Is a current of fresh water, flowing in a bed or channel from its source to the sea.

The term is appropriated to a considerable collection of waters, formed by the conflux of two or more brooks, which deliver into its channel the united streams of several rivulets, which have collected the supplies of many rills trickling down from numberless springs, and the torrents which carry off from the sloping grounds the surplus of every shower.

Rivers form one of the chief features of the surface of this globe, serving as voiders of all that is immediately redundant in our rains and springs, and also as boundaries and barriers, and even as highways, and in many countries as plentiful storehouses. They also fertilise our soil by laying upon our warm fields the richest mould, brought from the high mountains, where it would have remained useless for want of genial heat.

Being such interesting objects of attention, every branch acquires a proper name, and the whole acquires a sort of personal identity, of which it is frequently difficult to find the principle; for the name of the great body of waters which discharges itself into the sea is traced backwards to one of the sources, while all the contributing streams are lost, although their waters form the chief part of the collection. And sometimes the feeder in which the name is preserved is smaller than others which are united to the current, and which like a rich but ignoble alliance lose their name in that of the more illustrious family. Some rivers indeed are respectable even at their birth, coming at once in force from some great lake. Such is the Rio de la Plata, the river St Laurence, and the mighty streams which issue in all directions from the Baikal lake. But, like the sons of Adam, they are all of equal descent, and should take their name from one of the feeders of these lakes. This is indeed the case with a few, such as the Rhone, the Rhine, the Nile. These, after having mixed their waters with those of the lake, resume their appearance and their name at its outset.

But in general their origin and progress, and even the features of their character, bear some resemblance to the life of man. The river springs from the earth; but its origin is in heaven. Its beginnings are insignificant, and its infancy is frivolous; it plays among the flowers of a meadow; it waters a garden, or turns a little mill. Gathering strength in its youth, it becomes wild and impetuous. Impatient of the restraints which it still meets with in the hollows among the mountains, it is restless and fretful; quick in its turnings, and unsteady in its course. Now it is a roaring cataract, tearing up and overturning whatever opposes its progress, and it shoots headlong down from a rock; then it becomes a full and gloomy pool, buried in the bottom of a glen. Recovering breath by repose, it again dashes along, till tired of the uproar and mischief, it quits all that has swept along, and leaves the opening of the valley strewn with the rejected waste. Now, quitting its retirement, it comes abroad into the world, journeying with more prudence and discretion through cultivated fields, yielding to circumstances, and winding round what would trouble it to overwhelm or remove. It passes through the populous cities and all the busy haunts of man, tendering its services on every side, and becomes the support and ornament of the country. Now increased by numerous alliances, and advanced in its course of existence, it becomes grave and stately in its motions, loves peace and quiet; and in majestic silence rolls on its mighty waters, till it is laid to rest in the vast abyss.

The philosopher, the real lover of wisdom, sees much to admire in the economy and mechanism of running waters; and there are few operations of nature which give him more opportunities of remarking the nice adjustment of the most simple means for attaining many purposes of most extensive beneficence. All mankind seem to have felt this. The heart of man is ever open (unless perverted by the habits of selfish indulgence and arrogant self-conceit) to impressions of gratitude and love. He who ascribes the religious principle (defaced, though it be by the humbling abuses of superstition) to the workings of fear alone, may betray the slavish meanness of his own mind, but gives a very unfair and false picture of the hearts of his neighbours. Lucretius was but half a philosopher when he penned his often-quoted apophthegm. Indeed his own invocations show how much the animal was blended with the sage.

We apprehend, that whoever will read with an honest and candid mind, unbiassed by licentious wishes, the accounts of the ancient superstitions, will acknowledge that the amiable emotions of the human soul have had their share in creating the numerous divinities whose worship filled up their calendars. The sun and the host of heaven have in all ages and nations been the objects of a sincere worship. Next to them, the rivers seem to have attracted the grateful acknowledgments of the inhabitants of the adjacent countries. They have everywhere been considered as a sort of tutelar divinities; and each little district, every retired valley, had its river god, who was preferred to all others with a partial fondness. The expoliation of Naaman the Syrian, who was offended with the prophet for enjoining him to wash in the river Jordan, was the natural effusion of this attachment. "What! (said he), are not Abana and Pharpar, rivers of Damascus, more excellent than all the waters of Judea? Might I not wash in them and be clean? So he went away wroth."

In those countries particularly, where the rural labours, and the hopes of the shepherd and the husbandman, were not so immediately connected with the approach and recesses of the sun, and depended rather on what happened in a far distant country by the falls of periodical rains or the melting of collected snows, the Nile, the Ganges, the Indus, the river of Pegu, were the sensible agents of nature in procuring to the inhabitants of their fertile banks all their abundance, and they became the objects of grateful veneration. Their sources were sought out with anxious care even by conquering princes; and when found, were universally worshipped with the most affectionate devotion. These remarkable rivers, so eminently and so palpably benevolent, preserve to this day, amidst every change of habit, and every increase of civilization and improvement, the fond adoration of the inhabitants of those fruitful countries through which they hold their stately course, and their waters are still held sacred. No progress of artificial refinement, not all the corruption of luxurious sensuality, has been able to eradicate this plant of native growth from the heart of man. The sentiment is congenial to his nature, and therefore it is universal; and we could almost appeal to the feelings of every reader, whether he does not perceive it in his own breast. Perhaps we may be mistaken in our opinion in the case of the corrupted inhabitants of the populous and busy cities, who are habituated to the fond contemplation of their own individual exertions as the sources of all their hopes. Give the shoemaker but leather and a few tools, and he defies the powers of nature to disappoint him; but the simpler inhabitants of the country, the most worthy and the most respectable part of every nation, after equal, perhaps greater exertion both of skill and of industry, are more accustomed to resign themselves to the great ministers of Providence, and to look up to heaven for the "early and the latter rains," without which all their labours are fruitless.

extrema per illos

Numenque excellens terris religia fecit.

And among the husbandmen and the shepherds of all nations and ages, we find the same fond attachment to their springs and rivulets.

Fortunatis feneis, hic inter flumina nota

Et fontes sacros frigus captabitis opacum,

was the mournful ejaculation of poor Meliboeus. We hardly know a river of any note in our own country whose source is not looked on with some respect.

We repeat our assertion, that this worship was the offspring of affection and gratitude, and that it is giving a very unfair and false picture of the human mind to ascribe these superstitions to the working of fear alone. These would have represented the river-gods as seated on ruins, brandishing rooted-up trees, with angry looks, pouring out their weeping torrents. But no such thing. The lively imagination of the Greeks felt, and expressed with an energy unknown to all other nations, every emotion of the human soul. They figured the Naiads as beautiful nymphs, patterns of gentleness and elegance. They are represented as partially attached to the children of men; and their interference in human affairs is always in acts of kind affluence and protection. They resemble, in this respect, the rural deities of the northern nations, the fairies, but without their caprices and resentments. And, if we attend to the descriptions and representations of their River-Gods, beings armed with power, an attribute which slavish fear never fails to couple with cruelty and vengeance, we find the same expression of affectionate trust and confidence in their kind dispositions. They are generally called by the respectable but endearing name of father. "Da Tyberi pater," says Virgil. Mr Bruce says that the Nile at its source is called the abay or "father."—We observe this word, or its radix, blended with many names of rivers of the east; and think it probable that when our traveller got this name from the inhabitants of the neighbourhood, they applied to the stream what is meant to express the tutelar or presiding spirit. The river-gods are always represented as venerable old men, to indicate their being coeval with the world. But it is always a cruda viridisque senectus, and they are never represented as oppressed with age and decrepitude. Their beards are long and flowing, their looks placid, their attitude easy, reclined on a bank, covered, as they are crowned, with never-fading fedges and bulrushes, and leaning on their urns, from which they pour out their plentiful and fertilizing streams.— Mr Bruce's description of the sources of the Nile, and of the respect paid to the sacred waters, has not a frowning feature; and the hospitable old man, with his fair daughter Irepone, and the gentle priesthood which peopled the little village of Geeth, forms a contrast with the neighbouring Galla (among whom a military leader was called the lamb, because he did not murder pregnant women), which very distinctly paints the inspiring principle of this superstition. Pliny says (VIII. 8.) that at the source of the Clitumnus there is an ancient temple highly respected. The presence and the power of the divinity are expressed by the fates which stand in the vestibule.—Around this temple are several little chapels, each of which covers a sacred fountain; for the Clitumnus is the father of several little rivers which unite their streams with him.

At some distance below the temple is a bridge which divides the sacred waters from those which are open to common use. No one must presume to set his foot in the streams above this bridge; and to step over any of them is an indignity which renders a person infamous. They can only be visited in a consecrated boat. Below the bridge we are permitted to bathe, and the place is incessantly occupied by the neighbouring villagers. (See also Vitius Sequester Orbelini, p. 101—103, and 221—223; also Sueton. Caligula, c. 43. Virg. Georg. II, 146.)

What is the cause of all this? The Clitumnus flows (near its source) through the richest pastures, through which it was carefully distributed by numberless drains; and these nourished cattle of such spotless whiteness and extraordinary beauty, that they were sought for with eagerness over all Italy, as the most acceptable victims in their sacrifices. Is not this superstition then an effusion of gratitude?

Such are the dictates of kind-hearted nature in our breasts, before it has been vitiated by vanity and self-conceit, and we should not be ashamed of feeling the impression. We hardly think of making any apology for dwelling a little on this incidental circumstance of the superstitious veneration paid to rivers. We cannot think that our readers will be displeased at having agreeable ideas excited in their minds, being always of opinion that the torch of true philosophy will not only enlighten the understanding, but also warm and cherish the affections of the heart.

With respect to the origin of rivers, we have very little to offer in this place. It is obvious to every person, that besides the torrents which carry down into the rivers what part of the rains and melted snows is not absorbed by the soil or taken up by the plants which cover the earth, they are fed either immediately or remotely by the springs. A few remarkable streams rush at once out of the earth in force, and must be considered as the continuation of subterranean rivers, whose origin we are therefore to seek out; and we do not know any circumstance in which their first beginnings differ from those of other rivers, which are formed by the union of little streams and rills, each of which has its own source in a spring or fountain. This question, therefore, What is the process of nature, and what are the supplies which fill our springs? will be treated of under the word Spring.

Whatever be the source of rivers, it is to be met with in almost every part of the globe. The crust of earth with which the rocky framing of this globe is covered is generally stratified. Some of these strata are extremely pervious to water, having but small attraction for its particles, and being very porous. Such is the quality of gravelly strata in an eminent degree. Other strata are much more firm, or attract water more strongly, and refuse it a passage. This is the case with firm rock and with clay. When a stratum of the first kind has one of the other immediately under it, the water remains in the upper stratum, and bursts out wherever the sloping sides of the hills cut off the strata, and this will be in the form of a trickling spring, because the water in the porous stratum is greatly obstructed in its passage towards the outlet. As this irregular formation of the earth is very general, we must have springs, and of course rivers or rivulets, in every corner where there are high grounds.

Rivers flow from the higher to the low grounds. It is the arrangement of this elevation which distributes them over the surface of the earth. And this appears to be accomplished with considerable regularity; and, except the great desert of Kobi on the confines of China and Tartary, we do not remember any very extensive track of ground that is deprived of those channels for voiding the superfluous waters; and even there they are far from being redundant.

The course of rivers gives us the best general method of judging of the elevation of a country. Thus it appears that Savoy and Switzerland are the highest parts of Europe, grottoes of Europe, from whence the ground slopes in every direction. From the Alps proceed the Danube and the Rhine, whose courses mark the two great valleys, into which many lateral streams descend. The Po also and the Rhone come from the same head, and with a steeper and shorter course find their way to the sea through valleys of less breadth and length. On the west side of the valleys of the Rhine and the Rhone the ground rises pretty fast, so that few tributary streams come into them from that side; and from this gentle elevation France slopes to the westward. If a line, nearly straight, but bending a little to the northward, be drawn from the head of Savoy and Switzerland all the way to Solikamskoy in Siberia, it will nearly pass through the most elevated part of Europe; for in this track most of the rivers have their rise. On the left go off the various feeders of the Elbe, the Oder, the Wefel, the Niemen, the Duna, the Neva, the Dwina, the Petzora. On the right, after passing the feeders of the Danube, we see the sources of the Sereth and Pruth, the Dniester, the Bog, the Dnieper, the Don, and the mighty Volga. The elevation, however, is extremely moderate; and it appears from the levels taken with the barometer by the Abbé Chappe d'Auteroche, that the head of the Volga is not more than 470 feet above the surface of the ocean. And we may observe here by the bye, that its mouth, where it discharges its waters into the Caspian Sea, is undoubtedly lower, by many feet, than the surface of the ocean. See Pneumatics, n° 277. Spain and Finland, with Lapland, Norway, and Sweden, form two detached parts, which have little symmetry with the rest of Europe.

A chain of mountains begins in Nova Zembla, and stretches due south to near the Caspian Sea, dividing Europe from Asia. About three or four degrees north of of the Caspian sea it bends to the south-east, traverses western Tartary, and passing between the Tengis and Zai- zan lakes, it then branches to the east and south. The eastern branch runs to the shores of Korea and Kamt- chatka. The southern branch traverses Turkestan and Thibet, separating them from India, and at the head of the kingdom of Ava joins an arm stretching from the great eastern branch, and here forms the centre of a very singular radiation. Chains of mountains issue from it in every direction. Three or four of them keep very close together, dividing the continent into narrow slips, which have each a great river flowing in the middle, and reaching to the extreme points of Malacca, Cam- bodia, and Cochin-china. From the same central point proceeds another great ridge due east, and passes a little north of Canton in China. We called this a singular centre: for though it sends off so many branches, it is by no means the most elevated part of the continent. In the triangle which is included between the first sou- thern ridge (which comes from between the lakes Tan- ges and Zaizan), the great eastern ridge, and its branch which almost unites with the southern ridge, lies the Boutan, and part of Tibet, and the many little rivers which occupy its surface, flow southward and eastward, uniting a little to the north of the centre often men- tioned, and then pass through a gorge eastward into China. And it is farther to be observed, that these great ridges do not appear to be seated on the highest parts of the country; for the rivers which correspond to them are at no great distance from them, and receive their chief supplies from the other sides. This is re- markably the case with the great Oby, which runs al- most parallel to the ridge from the lakes to Nova Zem- bla. It receives its supplies from the east, and indeed it has its source far east. The highest grounds (if we except the ridges of mountains which are boundaries) of the continent seem to be in the country of the Cal- mics, about 95° east from London, and latitude 43° or 45° north. It is represented as a fine though sandy country, having many little rivers which lose themselves in the land, or end in little salt lakes. This elevation stretches north-east to a great distance; and in this track we find the heads of the Irtish, Selenga, and Tun- guikaia (the great feeders of the Oby), the Olenitz, the Lena, the Yana, and some other rivers which all go off to the north. On the other side we have the great ri- ver Amur, and many smaller rivers, whose names are not familiar. The Hoangho, the great river of China, rises on the south side of the great eastern ridge we have so often mentioned. This elevation, which is a conti- nuation of the former, is somewhat of the same com- plexion, being very sandy, and at present is a desert of prodigious extent. It is described, however, as inter- spersed with vast tracks of rich pasture; and we know that it was formerly the residence of a great nation, who came forth, by the name of Turks, and possessed themselves of most of the richest kingdoms of Asia. In the south-western extremity of this country are found remains not only of barbaric magnificence, but even of cultivation and elegance. It was a profitable privilege granted by Peter the Great to some adventurers to search these sandy deserts for remains of former opu- lence, and many pieces of delicate workmanship (tho' not in a style which we would admire) in gold and fil- ver were found. Vaults were found buried in the sand filled with written papers, in a character wholly un- known; and a wall was discovered extending several miles, built with hewn stone, and ornamented with corn- iche and battlements. But we are forgetting ourselves, and return to the consideration of the distribution of the rivers on the surface of the earth. A great ridge of mountains begins at the south-east corner of the Euxine Sea, and proceeds eastward, ranging along the south side of the Caspian, and still advancing unites with the mountains first mentioned in Thibet, sending off some branches to the south, which divide Persia, India, and Thibet. From the south side of this ridge flow the Eu- phrates, Tigris, Indus, Ganges, &c., and from the north the ancient Oxus and many unknown streams.

There is a remarkable circumstance in this quarter of the globe. Although it seems to be nearest to the greatest elevations, it seems also to have places of the greatest depression. We have already said that the Caspian Sea is lower than the ocean. There is in its neighbourhood another great basin of salt water, the lake Aral, which receives the waters of the Oxus or Gihon, which were said to have formerly run into the Caspian Sea. There cannot therefore be a great differ- ence in the level of these two basins; neither have they any outlet, tho' they receive great rivers. There is an- other great lake in the very middle of Persia, the Zara or Zara, which receives the river Hindendem, of near 250 miles length, besides other streams. There is an- other such in Asia Minor. The sea of Sodom and Gomorrha is another instance. And in the high coun- tries we mentioned, there are many small salt lakes, which receive little rivers, and have no outlet. The lake Za- ra in Persia, however, is the only one which indicates a considerable hollow of the country. It is now ascer- tained by actual survey, that the sea of Sodom is con- siderably higher than the Mediterranean. This feature is not, however, peculiar to Asia. It obtains also in Africa, whose rivers we now proceed to mention.

Of them, however, we know very little. The Nile Of Africa, indeed is perhaps better known than any river out of Europe; and of its source and progress we have given a full account in a separate article. See Nile.

By the register of the weather kept by Mr Bruce at Gondar in 1770 and 1771, it appears that the greatest rains are about the beginning of July. He says that at an average each month after June it doubles its rains. The caliph or canal is opened at Cairo about the 9th of August, when the river has risen 14 pecks (each 21 inches), and the waters begin to decrease about the 10th of September. Hence we may form a conjecture concerning the time which the water employs in coming from Abyssinia. Mr Bruce supposes it 9 days, which sup- poses a velocity not less than 14 feet in a second; a thing past belief, and inconsistent with all our notions. The general slope of the river is greatly diminished by several great cataracts; and Mr Bruce expressly says, that he might have come down from Sennaar to the cataracts of Syene in a boat, and that it is navigable for boats far above Sennaar. He came from Syene to Cairo by water. We apprehend that no boat would venture down a stream moving even five feet in a second, and none could row up if the velocity was three feet. As the waters begin to decrease about the 10th of Sep- September; we must conclude that the water then flowing past Cairo had left Abyssinia when the rains had greatly abated. Judging in this way, we must still allow the stream a velocity of more than five feet. Had the first swell at Cairo been noticed in 1770 or 1771, we might have guessed better. The year that Thévenot was in Egypt, the first swell of 8 pecks was observed Jan. 28. The caliph was opened for 14 pecks on August 14th, and the waters began to decrease on September 23rd, having risen to 21½ pecks. We may suppose a similar progress at Cairo corresponding to Mr Bruce's observations at Gondar, and date everything five days earlier.

We understand that some of our gentlemen stationed far up the Ganges have had the curiosity to take notes of the swellings of that river, and compare them with the overflowings at Calcutta, and that their observations are about to be made public. Such accounts are valuable additions to our practical knowledge, and we shall not neglect to insert the information in some kindred article of this work.

The same mountains which attract the tropical vapors, and produce the fertilizing inundations of the Nile, perform the same office to the famous Niger, whose existence has often been accounted fabulous, and with whose course we have very little acquaintance. The researches of the gentlemen of the African association render its existence no longer doubtful, and have greatly excited the public curiosity. For a farther account of its track, see Niger.

From the great number, and the very moderate size, of the rivers which fall into the Atlantic Ocean all the way south of the Gambia, we conclude that the western shore is the most elevated, and that the mountains are at no great distance inland. On the other hand, the rivers at Melinda and Sofala are of a magnitude which indicate a much longer course. But of all this we speak with much uncertainty.

The frame-work (to call it) of America is better known, and is singular. A chain of mountains begins, or at least is found, in longitude 110° west of London, and latitude 40° north, on the northern confines of the kingdom of Mexico, and stretching southward through that kingdom, forms the ridge of the neck of land which separates North from South America, and keeping almost close to the shore, ranges along the whole western coast of South America, terminating at Cape Horn. In its course it sends off branches, which after separating from it for a few leagues, rejoin it again, inclosing valleys of great extent from north to south, and of prodigious elevation. In one of these, under the equatorial sun, stands the city of Quito, in the midst of extensive fields of barley, oats, wheat, and gardens, containing apples, pears, and gooseberries, and in short all the grains and fruits of the cooler parts of Europe; and although the vine is also there in perfection, the olive is wanting. Not a dozen miles from it in the low countries, the sugar-cane, the indigo, and all the fruits of the torrid zone, find their congenial heat, and the inhabitants shelter under a burning sun. At as small a distance on the other hand tower aloft the pinacles of Pichincha, Corombourou, and Chemboracado, crowned with never melting snows.

The individual mountains of this stupendous range not only exceed in height all others in the world (if we except the Peak of Tenerife, Mount Etna, and Mount Blanc); but they are set down on a base incomparably more elevated than any other country. They cut off therefore all communication between the Pacific Ocean and the inland continent; and no rivers are to be found on the west coast of South America which have any considerable length of course or body of waters. The country is drained, like Africa, in the opposite direction. Not 100 miles from the city of Lima, the capital of Peru, which lies almost on the sea shore, and just at the foot of the high Cordilleras, arises out of a small lake the Maragon or Amazon's river, which, after running northward for about 100 miles, takes an easterly direction, and crosses nearly the broadest part of South America, and falls into the great western ocean at Para, after a course of not less than 3500 miles. In the first half of its descent it receives a few middle-sized rivers from the north, and from the south it receives the great river Combos, springing from another little lake not 50 miles distant from the head of the Maragon, and inclosing between them a wide extent of country. Then it receives the Yuta, the Yuera, the Cuchivara, and Parana Mire, each of which is equal to the Rhine; and then the Madeira, which flows above 1300 miles. At their junction the breadth is so great, that neither shore can be seen by a person standing up in a canoe; so that the united stream must be about 6 miles broad. In this majestic form it rolls along at a prodigious rate through a flat country, covered with impenetrable forests, and most of it as yet untrodden by human feet. Mr Condamine, who came down the stream, says, that all is silent as the desert, and the wild beasts and numblets birds crowd round the boat, eyeing it as some animal of which they did not seem afraid. The bed was cut deep through an equal and yielding soil, which seemed rich in every part, if he could judge by the vegetation, which was rank in the extreme. What an addition this to the possible population of this globe! A narrow slip along each bank of this mighty river would equal in surface the whole of Europe, and would probably exceed it in general fertility: and although the velocity in the main stream was great, he observed that it was extremely moderate, nay almost still, at the sides; so that in those parts where the country was inhabited by men, the Indians paddled up the river with perfect ease. Boats could go from Para to near the mouth of the Madeira in 38 days, which is near 1200 miles.

Mr Condamine made an observation during his passage down the Maragon, which is extremely curious and instructive, although it puzzled him very much. He observed that the tide was sensible at a vast distance from the mouth: It was very considerable at the junction of the Madeira; and he supposes that it might have been observed much farther up. This appeared to him very surprising, because there could be no doubt but that the surface of the water there was higher by a great many feet than the surface of the flood of the Atlantic ocean at the mouth of the river. It was therefore very natural for him to ascribe the tide in the Maragon to the immediate action of the moon on its waters; and this explanation was the more reasonable, because the river extends in the direction of terrestrial longitude, which by the Newtonian theory is most favourable to the production of a tide. Journeying as he did, did in an Indian canoe, we cannot suppose that he had much leisure or conveniency for calculations, and therefore are not surprised that he did not see that even this circumstance was of little avail in so small or shallow a body of water. He carefully noted, however, the times of high and low water as he passed along. When arrived at Para, he found not only that the high water was later and later as we are farther from the mouth, but he found that at one and the same instant there were several points of high water between Para and the confluence of the Madeira, with points of low water intervening. This conclusion was easily drawn from his own observations, although he could not see at one instant the high waters in different places. He had only to compute the time of high water at a particular spot, on the day he observed it at another; allowing, as usual, for the moon's change of position. The result of his observations therefore was, that the surface of the river was not an inclined plane whose slope was lessened by the tide of flood at the mouth of the river, but that it was a waving line, and that the propagation of the tide up the river was nothing different from the propagation of any other wave. We may conceive it clearly, though imperfectly, in this way. Let the place be noted where the tide happens 12 hours later than at the mouth of the river. It is evident that there is also a tide at the very mouth at the same instant; and, since the ocean tide had withdrawn itself during the time that the former tide had proceeded so far up the river, and the tide of ebb is successively felt above as well as the tide of flood, there must be a low water between these two high waters.

Newton had pointed out this curious fact, and observed that the tide at London-Bridge, which is 43 feet above the sea, is not the same with that at Gravelend, but the preceding tide (See Phil. Trans. 67.) This will be more particularly insisted on in another place.

Not far from the head of the Maragnon, the Cordileras send off a branch to the north-east, which reaches and ranges along the shore of the Mexican Gulf; and the Rio Grande de Sta Martha occupies the angle between the ridges.

Another ridge ranges with interruptions along the east coast of Terra Firma, so that the whole waters of this country are collected into the Oroonoko. In like manner the north and east of Brazil are hemmed in by mountainous ridges, through which there is no considerable passage; and the ground sloping backwards, all the waters of this immense track are collected from both sides by many considerable rivers into the great river Paraguay, or Rio de la Plata, which runs down the middle of this country for more than 1400 miles, and falls into the sea through a vast mouth in latitude 35°.

Thus the whole of South America seems as if it had been formerly surrounded by a mound, and been a great basin. The ground in the middle, where the Parama, the Madeira, and the Plata, take their rise, is an immense marsh, uninhabitable for its exhalations, and quite impervious in its present state.

The manner in which the continent of North America is watered, or rather drained, has also some peculiarities. By looking at the map, one will observe first of all a general division of the whole of the best known part into two, by the valleys in which the beds of the History river St Laurence and Mississippi are situated. The head of this is occupied by a singular series of fresh water seas or lakes, viz. the lake Superior and Michigan, which empty themselves into lake Huron by two cataracts. This again runs into lake Erie by the river Detroit, and the Erie pours its waters into the Ontario by the famous fall of Niagara, and from the Ontario proceeds the great river St Laurence.

The ground to the south-west of the lakes Superior and Erie is somewhat lower, and the middle of the valley is occupied by the Mississippi and the Missouri, which receive on both sides a number of smaller streams, and having joined, proceed to the south, under the name Mississippi. In latitude 37°, this river receives into its bed the Ohio, a river of equal magnitude, and the Cherokee river, which drains all the country lying at the back of the United States, separated from them by the ranges of the Appalachian mountains. The Mississippi is now one of the chief rivers on the globe, and proceeds due south, till it falls into the Mexican bay through several shifting mouths, which greatly resemble those of the Danube and the Nile, having run above 1200 miles.

The elevated country between this bed of the Mississippi and St Laurence and the Atlantic ocean is drained on the east side by a great number of rivers, some of which are very considerable, and of long course; because instead of being nearly at right angles to the coast, as in other countries, they are in a great measure parallel to it. This is more remarkably the case with Hudson's river, the Delaware, Patowmack, Rappahannock, &c. Indeed the whole of North America seems to consist of ribs or beams laid nearly parallel to each other from north to south, and the rivers occupy the interstices. All those which empty themselves into the bay of Mexico are parallel and almost perfectly straight, unlike what are seen in other parts of the world. The westernmost of them all, the North River, as it is named by the Spaniards, is nearly as long as the Mississippi.

We are very little informed as yet of the distribution of rivers on the north-west coast of America, or the course of those which run into Hudson's and Baffin's bay.

The Maragnon is undoubtedly the greatest river in the world, both as to length of run and the vast body of water which it rolls along. The other great rivers succeed nearly in the following order:

| Maragnon, | Amur, | |-----------|-------| | Senegal, | Oroonoko. | | Nile, | Ganges, | | St Laurence, | Euphrates, | | Hoangho, | Danube, | | Rio de la Plata, | Don, | | Yenisey, | Indus, | | Mississippi, | Dnieper, | | Volga, | Dniester, | | Oby, | &c. |

We have been much assisted in this account of the course of rivers, and their distribution over the globe, by a beautiful planisphere or map of the world published by Mr Bode astronomer royal at Berlin. The ranges of mountains are there laid down with philosophical discernment and precision; and we recommend it to the notice notice of our geographers, We cannot divine what has caused Mr Buffon to say that the course of most rivers is from east to west or from west to east. No physical point of his system seems to require it, and it needs only that we look at his own map to see its falsity. We should naturally expect to find the general course of rivers nearly perpendicular to the line of sea-coast; and we find it so; and the chief exceptions are in opposition to Mr Buffon's assertion. The structure of America is so particular, that very few of its rivers have their general course in this direction. We proceed now to consider the motion of rivers; a subject which naturally resolves itself into two parts, theoretical and practical.

PART I. THEORY OF THE MOTION OF RIVERS AND CANALS.

The importance of this subject needs no commentary. Every nation, every country, every city, is interested in it. Neither our wants, our comforts, nor our pleasures, can dispense with an ignorance of it. We must conduct their waters to the centre of our dwellings; we must secure ourselves against their ravages; we must employ them to drive those machines which, by compensating for our personal weakness, make a few able to perform the work of thousands; we employ them to water and fertilize our fields, to decorate our mansions, to cleanse and embellish our cities, to preserve or extend our demesnes, to transport from county to county every thing which necessity, convenience, or luxury, has rendered precious to man: for these purposes we must confine and govern the mighty rivers, we must preserve or change the beds of the smaller streams, draw off from them what shall water our fields, drive our machines, or supply our houses. We must keep up their waters for the purposes of navigation, or supply their places by canals; we must drain our fens, and defend them when drained; we must understand their motions, and their mode of secret, slow, but unceasing action, that our bridges, our wharfs, our dikes, may not become heaps of ruins. Ignorant how to proceed in these daily recurring cares, how often do we see projects of high expectation and heavy expense fail of their object, leaving the state burdened with works not only useless but frequently hurtful?

This has long been a most interesting subject of study in Italy, where the fertility of their fields is not more indebted to their rich soil and happy climate, than to their numerous derivations from the rivers which traverse them: and in Holland and Flanders, where their very existence requires unceasing attention to the waters, which are every moment ready to swallow up the inhabitants; and where the inhabitants, having once subdued this formidable enemy, have made those very waters their indefatigable drudges, transporting through every corner of the country the materials of the most extensive commerce on the face of this globe.

Such having been our incessant occupations with moving waters, we should expect that while the operative arts are continually furnishing facts and experiments, the man of speculative and scientific curiosity, excited by the importance of the subject, would ere now have made considerable progress in the science; and that the professional engineer would be daily acting from established principle, and be seldom disappointed in his expectations. Unfortunately the reverse of this is nearly the true state of the case; each engineer is obliged to collect the greatest part of his knowledge from his own experience, and by many dear-bought lessons, to direct his future operations, in which he still proceeds with anxiety and hesitation: for we have not yet acquired principles of theory, and experiments have not yet been collected and published, by which an empirical practice might be safely formed. Many experiments of inestimable value are daily made; but they remain with their authors, who seldom have either leisure, ability, or generosity, to add them to the public stock.

The motion of waters has been really so little investigated as yet, that hydraulics may still be called a new science yet in its infancy. We have merely skimmed over a few common notions concerning the motions of water; and the mathematicians of the first order seem to have contented themselves with such views as allowed them to entertain themselves with elegant applications of calculus. This, however, has not been their fault. They rarely had any opportunity of doing more, for want of a knowledge of facts. They have made excellent use of the few which have been given them; but it required much labour, great variety of opportunity, and great expense, to learn the multiplicity of things which are combined even in the simplest cases of water in motion. These are seldom the lot of the mathematician; and he is without blame when he enjoys the pleasures within his reach, and cultivates the science of geometry in its most abstracted form. Here he makes a progress which is the boast of human reason, being almost infurled from error by the intellectual simplicity of his subject. But when we turn our attention to material objects, and without knowing either the size and shape of the elementary particles, or the laws which nature has prescribed for their action, presume to foresee their effects, calculate their exertions, direct their actions, what must be the consequence? Nature shows her independence with respect to our notions, and, always faithful to the laws which are enjoined, and of which we are ignorant, she never fails to thwart our views, to disconcert our projects, and render useless all our efforts.

To wish to know the nature of the elements is vain, proper and our gross organs are insufficient for the study. To mode of investigation, suppose what we do not know, and to fancy shapes and sizes at will; this is to raise phantoms, and will produce a system, but will not prove a foundation, for any science. But to interrogate Nature herself, study the laws which she so faithfully observes, catch her, as we say, in the fact, and thus wrest from her the secret; this is the only way to become her master, and it is the only procedure consistent with good sense. And we see, that soon after Kepler detected the laws of the planetary motions, when Galileo discovered the uniform acceleration of gravity, when Pascal discovered the pressure of the atmosphere, and Newton discovered the laws of attraction and the track of a ray of light; astronomy, mechanics, hydrostatics, chemistry, optics, quickly became Theory came bodies of sound doctrine; and the deductions from their respective theories were found fair representations of the phenomena of nature. Whenever a man has discovered a law of nature, he has laid the foundation of a science, and he has given us a new mean of subjecting to our service some element hitherto independent: and so long as groups of natural operations follow a route which appears to us whimsical, and will not admit our calculations, we may be assured that we are ignorant of the principle which connects them all, and regulates their procedure.

