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will have a wide circulation, we shall content ourselves with extracting one that is at once extremely simple and highly useful.
"To determine the fall of the Water in the Arches of a Bridge. The effects of obstacles placed in a current of water, such as the piers of a bridge, are, a sudden steep descent, and an increase of velocity in the stream of water, just under the arches, more or less in proportion to the quantity of the obstruction and velocity of the current: being very small and hardly perceptible where the arches are large and the piers few or small, but in a high and extraordinary degree at London-bridge, and some others, where the piers and the sterlings are so very large, in pro. portion to the arches. This is the case, not only in such streams as run always the same way, but in tide rivers also, both upward and downward, but much less in the former than in the latter. During the time of flood, when the tide is flowing upward, the rise of the water is against the under side of the pièrs ; but the difference between the two sides gradually diminishes as the tide flows less rapidly towards the conclusion of the flood. When this has attained its full height, and there is no longer any current, but a stillness prevails in the water for a short time, the surface assumes an equal level, both above and below. bridge. But, as soon as the tide begins to ebb or return again, the resistance of the piers against the stream, and the contraction of the waterway, cause a rise of the surface above and under the arches, with a full and a more rapid descent in the contracted stream just below. The quantity of this rise, and of the consequent velocity below, keep both gradually increasing, as the tide con tinues ebbing, till at quite low water, when the stream or natural current being the quickest, the fall under the arches is the greatest. And it is the quantity of this fall which it is the object of this problem to determine.
Now, the motion of free running water is the consequence of, and produced by the force of gravity, as well as that of any other falling body. Hence the height due to the velocity, that is, the height to be freely fallen by any body to acquire the observed velocity, of the natural stream, in the river a little way above bridge, becomes known, From the same velocity also will be found that of the increased current in the narrowed way of the arches, by taking it in the reciprocal proportion of the breadth of the river above, to the contracted way in the arches ; viz, by saying, as the latter is to the former, so is the first velocity, or slower motion, to the quicker, Next, from this last velocity, will be found the height duc to it as before, that is, the height to be freely fallen through by gravity, to produce it. Then the difference of these two heights, thus freely fallen by grávity, to produce the two velocities, is the required quantity of the waterfall in the arches; allowing however, in the calculation for the contraction, in the narrowed passage, at the rate as observed by Sir I. Newton, in prop. 36 of the 2d book of the Principia, or by other authors, being nearly in the ratio of 25 to 21. Such then are the ele. ments and principles on which the solution of the problem is easily made out as follows.
Hutton's Course of Mathematics, Vol. IJI. Let b = the breadth of the channel in feet;
y = mean velocity of the water in feet per second;
ce breadth of the waterway between the obstacles. Now 2 : 21 :: 0:21, the waterway contracted as above. And 21: .6:: 0 :2.50, the velocity in the contracted way: Also 32" : 0 :: 16: 40', height fallen to gain the velocity v. And 32 : (2000) :: 16 : x stro", ditto for the vel. 25. Then (1956 is the measure of the fall required. Or Events – 1] x is a rule for computing the fall. Or rather 1-4260402 x n* very nearly, for the fall.
Exam. 1. For London-bridge. By the observations made by Mr. Labelye in 1746, 1 The breadth of the Thames at London-bridge is 926 feet; The sum of the waterways at the time of low-water is 236 ft; Mean velocity of the stream just above bridge is 34 ft. per sec. But under almost all the arches are driven into the bed great numbers of what are called dripshot piles, to prevent the bed from being washed away by the fall. These dripshot piles still further contract the waterways, at leastă of their measured breadth, or near 39 feet in the whole; so that the waterway will be reduced to 197 feet, or in round numbers suppose 200 feet. Then b = 926, c = 200, v = 35.
1.426 1217616_40000 Hence
-=.46. 64 X 40000
And v = 1046
: 62 Theref. •46 x 1075=473 ft. = 4 ft. 84 in. the fall required. By the most exact observations made about the year 1736, the measure of the fall was 4 feet 9 inches.
Exam. 2. For Westminster-bridge. Though the breadth of the river at Westminster-bridge is 1220 feet; yet, at the time of the greatest fell, there is water through only the 13 large arches, which amount to but 820 feet; to which adding the breadth of the 12 intermediate piers, equal to 174 feet, gives 994 for the breadth of the river at that time; and the velocity of the water a little above the bridge, from many experiments, is not more than 24 ft. per second.
And v?= =5746.
16 Theref. •01722 X 512 = .0872 ft. = 1 in. the fall required; which is about half an inch more than the greatest fall observed by Mr. La. belye.
And, for Blackfriar's-bridge, the fall will be much the same as that of Westminster.
On the whole, we cannot but congratulate the public upon the appearance of this volume, and the country upon that improved state of the Woolwich Academy, which renders such a volume necessary. The two ingenious authors seem to have been stimulated solely by a desire to compress into the smallest possible compass the greatest possible quantity of curious and useful matter. Both may be said to have studied the art of compression with the greatest success; though with Dr. Hutton it would rather seem to be a natural faculty, while in Dr. Gregory it appears like an acquired habit: for the former never deviates a single line from his purpose, while the latter appears fond (though in this work he has very seldom indulged that propensity) to make excursions into connected and surrounding subjects. Neither of them, however, manifests any inclination for parade. So far as their writings indicate their motives, it may be fairly inferred that they write, not for the purpose of catching admiration, but of imparting instruction. And this volume, especially, will, we doubt not, be found as striking and durable a memorial of their talents, as it seems intended to be of their friendship. We cannot always, permit anthors to adjust their own claims; but, on the present occasion, we think Dr. Hutton is fully justified in expressing his conviction, that, with the assistance of his friend and coadjutor in this supplementary volume, he has now produced a course of Mathematics in which a greater variety of useful subjects are introduced, and treated with perspicuity and correctness, than in any three volumes, of equal size, in any language.'. Art. Ill. Sermons, by Samuel Horsley, LL.D. F.R. S. F. A. S. Late Lord Bishop of St. Asaph.
