Hence, by an obvious and infallible deduction, the first term of the nth differences will be from which proposition, the author deduces the following corollaries, viz. 1. If, instead of the preceding series, we take the series x+o, x+p, x+q, &c., and arrange the several differences according to the several powers of x, the first term of the nth differences will then become m = d (7-1)d (= −2)d (—3) d, &c. (1) and which will be the constant first term of the several nth differences. n (n−1) (n—2), &c. 3. 2. 1. d", or = 1, 2, 3, &c. n.d"; which latter expression not only represents the first term of the nth differences of th powers, but exhibits the entire differences, which are therefore constant, and consequently the (n+1)th differences are all equal to Zero.-We do not attempt to give here the demonstration, or rather the explanation, of these deductions; because it would occupy too much room, and because we conceive that it is sufficiently obvious. Perhaps the author himself has been too prolix in this respect. The demonstration of the binomial theorem is drawn immediately from the two preceding corollaries, by substituting the series of natural numbers 1, 2, 3, &c. instead of q, in the general expansion, tÇ × wards the least. +, &c. viz. proceeding in an inverted order, to t being taken to represent the co-efficient of the n+1th term in m the expansion of (+q), and the form of which is required. Now, in the above series of arithmeticals, the common difference d= 1; therefore, the first term of the nth differences is equal to m which is obviously the same as the nth differences of the latter terms, or of the n+1th terms of the expansion, and which from cor. 2 is equal to 1. 2. 3. 4. ... n. tær ; whence, equating these two, we have for the value and form of t (that is, of the co-efficient of the +1th term of the expansion,) t = n (7—1) (——2) - - (———), the form required. We are aware that this demonstration may be too much contracted to appear quite perspicuous to some of our readers: but it must be remembered that we profess only to explain the principles of it; and not to give the demonstration itself, which appears to us to be perfectly satisfactory. The author concludes this essay by a demonstration of the Multinomial Theorem of De Moivre: but we cannot enter on any explanation of it in this place. Essay II. On finding per Saltum the several Orders of Differences. The formula for finding per saltum the several orders of differences in series, the author observes, have been usually deduced from a repeated arithmetical subduction. Thus, if the successive quantities are a, b, c, d, &c. then 1st diffs. a-b, bc, cd, &c. 2d diffs. a-2b+c, b—2c+d, &c. and universally, the nth differences, having their several co-efficients the same as those of a binomial raised to the th powers, will be represented by the formula a-nb + n (n−1) (n−2) d +, &c. 1. 2. 3. n(n-1) c I. 2. The object of the present essay is to generalize these formulæ ; and, as Mr. Burke states, to enlarge the analogy which has been observed in some cases to prevail between differences and fluxions. We cannot attempt to enter into the particular process of the author: but, as some of his propositions contain new properties, and others considerably extend old properties, it may not be amiss to point them out to the reader's notice; though for their demonstrations we must refer to the Essay itself. 1. The "th differences of the nth powers of quantities in arithmetical progression are constant, and equal to 1. 2. 3. 4. ... n. d", d being the common difference. This important property, we have seen, the author deduces as a corollary in the preceding essay, and it is only repeated here to preserve an uniformity in the present chapter. The invention of this formula is due to Briggs, he having been led to it in the construction of his logarithms: but its demonstration has been generally made to depend on the fluxional or differential calculus; whereas Mr. Burke, by making its demonstration rest on first principles, inverts the order of proceeding, and therefore renders the latter dependent on the former. 2. If there are a number of arithmetical series, whose common differences are respectively a, b, c, d, &c.; let all the different corresponding terms of the several series be multiplied together, the nth differences of the products will be constant, and equal to 1. 2. 3. . . . nx abcd &c.' 3. The series 1, +1 (n + 1 ) ( n + 2 ) ( n + 1 ) (n+2) (n+3) I. 2. I. 2. &c. will have the nth differences of its terms common, and equal to 3. unit, which is the definition of triangular (figurate) numbers of the ath order.' 