This is remarkably the case with several phenomena of the motions of fluids, and particularly in the motion of water in a bed or conduit of any kind. Although the first geniuses of Europe have for this century past turned much of their attention to this subject, we are almost ignorant of the general laws which may be observed in their motions. We have been able to select very few points of resemblance, and every case remains nearly an individual. About 150 years ago we discovered, by experience only, the quantity and velocity of water issuing from a small orifice, and, after much labour, have extended this to any orifice; and this is almost the whole of our confidential knowledge. But as to the uniform course of the streams which water the face of the earth, and the maxims which will certainly regulate this agreeably to our wishes, we are in a manner totally ignorant. Who can pretend to say what is the velocity of a river of which you tell him the breadth, the depth, and the declivity? Who can say what swell will be produced in different parts of its course, if a dam or weir of given dimensions be made in it, or a bridge be thrown across it? Or how much its waters will be raised by turning another stream into it, or sunk by taking off a branch to drive a mill? Who can say with confidence what must be the dimensions or slope of this branch, in order to furnish the water that is wanted, or the dimensions and slope of a canal which shall effectually drain a fenny district? Who can say what form will cause or will prevent the undermining of banks, the forming of elbows, the pooling of the bed, or the deposition of sands? Yet these are the most important questions.

The causes of this ignorance are the want or uncertainty of our principles; the fallacy of our only theory, which is belied by experience; and the small number of proper observations or experiments, and difficulty of making such as shall be serviceable. We have, it is true, made a few experiments on the efflux of water from small orifices, and from them we have deduced a sort of theory, dependant on the fall of heavy bodies and the laws of hydrostatic pressure. Hydrostatics is indeed founded on very simple principles, which give a very good account of the laws of the quiescent equilibrium of fluids, in consequence of gravity and perfect fluidity. But by what train of reasoning can we connect these with the phenomena of the uniform motion of the waters of a river or open stream, which can derive its motion only from the slope of its surface, and the modifications of this motion or its velocity only from the width and depth of the stream? These are the only circumstances which can distinguish a portion of a river from a vessel of the same size and shape, in which, however, the water is at rest. In both, gravity is the sole cause of pressure and motion; but there must be some circumstance peculiar to running waters which modifies the exertions of this active principle, and which, when discovered, must be the basis of hydraulics, and must oblige us to reject every theory founded on fancied hypotheses, and which can only lead to absurd conclusions; and purely absurd consequences, when legitimately drawn, are complete evidence of improper principles.

When it was discovered experimentally, that the velocities of water issuing from orifices at various depths under the surface were as the square roots of those of hydraulic depths, and the fact was verified by repeated experiments, this principle was immediately and without modification applied to every motion of water. Mariotte, Varignon, Guglielmini, made it the basis of complete systems of hydraulics, which prevail to this day, after having received various amendments and modifications. The same reasoning obtains through them all, though frequently obscured by other circumstances, which are more particularly expressed by Guglielmini in his Fundamental Theorems.

He considers every point P (fig. 1.) in a mass of fluid as an orifice in the side of a vessel, and conceives the particle as having a tendency to move with the same velocity with which it would issue from the orifice. Therefore, if a vertical line APC be drawn through that point, and if this be made the axis of a parabolic APE, of which A at the surface of the fluid is the vertex, and AB (four times the height through which a heavy body would fall in a second) is the parameter, the velocity of this particle will be represented by the ordinate PD of this parabola; that is, PD is the space which it would uniformly describe in a second.

From this principle is derived the following theory of running waters.

Let DC (fig. 2.) be the horizontal bottom of a reservoir, to which is joined a sloping channel CK of uniform breadth, and let AB be the surface of the standing water in the reservoir. Suppose the vertical plane BC pierced with an infinity of holes, through each of which the water issues. The velocity of each filament will be that which is acquired by falling from the surface AB†. The filament C, issuing with this velocity, will then glide down the inclined plane like any other heavy body; and (by the common doctrine of the motion down an inclined plane) when it has arrived at F, it will have the same velocity which it would have acquired by falling through the height OF, the point O being in the horizontal plane AB produced. The same may be said of its velocity when it arrives at H or K. The filament immediately above C will also issue with a velocity which is in the subduplicate ratio of its depth, and will then glide down above the first filament. The same may be affirmed of all the filaments; and of the superficial filament, which will occupy the surface of the descending stream.

From this account of the genesis of a running stream of water, we may fairly draw the following consequences:

1. The velocity of any particle R, in any part withdrawn from the stream, is that acquired by falling from the horizontal plane AN.

2. The velocity at the bottom of the stream is everywhere greater than anywhere above it, and is least of all at the surface.

3. The velocity of the stream increases continually as the stream recedes from its source. The depths EF, GH, &c., in different parts of the stream, will be nearly in the inverse subduplicate ratio of the depths under the surface AN; for since the same quantity of water is running through every section EF and GH, and the channel is supposed of uniform breadth, the depth of each section must be inversely as the velocity of the water passing through it.

This velocity is indeed different in different filaments of the section; but the mean velocity in each section is in the subduplicate ratio of the depth of the filament under the surface AB. Therefore the stream becomes more shallow as it recedes from the source; and in consequence of this the difference between LH and MG continually diminishes, and the velocities at the bottom and surface of the stream continually approach to equality, and at a great distance from the source they differ infinitely.

5. If the breadth of the stream be contracted in any part, the depth of the running water will be increased in that part, because the same quantity must still pass through; but the velocity at the bottom will remain the same, and that at the surface will be less than it was before; and the area of the section will be increased on the whole.

6. Should a sluice be put across the stream, dipping a little into the water, the water must immediately rise on the upper side of the sluice till it rises above the level of the reservoir, and the smallest immersion of the sluice will produce this effect. For by lowering the sluice, the area of the section is diminished, and the velocity cannot be increased till the water heap up to a greater height than the surface of the reservoir, and this acquires a pressure which will produce a greater velocity of efflux through the orifice left below the sluice.

7. An additional quantity of water coming into this channel will increase the depth of the stream, and the quantity of water which it conveys; but it will not increase the velocity of the bottom filaments, unless it comes from a higher source.

All these consequences are contrary to experience, and show the imperfection, at least, of the explanation.

The third consequence is of all the most contrary to experience. If any one will but take the trouble of following a single brook from its source to the sea, he will find it most rapid in its beginnings among the mountains, gradually slackening its pace as it winds among the hills and gentler declivities, and at last creeping slowly along through the flat grounds, till it is checked and brought to rest by the tides of the ocean.

Nor is the second consequence more agreeable to observation. It is universally found, that the velocity of the surface in the middle of the stream is the greatest of all, and that it gradually diminishes from thence to the bottom and sides.

And the first consequence, if true, would render the running waters on the surface of this earth the instruments of immediate ruin and devastation. If the waters of our rivers, in the cultivated parts of a country, which are two, three, and four hundred feet lower than their sources, run with the velocity due to that height, they would in a few minutes lay the earth bare to the very bones.

The velocities of our rivers, brooks, and rills, being so greatly inferior to what this theory assigns to them, the other consequences are equally contrary to experience. When a stream has its section diminished by narrowing the channel, the current increases in depth, and this is always accompanied by an increase of velocity through the whole of the section, and most of all at the surface; and the area of the section does not increase, but diminishes, all the phenomena, thus contradicting in every circumstance the deduction from the theory; and when the section has been diminished by a sluice let down into the stream, the water gradually heaps up on the upper side of the sluice, and, by its pressure, produces an acceleration of the stream below the sluice, in the same way as if it were the beginning of a stream, as explained in the theory. The velocity now is composed of the velocity preserved from the source and the velocity produced by this subordinate accumulation; and this accumulation and velocity continually increase, till they become such that the whole supply is again discharged through this contracted section: any additional water not only increases the quantity carried along the stream, but also increases the velocity, and therefore the section does not increase in the proportion of the quantity.

It is surprising that a theory really founded on a conceit, and which in every the most familiar and obvious circumstance is contradicted by facts, should have been met with so much attention. That Varignon should generally immediately catch at this notion of Guglielmini, and follow it the subject of many elaborate analytical memoirs, is not to be wondered at. This author only wanted, ed donner prise au calcul; and it was a usual joke among the academicians of Paris, when any new theorem was invented, donnons le à Varignon à généraliser. But his numerous theorems and corollaries were adopted by all, and still make the substance of the present systems of hydraulics. Gravesande, Muthenbroek, and all the elementary treatises of natural philosophy, deliver no other doctrines; and Belidor, who has been considered as the first of all the scientific engineers, details the same theory in his great work the Architecture Hydraulique.

Guglielmini was, however, not altogether the dupe of his own ingenuity. He was not only a pretty good one of the mathematician, but an affilious and sagacious observer, more ingenious. He had applied his theory to some important cases, which occurred in the course of his profession as inspector of the rivers and canals in the Milanese, and to tempted the course of the Danube; and could not but perceive to supply that great corrections were necessary for making the theory quadrate in some tolerable manner with observation; and he immediately saw that the motion was greatly obstructed by inequalities of the canal, which gave to the contiguous filaments of the stream transverse motions, which thwarted and confused the regular progress of the rest of the stream, and thus checked its general progress. These obstructions, he observed, were most effectual in the beginning of its course, while yet a small rill, running among stones, and in a very unequal bed. The whole stream being small, the inequalities bore a great proportion to it, and thus the general effect was great. He also saw that the same causes (these transverse motions produced by the unequal bottom) chiefly affected the contiguous filaments, and were the reasons why the velocity at the sides and bottom was so much diminished as to be less than the superficial velocity, and that even this might come to be diminished. Theory diminished by the same cause. For he observed, that the general stream of a river is frequently composed of a sort of boiling or tumbling motion, by which masses of water are brought up to the surface and again descend. Every person must recollect such appearances in the freshes of a muddy river; and in this way Guglielmini was enabled to account in some measure for the disagreement of his theory with observation.

Mariotte had observed the same obstructions even in the smoothest glass pipes. Here it could not be ascribed to the checks occasioned by transverse motions. He therefore ascribed it to friction, which he supposed to diminish the motion of fluid bodies in the same manner as of solids: and he thence concludes, that the filaments which immediately rub on the sides of the tube have their velocity gradually diminished; and that the filaments immediately adjoining to these, being thus obliged to pass over them or outstrip them, rub upon them, and have their own velocity diminished in like manner, but in a smaller degree; and that the succeeding filaments towards the axis of the tube suffer similar but smaller diminutions. By this means the whole stream may come to have a smaller velocity; and at any rate the medium velocity by which the quantity discharged is determined, is smaller than it would have been independent of friction.

Guglielmini adopted this opinion of Mariotte, and in his next work on the Motion of Rivers, considered this as the chief cause of the retardation; and he added a third circumstance, which he considered as of no less consequence, the viscosity or tenacity of water. He observes that syrup, oil, and other fluids, where this viscosity is more remarkable, have their motions prodigiously retarded by it, and supposes that water differs from them only in the degree in which it possesses this quality: and he says, that by this means not only the particles which are moving more rapidly have their motions diminished by those in their neighbourhood which move slower, but that the filaments also which would have moved more slowly are accelerated by their more active neighbours; and that in this manner the superficial and inferior velocities are brought nearer to an equality. But this will never account for the universal fact, that the superficial particles are the swiftest of all. The superficial particles, says he, acquire by this means a greater velocity than the parabolic law allows them; the medium velocity is often in the middle of the depth; the numerous obstacles, continually multiplied and repeated, cause the current to lose the velocity acquired by the fall; the slope of the bottom then diminishes, and often becomes very small, so that the force remaining is hardly able to overcome the obstacles which are still repeated, and the river is reduced almost to a state of stagnation. He observes, that the Rheno, a river of the Milanese, has near its mouth a slope of no more than 50°, which he considers as quite inadequate to the task; and here he introduces another principle, which he considers as an essential part of the theory of open currents. This is, that there arises from the very depth of the stream a propelling force which restores a part of the lost velocity. He offers nothing in proof of this principle, but uses it to account for and explain the motion of waters in horizontal canals. The principle has been adopted by the numerous Italian writers on hydraulics, and, by various contrivances, interwoven with the parabolic theory, as it is called, of Guglielmini. Our reader may see it in various modifications in the Idrostatica e Idraulica of P. Leechi, and in the Sperienze Idrauliche of Michelotti. It is by no means difficult either in its origin or in the manner of its application to the explanation of phenomena, and seems only to serve for giving something like consistency to the vague and obscure discussions which have been published on this subject in Italy. We have already remarked, that in that country the subject is particularly interesting, and has been much commented upon. But the writers of England, France, and Germany, have not paid so much attention to it, and have more generally occupied themselves with the motion of water in close conduits, which seem to admit of a more precise application of mathematical reasoning.

Some of those who have considered with more attention the effects of friction and viscosity. Sir Isaac Newton, with his usual penetration, had seen distinctly the manner in which it behoved these circumstances to operate. He had occasion, in his researches into the mechanism of the celestial motions, to examine the famous hypothesis of Descartes, that the planets were carried round the sun by fluid vortices, and saw that there would be no end to uncertainty and dispute till the modus operandi of these vortices was mechanically considered. He therefore employed himself in the investigation of the manner in which the acknowledged powers of natural bodies, acting according to the received laws of mechanics, could produce and preserve these vortices, and restore that motion which was expended in carrying the planets round the sun. He therefore, in the second book of the Principles of Natural Philosophy, gives a series of beautiful propositions, viz. 51, 52, &c., with their corollaries, showing how the rotation of an cylinder or sphere round its axis in the midst of a fluid will excite a vortical motion in this fluid; and he affirms with mathematical precision the motion of every filament of this vortex.

He sets out from the supposition that this motion is excited in the surrounding stratum of fluid in consequence of a want of perfect lubricity, and assumes as an hypothesis, that the initial resistance (or diminution of the motion of the cylinder) which arises from this want of lubricity, is proportional to the velocity with which the surface of the cylinder is separated from the contiguous surface of the surrounding fluid, and that the whole resistance is proportional to the velocity with which the parts of the fluid are mutually separated from each other. From this, and the equality of action and reaction, it evidently follows, that the velocity of any stratum of the vortex is the arithmetical medium between the velocities of the strata immediately within and without it. For the intermediate stratum cannot be in equilibrium, unless it is as much preëdited forward by the superior motion of the stratum within it, as it is kept back by the slower motion of the stratum without it.

This beautiful investigation applies in the most perfect manner to every change produced in the motion of a fluid filament, in consequence of the viscosity and friction of the adjoining filaments; and a filament proceeding along a tube at some small distance from the sides has, in like manner, a velocity which is the medium between those of the filaments immediately surrounding it. It is therefore a problem of no very difficult solution to assign the law by which the velocity will gradually diminish as the filament recedes from the axis of a cylindrical tube. It is somewhat surprising that so neat a problem has never occupied the attention of the mathematicians during the time that these subjects were so assiduously studied; but so it is, that nothing precise has been published on the subject. The only approach to a discussion of this kind, is a Memoire of Mr Pitot, read to the academy of Paris in 1726, where he considers the velocity of efflux through a pipe. Here, by attending to the comparative superiority of the quantity of motion in large pipes, he affirms, that the total diminutions arising from friction will be (ceteris paribus) in the inverse ratio of the diameters. This was thankfully received by other writers, and is now a part of our hydraulic theories. It has not, however, been attended to by those who write on the motion of rivers, though it is evident that it is applicable to them with equal propriety; and had it been introduced, it would at once have solved all their difficulties, and particularly would have shown how an almost imperceptible declivity would produce the gentle motion of a great river, without having recourse to the unintelligible principle of Guglielmini.

Mr Couplet made some experiments on the motion of the water in the great main pipes of Versailles, in order to obtain some notions of the retardations occasioned by friction. They were found prodigious; but were so irregular, and unsuceptible of reduction to any general principle, (and the experiments were indeed so few that they were unfit for this reduction), that he could establish no theory.—What Mr Belidor established on them, and makes a sort of system to direct future engineers, is quite unworthy of attention.

Upon the whole, this branch of hydraulics, although of much greater practical importance than the conduct of water in pipes, has never yet obtained more than a vague, and, we may call it, slovenly attention from the mathematicians; and we ascribe it to their not having taken the pains to settle its first principles with the same precision as had been done in the other branch. They were, from the beginning, satisfied with a sort of applicability of mathematical principles, without ever making the application. Were it not that some would accuse us of national partiality, we would ascribe it to this, that Newton had not pointed out the way in this as in the other branch. For any intelligent reader of the performances on the motions of fluids, in close vessels, will see that there has not a principle, nay hardly a step of investigation, been added to those which were used or pointed out by Sir Isaac Newton. He has nowhere touched this question, the motion of water in an open canal. In his theories of the tides, and of the propagation of waves, he had an excellent opportunity for giving at once the fundamental principles of motion in a free fluid whose surface was not horizontal. But, by means of some of those happy and shrewd guesses, in which, as Daniel Bernoulli says, he excelled all men, he saw the undoubted consequences of some palpable phenomenon which would answer all his present purposes, and therefore entered no farther into the investigation.

The original theory of Guglielmini, or the principle adopted by him, that each particle of the vertical section of a running stream has a tendency to move as if it were issuing from an orifice at that depth under the surface, is false; and that it really does so in the face of a dam when the flood-gate is taken away, is no less so; and if it did, the subsequent motions would hardly have any resemblance to those which he assigns them. Were this the case, the exterior form of the cascade would be something like what is sketched in fig. 3, with an abrupt angle at B, and a concave surface BEG. This will be evident to every one who combines the greater velocity of the lower filaments with the slower motion of those which must slide down above them. But this greater advance of the lower filaments cannot take place without an expenditure of the water under the surface AB. The surface therefore sinks, and B instantly ceases to retain its place in the horizontal plane. The water does not successively flow forward from A to B, and then tumble over the precipice; but immediately upon opening the flood-gate, the water wastes from the space immediately behind it, and the whole puts on the form represented in fig. 4, consisting of the curve AaPcEG, convex from A to c, and concave from cence forward. The superficial water begins to accelerate all the way from A; and the particles may be supposed (for the present) to have acquired the velocity corresponding to their depth under the horizontal surface. This must be understood as nothing more than a vague sketch of the motions. It requires a very critical and intricate investigation to determine either the form of the upper curve or the motions of the different filaments. The place A, where the curvature begins, is of equally difficult determination, and is various according to the differences of depth and of inclination of the succeeding canal.

We have given this sort of history of the progress which had been made in this part of hydraulics, that many of our readers might form some opinion of the many dissertations which have been written on the motion of rivers, and of the state of the arts depending on it; practice connected with it; and we may therefore believe, that since there was so little principle in the theories, there could be but very little certainty in the practical operations. The fact has been, that no engineer could pretend to say, with any precision, what would be the effect of his operations. One whose business had given him many opportunities, and who kept accurate and judicious registers of his own works, could pronounce, with some probability, how much water would be brought off by a drain of certain dimensions and a given slope, when the circumstances of the case happened to tally with some former work in which he had succeeded or failed; but out of the pale of his own experience he could only make a sagacious guess. A remarkable instance of this occurred not long ago. A small aqueduct was lately carried into Paris. It had been conducted on a plan presented to the academy, who had corrected it, and gave a report of what its performance would be. When executed in the most accurate manner, it was deficient in the proportion of five to nine. When the celebrated Desaguliers was employed by the city of Edinburgh to superintend the bringing in the water for the supply of the city, he gave a report on the plan which was to be followed. It was executed to his complete satisfaction; and the quantity of water delivered was about one-sixth of the quantity which he promised, and about one-eleventh of the quantity which the no less celebrated M'Laurin calculated from the same plan.

Such being the state of our theoretical knowledge multiplying (if it can be called by this name), naturalists began to be persuaded that it was but losing time to make any use of a theory so incongruous with observation, and that the only safe method of proceeding was to multiply experiments in every variety of circumstances, and to make a series of experiments in every important case, which should comprehend all the practicable modifications of that case. Perhaps circumstances of resemblance might occur, which would enable us to connect many of them together, and at last discover the principles which occasioned this connection; by which means a theory founded on science might be obtained. And if this point should not be gained, we might perhaps find a few general facts, which are modified in all these particular cases, in such a manner that we can still trace the general facts, and see the part of the particular case which depends on it. This would be the acquisition of what may be called an empirical theory, by which every phenomenon would be explained, in so far as the explanation of a phenomenon is nothing more than the pointing out the general fact or law under which it is comprehended; and this theory would answer every practical purpose, because we should confidently foresee what consequences would result from such and such premises; or if we should fail even in this, we should still have a series of experiments so comprehensive, that we could tell what place in the series would correspond to any particular case which might be proposed.

There are two gentlemen, whose labours in this respect deserve very particular notice, professor Michelotti at Turin, and Abbé Boffut at Paris. The first made a prodigious number of experiments both on the motion of water through pipes and in open canals. They were performed at the expense of the sovereign, and no expense was spared. A tower was built of the finest masonry, to serve as a vessel from which the water was to issue through holes of various sizes, under pressures from 5 to 22 feet. The water was received into basins constructed of masonry and nicely lined with stucco, from whence it was conveyed in canals of brickwork lined with stucco, and of various forms and elevations. The experiments on the expense of water through pipes are of all that have yet been made the most numerous and exact, and may be appealed to on every occasion. Those made in open canals are still more numerous, and are no doubt equally accurate; but they have not been so contrived as to be so generally useful, being in general very unlike the important cases which will occur in practice, and they seem to have been contrived chiefly with the view of establishing or overturning certain points of hydraulic doctrine which were probably prevalent at the time among the practical hydraulists.

The experiments of Boffut are also of both kinds; and though on a much smaller scale than those of Michelotti, seem to deserve equal confidence. As far as they follow the same track, they perfectly coincide in their results, which should procure confidence in the other; and they are made in situations much more analogous to the usual practical cases. This makes them doubly valuable. They are to be found in his two volumes intitled Hydrodynamique. He has opened this path of procedure in a manner so new and so judicious, that he has in some measure the merit of such as shall follow him in the same path.

This has been most candidly and liberally allowed him by the chevalier de Buat, who has taken up this progressive matter where the Abbé Boffut left it, and has prosecuted his experiments with great assiduity; and we must now add with singular success. By a very judicious consideration of the subject, he hit on a particular view of it, which saved him the trouble of a minute consideration of the small internal motions, and enabled him to proceed from a very general and evident proposition, which may be received as the key to a complete system of practical hydraulics. We shall follow this ingenious author in what we have farther to say on the subject; and we doubt not but that our readers will think we do a service to the public by making these diffusions of the chevalier de Buat more generally known in this country. It must not however be expected that we shall give more than a synoptical view of them, connected by such familiar reasoning as shall be either comprehended or confused in by persons not deeply versed in mathematical science.

Sect. I. Theory of Rivers.

It is certain that the motion of open streams must, in some respects, resemble that of bodies sliding down inclined planes perfectly polished; and that they would accelerate continually, were they not obstructed; but they are obstructed, and frequently move uniformly. This can only arise from an equilibrium between the forces which promote their descent and those which oppose it. Mr Buat, therefore, assumes the leading proposition, that

When water flows uniformly on any channel or bed, the accelerating force which obliges it to move is equal to the sum of all the resistances which it meets with, whether arising from its own viscosity, or from the friction of its bed.

This law is as old as the formation of rivers, and should be the key of hydraulic science. Its evidence is clear; and it is, at any rate, the basis of all uniform motion. And since it is so, there must be some considerable analogy between the motion in pipes and in open channels. Both owe their origin to an inequality of pressure; both would accelerate continually, if nothing hindered; and both are reduced to uniformity by the viscosity of the fluid and the friction of the channel.

It will therefore be convenient to examine the phenomena of water moving in pipes by the action of its own weight only along the sloping channel. But previous to this, we must take some notice of the obstruction caused by the entry of water into a channel of any kind, arising from the deflection of the many different filaments which press into the channel from the reservoir from every side. Then we shall be able to separate this diminution of motion from the sum total that is observed, and ascertain what part remains as produced by the subsequent obstructions.

We then shall consider the principle of uniform motion, the equilibrium between the power and the resistance. The power is the relative height of the column of fluid which tends to move along the inclined plane of its bed; the resistance is the friction of of the bed, the viscosity of the fluid, and its adhesion to the sides. Here are necessarily combined a number of circumstances which must be gradually detached that we may see the effect of each, viz., the extent of the bed, its perimeter, and its slope. By examining the effects produced by variations of each of these separately, we discover what share each has in the general effect; and having thus analysed the complicated phenomenon, we shall be able to combine those its elements, and frame a formula which shall comprehend every circumstance, from the greatest velocity to the extinction of all motion, and from the extent of a river to the narrow dimensions of a quill. We shall compare this formula with a series of experiments in all this variety of circumstances, partly made by Mr Buat, and partly collected from other authors; and we shall leave the reader to judge of the agreement.

Confident that this agreement will be found most satisfactory, we shall then proceed to consider very curiously the chief varieties which nature or art may introduce into these beds, the different velocities of the same stream, the intensity of the resistance produced by the inertia of the materials of the channel, and the force of the current by which it continually acts on this channel, tending to change either its dimensions or its form. We shall endeavour to trace the origin of these great rivers which spread like the branches of a vigorous tree, and occupy the surface even of a vast continent. We shall follow them in their course, unfold all their windings, study their train, and regimen, and point out the law of its stability; and we shall investigate the causes of their deviations and wanderings.

The study of these natural laws pleases the mind; but it answers a still greater purpose; it enables us to assist nature, and to hasten her operations, which our wants and our impatience often find too slow. It enables us to command the elements, and to force them to administer to our wants and our pleasures.

We shall therefore, in the next place, apply the knowledge which we may acquire to the solution of the most important hydraulic questions which occur in the practice of the civil engineer.

We shall consider the effects produced by a permanent addition to any river or stream by the union of another, and the opposite effect produced by any draught or offset, showing the elevation or depression produced up the stream, and the change made in the depth and velocity below the addition or offset.

We shall pay a similar attention to the temporary swells produced by freshets.

We shall ascertain the effects of straightening the course of a stream, which, by increasing its slope, must increase its velocity, and therefore sink the waters above the place where the curvature was removed, and diminish the tendency to overflow, while the same immediate consequence must expose the places farther down to the risk of floods from which they would otherwise have been free.

The effects of dams or weirs, and of bars, must then be considered; the gorge or swell which they produce up the stream must be determined for every distance from the weir or bar. This will furnish us with rules for rendering navigable or floatable such waters as have too little depth or too great slope. And it will appear that immense advantages may be thus derived, with a moderate expense, even from trifling brooks, if we will relinquish all prejudices, and not imagine that such conveyance is impossible, because it cannot be carried on by such boats and small craft as we have been accustomed to look at.

The effects of canals of derivation, the rules or maxims of draining, and the general maxims of embankment, come in the next place; and our discourses will conclude with remarks on the most proper forms for the entry to canals, locks, docks, harbours, and mouths of rivers, the best shape for the starlings of bridges and of boats for inland navigations, and such like subordinate but interesting particulars, which will be fugitively given by the general thread of discourse.

It is considered, as physically demonstrated (see Hydrosstatics and Hydraulics), that water issuing from a small orifice in the bottom or side of a very large hitherto vessel, almost instantly acquires and maintains the velocity which a heavy body would acquire by falling to the orifice from the horizontal surface of the stagnant water. This we shall call its Natural Velocity. Therefore if we multiply the area of the orifice by this velocity, the product will be the bulk or quantity of the water which is discharged. This we may call the Natural Expence of water, or the Natural Discharge.

Let O represent the area or section of the orifice expressed in some known measure, and b its depth under the surface. Let g express the velocity acquired by a heavy body during a second by falling. Let V be the medium velocity of the water's motion, Q the quantity of water discharged during a second, and N the natural expence.

We know that V is equal to $\sqrt{2g} \times \sqrt{b}$. Therefore $N = O \cdot \sqrt{2g} \cdot \sqrt{b}$.

If these dimensions be all taken in English feet, we have $\sqrt{2g}$ very nearly equal to 8; and therefore $V = 8\sqrt{b}$, and $N = O \cdot 8\sqrt{b}$.

But in our present business it is much more convenient to measure everything by inches. Therefore since a body acquires the velocity of 32 feet 2 inches in a second, we have $2g = 64$ feet 4 inches or 772 inches, and $\sqrt{2g} = 27.78$ inches nearly 27$\frac{1}{4}$ inches.

Therefore $V = \sqrt{772} \cdot \sqrt{b} = 27.78 \cdot \sqrt{b}$, and $N = O \cdot \sqrt{772} \cdot \sqrt{b} = O \cdot 27.78 \cdot \sqrt{b}$.

But it is also well known, that if we were to calculate the expence or discharge for every orifice by this simple rule, we should in every instance find it much greater than nature really gives us.

When water issues through a hole in a thin plate, the lateral columns, pressing into the hole from all sides, cause the issuing filaments to converge to the axis of the jet, and contract its dimensions at a little distance from the hole. And it is in this place of greatest contraction, that the water acquires that velocity which we observe in our experiments, and which we assume as equal to that acquired by falling from the surface. Therefore, that our computed discharge may best agree with observation, it must be calculated on the supposition that the orifice is diminished to the size of this smallest section. But the contraction is subject to variations, and the dimensions of this smallest section... are at all times difficult to ascertain with precision. It is therefore much more convenient to compute from the real dimensions of the orifice, and to correct this computed discharge, by means of an actual comparison of the computed and effective discharges in a series of experiments made in situations resembling those cases which most frequently occur in practice. This correction or its cause, in the mechanism of those internal motions, is generally called Contraction by the writers on hydraulics; and it is not confined to a hole in a thin plate: it happens in some degree in all cases where fluids are made to pass through narrow places. It happens in the entry into all pipes, canals, and sluices; nay even in the passage of water over the edge of a board, such as is usually set up on the head of a dam or weir, and even when this is immersed in water on both sides, as in a bar or keep, frequently employed for raising the waters of the level streams in Flanders, in order to render them navigable.

§ See Refl. ance of Fluids, p. 67.

Motion of filaments in various particular situations.

Fig. 5. A shows the motion through a thin plate. B shows the motion when a tube of about two diameters long is added, and when the water flows with a full mouth. This does not always happen in so short a pipe (and never in one that is shorter), but the water frequently detaches itself from the sides of the pipe, and flows with a contracted jet.

C shows the motion when the pipe projects into the inside of the vessel. In this case it is difficult to make it flow full.

D represents a mouth-piece fitted to the hole, and formed agreeably to that shape which a jet would assume of itself. In this case all contraction is avoided, because the mouth of this pipe may be considered as the real orifice, and nothing now diminishes the discharge but a trifling friction of the sides.

E shows the motion of water over a dam or weir, where the fall is free or unobstructed; the surface of the lower stream being lower than the edge or sole of the waste-board.

F is a similar representation of the motion of water over what we would call a bar or keep.

It was one great aim of the experiments of Michelotti and Bossut to determine the effects of contraction in these cases. Michelotti, after carefully observing the form and dimensions of the natural jet, made various mouth-pieces resembling it, till he obtained one which produced the smallest diminution of the computed discharge, or till the discharge computed for the area of its smaller end approached the nearest to the effective discharge. And he at last obtained one which gave a discharge of 983, when the natural discharge would have been 1000. This piece was formed by the revolution of a trochoid round the axis of the jet, and the dimensions were as follow:

| Diameter of the outer orifice | = 36 | |-------------------------------|-----| | Inner orifice | = 46 | | Length of the axis | = 96 |

The results of the experiments of the Abbé Bossut and of Michelotti scarcely differ, and they are expressed in the following table:

| Q for the thin plate fig. A almost at the surface | 6526 | | Q for ditto at the depth of 8 feet | 6195 | | Q for ditto at the depth of 16 feet | 6173 | | Q for a tube 2 diameters long, fig. B. | 8125 | | Q for ditto projecting inwards and flowing full | 6814 | | Q for ditto with a contracted jet, fig. C. | 5137 | | Q for the mouth-piece, fig. D. | 9831 | | Q for a weir, fig. E. | 9536 | | Q for a bar, fig. F. | 9730 |

The numbers in the last column of this little table are the cubical inches of water discharged in a second when the height \( b \) is one inch.

It must be observed that the discharges assigned here for the weir and bar relate only to the contractions occasioned by the passage over the edge of the board. The weir may also suffer a diminution by the contractions at its two ends, if it should be narrower than the stream, which is generally the case, because the two ends are commonly of square masonry or woodwork. The contraction there is nearly the same with that at the edge of a thin plate. But this could not be introduced into this table, because its effect on the expense is the same in quantity whatever is the length of the waste-board of the weir.

In like manner, the diminution of discharge through Diminution a sluice could not be expressed here. When a sluice is of discharge drawn up, but its lower edge still remains under water, a sluice, &c., the discharge is contracted both above and at the sides, and the diminution of discharge by each is in proportion to its extent. It is not easy to reduce either of these contractions to computation, but they may be very easily observed. We frequently can observe the water, at coming out of a sluice into a mill course, quit the edge of the aperture, and show a part of the bottom quite dry. This is always the case when the velocity of efflux is considerable. When it is very moderate, this place is occupied by an eddy water almost stagnant. When the head of water is 8 or 10 inches, and runs off freely, the space left between it and the sides is about \( \frac{1}{4} \) inches. If the sides of the entry have a slope, this void space can never appear; but there is always this tendency to convergence, which diminishes the quantity of the discharge.

It will frequently abridge computation very much to consider the water discharged in these different situations as moving with a common velocity, which we conceive as produced not by a fall from the surface of the fluid (which is exact only when the expense is equal to the natural expense), but by a fall accommodated to the discharges; or it is convenient to know the height which would produce that very velocity which the water issues with in these situations.

