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“Rich is that universal self whom thou worshippest as the Soul."-Vedas.

VOL. V.

FEBRUARY, 1888.

No. 2.

Periodicity-A Mathematical and Phasical Proof of the Immortality of Man.

NEVER, FOREVER! FOREVER, NEVER! Among the papers left by Dr. Ivan Slavonski, the Russian mathematician, whose death was announced in the St. Petersburg papers, was one paper in 1870, with the above title. For many years preceding his death, Dr. Slavonski had been Professor of Pure Mathematics in the Imperial University of St. Petersburg, and was recognized throughout Europe as one of the first mathematicians of his day. He discovered the formula for the reduction of "the irreducible case," in which Cardan's rule had failed, and thus rendered it possible to solve, by formulæ, cubic equations containing an imaginary expression - a task which had previously been considered almost as difficult as the solution of the problem, the "squaring of the circle." At his death he left a number of papers in the care of his friend and colleague Prof. Imanoff, expressing at the same time his wish that if anything of value should be found among them, it should be published. The first of these papers which has seen the light, is the one here, published and which was read by Prof. Imanoff before the Society of Physical Sciences at St Petersburg, and printed in the volume of its Transac tions published in April, 1870. It is impossible, and perhaps not desirable, to reproduce here the entire essay, as in some parts of it are calculations, the expression of which would require the use of many

cumbrous mathematical signs and formulæ, that would be unintelligi. ble generally, except to those who had made mathematics a special study. It is however both possible and desirable to give the gist of the thesis and make it comprehensible. It is evident at the outset that Dr. Slavonski intended this essay to be complete, and therefore he started at the natural beginning of his subject. To lose as little time as possible, let us also begin at the beginning.

PERMUTATION.

In how many ways is it possible to arrange two letters, a and B, as they are arranged in a printed line, such as this. A moment's thought will show that there are but two ways, namely: A B and B A; the three letters, A B C, give six permutations, which are as follows:

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In these permutations, observe that all three of the letters enter into each result, and that each enters into it but once. This is permutation according to law, or rather it is the law of permutation, for it is impossible to make more than six different arrangements of the three letters, A B C. Take four letters, A B C D, and arrange them in all possible ways, and you will find that but twenty-four different arrangements are possible as follows:

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The rule for finding the number of possible permutations is as follows: To find the permutation of two letters, multiply 1 by 2; of three letters, multiply I by 2 and the result by 3; of four letters, multiply 1 by 2, the result by 3, and this last result by 4. Therefore, for five letters, A B C D E, there will be 120 permutations; for six letters, 720 permutations; for seven letters, 5,040 permutations; for eight letters, 40,320 permutations; for nine letters, 362,880 permutations, and for ten letters, 3,628,800 permutations, and so on through the alphabet, which twenty-four letters of our alphabet can be permutated

620,448,401,733,239,039,360,000

ways and no more; and so on for any larger number of things we choose to take.

COMBINATION.

In combination, which is the disposition of any number of letters,

or objects, in all possible ways, we find two letters, A B, may be arranged by pairs in four ways, namely, A A, A B, B A, B B; three letters, A B C, may be varied by pairs nine ways, as follows:

A A, Therefore, to show the vast number of combinations of a few things taken by twos, by threes, by fours, and so on, it is only necessary to state that in this way the twenty-four letters of the alphabet would give 1,391,724,238,887,252,999,425,128,493,402,200

A B, A C, BA, BB, BC, CA, C B, C C,

combinations and no more. This result is definitely fixed by the law of their arrangement. This statement and exemplification of these laws, well known to tyros in mathematics, may be wearisome, but it is necessary for the full comprehension of the strange results which follow from it when taken in connection with other elements of the problem. Let us now proceed to something less dry and formidable.

MASS MOTION AND MOLECULAR MOTION.

In the present state of science it seems almost superfluous to speak of the changes which are constantly taking place in every part of the universe with which we are acquainted. Mass motion is shown on the largest scale in the whirling of planets and suns, and systems of suns. The geological eras descending by a series of colossal steps down into the darkness of the past eternity, show cycles upon cycles of changes in the earth we inhabit; nor does science rest here, for back in the night of time, the solar system was a nebulous ball whirling in space and throwing from it rings which afterwards broke up and produced the planets. The solar system is known to be in motion about some larger sun, to which it is but a planet, and its satellites are in turn but satellites. Molecular motion cannot be seen, but is known to be present among the molecules of all bodies. No scientist doubts that mass motion and molecular motion exist in accordance with law, and in no other way.

ATOMS. DO THEY EXIST?

