gan to be carefully investigated. No universal criterion for determining whether a given series is convergent or divergent was then known; nor do we possess such a one even to-day. A question naturally arising at this point of our inquiry is, whether the views of Cauchy and Abel and their co-workers met at once with general acceptance or not. As might almost be expected, they did not, but encountered firm opposition. The old combinatorial school in Germany would not surrender their orthodox views without a struggle. They obstinately defended every doctrine of their mathematical creed. Even such a man as Dr. Martin Ohm, who was really an enemy of the combinatorial school, and whose achievements in mathematics and physics place him among the coryphæi of science, was not willing to join Cauchy and Abel in calling divergent series fallacious. In an essay written by Ohm, entitled, The Spirit of Mathematical Analysis,* he admits that the great mathematicians of his day, as Gauss, Dirichlet, Jacobi, Bessel, Cauchy, do not employ demonstrations conducted with divergent series, while Poisson speaks decidedly against them. "But," says Ohm," that the series which are used and from which deductions are drawn ought to be always and necessarily convergent is a circumstance of which the author of this essay has not been able at all to convince himself; on the contrary, it is his opinion that series, as long as they are general, so that we can not speak of their convergency or divergency, must always, when properly treated, necessarily and unconditionally produce correct results." By a general series Ohm means one in which the letters represent neither magnitudes nor numbers, but are considered as perfectly insignificant (inhaltlos). Whenever the letters are made to represent magnitudes or numbers, then the series is no longer a general series, but is a "numeric" series, and in that case Ohm admits that an equality can exist between the function and its series only when the series is convergent. It is very difficult to see exactly what meaning shall be given to letters upon which algebraic operations are to be performed, when the letters represent neither magnitudes nor numbers. Nor is it easy to see in what way formulæ involving these empty, meaningless letters-these "ghosts of departed quantities "-can furnish rigorous methods in mathematical analysis. In fact, this theory of general series containing insignificant letters is one of the last shifts to which the opponents of the new school resorted; one of the last subterfuges before giving up a contest which had become entirely hopeless. If we pass from Germany to England we meet there with another mathematician who championed the old cause. I refer to George Peacock, who is well known to mathematicians for his Algebra and his Report, made in 1833 to the British Association, On the Recent Progress and Present State of Certain Branches of Analysis. Peacock states his views with more clearness than Ohm had stated his. He bases his argument on what he calls the "principle of the per *The Spirit of Mathematical Analysis and its Relation to a Logical System, by Dr. Martin Ohm; translated by Alexander John Ellis, London, 1843. manence of equivalent forms," which he considers to be the real foundation of all rules of symbolic algebra. According to this principle, all the rules and operations of arithmetic which have been established by numerical considerations are adopted without reference to relative magnitude; the symbols of algebra are taken to be perfectly general and unlimited in value, and the operations to which they are subject are equally general. To illustrate: In arithmetic we can subtract a smaller number from a larger, but we cannot subtract a larger from a smaller; that is to say, we can subtract 3 from 5, but not 5 from 3. In algebra, on the other hand, no limitation whatever is placed upon the relative values of minuend and subtrahend; there we can subtract 5 from 3 and give the answer a rational interpretation. By the principle of the permanence of equivalent forms every result obtained from mathematical operations must always be a correct result, no matter what the relative values of the quantities be upon which the operations are performed. Peacock applies this principle to the subject of infinite series. He says (p. 205, Report for 1833) that "the series (1+r)*=1*( 1+2x+ n (n 1) 202 + etc.) indefinitely continued, in which n is a particular value (a whole number), though general in form, must be true also, in virtue of the principle of the permanence of equivalent forms, when n is general in value as well as in form." Instead of being always a positive whole number, the exponent n may, therefore, be negative or fractional, and the above formula still holds true. Now, the principle of the permanence of equivalent forms laid down by Peacock is not self-evident, nor did it become known by intuition; on the contrary, it is merely an induction, and can, therefore, hardly be taken as a reliable basis upon which to settle a disputed question; for this very question may be one in which this law established by mere induction might fail. But even granting the principle of the permanence of equivalent forms to be generally applicable, does it really follow from it that infinite series are true, whether they be convergent or divergent? In order to discuss this point let us examine a series resulting from the division of the numerator of an algebraical fraction by 1 its denominator, such as 1-a From arithmetic we get the simple but general statement that the numerator of a fraction divided by its denominator is equal to the quotient plus the remainder (if there be any remainder). By the principle of the permanence of equivalent forms this must be true of fractions involving any quantities whatever. Now, if we divide 1 by 1 a we get 1+ a + a2 + a3 + If we carry the division further there is still a remainder. No matter how far the division proceeds it will not end, and a remainder will still exist. We may express this fact by writing 1 =1+a+ 1 - a an+1 1-a Now, if a has a value less than unity the 1 1 =1+ α a remainder approaches zero, and we may therefore write a+a2 + etc., ad infinitum. This infinite series is correct whenever a <1. But, according to Peacock, it would follow from the principle of the permanence of equivalent forms that, if this series is correct for a <1, it must be true for all values of a. Hence the series is true when a> 1, in which case the series is divergent. Now, this conclusion appears to be inadmissible, because Peacock does not examine the remainder. When a <1, the remainder approaches zero, and can therefore be neglected; but if a > 1, then we shall find that the remainder does not approach zero, and therefore cannot be neglected. To neglect it would be to violate the principle of the permanence of equivalent forms. This principle demands that whenever there is a remainder it shall always be considered and expressed, no matter how far the division be continued. If in the above series we take a = 2 and neglect the remainder, then we get which is an absurdity. But if the remainder be taken into account, then we have This equation is