And also, when the water is observed to be actually moving with a velocity \( V \), and we know whether it is coming through a thin plate, through a tube, over a dam, &c., it is necessary to know the pressure or head of water \( b \) which has actually produced this velocity. It is convenient therefore to have the following numbers in readiness. for the natural expense \( = \frac{V^2}{772} \)

for a thin plate \( = \frac{V^2}{296} \)

for a tube 2 diam. long \( = \frac{V^2}{505} \)

for a dam or weir \( = \frac{V^2}{726} \)

for a bar \( = \frac{V^2}{746} \)

It was necessary to premise these facts in hydraulics, that we may be able in every case to distinguish between the force expended in the entry of the water into the conduit or canal, and the force employed in overcoming the resistances along the canal, and in preserving or accelerating its motion in it.

The motion of running water is produced by two causes; 1. The action of gravity; and, 2. The mobility of the particles, which makes them assume a level position in confined vessels, or determines them to move to that side where there is a defect of pressure. When the surface is level, every particle is at rest, being equally pressed in all directions; but if the surface is not level, not only does a particle on the very surface tend by its own weight towards the lower side, as a body would slide along an inclined plane, but there is a force, external to itself, arising from a superiority of pressure on the upper end of the surface; which pushes this superficial particle towards the lower end; and this is not peculiar to the superficial particles, but affects every particle within the mass of water. In the vessel ACDE (fig. 6.), containing water with an inclined surface AE, if we suppose all frozen but the extreme columns AKHB, FGLE, and a connecting portion HKCDLG, it is evident, from hydrostatic laws, that the water on this connecting part will be pushed in the direction CD; and if the frozen mass BHGF were moveable, it would also be pushed along. Giving it fluidity will make no change in this respect; and it is indifferent what is the situation and shape of the connecting column or columns. The propelling force (MNF being horizontal) is the weight of the column AMNB. The same thing will obtain wherever we select the vertical columns. There will always be a force tending to push every particle of water in the direction of the declivity. The consequence will be, that the water will sink at one end and rise at the other, and its surface will rest in the horizontal position AOe, cutting the former in its middle O. This cannot be unless there be not only a motion of perpendicular descent and ascent of the vertical columns, but also a real motion of translation from K towards L. It perhaps exceeds our mathematical skill to tell what will be the motion of each particle. Newton did not attempt it in his investigation of the motion of waves, nor is it at all necessary here. We may, however, acquire a very distinct notion of its general effect. Let OPQ be a vertical plane passing through the middle point O. It is evident that every particle in PQ, such as P, is pressed in the direction QD, with a force equal to the weight of a single row of particles, whose length is the difference between the columns BH and FG. The force acting on the particle Q is, in like manner, the weight of a row of particles = AC—ED. Now if OQ, OA, OE, be divided in the same ratio, so that all the figures ACDE, BHGF, &c. may be similar, we see that the force arising solely from the declivity, and acting on each particle on the plane OQ, is proportional to its depth under the surface, and that the row of particles ACQDE, BHPGF, &c. which is to be moved by it, is in the same proportion. Hence it unquestionably follows, that the accelerating force on each particle of the row is the same in all. Therefore the whole plane OQ tends to advance forward together with the same velocity; and in the instant immediately succeeding, all these particles would be found again in a vertical plane indefinitely near to OQ; and if we sum up the forces, we shall find them the same as if OQ were the opening of a sluice, having the water on the side of D standing level with O, and the water on the other side standing at the height AC. This result is extremely different from that of the hasty theory of Guglielmini. He considers each particle in OQ as urged by an accelerating force proportional to its depth, if it is true; but he makes it equal to the weight of the row OP, and never recollects that the greatest part of it is balanced by an opposite pressure, nor perceives that the force which is not balanced must be distributed among a row of particles which varies in the same proportion with itself. When these two circumstances are neglected, the result must be incompatible with observation. When the balanced forces are taken into the account of pressure, it is evident that the surface may be supposed horizontal, and that motion should obtain in this case as well as in the case of a sloping surface; and indeed this is Guglielmini's professed theory, and what he highly values himself on. He announces this discovery of a new principle, which he calls the energy of deep waters, as an important addition to hydraulics. It is owing to this, says he, that the great rivers are not stagnant at their mouths, where they have no perceptible declivity of surface, but, on the contrary, have greater energy and velocity than farther up, where they are shallower. This principle is the basis of his improved theory of rivers, and is insisted on at great length by all the subsequent writers. Buffon, in his theory of the earth, makes much use of it. We cannot but wonder that it has been allowed a place in the theory of rivers given in the great Encyclopédie de Paris, and in an article having the signature (O) of D'Alembert. We have been very anxious to show the fallacy of this principle, because we consider it as a mere subterfuge of Guglielmini, by which he was able to patch up the mathematical theory which he had so hastily taken from Newton or Galileo; and we think that we have secured our readers from being misled by it, when we show that this energy must be equally operative when the surface is on a dead level. The absurdity of this is evident. We shall see by and by, that deep waters, when in actual motion, have an energy not to be found in shallow running waters, by which they are enabled to continue that motion; but this is not a moving principle; and it will be fully explained, as an immediate result of principles, not vaguely conceived and indistinctly expressed, like this of Guglielmini, but easily understood, and appreciable with the greatest precision. It is an energy common to all great bodies. Although they lose as much momentum in surmounting any obstacle as small ones, they lose but a small portion of their velocity. At present, employed only in consider- ing the progressive motion of an open stream, whose surface is not level, it is quite enough that we see that such a motion must obtain, and that we see that there are propelling forces; and that those forces arise solely from the want of a level surface, or from the slope of the surface; and that, with respect to any one particle, the force acting on it is proportional to the difference of level between each of the two columns (one on each side of the particle) which produce it. Were the surface level, there would be no motion; if it is not level, there will be motion; and this motion will be proportional to the want of level or the declivity of the surface: it is of no consequence whether the bottom be level or not, or what is its shape.

Hence we draw a fundamental principle, that the motion of rivers depends entirely on the slope of the surface.

The slope or declivity of any inclined plane is not properly expressed by the difference of height alone of its extremities; we must also consider its length: and the measure of the slope must be such that it may be the same while the declivity is the same. It must therefore be the same over the whole of any one inclined plane. We shall answer these conditions exactly, if we take for the measure of a slope the fraction which expresses the elevation of one extremity above the other divided by the length of the plane. Thus \( \frac{AM}{AF} \) will express the declivity of the plane AF.

If the water met with no resistance from the bed in which it runs, if it had no adhesion to its sides and bottom, and if its fluidity were perfect, its gravity would accelerate its course continually, and the earth and its inhabitants would be deprived of all the advantages which they derive from its numberless streams. They would run off so quickly, that our fields, dried up as soon as watered, would be barren and useless. No foil could resist the impetuosity of the torrents; and their accelerating force would render them a destroying scourge, were it not that, by kind Providence, the resistance of the bed, and the viscosity of the fluid, become a check which reins them in and sets bounds to their rapidity. In this manner the friction on the sides, which, by the viscosity of the water, is communicated to the whole mass, and the very adhesion of the particles to each other, and to the sides of the channel, are the causes which make the resistances bear a relation to the velocity; so that the resistances augmenting with the velocities, come at last to balance the accelerating force. Then the velocity now acquired is preserved, and the motion becomes uniform, without being able to acquire new increase, unless some change succeeds either in the slope or in the capacity of the channel. Hence arises the second maxim in the motion of rivers, that when a stream moves uniformly, the resistance is equal to the accelerating force.

As in the efflux of water through orifices, we pass over the very beginnings of the accelerated motion, which is a matter of speculative curiosity, and consider the motion in a state of permanency, depending on the head of water, the area of the orifice, the velocity, and the expense; so, in the theory of the uniform motion of rivers, we consider the slope, the transverse section or area of the stream, the uniform velocity, and the expense. It will be convenient to affix precise meanings to the terms which we shall employ.

The section of a stream is the area of a plane perpendicular to the direction of the general motion.

The resistances arise ultimately from the action of the plained water on the internal surface of the channel, and must be proportional (ceteris paribus) to the extent of the action. Therefore if we unfold the whole edge of this section, which is rubbed as it were by the passing water, we shall have a measure of the extent of this action. In a pipe, circular or prismatical, the whole circumference is acted on; but in a river or canal ACDQ (fig. 6.) the horizontal line \( aOe \), which makes the upper boundary of the section \( aCDe \), is free from all action. The action is confined to the three lines \( aC, CD, De \). We shall call this line \( aCDe \) the border of the section.

The mean velocity is that with which the whole section, moving equally, would generate a solid equal to the expense of the stream. This velocity is to be found perhaps but in one filament of the stream, and we do not know in which filament it is to be found.

Since we are attempting to establish an empirical theory of the motion of rivers, founded entirely on experiment and palpable deductions from them; and since it is extremely difficult to make experiments on open streams which shall have a precision sufficient for such an important purpose—it would be a most desirable thing to demonstrate an exact analogy between the mutual balancing of the acceleration and resistance in pipes and in rivers; for in those we can not only make experiments with all the desired accuracy, and admitting precise measures, but we can make them in a number of cases that are almost impracticable in rivers. We can increase the slope of a pipe from nothing to the vertical position, and we can employ every desired degree of pressure, so as to ascertain its effect on the velocity in degrees which open streams will not admit. The Chevalier de Buat has most happily succeeded in this demonstration; and it is here that his good fortune and his penetration have done so much service to practical science.

Let AB (fig. 7.) be a horizontal tube, through which the water is impelled by the pressure or head and DA. This head is the moving power; and it may be conceived as consisting of two parts, performing two offices. One of them is employed in impressing on the water that velocity with which it actually moves in the tube. Were there no obstructions to this motion, no greater head would be wanted; but there are obstructions arising from friction, adhesion, and viscosity. This requires force. Let this be the office of the rest of the head of water in the reservoir. There is but one allotment, appropriation, or repartition, of the whole head which will answer. Suppose E to be the point of partition, so that DE is the head necessary for impressing the actual velocity on the water (a head or pressure which has a relation to the form or circumstance of the entry, and the contraction which takes place there). The rest EA is wholly employed in overcoming the simultaneous resistances which take place along the whole tube AB, and is in equilibrium with this resistance. Therefore if we apply, at E a tube EC of the same length and diameter with AB, and and having the same degree of polish or roughness; and if this tube be inclined in such a manner that the axis of its extremity may coincide with the axis of AB in the point C—we affirm that the velocity will be the same in both pipes, and that they will have the same expense; for the moving force in the sloping pipe EC is composed of the whole weight of the column DE and the relative weight of the column EC; but this relative weight, by which alone it descends along the inclined pipe EC, is precisely equal to the weight of a vertical column EA of the same diameter. Everything therefore is equal in the two pipes, viz. the lengths, the diameters, the moving forces, and the resistances; therefore the velocities and discharges will also be equal.

This is not only the case on the whole, but also in every part of it. The relative weight of any part of it EK is precisely in equilibrium with the resistances along that part of the pipe; for it has the same proportion to the whole relative weight that the resistance has to the whole resistance. Therefore (and this is the most important circumstance, and the basis of the whole theory) the pipe EC may be cut shorter, or may be lengthened to infinity, without making any change in the velocity or expense, so long as the propelling head DE remains the same.

Leaving the whole head DA as it is, if we lengthen the horizontal pipe AB to G, it is evident that we increase the resistance without any addition of force to overcome it. The velocity must therefore be diminished; and it will now be a velocity which is produced by a smaller head than DE; therefore if we were to put in a pipe of equal length at E, terminating in the horizontal line AG, the water will not run equally in both pipes. In order that it may, we must discover the diminished velocity with which the water now actually runs along AG, and we must make a head DI capable of impressing this velocity at the entry of the pipe, and then insert at I a pipe IH of the same length with AG. The expense and velocity of both pipes will now be the same (a).

What has now been said of a horizontal pipe AB would have been equally true of any inclined pipe AB, A'B (fig. 8.) Drawing the horizontal line CB, we see that DC is the whole head or propelling pressure for either pipe AB or A'B; and if DE is the head necessary for the actual velocity, EC is the head necessary for balancing the resistances; and the pipe EF of the same length with AB, and terminating in the same horizontal line, will have the same velocity; and its inclination being thus determined, it will have the same velocity and expense whatever be its length.

Thus we see that the motion in any pipe, horizontal Analogy or sloping, may be referred to or substituted for the between motion in another inclined pipe, whose head of water, these pipes above the place of entry, is that productive of the actual demonstrative velocity of the water in the pipe. Now, in this case, led by De the accelerating force is equal to the resistance; we Buat. may therefore consider this last pipe as a river, of which the bed and the slope are uniform or constant, and the current in a state of permanency; and we now may clearly draw this important conclusion, that pipes and open streams, when in a state of permanency, perfectly resemble each other in the circumstances which are the immediate causes of this permanency. The equilibrium between the accelerating force obtains not only in general, but takes place through the whole length of the pipe or stream, and is predicable of every individual transverse section of either. To make this more palpably evident if possible, let us consider a sloping cylindrical pipe, the current of which is in a state of permanency. We can conceive it as consisting of two half cylinders, an upper and a lower. These are running together at an equal pace; and the filaments of each immediately contiguous to the separating plane and to each other, are not rubbing on each other, nor affecting each others motions in the smallest degree. It is true that the upper half is pressing on the lower, but in a direction perpendicular to the motion, and therefore not affecting the velocity; and we shall see presently, that although the lower side of the pipe bears somewhat more pressure than the other, the resistances are not changed. (Indeed this odds of pressure is accompanied with a difference of motion, which need not be considered at present; and we may suppose the pipe so small or so far below the surface, that this shall be insensible). Now let us suppose, that in an instant the upper half cylinder is annihilated: We then have an open stream; and every circumstance of accelerating force and of resistance remains precisely as it was. The motion must therefore continue as it did; and in this state the only accelerating force is the slope of the surface. The demonstration therefore is complete.

From these observations and reasonings we draw a general and important conclusion, "That the same pipe will be susceptible of different velocities, which it will preserve uniform to any distance, according as it has different inclinations; and each inclination of a pipe of given diameter has a certain velocity peculiar to itself, which will be maintained uniform to any distance what-

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(a) We recommend it to the reader to make this distribution or allotment of the different portions of the pressure very familiar to his mind. It is of the most extensive influence in every question of hydraulics, and will on every occasion give him distinct conceptions of the internal procedure. Obvious as the thought seems to be, it has escaped the attention of all the writers on the subject. Leechi, in his Hydraulics published in 1766, ascribes something like it to Daniel Bernoulli; but Bernoulli, in the passage quoted, only speaks of the partition of pressure in the instant of opening an orifice. Part of it, says he, is employed in accelerating the quiescent water, and producing the velocity of efflux, and the remainder produces the pressure (now diminished) on the sides of the vessel. Bernoulli, Bossut, and all the good writers, make this distribution in express terms in their explanation of the motion of water through successive orifices; and it is surprising that no one before the Chevalier de Buat saw that the resistance arising from friction required a similar partition of the pressure; but though we should call this good fortune, we must ascribe to his great sagacity and justness of conception the beautiful use that he has made of it: "suum cuique." whatever; and this velocity increases continually, according to some law, to be discovered by theory or experiment, as the position of the pipe changes, from being horizontal till it becomes vertical; in which position it has the greatest uniform velocity possible relative to its inclination, or depending on inclination alone.

Let this velocity be called the train, or the rate of each pipe.

It is evident that this principle is of the utmost consequence in the theory of hydraulics; for by experiment we can find the train of any pipe. It is in train when an increase of length makes no change in the velocity. If lengthening the pipe increases the velocity, the slope of the pipe is too great, and vice versa. And having discovered the train of a pipe, and observed its velocity, and computed the head productive of this velocity with the contraction at the entry, the remainder of the head, that is, the slope (for this is equivalent to EA), is the measure of the resistance. Thus we obtain the measure of the resistance to the motion with a given velocity in a pipe of given diameter. If we change only the velocity, we get the measure of the new resistance relative to the velocity; and thus discover the law of relation between the resistance and velocity. Then, changing only the diameter of the pipe, we get the measure of the resistance relative to the diameter. This is the aim of a prodigious number of experiments made and collected by Buat, and which we shall not repeat, but only give the results of the different parts of his investigation.

We may express the slope of a pipe by the symbol \( \frac{I}{s} \), I being an inch for instance, and s being the slant length of a pipe which is one inch more elevated at one end than at the other. Thus a river which has a declivity of an inch and a half in 120 fathoms or 8640 inches, has its slope \( = \frac{1}{8640} \) or \( \frac{1}{5760} \). But in order to obtain the hydraulic slope of a conduit pipe, the heights of the reservoir and place of discharge being given, we must subtract from the difference of elevation the height or head of water necessary for propelling the water into any pipe with the velocity V, which it is supposed actually to have. This is \( \frac{V^2}{505} \). The remainder d is to be considered as the height of the declivity, which is to be distributed equally over the whole length l of the pipe, and the slope is then \( \frac{d}{l} = \frac{1}{s} \).

There is another important view to be taken of the slope, which the reader should make very familiar to his thoughts. It expresses the proportion between the weight of the whole column which is in motion and the weight which is employed in overcoming the resistance; and the resistance to the motion of any column of water is equal to the weight of that column multiplied by the fraction \( \frac{1}{s} \), which expresses its slope.

We come now to consider more particularly the resistances which in this manner bring the motions to a state of uniformity. If we consider the resistances which arise from a cause analogous to friction, we see that they must depend entirely on the inertia of the water. What we call the resistance is the diminution of a motion which would have obtained but for these resistances; and the best way we have of measuring them is by the force which we must employ in order to keep up or restore this motion. We estimate this motion by a progressive velocity, which we measure by the expense of water in a given time. We judge the velocity to diminish, when the quantity discharged diminishes; yet it may be otherwise, and probably is otherwise. The absolute velocity of many, if not all, of the particles, may even be increased; but many of the motions, being transverse to the general direction, the quantity of motion in this direction may be less, while the sum of the absolute motions of all the particles may be greater. When we increase the general velocity, it is not unreasonable to suppose that the impulses on all the inequalities are increased in this proportion; and the number of particles thus impelling and deflected at the same time will increase in the same proportion. The whole quantity therefore of these useless and lost motions will increase in the duplicate ratio of the velocities, and the force necessary for keeping up the motion will do so also; that is, the resistances should increase as the squares of the velocities.

Or if we consider the resistances as arising merely from the curvature of the imperceptible internal motions occasioned by the inequalities of the sides of the pipe, and as measured by the forces necessary for producing these curvilinear motions; then, because the curves will be the same whatever are the velocities, the deflecting forces will be as the squares of the velocities; but these deflecting forces are pressures, propagated from the parts urged or pressed by the external force, and are proportional to these external pressures by the principles of hydrostatics. Therefore the pressures or forces necessary for keeping up the velocities are as the squares of these velocities; and they are our only measures of the resistances which must be considered as following the same ratio. Whatever view therefore we take of the nature of these resistances, we are led to consider them as proportional to the squares of the velocities.

We may therefore express the resistances by the symbol \( \frac{V^2}{m} \), m being some number to be discovered by experiment. Thus, in a particular pipe, the diminution of the motion or the resistance may be the 1000th part of the square of the velocity, and \( R = \frac{V^2}{1000} \).

Now if g be the accelerating power of gravity on any particle, \( \frac{g}{s} \) will be its accelerating power, by which it would urge it down the pipe whose slope is \( \frac{1}{s} \). Therefore, by the principle of uniform motion, the equality of the accelerating force, and the resistance, we shall have \( \frac{V^2}{m} = \frac{g}{s} \); and \( V \sqrt{s} = \sqrt{mg} \); that is, the product of the velocity, and the reciprocal of the square root of the slope, or the quotient of the velocity divided by the slope, is a constant quantity \( \sqrt{mg} \) for any given pipe; and the primary formula for all the uniform velocities of one pipe is \( V = \frac{\sqrt{mg}}{\sqrt{s}} \). Mr Buat therefore examined this by experiment, but found, that even with respect to a pipe or channel which was uniform throughout, this was not true. We could give at once the final formula which he found to express the velocity in every case whatever; but this would be too empirical. The chief steps of his very far-gracious investigation are instructive. We shall therefore mention them briefly, at least as far as they tend to give us any collateral information; and let it always be noted, that the instructions which they convey is not abstract speculation, but experimental truths, which must ever remain as an addition to our stock of knowledge, although Mr Buat's deductions from them should prove false.

He found, in the first place, that in the same channel the product of $V$ and $\sqrt{s}$ increased as $\sqrt{s}$ increased; that is, the velocities increased faster than the square roots of the slope, or the resistances did not increase as fast as the squares of the velocities. We beg leave to refer our readers to what we said on the resistance of pipes to the motion of fluids through them, in the article Pneumatics, when speaking of bellows. They will there see very valid reasons (we apprehend) for thinking that the resistances must increase more slowly than the squares of the velocities.

It being found, then, that $V\sqrt{s}$ is not equal to a constant quantity $\sqrt{mg}$, it becomes necessary to investigate some quantity depending on $\sqrt{s}$, or, as it is called, some function of $\sqrt{s}$, which shall render $\sqrt{mg}$ a constant quantity. Let $X$ be this function of $\sqrt{s}$, so that we shall always have $VX$ equal to the constant quantity $\sqrt{mg}$, or $\frac{\sqrt{mg}}{X}$ equal to the actual velocity $V$ of a pipe or channel which is in train.

Mr Buat, after many trials and reflections, the chief of which will be mentioned by and by, found a value of $X$ which corresponded with a vast variety of slopes and velocities, from motions almost imperceptible, in a bed nearly horizontal, to the greatest velocities which could be produced by gravity alone in a vertical pipe; and when he compared them together, he found a very discernible relation between the resistances and the magnitude of the section: that is, that in two channels which had the same slope, and the same propelling force, the velocity was greatest in the channel which had the greatest section relative to its border. This may reasonably be expected. The resistances arise from the mutual action of the water and this border. The water immediately contiguous to it is retarded, and this retards the next, and so on. It is to be expected, therefore, that if the border, and the velocity, and the slope, be the same, the diminution of this velocity will be so much the less as it is to be shared among a greater number of particles; that is, as the area of the section is greater in proportion to the extent of its border. The diminution of the general or medium velocity must be less in a cylindrical pipe than in a square one of the same area, because the border of its section is less.

It appears evident, that the resistance of each particle is in the direct proportion of the whole resistance, and the inverse proportion of the number of particles which receive equal shares of it. It is therefore directly as the border, and inversely as the section. Therefore in the expression $\frac{V^2}{m}$ which we have given for the resistance, the quantity $m$ cannot be constant, except in the same channel; and in different channels it must vary along with the relation of the section to its border, because the resistances diminish in proportion as this relation increases.

Without attempting to discover this relation by theoretical examination of the particular motions of the various filaments, Mr Buat endeavoured to discover it by a comparison of experiments. But this required some manner of stating this proportion between the augmentation of the section and the augmentation of its border.

His statement is this: He reduces every section to a rectangular parallelogram of the same area, and having its base equal to the border unfolded into a straight line. The product of this base by the height of the rectangle will be equal to the area of the section. Therefore this height will be a representative of this variable ratio of the section to its border (We do not mean that there is any ratio between a surface and a line; but the ratio of section to section is different from that of border to border; and it is the ratio of these ratios which is thus expressed by the height of this rectangle). If $S$ be the section, and $B$ the border, $S$ is evidently a line equal to the height of this rectangle. Every section being in this manner reduced to a rectangle, the perpendicular height of it may be called the hydraulic mean depth of the section, and may be expressed by the symbol $d$. (Buat calls it the mean radius). If the channel be a cylindrical pipe, or an open half cylinder, it is evident that $d$ is half the radius. If the section is a rectangle, whose width is $w$, and height $b$, the mean depth is $\frac{wb}{b + 2b}$, &c. In general, if $q$ represent the proportion of the breadth of a rectangular canal to its depth, that is, if $q$ be made $= \frac{w}{b}$, we shall have $d = \frac{wb}{q + 2}$, or $d = \frac{qb}{q + 2}$.

Now, since the resistances must augment as the proportion of the border to the section augments, $m$ in the formulas $\frac{V^2}{m} = \frac{g}{s}$ and $V\sqrt{s} = \sqrt{mg}$, must follow the proportions of $d$, and the quantity $\sqrt{mg}$ must be proportional to $\sqrt{d}$ for different channels, and $\frac{\sqrt{mg}}{\sqrt{d}}$ should be a constant quantity in every case.

Our author was aware, however, of a very specious objection to the close dependence of the resistance on the extent of the border; and that it might be laid that a double border did not occasion a double resistance, unless the pressure on all the parts was the same. For it may be naturally (and it is generally supposed) that the resistance will be greater when the pressure is greater. The friction or resistance analogous to friction may therefore be greater on an inch of the bottom than on an inch of the sides; but Mr D'Alembert and many others have demonstrated, that the paths of the filaments will be the same whatever be the pressures. This might serve to justify our ingenious author; but he was determined to test everything on experiment. He therefore made an experiment on the oscillation of water in syphons, which we have repeated in the following form, which is affected by the same circumstances, and is susceptible of much greater precision, and of more extensive and important application.

The two vessels ABCD, a b c d (fig. 9.) were connected by the syphon EFG g f e, which turned round in the short tubes E and e, without allowing any water to escape; the axes of these tubes being in one straight line. The vessels were about 10 inches deep, and the branches FG, f g of the syphon were about five feet long. The vessels were set on two tables of equal height, and (the hole c being flopped) the vessel ABCD, and the whole syphon, were filled with water, and water was poured into the vessel a b c d till it stood at a certain height LM. The syphon was then turned into a horizontal position, and the plug drawn out of e, and the time carefully noted which the water employed in rising to the level H K k b in both vessels. The whole apparatus was now inclined, so that the water run back into ABCD. The syphon was now put in a vertical position, and the experiment was repeated. — No sensible or regular difference was observed in the time. Yet in this experiment the pressure on the part G g of the syphon was more than six times greater than before. As it was thought that the friction on this small part (only six inches) was too small a portion of the whole obstruction, various additional obstructions were put into this part of the syphon, and it was even lengthened to nine feet; but still no remarkable difference was observed. It was even thought that the times were less when the syphon was vertical.

Thus Mr De Buat's opinion is completely justified; and he may be allowed to assert, that the resistance depends chiefly on the relation between the section and its border; and that \( \frac{\sqrt{mg}}{\sqrt{d}} \) should be a constant quantity.

To ascertain this point was the object of the next series of experiments; to see whether this quantity was really constant, and, if not, to discover the law of its variation, and the physical circumstances which accompanied the variations, and may therefore be considered as their causes. A careful comparison of a very great number of experiments, made with the same pipe, and with very different channels and velocities, showed that \( \sqrt{mg} \) did not follow the proportion of \( \sqrt{d} \), nor of any power of \( \sqrt{d} \). This quantity \( \sqrt{mg} \) increased by smaller degrees in proportion as \( \sqrt{d} \) was greater. In very great beds \( \sqrt{mg} \) was nearly proportional to \( \sqrt{d} \), but in smaller channels, the velocities diminished much more than \( \sqrt{d} \) did. Calling about for some way of accommodation, Mr Buat considered, that some approximation at least would be had by taking off from \( \sqrt{d} \) some constant small quantity. This is evident: For such a diminution will have but a trifling effect when \( \sqrt{d} \) is great, and its effect will increase rapidly when \( \sqrt{d} \) is very small. He therefore tried various values for this subtraction, and compared the results with the former experiments; and he found, that if in every case \( \sqrt{d} \) be diminished by one-tenth of an inch, the calculated discharges would agree very exactly with the experiment. Therefore, instead of \( \sqrt{d} \), he makes use of \( \sqrt{d} - 0.1 \), and finds this quantity always proportional to \( \sqrt{mg} \), or finds that \( \frac{\sqrt{mg}}{\sqrt{d} - 0.1} \) is a constant quantity, or very nearly so. It varied from 297 to 287 in all sections from that of a very small pipe to that of a little canal. In the large sections of canals and rivers it diminished still more, but never was less than 256.

This result is very agreeable to the most distinct notions that we can form of the mutual actions of the agreeable water and its bed. We see, that when the motion of water is obstructed by a solid body, which deflects the lines of the passing filaments, the disturbance does not extend to any considerable distance on the two sides of the body. In like manner, the small disturbances, and imperceptible curvilinear motions, which are occasioned by the infinitesimal inequalities of the channel, must extend to a very small distance indeed from the sides and bottom of the channel. We know, too, that the mutual adhesion or attraction of water for the solid bodies which are moistened by it, extends to a very small distance; which is probably the same, or nearly so, in all cases. Mr Buat observed, that a surface of 23 square inches, applied to the surface of stagnant water, lifted 7601 grains; another of 5½ square inches lifted 365; this was at the rate of 65 grains per inch nearly, making a column of about one-sixth of an inch high. Now this effect is very much analogous to a real contraction of the capacity of the channel. The water may be conceived as nearly stagnant to this small distance from the border of the section. Or, to speak more accurately, the diminution of the progressive velocity occasioned by the friction and adhesion of the sides, decreases very rapidly as we recede from the sides, and ceases to be sensible at a very small distance.

The writer of this article verified this by a very simple and instructive experiment. He was making experiments on the production of vortices, in the manner suggested by Sir Isaac Newton, by whirling a very accurate and smoothly polished cylinder in water; and he found that the rapid motion of the surrounding water was confined to an exceeding small distance from the cylinder, and it was not till after many revolutions that it was sensible even at the distance of half an inch. We may, by the way, suggest this as the best form of experiments for examining the resistances of pipes. The motion excited by the whirling cylinder in the stagnant water is equal and opposite to the motion lost by water passing along a surface equal to that of the cylinder with the same velocity. Be this as it may, we are justified in considering, with Mr Buat, the section of the stream as thus diminished by cutting off a narrow border all round the touching parts, and supposing that the motion and discharge is the same as if the root of the mean depth of the section were diminished by a small quantity, nearly constant. We see, too, that the effect of this must be infensible in great canals and rivers; so that, fortunately, its quantity is best ascertained by experiments made with small pipes. This is attended with another convenience, in the opinion of Mr Buat, namely, that the effect... The effect of viscosity is most sensible in great masses of water in flow motion, and is almost insensible in small pipes, so as not to disturb these experiments. We may therefore assume 297 as the general value of

$$\frac{\sqrt{mg}}{\sqrt{d} - o_1}.$$

Since we have $$\frac{\sqrt{mg}}{\sqrt{d} - o_1} = 297,$$ we have also

$$m = \frac{297}{g} \cdot \frac{\sqrt{d} - o_1}{362} = \frac{88209}{362} (\sqrt{d} - o_1)^2 = 243.7 (\sqrt{d} - o_1)^2.$$

This we may express by

$$n (\sqrt{d} - o_1)^2.$$

And thus, when we have expressed the effect of friction by $$\frac{V^2}{m},$$ the quantity $m$ is variable, and its general value is

$$\frac{V^2}{n (\sqrt{d} - o_1)}$$

in which $n$ is an invariable abstract number equal to 243.7, given by the nature of the resistance which water sustains from its bed, and which indicates its intensity.

And, lastly, since $m = n (\sqrt{d} - o_1)^2,$ we have

$$\frac{\sqrt{mg}}{\sqrt{n} g} (\sqrt{d} - o_1),$$

and the expression of the velocity $V,$ which water acquires and maintains along any channel whatever, now becomes

$$V = \frac{\sqrt{n} g (\sqrt{d} - o_1)}{X},$$

or

$$\frac{297 (\sqrt{d} - o_1)}{X},$$

in which $X$ is also a variable quantity, depending on the slope of the surface or channel, and expressing the accelerating force which, in the case of water in train, is in equilibrium with the resistances expressed by the numerator of the fraction.

Having so happily succeeded in ascertaining the variations of resistance, let us accompany Mr Buat in his investigation of the law of acceleration, expressed by the value of $X.$

Experience, in perfect agreement with any distinct opinions that we can form on this subject, had already showed him, that the resistances increased in a slower ratio than that of the squares of the velocities, or that the velocities increased slower than $\sqrt{s}.$ Therefore, in the formula

$$V = \frac{\sqrt{n} g (\sqrt{d} - o_1)}{X},$$

which, for one channel, we may express thus, $V = \frac{A}{X},$ we must admit that $X$ is sensibly equal to $\sqrt{s}$ when the slope is very small or $s$ very great. But, that we may accurately express the velocity in proportion as the slope augments, we must have $X$ greater than $\sqrt{s};$ and moreover,

$$\frac{\sqrt{s}}{X}$$

must increase as $\sqrt{s}$ diminishes. These conditions are necessary, that our values of $V,$ deduced from the formula $V = \frac{A}{X},$ may agree with the experiment.

In order to comprehend every degree of slope, we must particularly attend to the motion through pipes, because open canals will not furnish us with instances of exact trains with great slopes and velocities. We can make pipes vertical. In this case $\frac{1}{s}$ is $\frac{1}{1},$ and the velocity is the greatest possible for a train by the action of gravity; but we can give greater velocities than this by increasing the head of water beyond what produces the velocity of the train.

Let $AB$ (fig. 10.) be a vertical tube, and let $CA$ be the head competent to the velocity in the tube, which we suppose to be in train. The slope is $1,$ and the full weight of the column in motion is the precise measure of the resistance. The value of $\frac{1}{s},$ considered as a slope, is now a maximum; but, considered as expressing the proportion of the weight of the column in motion to the weight which is in equilibrium with the resistance, it may not be a maximum; it may surpass unity, and $s$ may be less than $1.$ For if the vessel be filled to $E,$ the head of water is increased, and will produce a greater velocity, and this will produce a greater resistance. The velocity being now greater, the head $EF$ which imparts it must be greater than $CA.$ But it will not be equal to $EA,$ because the uniform velocities are found to increase faster than the square roots of the pressures. This is the general fact. Therefore $F$ is above $A,$ and the weight of the column $FB,$ now employed to overcome the resistance, is greater than the weight of the column $AB$ in motion.