4. If the qth powers of the natural numbers, and the rth powers of the corresponding triangular numbers, which admit two orders of differences; and the corresponding sth powers of the triangular numbers having three orders, &c. be all multiplied as follows: these products will have as many orders of differences as there are units in q+2r+3s+, &c. and the last of these will be constant, and REV. MAY, 1814. D =2X 2 • Lemma. The quantities in any series can be expressed in a multinomial form, in terms of triangular numbers, and of the first of the several orders of differences of the quantities: viz. let P be the first quantity taken in a series, and Q, R, S, &c. be the first of the several differences, whose orders are one, two, three, &c. the following general formula will express the preceding and subsequent quantities, in the series; viz. P+ n Q+ n (n + 1) 1. 2. ceding quantity, and P―nQ + ing quantity.' R+ n ( n + 1 ) ( n + 2) Snth succeed 1. 2. 3. From this lemma, and the preceding propositions, Mr. B. draws his 5th and principal proposition; viz. 5. If P is one of the quantities in a series admitting several orders of differences, the first of the differences in the first, second, third, &c. orders of differences of those quantities being Q, R, S, &c. putting (p-d) + (d—e) +, &c. =p: and (p-d) + 2 (d—c) + 3 (e-ƒ) +, &c. orp +d+e+f, &c. = n. In all the nth dif ferences of the th powers of those quantities in the lemma, the m will be constant, or the same in all the differences of that order, and The above propositions, and their demonstrations, consti tute the first chapter of the present essay; and the second chapter contains their application to the method of finding per saltum all the constant parts of the differences of any order whatever, and the analogy between them and the co-efficients of the several orders of fluxions. Essay iii. On finding Divisors of Equations. This Essay is wholly occupied in investigating a method of finding the division of any proposed equation, in order to reduce it, if pos sible, to lower dimensions: but, as we have already extended this article to a greater length than we at first intended, we must avoid entering into any detail on the subject of the two remaining remaining chapters; and this we are the more inclined to do, because they relate to a problem of little practical utility. When an equation presents itself for solution, it is very doubtful whether, if we knew that it could be resolved into factors, it would answer our purpose to go through the whole operation necessary for that resolution: but, when we consider that, in all probability, in nineteen cases out of twenty, no such resolution can be accomplished, it is obvious that little practical advantage can be expected from any rule of this description. Before we conclude our account of these Essays, we ought to observe that they contain many new and original ideas, and certainly display a considerable share of mathematical genius: but we by no means admire the form and enumeration of several of the propositions; and the same remark applies to several of the demonstrations, these being in many cases too prolix, and in others rather unintelligible: yet, on the whole, it cannot be denied that they are highly creditable to the talents of their ingenious author. ΟΝ ART. IV. Mr. Eustace's Classical Tour through Italy. [Article concluded from the Rev. for March, p. 236.] N passing the river Po, its banks, on which the poplar is yet predominant, recall to mind the fable of Phaeton, so prettily told by Ovid. The approach of Pavia, which is situated on the Tesino, to which Claudian attached the epithet of beautiful, fills the mind of Mr. Eustace, as usual, with sweet and with bitter recollections. The antient name of this city was Tisinum, from the river Tisinus which bathes its walls: but, between the sixth and the eighth century, the antient name disappeared, and, under the appellation of Papia, softened by Italian euphony into Pavia, the town became a considerable city, and may perhaps be regarded as the first mother university. As the Tisinus was originally ennobled by the first battle between Hannibal and Scipio, so has Pavia witnessed the battle between the Emperor Charles the Fifth and his rival Francis, and the yet more bloody entrance of Bonaparte; who, enraged at the resistance made by its citizens, ordered all its magistrates to be shot. The university had been long on the decline, owing to the number of similar institutions in northern countries; and, before the recent events, which will restore life and health to all Italy, the city was on the eve of becoming desolate. Milan, Milano, or Mediolanum, is a great and splendid city, containing about 150,000 inhabitants. As the cathedral |