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. n (n+1` (n+2) §, &c.=nth pre

P+n Q+

n (n + 1 )

1. 2.

R+

R

I. 2.

I. 2. 3.

S,

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) + (de) +, &c. p: and (p-d) + 2 (de) + 3 (e−ƒ) +, &c. or p+d+e+f, &c. = n. In all the nth differences of the th powers of those quantities in the lemma, the

m

r

m

co-efficient of P

× Qp―d Rxdex S e−ƒ +, &c.

will be constant, or the same in all the differences of that order, and

(p-1)

dex 1. 2. 3.

e-fx, &c.'

The above propositions, and their demonstrations, constitute 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 possible, 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 itgenious author.

ART. IV. Mr. Eustace's Classical Tour through Italy.

[Article concluded from the Rev. for March, p. 236.] ON 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 univer sity. 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

is in dimensions, proportions, materials, and workmanship, confessedly the finest of Gothic structures, it naturally leads the author into a discussion on the comparative merits of the Grecian and Gothic styles, as they are exemplified in the two most famous structures of Rome and Milan. Among the statues crowded in and around this edifice, that of St. Bartholomew is the most esteemed. It represents the Saint holding his own skin, which had been drawn off like drapery from his shoulders but the irritated play of the muscles is shewn with an accuracy that rather disgusts and terrifies than pleases the spectator. Here, as in every other country in which theFrench have domineered, appearances of irreligion begin to strike the eye,-neglected churches and plundered hospitals,

"Edesque labentes Deorum, et

Fada nigro simulacra funo ;”

to which may be added the suppression of antient establishments, and the early depravation of youth thence resulting. Even the Ambrosian library has not been exempted from the ruin of the schools, churches, and hospitals: the works of Leonardo da Vinci have been tost into the common mass of French plunder; and the celebrated Last Supper of that consummate painter was used as a target by French soldiers. The heads,' says the author, were their favourite marks, and that of our Saviour in preference to others.' Ramparts, convents, and houses were involved in one common ruin, to make way for a projected forum of Bonaparte, intended doubtless to eclipse that of Trajan itself: but, worse than every other ravage, the simple and manly manners of the Milanese,

"fæcunda virorum

Ingenia, antiqui mores”

were in danger of being lost in one undistinguished mass of violence and fraud.

These injuries are the more to be deplored in Italian cities, because no people are so sensibly alive to their antient glory. Pliny was the benefactor of Comum. Could he return to the earth, with what complacency would he look on the honors paid to him in his beloved Como! "Tuæ meæque delicia," says he to his friend, speaking of this their common town and country; and, indeed, if in his time it was adorned with temples, statues, porticos, pillared gates, and splendid villas, Mr. Eustace is persuaded that, in these respects, the modern does not yield to the antient city. Its lake retains its antient dimensions; it is vast and deep; and the mountains, which border ir, enrich with their fertility the scene which they ennoble with 3

their

their stateliness. The schools, however, the literary establishments, and the religious houses, which diffused civilization and religion over these grand and mountainous tracts, had begun to share in the general ruin of all things that were subject to the continental despot.

At Lugano, the inhabitants were in open but ineffectual rebellion against French oppression. The author found the French engaged in making a road over Mount Sempione, in order to open a communication with Milan, and thus secure the dependance of the Italian republic. Here the face of nature resigns the warm features of Italy. Not far beyond the village of Gondo, the language also alters; and German, more conformable to the ruggedness of the situation, assumes the place of Italian.' On returning for the purpose of visiting Turin, Mr. Eustace institutes a comparison between the Italian and the British lakes; which is the more unreasonable because, by his confession, those of Britain almost sink into insignificance when compared to similar objects in Alpine regions, and to a traveller lately returned from Italy, Windermere appears a long pool, and Skiddaw shrinks into a hillock;' and the heavy lumpish form of Benlomond, its heathy sides and naked brow, with the lifeless masses around it, are very indifferent substitutes for the noble Alpine ridge that borders the Benacus, and presents every mountain-form, from the curve to the pinnacle, from the deep tints of the forest to the dazzling brightness of

snow.'

When to these conspicuous advantages we add the life and interest which such scenes derive from churches, villas, hamlets, and towns, placed as if by the hand of a painter in the most striking situations, so as to contrast with and relieve the horror of the surrounding picture, we describe the peculiar and characteristic features which distinguish the lakes of Italy, and give them an undisputed superiority.

"Adde lacus tantos te Lari maxime, teque

VIRGIL.'

Fluctibus et fremitu assurgens Benace marino." The greatest of the Italian lakes are in Gallia Cisalpina; which, as it had not become, in the time of the most distinguished poets, the seat of the Muses or of glorious achievements, was the country of scenery inanimate and mute.

The Roman name of Turin was Augusta Taurinorum. Its glories have been long secured to it by a succession of good sovereigns; and its vicinity to France has been its only misfortune. Its court was equally remarkable for politeness and regularity; and its academy enjoyed a long and merited reputation, since it answered the two great purposes of educating and introducing young men into elegant society. Here the

D 3

author

author endeavours to correct an error which is common to his countrymen, who are too much addicted to admire the Helvetian and to vilify the Italian character. A certain broad-faced exterior of simplicity has at all times introduced the Swiss into the menial employments of mercenary and porter to the great houses of France and Italy. They were formerly good private soldiers, and possibly, in spite of a certain self-interested leaven, not bad servants; whose good offices were secured by every high and transferred to every higher bidder. Point d'argent,' point de Suisse. In later days, they have patiently done the dirty work of France; have been very Arguses in the French police; and have, in this country, yet maintained their character of a simple people. The Italian is fitted to act in a higher sphere; has usually a character more confirmed; is superior in capacity; and, with a countenance that says more, and a mind better stored, has a heart more simple than his ultramontane neighbour. The present author's opinion of the comparative merits of the two universities of Geneva and Turin may be worthy of reflection:

A year passed in the latter, with the least application, enabled its members to prosecute their travels with advantage, not only by supplying them with the information necessary, but by procuring them such connections with the first families in all the great cities as might preclude the formalities of presentation, and admit them at once into the intimacy of Italian society. Without this confidential admission (which few travellers have enjoyed for many years past,). the domestic intercourse of Italians, and consequently the character of the nation, which is never fully and undisguisedly unfolded unless in such intercourse, must continue a mystery. Now, the academy of Turin, where the young students were considered as part of the court, and admitted to all its balls and amusements, placed this advantage completely within their reach, and was in this respect, and indeed in every other, far superior to Geneva, where the British youth of rank were too often sent to learn French and scepticism from the disciples of Rousseau, and familiarity, insolence, and sickly sentimentality, from the vulgar circles of citizens?'*

Turin, lately the capital of a large and populous territ zy, and long the residence of a race of active and magnanimous princes, furnished with all the establishments, literary and civil, that usually grace the seat of royalty, enlivened by a population of one hundred thousand souls, and frequented by crowds of strangers from most distant countries, was now degraded into the chief town of a French depart

*It should be remembered, however, that Geneva was a Protestant and Turin a Catholic seminary. · Rev.

ment,