In such cases, therefore, $\frac{1}{s},$ greater than unity, is a sort of fictitious slope, and only represents the proportion of the resistance to the weight of the moving column. This proportion may surpass unity.

But it cannot be infinite: For supposing the head of water infinite; if this produce a finite velocity, and we deduct from the whole height the height corresponding to this finite velocity, there will remain an infinite head, the measure of an infinite resistance produced by a finite velocity. This does not accord with the observed law of the velocities, where the resistances actually do not increase as fast as the squares of the velocities. Therefore an infinite head would have produced an infinite velocity, in opposition to the resistances: taking off the head of the tube, competent to this velocity, at the entry of the tube, which head would also be infinite, the remainder would in all probability be finite, balancing a finite resistance.

Therefore the value of $s$ may remain finite, although the velocity be infinite; and this is agreeable to all our clearest notions of the resistances.

Adopting this principle, we must find a value of $X$ which will answer all these conditions. 2. It must be sensibly proportional to $\sqrt{s},$ while $s$ is great. It must always be less than $\sqrt{s}.$ 3. It must deviate from the proportion of $\sqrt{s},$ to much the more as $\sqrt{s}$ is smaller. 4. It must not vanish when the velocity is infinite. 5. It must agree with a range of experiments with every variety of channel and of slope.

We shall understand the nature of this quantity $X$ better by representing by lines the quantities concerned informing it.

If the velocities were exactly as the square roots of the slopes, the equilateral hyperbola $NKS$ (fig. 10., n° 2) between its asymptotes $MA, AB,$ would represent the equation $V = \frac{A}{\sqrt{s}}.$ The values of $\sqrt{s}$ would be represented by the abscissa, and the velocities by the ordinates, and $V \sqrt{s} = A$ would be the power of the hyperbola. But since these velocities are not sensibly equal. equal to \( \frac{A}{\sqrt{s}} \) except when \( \sqrt{s} \) is very great, and deviate the more from this quantity as \( \sqrt{s} \) is smaller; we may represent the velocities by the ordinates of another curve \( PGT \), which approaches very near to the hyperbola, at a great distance from \( A \) along \( AB \); but separates from it when the abscissa are smaller: so that if \( AQ \) represents that value of \( \sqrt{s} \) (which we have seen may become less than unity), which corresponds to an infinite velocity, the line \( QO \) may be the asymptote of the new curve. Its ordinates are equal to \( \frac{A}{X} \) while those of the hyperbola are equal to \( \frac{A}{\sqrt{s}} \). Therefore the ratio of these ordinates or \( \frac{\sqrt{s}}{X} \) should be such that it shall be so much nearer to unity as \( \sqrt{s} \) is greater, and shall surpass it so much the more as \( \sqrt{s} \) is smaller.

To express \( X \) therefore as some function of \( \sqrt{s} \) so as to answer these conditions, we see in general that \( X \) must be less than \( \sqrt{s} \). And it must not be equal to any power of \( \sqrt{s} \) whose index is less than unity, because then \( \frac{\sqrt{s}}{X} \) would differ so much the more from unity as \( \sqrt{s} \) is greater. Nor must it be any multiple of \( \sqrt{s} \) such as \( q\sqrt{s} \), for the same reason. If we make \( X = \sqrt{s} - K \), \( K \) being a constant quantity, we may answer the first condition pretty well. But \( K \) must be very small, that \( X \) may not become equal to nothing, except in some exceedingly small value of \( \sqrt{s} \). Now the experiments will not admit of this, because the ratio \( \frac{\sqrt{s}}{\sqrt{s} - K} \) does not increase sufficiently to correspond with the velocities which we observe in certain slopes, unless we make \( K \) greater than unity, which again is inconsistent with other experiments. We learn from such canvassing that it will not do to make \( K \) a constant quantity. If we should make it any fractional power of \( \sqrt{s} \), it would make \( X = 0 \), that is, nothing, when \( s = 1 \), which is also contrary to experience. It would seem, therefore, that nothing will answer for \( K \) but some power of \( \sqrt{s} \) which has a variable index. The logarithm of \( \sqrt{s} \) has this property. We may therefore try to make \( X = \sqrt{s} - \log \sqrt{s} \). Accordingly if we try the equation \( V = \frac{A}{\sqrt{s} - \text{hyp. log. } \sqrt{s}} \), we shall find a very great agreement with the experiments till the declivity becomes considerable, or about \( \frac{1}{2} \), which is much greater than any river. But it will not agree with the velocities observed in some mill courses, and in pipes of a still greater declivity, and gives a velocity that is too small; and in vertical pipes the velocity is not above one half of the true one. We shall get rid of most of these incongruities if we make \( K \) consist of the hyperbolic logarithm of \( \sqrt{s} \) augmented by a small constant quantity, and by trying various values for this constant quantity, and comparing the results with experiment, we may hit on one sufficiently exact for all practical purposes.

Mr De Buat, after repeated trials, found that he would have a very great conformity with experiment by making \( K = \log \sqrt{s + 1} \), and that the velocities exhibited in his experiments would be very well represented by the formula \( V = \frac{297(\sqrt{d} - 0.1)}{\sqrt{s - L}\sqrt{s + 1}} \).

There is a circumstance which our author seems to have overlooked on this occasion, and which is undoubtedly of great effect in these motions, viz. the mutual adhesion of the particles of water. This causes the water which is descending (in a vertical pipe for example) to drag more water after it, and thus greatly increases its velocity. We have seen an experiment in which the water issued from the bottom of a reservoir through a long vertical pipe having a very gentle taper. It was 15 feet long, one inch diameter at the upper end, and two inches at the lower. The depth of the water in the reservoir was exactly one foot; in a minute there were discharged 27 cubic feet of water. It must therefore have issued through the hole in the bottom of the reservoir with the velocity of 8.85 feet per second. And yet we know that this head of water could not make it pass through the hole with a velocity greater than 6.56 feet per second. This increase must therefore have arisen from the cause we have mentioned, and is a proof of the great intensity of this force. We doubt not but that the discharge might have been much more increased by proper contrivances; and we know many instances in water pipes where this effect is produced in a very great degree.

The following case is very distinct: Water is brought into the town of Dunbar in the county of East Lothian from a spring at the distance of about 3200 yards. It is conveyed along the first 1100 yards in a pipe of two inches diameter, and the declivity is 12 feet nine inches; from thence the water flows in a pipe of 1½ diameter, with a declivity of 44 feet 3 inches, making in all 57 feet. When the work was carried as far as the two-inch pipe reached, the discharge was found to be 27 Scotch pints, of 103½ cubic inches each in a minute. When it was brought into the town, the discharge was 28. Here it is plain that the descent along the second stretch of the pipe could derive no impulsion from the first. This was only able to supply 27 pints, and to deliver it into a pipe of equal bore. It was not equivalent to the forcing it into a smaller pipe, and almost doubling its velocity. It must therefore have been dragged into this smaller pipe by the weight of what was descending along it, and this water was exerting a force equivalent to a head of 16 inches, increasing the velocity from 14 to about 28.

It must be observed, that if this formula be just, proves that there can be no declivity so small that a current of water will not take place in it. And accordingly none will have been observed in the surface of a stream when this did not happen. But it also should happen with respect to any declivity of bottom. Yet we know that water will hang on the sloping surface of a board without proceeding further. The cause of this seems to be the adhesion of the water combined with its viscosity. The viscosity of a fluid presents a certain force which must be overcome before any current can take place.

A series of important experiments were made by our author in order to ascertain the relation between the velocity at the surface of any stream and that at the bottom. These are curious and valuable on many accounts. One circumstance deserves our notice here, viz., that the difference between the superficial and bottom velocities of any stream are proportional to the square roots of the superficial velocities. From what has been already said on the gradual diminution of the velocities among the adjoining filaments, we must conclude that the same rule holds good with respect to the velocity of separation of two filaments immediately adjoining. Hence we learn that this velocity of separation is in all cases indefinitely small, and that we may, without danger of any sensible error, suppose it a constant quantity in all cases.

We think, with our ingenious author, that on a review of these circumstances, there is a constant or invariable portion of the accelerating force employed in overcoming this viscosity and producing this mutual separation of the adjoining filaments. We may express this part of the accelerating force by a part \( \frac{1}{S} \) of that slope which constitutes the whole of it. If it were not employed in overcoming this resistance, it would produce a velocity which (on account of this resistance) is not produced, or is lost. This would be \( \frac{A}{\sqrt{S}-L\sqrt{S}} \).

This must therefore be taken from the velocity exhibited by our general formula. When thus corrected, it would become \( V = (\sqrt{d}-c_1) \left( \frac{\sqrt{n}g}{\sqrt{s}-L\sqrt{s}+1.6} - \frac{\sqrt{n}g}{\sqrt{s}-L\sqrt{s}} \right) \). But as the term \( \frac{\sqrt{n}g}{\sqrt{s}-L\sqrt{s}} \) is compounded only of constant quantities, we may express it by a single number. This has been collected from a scrupulous attention to the experiments (especially in canals and great bodies of water moving with very small velocities; in which case the effects of viscosity must become more remarkable), and it appears that it may be valued at \( \frac{1}{0.09} \), or 0.3 inches very nearly.

From the whole of the foregoing considerations, drawn from nature, supported by such reasoning as our most distinct notions of the internal motions will admit, and authorized by a very extensive comparison with experiment, we are now in a condition to conclude a complete formula, expressive of the uniform motion of water, and involving every circumstance which appears to have any share in the operation.

Therefore let

\( V \) represent the mean velocity, in inches per second, of any current of water, running uniformly, or which is in train, in a pipe or open channel, whose section, figure, and slope, are constant, but its length indefinite.

\( d \) the hydraulic mean depth, that is, the quotient arising from dividing the section of the channel, in square inches, by its border, expressed in linear inches.

\( s \) The slope of the pipe, or of the surface of the current. It is the denominator of the fraction expressing this slope, the numerator being always unity; and is had by dividing the expanded length of the pipe or channel by the difference of height of its two extremities.

**Table containing the experiments from which the formula is deduced.**

| N° | Length of Pipe | Height of Reservoir | Values of \( s \) | Velocities observed | Velocities calculated | |----|----------------|---------------------|------------------|--------------------|----------------------| | | Inch. | Inch. | Inch. | Inch. | Inch. | | 1 | 12 | 16,166 | 0.7563 | 11,704 | 12,006 | | 2 | 12 | 13,125 | 0.9307 | 9,753 | 10,576 |

**Vertical Tube \( \frac{2}{3} \) of a Line in Diameter and \( \sqrt{d} = 0.117851 \).**

| Inch. | Inch. | Inch. | Inch. | |-------|-------|-------|-------| | 3 | 34,166| 42,166| 0.9062| | 4 | Do. | 38,333| 0.9951| | 5 | Do. | 36,666| 1.0306| | 6 | Do. | 35,333| 1.0781|

**Vertical Pipe \( \frac{1}{2} \) Lines Diameter, and \( \sqrt{d} = 0.176776 \) Inch.**

| Inch. | Inch. | Inch. | Inch. | |-------|-------|-------|-------| | 3 | 34,166| 42,166| 0.9062| | 4 | Do. | 38,333| 0.9951| | 5 | Do. | 36,666| 1.0306| | 6 | Do. | 35,333| 1.0781| ### Theory

#### The same Pipe horizontal.

| No | Length of Pipe | Height of Reservoir | Values of s | Velocities observed | Velocities calculated | |----|----------------|---------------------|-------------|--------------------|----------------------| | | Inch | Inch | Inch | Inch | Inch | | 7 | 34.166 | 14.583 | 25.838 | 26.202 | 25.523 | | 8 | Do | 9.292 | 4.0367 | 21.064 | 19.882 | | 9 | Do | 5.292 | 7.036 | 14.642 | 14.447 | | 10 | Do | 2.683 | 17.6378 | 7.320 | 2.351 |

Vertical Pipe 2 Lines Diameter, and \( \sqrt{d} = 0.204124 \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 11 | 36.25 | 51.250 | 0.85451 | 64.373 | 64.945 | | 12 | Do | 45.250 | 0.90388 | 59.605 | 60.428 | | 13 | Do | 41.916 | 1.0808 | 57.220 | 57.838 | | 14 | Do | 38.750 | 1.12047 | 54.186 | 55.321 |

Same Pipe with a slope of \( \frac{1}{13024} \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 15 | 36.25 | 33.500 | 1.29174 | 51.151 | 50.983 |

#### Experiments by the Abbé Bossut.

Horizontal Pipe 1 Inch Diameter \( \sqrt{d} = 0.5 \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 57 | 600 | 12 | 54.5966 | 22.282 | 21.975 | | 58 | 600 | 4 | 161.312 | 12.223 | 11.756 |

Horizontal Pipe 1.5 Inch Diameter \( \sqrt{d} = 0.5774 \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 59 | 360 | 24 | 19.0781 | 48.534 | 49.515 | | 60 | 720 | 24 | 33.0166 | 34.473 | 35.130 | | 61 | 360 | 12 | 37.5828 | 33.160 | 33.106 | | 62 | 180 | 24 | 48.3542 | 28.075 | 28.211 | | 63 | 1440 | 24 | 63.1806 | 24.004 | 24.023 | | 64 | 720 | 12 | 66.3020 | 23.360 | 23.345 | | 65 | 1800 | 24 | 78.0532 | 21.032 | 21.182 | | 66 | 2160 | 24 | 92.9474 | 18.896 | 19.096 | | 67 | 1080 | 12 | 95.8756 | 18.943 | 18.749 | | 68 | 1440 | 12 | 125.6007 | 16.128 | 15.991 | | 69 | 1800 | 12 | 155.4015 | 14.666 | 14.119 | | 70 | 2160 | 12 | 185.2487 | 12.560 | 12.750 |

Horizontal Pipe 2.01 Inch Diameter \( \sqrt{d} = 0.708946 \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 71 | 360 | 24 | 21.4709 | 58.903 | 58.803 | | 72 | 720 | 24 | 35.8062 | 43.130 | | | 73 | 360 | 12 | 41.2759 | 40.322 | 39.587 | | 74 | 1080 | 24 | 50.4119 | 35.765 | 35.096 | | 75 | 1440 | 24 | 65.1448 | 30.896 | 30.096 | | 76 | 720 | 12 | 70.1426 | 29.215 | 28.796 | | 77 | 1800 | 24 | 79.8487 | 27.470 | 26.639 | | 78 | 2160 | 24 | 94.7901 | 27.731 | 24.079 | | 79 | 1080 | 12 | 99.4979 | 23.806 | 23.400 | | 80 | 1440 | 12 | 129.0727 | 20.707 | 20.076 | | 81 | 1800 | 12 | 158.7512 | 18.304 | 17.788 | | 82 | 2160 | 12 | 188.5179 | 16.377 | 16.097 |

Mr Couplet's Experiments at Versailles.

Pipe 5 Inches Diameter \( \sqrt{d} = 1.1803 \).

| | Inch | Inch | Inch | Inch | Inch | |----|---------------|---------------------|-------------|--------------------|----------------------| | 83 | 84240 | 25 | 3378.26 | 5.323 | 5.287 | | 84 | Do | 24 | 3518.98 | 5.213 | 5.168 | | 85 | Do | 21.083 | 4005.66 | 4.806 | 4.807 | | 86 | Do | 16.750 | 5041.61 | 4.127 | 4.225 | | 87 | Do | 11.333 | 7450.42 | 3.154 | 3.288 | | 88 | Do | 5.583 | 15119.96 | 2.011 | 2.254 |

Pipe 18 Inches Diameter \( \sqrt{d} = 2.12132 \). ### Set II. Experiments with a Wooden Canal.

| No | Section of Canal | Border of Canal | Values of \( \sqrt{T} \) | Values of \( s \) | Mean Observed Velocity | Mean Calculated Velocity | |----|------------------|-----------------|--------------------------|------------------|------------------------|-------------------------| | | | | | | | |

#### Trapezium Canal.

| No | Inch | Inch | Inch | Inch | Inch | Inch | |----|------|------|------|------|------|------| | 1 | 18.84| 13.06| 120107| 212 | 27.91| 27.91| | 2 | 50.60| 29.50| 13.96| 212 | 28.92| 29.88| | 3 | 83.43| 26 | 17913 | 412 | 27.41| 28.55| | 4 | 27.20| 15.31| 13329 | 427 | 18.28| 20.39| | 5 | 39.36| 18.13| 14734 | 427 | 20.30| 22.71| | 6 | 50.44| 20.37| 15736 | 427 | 22.37| 24.37| | 7 | 56.43| 21.50| 16201 | 427 | 23.54| 25.14| | 8 | 98.74| 28.25| 18696 | 432 | 28.29| 29.06| | 9 | 100.74| 28.53| 18791 | 432 | 28.52| 29.23| | 10 | 119.58| 31.06| 19622 | 432 | 30.16| 30.60| | 11 | 126.20| 31.91| 19887 | 432 | 31.58| 31.03| | 12 | 130.71| 32.47| 20064 | 432 | 31.89| 31.32| | 13 | 135.32| 33.03| 20241 | 432 | 31.52| 31.61| | 14 | 20.83| 13.62| 12367 | 1728| 8.94 | 8.58 | | 15 | 36.77| 17.56| 14471 | 1728| 9.74 | 9.98 | | 16 | 42.01| 18.69| 14992 | 1728| 11.45| 10.17|

#### Rectangular Canal.

| No | Inch | Inch | Inch | Inch | Inch | Inch | |----|------|------|------|------|------|------| | 1 | 24.50| 21.25| 127418| 458 | 20.24| 18.66| | 2 | 86.25| 27.25| 177908| 458 | 28.29| 26.69| | 3 | 34.50| 21.25| 127418| 929 | 13.56| 12.53| | 4 | 35.22| 21.33| 128499| 1412| 9.20 | 10.01| | 5 | 51.75| 23.25| 149191| 1412| 12.10| 11.76| | 6 | 76.19| 26.08| 170021| 1412| 14.17| 13.59| | 7 | 105.78| 29.17| 190427| 1412| 15.55| 15.24| | 8 | 69 | 25.25| 165308| 9288| 4.59 | 4.56 | | 9 | 155.25| 35.25| 209868| 9288| 5.70 | 5.86 |

### Set III. Experiments on the Canal of Jard.

| No | Section of Canal | Border of Canal | Values of \( \sqrt{T} \) | Values of \( s \) | Observed Velocity | Calculated Velocity | |----|------------------|-----------------|--------------------------|------------------|------------------|-------------------| | 1 | 16252 | 402 | 63583 | 891 | 17.42 | 18.77 | | 2 | 11905 | 366 | 570302 | 11520 | 12.17 | 14.52 | | 3 | 10475 | 360 | 53942 | 15360 | 15.74 | 11.61 | | 4 | 7858 | 340 | 48074 | 21827 | 9.61 | 8.38 | | 5 | 7376 | 337 | 46784 | 27648 | 7.79 | 7.07 | | 6 | 6125 | 324 | 43475 | 27648 | 7.27 | 6.55 |

### Experiments on the River Haine.

| No | Section of River | Border of River | Values of \( \sqrt{T} \) | Values of \( s \) | Observed Velocity | Calculated Velocity | |----|------------------|-----------------|--------------------------|------------------|------------------|-------------------| | 1 | 31498 | 569 | 741974 | 6048 | 35.11 | 27.62 | | 2 | 38838 | 601 | 803879 | 6413 | 31.77 | 28.76 | | 3 | 39095 | 568 | 737032 | 32951 | 13.61 | 10.08 | | 4 | 39639 | 604 | 81018 | 35723 | 15.96 | 10.53 |

---

This comparison must be acknowledged to be most satisfactory, and shows the great penetration and address of the author, in so successfully fitting and appreciating the share which each co-operating circumstance has had in producing the very intricate and complicated effect. It adds some weight to the principles on which he has proceeded in this analysis of the mechanism of hydraulic motion, and must give us great confidence in a theory so fairly established on a very copious induction. The author offers it only as a rational and well-founded probability. To this character it is certainly entitled; for the suppositions made in it are agreeable to the most distinct notions we can form of these internal motions. And it must always be remembered that the investigation of the formula, although it be rendered somewhat more perspicuous by thus having recourse to those notions, has no dependence on the truth of the principles. For it is, in fact, nothing but a classification of experiments, which are grouped together by some one circumstance of slope, velocity, form of section, &c., in order to discover the law of the changes which are induced by a variation of the circumstances which do not resemble. The procedure was precisely similar to that of the astronomer when he deduces the elements of an orbit from a multitude of observations. This was the task of Mr de Buat; and he candidly and modestly informs us, that the finding out analytical forms of expression which would exhibit these changes was the work of Mr Benezech de St Honoré, a young officer of engineers, and his colleague in the experimental course. It does honour to his skill and address; and we think the whole both a pretty and instructive specimen of the method of discovering the laws of nature in the midst of complicated phenomena. Daniel Bernoulli first gave the rules of this method, and they have been greatly improved by Lambert, Condorcet, and De la Grange. Mr Coulomb has given some excellent examples of their application to the discovery of the laws of friction, of magnetical and electrical attraction, &c. But this present work is the most perspicuous and familiar of them all. It is the empirical method of generalising natural phenomena, and of deducing general rules, of which we can give no other demonstration but that they are faithful representations of matters of fact. We hope that others, encouraged by the success of Mr de Buat, will follow this example, where public utility is preferred to a display of mathematical knowledge.

Although the author may not have hit upon the precise modus operandi, we agree with him in thinking that nature seems to act in a way not unlike what is here supposed. At any rate, the range of experiments is so extensive, and so multifarious, that few cases can occur which are not included among them. The experiments will always retain their value (as we presume that they are faithfully narrated), whatever may become of the theory; and we are confident that the formula will give an answer to any question to which it may be applicable infinitely preferable to the vague guess of the most sagacious and experienced engineer.

We must however observe, that as the experiments on pipes were all made with scrupulous care in the contrivance and execution of the apparatus, excepting only those of Mr Couplet on the main pipes at Verfaillles, we may presume that the formula gives the greatest velocities which can be expected. In ordinary works, where joints are rough or leaky, where drops of foldery given by hang in the inside, where cocks intervene with defi- the formula client water-ways, where pipes have awkward bendings, contractions, or enlargements, and where they may contain sand or air, we should reckon on a smaller velocity than what results from our calculation; and we presume that an undertaker may with confidence promise of this quantity without any risk of disappointing his employer. We imagine that the actual performance of canals will be much nearer to the formula.

We have made inquiry after works of this kind executed in Britain, that we might compare them with the formula. But all our canals are locked and without motion; and we have only learned by an accidental information from Mr Watt, that a canal in his neighbourhood, which is 18 feet wide at the surface, and seven feet at the bottom, and four feet deep, and has a slope of one inch in a quarter of a mile, runs with the velocity of 17 inches per second at the surface, 10 at the bottom, and 14 in the middle. If we compute the motion of this canal by our formula, we shall find the mean velocity to be $13\frac{1}{2}$.

No river in the world has had its motions so much scrutinized as the Po about the end of the last century. It had been a subject of 100 years continual litigation between the inhabitants of the Bolognese and the Ferrarese, whether the waters of the Rheno should be thrown into the Tronco de Venezia or Po Grande. This occasioned very numerous measures to be taken of its sections and declivity, and the quantities of water which it contained in its different states of fullness. But, unfortunately, the long established methods of measuring waters, which were in force in Lombardy, made no account of the velocity, and not all the intricacies of Castelli, Grandi, and other moderns, could prevail on the visitors in this process to deviate from the established methods. We have therefore no minute accounts of its velocity, though there are many rough estimates to be met with in that valuable collection published at Florence in 1723, of the writings on the motion of rivers. From them we have extracted the only precise observations which are to be found in the whole work.

The Po Grande receives no river from Stellata to the sea, and its slope in that interval is found most surprisingly uniform, namely five inches in the mile (reduced to English measure). The breadth in its great freshes is 759 feet at Lago Scuro, with a very uniform depth of 31 feet. In its lowest state (in which it is called Po Magra), its breadth is not less than 700, and its depth about 105.

The Rheno has a uniform declivity from the Ponte Emilio to Vigaranano of 15 inches per mile. Its breadth in its greatest freshes is 189 feet, and its depth 9.

Signor Corrado in his report says, that in the state of the great freshes the velocity of the Rheno is most exactly $\frac{4}{3}$ of that of the Po.

Grandi says that a great fresh in the Rheno employs 12 hours (by many observations of his own) to come from Ponte Emilio to Vigaranano, which is 30 miles. This is a velocity of 44 inches per second. And, by Corrado's proportion, the velocity of the Po Grande must be 55 inches per second.

Montanari's observation gives the Po Magra a velocity of 31 inches per second.

Let us compare these velocities with the velocities calculated by Buat's formula.

The hydraulic mean depths $d$ and $D$ of the Rheno and Po in the great freshes deduced from the above measures, are 93.6 and 344 inches; and their slopes $s$ and $S$ are $\frac{1}{3}$ and $\frac{1}{3}$. This will give

$$\frac{307}{\sqrt{D - 0.1}} = \frac{307}{\sqrt{S + 1.6}} - 0.3(\sqrt{D - 0.1}) = 52,176 \text{ inches}$$

and

$$\frac{307}{\sqrt{s - 1.6}} - 0.3(\sqrt{d - 0.1}) = 46,727 \text{ inches}.$$

These results differ very little from the velocities above mentioned. And if the velocity corresponding to a depth of 31 feet be deduced from that observed by Montanari in the Po Magra 10 feet deep, on the supposition that they are in the proportion of $\sqrt{d}$, it will be found to be about 53$\frac{1}{2}$ inches per second.

This comparison is therefore highly to the credit of Highly to the theory, and would have been very agreeable to the credit M. de Buat, had he known it, as we hope it is to our the- readers.

We have collected many accounts of water pipes, and made the comparisons, and we flatter ourselves that these have enabled us to improve the theory. They shall appear in their proper place; and we may just observe here, that the two-inch pipe, which we formerly spoke of as conveying the water to Dunbar, should have yielded only 25$\frac{1}{2}$ Scotch pints per minute by the formula, instead of 27$\frac{1}{2}$, a small error.

We have, therefore, no hesitation in saying that this single formula of the uniform motion of water is one of the most valuable presents which natural science and the arts have received during the course of this century.

We hoped to have made this fortunate investigation of the chevalier de Buat still more acceptable to our readers by another table, which should contain the values of $\frac{307}{\sqrt{s - 1.6}}$, ready calculated for every declivity that can occur in water pipes, canals, or rivers. Aided by this, which supercedes the only difficult part of the computation, a person could calculate the velocity for any proposed case in less than two minutes. But we have not been able to get it ready for its appearance in this article, but we shall not fail to give it when we resume the subject in the article Water Works; and we hope even to give its results on a scale which may be carried in the pocket, and will enable the unlearned practitioner to solve any question with accuracy in half a minute.

We have now established in some measure a Theory of Hydraulics, by exhibiting a general theorem which expresses the relation of the chief circumstances of all such motions as have attained a state of permanency, in so far as this depends on the magnitude, form, and slope of the channel. This permanency we have expressed by the term Train, saying that the stream is in train.

We proceed to consider the subordinate circumstances contained in this theorem; such as, 1/2. The forms which nature or art may give to the bed of a running stream, and the manner of expressing this form in our theorem. 2d. The gradations of the velocity, by which it decreases in the different filaments; from the axis or most rapid filament to the border; and the connection of this with the mean velocity, which is expressed by our formula.

3d. Having acquired some distinct notions of this, we shall be able to see the manner in which undisturbed nature works in forming the beds of our rivers, the forms which she affects, and which we must imitate in all their local modifications, if we would secure that permanency which is the evident aim of all her operations. We shall here learn the mutual action of the current and its bed, and the circumstances which ensure the stability of both. These we may call the regimen or the conservation of the stream, and may say that it is in regimen, or in conservation. This has a relation, not to the dimensions and the slope alone, or to the accelerating force and the resistance arising from mere inertia; it respects immediately the tenacity of the bed, and is different from the train.

4th. These pieces of information will explain the deviation of rivers from the rectilineal course; the resistance occasioned by these deviations; and the circumstances on which the regimen of a winding stream depends.

§ 1. Of the Forms of the Channel.

The numerator of the fraction which expresses the velocity of a river in train has \( \sqrt{a} \) for one of its factors. That form, therefore, is most favourable to the motion which gives the greatest value to what we have called the hydraulic mean depth \( d \). This is the prerogative of the semicircle, and here \( d \) is equal to half the radius; and all other figures of the same area are the more favourable, as they approach nearer to a semicircle. This is the form, therefore, of all conduit pipes, and should be taken for aqueducts which are built of masonry. Ease and accuracy of execution, however, have made engineers prefer a rectangular form; but neither of these will do for a channel formed out of the ground. We shall soon see that the semicircle is incompatible with a regimen; and, if we proceed through the regular polygons, we shall find that the half hexagon is the only one which has any pretensions to a regimen; yet experience shows us, that even its banks are too steep for almost any soil. A dry earthen bank, not bound together by grass roots, will hardly stand with a slope of 45 degrees; and a canal which conveys running waters will not stand with this slope. Banks whose base is to their height as 4 to 3 will stand very well in moist soils, and this is a slope very usually given. This form is even affected in the spontaneous operations of nature, in the channels which she digs for the rills and rivulets in the higher and steeper grounds.

This form has some mathematical and mechanical properties which entitle it to some further notice. Let \( ABEC \) (fig. 11.) be such a trapezium, and \( AHGC \) the rectangle of equal width and depth. Bifect \( HB \) and \( EG \) by the verticals \( FD \) and \( KL \), and draw the verticals \( bB, eE \). Because \( AH : HB = 3 : 4 \), we have \( AB = 5 \), and \( BD = 2 \), and \( PD = 3 \), and \( BD + DF = BA \). From these premises it follows, that the trapezium \( ABEC \) has the same area with the rectangle; for \( HB \) being bifected in \( D \), the triangles \( ACF, BCD \) are equal. Also the border \( ABEC \), which is touched by the passing stream, is equal to \( FDIK \). Therefore the mean depth, which is the quotient of the area divided by the border, is the same in both; and this is the case, whatever is the width \( BE \) at the bottom, or even though there be no rectangle such as \( bBEe \) interposed between the flint sides.

Of all rectangles, that whose breadth is twice the best form height, or which is half of a square, gives the greatest of a channel mean depth. If, therefore, \( FK \) be double of \( FD \), the net trapezium \( ABEC \), which has the same area, will have the largest mean depth of any such trapezium, and will be the best form of a channel for conveying running waters. In this case, we have \( AC = 10 \), \( AH = 3 \), and \( BE = 2 \). Or we may say that the best form is a trapezium, whose bottom width is \( \frac{3}{4} \) of the depth, and whose extreme width is \( \frac{1}{2} \). This form approaches very near to that which the torrents in the hills naturally dig for themselves in uniform ground, where their action is not checked by stones which they lay bare, or which they deposit in their course. This shows us, and it will be fully confirmed by and by, that the channel of a river is not a fortuitous thing, but has a relation to the consistence of the soil and velocity of the stream.

A rectangle, whose breadth is \( \frac{3}{4} \) of the depth of water, will therefore have the same mean depth with a triangle whose surface width is \( \frac{3}{4} \) of its vertical depth; for this is the dimensions when the rectangle \( bBEe \) is taken away.

Let \( A \) be the area of the section of any channel, \( w \) its width (when rectangular), and \( b \) its depth of water. Then what we have called its mean depth, or \( d \), will be

\[ \frac{A}{w + 2b} = \frac{wb}{w + 2b}. \]

Or if \( q \) expresses the ratio of the width to the depth of a rectangular bed; that is, if \( q = \frac{w}{b} \), we have a very simple and ready expression for the mean depth, either from the width or depth. For \( d = \frac{wb}{q + 2} \), or \( d = \frac{qb}{q + 2} \).

Therefore, if the depth were infinite, and the width finite, we should have \( d = \frac{w}{2} \); or if the width be infinite, and the depth finite, we have \( d = b \). And these are the limits of the values of \( d \); and therefore, in rivers whose width is always great in comparison of the depth, we may without much error take their real depth for their hydraulic mean depth. Hence we derive a rule of easy recollection, and which will at all times give us a very near estimate of the velocity and hence of the expense of a running stream, viz., that the velocities are running nearly as the square roots of the depths. We find this confirmed by many experiments of Michelotti.

Also, when we are allowed to suppose this ratio of the velocities and depths, that is, in a rectangular canal of great breadth and small depth, we shall have the quantities discharged nearly in the proportion of the cubes of the velocities. For the quantity discharged \( d \) is as the velocity and area jointly, that is, as the height and velocity jointly, because when the width is the same the area is as the height. Therefore, we have \( d = b \cdot v^2 \). But, by the above remark, \( b = \frac{w}{v} \). Therefore, \( d = v^3 \); and this is confirmed by the experiments of Bouffet, vol. ii. 236. Also, because \( d \) is as \( v^3 \), when \( w \) is constant, and by the above remark (allowable when \( w \) is very great in proportion to \( b \)) \( v \) is as \( \sqrt[3]{b} \), we have \( d \) as \( \sqrt[3]{b} \), or \( b^{3/2} \), or the squares of the discharges. proportional to the cubes of the heights in rectangular beds, and in their corresponding trapeziums.