Chemical bodies which, upon being analyzed are found to be composed of different element, are compounded in definite proportions. Thus water consists of eight parts by weight of oxygen and one of hydrogen, and this proportion never varies. The elements are not only bound together in a quantitative order, in the stable condition of substances, but when a compound is broken up and its elements released from their affinities, they cannot escape the law of numerical destiny; they rush into new unions, but still in definite proportions. This is the fact, but the mind is not satisfied with facts. It seeks for the causes that determine combination in the fixed ways, and prevent it from taking place in others. The atomic theory assigns a cause for

this. An atom is a portion of matter exceedingly small and capable of being divided. This definition is good enough for present purposes, but another will be given presently to answer any metaphysical objection which may be raised against the existence of atoms. The old Greek philosophers loved to speculate about the infinite divisibility of matter, and held that we cannot conceive of particles so small that, if possessing weight, bulk, and figure, they cannot be still divided; but after all is said, the question is not about a possibility of conception, but of the facts of nature. The chemist assumes that in nature the divisibility of matter has a limit. As Liebig says:

"Without disputing the infinite divisibility of matter the chemist merely maintains the firm and immutable foundations of his science when he admits the existence of physical atoms as a truth entirely incontrovertible."

If then, for instance, water be formed by the combination of atoms of oxygen with atoms of hydrogen the atomic theory affords an explation of why its composition is definite and constant. It is not necessary to ask him who says that it is impossible to know whether atoms exist or matter is infinitely divisible-who says the fact is “unknowable "—it is only necessary to ask such a man whether or not there is the smallest existent portion of matter? To say that there is not, is to say that there is a portion of matter not smaller than itself, which is a self-evident absurdity.

ARE ATOMS INFINITE IN MATTER?

This question has also been relegated needlessly to the realms of the unknowable as Dr. Slavonski has shown, and shown by so simple a proposition as the following:

"We have shown by a propostiion, the opposite of which is unthinkable, that there are atoms-that is, there are the smallest existent particles of matter. By the same manner of reasoning it may be shown that the number of these atoms is limited. Thus there are as many of these smallest existent portions of matter as there are, for the simple reason that to state that there are more or less than this number of them, is to state an unthinkable proposition and to be guilty of stating a great absurdity."

The number of atoms that exist must be limited. To speak of an "unlimited number" is to be unintelligible.

THE RESULT.

Thus far four elements of the problem by Dr. Slavonski have been given. They are:

Firstly-That bodies which undergo change of place according to law, can be permutated and combined in a fixed number of ways.

Secondly-That everything in the universe is undergoing change according to law.

Thirdly-That atoms exist, and are not a metaphysical abstraction or a physical makeshift.

Fourthly-That the number of these atoms is limited.

It now remains to show the results of these four propositions when brought to bear upon each other. If there were but three atoms in the universe, and the law of their existence were that they could be changed in respect to each other only by taking position in contact, the first with the second, the second with the third, and so on in one line,

А В С, A CB, BAC, BCA, CAB, CBA,

it is evident that only six permutations would be possible. If taken in pairs there would be nine combinations,

A A, A B, A C, BA, B B, BC, CA, C B, с с.

The truth is, however, in regard to the actual sum of the atoms in the universe, that they can combine in so great a number of ways that men generally speak and think of their combinations as infinite; the word "infinite" in this case as in all others, being the means of covering the ignorance and the impossibility of the conception of vast numbers of units; in space units of miles, and in time units of years. If it were possible to cover the surface of the earth with numbers, the numbers thus written would not form the billionth part of the atoms of the material used to write them. Yet, in the universe, the number of atoms is limited by, and only by, the atoms themselves. These atoms are continually undergoing change of place-for atomic changes can only be changes of place and this change is according to law. There must then come a time when all possible changes and combinations shall have been exhausted. It is possible that when that time comes the universe will be at rest, as Spencer, Helmholtz, and Thompson also maintain, looking at the subject in a different light. Dr. Slavonski thought this was not possible, and for the following reason: the first place, heat and motion, or, if this expression be objectionable, heat or motion, are as much properties of matter as impenetrability is. No matter is without them, and properties of matter are nothing but

THE UNIVERSE ALWAYS IN MOTION.

In

what we ever find connected with it. If we cannot conceive of matter which is not impenetrable, neither can we think, nor have we any experimental scientific reason to suppose, there is or can be matter which does not move. Consequently, reasoning in the positive and scientific method from the known properties of matter, we arrive at the conclusion that there can be no rest for the atoms which are the component parts of the universe. Besides this, since time has no ob

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