(Concluded from page 877.) THE next discourse to which the course of our examina
tion conducts us, is on John xiii. 34, "a new commandment, &c, It must, before this time, have struck the
reader, that there is nothing of sentimental whining or affected pathos in the sermons of Bishop Horsley; and yet, when the subject has naturally adınitted' of descriptive and touching appeals, no one seems more completely in command of our feelings, or more capable of inspiring the consciousness, on after-reflection, that all those feelings were deserved. We have seldom read a finer specimen of appropriate and affecting exordium, than the following.
• In that menforable night, when divine love and infernal malice, had 'each their perfect work, the night when Jesus was betrayed into the hands of those who thirsted for his blood, and the mysterious scheme of man's redemption was brought to its accomplishment, Jesus, having finished the pascal supper, and instituted those holy mysteries, by which the thankful remembrance of his oblation of himself is con. tinued in the church until his second coming, and the believer is nourished with the food of everlasting life ;-When all this was finished, and nothing now remained of his great and painful under-, taking, but the last trying part of it, to be led like a sheep to the slaughter, and to make his life a sacrifice for sing in that trying hour, just before he retired to the garden, where the power of darkness was to be permitted to display on him its last and utmost effort, Jesus gave it solemnly in charge to the eleven apostles, (the twelfth the son of perdition was already lost; he was gone to hasten the execution of his intended treason),--to the eleven, whose loyalty remained as yet unshaken, Jesus, in that awful hour gave it solemnly in charge, sito love one another, as he had loved them.” And because the perverse wit of man is ever fertile in plausible evasions of the plainest duties, lest this command should be interpreted, in after ages, as an injunction in which the apostles only were concerned, imposed upon them in their peculiar character of the governors of the church, our great master, to obviate any such wilful misconstruction of his dying charge, declared it to be his pleasure and his meaning, that the exercise of mutual love; in all ages, and in all nations, among men of all ranks, callings and conditions, should be the general badge and distinction of his disciples.“ By this shall all men know that ye are my disciples, if ye love one another.” And this injunction he calls a new com. mandment-"&c. p. 255-7.
The R. R. author then proceeds to explain the novelty of this commandment because it was new in the practice of mankind in the lessons of divines and moralists, it was a topic' out of use and the disciples were required to love one another, after the manner, and if the frailty of human nature might so far aspire, in the degree in which Christ loved them. The remaining part of the sermon after the illustration of these topics, is occupied by some remarks on the perfection of the example of Christ; and gladly would we transcribe several pages of this truly eloquent and scriptural discourse, but for the length to which our notice has already extended. For
this reason, wè thust notice very brieħy what remains before us.
The twelfth serinon is an explanation of Matt.xvi. 28. " Verily I say unto you, there be some standing here, which shall not laste of death, till they see the son of man, coming in his kingdom.” Agreeably to the juterpretation of the phrase, the coming of the son of man," which he had already established, the Bishop considers this passage as an intimation, that a certain individual present when Christ spake these words, should not suffer the full punishment of his crimes, tilt the day of judgement; and the unhappy victim of this prophetic doom, was, in his opinion, the traitor Judas.
Not to taste of death' [says his lordship] is not to feel the bitterness of it. In this sense was the same expression used by our Lord upon other occasions. " If a man keep my saying he shall never taste of death.” The expression is to be understood, with referéncé to the intermediate state between death and the final judgenient, in which the souls both of the righteous and the wicked exist in a conscious state, the one comforted with the hope and prospect of their future glory, the other mortified with the expectation of torment. -It may be truly said of the wicked, that they have no real taste of death, till they taste it in the burning lake.:-p, 285,
Having ingeniously established the special reference of this passage to the betrayer of our Lord, he adverts to the doctrine of the eternity of future punishment; the justice of which he vindicates from rational views of the divine government, and the explicit assertions of Scripture.
The well-known declaration, recorded in Matt. xvi. 18, 19, on which the Romanists support the supremacy of St. Peter, is the subject of the next discourse, which was preached, in 1795, before the Society for the Propagation of the Gospel in Foreign Parts. The learned prelate differs from many protestant writers, in his interprétation of this passage; and in the following confident tone, scruples not to asserto his convictions.
St Peter, on this occasion answered only for himself :-the blessing i obtained was for himself singly, the reward of his being foremost
in the faith which he confessed to be the carrier of the keys of the kingdom of heaven--to loose and bind on earth, in any sense which the expressions may bear in this passages were personal dis. tinctions of the venerable primate of the apostolic college, appropriated to him in positive and absolute exclusion of all other persons, in exclusion of the apostles his contemporaries, and of the Bishops of Rome his sūccessors. We need not scruple to assert, that any interpretation of this passăge, or of any part of it, founded upon a