1. Knowing the mean depth and the proportion of finding the width and real depth, we can determine the dimensions of the bed, and we have \( w = q d + 2d \), and \( b = d + \frac{2d}{q} \).

2. If we know the area and mean depth, we can in like manner find the dimensions, that is, \( w \) and \( b \); for \( A = wb \), and \( d = \frac{lw}{w + 2b} \); therefore \( w = \frac{\sqrt{A^2 - 4d^2}}{2A} + \frac{A}{2d} \).

3. If \( d \) be known, and one of the dimensions be given, we can find the other; for \( d = \frac{wb}{w + 2b} \) gives \( w = \frac{2bd}{h - d} \) and \( b = \frac{wd}{w - 2d} \).

4. If the velocity \( V \) and the slope \( S \) for a river in train be given, we can find the mean depth; for \( V = \frac{297}{\sqrt{S - L\sqrt{S + 1,6}}} - 0.3 \) (\( \sqrt{d} - c,1 \)). Whence we deduce \( \sqrt{d} - c,1 = \frac{V}{297} - \frac{\sqrt{S - L\sqrt{S + 1,6}}}{0.3} \), and \( \sqrt{d} = \text{to this quantity} + o^{1} \).

5. We can deduce the slope which will put in train a river whose channel has given dimensions. We make \( \frac{297(\sqrt{d} - c,1)}{V + c,3(\sqrt{d} - c,1)} = \sqrt{S} \). This should be \( \sqrt{S} = \frac{L\sqrt{S + 1,6}}{c,3} \), which we correct by trials, which will be exemplified when we apply these doctrines to practice.

Having thus established the relation between the different circumstances of the form of the channel to our general formula, we proceed to consider,

§ 2. The gradations of velocity from the middle of the stream to the sides.

The knowledge of this is necessary for understanding the regimen of a river; for it is the velocity of the filaments in contact with the bed which produces any change in it, and occasions any preference of one to another, in respect of regimen or stability. Did these circumstances not operate, the water, true to the laws of hydraulics, and confined within the bounds which have been assigned them, would neither enlarge nor diminish the area of the channel. But this is all that we can promise of waters perfectly clear, running in pipes or hewn channels. But rivers, brooks, and smaller streams, carry along waters loaded with mud or sand, which they deposit wherever their velocity is checked; and they tear up, on the other hand, the materials of the channel wherever their velocity is sufficiently great. Nature, indeed, aims continually at an equilibrium, and works without ceasing to perpetuate her own performances, by establishing an equality of action and reaction, and proportioning the forms and direction of the motions to her agents, and to local circumstances. Her work is slow but unceasing; and what she cannot accomplish in a year she will do in a century. The beds of our rivers have acquired some stability, because they are the labour of ages; and it is to time that we owe those deep and wide valleys which receive and confine our rivers in channels, which are now consolidated, and with slopes which have been gradually moderated, so that they no longer either ravage our habitations or confound our boundaries. Art may imitate nature, and by directing her operations (which she still carries on according to her own imperceptible laws) according to our views, we can hasten her progress, and accomplish dreams our purpose, during the short period of human life. But we can do this only by studying the unalterable laws of mechanism. These are presented to us by spontaneous nature. Frequently we remain ignorant of their foundation: but it is not necessary for the prosperity of the subject that he have the talents of the senator; he can profit by the statute without understanding its grounds. It is so in the present instance. We have not as yet been able to infer the law of retardation observed in the filaments of a running stream from any found mechanical principle. The problem, however, does not appear beyond our powers, if we assume, with Sir Isaac Newton, that the velocity of any particular filament is the arithmetical mean between those of the filaments immediately adjoining. We may be assured, that the filament in the axis of an inclined cylindrical tube, of which the current is in train, moves the fastest, and that all those in the same circumference round it are moving with one velocity, and that the slowest are those which glide along the pipe. We may affirm the same thing of the motions in a semi-cylindrical inclined channel conveying an open stream. But even in these we have not yet demonstrated the ratio between the extreme velocities, nor in the different circles. This must be decided experimentally.

And here we are under great obligations to Mr de Buat. He has compared the velocity in the axis of a prodigious number and variety of streams, differing in size, form, slope, and velocity, and has computed in them all the mean velocity, by measuring the quantities of water discharged in a given time. His method of measuring the bottom velocity was simple and just. He threw in a gooseberry, as nearly as possible, of the same specific gravity with the water. It was carried along the bottom almost without touching it. See RESISTANCE OF FLUIDS, p. 67.

He discovered the following laws:

1. In small velocities of different proportions of fluids, the velocity in the axis is to that at the bottom in a ratio of considerable inequality.

2. This ratio diminishes as the velocity increases, and in very great velocities approaches to the ratio of equality.

3. What stream was most remarkable was, that neither the magnitude of the channel, nor its slope, had any influence in changing this proportion, while the mean velocity remained the same. Nay, though the stream ran on a channel covered with pebbles or coarse sand, no difference worth minding was to be observed from the velocity over a polished channel.

4. And if the velocity in the axis is constant, the velocity at the bottom is also constant, and is not affected by the depth of water or magnitude of the stream. In some experiments the depth was thrice the width, and in others the width was thrice the depth. This changed the proportion of the magnitude of the section. section to the magnitude of the rubbing part, but made no change on the ratio of the velocities. This is a thing which no theory could point out.

Another most important fact was also the result of his observation, viz. that the mean velocity in any pipe or open stream is the arithmetical mean between the velocity in the axis and the velocity at the sides of a pipe or bottom of an open stream. We have already observed, that the ratio of the velocity in the axis to the velocity at the bottom diminished as the mean velocity increased. This variation he was enabled to express in a very simple manner, so as to be easily remembered, and to enable us to tell any one of them by observing another.

If we take unity from the square root of the superficial velocity, expressed in inches, the square of the remainder is the velocity at the bottom; and the mean velocity is the half sum of these two. Thus, if the velocity in the middle of the stream be 25 inches per second, its square root is five; from which if we take unity, there remains four. The square of this, or 16, is the velocity at the bottom, and \( \frac{25 + 16}{2} \), or 20½, is the mean velocity.

This is a very curious and most useful piece of information. The velocity in the middle of the stream is the easiest measured of all, by any light small body floating down it; and the mean velocity is the one which regulates the train, the discharge, the effect on machines, and all the most important consequences.

We may express this by a formula of most easy recollection. Let \( V \) be the mean velocity, \( v \) the velocity in the axis, and \( u \) the velocity at the bottom; we have \( u = \sqrt{v - \frac{1}{4}} \), and \( V = \frac{v + u}{2} \).

Also \( v = (\sqrt{V - \frac{1}{4}} + \frac{1}{2})^2 \); and \( v = (\sqrt{u + \frac{1}{4}})^2 \).

\( V = (\sqrt{v - \frac{1}{4}})^2 + \frac{1}{4} \); and \( V = (\sqrt{u + \frac{1}{4}})^2 + \frac{1}{4} \).

\( u = (\sqrt{v - \frac{1}{4}})^2 \) and \( u = (\sqrt{V - \frac{1}{4}} - \frac{1}{2})^2 \).

Also \( v - u = 2 \sqrt{V - \frac{1}{4}} \) and \( v - V = V - u = \sqrt{V - \frac{1}{4}} \); that is, the difference between these velocities increases in the ratio of the square roots of the mean velocities diminished by a small constant quantity.

This may perhaps give the mathematicians some help in ascertaining the law of degradation from the axis to the sides. Thus, in a cylindrical pipe, we may conceive the current as consisting of an infinite number of cylindrical shells sliding within each other like the draw tubes of a spy-glass. Each of these is in equilibrium, or as much accelerated by the one within it as it is retarded by the one without; therefore as the momentum of each diminishes in the proportion of its diameter (the thickness being supposed the same in all), the velocity of separation must increase by a certain law from the sides to the axis. The magnitude of the small constant quantity here spoken of seems to fix this law.

The place of the mean velocity could not be determined with any precision. In moderate velocities it was not more than one-fourth or one-fifth of the depth velocity distant from the bottom. In very great velocities it was sensibly higher, but never in the middle of the depth.

The knowledge of these three velocities is of great importance. The superficial velocity is easily observed; hence the mean velocity is easily computed. This multiplied by the section gives the expense; and if we also measure the expanded border, and then obtain the mean depth (or \( \sqrt{d} \)), we can, by the formula of uniform motion, deduce the slope; or, knowing the slope, we can deduce any of the other circumstances.

The following table of these three velocities will save the trouble of calculation in one of the most frequent questions of hydraulics. The knowledge of the velocity at the bottom is of the greatest use for enabling us to judge of the action of the stream on its bed; and we shall now make some observations on this particular.

Every kind of soil has a certain velocity consistent with the stability of the channel. A greater velocity would enable the waters to tear it up, and a smaller velocity would permit the deposition of more moveable materials from above. It is not enough, then, for the stability of a river, that the accelerating forces are so adjusted to the size and figure of its channel that the current may be in train: it must also be in equilibrio with the tenacity of the channel.

We learn from observation, that a velocity of three inches per second at the bottom will just begin to work upon fine clay fit for pottery, and however firm and compact it may be, it will tear it up. Yet no beds are more stable than clay when the velocities do not exceed this: for the water soon takes away the impalpable particles of the superficial clay, leaving the particles of sand sticking by their lower half in the rest of the clay, which they now protect, making a very permanent bottom, if the stream does not bring down gravel or coarse sand, which will rub off this very thin crust, and allow another layer to be worn off; a velocity of six inches will lift fine sand; eight inches will lift sand as coarse as linseed; 12 inches will sweep along fine gravel; 24 inches will roll along rounded pebbles an inch diameter; and it requires three feet per second at the bottom to sweep along shivery angular stones of the size of an egg.

The manner in which uncared nature carries on some of these operations is curious, and deserves to be noticed a little. All must recollect the narrow ridges or wrinkles which are left on the sand by a temporary fresh or stream. They are observed to lie across the stream, and each ridge consists of a steep face AD, BF (fig. H.) which looks down the stream, and a gentler slope DB, FC, which connects this with the next ridge. As the stream comes over the first steep AD, it is directed almost perpendicularly against the point E immediately below D, and thus it gets hold of a particle of coarse sand, which it could not have detached from the reef had it been moving parallel to the surface of it. It easily rolls it up the gentle slope EB; arrived there, the particle tumbles over the ridge, and lies close at the bottom of it at F, where it is protected by the little eddy, which is formed in the very angle; other par-

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### Table of the three principal velocities

| Surface | Bottom | Mean | |---------|--------|------| | 1 | 0,000 | 0.5 | | 2 | 0,172 | 1,081| | 3 | 0,537 | 1,768| | 4 | 1 | 2.5 | | 5 | 1,526 | 3,263| | 6 | 2,1 | 4,050| | 7 | 2,709 | 4,854| | 8 | 3,342 | 5,67 | | 9 | 4 | 6.5 | | 10 | 4,674 | 7,337| | 11 | 5,369 | 8,184| | 12 | 6,071 | 9,036| | 13 | 6,786 | 9,893| | 14 | 7,513 | 10,756| | 15 | 8,254 | 11,622| | 16 | 9 | 12.5 | | 17 | 9,753 | 13,376| | 18 | 10,463 | 14,231| | 19 | 11,283 | 15,141| | 20 | 12,055 | 16,027| | 21 | 12,674 | 16,837| | 22 | 13,616 | 17,808| | 23 | 14,402 | 18,701| | 24 | 15,194 | 19,597| | 25 | 16 | 20.5 | | 26 | 16,802 | 21,401| | 27 | 17,606 | 22,393| | 28 | 18,421 | 23,210| | 29 | 19,228 | 24,114| | 30 | 20,044 | 25,022| | 31 | 20,857 | 25,924| | 32 | 21,678 | 26,839| | 33 | 22,506 | 27,753|

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### Table of the three principal velocities

| Surface | Bottom | Mean | |---------|--------|------| | 34 | 23,339 | 28,660| | 35 | 24,107 | 29,583| | 36 | 25 | 30.5 | | 37 | 25,827 | 31,413| | 38 | 26,667 | 32,333| | 39 | 27,51 | 33,255| | 40 | 28,345 | 34,172| | 41 | 29,192 | 35,096| | 42 | 30,030 | 36,015| | 43 | 30,880 | 36,940| | 44 | 31,742 | 37,871| | 45 | 32,581 | 38,790| | 46 | 33,432 | 39,716| | 47 | 34,293 | 40,646| | 48 | 35,151 | 41,570| | 49 | 36 | 42.5 | | 50 | 36,857 | 43,428| | 51 | 37,712 | 44,356| | 52 | 38,564 | 45,282| | 53 | 39,433 | 46,219| | 54 | 40,284 | 47,142| | 55 | 41,165 | 48,082| | 56 | 42,016 | 49,008| | 57 | 42,968 | 49,984| | 58 | 43,971 | 50,886| | 59 | 44,936 | 51,818| | 60 | 45,909 | 52,754| | 61 | 46,876 | 53,688| | 62 | 47,849 | 54,629| | 63 | 48,822 | 55,568| | 64 | 49 | 56.5 | | 65 | 49,782 | 57,439| | 66 | 50,751 | 58,376|

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### Table of the three principal velocities

| Surface | Bottom | Mean | |---------|--------|------| | 67 | 51,639 | 59,319| | 68 | 52,505 | 60,252| | 69 | 53,392 | 61,196| | 70 | 54,273 | 62,136| | 71 | 55,154 | 63,072| | 72 | 56,025 | 64,012| | 73 | 56,862 | 64,952| | 74 | 57,700 | 65,895| | 75 | 58,567 | 66,843| | 76 | 59,458 | 67,784| | 77 | 60,451 | 68,725| | 78 | 61,342 | 69,670| | 79 | 62,269 | 70,605| | 80 | 63,107 | 71,533| | 81 | 64 | 72.5 | | 82 | 64,883 | 73,441| | 83 | 65,780 | 74,390| | 84 | 66,651 | 75,325| | 85 | 67,568 | 76,284| | 86 | 68,459 | 77,229| | 87 | 69,339 | 78,169| | 88 | 70,224 | 79,112| | 89 | 71,132 | 80,066| | 90 | 72,012 | 81,006| | 91 | 72,915 | 81,957| | 92 | 73,788 | 82,894| | 93 | 74,719 | 83,839| | 94 | 75,603 | 84,801| | 95 | 76,451 | 85,755| | 96 | 77,370 | 86,685| | 97 | 78,305 | 87,622| | 98 | 79,192 | 88,596| | 99 | 80,120 | 89,565| | 100 | 81 | 90.5 | ticles lying about E are treated in the same way, and, tumbling over the ridge B, cover the first particle, and now protect it effectually from any further disturbance. The same operation is going on at the bottom of each ridge. The brow or steep of the ridge gradually advances down the stream, and the whole set change their places, as represented by the dotted line abdf; and after a certain time the particle which was deposited at F is found in an unprotected situation, as it was in E, and it now makes another step down the stream.

The Abbé Bossut found, that when the velocity of the stream was just sufficient for lifting the sand (and a small excess hindered this operation altogether) a ridge advanced about 20 feet in a day.

Since the current carries off the most moveable matters of the channel, it leaves the bottom covered with the remaining coarser sand, gravel, pebbles, and larger stones. To these are added many which come down the stream while it is more rapid, and also many which roll in from the sides as the banks wear away. All these form a bottom much more solid and immovable than a bottom of the medium soil would have been. But this does not always maintain the channel in a permanent form; but frequently occasions great changes, by obliging the current, in the event of any sudden fresh or swell, to enlarge its bed, and even to change it altogether, by working to the right and to the left, since it cannot work downwards. It is generally from such accumulation of gravel and pebbles in the bottom of the bed that rivers change their channels.

It remains to ascertain, in absolute measures, the force which a current really exerts in attempting to drag along with it the materials of its channel; and which will produce this effect unless resisted by the inertia of these materials. It is therefore of practical importance to know this force.

Nor is it arbitrary or difficult. For when a current is in train, the accelerating force is in equilibrium with the resistance, and is therefore its immediate measure. Now this accelerating force is precisely equal to the weight of the body of water in motion multiplied by the fraction which expresses the slope. The mean depth being equal to the quotient of the section divided by the border, the section is equal to the product of the mean depth multiplied by the border. Therefore, calling the border b, and the mean depth d, we have the section = db. The body of water in motion is therefore db s (because s was the slant length of a part whose difference of elevation is 1), and the accelerating force is db s × \frac{1}{s}, or db. But if we would only consider this resistance as corresponding to an unit of the length of the channel, we must divide the quantity db by s, and the resistance is then \frac{db}{s}. And if we would consider the resistance only for an unit of the border, we must divide this expression by b; and thus this resistance (taking an inch for the unit) will be expressed for one square inch of the bed by the weight of a bulk of water which has a square inch for its base, and \frac{d}{s} for its height. And lastly, if E be taken for any given superficial extent of the channel or bed, and F the obstruction which we consider as a sort of friction, we shall have \(F = \frac{Ed}{s}\).

Thus, let it be required to determine in pounds the resistance or friction on a square yard of a channel whose current is in train, which is 10 feet wide, four feet deep, and has a slope of one foot in a mile. Here E is nine feet. Ten feet width and four feet depth give a section of 40 feet. The border is 18 feet. Therefore \(d = \frac{40}{18} = 2\frac{11}{18}\), and \(s = 5280\). Therefore the friction is the weight of a column of water whose base is nine feet, and height \(\frac{2\frac{11}{18}}{5280}\), or nearly 3\(\frac{6}{7}\) ounces avoirdupois.

§ 3. Settlement of the Beds of Rivers.

He who looks with a careless eye at a map of the world, is apt to consider the rivers which ramble over its surface as a chance-medley disposition of the drainers which carry off the waters. But it will afford a most agreeable object to a considerate and contemplative mind, to take it up in this very simple light; and having considered the many ways in which the drenched surface might have been cleared of the superfluous waters, to attend particularly to the very way which nature has followed. In following the troubled waters of a mountain torrent, or the pure streams which trickle from their basins, till he sees them swallowed up in the ocean, and in attending to the many varieties in their motions, he will be delighted with observing how the simple laws of mechanism are made so fruitful in good consequences, both by modifying the motions of the waters themselves, and also by inducing new forms on the surface of the earth, fitted for re-acting on the waters, and producing these very modifications of their motions which render them so beneficial. The permanent beds of rivers are by no means fortuitous gutters hastily scooped out by dashing torrents; but both they and the valleys through which they flow are the patient but unceasing labours of nature, prompted by goodness and directed by wisdom.

Whether we trace a river from the torrents which collect the superfluous waters of heaven, or from the springs which discharge what would otherwise be condemned to perpetual inactivity, each feeder is but a little rill which could not ramble far from its scanty source among growing plants and absorbent earth, without being sucked up and evaporated, did it not meet with other rills in its course. When united they form a body of water still inconsiderable, but much more able, by its bulk, to overcome the little obstructions to its motion; and the rivulet then moves with greater speed, as we have now learned. At the same time, the surface exposed to evaporation and absorption is diminished by the union of the rills. Four equal rills have only the surface of two when united. Thus the portion which escapes arrestment, and travels downward, is continually increasing. This is a happy adjustment to the other operations of nature. Were it otherwise, the lower and more valuable countries would be loaded with the passing waters in addition to their own surplus rains, and the immediate neighbourhood of the sea would be almost covered by the drains of the interior countries. countries. But, fortunately, those passing waters occupy less room as they advance, and by this wise employment of the most simple means, not only are the superfluous waters drained off from our fertile fields, but the drains themselves become an useful part of the country by their magnitude. They become the habitation of a prodigious number of fishes, which share the Creator's bounty; and they become the means of mutual communication of all the blessings of cultivated society. The vague ramblings of the rivers scatter them over the face of the country, and bring them to every door. It is not even an indifferent circumstance, that they gather strength to cut out deep beds for themselves. By this means they cut open many springs. Without this, the produce of a heavy shower would make a swamp which would not dry up in many days. And it must be observed, that the same heat which is necessary for the vigorous growth of useful plants will produce a very copious evaporation. This must return in showers much too copious for immediate vegetation, and the overplus would be destructive. Is it not pleasant to contemplate this adjustment of the great operations of nature, so different from each other, that if chance alone directed the detail, it was almost an infinite odds that the earth would be uninhabitable?

But let us follow the waters in their operations, and note the face of the countries through which they flow; attending to the breadth, the depth, and the slope of the valleys, we shall be convinced that their present situation is extremely different from what it was in ancient days; and that the valleys themselves are the works of the rivers, or at least of waters which have descended from the heights, loaded with all the lighter matters which they were able to bring away with them. The rivers flow now in beds which have a considerable permanency; but this has been the work of ages. This has given stability, both by filling up and smoothing the valleys, and thus lessening the changing causes, and also by hardening the beds themselves, which are now covered with aquatic plants, and lined with the stones, gravel, and coarser sand, out of which all the lighter matters have been washed away.

The surface of the high grounds is undergoing a continual change; and the ground on which we now walk is by no means the same which was trodden by our remote ancestors. The flowers from heaven carry down into the valleys, or sweep along by the torrents, a part of the soil which covers the heights and steeps. The torrents carry this soil into the brooks, and these deliver part of it into the great rivers, and these discharge into the sea this fertilizing fat of the earth, where it is swallowed up, and forever lost for the purposes of vegetation. Thus the hillocks lose of their height, the valleys are filled up, and the mountains are laid bare, and show their naked precipices, which formerly were covered over with a flesh and skin, but now look like the skeleton of this globe. The low countries, raised and nourished for some time by the substance of the high lands, will go in their turn to be buried in the ocean; and then the earth, reduced to a dreary flat, will become an immense uninhabitable mass. This catastrophe is far distant, because this globe is in its youth, but it is not the least certain; and the united labours of the human race could not long protract the term.

But, in the mean time, we can trace a beneficent purpose, and a nice adjustment of seemingly remote circumstances. The grounds near the sources of all our rivers are indeed gradually stripped of their most fertile ingredients. But had they retained them for ages, the displayed sentient inhabitants of the earth, or at least the nobler in animals, with man at their head, would not have derived much advantage from it. The general laws of nature produce changes in our atmosphere which must ever render these great elevations unfruitful. That general warmth, which is equally necessary for the useful plant as for the animal which lives on it, is confined to the lower grounds. The earth, which on the top of mount Hæmus could only bring forth mosses and dittany, when brought into the gardens of Spalatro, produced pot-herbs so luxuriant, that Diocletian told his colleague Maximian that he had more pleasure in their cultivation than the Roman empire could confer. Thus nature not only provides us manure, but conveys it to our fields. She even keeps it safe in store for us till it shall be wanted. The tracts of country which are but newly inhabited by man, such as great part of America, and the newly discovered regions of Terra Australis, are still almost occupied by marshes and lakes, or covered with impenetrable forests; and they would remain long enough in this state, if population, continually increasing, did not increase industry, and multiply the hands of cultivators along with their necessities. The Author of Nature was alone able to form the huge ridges of the mountains, to model the hillocks and the valleys, to mark out the courses of the great rivers, and give the first trace to every rivulet; but has left to man the task of draining his own habitation and the fields which are to support him, because this is a task not beyond his powers. It was therefore of immense advantage to him that those parts of the globe into which he has not yet penetrated should remain covered with lakes, marshes, and forests, which keep in store the juice of the earth, which the influence of the air and the vivifying warmth of the sun would have expended long ere now in useless vegetation, and which the rains of heaven would have swept into the sea, had they not been, thus protected by their situation or their cover. It is therefore the business of man to open up these mines of hoarded wealth, and to thank the Author of all good, who has thus husbanded them for his use, and left them as a rightful heritage for those of after days.

The earth had not in the remote ages, as in our day, those great canals, those capacious voiders, always ready to drain off the rain waters (of which only part is absorbed by the thirsty ground), and the pure waters of the springs from the foot of the hills. The rivers did not then exist, or were only torrents, whose waters, confined by the gullies and glens, are searching for a place to escape. Hence arise those numerous lakes in the interior of great continents, of which there are still remarkable relics in North America, which in process of time will disappear, and become champaign countries. The most remote from the sea, unable to contain its waters, finds an issue through some gorge of the hills, and pours over its superfluous waters into a lower basin, which, in its turn, discharges its contents into another, and the last of the chain delivers its waters by a river into the ocean. The communication was originally begun by a simple overflowing at the lowest part of the margin. This made a torrent, which quickly quickly deepened its bed; and this circumstance increasing its velocity, as we have seen, would extend this deepening backward to the lake, and draw off more of its waters. The work would go on rapidly at first, while earth and small stones only resisted the labours of nature; but these being washed away, and the channel hollowed out to the firm rock on all sides, the operation must go on very slowly, till the immense cascade shall undermine what it cannot break off, and then a new discharge will commence, and a quantity of flat ground will emerge all round the lake. The torrent, in the mean time, makes its way down the country, and digs a canal, which may be called the first sketch of a river, which will deepen and widen its bed continually. The waters of several basins united, and running together in a great body, will (according to the principles we have established) have a much greater velocity, with the same slope, than those of the lakes in the interior parts of the continent; and the sum of them all united in the basin next the sea, after having broken through its natural mound, will make a prodigious torrent, which will dig for itself a bed so much the deeper as it has more slope and a greater body of waters.

The formation of the first valleys, by cutting open many springs which were formerly concealed underground, will add to the mass of running waters, and contribute to drain off the waters of their basins. In course of time many of them will disappear, and flat valleys among the mountains and hills are the traces of their former existence.

When nature thus traces out the courses of future rivers, it is to be expected that those streams will most deepen their channels which in their approach to the sea receive into their bed the greatest quantities of rain and spring waters, and that towards the middle of the continent they will deepen their channels less. In these last situations the natural slope of the fields causes the rain-water, rills, and the little rivulets from the springs, to seek their way to the rivers. The ground can sink only by the flattening of the hills and high grounds; and this must proceed with extreme slowness, because it is only the gentle, though incessant, work of the rains and springs. But the rivers, increasing in bulk and strength, and of necessity flowing over everything, form to themselves capacious beds in a more yielding soil, and dig them even to the level of the ocean.

The beds of rivers by no means form themselves in one inclined plane. If we should suppose a canal AB (fig. 12.) perfectly straight and horizontal at B, where it joins with the sea, this canal would really be an inclined channel of greater and greater slope as it is farther from B. This is evident; because gravity is directed towards the centre of the earth, and the angle CAB contained between the channel and the plumb-line at A is smaller than the similar angle CDB; and consequently the inclination to the horizon is greater in A than in D. Such a canal therefore would make the bed of a river; and some have thought that this was the real form of nature's work; but the supposition is a whim, and it is false. No river has a slope at all approaching to this. It would be 8 inches declivity in the mile next the ocean, 24 inches in the second mile, 40 inches in the third, and so on in the duplicate ratio (for the whole elevation) of the distances from the sea. Such a river would quickly tear up its bed in the mountains (were there any grounds high enough to receive it), and, except its first cascade, would soon acquire a more gentle slope. But the fact is, and it is the result of the imperceptible laws of nature, that the continued track of a river is a succession of inclined channels, whose slope diminishes by steps as the river approaches to the sea. It is not enough to say that this results from the natural slope of the countries through which it flows, which we observe to increase in declivity as we go to the interior parts of the continent. Were it otherwise, the equilibrium to which nature aims in all her operations would still produce the gradual diminution of the slope of rivers. Without it they could not be in a permanent train.

That we may more easily form a notion of the manner in which the permanent course of a river is established, let us suppose a stream or rivulet s a (fig. 13.) course of a river far up the country, make its way through a foil perfectly uniform to the sea, taking the course s a b c d e f, and receiving the permanent additions of the streams g a, b b, i c, k d, l e, and that its velocity and slope in all its parts are so suited to the tenacity of the soil and magnitude of its section, that neither do its waters during the annual freshes tear up its banks or deepen its bed, nor do they bring down from the high lands materials which they deposit in the channel in times of smaller velocity. Such a river may be said to be in a permanent state, to be in conservation, or to have stability. Let us call this state of a river its regimen, denoting by the word the proper adjustment of the velocity of the stream to the tenacity of the channel. The velocity of its regimen must be the same throughout, because it is this which regulates its action on the bottom, which is the same from its head to the sea. That its bed may have stability, the mean velocity of the current must be constant, notwithstanding the inequality of discharge through its different sections by the brooks which it receives in its course, and notwithstanding the augmentation of its section as it approaches the sea.

On the other hand, it behoved this exact regimen to commence at the mouth of the river, by the working of the whole body of the river, in concert with the waters of the ocean, which always keep within the same limits, and make the ultimate level invariable. This working will begin to dig the bed, giving it as little breadth as possible; for this working consists chiefly in the efforts of falls and rapid streams, which arise of themselves in every channel which has too much slope. The bottom deepens, and the sides remain very steep, till they are undermined and crumble down; and being then diluted in the water, they are carried down the stream, and deposited where the ocean checks its speed. The banks crumble down anew, the valley or hollow forms; but the section, always confined to its bottom, cannot acquire a great breadth, and it retains a good deal of the form of the trapezium formerly mentioned. In this manner does the regimen begin to be established from f to e.

With respect to the next part d e, the discharge or produce is diminished by the want of the brook l e. It must take a similar form, but its area will be diminished, in order that its velocity may be the same; and its mean depth d being less than in the portion ef below, the slope must be greater. Without these conditions we could not have the uniform velocity, which the assumed permanency permanency in an uniform soil necessarily supposes.

Reasoning after the same manner for all the portions \(a, b, c, d, e, f\), we see that the regimen will be successively established in them; and that the slope necessary for this purpose will be greater as we approach the river head. The vertical section or profile of the course of the river \(a b c d e f\) will therefore resemble the line SABCDEF which is sketched below, having its different parts variously inclined to the horizontal line HIF.

Such is the process of nature to be observed in every river on the surface of the globe. It long appeared a kind of puzzle to the theorists; and it was this observation of the increasing, or at least this continued velocity with smaller slope, as the rivers increased by the addition of their tributary streams, which caused Guigglini to have recourse to his new principle, the energy of deep waters. We have now seen in what this energy consists. It is only a greater quantity of motion remaining in the middle of a great stream of water after a quantity has been retarded by the sides and bottom; and we see clearly, that since the addition of a new and perhaps an equal stream does not occupy a bed of double surface, the proportion of the retardations to the remaining motion must continually diminish as a river increases by the addition of new streams. If therefore the slope were not diminished, the regimen would be destroyed, and the river would dig up its channel. We have a full confirmation of this in the many works which have been executed on the Po, which runs with rapidity through a rich and yielding soil. About the year 1600, the waters of the Panaro, a very considerable river, were added to the Po Grande; and although it brings along with it in its freshes a vast quantity of sand and mud, it has greatly deepened the whole Tronco di Venezia from the confluence to the sea. This point was clearly ascertained by Manfredi about the 1720, when the inhabitants of the valleys adjacent were alarmed by the project of bringing in the waters of the Reno, which then ran through the Ferrarese. Their fears were overcome, and the Po Grande continues to deepen its channel every day with a prodigious advantage to the navigations; and there are several extensive marshes which now drain off by it; after having been for ages under water: and it is to be particularly remarked, that the Reno is the foulest river in its freshes of any in that country. We insert this remark, because it may be of great practical utility, as pointing out a method of preserving and even improving the depth of rivers or drains in flat countries, which is not obvious, and rather appears improper: but it is strictly conformable to a true theory, and to the operations of nature, which never fails to adjust everything so as to bring about an equilibrium. Whatever the declivity of the country may have been originally, the regimen begins to be settled at the mouths of the rivers, and the slopes are diminished in succession as we recede from the coast. The original slopes inland may have been much greater; but they will (when busy nature has completed her work) be left somewhat, and only so much greater, that the velocity may be the same notwithstanding the diminution of the section and mean depth.

Freshes will disturb this methodical progress relative only to the successive permanent additions; but their effects chiefly accelerate the deepening of the bed, and the diminution of the slope, by augmenting the velocity during their continuance. But when the regimen of the permanent additions is once established, the freshes tend chiefly to widen the bed, without greatly deepening it: for the aquatic plants, which have been growing and thriving during the peaceable state of the river, are now laid along, but not swept away, by the freshes and protect the bottom from their attacks; and the stones and gravel, which must have been left bare in a course of years, working on the soil, will also collect in the bottom, and greatly augment its power of resistance; and even if the floods should have deepened the bottom some small matter, some mud will be deposited as the velocity of the freshes diminishes, and this will remain till the next flood.

We have supposed the soil uniform through the whole course: This seldom happens; therefore the circumstances which insure permanency, or the regimen of a river, may be very different in its different parts and in different rivers. We may say in general, that the farther that the regimen has advanced up the stream in any river, the more slowly will it convey its waters to the sea.

There are some general circumstances in the motion of rivers which it will be proper to take notice of just now, that they may not interrupt our more minute examination of their mechanism, and their explanations will then occur of themselves as corollaries of the propositions which we shall endeavour to demonstrate.

In a valley of small width the river always occupies the lowest part of it; and it is observed, that this is seldom in the middle of the valley, and is nearest to that side on which the slope from the higher grounds is steepest, and this without regard to the line of its course. The river generally adheres to the steepest hills, whether they advance into the plain or retire from it. This general feature may be observed over the whole globe. It is divided into compartments by great ranges of mountains; and it may be observed, that the great rivers hold their course not very far from them, and that their chief feeders come from the other side. In every compartment there is a swell of the low country at a distance from the bounding ridge of mountains; and on the summit of this swell the principal feeders of the great river have their sources.

