathematical analysis traces of its influence may be noticed.

[99] *Œuvres Complètes de Pascal* (1865), vol. iii. p. 302. Montucla states the name as De Gruières, *Histoire des Mathématiques*, vol. iii. p. 389.

*The Arithmetical Triangle.*

The Arithmetical Triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating. According to Montucla[100] “this triangle is in the theory of combinations and changes of order, almost what the table of Pythagoras is in ordinary arithmetic, that is to say, it places at once under the eyes the numbers required in a multitude of cases of this theory.” As early as 1544 Stifels had noticed the remarkable properties of these numbers and the mode of their evolution. Briggs, the inventor of the common system of logarithms, was so struck with their importance that he called them the Abacus Panchrestus. Pascal, however, was the first who wrote a distinct treatise on these numbers, and gave them the name by which they are still known. But Pascal did not by any means exhaust the subject, and it remained for James Bernoulli to demonstrate fully the importance of the *figurate numbers*, as they are also called. In his treatise *De Arte Conjectandi*, he points out their application in the theory of combinations and probabilities, and remarks of the Arithmetical Triangle, “It not only contains the clue to the mysterious doctrine of combinations, but it is also the ground or foundation of most of the important and abstruse discoveries that have been made in the other branches of the mathematics.”[101]

[100] *Histoire des Mathématiques*, vol. iii. p. 378.

[101] Bernoulli, *De Arte Conjectandi*, translated by Francis Maseres. London, 1795, p. 75.

The numbers of the triangle can be calculated in a very easy manner by successive additions. We commence with unity at the apex; in the next line we place a second unit to the right of this; to obtain the third line of figures we move the previous line one place to the right, and add them to the same figures as they were before removal; we can then repeat the same process *ad infinitum*. The fourth line of figures, for instance, contains 1, 3, 3, 1; moving them one place and adding as directed we obtain:--

Fourth line ... 1 3 3 1 1 3 3 1 -------------- Fifth line .... 1 4 6 4 1 1 4 6 4 1 ---------------- Sixth line .... 1 5 10 10 5 1

Carrying out this simple process through ten more steps we obtain the first seventeen lines of the Arithmetical Triangle as printed on the next page. Theoretically speaking the Triangle must be regarded as infinite in extent, but the numbers increase so rapidly that it soon becomes impracticable to continue the table. The longest table of the numbers which I have found is in Fortia’s “Traité des Progressions” (p. 80), where they are given up to the fortieth line and the ninth column.

THE ARITHMETICAL TRIANGLE.

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.