always true, no matter how great n may be; that is to say, no matter how far the division be continued. From similar considerations in other series it would appear that divergent series are false and absurd, except when written with the remainder. And yet not only Peacock, but even De Morgan was not willing to reject divergent series. Though De Morgan criticised the new school for the unconditional rejection of divergent series, he cannot be pronounced an enthusiastic supporter of the old school. In an article in the Transactions of the Cambridge Philosophical Society, Volume VIII, Part I, he says: "I do not pretend to have that confidence in series which, to judge from elementary writers on algebra, is common among mathematicians, not even convergent series." His views on this subject will be more fully elucidated by the following quotation from his article on "Series" in the Penny Cyclopædia: "A divergent series is, arithmetically speaking, infinite; that is, the quantity acquired by summing its terms may be made greater than any quantity agreed on at the beginning of a process. Nevertheless, as every algebraist knows, such series are frequently used as the representatives of 881-No. 3-24 finite quantities. It was usual to admit such series without hesitation; but of late years many of the continental mathematicians have declared against divergent series altogether, and have asserted instances in which the use of them leads to false results. Those of a contrary opinion have replied to the instances, and have argued from general principles in favor of retaining divergent series. Our own opinion is that the instances have arisen from a misunderstanding or misuse of the series employed, though sufficient to show that divergent series should be very carefully handled; but that, on the other hand, no perfectly general and indisputable right to the use of these series has been established a priori. They always lead to true results when properly used, but no demonstration has been given that they must always do 80." About the time when Peacock made his report to the British Association, Cauchy was developing new and valuable results on the subject of infinite series. With the aid of the integral calculus he was conducting a careful investigation of the conditions which must be fulfilled in order that a function be capable of being developed into a convergent infinite series. He found that four conditions must be satisfied: (1) The function must admit of a derivative. (2) The function must be uniform, that is, for any particular value for the function must have only one value. (3) The function must be finite. (4) The function must be continuous, that is, it must change gradually as the variable passes from one value to another. These results greatly strengthened the position held by the new school, and notwithstanding the adroit arguments brought forth by various mathematicians of the old school in favor of divergent series, the leading mathematicians of to-day have rejected the old views and adopted those of Cauchy and Abel. In the theory of functions, a branch of mathematics which is now assuming enormous proportions, the convergency of all series employed is carefully and scrupulously tested. In late years more reliable criteria have been invented for determining the convergency. Standard treatises on the subject devote the larger part of their space to the consideration of convergency. Whenever a series is divergent, then either the remainder is inserted or the series is unceremoniously rejected. Indeed, divergent series are now looked upon by our best mathematicians as being nothing more than exploded chimeras. Having briefly traced the history of infinite series in Europe, we shall consider the views on this subject held by American writers. Previous to the beginning of this century the text-books on algebra used in this country were all imported from abroad. About the only mathematical books published in America before 1800 were arithmetics and some few books on surveying. The earliest imported algebras came from Great Britain. The most important of them were the algebras of Maclaurin, Saunderson, Charles Hutton, John Bonnycastle, and Thomas Simpson. These writers belonged to what we have called the old school. As might be expected the subject of series was handled by them with the same looseness and recklessness as by the older school of mathematicians on the continent. Thus, in Hutton's Mathematics, which was a standard work in its day, considerable attention was paid to series, but the terms "convergent" and "divergent" were not even mentioned. The earliest American compiler of a course of mathematics for colleges was Samuel Webber. In 1801 he published his "Mathematics." The algebraical part was necessarily elementary in character, and of course contained no formal criteria for convergency. Whatever defects Webber's Algebra may have, it has also its merits. It is pleasing to observe. that as far as the author had entered upon the subject of infinite series he was on the right track. Speaking of a certain divergent series he says that "it is false, and the further it is continued the further it will diverge from the truth" (p. 291). This language possesses the true ring; it is free from the discords of error, and we regret that American writers of later date have not imitated it. In 1814, thirteen years after the publication of Webber's Mathematics, appeared the algebra of Jeremiah Day, of Yale College. All things considered, Day's Algebra is superior to Webber's, but on the particular subject of series it can hardly be said to excel. President Day points out, to be sure, that a certain series must converge in order to come nearer and nearer to the exact value of the fraction from which the series was derived, but he does not even hint at the insecurity or ab. surdity of divergent series. He gives no demonstration of the binomial theorem, but speaks of it as being universally true. 1 1-a Four years after the publication of Day's Algebra, John Farrar, professor of mathematics at Harvard, published An Introduction to the Elements of Algebra, Selected from the Algebra of Euler. On the continent of Europe Euler's writings were at that time justly considered as the most profound, and as affording the finest models of analysis. Yet his writings were not faultless. His views on series were those of the old school. The discussion of series as given in Farrar's Euler demands our attention, because subsequent American writers were doubtless greatly influenced by it. On page 76 of this book, the fraction is resolved by division into an infinite series. The following comments upon it are then made: "There are sufficient grounds to maintain that the value of this infinite series is the same as that of the fraction What we have said may at first seem surprising, but the 1. α consideration of some particular cases will make it easily understood. # If we suppose a=2, our series becomes =1+2+4+8+16+32+ 64, etc., to infinity, and its value must be which at first sight will appear absurd. if we wish to stop at any term of the 1 1 that is to say, 1 -1 =- - 1, But it must be remarked that above series we can not do so |