The name valley is given with less propriety to these immense regions, and is more applicable to tracks of champaign land which the eye can take in at one view. Even here we may observe a resemblance. It is not always in the very lowest part of this valley that the river has its bed; although the waters of the river flow in a channel below its immediate banks, these banks are frequently higher than the grounds at the foot of the hills. This is very distinctly seen in Lower Egypt, by means of the canals which are carried backward from the Nile for accelerating its fertilizing inundations. When the canals are opened to admit the waters, it is always observed that the districts most remote are the first covered, and it is several days before the immediately adjoining fields partake of the blesting. This is a consequence of that general operation of nature by which the valleys are formed. The river in its floods is loaded with mud, which it retains as long as it rolls rapidly along its limited bed, tumbling its waters over and over, and taking up in every spot as much as Theory. it deposits; but as soon as it overflows its banks, the very enlargement of its section diminishes the velocity of the water; and it may be observed still running in the track of its bed with great velocity, while the waters on each side are stagnant at a very small distance: Therefore the water, on getting over the banks, must deposit the heaviest, the firmest, and even the greatest part of its burden, and must become gradually clearer as it approaches the hills. Thus a gentle slope is given as the valley in a direction which is the reverse of what one would expect. It is, however, almost always the case in wide valleys, especially if the great river comes through a soft country. The banks of the brooks and ditches are observed to be deeper as they approach the river, and the merely superficial drains run backwards from it.

We have already observed, that the enlargement of the bed of a river, in its approach to the sea, is not in proportion to the increase of its waters. This would be the case even if the velocity continued the same; and therefore, since the velocity even increases, in consequence of the greater energy of a large body of water, which we now understand distinctly, a still smaller bed is sufficient for conveying all the water to the sea.

This general law is broken, however, in the immediate neighbourhood of the sea; because in this situation the velocity of the water is checked by the pausing flood-tides of the ocean. As the whole waters must still be discharged, they require a larger bed, and the enlargement will be chiefly in width. The sand and mud are deposited when the motion is retarded. The depth of the mouth of the channel is therefore diminished. It must therefore become wider. If this be done on a coast exposed to the force of a regular tide, which carries the waters of the ocean across the mouth of the river, this regular enlargement of the mouth will be the only consequence, and it will generally widen till it washes the foot of the adjoining hills; but if there be no tide in the sea, or a tide which does not set across the mouth of the river, the sands must be deposited at the sides of the opening, and become additions to the shore, lengthening the mouth of the channel. In this sheltered situation, every trivial circumstance will cause the river to work more on particular parts of the bottom, and deepen the channel there. This keeps the mud suspended in such parts of the channel, and it is not deposited till the stream has shot farther out into the sea. It is deposited on the sides of those deeper parts of the channel, and increases the velocity in them, and thus still farther protracts the deposition. Rivers so situated will not only lengthen their channels, but will divide them, and produce islands at their mouths. A bush, a tree torn up by the roots by a mountain torrent, and floated down the stream, will thus inevitably produce an island; and rivers in which this is common will be continually shifting their mouths. The Mississippi is a most remarkable instance of this. It has a long course through a rich soil, and disembogues itself into the bay of Mexico, in a place where there is no pausing tide, as may be seen by comparing the hours of high water in different places. No river that we know carries down its stream such numbers of rooted-up trees: they frequently interrupt the navigation, and render it always dangerous in the night-time. This river is so beset with flats and shifting sands at its mouth, that the most experienced pilots are puzzled; and it has protruded its channel above 50 miles in the short period that we have known it. The discharge of the Danube is very similar: so is that of the Nile; for it is discharged into a still corner of the Mediterranean. It may now be said to have acquired considerable permanency; but much of this is owing to human industry, which strips it as much as possible of its subduable matter. The Ganges too is in a situation pretty similar, and exhibits similar phenomena. The Maragnon might be noticed as an exception; but it is not an exception. It has flowed very far in a level bed, and its waters come pretty clear to Para; but besides, there is a strong transverse tide, or rather current, at its mouth, setting to the south-east both during flood and ebb. The mouth of the Po is perhaps the most remarkable of any on the surface of this globe, and exhibits appearances extremely singular. Its discharge is into a sequestered corner of the Adriatic. Though there be a more remarkable tide in this gulf than in any part of the Mediterranean, it is still but trifling, and it either sets directly upon the mouth of the river, or retires straight away from it. The river has many mouths, and they shift prodigiously. There has been a general increase of the land very remarkable. The marshes where Venice now stands were, in the Augustan age, everywhere penetrable by the fishing boats, and in the 5th century could only bear a few miserable huts; now they are covered with crowds of stately buildings. Ravenna, situated on the southernmost mouth of the Po, was, in the Augustan age, at the extremity of a swamp, and the road to it was along the top of an artificial mound made by Augustus at immense expense. It was, however, a fine city, containing extensive docks, arsenals, and other lofty buildings, being the great military port of the empire, where Augustus laid up his great ships of war. In the Gothic times it became almost the capital of the Western empire, and was the seat of government and of luxury. It must, therefore, be supposed to have every accommodation of opulence, and we cannot doubt of its having paved streets, wharfs, &c.; so that its wealthy inhabitants were at least walking dryfooted from house to house. But now it is an Italian mile from the sea, and surrounded with vineyards and cultivated fields, and is accessible in every direction. All this must have been formed by depositions from the Po, flowing through Lombardy loaded with the spoils of the Alps, which were here arrested by the reeds and bulrushes of the marsh. These things are in common course; but when wells are dug, we come to the pavements of the ancient city, and these pavements are all on one exact level, and they are eight feet below the surface of the sea at low water. This cannot be ascribed to the subduing of the ancient city. This would be irregular, and greatest among the heavy buildings. The tomb of Theodoric remains, and the pavement round it is on a level with all the others. The lower story is always full of water; so is the lower story of the cathedral to the depth of three feet. The ornaments of both these buildings leave no room to doubt that they were formerly dry; and such a building as the cathedral could not sink without crumbling into pieces.

It is by no means easy to account for all this. The depositions... depositions of the Po and other rivers must raise the ground; and yet the rivers must still flow over all. We must conclude that the surface of the Adriatic is by no means level, and that it slopes like a river from the Lagoon of Venice to the eastward. In all probability it even slopes considerably outwards from the shore. This will not hinder the alternations of ebb and flood tide, as will be shown in its proper place. The whole shores of this gulph exhibit most uncommon appearances.

The last general observations which we shall make in this place is, that the surface of a river is not flat, considered athwart the stream, but convex: this is owing to its motion. Suppose a canal of stagnant water; its surface would be a perfect level. But suppose it possible by any means to give the middle waters a motion in the direction of its length, they must drag along with them the waters immediately contiguous. These will move less swiftly, and will in like manner drag the waters without them; and thus the water at the sides being abraded, the depth must be less, and the general surface must be convex across. The fact in a running stream is similar to this; the side waters are withheld by the sides, and every filament is moving more slowly than the one next it towards the middle of the river, but faster than the adjoining filament on the land side. This alone must produce a convexity of surface. But besides this, it is demonstrable that the pressure of a running stream is diminished by its motion, and the diminution is proportional to the height which would produce the velocity with which it is gliding past the adjoining filament. This convexity must in all cases be very small. Few rivers have the velocity nearly equal to eight feet per second, and this requires a height of one foot only. An author quoted by Mr Buffon says, that he has observed on the river Aveyron an elevation of three feet in the middle during floods; but we suspect some error in the observation.

§ 4. Of the Windings of Rivers.

Rivers are seldom straight in their course. Formed by the hand of nature, they are accommodated to every change of circumstance. They wind around what they cannot get over, and work their way to either side according as the resistance of the opposite bank makes a straight course more difficult; and this seemingly fortuitous rambling distributes them more uniformly over the surface of a country, and makes them everywhere more at hand, to receive the numberless rills and rivulets which collect the waters of our springs and the superfluities of our showers, and to comfort our habitations with the many advantages which cultivation and society can derive from their presence. In their feeble beginnings the smallest inequality of slope or consistence is enough to turn them aside and make them ramble through every field, giving drink to our herds and fertility to our soil. The more we follow nature into the minutiae of her operations, the more must we admire the inexhaustible fertility of her resources, and the simplicity of the means by which she produces the most important and beneficial effects. By thus twisting the course of our rivers into 10,000 shapes, she keeps them long amidst our fields, and thus compensates for the declivity of the surface, which otherwise would tumble them with great rapidity into the ocean, loaded with the best and richest of our soil. Without this, the showers of heaven would have little influence in supplying the waste of incessant evaporation. But as things are, the rains are kept slowly trickling along the sloping sides of our hills and steeps, winding round every clod, nay every plant, which lengthens their course, diminishes their slope, checks their speed, and thus prevents them from quickly brushing off from every part of the surface the lightest and best of the soil. The flatter of our holm lands would be too steep, and the rivers would shoot along through our finest meadows, hurrying everything away with them, and would be unfit for the purposes of inland conveyance, if the inequalities of soil did not make them change this headlong course for the more beautiful meanders which we observe in the course of the small rivers winding through our meadows. Those rivers are in general the straightest in their course which are the most rapid, and which roll along the greatest bodies of water; such are the Rhone, the Po, the Danube. The smaller rivers continue more devious in their progress, till they approach the sea, and have gathered strength from all their tributary streams.

Every thing aims at an equilibrium, and this directs what even the ramblings of rivers. It is of importance to understand the relation between the force of a river and for man to resist the resistance which the soil opposes to those deviations performed from a rectilineal course; for it may frequently happen that the general procedure of nature may be inconsistent with our local purposes. Man was set down on this globe, and the task of cultivating it was given him by nature, and his chief enjoyment seems to be to struggle with the elements. He must not find things to his mind, but he must mould them to his own fancy. Yet even this seeming anomaly is one of nature's most beneficent laws; and his exertions must still be made in conformity with the general train of the operations of mechanical nature: and when we have any work to undertake relative to the course of rivers, we must be careful not to thwart their general rules, otherwise we shall be sooner or later punished for their infraction. Things will be brought back to their former state, if our operations are inconsistent with that equilibrium which is constantly aimed at, or some new state of things which is equivalent will be soon induced. If a well regulated river has been improperly deepened in some place, to answer some particular purpose of our own, or if its breadth has been improperly augmented, we shall soon see a deposition of mud or sand choke up our fancied improvements; because, as we have enlarged the section without increasing the slope or the supply, the velocity must diminish, and floating matters must be deposited.

It is true, we frequently see permanent channels where the forms are extremely different from that which the waters would dig for themselves in an uniform foil, and which approaches a good deal to the trapezium described formerly. We see a greater breadth frequently compensate for a want of depth; but all such deviations are a sort of constraint, or rather are indications of inequality of soil. Such irregular forms are the works of nature; and if they are permanent, the equilibrium is obtained. Commonly the bottom is harder than the sides, consisting of the coarsest of the sand and of gravel; and therefore the necessary section can be obtained only by increasing the width. We are are accustomed to attend chiefly to the appearances which prognosticate mischief, and we interpret the appearances of a permanent bed in the same way, and frequently form very false judgments. When we see one bank low and flat, and the other high and abrupt, we suppose that the waters are paffing along the first in peace, and with a gentle stream, but that they are rapid on the other side, and are tearing away the bank; but it is just the contrary. The bed being permanent, things are in equilibrium, and each bank is of a form just competent to that equilibrium. If the foil on both sides be uniform, the stream is most rapid on that side where the bank is low and flat, for in no other form would it withstand the action of the stream; and it has been worn away till its flatness compensates for the greater force of the stream. The stream on the other side must be more gentle, otherwise the bank could not remain abrupt. In short, in a state of permanency, the velocity of the stream and form of the bank are just suited to each other. It is quite otherwise before the river has acquired its proper regimen.

A careful consideration therefore of the general features of rivers which have settled their regimen, is of use for informing us concerning their internal motions, and directing us to the most effectual methods of regulating their course.

We have already said that perpendicular brims are inconsistent with stability. A femicircular section is the form which would produce the quickest train of a river whose expense and slope are given; but the banks at B and D (fig. 14.) would crumble in, and lie at the bottom, where their horizontal surface would secure them from farther change. The bed will acquire the form GcF, of equal section, but greater width, and with brims less inclining. The proportion of the velocities at A and c may be the same with that of the velocities at A and C; but the velocity at G and F will be less than it was formerly at B, C, or D; and the velocity in any intermediate point E, being somewhat between those at F and c, must be less than it was in any intermediate point of the femicircular bed. The velocities will therefore decrease along the border from c towards G and F, and the steepness of the border will augment at the same time, till, in every point of the new border GcF, these two circumstances will be so adjusted that the necessary equilibrium is established.

The same thing must happen in our trapezium. The slope of the brims may be exact, and will be retained; it will, however, be too great anywhere below, where the velocity is greater, and the sides will be worn away till the banks are undermined and crumble down, and the river will maintain its section by increasing its width. In short, no border made up of straight lines is consistent with that gradation of velocity which will take place whenever we depart from a femicircular form. And we accordingly see, that in all natural channels the section has a curvilinear border, with the slope increasing gradually from the bottom to the brim.

These observations will enable us to understand how nature operates when the inequality of surface or of tenacity obliges the current to change its direction, and the river forms an elbow.

Supposing always that the discharge continues the same, and that the mean velocity is either preserved or restored, the following conditions are necessary for a permanent regimen.

1. The depth of water must be greater in the elbow than anywhere else.

2. The main stream, after having struck the concave for a permanent regimen, must be reflected in an equal angle, and must then be in the direction of the next reach of the river.

3. The angle of incidence must be proportioned to the tenacity of the soil.

4. There must be in the elbow an increase of slope, or head of water, capable of overcoming the resistance occasioned by the elbow.

The reasonableness, at least, of these conditions will appear from the following considerations.

1. It is certain that force is expended in producing this change of direction in a channel which by supposition diminishes the current. The diminution arising from any cause which can be compared with friction must be greater when the stream is directed against one of the banks. It may be very difficult to state the proportion, and it would occupy too much of our time to attempt it; but it is sufficient that we be convinced that the retardation is greater in this case. We see no cause to increase the mean velocity in the elbow, and we must therefore conclude that it is diminished. But we are supposing that the discharge continues the same; the section must therefore augment, or the channel increase its transverse dimensions. The only question is, in what manner it does this, and what change of form does it affect, and what form is competent to the final equilibrium and the consequent permanency of the bed? Here there is much room for conjecture. Mr Buat reasons as follows. If we suppose that the points B and C (fig. 15.) continue on a level, and that the points H and I at the beginning of the next reach are also on a level, it is an inevitable consequence that the slope along CMH must be greater than along BEH, because the depression of H below B is equal to that of I below C, and BEH is longer than CMH. Therefore the velocity along the convex bank CMH must be greater than along BEH. There may even be a stagnation and an eddy in the contrary direction along the concave bank. Therefore, if the form of the section were the same as up the stream, the sides could not stand on the convex bank. When therefore the section has attained a permanent form, and the banks are again in equilibrium with the action of the current, the convex bank must be much flatter than the concave. If the water is really still on the concave bank, that bank will be absolutely perpendicular; nay, may overhang.—Accordingly, this state of things is matter of daily observation, and justifies our reasoning, and entitles us to say, that this is the nature of the internal motion of the filaments which we cannot distinctly observe. The water moves most rapidly along the convex bank, and the thread of the stream is nearest to this side. Reasoning in this way, the section, which we may suppose to have been originally of the form MbaE (fig. 16.) assumes the shape MBAE.

2. Without presuming to know the mechanism of the internal motions of fluids, we know that superficial waves are reflected precisely as if they were elastic bodies, making the angles of incidence and reflection equal. In as far therefore as the superficial wave is concerned in the operation, Mr Buat's second position is just. The permanency of the next reach requires that its axis shall be in the direction of the line EP which makes the angle GEP = FEN. If the next reach has the direction EQ, MR, the wave reflected in the line ES will work on the bank at S, and will be reflected in the line ST, and work again on the opposite bank at T. We know that the effect of the superficial motion is great, and that it is the principal agent in destroying the banks of canals. So far therefore Mr Buat is right. We cannot say with any precision or confidence how the actions of the under filaments are modified; but we know no reason for not extending to the under filaments what appears so probable with respect to the surface water.

3. The third position is no less evident. We do not know the mode of action of the water on the bank; but our general notions on this subject, confirmed by common experience, tell us that the more obliquely a stream of water beats on any bank, the less it tends to undermine it or wash it away. A stiff and cohesive soil therefore will suffer no more from being almost perpendicularly buffeted by a stream than a friable sand would suffer from water gliding along its face. Mr Buat thinks, from experience, that a clay bank is not sensibly affected till the angle FEB is about 36 degrees.

4. Since there are causes of retardation, and we still suppose that the discharge is kept up, and that the mean velocity, which had been diminished by the enlargement of the section, is again restored, we must grant that there is provided, in the mechanism of these motions, an accelerating force adequate to this effect. There can be no accelerating force in an open stream but the superficial slope. In the present case it is undoubtedly so; because by the deepening of the bottom where there is an elbow in the stream, we have of necessity a counter slope. Now, all this head of water, which must produce the augmentation of velocity in that part of the stream which ranges round the convex bank, will arise from the check which the water gets from the concave bank. This occasions a gorge or swell up the stream, enlarges a little the section at BVC; and this, by the principle of uniform motion, will augment all the velocities, deepen the channel, and put everything again into its train as soon as the water gets into the next reach. The water at the bottom of this basin has very little motion, but it defends the bottom by this very circumstance.

Such are the notions which Mr de Buat entertains of this part of the mechanism of running waters. We cannot say that they are very satisfactory, and they are very opposite to the opinions commonly entertained on the subject. Most persons think that the motion is most rapid and turbulent on the side of the concave bank, and that it is owing to this that the bank is worn away till it become perpendicular, and that the opposite bank is flat, because it has not been gnawed away in this manner. With respect to this general view of the matter, these persons may be in the right; and when a stream is turned into a crooked and yielding channel for the first time, this is its manner of action. But Mr Buat's aim is to investigate the circumstances which obtain in the case of a regimen; and in this view he is undoubtedly right as to the facts, though his mode of accounting for these facts may be erroneous. And as this is the only useful view to be taken of the subject, it ought chiefly to be attended to in all our attempts to procure stability to the bed of a river, without the expensive helps of masonry, &c. If we attempt to secure permanency by deepening on the inside of the elbow, our bank will undoubtedly crumble down, diminish the palliages, and occasion a more violent action on the hollow bank. The most effectual means of security is to enlarge the section; and if we do this on the inside bank, we must do it by widening the stream very much, that we may give a very sloping bank. Our attention is commonly drawn to it when the hollow bank is giving way, and with a view to stop the ravages of the stream. Things are not now in a state of permanency, but nature is working in her own way to bring it about. This may not suit our purpose, and we must thwart her. The phenomena which we then observe are frequently very unlike those described in the preceding paragraphs. We see a violent tumbling motion in the stream towards the hollow bank. We see an evident accumulation of water on that side, and the point B is frequently higher than C. This regurgitation of the water extends to some distance, and is of itself a cause of greater velocity, and contributes, like a head of flagrant water, to force the stream through the bend, and to deepen the bottom. This is clearly the case when the velocity is excessive, and the hollow bank able to abide the shock. In this situation the water thus heaped up escapes where it best can; and as the water, obstructed by an obstacle put in its way, escapes by the sides, and there has its velocity increased, so here the water gorged up against the hollow bank swells over towards the opposite side, and passes round the convex bank with an increased velocity. It depends much on the adjustment between the velocity and consequent accumulation, and the breadth of the stream and the angle of the elbow, whether this augmentation of velocity shall reach the convex bank; and we sometimes see the motion very languid in that place, and even depositions of mud and sand are made there. The whole phenomena are too complicated to be accurately described in general terms, even in the case of perfect regimen: for this regimen is relative to the confidence of the channel; and when this is very great, the motions may be most violent in every quarter. But the preceding observations are of importance, because they relate to ordinary cases and to ordinary channels.

It is evident, from Mr Buat's second position, that the proper form of an elbow depends on the breadth of the stream as well as on the radius of curvature, and that every angle of elbow will require a certain proportion between the width of the river and the radius of the sweep. Mr Buat gives rules and formulae for all these purposes, and shows that in one sweep there may be more than one reflection or rebound. It is needless to enlarge on this matter of mere geometrical discussion. It is with the view of enabling the engineer to trace the windings of a river in such a manner that there shall be no rebounds which shall direct the stream against the sides, but preserve it always in the axis of every reach. This is of consequence, even when the bends of the river are to be secured by masonry or piling; for we have seen the necessity of increasing the section, and the tendency which the waters have to deepen the channel on that side where the rebound is made. This tends to undermine our defences, and obliges us to give them deeper and more solid foundations in such places. But any person accustomed to the use of the scale and compasses will form to himself rules of practice equally sure and more expedient than Mr de Buat's formula.

We proceed, therefore, to what is more to our purpose, the consideration of the resistance caused by an elbow, and the methods of providing a force capable of overcoming it. We have already taken notice of the salutary consequences arising from the rambling course of rivers, inasmuch as it more effectually spreads them over the face of a country. It is no less beneficial by diminishing their velocity. This it does both by lengthening their course, which diminishes the declivity, and by the very resistance which they meet with at every bend. We derive the chief advantages from our rivers, when they no longer shoot their way from precipice to precipice, loaded with mud and sand, but peaceably roll along their clear waters, purified during their gentler course, and offer themselves for all the purposes of pasturage, agriculture, and navigation. The more a river winds its way round the foot of the hills, the more is the resistance of its bed multiplied; the more obstacles it meets with in its way from its source to the sea, the more moderate is its velocity; and instead of tearing up the very bowels of the earth, and digging itself a deep trough, along which it sweeps rocks and rooted-up trees, it flows with majestic pace even with the surface of our cultivated grounds, which it embellishes and fertilizes.

We may with safety proceed on the supposition, that the force necessary for overcoming the resistance arising from a rebound is as the square of the velocity; and it is reasonable to suppose it proportional to the square of the sine of the angle of incidence, and this for the reasons given for adopting this measure of the general Resistance of Fluids. It cannot, however, claim a greater confidence here than in that application; and it has been shown in that article with what uncertainty and limitations it must be received. We leave it to our readers to adopt either this or the simple ratio of the sines, and shall abide by the duplicate ratio with Mr Buat, because it appears by his experiments that this law is very exactly observed in tubes in inclinations not exceeding 40°; whereas it is in these small angles that the application to the general resistance of fluids is most in fault. But the correction is very simple, if this value shall be found erroneous. There can be little doubt that the force necessary for overcoming the resistance will increase as the number of rebounds.—Therefore we may express the resistance, in general, by the formula \( r = \frac{V^2 \cdot n}{m} \); where

\( r \) is the resistance, \( V \) the mean velocity of the stream, \( n \) the sine of the angle of incidence, \( n \) the number of equal rebounds (that is, having equal angles of incidence), and \( m \) is a number to be determined by experiment. Mr de Buat made many experiments on the resistance occasioned by the bendings of pipes, none of which differed from the result of the above formula above one part in twelve; and he concludes, that the resistance to one bend may be estimated at \( \frac{V^2 \cdot n}{3000} \).

The experiment was in this form: A pipe of 1 inch diameter, and 10 feet long, was formed with 10 bounds of 36° each. A head of water was applied to it, which gave the water a velocity of six feet per second. Another pipe of the same diameter and length, but without any bendings, was subjected to a pressure of a head of water, which was increased till the velocity of efflux was also six feet per second. The additional head of water was \( \frac{5}{10} \) inches. Another of the same diameter and length, having one bend of 24° 34', and running 85 inches per second, was compared with a straight pipe having the same velocity, and the difference of the heads of water was \( \frac{17}{10} \) inches. A computation from these two experiments will give the above result, or in English measure, \( r = \frac{V^2 \cdot n}{3200} \) very nearly. It is probable that this measure of the resistance is too great; for the pipe was of uniform diameter even in the bends: whereas in a river properly formed, where the regimen is exact, the capacity of the section of the bend is increased.

The application of this theory to inclined tubes and open streams is very obvious, and very legitimate and plied to infinite. Let \( AB \) (fig. 17.) be the whole height of the inclined tube, and \( BC \) the horizontal length of a flume, containing any number of rebounds, equal or unequal, but all regular, that is, constructed according to the conditions formerly mentioned. The whole head of water should be conceived as performing, or as divided into portions which perform, three different offices. One portion, \( AD = \frac{V^2}{505} \), impels the water into the entry of the pipe with the velocity with which it really moves in it; another portion \( EB \) is in equilibrium with the resistances arising from the mere length of the pipe expanded into a straight line; and the third portion \( DE \) serves to overcome the resistance of the bends. If, therefore, we draw the horizontal line \( BC \), and, taking the pipe \( BC \) out of its place, put it in the position \( DH \), with its mouth \( C \) in \( H \), so that \( DH \) is equal to \( BC \), the water will have the same velocity in it that it had before. N.B. For greater simplicity of argument, we may suppose that when the pipe was inserted at \( B \), its bends lay all in a horizontal plane, and that when it is inserted at \( D \), the plane in which all its bends lie slopes only in the direction \( DH \), and is perpendicular to the plane of the figure. We repeat it, the water will have the same velocity in the pipes \( BC \) and \( DH \), and the resistances will be overcome. If we now prolong the pipe \( DH \) towards \( L \) to any distance, repeating continually the same bendings in a series of lengths, each equal to \( DH \), the motion will be continued with the velocity corresponding to the pressure of the column \( AD \); because the declivity of the pipe is augmented in each length equal to \( DH \), by a quantity precisely sufficient for overcoming all the resistances in that length; and the true slope in these cases is \( BE + ED \), divided by the expanded length of the pipe \( BC \) or \( DH \).

The analogy which we were enabled to establish between the uniform motion or the train of pipes and of open streams, intitles us now to say, that when a river has bendings, which are regularly repeated at equal intervals, its slope is compounded of the slope which is necessary for overcoming the resistance of a straight channel of its whole expanded length, agreeably to the formula for uniform motion, and of the slope which is necessary for overcoming the resistance arising from its bendings alone.

Thus, let there be a river which, in the expanded course of 6000 fathoms, has 10 elbows, each of which has 30° of rebound; and let its mean velocity be 20 inches in a second. If we would learn its whole slope in this 6000 fathoms, we must first find (by the formula of uniform motion) the slope which will produce the velocity of 20 inches in a straight river of this length, section, and mean depth. Suppose this to be

\[ \frac{V^2}{3200} \text{ or } 20 \text{ inches in this whole length. We must then find (by the formula } \frac{V^2 \sin^2}{3200} \text{) the slope necessary for overcoming the resistance of 10 rebounds of } 30^\circ \text{ each. This we shall find to be } 6\frac{1}{2} \text{ inches in the 6000 fathoms. Therefore the river must have a slope of } 26\frac{1}{2} \text{ inches in 6000 fathoms, or } \frac{1}{18205} \text{; and this slope will produce the same velocity which 20 inches, or } \frac{1}{18205} \text{, would do in a straight running river of the same length.}

**Part II. Practical Inferences.**

Having thus established a theory of a most important part of hydraulics, which may be considered in as a just representation of nature's procedure, we shall apply it to the examination of the chief results of every thing which art has contrived for limiting the operations of nature, or modifying them so as to suit our particular views. Trusting to the detail which we have given of the connecting principles, and the chief circumstances which co-operate in producing the oftenest effect; and supposing that such of our readers as are interested in this subject will not think it too much trouble to make the applications in the same detail; we shall content ourselves with merely pointing out the steps of the process, and showing their foundation in the theory itself: and frequently, in place of the direct analysis which the theory enables us to employ for the solution of the problems, we shall recommend a process of approximation by trial and correction, sufficiently accurate, and more within the reach of practical engineers. We are naturally led to consider in order the following articles.

1. The effects of permanent additions of every kind to the waters of a river, and the most effectual methods of preventing or removing inundations.

2. The effects of weirs, bars, sluices, and keeps of every kind, for raising the surface of a river; and the similar effects of bridges, piers, and every thing which contracts the section of the stream.

3. The nature of canals; how they differ from rivers in respect of origin, discharge, and regimen, and what conditions are necessary for their most perfect construction.

4. Canals for draining land, and drafts or canals of derivation from the main stream. The principles of their construction, so that they may suit their intended purposes, and the change which they produce on the main stream, both above and below the point of derivation.

Of the effects of permanent additions to the waters of a river.

From what has been said already, it appears that to every kind of foil or bed there corresponds a certain velocity of current, too small to hurt it by digging it up, and too great to allow the deposition of the materials which it is carrying along. Supposing this known for any particular situation, and the quantity of water which the channel must of necessity discharge, we may wish to learn the smallest slope which must be given to this stream, that the waters may run with the required velocity. This suggests

\[ \frac{D}{V} = 2b^2. \]

For the area of the section is twice the square of the height, and the discharge is the product of this area and the velocity. Therefore \( \sqrt{\frac{D}{2V}} = b \) and \( \sqrt{\frac{2D}{V}} = \text{the breadth } b. \)

The formula of uniform motion gives \( \sqrt{s} = L\sqrt{s + 1.6} \)

\[ \frac{297(\sqrt{d} - 0.1)}{V + 0.3(\sqrt{d} - 0.1)}. \]

Instead of \( \sqrt{d} - 0.1 \), put it equal \( \sqrt{\frac{b}{2}} - 0.1 \), and every thing being known in the second member of this equation, we easily get the value of \( s \) by a few trials after the following manner. Suppose that the second member is equal to any number, such as 9. First suppose that \( \sqrt{s} \) is = 9. Then the hyperbolic logarithm of \( 9 + 1.6 \) or of \( 10.6 \) is 2.36. Therefore we have \( \sqrt{s} = L\sqrt{s + 1.6} = 9 - 2.36 = 6.64 \); whereas it should have been = 9. Therefore lay 6.64 : 9 = 9 : 12.2 nearly. Now suppose that \( \sqrt{s} \) is = 12.2. Then \( L_{12.2 + 1.6} = L_{13.8} = 7.625 \) nearly, and \( 12.2 - 2.625 \) is 9.575, whereas it should be 9. Now we find that changing the value of \( \sqrt{s} \) from 9 to 12.2 has changed the answer from 6.64 to 9.575, or a change of 3.2 in our assumption has made a change of 2.935 in the answer, and has left an error of 0.575. Therefore lay 2.935 : 0.575 = 3.2 : 0.628. Then, taking 0.628 from 12.2, we have (for our next assumption or value of \( \sqrt{s} \)) 11.572. Now 11.572 + 1.6 = 13.172, and \( L_{13.172} \) is 2.58 nearly. Now try this last value 11.572 - 2.58 is 9.008, sufficiently exact. This may serve as a specimen of the trials by which we may avoid an intricate analysis.

**Prob. II.** Given the discharge \( D \), the slope \( s \), and the velocity \( V \), of permanent regimen, to find the dimensions of the bed.

Let \( x \) be the width, and \( y \) the depth of the channel, and \( S \) the area of the section. This must be \( \frac{D}{V} \), which is therefore \( xy \). The denominator \( s \) being given, Practical given, we may make \( \sqrt{s} - L\sqrt{s + 1} = \sqrt{B} \), and the formula of mean velocity will give \( V = \frac{297}{\sqrt{B}}(\sqrt{d} - 0.1) \)

\( - 0.3 (\sqrt{d} - 0.1) \), which we may express thus: \( V = \frac{297}{\sqrt{B}}(\sqrt{d} - 0.1)(\sqrt{B} - 0.3) \), which gives

\( \frac{V}{\sqrt{B}} = \frac{297}{\sqrt{B}} - 0.3 \)

\( \sqrt{d} - 0.1 \); and finally, \( \frac{V}{\sqrt{B}} = \frac{297}{\sqrt{B}} - 0.3 + c_1 = \sqrt{d} \).

Having thus obtained what we called the mean depth, we may suppose the section rectangular. This gives \( d = \frac{xy}{x + 2y} \). Thus we have two equations, \( S = xy \)

and \( d = \frac{xy}{x + 2y} \).

From which we obtain \( x = \sqrt{\left( \frac{S}{2d} \right)^2 - 2S} + \frac{S}{2d} \).

And having the breadth \( x \) and area \( S \), we have \( y = \frac{S}{x} \).

And then we may change this for the trapezium often mentioned.

These are the chief problems on this part of the subject, and they enable us to adjust the slope and channel of a river which receives any number of successive permanent additions by the influx of other streams. This last informs us of the rise which a new supply will produce, because the additional supply will require additional dimensions of the channel; and as this is not supposed to increase in breadth, the addition will be in depth. The question may be proposed in the following problem.

**Prob. III.** Given the slope \( s \), the depth and the base of a rectangular bed (or a trapezium), and consequently the discharge \( D \), to find how much the section will rise, if the discharge be augmented by a given quantity.

Let \( b \) be the height after the augmentation, and \( w \) the width for the rectangular bed. We have in any uniform current \( \sqrt{d} = \frac{V}{297}(\sqrt{B} - 0.3) \)

Rafting this to a square,

\( \frac{w}{b} = \frac{D}{w + 2b} \)

and putting for \( d \) and \( V \) their values \( \frac{w}{b} \) and \( \frac{D}{w + 2b} \),

making \( \frac{297}{\sqrt{B}} - 0.3 = K \), the equation becomes \( \frac{w}{b} = \frac{D}{w + 2b} \)

\( = \left( \frac{D}{w + 2b} \right)^2 \)

Rafting the second member to a square, and reducing, we obtain a cubic equation, to be solved in the usual manner.

