This noble structure was opened yesterday for the public accommodation, with as much splendour and dignity as it is possible to give to a ceremony of this description. The bridge, as our readers know, was originally named "The Strand Bridge;" but the natural and patriotic desire of commemorating, in the most noble public manner, the ever-memorable victory of Waterloo, afforded a fine opportunity for changing its appellation from that of the street merely into which it opens. No mode of perpetuating great deeds by works of art is more consistent with good taste than where such works combine, in a high degree, what is ornamental with what is useful. Monuments of this kind have stronger claims on public respect than the costly construction of pillars, obelisks, and towers. There are many instances of public works having received their names from events honourable to the country in which they were erected. In late times, Buonaparte, who, with all his vices, had a very shrewd insight into human nature, and the external means by which it is worked upon, took advantage of this principle, not simply by his triumphal columns or arches to the honour of Dessaix, of himself, and of his army; but also in giving to two new bridges the names of Jena and Austerlitz, where he had gained two decisive victories. But those bridges, however elegant and convenient, are but trifles in civil architecture and engineering when compared with that which was opened yesterday; and the general appearance of which in its progress attracted particularly the admiration of the Emperor of Russia on his visit to this country.

The intended ceremony excited the public curiosity to a great extent yesterday forenoon, which was much increased by the remarkable fitness of the weather, alike favourable for standing on the bridge, viewing the procession from the banks, or making excursions on the river. The banks of the Thames, from Blackfriars-bridge to Whitehall, were immensely crowded by noon with all descriptions of persons of both sexes, from the curious of the lowest order up to the elegance of the highest fashion. The Temple-gardens, the terraces and roof of Somerset-place, the Adelphi-terrace, York-buildings-terrace, and the gardens of Fife-house, and others in Scotland-yard, were particularly filled. On the south side of the river the crowd fully corresponded in numbers. Seats in stages were erected and let out in all the yards belonging to the various wharfs. The Thames itself seemed covered with barges and boats of all kinds. Some, in the line where the coal and corn barges generally lie, were fitted up in a temporary way to accommodate the numerous spectators. The little pleasure-vessels of the amateurs of aquatic pleasures enlivened the scene by the neatness and facility of their movements. Colours were hoisted on the steeples of several churches, and on the yards of wharfingers, and on many private boats. The Navy standard waved on the centre of Somerset place. A party of horse-guards, who had, we understood, been present at the battle of Waterloo, and many of whom bore on their brave breasts the trophies of their valour, went upon the bridge about ten o'clock in the morning. A party of foot-guards also attended with their band; and a detachment of the Royal horse artillery, with 20 field pieces. The bridge was decorated with 18 standards elevated. In the centre, and at each end, were to Royal Standards of Great Britain; there were between these, standards of Russia and the Netherlands, and the Orange flag, thus representing the nations the success of whose combined armies occasioned the appellation of Waterloo-bridge. The eastern side of the bridge was railed off, and temporary benches were placed to accommodate the spectators. Divisions of foot guards were stationed near Whitehall, and a Captain's guard was drawn up in the area before Fife-house.

Some time after three the Prince Regent arrived at the Whitehall stairs in his private carriage, whence he embarked on board the Royal barge, bearing the Royal standard. This barge was followed by the Lord Mayor's barge, which attended with his Lordship, and a full company, to conduct the Prince Regent to the bridge. Other barges belonging to the Admiralty, the Navy, and other public offices succeeded, bearing union flags, or the appropriate flags of the respective departments. The discharges of the artillery commenced on the Regent's embarking, and continued till he landed at the bottom of the flight of steps on the south-east of the bridge, which he ascended. His Royal Highness was received in the most respectful manner by the Committee, and then walked along on the western side of the bridge, between the Duke of York and the Duke of Wellington, followed by a number of military officers, officers of State, and persons of distinction, and attended by a military guard of honour. Arrived at the north end of the bridge, he descended by the north-west stairs to the Royal barge. The firing then re-commenced, and did not terminate till his Royal Highness had landed at Whitehall watergate, and returned to Carlton-house. The City barge continued on the river for some time after; and the other boats remained a considerable time rowing or sailing backwards and forwards. We scarcely recollect an occasion of public gaiety on which a greater number of persons of all descriptions appeared in the streets, and particularly on the Surrey side of the river. Here all the roads leading towards the bridge were literally crowded. Among other contrivances to increase, and to profit by the general popular festivity, a fair was opened a little to the south-east of the bridge, and called, of course, Waterloo Fair, in which, in the true style of the holydays at Greenwich, was a sufficient assortment of different swings, and other vehicles for popular amusement, together with a plentiful supply of cakes, gingerbread, beer, and gin. Seats were let in the upper floors, and on the roofs of houses at too great a distance to admit any satisfactory view of the proceedings, at prices up to five shillings per head. All seemed well disposed and to rejoice and make merry, as might naturally be expected, and there were not wanting many instances of the usual effects of vulgar hilarity. Decorations of laurel were worn by the soldiers, with "Waterloo, 18th June, 1815," in gold letters. A bough of laurel, with a similar inscription, was displayed at the suttling-house in the Tilt-yard. A new banner was hoisted on the spire of St. Martin's church. The firemen of different insurance companies appeared in their full dress. A large cannon, taken at the battle of Waterloo, was placed on the parade in St. James's-park on Tuesday.