But the solution would be extremely complicated. We may obtain a very expeditious and exact approximation from this consideration, that a small change in one of the dimensions of the section will produce a much greater change in the section and the discharge than in the mean depth \( d \). Having therefore augmented the unknown dimension, which is here the height, make use of this to form a new mean depth, and then the new equation \( \sqrt{d} = \frac{D}{w + 2b} \)

\( = \frac{297}{\sqrt{B}}(\sqrt{B} - 0.3) + c_1 \) will give us another value of \( b \), which will rarely exceed the truth by \( \frac{1}{5} \). This serves (by the same process) for finding another, which will commonly be sufficiently exact. We shall illustrate this by an example.

Let there be a river whose channel is a rectangle 150 feet wide and five feet deep, and which discharges 1500 cubic feet of water per second, having a velocity of 20 inches, and slope of \( \frac{1}{180} \), or about \( \frac{1}{180} \) of an inch in 100 fathoms. How much will it rise if it receives an addition which triples its discharge? and what will be its velocity?

If the velocity remained the same, its depth would be tripled; but we know by the general formula that its velocity will be greatly increased, and therefore its depth will not be tripled. Suppose it to be doubled, and to become 12 feet. This will give \( d = 10,344.83 \),

or 124,138 inches; then the equation \( \sqrt{d} - 0.1 = \frac{D}{w + 2b} \)

\( = \frac{297}{\sqrt{B}}(\sqrt{B} - 0.3) + c_1 \)

in which we have \( \sqrt{B} = 107.8, D = 4500; \sqrt{d} - 0.1 = 11.0417 \), will give \( b = 13,276 \); whereas it should have been 12. This shows that our calculated value of \( d \) was too small. Let us therefore increase the depth by 0.9, or make it 12.9, and repeat the calculation. This will give \( \sqrt{d} - 0.1 = 11.3927 \), and \( b = 12,867 \),

instead of 13,276. Therefore augmenting our data 0.9 changes our answer 0.409. If we suppose these small changes to retain their proportions, we may conclude that if 12 be augmented by the quantity \( x \times 0.9 \), the quantity 13,276 will diminish by the quantity \( x \times 0.409 \). Therefore, that the estimated value of \( b \) may agree with the one which results from the calculation, we must have \( 12 + x \times 0.9 = 13,276 - x \times 0.409 \).

This will give \( x = \frac{1276}{13,299} = 0.9748 \), and \( x \times 0.9 = 0.8773 \);

and \( b = 12,8773 \). If we repeat the calculation with this value of \( b \), we shall find no change.

This value of \( b \) gives \( d = 131,883.6 \) inches. If we now compute the new velocity by dividing the new discharge 4500 by the new area 150 \times 12,8773, we shall find it to be 27.95 inches, in place of 20, the former velocity.

We might have made a pretty exact first assumption, by recollecting what was formerly observed, that when the breadth is very great in proportion to the depth, the mean depth differs infinitely from the real depth, or rather follows nearly the same proportions, and that the velocities are proportional to the square roots of the depths. Call the first discharge \( d \), the height \( b \), and velocity \( v \), and let \( D, H, \) and \( V \), express these things in their augmented state. We have \( v = \frac{d}{w + 2b} \)

\( V = \frac{D}{w + 2b} \), and \( v : V = \frac{d}{H} : \frac{D}{H} \), and \( v^2 : V^2 = \frac{d^2}{H^2} : \frac{D^2}{H^2} \).

But by this remark \( v^2 : V^2 = b : H \). Therefore \( b : H = \frac{d^2}{H^2} : \frac{D^2}{H^2} \), and \( \frac{bD^2}{H^2} = \frac{Hd^2}{b^2} \), and \( b^3D^2 = H^3d^2 \),

and \( d^2 : D^2 = b^3 : H^3 \) (a useful theorem) and \( H^3 = \frac{b^3D^2}{d^2} \), and \( H = \sqrt[3]{\frac{b^3D^2}{d^2}} = 12.48 \).

Or we might have made the same assumption by the remark remark also formerly made on this case, that the squares of the discharges are nearly as the cubes of the height, or \(150^2 : 450^2 = 6^3 : 12.48^3\).

And in making these first guesses we shall do it more exactly, by recollecting that a certain variation of the mean depth requires a greater variation of the height, and the increment will be to the height nearly as half the height to the width, as may easily be seen. Therefore, if we add to 12.48 its \(\frac{6}{15}\) th part, or its \(\frac{24}{15}\) th part, viz. 0.52, we have 1.3 for our first assumption, exceeding the truth only an inch and a half. We mention these circumstances, that those who are disposed to apply these doctrines to the solution of practical cases may be at no loss when one occurs of which the regular solution requires an intricate analysis.

It is evident that the inverse of the foregoing problems will show the effects of enlarging the section of a river, that is, will show how much its surface will be sunk by any proposed enlargement of its bed. It is therefore needless to propose such problems in this place. Common sense directs us to make these enlargements in those parts of the river where their effect will be greatest, that is, where it is shallowest when its breadth greatly exceeds its depth, or where it is narrowest (if its depth exceed the breadth, which is a very rare case), or in general, where the slope is the smallest for a short run.

The same general principles direct us in the method of embankments, for the prevention of floods, by enabling us to ascertain the heights necessary to be given to our banks. This will evidently depend, not only on the additional quantity of water which experience tells us a river brings down during its freshes, but also on the distance at which we place the banks from the natural banks of the river. This is a point where mistaken economy frequently defeats its own purpose. If we raise our embankment at some distance from the natural banks of the river, not only will a smaller height suffice, and consequently a smaller base, which will make a saving in the duplicate proportion of the height; but our works will be so much the more durable nearly, if not exactly, in the same proportion. For by thus enlarging the additional bed which we give to the swollen river, we diminish its velocity almost in the same proportion that we enlarge its channel, and thus diminish its power of ruining our works. Except, therefore, in the case of a river whose freshes are loaded with fine sand to destroy the turf, it is always proper to place the embankment at a considerable distance from the natural banks. Placing them at half the breadth of the stream from its natural banks, will nearly double its channel; and, except in the case now mentioned, the space thus detached from our fields will afford excellent pasture.

The limits of such a work as ours will not permit us to enter into any detail on the method of embankment. It would require a volume to give instructions as to the manner of founding, raising, and securing the dykes which must be raised, and a thousand circumstances which must be attended to. But a few general observations may be made, which naturally occur while we are considering the manner in which a river works in setting or altering its channel.

It must be remarked, in the first place, that the river will rise higher when embanked than it does while it was allowed to spread; and it is by no means easy to conclude to what height it will rise from the greatest height to which it has been observed to rise in its floods. When at liberty to expand over a wide valley; then it could only rise till it overflowed with a thickness or depth of water sufficient to produce a motion backwards into the valley quick enough to take off the water as fast as it was supplied; and we imagine that a foot or two would suffice in most cases. The best way for a prudent engineer will be to observe the utmost rise remembered by the neighbours in some gorge, where the river cannot spread out. Measure the increased section in this place, and at the same time recollect, that the water increases in a much greater proportion than the section; because an increase of the hydraulic mean depth produces an increase of velocity in the duplicate proportion of the depth nearly. But as this augmentation of velocity will obtain also between the embankments, it will be sufficiently exact to suppose that the section must be increased here nearly in the same proportion as at the gorge already mentioned. Neglecting this method of information, and regulating the height of our embankment by the greatest swell that has been observed in the plain, will assuredly make them too low, and render them totally useless.

A line of embankment should always be carried on by a strict concert of the proprietors of both banks through its whole extent. A greedy proprietor, by advancing his own embankment beyond that of his neighbours, not only exposes himself to risk by the working of the waters on the angles which this will produce, but exposes his neighbours also to danger, by narrowing the section, and thereby raising the surface and increasing the velocity, and by turning the stream athwart, and causing it to shoot against the opposite bank. The whole should be as much as possible in a line; and the general effect should be to make the course of the stream straighter than it was before. All bends should be made more gentle, by keeping the embankment further from the river in all convex lines of the natural bank, and bringing it nearer where the bank is concave. This will greatly diminish the action of the waters on the bankment, and insure their duration. The same maxim must be followed in fencing any brook which discharges itself into the river. The bends given at its mouth to the two lines of embankment should be made less acute than those of the natural brook, although, by this means, two points of land are left out. And the opportunity should be embraced of making the direction of this transverse brook more sloping than before, that is, less athwart the direction of the river.

It is of great consequence to cover the outside of the dyke with very compact turf closely united. If it admits water, the interior part of the wall, which is always more porous, becomes drenched in water, and this water acts with its statical pressure, tending to burst the bank on the land-side, and will quickly shift it from its seat. The utmost care should therefore be taken to make it and keep it perfectly tight. It should be a continued fine turf, and every bare spot should be carefully covered with fresh sod; and rat holes must be carefully closed up. Of straightening or changing the course of rivers.

We have seen, that every bending of a river requires an additional slope in order to continue its train, or enable it to convey the same quantity of water without swelling in its bed. Therefore the effect of taking away any of these bends must be to sink the waters of the river. It is proper, therefore, to have it in our power to estimate these effects. It may be desirable to gain property, by taking away the sweeps of a very winding stream. But this may be prejudicial, by destroying the navigation on such a river. It may also hurt the proprietors below, by increasing the velocity of the stream, which will expose them to the risk of its overflowing, or of its destroying its bed, and taking a new course. Or this increase of velocity may be inconsistent with the regimen of the new channel, or at least require larger dimensions than we should have given it if ignorant of this effect.

Our principles of uniform motion enable us to answer every question of this kind which can occur; and Mr de Buat proposes several problems to this effect. The regular solutions of them are complicated and difficult; and we do not think them necessary in this place, because they may all be solved in a manner not indeed so elegant, because indirect, but abundantly accurate, and easy to any person familiar with those which we have already considered.

We can take the exact level across all these sweeps, and thus obtain the whole slope. We can measure with accuracy the velocity in some part of the channel which is most remote from any bend, and where the channel itself has the greatest regularity of form. This will give us the expense or discharge of the river, and the mean depth connected with it. We can then examine whether this velocity is precisely such as is compatible with stability in the straight course. If it is, it is evident that if we cut off the bends, the greater slope which this will produce will communicate to the waters a velocity incompatible with the regimen suited to this soil, unless we enlarge the width of the stream, that is, unless we make the new channel more capacious than the old one. We must now calculate the dimensions of the channel which, with this increased slope, will conduct the waters with the velocity that is necessary. All this may be done by the foregoing problems; and we may easily accomplish this by steps. First, suppose the bed the same with the old one, and calculate the velocity for the increased slope by the general formula. Then change one of the dimensions of the channel, so as to produce the velocity we want, which is a very simple process. And in doing this, the object to be kept chiefly in view is not to make the new velocity such as will be incompatible with the stability of the new bed.

Having accomplished this first purpose, we learn (in the very solution) how much shallower this channel with its greater slope will be than the former, while it discharges all the waters. This diminution of depth must increase the slope and the velocity, and must diminish the depth of the river, above the place where the alteration is to be made. How far it produces these effects may be calculated by the general formula. We then see whether the navigation will be hurt, either in the old river up the stream, or in the new channel. It is plain that all these points cannot be reconciled. We may make the new channel such, that it shall leave a velocity compatible with flability, and that it shall not diminish the depth of the river up the stream. But, having a greater slope, it must have a smaller mean depth, and also a smaller real depth, unless we make it of a very inconvenient form.

The same things viewed in a different light, will show us what depression of waters may be produced by rectifying the course of a river in order to prevent its overflowing. And the process which we would recommend is the same with the foregoing. We apprehend it to be quite needless to measure the angles of rebound, in order to compute the slope which is employed for sending the river through the bend, with a view to supersede this by straightening the river. It is infinitely easier and more exact to measure the levels themselves, and then we know the effect of removing them.

Nor need we follow Mr de Buat in solving problems for diminishing the slope and velocity, and deepening the channel of a river by bending its course. The expense of this would be in every case enormous; and the practices which we are just going to enter upon afford infinitely other methods of accomplishing all the purposes which are to be gained by these changes.

Of Bars, Weirs, and Jettyes, for raising the Surface of Rivers.

We propose, under the article WATER-WORKS, to consider in sufficient practical detail all that relates to the construction and mechanism of these and other erections in water; and we confine ourselves, in this place, to the mere effect which they will produce on the current of the river.

We gave the name of weir or bar to a dam erected across a river for the purpose of raising its waters, whether in order to take off a draft for a mill or to deepen the channel. Before we can tell the effect which they will produce, we must have a general rule for ascertaining the relation between the height of the water above the lip of the weir or bar, and the quantity of water which will flow over.

First, then, with respect to a weir, represented in fig. 18, and fig. 18, n° 2. The latter figure more resembles their usual form, consisting of a dam of solid masonry, or built of timber, properly fortified with hoards and banks. On the top is set up a strong plank FR, called the waferboard, or wafer, over which the water flows. This is brought to an accurate level, of the proper height. Such voiders are frequently made in the side of a mill-course, for letting the superfluous water run off. This is properly the WASTER, VORDER: it is also called an OFFSET. The same observations will explain all these different pieces of practice. The following questions occur in course.

PROB. I. Given the length of an offset or waferboard, made in the face of a reservoir of stagnant water, and the depth of its lip under the horizontal surface of the water, to determine the discharge, or the quantity of water which will run over in a second?

Let AB be the horizontal surface of the still water, and F the lip of the waferboard. Call the depth BF under the surface b, and the length of the waferboard l.

N.B. The water is supposed to flow over into another basin or channel, so much lower that the surface... If the water could be supported at the height BF, BF might be considered as an orifice in the side of a vessel. In which case, the discharge would be the same as if the whole water were flowing with the velocity acquired from the height \( \frac{3}{2} BF \) or \( \frac{3}{2} b \). And if we suppose that there is no contraction at the orifice, the mean velocity would be \( \sqrt{\frac{7}{2} \cdot \frac{3}{2} b} = \sqrt{\frac{7}{2} \cdot \frac{3}{2} b} \), in English inches per second. The area of this orifice is \( lb \). Therefore the discharge would be \( lb \cdot \sqrt{\frac{7}{2} \cdot \frac{3}{2} b} \), all being measured in inches. This is the usual theory; but it is not an exact representation of the manner in which the efflux really happens. The water cannot remain at the height BF; but in drawing towards the waferboard from all sides, it forms a convex surface AIH, so that the point I, where the vertical drawn from the edge of the waferboard meets the curve, is considerably lower than B. But as all the mass above F is supposed perfectly fluid, the pressure of the incumbent water is propagated, in the opinion of Mr de Buat, to the filament falling over at F without any diminution. The same may be said of any filament between F and I. Each tends, therefore, to move in the same manner as if it were really impelled through an orifice in its place. Therefore the motions through every part of the line or plane IF are the same as if the water were escaping through an orifice IF, made by a sluice let down on the water, and keeping up the water of the reservoir to the level AB. It is beyond a doubt (says he) that the height IF must depend on the whole height BF, and that there must be a certain determined proportion between them. He does not attempt to determine this proportion theoretically, but says, that his experiments ascertain it with great precision to be the proportion of one to two, or that IF is always one-half of BF. He says, however, that this determination was not by an immediate and direct measurement; he concluded it from the comparison of the quantities of water discharged under different heights of the water in the reservoir.

We cannot help thinking that this reasoning is very defective in several particulars. It cannot be inferred, from the laws of hydrostatical pressure, that the filament at I is pressed forward with all the weight of the column BI. The particle I is really at the surface; and considering it as making part of the surface of a running stream, it is subjected to hardly any pressure, any more than the particles on the surface of a cup of water held in the hand, while it is carried round the axis of the earth and round the sun. Reasoning according to his own principles, and availing himself of his own discovery, he should say, that the particle at I has an accelerating force depending on its slope only; and then he should have endeavoured to ascertain this slope. The motion of the particle at I has no immediate connection with the pressure of the column BI; and if it had, the motion would be extremely different from what it is: for this pressure alone would give it the velocity which Mr Buat assigns it. Now it is already passing through the point I with the velocity which it has acquired in descending along the curve AI; and this is the real state of the case. The particles are passing through with a velocity already acquired by a flopping current; and they are accelerated by the hydrostatical pressure of the water above them. Practical inferences. The internal mechanism of these motions is infinitely more complex than Mr Buat here supposes; and on this supposition, he very nearly abandons the theory which he has so ingeniously established, and adopts the theory of Guglielmini which he had exploded. At the same time, we think that he is not much mistaken when he affirms, that the motions are nearly the same as if a sluice had been let down from the surface to I. For the filament which passes at I has been gliding down a curved surface, and has not been exposed to any friction. It is perhaps the very case of hydraulics, where the obstructions are the smallest; and we should therefore expect that its motion will be the least retarded.

We have therefore no hesitation in saying, that the filament at I is in the very state of motion which the theory would assign to it if it were passing under a sluice, as Mr Buat supposes. And with respect to the inferior filaments, without attempting the very difficult task of investigating their motions, we shall just say, that we do not see any reason for supposing that they will move slower than our author supposes. Therefore, though we reject his theory, we admit his experimental proposition in general; that is, we admit that the whole water which passes through the plane IF moves with the velocity (though not in the same direction) with which it would have run through a sluice of the same depth; and we may proceed with his determination of the quantity of water discharged.

If we make BC the axis of a parabola BEGH, the velocities of the filaments passing at I and F will be represented by the ordinates IE and FG, and the discharge by the area IEGF. This allows a very neat solution of the problem. Let the quantity discharged per second be D, and let the whole height BF be \( b \). Let \( zG \) be the quantity by which we must divide the square of the mean velocity, in order to have the producing height. This will be less than \( zg \), the acceleration of gravity, on account of the convergency at the sides and the tendency to convergence at the lip F. We formerly gave for its measure \( \frac{7}{2}6 \) inches, instead of \( \frac{7}{2}2 \), and said that the inches discharged per second from an orifice of one inch were \( \frac{26}{49} \), instead of \( \frac{27}{18} \). Let \( x \) be the distance of any filament from the horizontal line AB. An element of the orifice, therefore, (for we may give it this name) is \( \frac{1}{2}x \). The velocity of this element is \( \sqrt{\frac{z}{2}Gx} \), or \( \sqrt{\frac{z}{2}G} \times \sqrt{x} \). The discharge from it is \( l \sqrt{\frac{z}{2}Gx} \), and the fluent of this, or \( D = \int l \sqrt{\frac{z}{2}Gx} \), which is \( \frac{2}{3}l \sqrt{\frac{z}{2}Gx^{\frac{3}{2}}} + C \).

To determine the constant quantity C, observe that Mr de Buat found by experiment that BI was in all cases \( \frac{1}{2}BF \). Therefore D must be nothing when \( x = \frac{1}{2}b \); consequently \( C = -\frac{2}{3}l \sqrt{\frac{z}{2}G} \left( \frac{b}{2} \right)^{\frac{3}{2}} \), and the completed fluent, will be \( D = \frac{2}{3}l \sqrt{\frac{z}{2}G} \left( x^{\frac{3}{2}} - \left( \frac{b}{2} \right)^{\frac{3}{2}} \right) \).

Now make \( x = b \), and we have

\[ D = \frac{2}{3}l \sqrt{\frac{z}{2}G} \left( b^{\frac{3}{2}} - \left( \frac{b}{2} \right)^{\frac{3}{2}} \right) = \frac{2}{3}l \sqrt{\frac{z}{2}G} \left( 1 - \left( \frac{1}{2} \right)^{\frac{3}{2}} \right)b^{\frac{3}{2}}. \]

But \( 1 - \left( \frac{1}{2} \right)^{\frac{3}{2}} = 0.6465 \), and \( \frac{2}{3} \) of this is 0.431; Therefore, finally, D = 0.431 (\sqrt{2G} b^{3/2} \times l).

If we now put 26.49 or 26\(\frac{1}{2}\) for \(\sqrt{2G}\), or the velocity with which a head of water of one inch will impel the water over a weir, and multiply this by 0.431, we get the following quantity 114.172, or, in numbers of easy recollection, 114\(\frac{1}{2}\), for the cubic inches of water per second, which runs over every inch of a wateboard when the edge of it is one inch below the surface of the reservoir; and this must be multiplied by \(b^{\frac{3}{2}}\), or by the square root of the cube of the head of water. Thus let the edge of the wateboard be four inches below the surface of the water. The cube of this is 64, of which the square root is eight. Therefore a wateboard of this depth under the surface, and three feet long, will discharge every second \(8 \times 3 \times 114\(\frac{1}{2}\)\) cubic inches of water, or 114\(\frac{1}{2}\) cubic feet, English measure.

The following comparisons will show how much this theory may be depended on. Col. 1. shows the depth of the edge of the board under the surface; 2. shows the discharge by theory; and, 3. the discharge actually observed. The length of the board was 18\(\frac{1}{2}\) inches.

N.B. The number in Mr Buat's experiments are here reduced to English measure.

| Depth | D Theor. | D Exp. | E | |-------|----------|--------|---| | 1 | 0.403 | | | | 2 | 1.140 | | | | 3 | 2.095 | | | | 4 | 3.225 | | | | 5 | 4.507 | | | | 6 | 5.925 | | | | 7 | 7.466 | | | | 8 | 9.122 | | | | 9 | 10.884 | | | | 10 | 12.748 | | | | 11 | 14.707 | | | | 12 | 16.758 | | | | 13 | 18.895 | | | | 14 | 21.117 | | | | 15 | 23.419 | | | | 16 | 25.800 | | | | 17 | 28.258 | | | | 18 | 30.786 | | |

The last column is the cubic inches discharged in a second by each inch of the wateboard. The correspondence is undoubtedly very great. The greatest error is in the first, which may be attributed to a much smaller lateral contraction under so small a head of water.

But it must be remarked, that the calculation proceeds on two suppositions. The height IF is supposed \(b\) of BI; and 2 G is supposed 726. It is evident, that by increasing the one and diminishing the other, nearly the same answers may be produced, unless much greater variations of \(b\) be examined. Both of these quantities are matters of considerable uncertainty, particularly the first; and it must be farther remarked, that this was not measured, but deduced from the uniformity of the experiments. We presume that Mr Buat tried various values of G, till he found one which gave the ratios of discharge which he observed. We beg leave to observe, that in a set of numerous experiments which we had access to examine, BI was uniformly much less than \(b\); it was very nearly \(b\); and the quantity discharged was greater than what would result from Mr Buat's calculation. It was farther observed, that IF depended very much on the form of the wateboard. When it was a very thin board of considerable depth, IF was very considerably greater than if the board was thick, or narrow, and set on the top of a broad dam-head, as in fig. 18. n° 2.

It may be proper to give the formula a form which will correspond to any ratio which experience may discover between BF and IF. Thus, let BI be \(\frac{m}{n} \cdot BF\).

The formula will be \(D = \frac{1}{2} l \sqrt{2G} \left(1 - \left(\frac{m}{n}\right)^{\frac{3}{2}}\right) b^{\frac{3}{2}}\).

It is hoped that this and some other fundamental facts in practical hydraulics will soon be determined by accurate experiments. The Honourable Board for Fisheries and Improvements in Scotland have allotted a sum of money for making the necessary experiments, and the results will be published by their authority. Meanwhile, this theory of Mr de Buat is of great value to the practical engineer, who at present must content himself with a very vague conjecture, or take the calculation of the erroneous theory of Guglielmini. By that theory, the board of three feet, at the depth of four inches, should discharge nearly 3\(\frac{1}{2}\) cubic feet per second, which is almost double of what it really delivers.

We presume, therefore, that the following table will be acceptable to practical engineers, who are not familiar with such computations. It contains, in the first column, the depth in English inches from the surface of the stagnant water of a reservoir to the edge of the wateboard. The second column is the cubic feet of water discharged in a minute by every inch of the wateboard.

When the depth does not exceed four inches, it will not be exact enough to take proportional parts for the fractions of an inch. The following method is exact.

If they be odd quarters of an inch, look in the table for as many inches as the depth contains quarters, and take the eighth part of the answer. Thus, for 3\(\frac{1}{2}\) inches, take the eighth part of 23.419, which corresponds to 15 inches. This is 2.927.

If the wateboard is not on the face of a dam, but in a running stream, we must augment the discharge by multiplying the section by the velocity of the stream. But this correction can seldom occur in practice; because, in this case, the discharge is previously known; and it is \(b\) that we want; which is the object of the next problem.

We only beg leave to add, that the experiments which we mention as having been already made in this country, give a result somewhat greater than this table, viz., about 1\(\frac{1}{2}\). Therefore, having obtained the answer by this table, add to it its 16th part, and we apprehend that it will be extremely near the truth.

When, on the other hand, we know the discharge over a wateboard, we can tell the depth of its edge under the surface of the stagnant water of the reservoir, because we have \(b = \left(\frac{D}{l}\right)^{\frac{2}{3}}\) very nearly. We are now in a position to solve the problem respecting a weir across a river.

**Prob. II.** The discharge and section of a river being given, it is required to determine how much the waters will be raised by a weir of the whole breadth of the river, discharging the water with a clear fall, that is, the surface of the water in the lower channel being below the edge of the weir?

In this case we have \( z = \frac{D}{S} \) nearly, because there will be no contraction at the sides when the weir is the whole breadth of the river. But further, the water is not now stagnant, but moving with the velocity \( \frac{D}{S} \), \( S \) being the section of the river.

Therefore let \( a \) be the height of the weir from the bottom of the river, and \( b \) the height of the water above the edge of the weir. We have the velocity with which the water approaches the weir \( = \frac{D}{l(a+b)} \), \( l \) being the length of the weir or breadth of the river. Therefore the height producing the primary mean velocity is \( \left( \frac{D}{l\sqrt{2g}(a+b)} \right)^{\frac{3}{2}} \). The equation given a little ago will give \( b = \left( \frac{D}{0.431\sqrt{2g}(a+b)} \right)^{\frac{3}{2}} \), when the water above the weir is stagnant. Therefore, when it is already moving with the velocity \( \frac{D}{l(a+b)} \), we shall have \( b = \left( \frac{D}{0.431\sqrt{2g}(a+b)} \right)^{\frac{3}{2}} - \left( \frac{D}{l\sqrt{2g}(a+b)} \right)^{\frac{3}{2}} \). It would be very troublesome to solve this equation regularly, because the unknown quantity \( b \) is found in the second term of the answer. But we know that the height producing the velocity above the weir is very small in comparison of \( b \) and of \( a \), and, if only estimated roughly, will make a very insensible change in the value of \( b \); and, by repeating the operation, we can correct this value, and obtain \( b \) to any degree of exactness.

To illustrate this by an example. Suppose a river, the section of whose stream is 150 feet, and that it discharges 174 cubic feet of water in a second; how much will the waters of this river be raised by a weir of the same width, and 3 feet high?

Suppose the width to be 50 feet. This will give 3 feet for the depth; and we see that the water will have a clear fall, because the lower stream will be the same as before.

The section being 150 feet, and the discharge 174, the mean velocity is \( \frac{174}{150} = 1.16 \) feet, \( = 14 \) inches nearly, which requires the height of \( \frac{1}{4} \) of an inch very nearly. This may be taken for the second term of the value of \( b \). Therefore \( b = \left( \frac{D}{0.431\sqrt{2g}l} \right)^{\frac{3}{2}} - \frac{1}{4} \). Now \( \sqrt{2g}l \) is, in the present case, \( = 27,313 \); \( l \) is 600, and \( D = 174 \times 1728 = 300672 \). Therefore \( b = 12,192 - 0.25 = 11,942 \). Now correct this value of \( b \), by correcting the second term, which is \( \frac{1}{4} \) of an inch, instead of \( \left( \frac{D}{\sqrt{2g}l(a+b)} \right)^{\frac{3}{2}} \), or 0.141. This will give us \( b = 12,192 - 0.141 = 12,051 \), differing from the first value about \( \frac{1}{5} \) of an inch. It is needless to carry the approximation farther. Thus we see that a weir, which dams up the whole of the former current of three feet deep, will only raise the waters of this river one foot.

The same rule serves for showing how high we ought to raise this weir in order to produce any given rise of the waters, whether for the purposes of navigation, or for taking off a draft to drive mills, or for any other service; for if the breadth of the river remain the same, the water will still flow over the weir with nearly the same depth. A very small and hardly perceptible difference will indeed arise from the diminution of slope occasioned by this rise, and a consequent diminution of the velocity with which the river approaches the weir. But this difference must always be a small fraction of the second term of our answer; which term is itself very small; and even this will be compensated, in some degree, by the freer fall which the water will have over the weir.

If the intended weir is not to have the whole breadth of the river (which is seldom necessary even for the purposes of navigation), the waters will be raised higher by the same height of the wasteboard. The calculation is precisely the same for this case. Only in the second term, which gives the head of water corresponding to the velocity of the river, \( l \) must still be taken for the whole breadth of the river, while in the first term \( l \) is the length of the wasteboard. Also \( \sqrt{2g}l \) must be a little less, on account of the contractions at the ends of the weir, unless these be avoided by giving the masonry at the ends of the wasteboard a curved shape on the upper side of the wasteboard. This should not be done when the sole object of the weir is to raise the surface of the waters. Its effect is but trifling at any rate, when the length of the wasteboard is considerable, in proportion to the thickness of the sheet of water flowing over it.

The following comparisons of this rule with experiment will give our readers some notion of its utility.

| Discharge of the Weir per Second | Head producing the velocity at the Weir | Head producing the velocity above it | Calculated Height of the River above the Wasteboard | Observed Height | |---------------------------------|----------------------------------------|-------------------------------------|--------------------------------------------------|----------------| | Inches | Inches | Inches | Inches | Inches | | 3888 | 7.302 | 0.625 | 6.677 | 6.583 | | 2462 | 5.385 | 0.350 | 5.035 | 4.750 | | 1112 | 3.171 | 0.116 | 3.055 | 3.166 | | 259 | 1.201 | 0.0114 | 1.189 | 1.250 |

It was found extremely difficult to measure the exact height of the water in the upper stream above the wasteboard. The curvature A I extended several feet up the stream. Indeed there must be something arbitrary in this measurement, because the surface of the stream is not horizontal. The deviation should be taken, not from a horizontal plane, but from the inclined surface of the river.

It is plain that a river cannot be fitted for continued navigation by weirs. These occasion interruptions; but a few inches may sometimes be added to the waters of a river by a bar, which may still allow a flat-bottomed lighter or a raft to pass over it. This is a very frequent practice in Holland and Flanders; and a very cheap Practical cheap and certain conveyance of goods is there obtained by means of streams which we would think no better than boundary ditches, and unfit for every purpose of this kind. By means of a bar the water is kept up a very few inches, and the stream has free course to the sea. The flood over the bar is prevented by means of another bar placed a little way below it, lying flat in the bottom of the ditch, but which may be raised up on hinges. The lighterman makes his boat fast to a stake immediately above the bar, raises the lower bar, brings over his boat, again makes it fast, and, having laid down the other bar again, proceeds on his journey. This contrivance answers the end of a lock at a very trifling expense; and though it does not admit of what we are accustomed to call navigation, it gives a very sure conveyance, which would otherwise be impossible. When the waters can be raised by bars, so that they may be drawn off for machinery or other purposes, they are preferable to weirs, because they do not obstruct floating with rafts, and are not destroyed by the ice.

**Prob. III.** Given the height of a bar, the depth of water both above and below it, and the width of the river, to determine the discharge?

This is by no means so easily solved as the discharge over a weir, and we cannot do it with the same degree of evidence. We imagine, however, that the following observations will not be very far from a true account of the matter.

We may first suppose a reservoir LFBM (fig. 19.) of stagnant water, and that it has a waterboard of the height CB. We may then determine, by the foregoing problems, the discharge through the plane EC. With respect to the discharge through the part CA, it should be equal to this product of the part of the section by the velocity corresponding to the fall EC, which is the difference of the heights of water above and below the bar; for, because the difference of Ea and Ca is equal to EC, every particle a of water in the plane CA is pressed in the direction of this stream with the same force, viz., the weight of the column EC. The sum of these discharges should be the whole discharge over the bar; but since the bar is set up across a running river, its discharge must be the same with that of the river. The water of the river, when it comes to the place of the bar, has acquired some velocity by its slope or other causes, and this corresponds to some height FE. This velocity, multiplied by the section of the river, having the height EB, should give a discharge equal to the discharge over the bar.

To avoid this complication of conditions, we may first compute the discharge of the bar in the manner now pointed out, without the consideration of the previous velocity of the stream. This discharge will be a little too small. If we divide it by the section FB, it will give a primary velocity too small, but not far from the truth. Therefore we shall get the height FE, by means of which we shall be able to determine a velocity intermediate between DG and CH, which would correspond to a weir, as also the velocity CI, which corresponds to the part of the section CA, which is wholly under water. Then we correct all these quantities by repeating the operation with them instead of our first assumptions.

Mr. Buat found this computation extremely near the truth, but in all cases a little greater than observation exhibited.

We may now solve the problem in the most general terms.

**Prob. IV.** Given the breadth, depth, and the slope of a river, if we confine its passage by a bar or weir of a known height and width, to determine the rise of the waters above the bar.

The slope and dimensions of the channel being given, our formula will give us the velocity and the quantity of water discharged. Then, by the preceding problem, find the height of water above the waterboard. From the sum of these two heights deduct the ordinary depth of the river. The remainder is the rise of the waters.

For example:

Let there be a river whose ordinary depth is 3 feet, and breadth 40, and whose slope is 1½ inches in 100 fathoms, or \( \frac{1}{8} \) feet. Suppose a weir on this river 6 feet high and 18 feet wide.

We must first find the velocity and discharge of the river in its natural state, we have \( l = 480 \) inches, \( b = 36 \), \( \frac{1}{s} = \frac{1}{8} \). Our formula of uniform motion gives

\[ V = 23.45, \quad D = 405216 \text{ cubic inches}. \]

The contraction obtains here on the three sides of the sluice. We may therefore take \( \sqrt{\frac{D}{2G}} = 26.1 \).

N.B. This example is Mr. Buat's, and all the measures are French. We have also \( a \) (the height of the weir) 72, and \( 2g = 724 \). Therefore the equation \( b = \left( \frac{D}{c_0 + \frac{1}{4} \sqrt{\frac{D}{2G}}} \right)^{\frac{3}{2}} - \left( \frac{D}{4 \sqrt{2g}(a+b)} \right)^{\frac{3}{2}} \) becomes 30,182.

Add this to the height of the weir, and the depth of the river above the sluice is 102,182, = 8 feet and 6,182 inches. From this take 3 feet, and there remains 5 feet and 6,182 inches for the rise of the waters.

There is, however, an important circumstance in this rise of the waters, which must be distinctly understood before we can say what are the interesting effects of this weir. This swell extends, as we all know, to a considerable distance up the stream, but is less sensible as we go away from the weir. What is the distance to which the swell extends, and what increase does it produce in the depth at different distances from the weir?

If we suppose that the slope and the breadth of the channel remain as before, it is plain, that as we come down the stream from that point where the swell is insensible, the depth of the channel increases all the way to the dam. Therefore, as the same quantity of water passes through every section of the river, the velocity must diminish in the same proportion (very nearly) that the section increases. But this being an open stream, and therefore the velocity being inseparably connected with the slope of the surface, it follows, that the slope of the surface must diminish all the way from that point where the swell of the water is insensible to the dam. The surface, therefore, cannot be a simple inclined plane, but must be concave upwards, as represented in fig. 20, where FKLB represents the channel of a river, and FB the surface of the water running in it. If this be kept up to A by a weir AL, the surface will be a curve FIA, touching the natural surface F at the beginning of the swell, and the line AD which touches it in A will have the slope S corresponding to the velocity which the waters have immediately before going over the weir. We know this slope, because we are supposed to know the discharge of the river and its slope and other circumstances before barring it with a dam; and we know the height of the dam H, and therefore the new velocity at A, or immediately above A, and consequently the slope S. Therefore, drawing the horizontal lines DC, AG, it is plain that CB and CA will be the primary slope of the river, and the slope S corresponding to the velocity in the immediate neighbourhood of A, because these verticals have the same horizontal distance DC. We have therefore CB : CA = S : s very nearly, and S = CB - CA : CA, = AB (nearly) : CA. Therefore CA = \(\frac{AB \times s}{S - s}\), = \(\frac{H_s}{S - s}\). But DA = CA × S, by our definition of slope; therefore DA = \(\frac{H_s \times s}{S - s}\).

This is all that we can say with precision of this curve. Mr Buat examined what would result from supposing it an arch of a circle. In this case we should have DA = DF, and AF very nearly equal to 2 AD; and as we can thus find AD, we get the whole length FIA of the swell, and also the distances of any part of the curve from the primitive surface FB of the river; for these will be very nearly in the duplicate proportion of their distances from F. Thus ID will be \(\frac{1}{2}\) of AB, &c. Therefore we should obtain the depth ID of the stream in that place. Getting the depth of the stream, and knowing the discharge, we get the velocity, and can compare this with the slope of the surface at I. This should be the slope of that part of the arch of the circle. Making this comparison, he found these circumstances to be incompatible. He found that the section and swell at I, corresponding to an arch of a circle, gave a discharge nearly \(\frac{1}{2}\)th too great (they were as 405216 to 492142). Therefore the curve is such, that AD is greater than DF, and that it is more incurved at F than at A. He found, that making DA to DF as 10 to 9, and the curve FIA an arch of an ellipse whose longer axis was vertical, would give a very nice correspondence of the sections, velocities, and slopes. The whole extent of the swell therefore can never be double of AD, and must always greatly surpass AD; and these limits will do very well for every practical question. Therefore making DF \(= \frac{9}{10}\) of AD, and drawing the chord AD, and making DI \(= \frac{1}{2}\) of DI, we shall be very near the truth. Then we get the swell with sufficient precision for any point H between F and D, by making FD² : FH² = ID : HB; and if H is between D and A, we get its distance from the tangent DA by a similar process.

It only remains to determine the swell produced in the waters of a river by the erection of a bridge or cleaning sluice which contracts the passage. This requires the solution of

**Prob. V.** Given the depth, breadth, and slope of a river, to determine the swell occasioned by the piers of a bridge or sides of a cleaning sluice, which contract the passage by a given quantity, for a given length of channel.

This swell depends on two circumstances.

1. The whole river must pass through a narrow space, with a velocity proportionably increased; and this requires a certain head of water above the bridge. 2. The water, in passing the length of the piers with a velocity greater than that corresponding to the primary slope of the river, will require a greater slope in order to acquire this velocity.

Let V be the velocity of the river before the erection of the bridge, and K the quotient of the width of the river divided by the sum of the widths between the piers. If the length of the piers, or their dimension in the direction of the stream, is not very great, KV will nearly express the velocity of the river under the arches; and if we suppose for a moment the contraction (in the sense hitherto used) to be nothing, the height producing this velocity will be \(\frac{K^2 V^2}{2g}\). But the river will not rise so high, having already a slope and velocity before getting under the arches, and the height corresponding to this velocity is \(\frac{V^2}{2g}\); therefore the height for producing the augmentation of velocity is \(\frac{K^2 V^2}{2g} - \frac{V^2}{2g}\). But if we make allowance for contraction, we must employ a \(2G\) less than \(2g\), and we must multiply the height now found by \(\frac{2g}{2G}\). It will then become \((\frac{K^2 V^2}{2g} - \frac{V^2}{2g}) \frac{2g}{2G} = \frac{V^2}{2G} (K^2 - 1)\). This is that part of the swell which must produce the augmentation of velocity.

With respect to what is necessary for producing the additional slope between the piers, let \(p\) be the natural slope of the river (or rather the difference of level in the length of the piers) before the erection of the bridge, and corresponding to the velocity V; \(K^2 p\) will very nearly express the difference of superficial level for the length of the piers, which is necessary for maintaining the velocity KV through the same length. The increase of slope therefore is \(K^2 p - p = p(K^2 - 1)\). Therefore the whole swell will be \(\frac{V^2}{2G} + p(K^2 - 1)\).

These are the chief questions or problems on this subject which occur in the practice of an engineer; and attention to the solutions which we have given may in every case be recommended as very near the truth, and we are confident that the errors will never amount to one-fifth of the whole quantity. We are equally certain, that of those who call themselves engineers, and who, without hesitation, undertake jobs of enormous expense, not one in ten is able even to guess at the result of such operations, unless the circumstances of the case happen to coincide with those of some other project which he has executed, or has distinctly examined; and very few have the sagacity and penetration necessary for appreciating the effects of the distinguishing circumstances which yet remain. The society established for the encouragement of arts and manufactures could scarcely do a more important service to the public in the line of their institution, than by publishing in their Transactions a description of every work of this kind executed in the kingdom, with an account of its performance. This would be a most valuable collection of experiments and facts. The unlearned practitioner would find among them something which resembles in its chief circumstances almost any project which could occur to him in his We shall conclude this article with some observations on the methods which may be taken for rendering small rivers and brooks fit for inland navigation, or at least for floatage. We get much instruction on this subject from what has been said concerning the swell produced in a river by weirs, bars, or any diminution of its former section. Our knowledge of the form which the surface of this swell affects, will furnish rules for spacing these obstructions in such a manner, and at such distances from each other, that the swell produced by one shall extend to the one above it.

If we know the slope, the breadth, and the depth of a river, in the droughts of summer, and have determined on the height of the flood-gates, or keeps, which are to be set up in its bed, it is evident that their stations are not matters of arbitrary choice, if we would derive the greatest possible advantage from them.

Some rivers in Flanders and Italy are made navigable in some fort by simple sluices, which, being shut, form magazines of water, which, being discharged by opening the gates, raises the inferior reach enough to permit the passage of the craft which are kept on it. After this momentary rise the keeps are shut again, the water sinks in the lower reach, and the lighters which were floated through the shallows are now obliged to draw into those parts of the reach where they can lie afloat till the next supply of water from above enables them to proceed. This is a very rude and imperfect method, and unjustifiable at this day, when we know the effect of locks, or at least of double gates. We do not mean to enter on the consideration of these contrivances, and to give the methods of their construction, in this place, but refer our readers to what has been already said on this subject in the articles CANAL, LOCK, NAVIGATION (Inland), and to what will be said in the article WATER-WORKS. At present we confine ourselves to the single point of husbanding the different falls in the bed of the river, in such a manner that there may be everywhere a sufficient depth of water; and, in what we have to deliver on the subject, we shall take the form of an example to illustrate the application of the foregoing rules.

Suppose then a river 40 feet wide and 3 feet deep in the droughts of summer, with a slope of 1 in 4800. This, by the formula of uniform motion, will have a velocity $V = \frac{23}{5}$ inches per second, and its discharge will be 405216 cubic inches, or 234$\frac{1}{2}$ feet. It is proposed to give this river a depth not less than five feet in any place, by means of flood-gates of six feet high and 18 feet wide.

We first compute the height at which this body of 234$\frac{1}{2}$ cubic feet of water will discharge itself over the flood-gates. This we shall find by Prob. II. to be 30$\frac{1}{2}$ inches, to which adding 72, the height of the gate, we have 102$\frac{1}{2}$ for the whole height of the water above the floor of the gate; the primitive depth of the river being 3 feet, the rise or swell 5 feet 6$\frac{1}{2}$ inches. In the next place, we find the range or sensible extent of this swell by Prob. I. and the observations which accompany it. This will be found to be nearly 9177 fathoms. Now since the primitive depth of the river is three feet, there is only wanted two feet of addition; and the question is reduced to the finding what point of the curved surface of the swell is two feet above the tangent plane at the head of the swell? or how far this point is from the gate? The whole extent being 9177 fathoms, and the deviations from the tangent plane being nearly in the duplicate ratio of the distances from the point of contact, we may institute this proportion $\frac{66\frac{1}{2}}{24} = \frac{9177^2}{5526^2}$. The last term is the distance (from the head of the swell) of that part of the surface which is two feet above the primitive surface of the river. Therefore 9177—5526, or 3651 fathoms, is the distance of this part from the flood-gate; and this is the distance at which the gates should be placed from each other. No inconvenience would arise from having them nearer, if the banks be high enough to contain the waters; but if they are farther distant, the required depth of water cannot be had without increasing the height of the gates; but if reasons of convenience should induce us to place them nearer, the same depth may be secured by lower gates, and no additional height will be required for the banks. This is generally a matter of moment, because the raising the water brings along with it the chance of flooding the adjoining fields. Knowing the place where the swell ceases to be sensible, we can keep the top of the intermediate flood-gate at the precise height of the curved surface of the swell by means of the proportionality of the deviations from the tangent to the distances from the point of contact.

But this rule will not do for a gate which is at a greater distance from the one above it than the 3651 fathoms already mentioned. We know that a higher gate is required, producing a more extensive swell; and the one swell does not coincide with the other, although they may both begin from the same point A (fig. 21). Nor will the curves even be similar, unless the thicknesses of the sheet of water flowing over the gate be increased in the same ratio. But this is not the case; because the produce of the river, and therefore the thicknesses of the sheet of water, is constant.

But we may suppose them similar without erring more than two or three decimals of an inch; and then we shall have $AF : AL = fF : DL$; from which, if we take the thickness of the sheet of water already calculated for the other gates, there will remain the height of the gate BL.

By following these methods, instead of proceeding by random guesses, we shall procure the greatest depth of water at the smallest expense possible.

But there is a circumstance which must be attended to, and which, if neglected, may in a short time render useless all our works useless. These gates must frequently be open in the time of freshes; and as this channel then has its natural slope increased in every reach by the great contraction of the section in the gates, and also rolls along a greater body of water, the action of the stream on its bed must be increased by the augmentation of velocity which these circumstances will produce. Practical and although we may say that the general slope is necessarily secured by the cills of the flood-gates, which are paved with stone or covered with planks, yet this will not hinder this increased current from digging up the bottom in the intervals, undermining the banks, and lodging the mud and earth thus carried off in places where the current meets with any check. All these consequences will assuredly follow if the increased velocity is greater than what corresponds to the regimen relative to the foil in which the river holds on its course.

In order therefore to procure durability to works of this kind, which are generally of enormous expense, the local circumstances must be most scrupulously studied. It is not the ordinary hurried survey of an engineer that will free us from the risk of our navigation becoming very troublesome by the rise of the waters being diminished from their former quantity, and banks formed at a small distance below every sluice. We must attentively study the nature of the foil, and discover experimentally the velocity which is not inconsistent with the permanency of the channel. If this be not a great deal less than that of the river when accelerated by freshes, the regimen may be preserved after the establishment of the gate, and no great changes in the channel will be necessary; but if, on the other hand, the natural velocity of the river during its freshes greatly exceeds what is consistent with stability, we must enlarge the width of the channel, that we may diminish the hydraulic mean depth, and along with this the velocity. Therefore, knowing the quantity discharged during the freshes, divide it by the velocity of regimen, or rather by a velocity somewhat greater (for a reason which will appear by and by), the quotient will be the area of a new section. Then taking the natural slope of the river for the slope which it will preserve in this enlarged channel, and after the cills of the flood-gates have been fixed, we must calculate the hydraulic mean depth, and then the other dimensions of the channel. And lastly, from the known dimensions of the channel and the discharge (which we must now compute), we proceed to calculate the height and the distances of the flood-gates, adjusted to their widths, which must be regulated by the room which may be thought proper for the free passage of the lighters which are to ply on the river. An example will illustrate the whole of this process.

Suppose then a small river having a slope of 2 inches in 100 fathoms or \( \frac{1}{50} \) inches, which is a very usual declivity of such small streams, and whose depth in summer is 2 feet, but subject to floods which raise it to nine feet. Let its breadth at the bottom be 18 feet, and the base of its slanting sides \( \frac{1}{3} \) of their height. All of these dimensions are very conformable to the ordinary course of things. It is proposed to make this river navigable in all seasons by means of keeps and gates placed at proper distances; and we want to know the dimensions of a channel which will be permanent, in a foil which begins to yield to a velocity of 80 inches per second, but will be safe under a velocity of 24.

The primitive channel having the properties of a rectangular channel, its breadth during the freshes must be \( B = 30 \) feet, or 360 inches, and its depth \( b = 9 \) feet or 108 inches; therefore its hydraulic mean depth

\[ d = \frac{Bb}{B + 2b} = 61.88 \text{ inches}. \]

Its real velocity therefore, during the freshes, will be 38,9447 inches, and its discharge 1514169 cubic inches, or 8764 cubic feet per second. We see therefore that the natural channel will not be permanent, and will be very quickly destroyed or changed by this great velocity. We have two methods for procuring stability, viz., diminishing the slope, or widening the bed. The first method will require the course to be lengthened in the proportion of \( \frac{24^2}{3988^2} \), or nearly of 36 to 100. The expense of this would be enormous. The second method will require the hydraulic mean depth to be increased nearly in the same proportion (because the velocities are nearly as \( \frac{\sqrt{d}}{\sqrt{s}} \)). This will evidently be much less costly, and even to procure convenient room for the navigation, must be preferred.

We must now observe, that the great velocity, of which we are afraid, obtains only during the winter floods. If therefore we reduce this to 24 inches, it must happen that the autumnal freshes, loaded with sand and mud, will certainly deposit a part of it, and choke up our channel below the flood-gates. We must therefore select a mean velocity somewhat exceeding the regimen, that it may carry off these depositions. We shall take 27 inches, which will produce this effect on the loose mud without endangering our channel in any remarkable degree.

Therefore we have, by the theorem for uniform motion,

\[ V = 27 = \frac{297(\sqrt{d} - 0.1)}{\sqrt{s} - L\sqrt{s} + 1.6} = 0.3(\sqrt{i} - 0.1) \]

Calculating the divisor of this formula, we find it

\[ = 55,884. \]

Hence \( \sqrt{d} - 0.1 = \frac{27}{55,884} = 0.3 \)

\( 5,3843 \), and therefore \( d = 30.1 \). Having thus determined the hydraulic mean depth, we find the area \( S \) of the section by dividing the discharge 1514169 by the velocity 27. This gives us 56080368. Then we get the breadth \( B \) by the formula formerly given,

\[ B = \sqrt{\left(\frac{S}{2d}\right)^2 - 2S} + \frac{S}{2d^2} = 1802,296 \text{ inches, or } 150,19 \text{ feet, and the depth } b = 31,115 \text{ inches.} \]

With these dimensions of the section we are certain that the channel will be permanent; and the cills of the flood-gates being all fixed agreeable to the primitive slope, we need not fear that it will be changed in the intervals by the action of the current. The gates being all open during the freshes, the bottom will be cleared of all deposited mud.

We must now station the flood-gates along the new channel, at such distances that we may have the depth of water which is proper for the lighters that are to be employed in the navigation. Suppose this to be four feet. We must first of all learn how high the water will be kept in this new channel during the summer droughts. There remained in the primitive channel only 2 feet, and the section in this case had 20 feet 8 inches mean width; and the discharge corresponding to this section and slope of \( \frac{1}{50} \) is, by the theorem of uniform motion, 130,849 cubic inches per second. To find find the depth of water in the new channel corresponding to this discharge; and the same slope, we must take the method of approximation formerly exemplified, remembering that the discharge D is 130849, and the breadth B is 1760.8 at the bottom (the flint sides being \( \frac{3}{4} \)). These data will produce a depth of water \( = 6\frac{1}{2} \) inches. To obtain four feet therefore behind any of the flood-gates, we must have a swell of \( 4\frac{1}{2} \) inches produced by the gate below.

We must now determine the width of passage which must be given at the gates. This will regulate the thickness of the sheet of water which flows over them when shut; and this, with the height of the gate, fixes the swell at the gate. The extent of this swell, and the elevation of every point of its curved surface above the new surface of the river, requires a combination of the height of swell at the flood-gate, with the primitive slope and the new velocity. These being computed, the stations of the gates may be assigned, which will secure four feet of water behind each in summer. We need not give these computations, having already exemplified them all with relation to another river.

This example not only illustrates the method of proceeding, so as to be ensured of success, but also gives us a precise instance of what must be done in a case which cannot but frequently occur. We see what a prodigious excavation is necessary, in order to obtain permanency. We have been obliged to enlarge the primitive bed to about thrice its former size, so that the excavation is at least two-thirds of what the other method required. The expense, however, will still be vastly inferior to the other, both from the nature of the work and the quantity of ground occupied. At all events, the expense is enormous, and what could never be repaid by the navigation, except in a very rich and populous country.

There is another circumstance to be attended to.—The navigation of this river by sluices must be very defective, unless they are extremely numerous, and of small heights. The natural surface of the swell being concave upwards, the additions made by its different parts to the primitive height of the river decrease rapidly as they approach to the place A (fig. 20), where the swell terminates; and three gates, each of which raises the water one foot when placed at the proper distance from each other, will raise the water much more than two gates at twice this distance, each raising the water two feet. Moreover, when the elevation produced by a flood-gate is considerable, exceeding a very few inches, the fall and current produced by the opening of the gate is such, that no boat can possibly pass up the river, and it runs imminent risk of being overset and sunk, in the attempt to go down the stream. This renders the navigation defective. A number of lighters collect themselves at the gates, and wait their opening. They pass through as soon as the current becomes moderate. This would not, perhaps, be very hurtful in a regulated navigation, if they could then proceed on their voyage. But the boats bound up the river must stay on the upper side of the gate which they have just now passed, because the channel is now too shallow for them to proceed. Those bound down the river can only go to the next gate, unless it has been opened at a time nicely adjusted to the opening of the one above it. The passage downwards may, in many cases, be continued, by very intelligent and attentive lockmen, but the passage up must be exceedingly tedious. Nay, we may say, that while the passage downwards is continuous, it is but in a very few cases that the passage upward is practicable. If we add to these inconveniences the great danger of passage during the freshes, while all the gates are open, and the immense and unavoidable accumulations of ice, on occasion even of slight frosts, we may see that this method of procuring an inland navigation is amazingly expensive, defunctory, tedious, and hazardous. It did not therefore merit, on its own account, the attention we have bestowed on it. But the discussion was absolutely necessary, in order to show what must be done in order to obtain effect and permanency, and thus to prevent us from engaging in a project which, to a person not duly and confidently informed, is so feasible and promising. Many professional engineers are ready, and with honest intentions, to undertake such tasks; and by avoiding this immense expense, and contenting themselves with a much narrower channel, they succeed, (witness the old navigation of the river Mersey). But the work has no duration; and, not having been found very serviceable, its cessation is not matter of much regret. The work is not much spoken of during its continuance. It is soon forgotten, as well as its failure, and engineers are found ready to engage for such another.

It was not a very refined thought to change this imperfect mode for another free from most of its inconveniences. A boat was brought up the river, through locks, one of these gates, only by raising the waters of the inferior reach, and depressing those of the upper; and it could not escape observation, that when the gates were far asunder, a vast body of water must be discharged before this could be done, and that it would be a great improvement to double each gate, with a very small distance between. Thus a very small quantity of water would fill the interval to the desired height, and allow the boat to come through; and this thought was the more obvious, from a similar practice having preceded it, viz. that of navigating a small river by means of double bars, the lowest of which lay flat in the bottom of the river, but could be raised up on hinges. We have mentioned this already; and it appears to have been an old practice, being mentioned by Stevinus in his valuable work on sluices, published about the beginning of the last century; yet no trace of this method is to be found of much older dates. It occurred, however, accidentally, pretty often in the flat countries of Holland and Flanders, which being the seat of frequent wars, almost every town and village was fortified with wet ditches, connected with the adjoining rivers. Stevinus mentions particularly the works of Condé, as having been long employed, with great ingenuity, for rendering navigable a very long stretch of the Scheldt. The boats were received into the lower part of the fosse, which was separated from the rest by a stone batardeau, serving to keep up the waters in the rest of the fosse about eight feet. In this was a sluice and another dam, by which the boats could be taken into the upper fosse, which communicated with a remote part of the Scheldt by a long canal. This appears to be one of the earliest locks.

In the first attempt to introduce this improvement in the navigation of rivers already kept up by weirs, which gave a partial and interrupted navigation, it was usual to avoid the great expense of the second dam and gate, by making the lock altogether detached from the river, within land, and having its basin parallel to the river, and communicating by one end with the river above the weir, and by the other end with the river below the weir, and having a flood-gate at each end. This was a most ingenious thought; and it was a prodigious improvement, free from all the inconveniences of currents, ice, &c. &c. It was called a Schleusen, or lock, with considerable propriety; and this was the origin of the word sluice, and of our application of its translation lock. This practice being once introduced, it was not long before engineers found that a complete separation of the navigation from the bed of the river was not only the most perfect method for obtaining a sure, easy, and uninterrupted navigation, but that it was in general the most economical in its first construction, and subject to no risk of deterioration by the action of the current, which was here entirely removed. Locked canals, therefore, have almost entirely supplanted all attempts to improve the natural beds of rivers; and this is hardly ever attempted except in the flat countries, where they can hardly be said to differ from horizontal canals. We therefore close with these observations this article, and reserve what is yet to be said on the construction of canals and locks for the article WATER-WORKS.

We beg leave, however, to detain the reader for a few moments. He cannot but have observed our anxiety to render this dissertation worthy of his notice, by making it practically useful. We have on every occasion appealed, from all theoretical deductions, however specious and well supported, to fact and observation of those spontaneous phenomena of nature which are continually passing in review before us in the motion of running waters. Resting in this manner our whole doctrines on experiment, on the observation of what really happens, and what happens in a way which we cannot or do not fully explain, these spontaneous operations of nature came insensibly to acquire a particular value in our imagination. It has also happened in the course of our reflections on these subjects, that these phenomena have frequently presented themselves to our view in groups, not less remarkable for the extent and the importance of their consequences than for the simplicity, and frequently the seeming insignificance, nay frivolity, of the means employed. Our fancy has therefore been sometimes warmed with the view of something; an

Ens agitans molem, et magno se corpore miscens.

This has sometimes made us express ourselves in a way that is susceptible of misinterpretation, and may even lead into a mistake of our meaning.

We therefore find ourselves obliged to declare, that by the term NATURE, which we have so frequently used con amore, we do not mean that indefinable idol which the self-conceit and vanity of our neighbours in France have set up of late, and ostentatiously stand on tiptoe to worship. This ens rationis, this creature of the imagination, has long been the object of cool contemplation in the closet of the philosopher, and has shared his attention with many other play-things of his ever-working fancy. But she has now become the object of a sincere and fond idolatry, being held forth by her zealous high-priests to the refined vanity of man as a sort of mirror, in which he may behold his own cherished features, and admire a beauty of his own composition, painted with the most delicate glow of humanity, and decked out with every ornament with which the courtly fancies of a Voltaire, a Diderot, a Mirabeau, could contrive, to smooth over or to hide all traces of created imperfection. We leave this idol to the worship of her intoxicated and unfortunate votaries. The solemn farce in the church of Notre Dame at Paris was an adoration every way worthy of the Divinity; and our horror in reading the description of the ceremonial was not without some alloy of pleasure, when we saw among her most active priests an artist, whom we had seen a few years before the machiniste de l'opéra at St Peterburgh, and grand-master of the lodge des Mouffet. We hope to be forgiven the pun, when we say that the ancient fabric which was that day profaned by the abomination of desolation, was then in reality the temple de Notre Dame. Mr Brignonzi was, by his profession, a fit successor in the priesthood to those fages de la France (such was the appellation that they gave each other), whom we have just now named; and his Tours de Théâtre, for which we have frequently admired his talents, were a very proper accompaniment to the finery and ruse of these soi-disant philosophers, who, under the mask of the most refined humanity, habitually practised arts of dishonesty which would have ruined the character of the meanest pedlar. No one will think that we express ourselves too strongly who reflects on the many infamous tricks played by Voltaire to his bookellers. No one will think the charge too harsh, when he learns that Diderot, after having pretended to the possession of an immense library, and sold it to the empereurs of Russia for an enormous sum, had to ramble the warehouses of the bookellers of Paris and throughout all Germany, in order to fill his shelves. As for Mirabeau, he surpasses eulogy.

Most affluous were those apostles in spreading this fanaticism, of which they enjoyed the courtly profits; and we imagine that the employment was as agreeable as it was lucrative; for we cannot suppose that Le Kain had more enjoyment, when fascinating his Parisian audience in the character of Voltaire's Mahomet, than its author felt in the side-box, when grinning to himself, and conscious what a sordid and envious wretch he was, he found himself crowned by the first actres, and worshipped by the audience as the apostle of philanthropy and universal benevolence.

Such was the worship, such were the priests, of this Gallic idol; and, like their predecessors the Druids, they have made human sacrifices a customary oblation at the shrine. We wonder at these things, and are surprised that any thing which can even be nicknamed philosophy can produce such effects. But the task of this apostlelele was as easy as it was agreeable. It was not the work of a day; it was the completion of a studied corruption of principles, which is now above a century old. We may say that it began under the clever but infamous Dubois; who from being the valet de chambre of an infirm bishop, became cardinal, and sovereign of the Gallic church, and almost of the state. When objected to by the bigotted Louis XIV. (on a pretention for preferment) as a Jansenist, "Ob qui non," said the duke of Orleans, "Ob, Sire, qui non, il n'y a qu'ailleurs." He was at the utmost pains to bring into the court every man of eminent talents in gay literature, and of licentious principles in religion and morals, whom he employed in corrupting the minds of the young courtiers, and giving them favourable impressions of the indulgence which they might expect from him when he should have the sole direction of affairs. This system was most assiduously pursued during that most licentious and dissolute administration of the regent Orleans, who was himself a specimen of elegant sensuality not to be matched in the annals of the world. Long before the present day, all thinking men in France saw the mummery of the church, and groaned under its oppression; and having no other notions of religion but what they were accustomed to from their cradle, no wonder that they discarded the principle along with those detestable accessories. The nation, therefore, being greedy of flattery, buoyed up by a self-conceit, in which even the ancient Greeks have not surpassed them, and having been thus studiously corrupted, and long immersed in a luxurious and refined sensuality, of which we in this nation have not yet acquired an adequate idea, was fully prepared for feeling all the effects of this fanaticism of NATURALISM.

But this idolatry we abhor. It shocks our reason; and, although it may at first seem to flatter our thoughtless vanity, it really debases our nature, by taking from us our intellectual kindred to the mind of perfect wisdom. Who would not feel pleasure in being the relation of a Bacon, of a Newton, or would thank the man who detected the false pedigree? It puts an end to our fond hopes, that the day will come when we shall surpass in understanding, in worth, and in felicity, the wisest, the best, and the most fortunate of our species.

We cannot but lament the appearances, however faint, of this fanaticism among ourselves. We cannot but observe, that some of the hired directors of public opinion in matters of taste and science have of late shewn a wonderful tenderness for the bold and licentious opinions in religion, morals, and politics, which are daily pouring in upon us from the presses of Paris. Perhaps they may be incited to this conduct by the success of their brother journalists in that profligate metropolis; and may hope to be one day, like them, the directors of the public councils and the sovereigns of the nation. We trust, however, that the better part of the reflecting natives of Britain will not allow themselves to be sneered out of their highest boast and their sweetest comforts; namely, that they are not the chance fragments of a fatal chaos, but the beautiful productions of a wonderful Artist, and the darling objects of his care; and we assure ourselves that ten thousands of our countrymen are ready to rally under the banners of true religion and sound philosophy, and to follow the steps of a Clarke, a Butler, a Newton, and a Boyle, who so eminently distinguished themselves in the cause of Nature's God.

By NATURE, then, we mean that admirable system of general laws, by which the adored Author and Governor of the universe has thought fit to connect the various parts of this wonderful and goodly frame of things, and to regulate all their operations.

We are not afraid of continually appealing to the laws of nature; and as we have already observed in the article PHILOSOPHY, we consider these general laws as the most magnificent displays of Infinite Wisdom, and the contemplation of them as the most cheering employment of our understandings.

Igneus est illis vigor et caelestis origo Semibibus.

At the same time we despise the cold-hearted philosopher who stops short here, and is satisfied (perhaps inwardly pleased) that he has completely accounted for everything by the laws of unchanging nature; and we suspect that this philosopher would analyse with the same frigid ingenuity, and explain by irresistible arguments, the tender attachment of her whose breast he fucked, and who by many anxious and sleepless nights preserved alive the puling infant. But let us rather listen to the words of him who was the most sagacious observer and the most faithful interpreter of nature's laws, our illustrious countryman Sir Isaac Newton. He says,

"Elegantissima haec rerum compages non nisi confilio et domino entis sapientissimi et potentissimi oriri potuit. Omnia, similis constructa confilio, suberunt unus dominio. Hic omnia regit, non ut anima mundi, sed ut universorum dominus. Propter dominium suum dominus deus, παντοκράτωρ nuncupatur. Deus ad fervientes recipit, et dellos eft dominatio dei, non in corpus proprium, uti fertunt quibus deus eft eft natura seu anima mundi, sed in servos. Deus summus eft ens eternum, infinatum, absolutae perfectum. Ens uterque perfectum, at fine dominio, non eft dominus deus.

"Hunc cognoscimus, folummodo per proprietates ejus et attributa. Attribuuntur ut ex phenomenis dignofeuntur. Phenomena sunt sapientissimi et optimae rerum structurae, atque causae finales.—Hunc admiramus ob perfectiones; hunc veneramus et colimus ob dominium" (B).

(b) Our readers will probably be pleased with the following list of authors who have treated professedly of the motions of rivers: Guglielmini De Fluvii et Conflellas Aquarum—Danubius Illustratus; Grandi De Castelli; Zendrini De Motu Aquarum; Frisius De Fluvii; Leechi Idrostatica i Idraulica; Michelotti Speranza Idruliche; Belidor's Architecture Hydraulique; Boffius Hydrodynamique; Buat Hydraulique; Silberichlag Theorie des Fleuves; Lettres de M. L'Epinasse au P. Frisi touchant sa Theorie des Fleuves; Tableau des principales Rivieres du Monde, par Genette; Stevin sur les Ecluses; Tracté des Ecluses, par Boullard, qui a remporté le Prix de l'Acad. de Lyons; Bleiswyck Differatio de Aggeribus; Boffius et Viallet sur la Construction des Diques; Stevin Hydrostatique; Tichman van der Horst Théâtreum Machinarum Universale; De la Lande sur les Canaux de Navigation; Racolta di Autori chi Trattano del Moto dell' Acqua, 3 tom. 4to, Firenze 1723.—This most valuable collection contains the writings of Archimedes, Albizi, Galileo, Caffelli, Michelini, Borelli, Montanari, Viviani, Caffini, Guglielmini, Grandi, Manfredi, Picard, and Narducci; and an account of the numberless works which have been carried on in the embankment of the Po. River-Water. This is generally much softer and better accommodated to economical purposes than spring-water. For though rivers proceed originally from springs, yet, by their rapid motion, and by being exposed during a long course to the influence of the sun and air, the earthy and metallic salts which they contain are decomposed, the acid flies off, and the terrestrial parts precipitate to the bottom. Rivers are also rendered softer by the vast quantity of rain-water, which, passing along the surface of the earth, is conveyed into their channels. But all rivers carry with them a great deal of mud and other impurities; and, when they flow near large and populous towns, they become impregnated with a number of heterogeneous substances, in which state the water is certainly unfit for the purposes of life; yet, by remaining for some time at rest, all the feculencies subside, and the water becomes sufficiently pure and potable.