Having noticed the particulars connected with yesterday's ceremony, it is an agreeable task to say something of the bridge itself, which we consider to be a very high testimonial of the great abilities of Mr. Rennie, the architect, and of the liberality of the great body of shareholders who have provided the funds for its erection. We believe that there is no bridge in any of the European capitals (and certainly none elsewhere) which is equal, as a great work, to either of the bridges at Westminster or Blackfriars. They are superior works to any of the kind at Paris, Petersburg, Madrid, Vienna, or Dresden, or in the Italian States. But then, the Waterloo-bridge is superior to the others which bestride the Thames, and is consequently, the finest in the world. We do not think that any of the great Roman bridges, such as that over the Danube, can be considered with any comparative disadvantage to it. It was unquestionably wanted, as a work of public utility, though we fear it can never become a source of profit to the shareholders. Perhaps we need not mention that it is built of granite, and within the walls is filled up with an inferior stone (Merstham stone we understand). The departure from the old custom of curved bridges, and adopting the straight line, has removed any doubts which existed among lovers of the arts, as to the relative beauty of the different forms. A view of this new bridge, however, shows at once that the form is not only more classical, but more simple and more striking in its effect. The choice of a straight line was, probably, influenced by the height of the pavement in the Strand, and a desire to avoid a disagreeable ascent and descent in the short space between that street and the river. Some have objected to the balustrades being used instead of what is called a plain blocking course, a parapet; but this is an inferior matter of taste. If the balustrades do not add to the appearance of solidity, they confer a considerable share of lightness and elegance, and are also more agreeable to the eye of the passenger. The coupled Doric pillars on the piers have likewise been the subject of criticisms, but we have not seen the inappropriateness of their position proved. If they were not introduced, some sort of buttress must have been constructed, or no projections at all. We know no sufficient reason why a bridge should, unlike all other edifices, be stripped of architectural ornament. The lowness of the ground on the Surrey-side, has rendered necessary a long series of arches to support the southern avenue, and to make the descent towards St. George's-fields as easy as possible. This convenience has, at a very great expense, been attained. If there be any parts of this great work to which objections can be made, we should think they might apply to the curvature and narrowness of the stairs leading to the landing places, which we fear may expose them to nuisances. Possibly this may be ascribable to the want of a sufficient expansion of opening at the ends. The four toll-lodges are next appropriate Doric structures. We observed a clever contrivance at each lodge for the purpose of checking and preventing their dishonesty to the trust. The kind of iron turn-stiles, which admit of only one person passing at a time, touch some machinery which communicates with a clock locked up in an oak box in each toll-house, the index of which is thereby moved, so that on looking at it the number of those who have passed is directly seen. Some machinery for a similar object is to be applied to the horse and carriage gates. We understand that, when the lamps are completed, the gas lights will be introduced. The situation of this bridge is remarkably fine. It gives the grandest view we have of the river in its beautiful meander, displays the rising crescent of buildings on the north side, and brings out Somerset-terrace in the most favourable way; while on the south it opens the view of the Surrey hills. We congratulate the metropolis on the completion of this additional ornament and convenience to the first city in the world. We incline to think that something more will yet be requisite to complete the northern avenue. The street should cross the Strand to the bottom of Charles-street; and from the north of Bow-street, a desirable opening might be made, so as to make a clear communication with Tottenham-court-road, Marylebone, Bloomsbury, &c.

There are 320 piles driven into the bed of the river under each pier; the length of each pile from 19 to 22 feet, and the diameter about 13 inches. There is one pile to every yard square.

The scientific manner on which the centres were constructed was admirable; and as all the arches are of the same size, the centres were removed from those that were finished, and placed on the piers where the arches were not yet thrown; this was an operation that required great skill and care, and was very ably executed.

When the centres were removed, so solidly and well was the masonry constructed, that in the middle they only sunk about one inch. Those of the Pont de Neuilly in France, six miles from Paris, which are nearly similar, sunk about 18 inches in the middle after the centres were taken away.

The scientific principles on which the centres were constructed, which does great credit to Mr. Rennie the engineer, was that of the longitudinal incompressibility of timber. The strongest and largest beams of wood bend and yield when pressed upon laterally, and by that means the form of a centre constructed in the usual manner, is different when loaded from what it is when not loaded; but as no weight that men are acquainted with, when acting gradually, will shorten the length of the beam, it was so contrived that the pressure acted always longitudinally or lengthways, and not laterally or sideways, so that those centres remained in form unchangeable, as much as if they had been one solid mass of matter, the two extreme points resting on the firm and well constructed piers.

In circular arches, such as those of Westminster or Blackfriars bridges, the pressure on the centres before the key stones are put in place, is not near so great as in eliptical arches like those of Waterloo.

The bridge was only six years in building; it is exactly on a level with the Strand where it joins, and is fifty feet above the surface of the water of the river Thames.

The Prince Regent was attended by Admiral Sir E. Nagle and Lord Forbes, as his aides-de-camp. On this occasion the guards wore their new pantaloons, of a dark mixture, striped on the outsides with scarlet. Lord and Lady Liverpool received the Regent at the door of Fife-house. In a similar manner they received the Duke and Duchess of York. The Duke was accompanied by Sir H. Calvert, Sir W. Gordon, Sir H. Torrens, &c. The Dukes of Cumberland and Glocester arrived afterwards with their suites. Most of the Foreign Ambassadors and Cabinet Ministers, the Lord Chamberlain, the Lord Steward, &c. attended. Most of the company wore the Windsor uniform. The Duke of Glocester's band played. On the return to Fife-house there was an elegant collation provided. After the Regent had changed his dress at Carlton-house, he went in his travelling carriage, with Sir B. Bloomfield, to Windsor, to attend the grand ball to be given by the officers of the Royal Horse-guards.